![BIBLIOGRAPHY 11
efiicient ano which produces results with tht- desired degree of acC'.lr.lcy. The only methods
which currently sathi), these requirements are the level 2 method;;. although analysis by Montf.-·
Cilflo simulation is sometimes feasible. In this book emphasis is pJ.acec. on tn.? theory and appli·
cation of le'el 2 methods and their use in the design of le'ell structurAl codes.
BIBLIOGRAPHY
(1.11 CIRIA: Rationalisation of Safety and Seruiceability Factors.in Structural Codes. Can·
struction Industry Research and Information Association, Report. No. 63, 1977.
(1.2J Cornell, C. A.: Bayesian Statistical Decision Theory and RelUlbility.Based Desig/!. Inter·
national Conference on Struct~iaI Safety and Reliability, Wa;hington, 1969 .. (pub.
Pergamon 1972):_.
11.3] Ditlevsen, 0.: Uncertaint:r /I~od,!lillg. Mcqraw Hill,19B1.
(1.4J - Freudenthal. A. M . .-Garretts, J. M. and Shinoz.uka, M.: The Analysis of Structural Safety.
Jownal of the Structural Division, ASCE, Vol. 9~, ~o. STi, Feb. 1966.
11.5)- Joint Committee on Structural Safety, CEB.- CECM • CIB· FIP . L0U3SE· lASS· RILEI"II:
First Order Reliability Concepts for Design Codes. CEB Bulletin No. 112, July 1976.
11.6J Joint Committee on Structural Safety, CEB· CECM· CIB· FIP· lA-BSE • lASS . RILEM:
Interlltltiolltll System of Unified Standard Codes for Structures. Volume I: Common Uni·
fied Rules for Different Types of Construction and Material. ~EB/~IP 1978.
11.7] Joint Committee on Structural Safety, CEB • CECt-,·j • CIB· FIP - L-.BSE . lASS· RILEM:
Genera! Principles on Reliabilit)' for Structural Design. Intemational.Association for
Bridge .and Structural Engjneering·,·1981.
[1.8J Leporati, L: The Assessment of Structural Safety. Researeh StUdies Press, 1979.
11.9J Nordic Committee on Building Regulations: Recommendation for Loading and Safety
Regulatiolls for StructurtJI Desigll ... NKB.Report No. 36, No .... 1S7S.](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-23-320.jpg)
![2. f'UNDA.IE?o.'TALS or PROBABILITY THEORY
'-" - " ' - r ! P2
, , B, ,. ~ ' :3" . '7777. , :>I' , ' . ,
A
Figure 2.1
This sample space is discrete. Further, it ha,s a finite number oC sample points. There·
fore, it is caUed a finite sample space. A s:unpl~ space with the countable infinite num·
ber of sample poinu is called .an infinite 5:Imple spacp.
?ot.e that the sample spaces for the loads P1 and P2 are 0 1 - ·{4·,.S,·6} ,"and:n 2 ~ . ~
'{3, 4}, respedivel)'. Also note that n ~.nl X: fi2,' ~~o~ as.ll~ e~~.rc,isf. ~!'Iat.~,~~mple
space for the reacuon RA in point A.is ~A .. P]/~~ 121.3. 13/3, 1~/3, ' 1513; 16la}.
> . , .
A subset of a sample sp3ce is called an· ~~en/. ~~yent is therefore a se~ or sample'points. If
It contains no sample paints. it is called an l~po~ble e~~nt, A cert(Ji~' e'uenf co'nt:ains ali the
sample points in the sampJe ~pace • that is, a c~~ eye~t is equal t'o'the sample space itself.
~ . . - '
Example 2.2. Consider again the be:lm in ligure 2.1. The sample space fo ... ·the reaction
RA is Sl A - {lIla, 12/3.· -13/3. 14/3, 1513.: 16/3). The. subset {IS/3, ,16/3) is the
eventlh'at R . .a.' is· equal to 1513 or 1613.
Let £1 and £.2 be tWO events. The Imion of El and E2 ls an event d,e~,o~ ~~ ' ~1? £2 and it
Is the_subset. of sample points,~a~ ~Iong to E1 ,and/or E2, Tne interseclion of El and E:? Is an
event denoted by £l' f'I £2 and,is ~he sl;lbse1..oi ~ple"po~U::~I~nglng to b'ot.1l E} 'arid E2.
The tWO events ~1 and £2 are said t~ be mUluall." ex'?-lusilJe if the)~ ace disjoint. ihlit'is if they
ha"e no sample poinu in common. In this c~ £1 f". E2 '" C!'-,: where 0 'fi titi Impossible event
(an-emplY set},' .• -- ..... r .. ';. , .. , - ~. .. •.. ..
. Let P. -bf-nample-'.ipace and E an event, The event com.aining all the sample'poini,s in n that
. "an- o;'t in E is called the camp/eruen tary ·ellent and is denoled by E. Obviously. E u E • n
an·dE()E-0. ' . • . .. ,
Il is easy to show thai the 'i~ter$ectlon and union operations obey the following commutative,
associative. and distribuUye·hiws
.. " ~ .
El n (E2 n E31 '" (E1 n E2) n E3 }
}
(2.2)
12,3)
(2.-11
i
~
"](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-26-320.jpg)
![2.3 AXIOMS ASD THEOREMS 0: PROBABrL1n' TkEORY 15
Due to these Jaws it mtlkes sensE' to t'onsicier the intersection or the union of tOI!' events
E1, E2 •...• En' These new l''ents are denoted
and
n• E] - El '" E::!. n ... . "1 En
i-I
U• Ej - El U E2 U ... 'J En
i-I
Exercise 2.1. Prove thl: so-called De Morgan's laws
ElnE2c~
2.3 A."{IOMS AND THEOREMS OF PROBABILITY THEORY
(2.5/
12.6)
(2."} )
(2.8)
In this section is shown how a probability measure can be assigned to any e·ent. Such a proba.
billty measure is a set function because an event is a subset of the sample space. Further. the
prob:lbility of the certain e'eot (the sample space itself) is unity. finally, it is reasonable to as·
sump that the prohability o~ the union of mutually exclusive events is equal 10 the sum of the
probability of the individual events. These assumptions .re given a mathematical, precise formu·
lation by ·,he following funciament31 axioms of probability theory.
Axiom 1
For any event E
0< P(E) < 1 (2.9)
where lhe IUnction P is the probabilit), measure. peE) is the probabillty of the event E.
Axiom 2
Let the sample space be Po. Then
P(O) - ' 12 .10)
Axiom 3
U £1' E2~ ...• En are mutually exclusive events then
• •
P( U Eil"l~p(Ei)
i-I j_ )
12.11 )](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-27-320.jpg)
![26 2. FUNDAM£NTALS C?F PROBAIUUTY THEORY
41 .(x)· ' ' r1n : eXPl-,?" ldt
X v~:: -
'-00 . ' ..
(2.49)
The corresponding $tand"rd normal density functioll ill
(2.50)
Due to the important relation (2.4S) only a standud normal table is neces~·. The functions
';x and 4lx are ",own in figure 2.8.
f.;x tx, A -3-2-1 123
, 1.0 ---. .:::-:;-~-
x
Figure 2.8. .: , ; , '
r .
. Let the random variable Y .. tnX De normally distributed N{~y. 11~)' Then the"l'lndom variable
X isS3.id to follow a Joprjthmic normal distribution with th~ paiimeters ~rE R and Oy > O. The "
IOjl:.normal density function is . . .; -. .,.
1 1 1 inx-,uy 2
'x(X.)~ ay$ xexP[-I<--,,-.-) J (2.51)
where x;:' O.
: Exezcili<! 2.'7. Derive the lo&,-normal densit)' function'·{2.51),by the USE' of equatioD (2.38).
Let X ~ ]og·non:uii1ly distributee. with the parameten .u y' and ~y: Note that lAy and 0y are not
equal to.ux :md "x:ll can be shown that
(2.52)
EsetcisC! 2.8. Lei X be log·normally distributed with the parameters Ily and ay. Show
!~t .
. lnx-lJy
Fx (xl· P(X .;; x)· 4>(--,,-.- ) (2.54)
wh"ft' .]. is the 5c:mciard no~ttl cii3tribution function.](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-38-320.jpg)
![:. -
2.9 FUNCTIONS OF RAl'Dm.t VARiABLES 33
It can be shown that
" (2.16)
where i ., (xl' ...• xn) and f xCi) is the probability density function for the random vector x.
Exercise 2.10. Show that
" " E(l";(Xill - IE!f;(X;lI (2.77)
i-I i-1
so that the'op'eration of expectations and summations can commut~'~~i£ompare with
exercise 2.5.
Exercise 2.11. Show that
EliZ" ';(X;J1~ II" E!f;(X;)] (2.78)
i-I i-I
when Xl' , , , • X:1 are independent random variables,
_ ~et Xl and X2 be two random -ariables with the expected values E[X1 ] ., Jlx
1
and E[X2, :a
IlX2
, The mixed central moment defined by
, (2.79)
-: '~ ,"!to ...... <':" .. - ••
"'-"" '.
is called the cOlJariancl! of Xl and X2. The ratio,
Cov(Xt , X.,]
PXl X~ " aX
t
aX
2
-
r, 'r;
(2.80)
: •. "
where aX t and aX
2 ~ the standard deviations'of the ran'dom variables Xi and X2'.' is called
the correlation coeffici.~nt. It can be used as a measure of mutual linear dependence between
a pair of random variables. It can be shown that - ~,;;; .oX1.X2·-';;; 1.
Two random variables Xl and X2 are said to be uncorrelated if .oX!::<2 .. O. It follows from
equation (2.76) that .
(2.81)
Therefore. for uncorrelatec random variables Xl and X2 wt' ha'e
E(X,·X,] = PX, JE(X,J (2.82)
!](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-45-320.jpg)
![I 2, FUNDAMENTACs OF' PROBABILITY THEORY
I !
,
!
1: is imponant to note th:n indcpentlem random variables 3n~ uncorrelat!d. but uncorrelated
'anables art' not ir. i!eneml independem .
. :Note that
(2.83)
Therefore. the mutual correlation between random variables Xl' X! •...• Xn can be expressed
by the so-called COt'Griancc mcurix C defined by
.;', .
.. .. . . • . . . . , c. ·..',Co,jX,.X,:.) 1
..• " • • ,.: .: • ... ~ CovIX2
•Xnl
. .
I,· •. , ·Var!X21
.. ....... ':. V..{X,I J ....
Exercise 2.1.2. Let the random 'ariable Y be defined by
where Xl ' X:: are random variables and ai' 32 constants. Show that
Varl '1 .. ai VarlXl J + 3i Var!X2 J' + 2a132 C~'[Xl' x:!]
2.10 MULTIVARl.-TE DISTRIBUTIONS
(2.84)
(2.85)
(2,86)
The most imponan~ joint density function of two continuous random variables Xl and X2 is
the biLoariate norln~! d.~ns~t~.fu'}.rr.tion , j!:i'en ,br
(2.87)
,· .. here - ... " ~1 " ... - ... ..:;; x2 .:;: -. and,.: ';:2 are {h.e-means, 0l:~ 02 · ~he standard deviations
and p the correlation coefficient of Xl <lnd X2 , .. .... . .; . "1"," ' .:,
Exercise !U3, Show that the mo.r~inal density functio~li ~Xl (Xl) for (2,8?1 are
1 lx,-~~ f X (Xl) c -=- exp! - - (----.l) J (2.88)
• I , / 2:: a
l
2 0 1](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-46-320.jpg)
![BIBLIOCRAPHY
.. ' . ,;.
The multiloariale normal ciensilY {uncllon i;; defined aJ;
," .. '
" ',." ;~
BIBLIOGRAPHY
12.11
12.21
12.31
" ..
12.41
' (2.5]
12.61
12.7J
An~, A. H·S, & W. H. Tan(/:: Probabili~y Concepts in Enginacring Planning and Design.
Vol.l, Wiley, N. Y.,197S.
Benja~in. {R: & C. A. Corn'ell: Probability, Statistics and Decision for Civil Engi.
neers. Mcci~a~.Hii1: N.Y' .. 1970.
Bolotin. V. V.: Statistical Methods in Structural Alecharlics: Holden·Day. San Fran·
cisco,1969.
Ditlevsen, 0.: Uncertainty Modeling. McGra ..... ·HiII. N.V .. 1981.
Lin, Y. K.:'Probabilistic Tneory of Structural Dynamics. McGraw-HilL N.Y., 1967.
Feller, W.: A~ Jntrod~ction to Probabilit)· Theory and ils Applications. Wiley, N.Y.,
Vol. I, 1950, '01. 1l,1966,
Larson, H. J. & 6. O. Shubert: Probabilistic Models in Engjllacring Sciencc •. T. Wiley
& Sam. N.)' .. Vol. I 6: fl. 1979.
35](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-47-320.jpg)
![42 3. PRODAB.ILlSTIC MODELS FOIl LOADS AND RESISTANCE VARIABLES
1
with mean given by
(3.15)
Type I asymptotic distribution of the smallest extreme: This is of rather similar form to the,
Type I maxima distribution. but will not be discussed here. The reader should refer to one of
the standard texts·5ee (3.81.13.111 or 13.51. '
3.3.2 Type U extRme-value distributions
As with the type I e:.;:trem~va1ue distribullons, the type II distributions 'are of two types. Oldy
tbe type II distribution of the largest extreme will b.,e conside~ed here. Its di~tributionfu~ction
Fy is given by
Fy(Y) s::: &p(- (u/y)") y;o. O. u > 0, k > 0 (3.16)
where the ~eterS u and k are related tO,the mean and ~~~~dde~iat~on by ..
#J.~ =ur(i~l/k)
"',1.
(3.17.) .
,
0y - u{f{l- 2/k) - r 1 (1 - l/k)]2 with k> 2 (3.18) .
where r is the gamma function defined by
,- -11 11:-1
r(k): e Il. du
. ·0
(3.19)
It should be noted that for k '" 2, the standard deviation Oy is not defined. It is also of i~terest
that if Y is" type II maximi'distributed, then Z;;. .l!ny is type I ma.xima·di·~tributed~ .
Elo:etcise 3.2. Let Y be type II maxima d.istribuied with ~istribu~~.o~,~uncti~~l'Y and
'. coefficient o!nriation ay/Jl.y' Show that the variable representing the largest extreme with
distribution function (Fy(y»n has the same coefficient of variation.
The type II_~~ d~tribu~ion is freqlJ.~ntly,used in modelling extreme.hY,drological and meterologica,
l, events. ~F ~~.as the limiting distri,bution of the largest valLIe .of manY.independent
ident.ically distribute~ .~.9P~_var~ables,_ whe~. the parent distribution_is limited to ,values greater
than zero and bas an infmite tail to the right of the form
3.3.3 Type III exbeme--'alue distl'ibutions
In this case only the t)'pe 1IJ asymptotic distribution of the smallest extreme will be considered.
It arises when the parent distribution 15 of the form:](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-54-320.jpg)
![3. PROBABILISTIC ~IODELS FOR LOADS A~D RESIST.-:CE VARIABLES
with
13.26)
'nd
(3.27)
Comparisons of the type I maxima and type II maxima distributions with the normal and lognormal
distributions are shown in figure 3.4. The random variables in each case have the same
mean and standard de;ation. namely 1.0 and 0.2.
3.4 ~IODELLING OF RESIST_-"'~CE VARIABLES - MODEL SELECTION
3.4.1 General remarks
In this section some general guidelines are given for the selection of probability distributions to
represent the physical ,uncertainty in variables which affect,the.strength of structural components
and complete structures - for example, dimensions, geometrical imperfections and material
properties. Since each material and mechanical property is different. each requires individual
attention. Nevertheless. a number of general rules apply. Attention will be restricted here
to the modelling of continuously distributed as opposed to discrete quantities.
The easiest starting point is to consider the probability density function fX of a random variable
X as the limiting case of a histogram of sample observations as the number of sample elements is
increased and the class interval reduced. However. for small sample sizes, the shape of the histogram
varies somewhat from 'sample to sample. as a result of the random nature of the variable.
Figure 3.5 shows two sets of 100 observations of the thickness T of reinforced concrete slabs
having a nominal thickness of 150 m!", which illustrates this point. These data wer.e not. in fact,
obtained by measurements in rea] structures but were randomly sampled from a logarithmic
normal distriJution with a mean JlT"=' 150 mm and a coefficient of variation VT ". 0.15 (see appendix
A). The corresponding density function fT is also shown.in figure 3.S.
For comparison. figure 3.6 shows data obtained from'a real construction sit~.
A clear disti!lction mus~ be made. however~ betwe-en a histo~am or a relative freguency diagram
on the one hand and a probability density function on the other. Whereas the former is
Simply a record of obsen-ations. the latter is intended for predicting the occurrence of future
events· e.g. a thickness less than 100 mm.
If the probability den,;ity function fx of a random variable X is interp_ret~d as the limiting case
of a histogram or re!ati'e frequency dia~m as the sample. size i~[Ids tojnii~ity. the probaoiiity
P given by
,x:!
P=>P(:<'l<X':;X:!:I= ':x(xldx
• x! I
13.231](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-56-320.jpg)
![3,6 ES11~lATlO!l: OF DlSTRIBt'TIO:-'; PARAMETERS
It cannot 'be e~phasised.tou 5lrongly that the blind apPlic~tion of ~~atiSUCal p:ocedures can
lead to ,,'ery misleading results amI that an initial inspection of the a'ailable da~a should always
be undenaken before any formal calculations are made.
Let I.lS consider the practical problem of estimating·the parameters ofa single disi'rib"ution func·
tion fr.om a single sample of experimental·data. The first. step is to check the dau for obvious
inconsistencies and errors. Manually recorded or copied data have a high probability of contain·
ing at keast.some transcription errors. These should be eliminated if possible. The second step
is to'plot the data in the form of a histogram to check for outliers and to confi.lin tbat its shape
does o"?t devia.te markedly from.the shape of.the density function being fitted. If the histogram
is" clearl}' bi·modal when a uni·modal distribut.ion is being fitted to the data or if the sample ap·
peals'Co be truncated wnen the variable is ~sumed t~ be unbounded. checks 00 the data source
. ~. arecleady'requii-edjri1consistenCies are often found to aris'e'when the'~~'t'of data h$ been ob·
"', t~ned from.experimental test programmes in more than one iaboiator,:,i. Such lumping of data
is often necessary when the sample sizes would otherwise be very small. but this should'be a~::
voided if possible. Checks on the consistency of the means and variances of the v~ous sub·
. samples (see for exampl~ [3~5l): should generally be under~k~'n wh~n practic:abl~., .'" .
The next step is to estimate the parameters of the selected distribut!on using one or more of the
tecbruques described in section . -". 3.6.1 below. The baSic methods are :.. ,.
•
the method of moments
•
the'method of maximum likelihood
•
virious gra~hi~"ai procedures
•
use of order statistics.
The last step is to check that the sample data are well modelled by the chosen distribution and
parameters. ~lethods fordoing,this are briefly reviewed in section- 3.6.2.
3.6.1 Techniques for parame.ter estimation .
This! i;;~~ larg~ subject in itself andonl)'. a brief description ,is p~~sible here. Readers unfamiliar
w.i,th ~he :~,i~~S co.n~epts s~~~ld~lso study a specialist text (3.11J. 13.5],13.8].
It is assufliea in the ,f911o)'ing that the distribution function is known or has b-:en postulated
.and that its parameters are now to be estimated, Depending on the distribution type. one; two.
three or more parameters will be involved. The general procedure is to obtain estimates of
the~ 'unknow~' parame~er~ in terms of appropriatE.' functions of the sample '8lues. The word
estimale"is used in this contest advisedly. It should be clear that because of the random nature
of the variable no sample. howe'er large. is completely representative of the source from which
it deffi'e5;and indeed, small samples may be markedly unrepresentative. For example. in a ran·
doni'sample of 10 independent Ob~~~'~r,j,~~~ drawn from _~.~ormal distrib~tioo. there is a prob·
ability of approximately 1:1000 that by chance all obsen'ations will be grean:!' than the mean,
Any attempt ~" p.:>timate the parameters (p, oj of the parem dis.tribUtion from this particular
sample will result in cOll"idt'r3ble error. This difficulty cannOl: be eSl':l.ped. but the probability
of Jargi> errors occurring decreases as llic s:'lmple size increases.](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-71-320.jpg)
![60 3. PROBABILISTIC ~10DEtS FOR LOADS A;.IDRESISTANCE VARIABLES
In essence there are two types of estimau~s for distribution p3Iame~ers that can be obtained
. point estimates and in!ert.'al estimates: A point estim3.te is a single 'estimate 0"( the 'pa~meter
whereas intetval.estimat.es allow certain additional confidence or prob~bility' s~temen'ri t~ be
made. In this s:;:ction only-point estimates will be discussed.
The different techniques of parameter estimation su~m:irised below correspond to the use
of dif!erent functions of the sample data and give rise to different estimator~ 'ior'th~ para~~ters .
.- number of desirable properties which characterize IIg00dll estimatorS·areiunbi~sedness. efficien.
cy and consisten.cy. (For a precise definition of these terms, see .f or exa~ple. r3.ill). No ~sti· .." . ,' . ~
mator. however, has all th~e properties and in practice the choice' or estimator is iov~~,n:~ .by
the particular requirements of the problem, or expediency .
. Jethod of moments: Let'the variable of interest X have a probability de~~it~ iun~tion..rX' with
parameters 81,92, ::". 6k . From equation (2.35) the jth ro?~ent o~ X is given by
'ij .. E[Xi] ... r xiixex)dx
Since fx is a'functlon of the k parameters 8'1' 6~ ....• Ok' the ~ght han~ side of eql:1at!9n
(3046) is also a function o~ the same k parameters and tj may he ~xpressed.as
" ..•. . " (3.47) .
Usin~ equation (3...16) to generate the first k momentS t. we obtaln k equations in the k un-
J , ..• • " . .. ' . '.
known distribution parameters 8j .
If we now -consider ~ 'iandom sample of' th~ ~ariable X of size n '~ith ,values (xl' -Xa t.· ••• xn)
the equivalent 'sample mOrl:lc'nts ru-e given b~
: ; . .
(3.48)
Finally. the moment estimators 8J. i'" 1, .... k for thek unkna'~. d~ttibU~iO~ p~me~fS 8, ,'
may be obtained by equating the momen~~'~,~ -:f. ~~d the s~ple,momen~ ro, ~:'r' , ".:
j -1. .. : I k (3.49)
Example 3.12. Let X be a normally' distributed random ·aria.b·I~: having parameters jJ. and
(1. The ~e~sity iunctio:n,given by equation (2.45) is ~ I • • ' •
':"-.
(3.50)" . .
Assume that a random sample of n 'observations of X has b~{(obi3ined.iil' x2, ~" ,' X~ ')~
The moment estimators for Jl and a= are now determined ;;.:; follows ... , J' •
:,: ;'.: ,.;.:.,,:
p.51) .,,;](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-72-320.jpg)
![68 4. FUNDA.IENTALS OF STRUCTURAL RELIABILITY THEORY
Reliability {unction: Typically. howe'er, the probability of failure of a system or component
is a runctio~ of,ol>enlting or exposure time: so thnt ,the reliability.may, ~e expressed in tenns
of the distribution function FT of the 'aryable T. the rand,om timeit?Jailure: The reliabil!ty
tunc.cion 4tr ~hich is t~'e probability th~t t~e sy~ will, ~t~!l_~ C?~.rational at tim~_t is gl,ve,n
by : .;~ . .
(4.H.
If the density function (T of the time to failUre is Imo~n .~h~n ~T' ~~y')e e~pressed ~
llT(t) .. 1 - rt fT(,)dT .. r~ fT(T )d, . lo It · (4.~1 .
In some circumstances there 'are good a priori'rew'ns Cor sel~ting a particular form r~r CT - .
for ~xample. a' Weibuil distribution, whosc' distributiori lunction takes the f~rm ' .
, ;'. ;)" .•.. ; .
FT(t)'" 1 - exp[- (tiaJ ' .... . '
. ~ .,,; . . .. ' ~: ' . (4.31
Substituting in equation (4.1) glves '
. .;
t" 0 (4.41
' .'.',
Expected.life; , ,A further property of a system or component which is 'orinterest is the ex·-·
peeted life or the expected,time for:which the syStem or component ~an 00 expected to o!'er.
ate satisfactorily, Referring to chapter 2. this is airen by- .
EIT; .~-.. 7'fT(r)dr .. r 6tT(t)dt
. 0 0
. (4.5)
since, integrating by parts,
roo 6t T (t)dt .. (tlRT(t)J" + roo ttr(t)~t .. E[TI Jo 0 Jo
(4.6)
provided lim [tlRT{t)] .. O. ,--
Failure rates and hazard functions: Also of Interest- is a knowledge or how the rate' of failure "
changes with time for any particular fonn of reliability function. The probability of failure
within any given time interval I t. t + oS tl is the probability that the actual life T lies' in the
range t to t ... -} t and is given by
".,.,](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-80-320.jpg)
![01,3 STRUCTURAL RELIABILITY ANALYSIS
]n'c'ontrast to electronic/mechanical systems. structural syi~ems tend not to deteriorate, ex·
cept by the mechani.5;nS'()f corrosion and fatigue, an'd in some cases may even get stronger-
71
for example: the incrEia.se in' the strength of concrete with timt, and the increase in the strength
of soils as a result of consolidation. Whilst basic dat.a are a-ai.lable for the time to failure of
electronic and mechanical components, no such information is avaiJable for structura,l componentS,
because in general they do not fail in service. The probl~in is of a different nature.
Structures or structural componenLS IIfailn when they encounter an extreme load, or when a
combination of loads caus~s" an 'extreme lo~d ef(~ct of sufficient magnitud~ for the structure
to attain a .failure state»; this may be an ultimate or a servicea~iJjty co~ition. Pan. of the
problem, therefore, is to predict the magnitude of these extreme events, The other part is to
predict the strength or toad..()e~e<:tion charactepstics of eacry ~tr')Jctural component from the
info·~ation available at the ·de.si~ ~tage: I~ ·other "words, it is n~c.~~ary to synthesise probabi·
listie models for the two parts ~~r the p;obl~~~ ·i'~~l~din~, on the one· h~d, all the unce;tain.
ties affecting the loading, and, on the other, all the uncertainties aff~cting the streneth .or resistance
of the structures. As discussed in chapter 3, statistical data may be available to quantify
some of the sources of uncertainty and this information ca~ be included in a suitable
form. Other sources of uncertainty may have to be assessed subjectively,
A further difference between electronic/mechanical systems and suuctural systems is that
whereas most structural.systems aIe aone-offl., all but the IllOst complex electronic and mechanical
systems are produced in considerable numbers and can be assumed to be nomina.lly identical,
Th~ . exi5tence of this underlying population of nominally identical systems or ·compo"
Mnts ~eans that it is possible t.o interpret failure .probabillties in terms of relative frequencies,
This is less justified for civil engineering syiOtems, and the probabilities· that are determined
should not be thoue;ht of in this way. They should be thought of as intermediate results in a
decisia:n·making process and with liule or no absolute meaning of their own • .As discussed in
chapter I, the calculated reliability or failure probability for s.particuiar structure is not a
unique property of that structure but II function of the reliability analyst's bck of know-ledge
of the properties of the structure and the uncertain Illlture of the loading to which it,
will be /i.luj=1.<;:d i.1 .~he future.
4,3.2 The Fundamental Case
For a simple structural member selected at random from a population with II known distribu'
tion function FR of ultimate strength R in some specified mode of failure, the probability of
fail~r~ Pr under t.he action of a single known load effect sis
::.. I , ;. '
P, c P(R,..., 0;;. 0) = FR (,) = P(R/'" 1) (4.13)
If the load effect S is also a random variable, with distribution function F S' equation (4,13.)
is replaced by
IU4)](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-83-320.jpg)
![·- __ 4JW'DAMEl'!.TALS OF STRUCTURAL RELIABILITY THEORY
M<O
Failure
,
o 'JIM m
Figure 4.3. Illustration O{t?,~ rel,iabUity i~dex Il.
: '!
and L' "
(4,24)
giving ,
oM .. (o~ + CJ§i'2 (4,25)
Since Rand S are normal, M. a linea:" function of Rand S, is also normally distributed
. _:_and (M - /-1M )/°14 is unit standard normal, giving
- 0 ' ',;',,;: C';;,: _,~ -,;~" 1(. .' :",:,'::- "
Pr-<fl( :P.!~)cq,LJJ~~JJ~,]' -'",-
M ,_,:,-ca~ + (J~)2J
(4.26)"·
where <fl' is the standard normal distribution funct.ion and I1s.I1R'. as 'ana oR are'respecti'vely
the means and standard deviations of random variables S and R.
The r:e1iabilit)' index ~. may now be define,d as ine ratio JJM/oM or the number of standard
deviations by which 11;'" exceed~ ~er?~,S~~ figure 4.3.
Hence •
(4,27)
For the more general case where R and S are jointly normally distributed with correlation
coefficient p. e9uation (4.27) still holds but oM is given by
(4.28)
Example 4.3. If Rand S are !:loth log-normt>,lIy d:stributed. Pr may be expressed ,as
Pr - P(M' '" 1) (4,29)
where
11' - RIS (4,30)](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-86-320.jpg)
![' .. ,
76 ~ . FUNOA.t£~TALS·OF STRUCTURAL RELIABILITY THEORY
. ',:,! ~ " -,. '
out owing to the nature of the end restraiols .. ,there is consid<!rable uncertainty in itS 'effective
'length '~. ihis uncf~nt); m:ly be repr~ented by, me:deUini L as a lognorm.u ,,ari.
able 'vfth a' rri'ean 'o{'4 m and 3. coeWciem' of variation VL · 0.15. The other quantities C.
the model uncertainty, and E, Young's modulus, may also be assumed to be lognormally
distributed with the following parameters ". ."" .•.• ,:
/J1;" 0.9 '
;"
, #J.E ,- 20~ ~N/mm.~ VE ,. 0.025 . . "~_ , ., .', .
Assuming'that 'there is no 'uncertainty associated '~vith the diameter D:'~~d th.e required
value of D such that the column has a reliability of 0.9999 for a 50 yea~ reference period,
This problem may be solved as follows. If the applied load is p. failure' occurs wlte.n
Cn- 1 ED~
,: p' :::Pc --~·", !;'
or 're-~'ngin'g',' ,'vhen'
",._~ , ,r- '_" 1 ",
0 - (64 Ll p)'i , ._,"
C:rlE
'" ',.' ~
Taking logarithms t.o the_ base e give,s
-.- " • -' .. <,-, ~' , ..
' .. (.,', " , ,, , ~
.,' " . . .'" " .: (~
.. ' ,')
1 .
~nID) - '4 {£nP + I!n 6-1 + 21!nL - enc - 3.enrr -l!nE)
neplacin~ 0, the actual diameter. bye, the variable diameter at'which failure will just
occur. ;md using the rules of expectation and variance for linear functions gives
1 . . ." . . .. .. .. . . . .
~1~nel-:t(E[enPI '~ E[2n6·lj' + '2EC2nLJ- Elinel..;.. 3E[Qmfl- E(QnEI) ,
and
- ,,: " ', , .. " '. ' : ri,:; i:.: : 'f;',-"" ' " '" ,
,~; : '
" Var[2n9 J . ·11S'(Varl2nPJ ·+k1 VarlfnLf+ :VarUnC," + Vai{2~I)I - ':1:':
:: . . :. ~ ,' .. . .1_ .. .:"-,
But from equation ~4:_3~):: _.. ....; : -;
': ~", .:,', V:uihPJ .. tn{Vp ,~~ ~J __ ~:~.~O.251' 4- 11 :~ 0.0606 .
Similari>', V'ar (inCI:I O.q0995, Var(2nEj" 0 ,001)62 and Ver{tnLJ "" 0,02225,
Substituting these values yields
Varlene]" 0,010013
and thus "~ll ~ :::0 0,10006.
From e~uations (-I.3-1) :lOd I·L35) , . ... . . :,, ',
ElenP] = ~nJ.lp -t (1~ = ~n1250i - 0.5:< (0.0606)"" 5A9115](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-88-320.jpg)
![82 5. LEVEL 2 METHODS
j
In a leve13 method knowledge of the joint probability density {unction fx is required. but in
the level 2 metho,d presented in this chapter only the expectations
i '" 1, ... , n (5.1J
and the covariances
i,j -1. 2 •... , n (5.2)
J is equal to Cov[ XI' XI]. . I ,- ,
are used. Note that the variance Vatl XI J - Q j
In t~e {ollo~'ing ltjs assumed that a set of basic v~a~le,~.X 'r. (Xl"'," Xn) is c;:hosep in, such
a ~'ay ,that a failure surfa,ce .. (or I!':'1i{ state sp.r(ace} ,~n .be .definecJ in . th~, n1i~ensi(:mal bas~c
variable;$ pa~ w ~ ,A fail':lr~ ~urfa~ .. i.s a. su~[ace _di~ding the bll$ic . vlI!i~a~le spac~ _i.~.~. two regions
namely a failure region w(.and a $Ore .regio,! w,' The failure region contains all reali:z.ations of
X that would result in failure, and the safe region contains all realizations of X that would not
result in failure. It is convenient to' descri~ the failure surface by an equation of the form
, ' : ,; ~,
(5.S)
in such a w~y that positive values o'r i indicate safe sets of basic variables (the sire re!!ion) and
non:po'sitive values of r indi~te uniare sets of 'ariables {the failure' re~ion}, i.e.
when x E w,
when~. S WI
A 2<iimensiomi.l case is illustrated in figUre 5.1:The function f: wl'""'t R is called the failure
function. It is important to note th3t the same failure surface can be described by a number
or equivalent failure functions.
---:----:1----'6~----",
Firure 5.1.](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-94-320.jpg)
![". . ..... . . ~,' c .~ ;·~ ."· .... ; . ::.· c·-;'. '~c~ --. .. -,;., : -' ~ "!'.:'.-
5.3 UNtAR FAILURE FUNCTIONS AND NORMAL 8ASI~ ~~jUABLES" .• ~ :. ' '-'. ) '.~. 83 ., ...
Let f ~ a failure function . The random variable M '" r(X) is then called a safet)' margin.
,"];' .. . '
Eiampl~ ·5.1 : COnuder the fundamental case with only t~o basic,"ariables (a l~d variable
S and a strength paruneter R) and a failure function, (1 :,~~tjR" ~here .. '. .
" (5.5) .. . -- The failure 'surface. the'failure region and the sale region are shown in figure 5.2. The
col1'esponding.safet,.· margin M I Is g; ~e~ by ' -" . . .,. t
lof1 ·R·..;... ·S (5.6)
An.eq~~al~nt fah~re functi~n is 'f2': Rl~R~ where
.'. - .. ; ! ,' •. .• , : .. :~. : , . , • . ' ' :! . ,... .
. f 2{r •. sr= 2n1-," 2nr~,.i.n5 ~
.,; ..
. (5.7)
~~th the. safety margin
11.12 ~' in ~ • ~n R - 2n~ -. ' (5.8)
failure n,ion
... re rc~lon
FifU'~ 5 .2
5.3 RELIABILITY INDEX FOR LINEAR FAILURE FUNCTIONS AND NORMAL BASIC
VARlABLES
For the case of a line'ar safety margin M and normal basic variables, the reliabilit.y index IJ is
deftned by
(5.9)
where $1M is the mean of M and aM is the standard deviation of M. This definition o[ the relia·
billty index 13 was used by Cornell 15 ,11 as early as 1969.](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-95-320.jpg)
![','~, ~A HASOFER .... ~O tINO'S RELJ.-BIUTY I~OEX 89
The reliability index iJ as defined by equation (5.9) will coincide with the :elio.bility index) de.
(ined above, when the failure surface is linear (a hyperplane), This -is ~hown (or the two-riimen.
sional case at the end of last section. but can easily be proven in the gene:-J.1 n·dimensional COlle.
Therefore. in this esse the important r~elatiori -(5.28) between {J and thl! probability of fuilure Pc
can also be Cormulated. provided the basic varitibles Xi' i-I. 2: .... n are normally distributed
(5,33)
In the previous section it was shown how an approximate calculation of the reliahility index i1
defined by eq. (5.9) can be obtained when the safety margin ~ is non·lineu by expanding ~1 in a
Taylor Series about the point ,XI' ': . " Xn) .. (~l ' _ .. , '~n J. It is no~ obious.crom lh~ above
reIl)o~rks:that the twp"definitions of ~ will coincide if this expansion is mace 3bout t!le de~ign
point. Th!s corresponds to approximating the non-linear failure surface b~' its tangent plane at
the design point (A in figure 5.5). · .
The definition of the reliability index IJ by Hasofer and Lind can be formt:.::lted in the following
way
15.3-1)
where aw is the (allure surface in' the 'z<:oordinate·s~~tem . The calculatio~ ot ~ can be unaer~
ken in a number of different ways (see also page 227). In the general c~~ where the failure
, 1urf3ce is non·linear an iterative method must be u;ed. Here an iteration :ll!thod. with fast rate
of convergence is given. The procedure will be illustrated by a number oi ii.rople examples
where the failure lunction is assumed to be differentiable. In this case the diStance jj and the.
unit vector Q '" (0.1' ..•• an) given by OA .. ~Q. ~here A is the design paine. can b~ determined
by solving the fonowing n "+' 1 equations -iteratively . . ...., _.
(f·,ll. (pa)2 fk
k-l .. Qzc ',' . '
1()1a 1• /10 2, • . •• po .... ) .- 0
where the failure sucface is give'n~b; ,
,1'" 1. 2 •. , •• n
(5.35)
(5.36)
Example 5.5. Consider th~ $;Im t' tie-am as in ",:xampl·c'"5.3 but now w: th the following de llec.
tion _f~!~r!_critQrion ;JI ,:' .
~ i~ J.. . .
umax -18 ti ;;:. 30 ~ ,.,),3. I
where umax is the ma.ximum deflection. e the mOllulu ~ 01 i']a5,icity .:...,d i the reie".-ant mo·
ment of inerti3. Further. let urn"'" ~ p. ~. e and i .~!.e reaiizations 01 un':·Jtrela.ted r:l.!:aom ·3ri·
abies Urnu ' P. L. E.:lI1Q J with](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-103-320.jpg)
![~QB"'10kN
JJl..r . = 75 kNm
O'Qs -lA142 kN
"11.1803 kNm
':::mple 6.1 and exercise 6.1 random variables QB and M:s both depending on the same
lbles PI and P2 are considered. Therefore, QB and MB a;:. c~rrelated although PI and P2
_' uncorrelated.
;ery often one needs to calculate the mean and variance fo~:!. random variable Y which is a
:linear function of a number of random variables Xl' __ .• X::
Tne expected value ElY] is then given by
•
E[YI:::: ao ..,. I ajE[Xjl
j"l
l::ld the 'ariance Varl Yl by
• • •
VariYI" IafVar[Xj]+ I I ajajCov[Xj.Xjl
j"l i * j
V ... ere equation (G.5) is a generalization of equ~tion (2.86), SH- ~1so (5.15).' .
:.et the random ·ariables Y1 and '12 be linear functions of Xl' X2 · ...• Xn
, .. YI :: IajXj
i-I
•
Y2 '" IbiXi
j .. l
:: '!::an then be shown that·
, n .. " .,.' Il , n
Cov[YI • Y21= IajbjVarIXi-] + I I ajbjCov(Xj.Xjl
j"l i' j
(6.3)
(6.4)
, (6.5)
:'.'1
(6.6)
(6.7)
(6.B)
Example 6.2. Consider again the probiem solved in examp:e 6.1. The covariance between
QB and MB can now more easily be calculated by equatio:. !6.~!](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-110-320.jpg)
![. ··· ·r··
i ,
Consider aE!.:Iin :n~ ~J of n r3~do~ ~a~iables X ~ (Xl' .... An I with eXpfc;fld "aJue~ Ef XI i.
i ::::I 1. 2 ..... n a.'ld :ariance and covariance collected in the CD"arianc.e matrix ex I.deiincf,
ly equation 12.$'; 'I
CO"I~I : Xn l I" (6.91
CovfX . X2J' ... : ....... n... , , . i .:
j !
VariX.] . J
Clearly. no correlation between any pair .oC random variables will exist if this matrix ex is a
dia!lonai matrix.. It will now be shown how a (new) set oC random variables Y = (Y, .... , Ynl..
where Yj , j ". 1, 2 .... , n are linear functions of Xl' .... Xn. can be constructed so that the
co~ponding CO"ariance matrix Cy is a diagonal matrix, that is
16.10)
According to wel:-known theorems in linear algebra such a set of uncorrelated variables can be
obtained by the transformation
~_AT~
',, '1.
(6.11)
where A.is an o.nhogona matrix ,~ith '~~iu~n 'vectors equal to the orthonormal eil1en'vector.s.
of eX' By this tran~for~aiio~ " . ' . .. .
:. (6.121
(6.131
The dia~onal elements of Cy . i.e. Varl "il. i =: 1, ... , n are equal to the eigenvalue$. of eX.
Example 6.3. Consider two correlated random ~riabJes Xl 2I1d X2 with the mean vector
EIXI- (EIX,I. EIX, II ' (2. 3)
and tnt> COl·.;:riance- matrix](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-111-320.jpg)
![le'':' 6, EX~;DED LE'EL ~ METHODS
W~;f!e /.I XI and aX; are the mean and the standard deviation of Xi' By :.:.~ linear mapping (6.15!
t,;;. failure surface in th(' x-coordinate system i; mapped into a failuTe F..:rCace in the z·coordinate
~yi":~m. The reliability index (J is then defined as the shortest distance ::='om the origin to the
f<.. .. :.m.' surface. in the normalized z-coordinate system.
W:.-?n the basic variables X ... (Xl • ...• Xn) are correlated the first ste;: is to obtain a set. of
w:.:orrclaled uariablc~ r r: (Y1, . _ .• 1'>1) as shown in the last section. 7ne next step is:then to
ncmalize the un correlated variables and thereby obtain a set of nor~ized and uncorrelated
l',;:-..::rbles Z '" (Z] • ...• Zn)' Fina1I~·. the reliability index can be define:' in the z-coordinate
s;,',o:em as shown above. This procedure is illustrated in example 6.4.
Example 6.4. Consider the simply supponed beam shown on flg-..:::-e 6.3. The beam is loaded
with LWO concentrated loads Xl and X2 with the means EIX l J ,. E!X2} ,. ~ and the covariance
matrix
ex;:: I Var[XlJ
LCov/X"X,/
wherc-l < p <1.
Uncorrelated variables Y land 1'z are acc~r~ing to equation (6.1:, determined by
Y=ATX-where
the transformation matrix i~,""determined b:-' the eigenvectc:-; for Cx< The eigenvalues
A1. AZ and eigenvectors vI' v2 for- ex are easily determined in tn:! simple case
V}, --1 (1,1)
"2 =,q. (1, -1)
The transformation A is thererore gil-en by
F='7111J II -1
and. according to the equations (6.11) - (6.13), _ ~T"'{. Y" = 4 (X,. + X,)
Y=A X-Y
.>1%. (X -X)
,2 2 1 2](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-114-320.jpg)
![lOS Ij, EXTI::~DEO LEVEL:! ~I£1'HODS
I .• "I
Exercise SA. Determine the reliubilit)' index p for the .iar~[y problem ionnuiateu in (>xam·
pie 0.7.
6.4 NON ·-SORMAL" BASIC VARIABLES :
Until now only second order information hIlS been taken into uccount when ev.duating the r~lia·
bility of·n: mucture. Let the failure .iurfa'ce be lin~ar ~~d let ih~ b~fc' ~:U:iahles be normally distributed.
Then the' following rel3tio"n hetween the prcibabil"riy 'or" fail~;~ P; :lnd the reliability
.,. . -.' . .
.... -, ": ,; . ' -,".: . , .;~
-"', (6.34'
: ; ~! .: .. ;;'1
. " ,~.
where .,. is the stanclard!z,ed .nperna! distri~ution fllnc~ion~ _,', ' ~; ;.' .:1 .. :"
It is -im'~rtlnt to r'~~ember that no information regardio,," the prob:lhilitV cii [llilure can he (lh·
tain~ when the distributions of the basic.variables are .unknown. IUs 'reasonable in some situations
t~' ~X~t that a gi~e~' ~~d~m variabl~ c~ b!! considered nonnaJiy distributed as a gaol!
appro':<imatj:o'n. but oiten'such at'~sum'ption is q~ite unre3Sonable. Consider the yield stre5S
oi a steel bar. For such a random 'ariable negative values cannot occur. It is therefore more
reoJistic to assume lag-normal distribution in this case. To overcome the problem that the definition
space ior the set of basic -ariables Xl' ...• Xn is not the whale Rn $pace. one can apply
one·to-one transformations to the relevant variables. For the yield stress mentioned above
this transformation should map R_ on R and be continuous.
In this chapter the transformation ¢ven by Rackwitz and Fiessler [6.-11 will be usect. This transformation
is chosen in such a way that the values of the original density functions fXI and the
original distribution function FX, (or the r.mdom variahles Xi are equal to the corresponding
values of the density function and the distribution (unction for a normally distributed variable
at the design ]:lOint A (see figure 5.5). i.e.
16.35)
16.361
wherethed~3ignpoinl.-" (xi,'.' .xj,. _ .. x~) and where,ux; andux, are the tunknownJ
mean and :itandard devi:tth:m of the approximate normal distributiCln. Solving (6.35' and 16.36)
wilh re!!:ard:o Il~t and a XI we havE'
.:d .-Llf.lxtij) {'Xi:: :
- I'xr"j i
16.37)
i6.381](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-120-320.jpg)
![."
Uo '. RElI.~BILI'fY ,OF ~-rRl!e!;t:ftAL SYSTEMS
. will re~ult in differen~ loa~s .(st~sses) .~':' ~h~ ~in;gl~ ~~em.ents, ri~fe ';' ,5 is usefu].}n ~~lc~.liatini
the diStribution function FR for the strength R of the series ~ystem. Let FR be .the qistribution
. , . . . . . . .. . _" .. _ . .. , ... .. .", I . .
function for the strength Rj of element i; then
FR(r)<=P(RC r) - 1-p(R>rJ - 1-P(R1 > [1 ()R:> rzr"l· ·.· nRn.,?'rn) . . .- ' '', ' : ..' . "::;; ' :' " " ','- .:. . ".. ' . ..
. • _ . _ :... • '. ... .-, : : ~ ,. n · .",
• ,1,- (1 -: E.'RI.cr~p~l - FR:! (rzV : ' . (1 ,- , FR~ (rn» ~, l-::- !1 (1- FR/rj )) (7,3)
;' . .. I . ~ ... ,', , :~. . • :I,~ l ,
where it is ~med ~h:at - th;e stre~gths oOhc eleme:ntsareinciependent.
;,' .' .(':. --:" '. , ; .':: ;, ~ , ,', '
£:tample 7~1.' Consider 'a"simple'structure ciins:isting of two tchsiii b-~ as shown in
figure i-.6. Let the strength --R"~ oc ieac'f· ten:siie bar b-e ~ ~d6'm(:;ariO:bhi witti the density
function fRe,:showri"in' !the 'Sa~-e"{rgure, Lefthe'siiilctiir{bEi ib'aded"by n '~ingie tensile
force S '" 1.1 kN, It is reasonable to model this structure as a ~rie5 system with two
elements. The distribution function FR for the strength R of the system can then be de·
rived by equation (7.3) if the strength of tW9 elements can be assumed i~dependent, One
obtains
wher.e -. ~.
{ 0 for, ,< 1
FR, If):a t,-t -'';'; for 1.,!, 1<,3 ~ , ,:'
1 (0' ,,. 3
."
1 Ill. '
(r.l
1.0 0.1~, , .._ ~ ~~,kN 3
FiGure j ,6](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-128-320.jpg)
![120 '. REtl.-BILITY OF STRUCTURAL 5'ST£.I$
.,
,.' . "
Filun 1.9
P:uallel ductile systems are relatively easy to deal }Yith because of the simple relation (7.5) between
the strength R of the system and the ~trencths R;. i '" 1.2. , ..• n of the individual elemenJ,s .
• - parallel system with perfectly brittle elements is shown in figure 'j .10, IC:m element in such
a system tails. iu load.bearing capacity is compl.etely .. e~~a~.5te~ .~he ,~ther 'elements mayor
may not be able to prevent the g)'stem from failing'cO'mpletely .• ~y_ redistribution o[ the loads.
In real structures' with a low degree of statical indete~;;;inacy, the . brittl;. iail~re· of one element
will usually result in the subsequent i.lilure of other eiemet:!'r;; because of this.redistribution. If
this is the case, the system behaviour is like a series system. ,However, Cor structures with a high
degree of statical indeterminacy and a relatively high safety factor the system in figure 7.10 may
be a reasonable model because in ;;,uch situations there will often be enough reserve strength
capacity to c3.ITy the load after a brittle element (ailu(e.
Let r t' r2, ... , I", where r < r~ < .. , < rn, be the strength of the n elements shown in figure
7.10. The strength r o( the system is then given by-r"
max(nr1, (n -1)r2, .... 2r,,_I' rll
) (7.8)
It can be shown that under cenain conditions r is a realisation of a random variable R which
for large n approaches a normal distribution.
.." - '
Fi~re] . ln ----](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-132-320.jpg)
![~A SYSTEMS WITH EQt.::ALLY CORRELATED ELJIE::"'TS . !
{I.012
I i !;:. -3.0 i --- I,~
! I
,I ~
i
I I '-....
I 'F=-C ~."". I ..,
~. OlO
{I.OOb
(1.006
I '.'-l ~
0.004
0.002
(r.000
0.0
. 'Ffrun! 7.14 ' '1 .
o.
o
Ficurt ~ .15
0.2
0.' 0 ..
' . .. 3.0
~ •• 2.11
0.' 0.6 0.'
n-1'/ ! ,
0.6 0.' 1.0 P
1.0
'.'1":
~ote thal. as expected. the probability of tallure Pr decreases with the correlation coefficient
p and increa~es ~dth the number oC clemel]ts n.
A formal reliabiiity index Ps for the aeries $~'stem can now be calculated by
(7.12)
The dependence of PSIP t on the correlation coeificientP for ~t - . 3.0 and Pe.- 2.0 and for
n -1. 2. 5. and lOis shown in figure i.15 (lake~ from the paper by Grigoriu & Turkstra).
". . ' ., . I· · " i _](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-135-320.jpg)
![,. .. ',. ..,, ' ,. ...
126 i. R!::LIABILln· OF STRUCTURAL SYSTEMS
Finally
p_.E IRP1Rfl2 1-EIRp_F _4(._oa: _-11 '1-4u: .. ..=2t;J. ...
p VarlRpl 2(1+p)a:< . . 1+.0-" (7.20)
The probability of failure Pf for th@system is therefore .;._
· .. ap+.JP;t]
I
P,' I - I"{ ,fI=P.. )j o{t)dt
.__ I-pp
(7.21)
where Pp is given by (7.17, and Pp by (7.20).
It is obvious from example 7.5 that the only calculation problem in estimating the:'probabm.
ty of failure for structural systems satisfying the assumptions mentioned above is tlJ;e integral
in equation (7.11). For De - 3.00 values of Pr are tabulated in Table i.1 for n :: 1~' 2 •.... 10
and.o = [0 ; 1.001. These values have been calculated by numerical integration using Simpson's
formula and 200 intervals.
1 2 3 4 ·
"
7 8 , 9 I 10
0.00 13.50 26.98 40.44 53.89 67.31180.72 94.11 107.4 120.8 134.2
0.05 13.50 26.97 40.41 53.81 67.19 80.54 93.86 107.1. 120.4 133.6
0.10 13.50 26.95 40.35 53.iO 67.01, 80.27 93.48 106.6 119.8 132.8
0.15 13.50 26.92 40.27 53.74 66.75 79.88 92.93 105.9 118.8 131.7
0.20 13.50 26.88 40.16 53.32 66.38 79.34 92.19 .104.9, 117.6 130~ .
0.25 l3.S() ' 26:83' 40.00. "53.02 65.88 78.61 91.20 -103.6 116.0 ·128.2·
0.30 13.50 26.76 39:80 52.'6'2 65.24 77.68 89.93 102:0 113.9 125.7
0.35 13.50 26.67 39.53 52.10 64.42 76.49 88.34 99,96 111.4 122.6
0.40 13.50 26.54 39.18 51.45 63.39 75.03 86.39 97.49 108'.3 119.0
0.45 13.50 26.38 38.74 50.64 62.13 1 73.26 84.05 94.55 104.8 114.7
0.50 13.50 26.18 38.19 49.64 60.61 71.14 81.13 91.12 100.6 109.9
0.55 13.50 25.92 37.52 4S.H 58.79 68.66 78.11 87.19 95.93 104.4
0.60 13.50 25.60 36,69 46.99 56.66 65.78 74.45 82.72 90.65 98.26
0.65 13.50 25.20 35.69 45.28 54.16 62A7 70.30 77 •. 71 84.76 91.50
0.70 13.50 24.70 34.48 43.26 51.28 S8.69 65.61 i2.11 78.25 84.09
0.75 13.50 24.07 33.01 40.87 ' 47.94 54.39 60,34 65.89 71.10 76.01
0.80 13.50 23.28 3}.24 38.05 44.07 49.49 54.44 59.00 63.47 67.23
0.85 13.50 22.26 29.05 34.69 39.56 43.89 4i .i7 51.32 54.59 57.63
0.90 13.50 20.89 26.27 30.58 34.19 37.33 40.11 42.62 44.90 "47.00
10.95 13.50 18.91 22.49 2: ':oI2:.~0 29.~6 30.87 32.29 33.56 3'4.72
11.00 13.50 13.50 13.50 1 .... .,0 , ]" . .,0, 13 . .,0 13.50 13.50 13.50 13.50
T3blei.1. PrY-lOa after(i.ll)whh;ie " 3.00](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-138-320.jpg)
![S.2 SI~IPLE BOUNDS 131
or
" SsIF)-II F, .
;-1
(S.9)
Return now to a single element i. From the definition of S; and F; it follows lha: the eXre<"ted
'alues EfSjl and ElF;] are associated with thr. probability of failure P(F; ·1) [or element i. in
the (oliowing way
(8.10)
and
(8.11)
The probability of {ajlure Pr for the system is ~en by analogy
(8.12)
As mentioned earlier. determination of the probability of failure PI for a system on the basis
of equation (8.12) implies the calculation of I"l-dimensional multi.inte~a ls, However, very simple
bounds for Pr can be derived lor a series s~'stcm with positive correlation between the stren~h
of the elements.
Assume that
1 ... 1 ;
p(n Sj -l» PISj.q :::l)p(n Sj " l J (S.131
;-1 .j" l
for aliI" i "" n -1. The condition (8.13) expresses that the probability of non·!.allure for the
system consistin:;! of the elements j .. 1. 2 •.... i..l...1 is greater than or equal to the product o~
the probability of non·failure for the systr-m l·{'nsisting or the elements j .. 1. 2 ...•• i and the
probability of non-f(iilure for element j+llsee figure S.l),
j-l
"n1 ... . S-1) ,-, ,](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-143-320.jpg)
![136 S. RELIABILITY BOU~DS fOR STRUCTl:RAL SYSTE~IS
Assume Ihat the a~sllmptions Ill: (2) and r 31 abo ... e ure fulfilled, with Pe '" 3.50 and let
the .:orrelation matrix be
l 0.5 0.2 0.1 ;,] 0.5 1 0.5 0.2
.'. ~o~ 0.5 0.5 0.2
0.2 0.5 1 '0.5
:.- 0.1 0.2 0.5 1
The average .:orreiation coeificient is
The reliability index for the system is therefore
". '·c.. - I 5' j:'-.
Ps .· 3.50.y 1 + 0.28 '; 4 - .;:).38 fS.2S)
Exercise 8.1. Compare the exact result (S.2S1 with the upproximate results obtained by
. :lSSllmiiig'nocorrelation Pjj '" 0 and p~riect correlation /.Iij · 1.
S.5 SERl.ES SYSTE~,IS W.ITH UNEQUALLY CORRELATED ELE~lENTS
In this section series sys~~mswit.h unequal correlation between their elements are considered.
An exact determination of the probability of failure Pc for such a system will, as mentioned
earlier, imply extensive nu~erical calculations of multi-integrals. In section 8.2 the following
very simple bounds were'derived for a series system with n elements
" " max PIFj '" 1)';;; Pf ';; 1-111l-P(Fj '" IJ) (8.16)
iOo1. n i'"
where P(Fj '", IJ is the probability of failure for element i. ~luch ~e"iter upper'and lower bounds
- a~e .the Ditlevsen bounds 18.23) and (8.24).
In some special cases. the exact probability of failure Pr can be calculated relati'ely easily. This
has been shown in section 7.-1 under the following assumptions:
f 1) the loading is deterministic and constant in time
d:) . the strength R,. i · t, 2, ... ,n of the members IS id~nticallY ncirrilaJly disrnbU[ed ~(p., (J)
13) all eleml:!nt5 :.lfe desig-ned to have a common fl:!liability index:3~
I..}) common correlation coefficient p hetwt:>cn any 'pair of elements.
t"nder Ihl'st,l assumpt'lons the probability of failurt:> P:. Li~i'enby !i.ll)
r'" S, • ..:... ,/fj t n d
Pf - l-J 1'/'1. II :10 t __ vl-p
(i_Il)](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-148-320.jpg)
![146
,r",r-~-~' '- vq'iv(--VVv!.:. X(l,J-;-- ---- I I
I 1 I I
[. 1 :] . , ',',' ~ '. I
,I I tl I t~ I
I I ,I ~ --- 1 l 1
I
I
.t T
Fillure 9.1
xC'. 'i
•
I
•
• > realizatlon~
, • • • • • • • • • •
Figure 9.2
For a fixed sample space n the outcome of the set of stochastic variables form an o,rdinary
fu.nctioncalled a realization ,(~ee firure 9.1). A realization might be thought of as the outcome
- '- -, .
of an experiment. If th,e experiment is repeated, the new realisations will not be the same as in
figure 9.1, but its p;obabilistic contents will b~ the same. In figure 9.2 values of X(t1l are ,
shown.for a number of realizations and also ~he associated de~s~ty functi~n fix} (x; h).
Given two instants 'of time tl and;'2 some correlation ,between x(t1) and ::dt,2),wlli usually. exist,
especially when the time.cJifference ltl - 121 is small. This is taken into consideration through
the joint distribution function F {x :.rx} " x2 : L1" t 2) defined (see. (2.58)) by
(9.1)
This joint distribution function for arbjtrary.(t1., t2 J E Tl .is called the joint d~bution (unction
of.order 2,The_ corresponding joint density f~!lction of order 2 is given by
(9,2)](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-158-320.jpg)
![148 9. INTRODUCTION TO.STOCHASTIC PROCESS THEORY AND ITS USES
In practical applicatiol,ls, the modelling of a physical quantity by a stochas~ic process must
often be based on a single realisation of a stationary Pf'?Cess. If only one realisation is a~. hMd
it. is natural to estimate the mean value in the following way
1 rT
Jl 2 if J x(r)dr
o
(9.8)
If this time average approaches Jix for T ..... - the process is said to be ergodic in the mean
value. In the same manner a process is ergodic in correlation if
R(T):: T ~ T ~T-1'" x(t + r)x(t)dt
o
(9.9)
approaches Rxx (1) for T - .... If this property holds Cor all moments, the process is called
ergodic.
Note that stationarity is an assumption behind the definition of an ergodic process so that
any ergodic process is stationary but not. vice versa.
9.3 GAUSSIAN PROCESSES
. In this section so-called Gaussian processes u.re treated. It has been stated several times that a
linear transformation of a set of Gaussian (normal) random variables result in a new set 0'[
Gaussian random variables. This important property of Gaussian random variables is the molln
reason why they ore used for modelling whenever it can be juseified. In a similar manner, it
can be shown that linear operations on a Gaussian process results in another Gaussian process.
A process (X(t), t E T) is Gaussian it the random variables X(t1l, X(t2l •... ,X(tnl nre jointly
normal for any n, t 1, t 2 , .•. ,tn' The probability density function for the corresponding n·dj·
mensional nth order distribution is then given by (see (2.89))
f{x){Xl'····Xnit.l·····tn) ·
(9.10)
where C is the autocovariance mat.rix
CXX(',",) ......... CXX:".'O)]
CXX(tn , t.2) ........... CXX(tn• tn)
(9.11)](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-160-320.jpg)
![9.3 GAUSSIAN PROCESSES .I. ....
and Mij Is the i. jth element in C - I • It is clear from the definition (9.10) that a G:mssian
process is completely determined by the mean value [unction Jlx (t) and the autocovariance
function Cxx (tl • t2). Therefore. a stationary Gaussian process is always strictly stationary.
An important property of a Gaussian process (X(t)} is" that its derivative process (X(t)) is
also a Gaussian process. Let x(t) be a realization of (X(t)} and let
x• (t ) "(fdt x(t) (9.12)
be meaningful. The derivative process {X(t.)} is then determined by the realizations x(t) when
almost all realizations :t(t) or (X(t)} are considered.
Ex.ample 9.1. Consider two independent nonnal random variables Xl and Xz with Jlx =
JlX2 "" 0 and ail · a?cz = a1
• Let a stochastic process {XCtl} be given by )
X(t) "" Xl cos(wt) + X2sin(wt.) (9.13)
where w is a constant. The random variables X(~). ti E T are clearly jointly normal and
their ~tatistics are dc:ermined by the mean and autocorrelation of the process {X(t)}.
By (9.13)
"x(t) - E(X(t)] = 0
and by the definition (9.4).
RX1 Xz (e l , t2) ;; E[(X1 coswtl + X2sinwtl )(X1 cosw t2 + X2sinw t2)}
"" E[Xllcoswtl coswt2 + E[X2]sinw tl sinwt2 :: 0 1 COSW (tl - t2)
since E(X1 X2) • O. From (9.15)
oi (t) .. Rxx (t, t) - Jli (t) = 0
1
(9.14)
(9.15)
(9.16)
The process {X(t)} is therefore a stationary Gaussian process with zero mean and variance
cr'.
Example 9.2. Consider the same process {X(t)} as in example 9.1. The autocorrelation
coefficient is
_ !lx. (t!. t.) _ _
PXX(tl,t2) 0x(t
1
)a
X
(t
2
) -cosw(t1 t 2)
so that the joint distribution density function is given by
:x:~ - 2x1 x 2cos...,,. + x;
2a'Cl cos' ... ,)
(9.17)
(9.18)](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-161-320.jpg)
![1~4 9, I!'TRODUCTION TO STOCHASTIC PROCESS THEORY AtJD ITS USES
where N~ (t) is the rate of positive ttossings of the barrier ( and !'~ U, is tne pte of negative
crossings of t.h,e barr,i~r ~.
Note that. for p6~it.i'e crossings:i: > 0 so t~~t ~rom (~ .30)
(9.34)
and similarlY for negative crossings. Hence, for stationary processes. the joint,density function
fxx is an even [unction in the variable i., The fundamental formula (9.34.> is ealled Rice', for.
milia,
Example 9.4. Let {X(t)} be a stationary Gaussian process with zero mean. The joint
density function fxX is then given by (9.21). From Rice's formula (9.34)
(9.35)
For ~ - D.the expected rate of positive zero crossings is
E[N' (0)).1-''2>. .,. 2:; Ox (9.36)
Example 9.5. Consider a statiOnalY non·Gaussian'process {X(t)} with the (ollowing
joint. densit.y function
fXX (x, x)"
,
Fit:1m: 9.6
r 1. (1 ~ x) 1I :2( 1-X)
for (x, x) E r~ 1: 0] XI-I: 11
for (x, xl E [0: 1) X [-1; 11 (9.37)
othervdse
.....](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-166-320.jpg)
![156 9. IXTROOl."CTION TO STOCHASTIC PROCESS THEORY ~:-;D ITS USES
9.5 PEAK DISTRIBUTION
The results derived in section 9.4 can be used to investigate the statistics of the peak distribu·
tion of::l stQCh:l.Stic process {X(t}}, because peaks or troughs (e:urema in (X(t))) occur when
the stochastic process {X(t)} has a zero crossing. The number of zero crossings of {Xft)} is
equal to the number oC extrema in {X(t)}. The formulas derived in section 9.4 can there·
fore be used when ~X(t)} and iX:(t)} :lre substituted for {X(t)}and {XC;)}. ;.
When the process {X(t)} IS 'a narrow.wnd Gaussian process the distribution of the peaks can be
chherm'i~ed in a.very si'mpie way.'A r~liz;'ti6~ ~'( a' ~:mow.banc!- ,~~oc.~ss· is' sh~.vn in figure 9.8.
It. is similar to a sinusoid, but. the amplitude and phase are slowly' varying. The stationary response
of a lightly damped' linear syste'rri wiil'often'be narrow.banded. when the input procesS is a
broad-banded G;sU:I$ian process,"such as an eaithquake excitation.
In this case the expected number o[ pe3ks abo;e the level U~ > oj per unit time is. with good
approximation, equal to the expected rate of crossings of the'blirrier t, Le. equal to E[X~ (tl].
SimiJar.ly. •. the expected total number oi peaks' per unit time is equal to the expected rate of zero
crossings E[N ~.(O)/. Therefor~, (he expected relative number of peaks above ~ per unit time is
E(N~{nl - ~1~:x
EIN~ lOll' ,
whete the formulas (9.35) :lnd (9.36) h·.il.ve been used.
, ,
The distribution function F:! (n for the 'peak ma~itude (t > 0) is then given by
t' -2"x F,CO- 1-,
nod the density function fxCt) by
t'
f:;:(~) ""!r- e -2~~ 0<, ~ < ..
x
This distribution is the so-called Rayleigh distribution_
19 .. 2)
19.43)
19.-14)](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-168-320.jpg)
![I
mBLIOGRAPHY 159
I
and calculate the probability of obtaining peak values g-,'eater than 5.
(Answer: P(~ > 5) c: 0.075).,
BmLIOGRAPHY
19.11
.. [9.21
" ," "" ' ;" - -, ",:' .. ,
Crandall. S. H. & W. D. Mark: Random Vibration in Mechanical S~·stem5. Academic
Press. N. y" 1973 .
Krenk, S.: First-Passage_ Times_and Ext,:,em.es of Stochastic Processes .. Lectures on Strnc· .
tura} Reliability (e~LP. ThoCt-Chri,sJensenl, Aalborg University Centre. Aalborg. Den·
mark, 1980. pp. 75-98.
[9.3] . Lin, Y. K.:Prob~.~ilistic Theory.of StructuratDynamics. McGraw.Hill, N.Y. 1967.
{9A] Nielsen, S. K.: Strength- and Load Processes. Lectures on Structural Reliability (ed,
. P. Thoft·Christensen), Aalborg University Centre. Aalborg, Denmark, 1980;pp. 39-73.
[9.51 Papoulis, A.: Probability. Random Variables and Stochastic Processes. McGraw.HiIl,
N. " .• 1965.](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-171-320.jpg)
![·" I~!!· ~~ '~~A~'~~;1~,'~~:;t~~;; ~B~i.~~,'· :" .. ': .;..~
I
~
I
I PrmoxXrtl>l. t~IO,Tlr=
- P(X{O) > ~ I + P(one or more upcrossings of ~ I X(O) < ~) (10.21
where P{X(O) > 0 is the probability that the process {X(t)} h:ls a value greater than ~ lit t., a
and the last. term in (10.2) is approximately equal to .-- Plane or more upc::ro,sing~ of t)· L P (n upcrossings of~)
n"'1 .
(10.3)
The expected number of upcrossings (positive crossings) of a level ~ per unit. time ''for a sta1ionary
... process Is denoted E[N; (Hl in sect.ion 9.4. It is convenient to use a. shorter notation ~'x(n here.
When £·X eu - EI N; en) is known, then the expected number of upcrossings in the time interval
10j TJ is equal to "x(U·T. i.e.
""-
E!number of upcrossincsj'" "",(O'T" I n-P(n upcr05sings of tl (l0.4) .-1
It. follo's from (10.2). (l0.3) and (10.4) that
P(maxX(t) > ~. t E [0: Tll" P(X(O) > ~) + l'XCO·T (10.5)
In general P(XIO) > H « I'X (~). T and for most practical reliabllit~' problems "x(~ I' T <c 1. In
such cases "x(~)' T is a gooO approxi~alion of P(max X(t) > ~, tEl 0: T]l. i.e.
P(max (X(t) > L t E 10: Til'" t'x(U'T (10.61
The left hand side of l10.6) i~ equal to 1 - F :;:(t). where F~ It:- the dist.ribution function of the
ma.'<imum value of the stochastic process {X(t)} in the time interval (0: TI. Therefore.
(10.7)
where "'x<t I' T < 1. By (10.;.1 the problem of calculating the distribution function F 1: for the
maximum value oC {X(t)} .. -:X 10-) + X2tt») is reduced to that. of determining the rate of up·
crOSSings (the expected number of positive crossings) "X(U - EIN~ em for {X(t.)). Unfortunately,
exact expressions for "x (~l are only kno'n for some special kind' of processes, An obvious
way of calculating "X(£) is to use Rice's fonnula (9.34)
(10.8)
where 'xx i~ the joint demity (unction for the prOCe5$ {Xft)} and its derh'ative process {Xn)}.
The joint density (un.:t:on ex * can be derived by the l'G-caJled convolution integral](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-175-320.jpg)
![168 10. LOAD COMBINATIONS
10,4 CO.IBINATION RULES
' '' . r _ - .. :, " , • '; ' • • ' . ' '. ' • .1:> :- " ~', ,'" " . ;~!
It ilas ~en empil:lSized earlier that two loading processes willusuilliy not reach their maximum
value in a gi~'en rcierence period T at th~ ,same instant of time. It Is therefore too conservative
to replace ffi3.1: {Xl (t) + .. . + Xr(tj} by max (Xl (tl)+ ... + max (X2(t.)}. On the other hand •
. . ' .,. ' T, " . . "_ :'. T, ,. ".' _: ,_~ T. "il." i" , " , "~,,~ '
m~ {Xl (1) + .' . . + ~(t)} is :l ve'ry corri~~.ica~d:sio e hasti~ v:U"iabl~ t~ ~~~ in practice, so some
kind of 3pproximation must be made.
Using Turkstnzs ru.le, m.r' (X1(t) + .. ' + Xr(t)} Heplaced bY ','stochastic variables,
namely
.,;, (10.19)
where t- i5 an arbitrary point in time. By this'rule the reliability ?f a structure is only checked
at those point5 in time where the individual load pr~esses reach their maximum value. There,
fore, the reliability of 11 stNCture will be oyerestim~ted. However, it has been shown that this
overestimation is usually very small.
:. more refined Nle has been formulated ,in connection with the Ferry Borges, Castanheta load
model presented in section 10.3, tn this model the loading processes {Xl}' {X2}. :', . , {Xr} are
rectaniU1ar load processes with "n{, °2, .. , • nr repetitions in the 'r~ference period T. where
01 C; n2 ';;; ••• < nt •
FO,r r" 2, the rule gives the following 2 combinations for the loads:
.. .. ,! " ' ..
Combinau(]n I ;:':0. of repelitions vi loud
Xo. I 1 2
1 I n, n2101
2 1 'n2
: ' ~:
Table 1~>.1
For r - 3, the rule gives the following -l combinations:
• ': !:! ' !':.I,:':
.. r"](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-180-320.jpg)
![10.4 CO~tBINATIO=' Rt:LES 171
(10.27)
.....
In the normalisecj coorcinat(' system t~e failure surface is th~n given by
.r : • ,. ..,
(12.;•) -. -,,.'1 4 'xl' • --.,....( ~X,: !; + "o X;:'':.: ! 'J'oO (10,28)
:. t" :.: ' - .'
: The reliability index . ~ can ~ow be calcula~d by the same Iterative technique as used in
'. ex~mple 6.iL With the usual notAtion' ' ..
2.511;" -12.5 ' . . .
(J::I 2. • , (10.29)
. 2.14 a1·· . . 2.tJ0x2. Q2
Q'] E -i-' 2.14
° ,I 2.' 2 ....... k' ·;)(lx!!
where (lX2
<md IJ'X2 nre given by (10.22) and (10.23) wlih
xi -2
0.2
:.. · 1 stirt ·f-I-,--..,._I,,"'_"'_t_;o_n_N-,-°r·""·_'-:--I , , ....... ::
, - 1 -'I 2 '1 3"
3.00 ! 3.87 r' 3.11 "'rs.li- -I .:-,
-0.717;!I-o.989 ·,I-O.9g~:. ~~.~.9/·
'. '0.717 i ",,o:'48! 0.141 0,138 ,-,' .."
2.151 : 0_573 0.439 .~:~:9 1
'0281 0:121 0.1 is 0.1191
2.31 j 2.31 1 2.32 2.32 L
" (x: -2) I --o.z
Table 10.3. The reliabi1ity index G/3 = 3.11:
: - : '
(10.30)
(10.31)
(10.32)
-,
Exercise 10.2. Show that the reliability index for the structure in example 10.3 is is = 3.19
Ir the number of repetitiom"1 is equal to 6 (and not 12) but with aU other data unchanged.
Examp1e 10.4. Consider the same structure as in examples 10.2 and 10.a: nie variation or
the reliability index e with the number of repetitions n" for the load process {P.,(t)j is
shown in figure 10.8. ~ ..
. r:
" -- -...](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-183-320.jpg)
![174
..... ---~-.-, ... -.--.-,- -
10, LOAD COMBL~ATlONS
• '., f
Let X .. = rna..... [X<.i be approxima~ed by a normal distribution Xu,:X ,ax) and let:
i"':.. .... 30 •. 3 -3 '~
X_ = X.,..o:-' X •. X5' wil: then be normally distributed' N(- 0.20 + P.x . .J0.40:'';' 'C X J2 J:
;) - , _ ,.,' ,'., ~,.C ' .:> -J "3 7
Finally, Ie:. Xa - max [X5i l be approximated by a norml'l1 'i:::. ... ibo.iuon N(pX ,eX ). ,
i~l_,: __ .. ~Q , .' , ,. . , .' . ,'" '" b b:
.~1 appi~~~~iori~.a,i~ m~de a.i·t~,~;1~si~::p?~n~,pX;':+:,Xj,.~,~/:1~:tri~·~?,rmalised coordin~te
system thef~lure_s,~rfacels theri.~~,el]-'-I:>Y , C" .~ "
-' , " " .. 95," '.' -I' 'j .'-
(10.93~+ 1.3975~·1)'--8 (PX
b
+ 0xbXa)"'O
and the iur.ition formUlas are
. ",'
25. f.; .:
"rr= . "8 pX~;~1~.935
. i.:a975"1 ~.~ G~~Ct'6
--..1.13975 ttl k
1
'
1 25 ,
0:6 =k
1
SOXb
_ 1
"2 -~ 0.40
'1 •
0:'=k
2
0X3
13 __ P.x5 + 0.20 - -;-.3 .;. ,G0:60X ~
";0.--10: - tax )~
_"._. . . 3 _ .-
Tile iteration sc~eme ~ ~ follows:
, !:
, .
,
Iteration No. t Star-
1 2 "3. c 4
-. °X3 L. 1.000 . C.503 0.513_ 0:7Ci1 .. . 0:783
1,
PX3
j-2.000 -0.005 -0.008 -0.233 -0.435
°x~ ~ 1.07'i'. 0.358 0.541 0.698 0.777
t-2.200
.- I I
..
"'X~ 0.557 0.442 0'.302 0.121
I , J
p. j 3.000 00450 2.951 3;386 3;662 -
}
, }
~ , .' . 1
• :" 5- I,
0.811 -0.521 I
)
0.802
,
!
j
0.045
'3.738 .. -,
", J-0.707 :-0.781 ':7Q..€!~L .=.o .. :§39 _ ,=Q.',~Ht9 " -:Q.:487
.,
", 1 0.707' 0.625 0.771 0~84.2 0.8S7 o.8ul
I I
. I ., 0.707
I
0.322 0.615 0.496 0.455 0.442 ,
j
,
!
". 0.707 0:83 0.7$9 0.868 0.891 I 0.897
I , I ,
i I , ! 3.00 , 0.: 7 4.45 3.86 3.77 I 3.76 ,
I , ,
Tatt!e 10.6
(10AO)
(10041)
(10.42)
(10,43)
(10.44)](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-186-320.jpg)
![180
• .; specify appr9pri3te quali~y COf).tr9J.r~e.asures and aeeeptance:critcria,Coc the manufacture
and fabrication of basic materials and components. ' "",1".
• determine the parameters of the relevant models from loading data and from materials
data obtained under tr.e specified standards of quality control and inspection,
• select a suitable safety format· the number of partial coefficients and their position in the
design equations (i.e. the variables associated with partial coefficients), etc ..
• select appropriate representative values of all basic random variables (e.g. nominal. charac·
teristic or mean values I to be used as flXeci"detenninistic quantities in the code,
• determine the magnitude of the partial coefficients to ,be used in conjunction with the
above representative values to achifl"e the required standards oC reliability. ,
Procedures such as this h!l'e already been used in the application-ofstructural -reliabUit.y theory
,to practical level 1 codes, e.g,lll.6], (11.101. [11.121. Some of these st.eps have'already been con.
sidered in some detail. e.g .• he modelling oC load and resistance variables, and others. e.g. quality
. cont.rol·procedures, are: be~'ond the scope ol this book: In~ the ~~~~d'~j:;~c=th~'~hapte~ we shall
concen'trate 'on the questlon'oC choosin'g'suitable saeeiy '(annals [or 5truetu~ co~e5 and on the
01 ; ';~,
ealculation of panial coeff!cients.
11.3 RECOMMENDED SAFETY FORMATS FOR,LEVEL 1 CODES
The safe~y format of a code is defined as the way in which the various clauses of the code regulate
the_ degree of safety, .or more.gene~y' the reliability, of structures desig~ed.to the code.
In p~ic'-:llar, it. concems:.the numbeq)[,partial coefficienls or other safety element.s to be used.
UJeir positions in the de~l gr. equatio~ •. an~ rulesJor load combinations.,
. The following recommenClations for leyel l ' codes are based' on the work of the Interriational
Joint Committee on Structural Safety [11.71. [11:81. and ate likely'to fonn tile bastii'of a new
international standard to replace ISO 2394: General principles for ·the· verification of the safety
of strudures.
11.3.1 Limit state functioQS and cbeckiDg equations
As discussed In chapters 4 and 5,. the genetaI conditions for a limit state' not to be'exceeded may
be expressed as ':'
(lLl)
where
X are the' n basic random 'ariables whieh influence the'limit state:~'~d
is the limit stat~ func~:~~ (t'ail~re·t~n~ti~;n-)_
· ,,_ '. ' ....... . , . .. , . . _ :.,1"' '.::..
The varia~les X ~ay be su~i~ded i.nto 'ar~ibl: loads and .actions. Q. p':fO"!~ent !.?~~~~. 9. rna·
teriaJ properties E. geomel::cal parameters D . .md model uncertainties X (see equation (1.1 l),
. . ~!:. ' ,':. ' : '. . . . m ' _.'" ,,',' .
In addition. each limit stiltt function is likely to involve one or more constants c, EQl.;Iation
(11.1) may thereiore be re",'ritten as](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-192-320.jpg)
![,11.3. RECOMMENDtDSAFETY FORMATS FOR LEVEL 1 CODES 183
The characteristic value xk of a basic random variable X Is defined as the pth fractlle of X
given by . . -. .
"' ~, .
(11.7)
where
Fx is the inyerse distribution function of X. and
p is a probability which'depends on the type of variable being considered (i.e. a load or
a strength). "
The'selection of the probability p is to a large extent arbitrary but is influenced by the follow·
ing considerations "';'.-. ",
• characteris!:ic ~aiue~ C?f loads: and other actions are values which sh~uld rarely. b~ exceeded.
• characteristic values of ~aterial strength properties should normally .be exceeded by actual
properties, . .
• the values of p should neither be so large nor so small that the values x ... are not occasionally
encountered,
• it is often sensible to use previously adopted nomina] values as specified chartlcteristic
values" x'P.
The distinction between characteristic vo/ue and specified ctuJracterisric ua/Ill! (specified value)
should be made clear. The former is a fractlle of a random variable, vhereas the latter is some
specified single value'of the same quantity ~ a constant. For practical reasons it is generally
necessary for the user of a level 1 code to work ,with specified valuas of all the design variables
rather than with actual characteristic values, some of which will not be known at. the design
stage. For example" the actual characteristic value of the 28-day cube or cylinder strength of
concrete is likely to depend on the particular supplier or contractor and is not known in advance_
In this case it is necessary for the quality control procedures specified by the code
wrirers to be such that the actual characteristic.strength of the material exceeds the specified
strength by an appropri.:1te margin or with a stated probability. Similarly, the user of a code
should normally work with specified deterministic values of loads and other actions; it is the
responsibility of the code writers to relate tnese values to the distributions of the actual loads
and actions, ..... .:l t:- recommend associated partial coef!jcients or other safety elements.
11.3.3 Treatment at geometrical variables
Geometrical varia~les ate of t:wo main types· structural, dimensions (e.g. the depth of a beam)
and geometncal imperfections (e.g. the out·ot-straightness of a column) .
. ' 'J.
Structurc( dimensions: 'fhe uncertainties in most structural dimensions D are generally small
ani::i for fhis rea'sort the mean value PD maY'b.e taken as"th"e characteristic value (i .e. d~ .. lAD) ·
Tolerance limits are specifica in codes for most struct.u ia,.i dimensions, and if these are of the ' . ' ~ .
form
I11.SI](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-195-320.jpg)
![11.4. :-'1ETHODS FOR THE EVALUATION OF PARTIAL COEFFtCIE!>lTS 189
, J:h~ PC!!~,~P.~ t,~,~ ~~.i.l~r~ s]Jrfl'c.e:. .. . .. hlch is.closest to the origin is' referred to as the' design point
(see figure 5 •. 5), :m{has c.o:.C!~l;Iinat~~ . WCJ:l' QCJ:2• '.' .• ~Qn) .. 'where (s3e (5.35))' ,'~ ':
' " , .....
. (11.21)
with
(11.22)
andz·"'tJii.
By using the inverse mapping
.' . "
i=1.2 •...• n (11.23)
we obtain the set of values x- for th~ original basic "az:iables ~ co.rresponding to th.c. design
pol~'t z· . tr·the variables X are all "~~m:illY di.~tributed.·then .th~ set of.valu~ x~ .are the values
of the variables at which failu~:'i's most Iik'~ly to ~~ (U-this ev~nt were to happ~n). Le .
. "
i "' .. 1. 2 . ... : . .' n (11.24)
'j, " , ,;'
~here (.r~ Is t~efaii~~~ re~on .... .. :~: ... ~': . ... ,. .;.' .. "
rr x,_~·n~~~:~·~.~~:~~:.'~~~~:~;;~·~~,~~~~~p &~V~ _~~I~,~I?~r~x!mate. . ~'.j • .
It ~an now ~.:s.e~~' ~~~lir ,~~ft.'(a!u~~;~~.w~r,e;.to .. be used as the design values xd:in a de~ermiiiistic
level ~: desi~;cai7.~,!~F?~"tJl~ :r;~!Hl1p_tn.!c,tu.re;would have. a .reliability;index IJ-and a relia:'
b~iity, ~, . ", ~~ ,-.:~)(,-::, (1)'T~1-ls., i( ~ is.M .acsep,!able, reliability (or the structure.; a satisfactory ,
set of partial coeWcients is given by .. ... :, .. ;; .,. ...
":! .
' . '.... !":t ;
w'here ~ '" is the sPeCified yalue.of the. re3i$tance variable XI' and by - 'I
,,. , .'PI ·;; ·. ' .' _ . . , . " .
:oed " x-! :fx('~(zj-» ': r· ."'"
,. _ _ L _ .::L = ~,,--'-
' .' ,. . ...
J X'PI xSPJ x~P~ ~., •. ;, .... ; .• ,: .. ; . .• .. : , _, . . " .: •. , .... : ~'":.:.h~." .: .. , .
where x is the speci'fjed value oC the loadlng,varioble X·-: t ,·
.~PI' :,: ' . . ,' .. ., .; ,,,.. ',' , . ... - .. " '. '"" ' - J
• . .' ::':. '", ':')::l.i:"'! .;. ,] .. i .' . "'. ' :::. . . ;;.:
. Ex~ple 11:.~; '!f ~j i~ ~_~?r!1]~~'r dj~tri_~.t~,d: l.oading ,'3riable. then
Fit ('litz:)) J.lx + (lpiox
." - I '.' , . !
1, x
XJ PI "PI
, "1. .•
(l1.25] ,
(11.26)
i
11r.27](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-201-320.jpg)
![19S
11.5.2 Results of calibration
. . Figure 11.5 shows. l.he sC3ttl~r i~ the .c~mpu~ed failUre pr;o,baJ:?ilities ior the major itrpctural com·
" ~-nent:s desi~ed to "the umiu of BS 153 which were)n,cluded in th~ caliocationcalc'ulations.
The failure prqbabilities exhibit very wide sca.t~c:r vary'U)g. ~.Yt:~ man)' oniers,ofmagnitude. In
addi~ion: th~i~ a;.~ $igniricM~' di!cerenc~s' i~ 'th~'av~~;g~"';;li-~biii~y ~'r &ii,,;;~~ tYPes of co~·po.
nent. Neither of these facts is surprisinc: since the code was originally based on det.enninistic
concepts with no regard for the relative magnitude of the various uncerta.b1ties. It should be
noted that the modellint: did not allow for-the possibilitY '9f"gross elTGIS in deSIgn or ,construc·
tion and for" this reason the' probabmiie's :shci~ld be';h:;ldq;;~ied as a ~~ of ~ia'tive safety
• ' . "'" •. • ,"' ~. " I" " ' "" " " ' "
and not as failure frequencies.
The target failUre probabUitY~PfL" for the new code as 5400; Part 3 was determined as the
"'eighted average of thdailure probabilitiedor components -designed to BS 15~ and was
0.63 X 10 .... In caJculating:UltS 'valu'e";"'s"tiffened compression flanges aDd unw~ plate panels
were excluded, the former because they had not been"shown to behave" satis!!:ctorllY in service
and the latter because the data o~ model uncertainty were coruidered ~gua&e':;": "
:,"," " " .
The partial coefficients for the new code were determined for use with checking equations of
the form
Ihm2 funct.ion(fyhml. other parameters) - ef~ects of t'YfOI GI • 'Y!G2G2' "lfQQ] > 0
(11.42)
where
fy iJ the yield-stress of the steet,
Gl is structu;a(self w~ight. '
G2 is superimposed permanent load,
Q is traffic loading.
".1Ql is a partial coefficient on yield stress which applies throughout the code,
)"m2 is a partial coefficient on the~computed resistance which varies "itb type of component, and
lfGl.-.rrC2 and 'trQ are pa.~ial ~~fficients o~ loads.
~aiues.of the partial coefficie_nts obta:!~ed by mi~imising~e_" quantity SdeCined by equation
(11.38), subject to the constr~nt given by equatio;n (11.39), arc listed in column 1 of table 11.4.
TIle other columnS in this table show the values of partial c~!(icients 'm2 when other constraints
are introduced: For example, -colu;n~ 3 shows the effect of setting 'rm} ~ 1.0, lfQ • 1.5 and 'reG
=~~.l~ _(given ~er_~. ~ a ~ei,h,~" average of 'rIGI ~nd ,7m2)' an~ t~us effec?vely eliminating "ml
from the code.
r.ne quantity S is given in :he ~ penultimate'row of iable "11.4 fof~ach" of the' SetS~ or partiiu coef·
ficients calculated and car: be seen to increase as additional constraints are introduced. The,
ql!a.. .. nity I WjBj given ir. :he 18st row of the Lable is the ratio of the ~mount of steel used when](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-210-320.jpg)
![203
Chapter 12
APPLICATIONS TO FIXED OFFSHORE STRUCTURES
12.1 INTRODUCTION
In the previous chapter I structural reliability theory was found to be a valuable tool in the de·
velopment of rath:lnal safety formats f~r design codes and for the evaluation of par~ial coefiicien~.
This is a major a~ea of application. However, the main purpose of codes is to ens.~re
that structures are designed and built in such a way that they can safety resist the (generally uncertain)
loads to which they will be subjected; and it is therefore logical that reliability theory
should be used directly as part of the design/analysis process, when appropriate. The main pote~
ti'al for ?irec:'t application is with s~ctures baving large failure conseque.nces 'or where the
use gives immediate savings in construc,tion costs. Other areas of application are in the assessment
of existing structures and in research.
The examples given in earlier chapters are mainly of a simple nature and are included to illustrate
the various calculation methods. The purpose of the present chapter is to discuss more
complex structures where the direct application of the theory could be used in the process
of safety assessment. Offshore structures have been ch'osen because of theirecon~mi·c importimce
and because they are toe subject of much othe.r res"~arch. Consideration will be reo
stricted, however, to steel jacket structures, on which a number of detailed reliability studies
have been carried out, e.g. (12.21, [12.26J. Nevenheless. many of the principles are equally
applicable to other types of installation.
In the following, attention is focussed on the reliability ot offshore structures under the action6fextreme
environmental loads (wind, waves and currents). Various aspects of the 11)0-
"delliiig of:iacket stnictures are discussed. but not the broader features of oUshore safety such
as bl~";"'.outs, fires, E'xplosions. collisions and other h~~ds. For information on these topics,
see, for: example, 112.15]. 112.27].
12.2 MODELLING THE RESPONSE OF JACKET STRUCTURES FOR RELIABILITY
ANALYSIS
Jacket structures are currently by iar the commonest type of fl . ....:edoffshore platform and are
likely to continue to be used for oil and gas production in moderate water depths (up to 200](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-215-320.jpg)
![12,!! MODELLING THE RESPONSE Of' JACKET STRUCTURES FOR RELIABILITY ANALYSIS 211
Praclical spectral {orms (or water sur{ace elevation, TI(t) : Research by 'arious oceanol!raphers
has led to a number oC semi-empirical expressions for the (orm of the spectra 5.,,, (w I of waler
surface elevation TIlt), (i.!enerally called wa'e spectra). Two commonly used SpecLJ"d are the
Pierson-Moskowitz (P·M) 112.311 and the JO~SWAP 112.171,
pjerson and Moskowitz found that for fully developed seas (Le. those that occur when the
wind blo ..... s uninterrupted over an effecth'e1y unlimited fetch) the wa'e spectra approximate
to a single form which is dependent only on "dnd speed and two dimensionless constants Q
and /1. In its original Corm, the P-M spectrum was given by
(12.13)
where
w is frequency in radians/second
Wo .. glu19.S
g is acceleration due to gravity (9.81 m/s' )
u19.s is-wind speed in mls at a heigh~ oC 19.5 m above mean still water level
Q .. 0.0081 (Phillip's empirical constant)
~ c 0.74
Making use of equation (12.10), the P·M spectrum may be expressed as a function of cyclic
frequency f, as follows
O"C<- (l2.14)
The spectrum is shown in figure 12.5 tor ... mollS values at wind speed ul~. 5' This illustrates
the increase in wave energy with increase in ,ind speed and the corresponding decrease in the
Crequency of the spectral peak (increase In dominant wave perioJ). Th~ frequency of the peak
fp is obtained by difterentiating SI)r, (C) 'with respect to r and equating to zerO, giving
r = (O.8/1)O.~S -"P
211'u19•5
'Finally, combining equations (12.14) and (12.15) gives
o<;C,,-
. ;"
(12.15)
(12.16)
The JONSWAP spectrum was developed from the results (Jf the,Joint North Sell Wa~c Project
112:17] and" ~pplie$ to wind genemted ..... aves under.conditions.oflimited fetch :md homogene·
ous wind fields .. 1t is gh'en by](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-223-320.jpg)
![2H
I
extreme wind speed are known. then thi~ appro;Jch can be adopted (see also section 12.3.1'.
However •. in. .. s~~,e. ." a.!!~ s.uff~~i~~t st3.tist.i~al, ~~_ta o~, w~ve. . ~ei~h~~. ~~,d ~.~,';1~ ~a~:~.~7 ~~'~ilab!e
to allow the. reliability anal'st lO fonnulate the problem d:rectl)' in .terms of the probabiljw
" '. "' , "~,,, . . ' .. ""'~,., , ' " " '.,,, ,,,· ~ 'i": '-; "1<7 ••• -.,;:
disuibu.tions. 9f,th~_,~~~:~t<~;t~ p':arame.te~",~.g~ Hi ,and Tz ' l~ , s.~~? "8:5es." i~,.~.s, ~~.c:s.~a;.~ ,~~, ~~.Iate
the wave spectra t.o. thes~ sea,st8t~ . p~meters, ~ .. ,;.~! . ~ i-. ,': " :
If fl( t) ,is assume~ to be a nati.onary zero·mean Gaussian pr.9ce,~s I(see,. chapter 9) it has been
shown by Ri~~ . (12 ~3·2} (12,33j that the expected 'n~mber c'C up:~'rossi~~~ t;". '1(~J' th~~ugh tbe '.. . • . '. . . . . ,' , . ' .;-,.J-;.o::: •. .; .. ' . .' ,
le~'el zer~ pe~ .un~t; ~i.!"~. is giv~n by : ".
, • _ . 1
"0 - E[~~ (0)1 ~ (m2/moji,
where mo and m2 are r~spectlvely the z~r.oth and second m,o ments of the spectrum S (t);" ~,
de£aned. f.or the g~neral case, by
, . ,~
Hence, Tz:the mean iime '~tween 'zero up-crossings. is given by
,:- . ':. -. ::.. ,:. ,
The spectral bandwidth parameter f is defined by'
,
o!~ • mom.j -m~
'., · mom.j
. (12.22)· ·
~ . ' . _ ..
(12.23)
.(.1.. 2.24)
and 'Ii'~s'i~ th~-~~nie a < ~;.~ 1. 'hen ~ is close t~ zero, the spectrum co~sis~ of a nanow band
of f;~-~~~.n.'.~.ie s'~~d" ;he.- s~tru~ 'is termed tw,,:ow.band. -, ' , .
Consider now the heights ~ of all the wave peaks (maxima in 7f{t» measured from the mean
still· ..... ater level (7f(t) ;: 0) • see figure 12.4, Canwright and Longuet·Higgins have shown (see
figure 4 of 12,10]) that f.or reasonably narrow·band spectra (t < 0.4) the mean height of the
hignest one.third. of_all pe~~s, ~l/3' is gi'en by
(12.25)
Furthermore, it. can be shown that, for reasonably narrow·band spt!Ctra, the largest depth of
a trough between zer.o cr.ossings is almOst equal to the height of the wave crest which preceeded
it. so that. the wave h'eight H!:1 2~. Thus the significant wave height Hs may be determined
from
(12.26)
Tht last qu.anlity .. that will be .d~5C.ribed h~'r~ is'Tp' the eX~l.ed time b.etween successive peaks
(ma.'Cma) in the process '1(t). FO,t a stationary zero·mean Gal~ian proCi!SS, Tp can be shown](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-226-320.jpg)
![12.2 MODELl.INC THE RESPONSE OF JACK.&T STRUCTURES FOR RELIABILITY ANALYSIS 215
112.91 to be related to m2 and m~ by
(12.2"
Hen<.:e .In: ra~io or the number oC peaks to the number zero crossings. which is a measure of tho;
depee oC inegularily oC '1(t), is given by .
,
r - Tm':, C m2J(mOm~)2 (12.28)
with 0 < r< 1. For narrow-band spectra r is close to u.nity, and the number of ma..ximo. is not
significantly SreaLer than the number of zero up-crossings.
Use may now be made oC equations (12.23) and (12.26) to relate the sea·state parameters Hs.
and T, to the moments of the spectrum S"" (fl, and hence to obtain expressions for the spectral
paramet,ers an terms of H& and T;. Houmb and Overvik: 112.19) have undert.:J.ken such a para·
rpeterization of the JONSWAP spectrum (equation (12.17) to determine the parameters 0:, i
and fm for different sets of values of H, and T,. This inConnation can. be used, together wiLh
the Ilppropriate extreme joint. distribution function of Hs and T, for a particular offshore location
whet. this is known, as the input for a structural reliability analysis.
12.2.2 Wal'e model
Wave theory is a subject in its own right and cannot be discussed in detail in a book on s(fIJCtural
reliability theor)~. Attention will be focussed therefore on two-dimensional smaIl-amplitude
simple harmonic waves, generally known as Airy waves. They are two·dimensional in the
sense that the wave creslS are assumed to be paraUel and to e~tend to infinity, and of small am·
pliwde in comparison with .wave·length and water depth. Airy waves cannot really be considered
to octer in practice (actual deep-water waves correspond more closely to such wave
theOlies us Stokes' 5th order) but :lre imponant because of their central role in spectral analysis
in which the sea state is represented by an infinite number of Fourier components: For a
thorough presentation of Airy and other wave theories the reader should study a specialist text,
e.g, Kinsman 112.21],
The assumptions on which linear, or Airy, wave theory ar~ based are as follows:
the flow is two-dimensional, Irrctational and incompressible,
the pressure at the free surface is negligible,
the non·linear terms in the equations of motion are negligible compared with the linear term:>,
the water surract displacementS art~ smail so that th~ (velocilY), te"rms are lIegli Rible.
Given these assumptions, it can be shown 112.21J that the water surfuce elevation 11 above still
water lavel may be expressed as a function of time t and of horizontal distance x in the direction
normal to the crest
llX. U'" acos(kx - wt)](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-227-320.jpg)
![218 .. ~ .~
12. APP~ICATIJNS TO FlxtD OFFSHORt STRUCTURES
--- --..• . .... .. _- -_ ..
a drag Corce proporlionallo the sql,lare 0: the water fankle 'eIOCit)'~.w~ichmay ~.e repre·
sented by a drag coeCCicient having subs"tantially the same value as for steady flow;and
an inertia f!Jrce proportional to the horizontal component of the accelerative force exerted
·on the virtual maiS of the water displaced by the body.
It.should be noted,thatthe virtual mass governing·the inertia force is ·itself composed of two
.parts. the. mass or water displaced by the actual ·v.olume of the.slender-body and an (external.
Iy) entrained mass or water which moves with the bodr. The latter is known as added mass
and depends on the shape of the body and ·its surface roughness.
~ 'Marison's assuJ'!Iptians can be lal(en to be valid provide(! that t.he body does not significantly
affect the wave·pattem. ·This is ·iNe if the width of the body D is less than about 20% of the
relevant wavelength. The·total force at any instant w time't acting on an incrementallen·gth
of the body dz is then given by
, ;
(12.3S)
where
p is the water density
up . is the instantaneous W8!er particle vel~ity nonnal to the 10ng1tudinal.~is of the body
. ..
up is .the co.rre~pqnding water panic.1c ,,:cceleratioll
A.: . is the cross·sectional area of the bod)'
Cd is a drag c?,!fficient.
em is an inertia coeffiCient .
The total horizontal Corce on the vertical rigid body can be obtained Cram
.:t .. 0
,P(t) ". dP(t)
.... - d
.. ~ !
. (12.36)
To distinguish between the co~tributions of the actual displaced mass per unit length pA and
the added mass (em -l)pA. equation (12.35) may be re-written as. ! .
(12.37)
J:.
However, most fixed offshore structures cannot be assumed to be rigid and;·indeed,·those in
deep ..... ater may exhibit appreciable dynamic response. In these circumstances, thfdnduced'
structural velocities Us and accelerations il• maybe·significant in comparison with up and U·p
necessitating a modification to equation (12.35). Following the discussion by Chakrabanl
.112.1).1 on,a pap~r by Malhotra and Penzien f12.25] it can be sho ..... n that the revised form
. of. ~~o~on's eq'"!~tio~ to allow for structural motion is
(12.38,](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-230-320.jpg)
![224 12. APPLICATIONS TO FIXED OFFSHORE: STR{]CTL-'RES
if,.!)!)
I
u.9999g2
0.99!t9 E
0.999 r 0.99
r 0 .90 r Wcibull
0.80
~
sC31e
0;;0
0.60
0 .50
1)..;0
1).:).0
t 0.20
•
O.t.from 1972·74
I • ruo ,
!
1) .03 I'- --'---L-L•.C 'J ."...,L -'-__' ----'_L L LL.L.u.._~ , , , 5 7 8 9 10 2:0 :30"'0 5060 80 100
fh,:urc 12.9. ~tean·o:iaily wind s()Ced rrorn M.V. Famit. (57" 30·:-<. 3'El
(US +-___________________ --.
1).14
0.12
0.10
.·mean·hourly
parent
}t . y (m/s)
o+-----~----~~=-~--~--+_----4_--
10 20 30 .. 0 50
filure 12.10. Distrib~tiQns or pan:n~ Inti annual eir.tr'cme winds (or Lcrwlek;
.:':./:'
' . ~
Taklnc the useful life of an offshore platform as 25 years. the assumption that th~ 25' year maximum
mean·hourly wind speed has this distribution is even more reaSonable, unless there are
other influences ';.,·hich begin to dobunate at these low probability Ie!,·els. Such an innuence·
eould be the presence oC a numbcriol dillerent storm generating meehan.isms (e.g. tr~pical cYclones
in addition to fully developI{d pressure systems). However. lor areas such as the North
Sea. the major stonn generating mechanisms can be assumed to be of a single type. I](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-236-320.jpg)
![I 226. 12. AP~t;~.AT~qNS TC? t:lXED OFFSHOR~ STR!)enJRES ,
,j- - --.
1~ . 7~~~~ast to. ~he above are som~ ~~s~!~s .t?bttined".bY .~~~_~ f?;?~ .~p.ll.s~a1e sec,tion of a 325
mm diame~!' pile expo.::ed too ~'~ve loading, ~ reported b): ~ and Hibbard (12.20]. This I
: ', ,_.:.' " .:.0 .' , ' .' . '. , •• 1 •••• :"" '" __ _ '1 _. ,... _"
~r","?ilte,,~ an.~.s~s,men~ o(.~~o~~on~s~,~o.t:({i.?~:~.ts, .~~~~pe,n~,~~~.?~ ~y wave .. lh,:O?:. The ~~an
~al~e, .O~taifled for Cd :n.'~ 0.61.W~lh_~,c3e,~fi~~~~t ~rV_~a?on of 0.24, wh~r~~ for em' the
I cC?rre~IX?~di?s: ~ean and ~oefficienl ~f "v~!~~i~~;~!;~~ ~~9. ~ .0~2 .l'r!.?, si~ifi can,t ~jf(e,re.nces
were found for se,as wi.th sign.ifieant w~ve heights ~~,~om 0.8 to 3.0 m. , :"
Experi,mental, da~give no c.I~~, !n~~~.H~n o~ ,~h~t ,ar~ app~p~~e. val~.e.s ~o take,for Gd and Cm
for off.sho:~e ~~P~,cati~:ms : Although t~se force coefticie~u may .weU be co~pl~x detenninislic
fWlctions of basic parameters such as viscosity,.~ensitY,,';Uld Dow geome.U)·._these relationships
are not k.n,?~~. What is m:or~~ ~ven if t.hey were ~~~wn for s~ple is"~~~ d . ~il.e.s " i?teraction ~!feeLS
betwet:~, i~,d~~'~,:,a1 pile:. ~~ a jacke~.~h;, ,eff:~Ls o~ ~~, m~~ gr.~~t.~ and th~~fn~raH~ion
of waves and 'currents would .render any simpl~ th~ory invalid when applied to real structur~s,
The're is thetl!r~iel $~'rY :~o~si-derabl}' uncerLai~~;:'in' the c~~~~~ i~" is appropriate to c~~-
. " .... , _ .. ,., 1 ' " .' " " '~'" .' " " . - ,'" .J." • 'J" ... ' . •• ' .
sider Cd and' C;., "as random yariables wh'ere tlietolfl:l uncertai.rltY ~ fr.o~.the. cOmbined .ef-fec'i
of a large n~'mbe; ~/f mainl)' unkn~wn ·,in~,:,:en~~s. M~~ vaiues' for Cd ,and C~. of 0.75' and
1.8. respectively. with coefficients of variation of tictweeD 2.5~ and 3;;;% provides a reasonable
pr?babilistic model for these force coefficients for members of typical jacket structures, The
coefficients can be sensibly assumed to be nonnally distributed. " .(.. . ' r, ... , :" ,·, . . ,
12.4 METHODS OF RELIABILITY ANALYSIS
.12.4.1 ~enen1 : , e'" ' ; _ i .~.' ,
The basic prir.~ipl~ and meth.o,ds,of §tructural re~ia~~litytheo~)' " have, been_discussed in.chapters
4, ;; and ~ •. bu; ~ .. nll.mbe:r of &C:lditionai topAcs,need to be_considered here in connection with-offshore
structures.
In ch~p~rs _ 4 and ,5 it was seen that the key s~p.in . any analysis is the identification of a suitable
ma~h~_l!1:atical model which defines survival· in tenus-of a function·
(12.51)
where X is,a 'cctor of basic random variables (e.g. wind speed, material properties. dimensions,
etc.J . ln most of the simple structural examples given in earlier chapters. the form oUhe failure
func~io n f is es.plicit and is easily determined, as-for.example in the case of-the propped cantilever
beatp. gi" en in, example 5.3. However. for complex structural systems this is 'not the case.
For dynamically sensitive· offshore structures.the [unction_fincorporates ail the'models [or
loading and response given in section 12.2', .. together with the.fallure criterion far the structural
component under consideration., e.g. the buckling oC a stiff~n..ad. tlilbular member under the
combined aClion of end loads and, moments. The only sensible.form of expression for f in this
case is a camp1.!ter program.
Reg3,f~less 91 tne . c~mp:icx ity of the.(uncti9n .~. ,~e , princip_IC$.o.f reliability analysis are 'the same.
pro1d~d that the basic randam 'uiables do not have p~~et.e1S which themselves change with
, 10- '"". , ,_ . .. ,,' ,"](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-238-320.jpg)
![236 12. APPLICATIOl'S TO FIXED OFFSHORE STRUCTURES
[12:211 Kinsman. B.: Wind Vaves. Their Generation and Propagation on the Ocean Surface.
Prentice-Hall. Englewood Cliffs, :-;. J •• 1965,·.
[12.22] Laboratorio Nacional de Engenharia Ci;!: Wind in Western Europe. Results of an
Enquiry. Proc. 34/13{4047. LNEC. Lisbon. Feb. 1973.
[12.23] L. E. A. Offshore Management: Permanently Located Offshore Structures • Jacket
Sensitiuity Study. 1979 (unpublished).
[12.24 J Malhotra. A. K. and Penzien, J.: Response of Offshore Structures to Random Waue
Loads. Journal Str. Div. ASCE, STIO. Oct. 1970.
[12.25] ~Ialhotra. A. K. an? Penzien. J.: Nondeterministic Analysis of Offsllore Structures.
Journal Eng. _~I_ech};>.i~~ ASCE. EMS, Dec. 1970.
[12.26] Marshall, P, W. ap.d ,Bea, ~, G.: Failure .Uodes of Offshore Platforms. Proc. lst Inter·
national Conference on,the Behaviour of Offshore Structures. Vol. 2, Trondheim.
1976.
[12.271 Moan. T. and Holand. I.: Ris/t Assessment or Fixed Offshore Structures· Experience
and Principles. Proc. 3rd International Conference of Structural Safety and Relia·
bility. Trondheim, 1981.
(12.28J )'-Iorison, J. R .. O'Brien. ~L P .• Johnson. J. W. and Schaaf. S. A.: Tile Forces Exerted
by Surface Waves on Piles. Petroleum Transactions, AThIE, Vol. 189, 1950;
[12.29] Moses. F.: Reliability Analysis Format for Offshore Structure~. Proe. Offshore Tech.
Conf. Paper 3046, Houston', 1978.
[12.301 Newland. D. E.: An Introduction to Random Vi!:Jrations and Spectral Analysis. Long-man,
London, 1975.
[12.311 Pierson, W. J. and Moskowitz, L.: A Proposed Spectral Form for Fully.Develop~d
Wind Seas based. on the Similarity Theory of S. A. Kitaigordshii. Journal or Geo;
physical Researe~,.vol. 69, No. 24. pee. 1964.
{12.321 Rice, S. q.: .i.~Jathematical Analysis. of Random Noise. Bell Systems Tech. Journal •
l12.331
- [12-.34]
(12.351
[12.361
. Vol. 23, 1944. Reprinte.d in Wax, N. (ed.):Selected Papers onNoise and Stochastic
Processes .. Dov:er, N. Y .• 1954.
Rice, S. O.: .. Hathematical Analysis of Random Noise. Bell Systems Tech. Journal;
Vol. 44, 1945 (Reprinted with [12.32}) •
. Ro:::enblatt. M.: Remark~ o:n.·a Mult(-Variate Transformation. Annals of MaHil!matical
Statistics, Vol. 23, 1952.
Sarpkaya, T.: The Hydrodynamic Resistance of Roughened Cylinders in Harmoliic
Flow. RoyallNst. :--.iaval Architects. S!Jring ).[eeting,-1977, Paper No.4.
5tuueroep Problematii!k ,-,on Offshore Constnleties .Stu POC): Probabilistic Refia·
6ifity' Analysis for Offshore S}ructlfres. StuPOC·V-5·6. Netherlands Industrial Coun·
<:il for Oceanolo~" English El~ition. 1980.
,I](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-248-320.jpg)
![91BLlOGRAPHY 237
[12.37] Van der Hoven. I.:Pou;er S.occtrr<m of .1'orizontal Wind Speed in the Frequency
Range from 0.0007 to 900 Cycff)s per .=iour. Journal oC :-'Ieteorology, Vol. 14. 1957.
(12.38J Wiegel. R. L.: lVaVf)S and Wove Spectre ,;nd Design Estimates. Con!. Deep-Sea Oil
Production Structures. University of Cilifomia. Berkeley. 1978.](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-249-320.jpg)
![Let the initial cost of the structare be given by
'; :.
245
(13.2)
where a is a constane. and let the consequential cost of Cailure, should it occur, be
(13.3)
The probabiliey oC fallure, given a gross error of magnitude g and a model uncertainty of
magnitude k. may be expressed as
Pri g, k - P(gR- kS" 0)
and the expected conditional total c;ost as
ErCrlg,-kP' c1 ~ ·c; p~ig. k ."
(13.4)
. -.....:: :: (13.5)
Assuming that the model uncen~nty K can take the following discrete· values with prob.
ability mass p(ki )
k; 0.4 0.5 0.6
p(ki ) 0.2 0.6 0.2
Table 13.3
the expected total cost, given a ;ross error g. E( CT 19l. is
3 . ~ ' .. .
E[CTIgJ 3 I (e, +" 'ciPrlgJP(k)
i-I ·.,
(13.6)
The expected total cost given that .there is a &!ioss ~rror. g~it.h a probability pIg). and no
error (g . 1) with a probabilit.y 1 - p{g) is thus ' " "
3 3 ···
E[CT(g)] '" .E (c, + crPrlg)p(k)p(g) + X (el 4> crPr)p(kl(l- p(g» (13.7)
i-I i - I
Substituting for c} and cr from equations (13.21 and (~3,3) gives
3 3
EICT(g)]:cI !.,~ QJ.la(l + JP!ig)p(k)p(g) + SaPR (1 + ~Pr)p(k)(l-p(g» (13.8)
i-I · i-I
Variable • I ,I.
R - I 0.07
S 100 I 0.10
(I" - 10 eost units
~ ::::: 20 I
Table 13A](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-257-320.jpg)
![2.,l,8
[13.3[
[13.4 [
[13.5]
[13.6[
[13,7[
[13.8[
(13.91
13. RELIABILITY THEORY AND QUALITY ASSUR.A~CE
Blackley, D. I.: The Nature of Structural Design and Safety. Ellis Horwood.
Chichester, 1980.
Bosshard, W.:Structural Safety • .-I .. Matter 0; Decision and Control. IABSE Surveys
S 9/19, LOBSE Periodica 2/1979.
CIRL!.: Rationalisation of Safety and Serviceability Factors in Structural Codes.
Construction IndustrY Research and Information Association. Report :-10 .. 63,
1977.
Ditlevsen. 0.: Formal and Real Structural Safety. Influence of Gross Errors.
Lectures on Structural Reliability (ed. P. Thaft·Christensen), Aalbarg University
Centre, Aalborg, Denmark. 1980, pp. 121·147.
Feld, J.: Lessons from Failures of Concrete Structures. ACI ~Ionograph No. I,
1964.
Institution·of Structula!. Engineers:Structural Failures in Buildings. Symposium.
London, April 1980. "'
Joint Committee on Structural Safety, ·CEB • CEC:-l . CIB • FIP • IABSE • IASSRILEM:
Gener~l Principles on Quality Assurance for Structures. IABSE. Reports
of the Working Commissions. Vol. 35, 1981.
[13.10] :Vlatousek, :'01.: Massnahmen gegen Fehler im Bauprozess. Dissertation );0. 6941.
Eidgenossische Technische Hochschule (ETH), ZUrich. 1981.
[13.11] i¥latousek. M. and Sch.r:te~der, J.: Untersuchungen zur Struktur desSicnerheitsPr:oblems
bei Bauwerken. Sericht ~r. 59, Institut fUr Baustatik und Konstruktion.ETH,
ZUrich, 1976.
[13.121 Melchers, R. E.: Selection of Control LelJels for Maximum Utility of Structures.
Proc. 3rd International Conference on the Application of Statistics and Probability
in Soil and Structural Engineering. SydneY,1979.
[13.131 :-Ioan, T. and Holand.l.: Risk Assessment of Offshore Structures· Experience and
Principles. Proc. 3rd International Conference on Stru't;tural Safety and Reliability,
Trondheim, 1981.
[13.14] Pugsley. A. G.: The Safety of Structures. Arnold, London. 1966.
[13.15] Schneider. J.: Organisation and J./anagement of Structural Safety during Design.
Construction and Operation of Structures. Proc. 3rd International Conference on
Structural Safety and Reliability. Trondheim. 1981.
(13.161 SIA: Weisung fUr die Koordination des Normwerhs des SiA im Hinblick auf Sicher·
heit und·Gt!orauchsfiilligkeit von Tragwerken. SIA 260. Schweiz. Ingenieur· und
Architekten-Verein ISlA), ZUrich. 1981.
[13 .. 17] Sibly. P. G. and WaL1.;:er .• . C.: Structural A.ccidenfs and their Causes. ?roc. Insti·
tution of Ch'U Engi .... leers. Part 1. Vol. $2. :'ofay 19"ii.
[ 13.1B I Tveit. O. J. and Eva.'ldt. 0.: EXperiences with Faiipres and Accidents of Olfshore
Structures. Proc. 3rd International Conference onlStructural Safety and Reliability.
Trondheim. 1981.](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-260-320.jpg)
![255
Appendix B
SPECTRAL ANALYSIS OF WAVE FORCES
.... ;
-. : ' ,; ....
1. INTRODUCTION
The purpOse of this appendix is: to d~ri"e a 'r~lationshiP "between t~e sp~trum of water surface
elevation' 5 1)1) (w) aod the spectru'm oC structural displacements SSS( w) COt a typical multipile
jacket structure in a given normal response mode. The theory is derived from the work of
Borgman lB.II and Malhotra and P~nzien IB.3]. For a discussion·oC the general theory of ran" "
dam vibr8:Uo~ and sP'~iia:1 ~alysiS See, for ex'amPle, (B.21 or IB.H.
2. GENERAL EQUATIONS OF MOTION '
.ossuming tha); ihe/sirucbire can be id~alised as a lumped mass system (see figure B.l). the equa.
tions of motion 'may t>e expressed in the well-known form
(B.l)
. , : ! .dl1
" / "
Fi~urc B. l .](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-267-320.jpg)
![258 APPENDIX B. SkCTRAL ANALYStS or WAVE F~RCES
. .. .< l.. ..' '.' ., ,.. .. " .,
,- .•.. -. - . .. - I!
EQuation (B.II) consists of a set of unco:;.pled modal equations which can be used to obtain
the structural ~es.~nse. ~ ,~ach lIno~a~,. mode separatel~·. tfeating ~~~h.. ~ a one. degree of ftee·
'dom system: ' , .... -." .'. .... . , ... ,. "., i . .: '. ,
4. SOLtJI'lON STRATEGY
. -., (: .
Consideration will be restricted to response in a single mode. Thus, equation (B.1.1) may be re.-
written for mode j as (1.1 :::t Mjj• C - Cjj• K" Kjjl
(8.12)
or ,
. " .- : u ·,
Mij + CXj + K:r:J • L:"'JiPj(t)
j .. l
".'
' . .. ... . ,.-. . ' : . (B.13)
where n is the nUlDber of lumped masse~ Il?d. Pi(t),. cr,il ~p,[ tFb.lI~pi'
. '.' . ~ "' Usltlg complelt: number representation (I ... ,I-I), let
• . ,-. 'I ." ': : " ~ . . ~r~'> " ': ' I. ', ., ..
. Substitution {or' ~, and Xj in equ~tion (B,l.S) gives
.. . . : " _" :':: : " ',J ' ... , ... . '.' , • ' . . .
So
, "
(-w~M+ ICo)C+ K}:!j '-":L' -"'jjPj(t) 1-,
H~I~') t ~'liPj(t)
• 0 i"l .
. , "
where H(lw) is kno .... -n as the complex freq!ency response (unction.
Thus
. • . n
, X·j(w)",wxj - c.wH("..I) .l'+jll(t)
i"1 :-.' ;: . . ~
,; "
" ; ' ." ' ) ~ ' :'(B.14)
(B.15)
(B.17)
! -.
(B.lS)
(B.19)
ancHhe ~'i~ctural velocity for each mass j r~po[,,#tig in mode·j is iiven·Crorn'equation·(B.9) as
' ,'. •• v' :r,1 '.. .' - , . '
. ~., '.';.', -:,. " .;."]](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-270-320.jpg)
![APPE~DIX B. SPECTRAL ANALYSIS OF WAVE FORCES 261
5. ~IULTlPLE PILES
Tho preceding theory applies to a wave force acting on :1 singh! venical member or pile. or to
a br3ced frame lying parallello the wave front. However, if the structure consists of two Of
mote members or fnunes. the wave forces acting on the separate parts of the structure are dif·
ferent, and depend in part on the ratio of wave length to the distance between the frames. The
concept of a multiple pile tr3nsfer function IB.II can be used to modify the drag and in~rtia
coe!ficients to allow for this effect. This approach t.nkes account of the relative positions of the
members with respect to 'Wave length at each component of frequency. The modified coefficients
c~ and ct, are gh'en by
(B.3S,
" . ,
cb DCb ( I I cosk(xn -xm))l IB.361
m .. t ft-1
where (:tn - 1m.) is the horizontal distance betwet!n the nth Qnd mth pile. k is the wa'e num·
ber (see page 216), n' is the number uf piles or frames in the direction normal to the ..... ave
front. and c. and cb c:m be obtained from the equations on page 256.
In pt:lcticai applications, it ls clear that there may be considerable uncertainty in the total.
force acting on the structure due to shielding effects. wave slam. the'effects ot' members in·
clined to the vertical, but. at present these innuences cannot be allowed foe in a rigorous way.
It is suggested, however, that some account of this is taken when m9delling ~Iorison's coer·
ficients Cd and em'
6. COMPUTATIONAL PROCEDURE
The calculations described above require considerable computational e£{on· particularly when
they are us~d within an iterative level '2 reliability anat;sis. In particular, the areas under the
various speetra n~ to be evaluated by numerical integration, and careful selection of frequency
points is required to obtain sufficient accuracy without undue loss of comput.'ltionai eflicieocy_
BIBLIOGRAPHY
{B.l] Borgman, L.E.: The Spectral Density {or Ocean Waue Forces. Coastal Eng. Can!..
ASCE.1965.
lB.2J ~ewl3nd. D. E.: An Introduction to Random Vibrations and Spec.!ral Anaiyois.
Longman. Lonaon. 1915.
(B.31 ;"Ialhou3. A. ; ~. and Penden. J.: .'1on·Determinis tic Analysis of Offshore Structures.
Joumo.l Eng •• lech. Div., ASCE. E~16. Dec. 197q: ' ~
!BAI Lin. Y. K.:Probabifistic Theory of Slructural-Dynamlcs. )1cGraw·!iill. :-l. Y .. 1967.](https://image.slidesharecdn.com/1-structuralreliabilitytheoryanditsapplications-141113070332-conversion-gate02/85/1-structural-reliability-theory-and-its-applications-273-320.jpg)
1 structural reliability theory and its applications
- 1. Palle Thaft-Christensen Michael 1. Baker Structural Reliability Theory, and Its Applications With 107 Figures .: ~. Springer-Verlag Berlin Heidelberg New York 1982
- 2. - /~. "'- .. P.-LLE THOIT-CHRJSTENSEN, Pr of.:sso r. Ph, 0 , Institute ofSui!ding T-:chnology and Struc'luml Engineering .. ~~:~~~.~~::~t C~nlr~ .~t;;!;r1~X $I.S. ~ ~:~> r J .>~",;," :- .:' ~ i:' .' "f;"' .,,::, .::;' MICHAEL), BAKER. ii,s~, (Eng) " Deparlment ol"Ci'ii Eng'in~erillg - •. ," 'Imperial CoUege ofSdenciarid Te.:hno!ogy: .. ·'L?ndon'~ ~-~gi~~.;,Lt:. ~;.;.: H~~ .~~ ~. ~~ .>~ } ,~ / <,~.: :::);~ i~i;' ~- F; ISBN 3-S-r.O-U73J-S Springe r~"er!ag Berlin Heidelberg New York ISBN 0-387·1l731-8 SpringerNerlag New York Heidelberg Berlin .:~ rhi. ,,~!1; i:i l ubje.'1 10 .Vl) ",N, :lIn,lItj Mt rU(f"(l. ... MtlltnM .1I<>Ie ur I'"m ~t"tht m:lLtli:&1 iS ~"ln..~m.~ . ,~ c",Li.~Uf 1111»<: "itnrublllXl. ;'I»"'nllrI __ I~'~U "iiUu>!l:IlIuru. bro.W ....... inlf., r~I»"l<JII(lIon b1 p/lo""'Q~in~ III:1~n,nt «so:nol .... manl, ~nJ >!II~ '" .a.t~ I13nh. l:n""r ~ ~ 4 ui Ibc: Gmll'.OQ CUl'!'ntni U,", ,,"lkCC """ies ~.c "",de fu rOiMf Ih~n ~" .al" ...... ~ Kt i~ ~ablor I') ·'~ "'tnun¥.,.ul1.d:~I Won-, ~I~";';h. ,,; SlHln~tr,·~rt~~ Bc:tlin.llddcixr> I~~ l'nm~.J!1I vcrm~n- " n~c u"" "I' r~~III.red "~m~" ,~~;,I~~I~', ~t •. ,n In .. :(lu"h,·~':"n .toe: ""I 'nl~IY: :"~n i~' ,iic ai:>..:n,·~ ot"" 'I'~,;,li,; ;UI~"l~n" :n.OI.~.n n.",co.l: C'.:n'rl ll<Hn Ln~ rcl"'~nt rrUL~';LI'~ j~ .... nu r~"u!.lLiun~ "n;; :MlClUr" . I('~ hlL"~""cr.i u,". ~)lr"' lrronlon ... : IOLO.Op ... ,lh;,m "' c,hen ... t;.IlJrnhL~~L Il""l.bll1~II1~: IlIm.~ frill,"". 'UUl)ul¥ ~Uo'>I/.()~J_~H2lo)
- 3. pREFACE Structural [.'liability theory is con~meci with the r:ltjonalll1!3tment of uncert:lintie~ in Hrue· (urOlI engin~ring and with the methods :'or a;sessing the :>nitty and sel":iceability of ch·j! ~ n ' '.lin~erin~ and other structures. It i.3 :J. subject which has grown r3pidly during" the l:m de-::lde and has t!'oh'ed from bt.'ing :J. topic {or :lC'3.riemic research to:1 set of well·developea or ::t'elop· ing nl'"'thuuologies with a wide r3.llge of ?tactical applications. L'm:ertainties exist in most :ueas ot d;"jj :md structural engineeri!'l~ nnd rational desi'¥l c~isicns cannot LJe ma<.le without modelling them.:utd taking them into account. :Iany struClur.J.! ~n · Kineers are shielded irom ha'ing to think about such problems. ulleast when designing: ::.-nple' s tructu~:>.lkoc:J.use ui lh~ prescriptive and essenti<lUr deterministic nature oi most .;odes of pr:l!:tk~. This is an undesir.1ble situation. :lost loads and other structural design p:l.r:lm.!:~rs art rardy known with c~rtaint>" and should be regarded as r.mdoro v:1riables 0 stochastic prxC!5slIs. I:wn if in u('si!J:" calculations tll~y are c'entually treated as deterministic. Some prohl~1lU such , ~IS the <lnulysis of load combimuions c:mnot even be fonnulated without re..:ourse to prc!)abili~tic Tt':lsoning. Th ... re is :J. Ilet'u for all stru.;tural en)l:ineers to ue'elop an underst3nding of ~tructurJl rel ~10il ity th'ur'Y ami for this to he appli<.>d in desi~ and construction. "'hher indirectly through CO..les ;.J f. by diflX"t :tpplil'ation in [he .;:lSe of sped.3i structures ha..-ing large failure consequences. :::e aim in hoth c:tses l>t!ing to achie'e (>(-onomy together with :tIl appropri:ne d~ree of safety. T:;e sub· jl..'ct is n~w ~ufficicntly well clevelopoo for it to be included a5 a formal part of the traim::'l oi all ch·i[ anr.! structural engineers. both at !,mdergr.1duate :lnd pOH·gr:lduate It!'ek Cours!1-1 on "tructural sai.,ty have lll!en ~h'en:1I some- uni'erst[ie~ [or a number uf Yl':l,rs. In wruin~ thi~ book w.~ haw tri~d co brini! :1}1lC'ther u::.d~r ont' ~'o"t~r ti,le mqor cOr.:!;:lOn!':':ts ,If stnh:lUral wliahdity lh~ory with lh¥ :urn of makine Ie pU~iib!c (or;~ ne''omt!r til see ':"'1r.!
- 4. V/ stt:dy the suhject as a whole. The hook should be of value to those with no prior knowledge oi reliability theory. but it shoult1:also be of interest to thO£e enginee!S in'otved in the deveiopment of structur31 and loading codes and to those concerned with the safety assessment oi compl~x s~ruc:ures. The .b~k does not try to caver all aspects of structural safety and no at~mpt is made. for example. to discuss structural (:tilures except in generaistatisticai terms. It ',"'as the intention to make this book moderately self.(;ontained and ro~ .thi5 reason chapter 2 is de'oted to the essential fundamentals of probability theory. H~we,:ei. readers.who have had no training in this branch of mathematics would be well ad1sed to tudy a more general" text in addition. Topics such as the statistical theory of extremes. me~hqds_of parameter estimation <lnd stochastic process theory ,ue inttoduced in later-chapters as and when they are required. The mai~ core of the book is devoted to the so-called-leveI2 methods of analysis which have provided the key to fast computational procedures ror -structUral reliability calculations. Other chapters cover the reliability of structural systems. load combinations. iTOSS errors and ~ome major areas of application. , :'. ree work is set out in the form of a textbook with :t number of clCarnples and simple exercises. The purpose of these is to illustrate the important principles and methods and to extend the scope of the main" text with economy of space: The readeds ·.Yarned igainst n too literal in terpretation of some of the simple examples as these were not inc!uded to provide insight into particular pructical problems. In $ome examples. the parameters of t~e probability: distributions used in the calculations h<lve been chosen quite arbitra.rily . ~~ in ~uch _il" way,as to demonstrate the calculation procedure with maximum effect. This doe~ ~~t mea':!.that"th"e practical aspects of nructural reliability theor~' have been overlooked - indeed. the theory ,":ould bEl of little value if it could not be applied. Chapter 11, on the application_ oC reliabil,ity theory to "the de· . velopment of level 1 codes. attemptS to address m;tny of the pr~~~I~:it "p~obI~~s faced by code "!'iters in the selection of panial coefficienu (partial factors); ~d' in ~ha"pter 3. the modellinlJ of load and resistance variables has been approachcd with' ap~iicati6ns strongly in mind. However. 3 complete book would be required to cover this subject-in ~'pth_ Chapte'r 12 on of{shore structures should be of interest to those working in this field. In compiting the bibliography OUt approach has been to list only a selection of the more im· portant works 1n each subject area. 'along with other works to which specific reference is made. Whilst many important contributions to the literature are tnus omi~ted_ it is considered that this selective approach will ha of more help to the new reader:" ''''e should like to acknowfedge the major contributions to the field of structural reliability theery that have been made by a relati'ely small number "ot peopi~.mainly during the last 10 to 15 years. and without which this book would not ha·e·"~~~ " p"asSi.b!~~ Th~ subject has benef:;:: ecl from a large degree of intermltional co·operation which lias p~en stimulated by various ~ociie~ - i;, particular. the Joint Committet' on Sttuctura(S3ie~·~:"~nder. the chairmanship of J. :~:TY Borges. The respon~ibility lor thi3 hook must. nowo;!vet. rest vith the authors ,lnd we ir.OU;U be plt'lSed to receh"e nOtiiicltion of l'orrections or omissions of any nature.
- 5. VII Thanks are due to our respective culleagues in Aalborg :md Lonuon for tnt-Ir helpful comments and contributions and in pnrticulu to MI'$. Kirsten Aakjrer :md :'-Irs. Norm3 Hornung who h,we undertaken the type.setting and drawing of fifPlres. respectively, with such skilJ and efficiency. We conclude with some words of caution. Structural reliability theory should nol be thought of as the solution to all safety problems or as a procedure w~ich t."1n be applied in :l. mecha.ni~l fashion. In the right bnnds it is a.powerful tool to aid decision m3king ill matters of structural safety. but like other tools it c!l.n be misused. It should not be thought of !is an atemat{'c to more tr:aditional methods of safety analysis. because all the information that is currently used in other approaches can and should be incorporated within :d reliability annlysis. On OC(;nsions the theory may gi'e results which seem to contradict '~xperience". In lhi~ case. either IICxperi. ence)t will be found to have been incorrectly interpreted or SOITll! part of the rnliahility analysis will be at fault. generally the modelling. The resolulion of these real or app:nent contr.lI..lictions will often pro;de considerable insight. into the nature of the prohlem being examined. which can only be of benefit. ~1;uch. 1982 P:lUe Thoit·Christensen Institute of Building Technology and Structural Engineering .~alborg Unive~ity Centre AaJborg, Denmark :'o.lichuel J. Baker Department. of Civil Enl;;ineering imperial Colle~e of Science and T8(:hnoto~y London. England
- 7. CONTENTS 1. THE TREAiMENTOF UNCERTAINT1ES IN STRUCTu'RAL ENGINEERIN'G .• ',. 1.1 INTRODUCTION ..••••.•....•................................... 1.1.1 Current risk levels, 2 1.1.2 Struciural codes, 3 IX 1 1.2 UNCERTAINTY •• ,'....................................... ........ 4 1.2.1 General. 4 1.2.2' Basic variables, 5 1.2.3 Types of un~ty. 6 1.3 STRUCTURAL RELIABILITY AN.-LYSISAND SAFETY CHECKING .•... ' 7 1.3.1 Structural reliability I 8 1.3.2 Methods of safety checking, 10 BIBLIOGRAPHY •..•.. , .. .... .............. .'. . • . . . . . . • • • . . • .. • . . .. .. 11 2. FUNDAMENTALS 9F PROBABILITY THEORy...... . .................... 13 2.1 INTRODUCTION...... ............... ................ . . . ... . . ... 13 2.2 SAMPLE SPACE •.•..•. •. ..... , ................ ..•....•..••.... .. 13 2.3 AXIO~IS AND THEOREMS OF PROBABILITY THEORy ... ,.......... .. 15 2,4 RANDOM VARIABLES." .. ,-" ...... ,.,., .. ,., ... ... , .... , .••... ,. 19 ·2.5 l10:!l.1E:'TS ......••..•...... , . ....• , • •.•. "., . • .. ,., ... • . , .. . .. ·. 22 2.6 UNIVARIATE DfSTRmUTIONS ......................... . . .... .. ... ' 25 2.7 RANDOl! VEC~ORS •....... , •.. .. : . . . . . • . . . . . . • . . . . • . . . • • • • . . . .. 25 2.8 CONDITIONAL DlSTRIBUTlO:-.lS ............. .. .. . ........ . . . _. . . .. 31 2.9 FUNCTIONS OF RAl1)O:1 VARIABLES ..... _ .... . ... . .. ' • ••• • ..... 32 BIBLIOGRAPHy •...•...... .... " ...... , .•. , .....•..... , .... •• .. . . .. 35 3. PROBABILISTIC MODELS FOR LOADS AND RESISTANCE VARIABLES. . . • .. 3i 3.1 INTRODUCTION ............................... , ....•. ,......... 3i 3.2 STATISTICAL THEORY OF EXTREiIES ....... ,..................... 37 3.2.1 Derivation of the eumulaLive distribution of the jth smaJlest value of n identically distributed independent randont variables Xi' 38 3.2.2 Normal exttemes. 39 3.3 ASYMPTOTIC EXTRE.IE·YALUE DISTRIBUTIONS ... , ... , • . . • . . 40 3.3.1 Type I extT'eme~alue distributions (Cumbel dinributions), 40 3.3.2 Type II exttem€-value distributions. 42 3.3.3 Type 1II exueme-value dif.!!ibution •. ;-1.2
- 8. x 3.4 ~IODELLING OF RESISTANCE VARIABLES· ~IODELSELECTION ......• 44 3.4.1 General remarks. H 3.4.2 Choice of d~tributioru 'for resisUince variables. 52 3.5 ~[ODELLING OF LOAD VARIABLES· MODEL SELECTlQN; .• :.... ... .. 54 3,5.1 General rem:1rks. 54 . 3.5.2 Choice or distributions of loads and other actions, 58 3.6 ESTIMATION OF DISTRIBUTION PARAMETERS ...... .. ... . .... .. ... 59 3.6.1 Techniques (or parameter estimation, 59 3.6.2 ~I od el verification, 63 3.7 INCLUSION OF STATISTICAL UNCERTAINTy............ . . . .. .. .... 63 BIBLlOGRAPHY ............... : ..•... '::;:: ........ , ..• ~ . ;': .... • ..••.... 64 4, FUNDAMENTALS OF STRUCTURAL R;ELIABI~lTY TJ,-E;ORY .• , '... . . ........ 6i 4.1 INTRODUCTION ... : .........•. >:: :,,;;r .. ~ · : . . .. . ;;: ..... :: . ......... Gi 4.2 ELEMENTS OF CLASSICAL RELIABILITY THEORY ~ ; .. ~ '.: . • • • , • .. , ,. 67 4..3 STRUCTURAL R~LIABILITY .-NALYSIS .•..•.. •• .. _ ~: ... " •. • ,...... 70 .t.3.1 General. 70 4.3.2 The fundamental case.;1 4.3.3 Problems reducing to the (un'damental case. i5' 4,3.4 Treatment of a single time,varying load. 77 4.3.5 The ~eneral case, 71 4.3.6 Monte·Carlo methods. 79 BIBLIOGRAPHy . .. , . .. , ....... ,........... . . . ...... .. ...... . . . . .... 80 a. LEVEL 2 METHODS .••.•.••.. , , .••..••...•. '.' ••• : •••••••• -•••• •• .• , : . 81 5.1 INTRODUCTION ................. ............. ........... . ...... 81 5.2 BASIC VARIABLES AND FAILURE SURF ACES .. : . . .. . . . . . . . . . . . . . . .. 81 5.3 RELIABILITY INDEX FOR LINEAR FAILURE Fl.i'NCTlONS AND NOR· ; , ~ ._.: . ~1AL BASIC VARIABLES .....•...•......••....•.•.•......•.•.. : .. 83 5.4 HASOFER AND LL.'lO'S RELL-BILITY lNDEX ..... ....•...•..•• , _ . • .. 88 BIBLIOGRAPHy ..•........... , ... ,. ..... . .. ...•...•..•...•...•... • . 93 6. EXTENDED LEVEL 2 METHODS ••...••.•.• • •••.•• , •••• •• • • • • • • • . • • • • •• 95 6.1 INTR.ODUCTION................................... . ............ 95 6.2 CONCEPT OF CORRELATlO:s,.................................... 96 6.3 CORREL.-TEO BASIC VA RIABLES ...................... , .......... 101 6A XON·;-';OR:<'IAL BASIC VARIABLES .........• ',' '.': :..'.! •••••• •• ••••• • • 108 BIBLIOGR.-PHY ...... .. ..... ....... , ......... _ ............• • ....... 110
- 9. 7. RELLBILITY,OF STRUCTURAL SYSTE~(S ..•.•..•••...• , .•. .••• " ••...•• 113 7.1 L'''TRODUCTlON ......... ::' .......................... ; .. ~ ....... 113 7.2 PERFECTLY BRITTLE AND PERF:ECTJ,.Y Dl:CTILE·ELEMEN'TS .•.•.••. 114 7.3 FUNDAMENTAL SYSTEMS :.:: •. : ....... ... . : ....... ~ .• : ............... 115 7.4 SYSTEMS WITH EQUALLY· CORRELATED ELEMENTS .:: •..••. : •.•... 122 BIBUOCRAPHY .. · ..... . .. : ... '; ... . : .. : ....... : ..... : .. :. ;·.::-.'l ~ .~I.; .. ... 127 -. . " ~ 8. REL~~~.~"BOUNDS ~(}R STRUcrURAL SYSTEMS' • .• '. ~~: ~ •••.•••••••• 129 8.11):TROO~PTIO~ .. ~ •. ,,',, ................. : .. •.. ~ .. ; ., ... .... : . : ... 129 8.2 $[lPLE BOUNDS ...... .. .... . ............ ; .. , ' ..•. : .' •..• '. : .. • ~ •.. 130 8.3 OITLEVSEN BOUNDS .......• ' ......... ;': '. ' ..... :.::.:: ............ 133 8.4 PARALLEL SYSTEMS WITH UNEQUALLY CORRELATED EiE~IE~"S .. 134 8.5 SERlES SYSTEIS WITH UNEQUALLY CORRELATED ELEME:-..-rs . •••• • 136 BIBLIOGRAPHY ... .-:. ; .' ..... >.:; ; .:. : :,; .. ;'. ' .•. ': -~;. : .. -.... . : : .,;.: ,'. -;" ' . .. '~. '; 143 9. L~TROblicTION~TO STOCHASTIC PROCESS TH£bk'i" :~~~ Irs' 4SES ........ 145 9.11~TROOUCTION .. . ..................... ';- •....... ;' ......... : .. .... 145 9.2 STOCHASTIC PROCESSES ......... , . '.' ...••... . :: ..•.. ',' •.•.•.... 145 9.3 GAUSSIAN PROCE,SSES ...... ...... .. . .... .. . ... :. ~. '. .. .. ~ ; ....... 148 9.4 BARRIER CROSSING PROBLE~I. ... .. . . .. .......... ... .... . ........ 150 9.5 PEAK DiSTRIBUTION ....... ....... . . .... .. . .... ......... :: ....... 156 BIBLIOGRAPHY •......... , ..... , ..•.•.• •. .•.. : . :' ••.. -' .. .' .•.•••..•. 159 '.,' 10. LOAD COMBINATIONS ....... 0 .............. : ... :' .. :~~':.; .. . " ...· 0 .... 161 10.1 INTRODUC!ION .......•.....•.......•....... '·' , o' ., ;.~., ; .• ;.: .... ' ... 161 10.2 THE LOAD COMBINATION PROBLEM .............. ~';':; .• '. :., . : .•.... 162 10.3 THE FERRY BORGES·CASTANHETA LOAD ~IODEL ~.: .• ' • .• 0 ; ••••••••• 166 10.4 CO~;8INATION RULES ....... " ......... ::-: ...... ..... -:.: . : ..• : : ...... 168 BIBLIOGR .. PHY ............ .... ..... . . .............. ... . ::f·. '~'. '~ ~ . .' ..... 175 11. APPLICATIONS TO STRUCTURAL CODES ............ ,." ............... 177 1l.II!'lTRODUCTION ..... ....... ..... : .............. . ... .;: .... . ; .. ...... 177 11.2 STRUCTURAL SAFETY AND LEVEL 1 CODES .. . ...•..•. -.. . : ... , ...... 178 "
- 10. , XII 11.3 mjcO;I:-.tENDED SAFETY F O R~t:.l,.TS FOR LEVEL 1 COnES, .. . . • . .... 180 lli3.1 Limit ;;uu' (ullction; :md checkin,? equ:nions. 180 11.32 . Characteristic ':Uu(',;. of basi" '3.r1ables. 1 S2 11.3.3 T~atment of geomeuical 'ariables~ IS3 11.3.4 Treatment of material propenies. 185 11.3.5 Trea.tment of loads and ,other'actions. 185 11,- :-.IETHODS FOR THE EVAL-VATIO!" OF PARTIAL COEFFICiENTS.; .•.. 188 11A..1 Relationship of parcbJ coefficients to level 2 design point. ISS 11.·L2 Approximate direct method for the e'aluotion of panial coem· d ents.190 .. 1';" :~. 11.-1.3 General metho"d for ihe evaluation of parti:ll coefficients. 194 11.5 A." EXA:.iPLE OF PROB.-BILlSTIC CODE CALIBRATION" .. .. ,.< ....... 196 11.5.1 Aims of calibration.l96 '-. 11.5.2 Results of calibration. 198 ~IQLlO.GR.~P~Y .. • ...... . •.. . .. . ......• : . ..•. :.: ' _" ':' :;.:: . ..•.• • : . • d ••• 201 12. APPLICATIONS TO FIXED OFFSHORE STRUCTURES ... .- ••••... • . ~ ..•••. 203 12.1 INTRODUCTION ........ . ...................................... 203 12.2 M.OI?~LLIfJ.9. THE . ~E~.P9~~E ~F J ~-'CKET ~-Z:~UCTURES. FOR RELIA-BlUTY AN.-!. YSIS ..... ...... . . ... .. .. .. , . .. . .. . . . • . . ... .....•. 203 1~2.l $ea-5late model. ~O; 1~.2.2 Wa'e model, 215 12.2.3 Lo.3.di.~g model. 217 12.2A Natural frequency model. 219 12.2.5 E'aluationof structural r~sponse. 219 12.:!.a E'aluation ·,f pe3k response. 220 12.2.7 Oti1er models, 2~2 12.3 PROBABILITY DlSTRI3lJTIO ~S FOR L:IPORTANT LOADING ' ARI· _-BLES . . ..•........ . .. ... ......... . .. . . .. .......• .. . . . . ...... 223 12.3.1 Wind speed, 223 ',' . ,; ,J" 12.3.2 Morison':; coefficientS. 225 12.4 ;IETIIODS OF RELIABILITY A:-.'ALYSI5 .. ' . . . ... :-: :.":; ;.: :.- . ".' : . ~ . ••... 226 12.4.1 Geneal.226 . 12.-1.2 Lelle12 method. 227 .~ . 12.550)[£ RESL'L TS FRO)} THE STUDY or .- JACKET SrRUcrURE . . . .... 232 BIBLIOGR.-PHY ............ ... ............. . ........•. '" . • . • ' ... ' .... 234 13. RELIABIL!TY THEORY AND QUALlTY ASSUR.-NCE .••.•.•.....••••.••• 239 lS.1 )~"RODUCTION ...... ..... ... ~ . . .. ... ..... . . : .. . . . . . ... ~:. :.' .!: .... !239 lS.2GROSSERRORS .... ..... ........... . .. . . .. : .. -~ ~ .. : •. :; ,': ....... 239 13..2.1 General. 239 13.2.2 Classification of gro$3 error~. 2·11
- 11. XIll 13.3 I:TERACTlOX OF RELIABILITY A~D QL'ALlTY ASSUR.-XCE ... ' " .. 2-13 13.3.1 General. 243 1Z.3.2 The effect of gross errors on the choice of p:ll'tiaJ coefficients. 244 ... 13..1 Ql}.-LITY .-SStJR.-~CE ...................•.................... ' 247 BIBLIOGRAPHy .....................................•......•...... 247 APPENDIX A. RANDOM NID1BER GENERATORS .•...•....•.•...••.••.•...• 249 1. GENERAL .....•.••...•.••.•..•.........•....•................ 249 2. UNIFORM RANDO~l NU?l.1BER GENERATORS: ........••.•......... 249 3. MULTIPLICATIVE CONGRUENCE METHOD •.•.........•.•......... 250 4. GENERATION OF RA..~DOM DEVIATES WITH A SPECIFIED PROB· ABILITY O'ISTRIBUTION FUNCTION Fx . . . • . . . . . . •. . . . . . . . . . . . . . .. 251 5. SPECIAL CASES: GENERATION OF RANDO:"! DEVIATES HAVING NORMAL AND LOG·NOR!o.lAL DISTRIBUTIONS ..................... 252 BIBLIOGRAPHY ...••................••....•.•....•..•.......••.... 253 APPENDIX B. SPECTRAL ANALYSIS O~ WAVE FORCES •••••••..••.•.••...• 255 1. INTRODUCTION ............................................... 255 2. GENERAL EQUATIONS OF MOTION .............................. 255 3. MODAL ANALySIS ............................................. 257 4. SOLUTION STRATEGy .......................................... 258 5. MULTIPLE PILES ........................ . ..................... 261 6. COMPUT.-TIONALPROCEDURE .................................. 261 BIBLIOGRAPHY .....•••.•....................•....•...•.•......... 261 , INDEX •.• ~ •• ' ...•••.•.•••..•..•..••••••...•••••••••.•••.•••..•••..••• 263
- 13. Chapter 1 THE TREATMENT OF UNCERTAINTIES IN ST~WCTURAL ENGINEERING ...•. ' ;. " 1.1 INTRODUCTION:· Cntil fairly rec'entlY there has- been 3 tendency Cor Structural engineering to be dominated by ~eterministic thinking, characterised in design calculations by the use of s~ified minimum ml1terial properties. specified load intensities and by pres<:ribed. procedures f~ computing $tresses :!nd defle<:tions. This deterministic approach has almost certainly been reinforced by the ';ery large extent to which structural enginccrin~ design is codified and t,M !ac~ of Ceedback about the actuo.l performance of structures. For exo.mpl~ .. actual stresses are rarely known . . deflections are rarely observed or monitored, and since most structures do not collapse the real reserves of strengths arl! generally not known. In contrast, in the field-of hydr:lUlic systems, much more is known about the actual performance of, say. pipe networks::wtin. spi1lways etc., 3S their performance in service c:m be relatively easily observed Dr determined.' The lack oC inCormation about the actual behaviour of structures combined ~h the use of codes embodying rel:1tively high saCety factors can lead to the ;ew, 'still heid by some engi' neers as well as by some members of the general public. that absolute safety C:an De achieved. Absolute safety is of course unobtainable: ;lnd such a go:!1 is also ~ndesirnble, since absolute S3J'ety could be achieved only by deplo~'ing infinile resources'. ..~', ~ " It is now widely recognised. however. that some risk of un;:l.~ceptable st~ct~nal' performance must be tolerated. The main object of structural desio;::n is therefore. to ensure;'at an acceptable le'el of prohability. thnt each structure will not become unfit for its intended'purpose at any cime during its sped£ied desi,~n lifE. )!ost structures. howc'er. hl1ve mUltiple ptrformancc ret;: Jirements. commonly espressed in terms of a :set of serviceahility and ul.ornate limit states. :nClst of which are not inuependent; and thus the problem is much-mort! cj:"tlpli:x. than the spe- " . c:fication of just il sin!;Ic' probah~l.ity .
- 14. I nn: TREA nl~;":T OF L1:-':CcnT,!:-;TI~:$ 1:-: STRUCTURAL ENGI!'IF.t~RI~G 1".1.1 Current Risk Levels . .l,.s 01 me::uis of as':('ning Ini' rel:llh'f' imponanC'(' of structural f:lilun'~ a5 a caU~l' of Ol'alh. !;Olnt' comparati'~ SJ,atistics for the U.K. nre ~;'en in tanl(> 1.1 for a numher of caus~~. 'ilws(' iiJrurf'f show that, at least for a typical Western Euwp,~al1 cou:1try. lh(' ri,;.k lO Hrl' from ,;.trUl:tur.. . l failures is nCllligihle. For the 3 yr3l' period repont-d. Ihe :11'cr.Jj!l' numh«!! or cle;lth~ per annum directly auributablt> 10 structural £3;lul(, I'.u l~. divided almost L'qually between failures occurring during conruuction and the failures of completed StruClures. Other structur.ll failures occur in which there are no deaths or personal injuries: but data on such railures arc more difficult to assemble because in many counuie3 they do not have io be reported. In comparine: th~ reilth'e risks given in tabie 1.1, account should be taken of difference~' in ·~~"po$Ure time i)-pical Cor the '~ious activities. For example. although air tra ... el S as.sd~iat.ed with 3 high ruk per hour, a typical passenger rna)' be exposed fur betwee~ only, 53)' ,,10·100 hours per year,leading to a risk of death of between 10" and 10'" per year. ii.e. between 1 in 10J and 1 in 1<rl ). In contrast. most pt'oplf' spend alleast 70% of their life indoon; and are therefore e.'{posed to the possible errects of structural failure: but this leads to an average annual risk ~ person of oLlly 10·~. Ne·ertheless. the only fair basis for comparing this risk is comparison ith the inescapable minimum risk that has to be accepted by an indio vidual member oC society as beyond his cOlltrol nnd for which 110 blame can be attributed to other people. for example, the risk of d.cath by disease. Many people attept 'oluntary risks ., many orders of magnitude hi~ner. but these 3ho~ld not b~ take~ into account when consider-ing structural safety. !!10untaineeting (International I .j Air, travel (Intemation31J 'Deep water leu'ling ,; j Car travel ! Coal ~i.ni_n,g . ... : - . ! Cons~ru~.t1pn .. ~hes._ . : t.lanuf3!'!turing . Accidents al.·~~~~ ,:alii. Accidents at home (able.-b~died persons) Fire at home Naturnl causes 13'eraee. 31 ages) Males aged 30.{alJ causes) Females aged 30 (all'c:lllses) ,.! ~b.le!O aged 50 i,all causes I ! femaJes aged 50 Iail (';J.meH ; J:fumber oC deaths pl!r hour pet 101> persons 2iOO 120 59 56 21 7:1 2.0 2.1 0.7 ,I 0.1 I , 0.002 129 15 13 8' 51 Tachr 1.1 Compan~~ Q~.th rilk. !Ayer~Gr li1:{hlS;3 ill t:.K. boiSed on CC'HflI SI"listic~ Off~t. Abnr:t.c; 19:..11. I I,
- 15. I.;.:? STRCCTl:RAL CODES 3 In assessing the imponance of structural failurei. account should also he- taken a! lbe economh: consequences of collapse and unserviC't:ability. In fact the- economic aspects m:JY be' rel!arded as dominanlsinct' the marrinal returns in terms oC Ih'es 5a'ed for each additional £ 1 r.1i11ion in· vested in impro'ln~ the sarety of structures may be small in comparisor. willI the benefits of investint: the same sum in. say, road safet)' or health care. However. structures should. where possib!e. be designed in such a way thllt there is ample warning of impending failure and with brittle failure modes having l~er safety maTiins than ductile modes (Le. higher reliabilities). 1.1.2 Structural Codes Most structural design is undenaken in accordance with codes of practice. which In many coun· tries have legal status: although in the U.K., structural codes for buildings are simply Ildeemed to satisfYI) the building regulations. meaning that compliance with the code automatically en· sures compliance with the relevant clauses of the building laws. Structural codes typically and properly have a deterministic fonnat and describe what are considered 10 be the minimum standards for design, construction. and workmanship for each type of structure. Most codes can be seen to be evolutionary in nature, with changes being introduced or major re'isions made at intervals of 3 • 10 y~s to allow for: new types of structural form (e.g. reinforced masonry). the effects of improved understanding of structural behaviour le.g. of nif[ened plated structures), the effects of changes in manufacturing tolerances or quality control procedures. 3 better knowledge of loads, etc. Until recently. structural codes could be considered to be documents in which current good practice was codified; and these documents could be relied upon to produce sate, if not economic, structures. These high standards of safety were achieved for the majority of structures. not from an understanding of all the uncertainties that afff!('l the loading and response, but by codifying practice that was known by experience to be satisfactory. The recent generation of structuraJ codes, including the Euro-codes and the associated model cooes for steel and can· crete, are however more scientific in nature. They typically cover a wider ra:lf!' of structural elements and incorporate the results of much experimental and theoretical research. They are also more complex documents to assimilate and to use and the associated desip1 costs are can· s('quenl-ly highet, as are the risks of errors in interpretation. The benefits of these new codes mU51lherefore lie in the possibility of: • increased O'er~1I safety for the same construction COStS; • the same or more consistent levels of safety 'with reduced construction COStsj • or, a combination of these two. A rurther aim should be the trend. where appropriate. towards design procedures which can be applied with confidence to completely new forms of construction without t~e prior need fOI prototypt> ,esting.
- 16. 1. THE TREATIIEXT OF L:N<.:ERT,lSTJES 1:-: STRlil.Tt.:R.L ENGINEERING Tb.e aims and benefits de~cribed above can only be achieved b}' a rutiollal .t5$essment ot the various ~ncert.1irties :lsSocitlted with each type of structure tlnd :!.. study of their interactions. This is the essence of structural reliability analysis - the CundamC!ntals of which will .be described in the following ch~pte;';;: along with some recent applications. 1.2 UNCERTAINTY 1.2.1 General Structural reliabili.ty an~lysls is concer'ned with the rational treatment pf uncertainties in struc· tural engil?-eering desi;" and the associated problems'of rational decision making. Consider the (ol/oIVing statement: " " • All quantities (except physicaI and mathematical constants) that currently enter .. into en· gineerinl1 cnlcul3tions are in reality associated with some uncertainty. This fnct hos been implicitly recognised in .current and previous codes: If t.his· were ~ot 'the case.';' »S3iety tactor"1t only, s~ightly in excess t;)f unity would suffice in 'all circumstahces. Thl! 'detf!rminatlon of appropriate standards o"{ safety requires the quantification of these unc:eflainties by some' appropriate means and a study of their interaction 'Cor the structur~ u.nder can- . sideration. ,'.!. Before continuing. it 15 worth noting that the argument is sometlmes ad~ance~ t~at the magni· - tud'es of all variables are either bounded or can be restricted within specified ,1.imits-bY ~:<ercising appropriate sundards of control. and that these bounding.v.alues shoul?, qe used as the basis for design. In structural engineering. however, such allUments are inap'propriate .~or a number of reasons: • . upper limits to individual loads and lower limits to material strength are not cas!l}' identi· fied in practice le.g. building occupancy}oads. wi~ !oads. t~e yield slress oC.steel. the cube or cylinder strength of concrete): - . . . , ; .r~ ;'. • even if such natural limits exist. their.?irect use in design is likely. to be extremely un· econom.ic; • limits Imposed by quality control and testing can never be completely ef(~tive. particu. larly in the case or properties whkh can be measured only by destructive tests or in circumstances in which changes in the potential properties take place between sampling and use oC the material (e.g. concretel: • even if recognisable limits do exist. their use may not nlways be r~tional. ." : • > ' • ' .; : ~ • 1 . , Example 1.1. Consider:l column su~portinJfn' flc)ors of:l buildin~' on which the loads are known to vary independently with time. Anumin" that the load on ea:h noor is physi· cally restricted b~: .so me hypothetic:tl fai l-safe devJce so th:n unrier no circumst:lnces can it exceed some speci(ied m~imum t':lIue, an? gi':en that each load stays (It this maximum "alue for. say. 1::' of ~h~ eime. the "'tional desi~n load Cor the column can l1ener:ll1y he shawl': to be lesi than the sum of the ma:<imum loads. This desu:!n ioad will. of course. depend on t!le number of storeys supported and the design liCe DC the structure.
- 17. ' :~, 1.2:2' BASIC VARIABLES , ':, : 8§"' '1 I " -I " , ,."',, , ,-' , I, ' . .;;, 1-: 0 . " •.. 15 "''';1 ,'' , .. ;, ,,' , SO. ,Qr .t~~~·S ,~u"p~rt~,..;, ',,', Fi~re 1.1 shows Ihe probabilitieS' p'that the ma.'I>lmum cOiu~~·l~,~d:l.i. ground' t~'el will reach the sum of the ma....:fina of the individual floor'io:1ds ' i:e. the ma....:imum''pOssible ' column'loadra:t so'ine' time 'duiing a 50 year perioa. o'n'the MSumpticri tharthe ,(ioor loads are mutu:llly independe nt ~, that.,they remain con.Slant_for an hour and then change to some new random value. and that each has a 1% chance fp =- 0.01) of being at its mu:imum 'alue 'after each renewal (Le. each Ooor is loaded tJ its maxi!Dl.~ '~~!IU~ (ot:,!lppro:<im~tely 1% of the time). n~e figure shows that eyen ;th only !i:<: floors. :he probability that the maxi- . l' ,. ', mum pos'sible' fcilumn Imid oceurs is"iifsn1all as :1b~ in: 50 ~·ears. 'Even 'if each floor is loaded to its ma.,,<imum value for about' 10% ofthe time (p .. 0.1); the probability 'of the maxImum possible column load occurring Is still very small if the number of floors supported is 10 or more. ,',I ' . ' :,' .. ' " ' , ' :~ l ,.,;:: ' ,';,.- , .. " " In such cases, it would be irrational and uneconomic to design for this Vorst possible con-dition, However, it should be noted th'at in practice the i:f~ee of conservatism depends on whether the_ individual floor loads an In fact inde~ndent: .lnd the :l:ceeptable..risk level IS " ~ '_ ' " - " r'" ,! " • " ', ", ,y, " " ", ' _ " " govemea, bY,the cons~qiIen'ces oC faihire and the rapidit;· with which fallure OCCUrs. Some ". ' 1.."'nowledge"of th'EFp'rob'abilities 'C;! oc~urrence 'olto'ads Ie's:! than the inaXimum'l,.vould also be . : ' required' (Or each noor for a rational uesign: ,' ;",;' " '::::-:.i":;: ,', ' ' ,The precedini example demonstrates that although .. there is no (strictly, negligible) uncertainty ) n the magnitude o[ the,ma.,,<imum load intensit~' :on;each floor. ~ere may be appreciable un- ,- .certainty in the magnitude of the m:t.ximum combined load carried by the column. 'We retum:now to the 'quesiiori o( cl:lssif~;ins't1e 'aI;OU5 't)-p'es of uncertainty:that can arise in structural reliobility analysis, Howe'er. beiore these.:ue discussed it is helpful to' introduce the ,yorc~p,t, ,()r~as!C ',;ar!ahles, Th,is..concept ii, tre~,ted in ::10~e: ~e,t~ in chapler, 5 . ,~ , " . " .1 > ,·,' For ,the! purp9~es of quantifyin;r UI:cert~i~ties in the :Ield o[str..:ctural el')gineE¥'ing. ilnd for sub,~ eRu~wnt .re1~ll.iJ,i,~~; ~ry~,[y~is i~ is nen.'553ry, ;Cl define. a :e.t ,o~.ba$ic z,·ariabfe,$. The~ ;u:e defined
- 18. 6 1. TH£TREAT:.IE... .. ,.Of UNCERTAINTIES IN STRUCTURAL £NGlSEERING as lhi set of basic quantities go'erning the s:.atic or dynamic response of t;he structure. Basic 'aria~les are quantities such as mechanical propenie! of materials. dimensions, unit. weights, e:lvironment~H.OOIds-;.etc. They are basic in the .sen~e that they are the most fundamental quan· ;:~ tit.ies normallY:recognised and use<! by designers 1ulc1 .an:i.l yslS in structural calculations. Thus, the yield stress~o( steel can be considered as a ba5lc variable, althou!':h this property is iUielf depende~i on chemical composition and·variOUs"rili,CrO-5tructuraJ p:ara~~ters. Mathematical models involvina these latter parameters are often use~ by steel producers (or predicting the mechanical p~opertie.s of structural steei.:1i and for .the purposes of quali~y control. HoweVer • It is generall)' seDsible to treat the mechanical propenies as basic variables for the purPoses of • r '; Structural reliability.analysis. One justific3tion is that more st:ltisticai data are available for the l-mechanicaiproperi'r~ s of, say, steels than for the mor~" basic metallurgical properties. • .~ -... " " . .... " :.: It should also be,mentioned that it is genernlly impractic3bl"irlo try to obtain sufficient statisti- L; v cal 'data to model the variations in the saength of complete structural components directly. Re- ;~ li.:lnce must be plact!d on the abilit)~ of the analyst to synthesise this higher level information ~. 'when required. Ideally, basic variables should be coasen so that they are statistically independent quantities, 'Ho;,vever, this ni'a):iot ahy"1iys be possible if the' s'trerigth '~f a structure is k'no'wn to be depend- " em "an, fo~ example, any t~~·~ ~echanical pro~erties 'that are known to be co~elated, e.g. the . .~: · tensile strength and the compressive suength of a batch of concrete.· "1.2.3 Typei of Un'cer1:aiht), ;. · For the purposes 'of 'si~ctural re~~il!~~' :~.n~:-.ss ,it is' necessB.l); .~o d.istinguish between at least It three types of uncenainty - physical uncena.! . .''lry. statistical uncertainty and model uncertain-ty. These are now described. It should be nOled that in the foUowing. ramjom 'ariables will be den~ted by ~pper case letters~ ' Physida'/ u·ncertainty: Whe'tllet';or"riot'a suuet:o.lre 'or ~irUcturi.'l element: fails ;hen loaded depen~'~ , ~ p~ 'on the actua~ :~l~~; ?~~the, ,~ete'.~t m:neri'al prop·~.rii~.s.that gO'~Fn. i~~·~t.rength. The reo : : liability analyst must therefore be concemec:. with the nature of the actual variability of phys: } ical quantities. such as loads. material propen:ies and dimensions. This 'ariabilicy can be de· · scribed in terms or probability distributions or stochastic proCesses and some typical examples '~ are discussed in detail in chapter 3. However. physical variabili.ty can be qUlntified only by .~ examining sample data; but. since sample sites are limited by practical and economic consider· ~.~ ations. some uncertainty. must remain. This .p~cticallimit gives rise to so~alled statistical un· . . :.::, certain.ty •. Statistical uncerla/my: As-,yill be discussed ~ iSler chapters, statisi.ics. ss'bpposea to proba~i: . . ~ .;.;; ity, is concerned with inference. and in particular with tile inferences that can be drawn from ) s:!mpie observations. Data may be collected (or the purposes of buildi:-ig' -~ probabilistic model :J of the physical variahiliiy of s'ome quantify ,,'hieh will' irl'o'ol'e , firstly the 'seiection of an ap· H propri3te p~obatiil:ty distribution t)'pe. and !!len dete::ninarion 'of numeiYc'ai''alues for its pa· "3
- 19. :~~~i.2:3 TYPES OF UNCERTA1NTY 7 r...r.lc~h. <.:ommon probability disuibutions h:l'e bet.ween one an(! four parameters which immediately placts a lower bound on the sample size req~ired. but in practice very large sam· o pies are required to establish reliable estimate!. of the :'J1:1fr!.erical ValU~5 of parameters. Fpr 11 given sCt of data. Iherefore. the distribution parameters may th~mStlves be considered (0 be random variables. the uncenuint~· in 'which is dependent ~n ~he ~~ount of sample data· or. in general, on the amount of data and any prior knowledge. This uncertainty is termed sta· tisticoll.mcertainly and, unlike physical variability. arises solely as a result!lf I.ack of infor· mation. Model uncertainty: .Structural design and analysis make use of mathemati~al' 'm'odels relating desired output quantities (e.g. the deflection at the centre o[ a re!nforc~ 'con~rele beam) to ~3 ': the values o[ a set of input quantities or basic variables (e.g. load intens!~ies; modulus of elastf~ ' icity, duration of loading, etc.). These models are generally determitiistic in form (e.g. Iinear~'.~ ·-;iasLiC structural analysis) although they lXIay be probabilistic.(e.g. calculation of the peak re- " .sporu.e of an oUshore structure to' stochastic wave ioadina:). Furthermore. they may be based on an intimate understanding of the mechanics of the problem (e.g. plastic collilpse analysis oh steel portal irame) or they may be ~i8hly empiric~ (e.g. punching'shear at tubular joint connections in offshore jacket stNctures). However. with very few exceptions, it is rarely pos· sible to make hiehly accurate predictions about the magnitude of the respons~ of typical civil engineer~ng structures to loading even when the governing input quantities are known exactly. In other words; the response of typical structures and stntctural elements cont-ains a ~ampo· nem. of uncertainty in addition to thai(! components arising {rom uncertaint!.es in t~e values of the basic loadln~ anc. strength variables. This additional source o~ un.~ertaint~' is termed model uncertaint)' aud occurs as a result of simplifyins assumptions, unknown boundary can· ditions and as a result of the unknown effects of other variables and their interactions which are not Included in the model. For example. the shear Strength of nominally similar reinforced concrete beams exhibits considerable scatter even when due allowance has been made for the various known dif!erences between iest specimens. ' r The model uncertuinty associated with a particular mathematical model may be expressed in terms of the probability distribution of a 'ariable Xm defined as x . " actual strength (response) m predicted strength (response) using model (Ll) In many components and structures, mod!!l uncertainties have a large effect on structural reo iiabiliL), anc! should not be neglected. 1:3 STRUCTURAL RELIABILITY _1!ALYSlS AND SAFETY CHECKING .. ~i'ne F ",:·" linl!. rema.rks were concerned ..... ith th~ '~~riou5 types and sources of ~ncert.:.int)' thot n('ed to be t:iken into acco .. nt in predicting the abilit.y' of D structure to withs~3.nd the actual b~t u:lknown 10Jds th3.1·will h,: :).ppl.ied ·~"· it. . ..., tll"n no ..... comider the various WJ::s in which lhe~ prt'dictions c:m bE' made and .the usc tnat can be- IIItlcle. 01 t j,.;.:I' . But first some definiliuns und prelimir.:nies.
- 20. s 1. THE TRE.nJ~:~T OF UNCER:rAINTIES I~ STRCCTCR.L E:-:GTNEERING 1.3.1 Structural Reliability The term :JCnlcturai reii(Jbility should !Je clJnsiclered as having two meaninll5 -3 gl?nerai one and :1 mathematical one. • tn the most general sense. the reliD.bility of a structure is its ability to fulfil ;[5 design purpose ror some specified time. • In 3 narrow sense it is the probDbility that a structure will not attain crlch specified limit state (ultimate or serviceabUity)'during a specified ~e{erence period. . -'-" '-Cn this book we shall be con~e.med_~.th st~.ci~~r re~~~bil.ity in "the narrow sense and sho.ll gen-erally be treating each limit state or [:lilure mode separately o.nd explicitly. HOWever. most structures and structural elements have a number of possible failure modes. and in determining the overall reliability at a stru~tural system this must t>e ·taken·into 3Ccount. makinf due allowance for th~ .correl:ltions ads'lng Crom'common sourees of loading and common m:lterial J:lroperties. These aspects of the problem are covered in chapters 7 and 8. However, Glthou~h" the ~defi~ition' above may seem clear. it is necessary to e;<a"i1!l1e ~vit~ care exactly what-is meant by »the probability that a structure will not attai.t:i eac~ i;w=ified limit state durin'g a specified reference periodll. , f . • . ' ,. Consider .CJr,st. t.~e need for defining a reference period. 8«nuse the.majority of structural loads 'nry with time in an uncertain manner." the probability that any selec~e? load btensitr will be exceeded in a fL.. .. ed interval of time is II function of the length of that intetval,'.ll1d possibly the time at 'which it begins). Hence. in general, struc~ural reliability is dep~ndent o,n time of expo-sure to the loading environment. It is also affected it material properties chan3e with time. Only (or the rare cues, when lauds and strength are constant. can the referent:'t period be ignored , In .such cas:es the loads are applied 'once and the structure either does ·O! does not fail (e.g, when the structure or component is loaded entirely by hs own self weightl. The second que~tion is mo~ d~ding. 'What is meant by lithe probability tn3t .. . 11? This is best explored by a simple example. Example 1.2. Assume .that an offshore structure is idenlised as a unifon:l 'erticol cantilever rigidly connected to the sea bed. The structure will fail when the ::1Clment 5 induced at the root of the canlilever exceeds the flexural strength R, Assu:ne fUrther that Rand S are random variables whose statistic.:LI distributions ure known ret)' precisely :lS a result o[ a very long: series of measurements. R is a variable representin~ the varilltions in strength between nominally identical structures. whereas S represent> the mu.imum IOlld effects in 5UCcessn-e T year periods. The distributions of R .md S.:l..'"e both assumed to be stationary with time. Under these assumptions •. the pr",balJiiity U::1t the StruCture will collapse duri.n.II: an'l'1'f¥ronce }Jt:rtocJ of duouion T years will be siu:wn in chapter -I ~ c be given tiy Pf - PDI.o;Ol - -· F'!'I.lxlfS,xklx t l.:!l ~-.- where
- 21. .': 9 )'1 c R-S . :md where FR is the ,t:'~o.b.abiiity distFibution ('t~c:i,~n' ~:C Ii m~~ .Cs i~e'p:o~~bility density , . . (unction.of .S, These term&:u? defined more Cu.Uy iri chapter 2. " .. . . . " .' • • .. , ~ ,.,1.1 " .- ' .... . ... ,. Because. pt thedeCinition of 'R d~ci S i~', ~e~~ ,of Crequentisl ~ro"babiliti~s. tne p,robability detCfmi'~ ed i~o~' ~q~atio~ I 1.2) m3~' be i~te~r~~ed as 3 10n~~~.~~·~aiiure fre.iuenCY:' Simil.arly the reliability ~, defined as ", . .Il = I-Pr ", .. " " -. ' m~y be interpreted 'as along-run-survival frequency or long·ntn reliabili.ty Jlnd is the percent- . age of a' ~~tionalJy Infinite set of :tominany~ldentical ·5tiuCtures which survive Cor tlte durn· tion of the reference period T.'~ :::.ay therefore be c.alled a frequentisl reliability. If. however. 'we are Carced to focus 'our 3Uention'on'one'particular structure (and this is genernlly the C3St! Cor Ii~.~~.offll ~ivij'~n~nee~ing,' structures):~ ina~' 0.150 be interpreted is a me;l5l1re of the relia· bility' of t.hat p~rticular struct~r~:': : . : p • This interpretation of reliability ii (undnment3.lly different from that given above. becnuse. a1. though ttlc ~'tructure may 'be' s':lf~'p:ed at random Crom th'e' th~oreticnlly infinitc population de~ cribed b~Hhe random' varillb'le R, ',mce'ltle p:irticula'r ·structu'te bas been· selected (and. in proc· tice, constructed) the reliability b,;,:omes the probability that the fixed. but unknown, resistance r ,,,ill be exceeded by the 3S yet' ~un~sampled .. reference period extreme 10!ld eH~ct S I note that lover case r is used here to denote,the outcome of R J. ~The numerical value ,pf U~~ . failure probability.remains the same but ~ now dependent upon two rndically different types of un· certainty .• ·ii~s~iy. the physicaJ vat~bilty'of the' e~tre~e'!o~d ~r"f~t·,:-3nd.·se~ondIY;lack of knowledge about the true value oi,the f~~" b~~'~ '~.tnkJ!.~~n .(~~~t.a~ce·. Tbi~- type of probability does not h.we a rellldve frequency interpretation and is commonly.c:dled ,a.subjecliV1! plI!babil. ity. The associated reliability C:1n be called a subjectl,-c or Bayesian reUability. ror a particular stru~t~;~:':the i1Ume~jc:iI';'alu~ c;r~his 're'iiabii'ityttiimies 3S the state of krlowledge 3bOut'~he mUfture' changes - for ex~~ple'-if :iori~~srruai'e tests were to be'c3i'ried'out on the structure to estimate the magnitude or r.rn:he·,iriiit when i:b'ecomes known'e;,(3ctly, the probability or r .. ilure given hy equation i 1.2 ) cha.'ges to .r - '. . (1.5) This special case may also he inte<,lreted as 'a conditional failure probability .wilh 3. relath'e fre· . ql~ncy interpretation. i.e. :/,' il1.6' The symhol i may he read as .gl~:'1 thUll •.
- 22. l~ ! I. THE TRI::ATMENT OF L.'NCERTAI~TIES IN STRUCTURAL.£NGlNEERING 1,~,2 Methods of S3!ety Checking tlany de'elopments ha'e Laken place in tne field of stNctur:d reliabi.Jir.)." an;uysis during Ihe last 10 years ~rid to the n'e~comer the Jiter",ture may seem confu$ing. To help clarifying the situation. the Joint Co~mi~tee an 'Structuia! Safety j'an inie~ational bOdy sponsored by such international organis3tions as CEB, CIB, CECM, IABSE, lASS, FCP and RILEM) set up a sub· commJttee in 1975.to prov!de a broad classification system for the diCierent method$ then being p~oposed fO,r -cheCking Ui'e safety of structures and 'to establish tne main differences be-tween them~ This cl~ifica'~ion' is ~till usef;tl. ,- , Methods of structural reliability analysis can be dh'iried into two broad classes. These are: Leuel3: Methods in which calculations aP. made to determine the JreX!lcu. probability of faillolre [or a structure or Structur.ll component, making use of a full probabilistic description of the joint'occurrence of the 'arious,quantities,which affect the reo sponse of the st~cture and takiag into accou,nt t~e true nature of ~~e failure do, main. Level 2: . Methods involving certain appronmate iterative calculation procedures to obtain . an,npproximatic!'.l to ,the failure pIObability of a structure or structuRt sy~tem. generally requiring an ide31isation of faUW'e domain -and often'associated with a simplified representation of the joint probatiility distribution of the variables. In theory, both level 3 and Jevel 2 methods can' be used for checking the safety of a design or direcdy in ,the design p;~cess. provided,a ~get reliabi,I,I~y'~r re:lia.bii.r~~:inde~ 'has betn spe- f ,clfied, For the sake 'of completeness, some mention should also be made of level 1 methods at this stag;:: The~' are not 'm.!thocls of reliability anaJ}'sii, but are methods of design or safety checking. ,- ,Level,1: Desi~n "~e~h'~s in which appr~Priate depees of litrUClllraJ reliii6ilit}; 'areprovided on ~:. , 1' stnid'uial elemern basis (occasionally on a structuri.! bash) tiy the use of a number ~ , :' "';o! partial 'o.fety'factotS~' or pania! coefficients, related to pre·defined characteristic of'! "(~ :norriinaJ values-of the major stnJ:Ctura! and loading 'ariabies, .j; ' I I •• , • • _ .,0< ~~ A level) struc!ural deSIgn, with the explicit ~o.n,s lderatlon of a number or'separate limit states, ~~ , - ' ,-" ~. is what is now commonly ~alled limil·stat~ de~ign. It is pre!t!rablc. however, that this should be,::'': called le!:,cl1 design,. an~ that tbe ,term limit.st:ate should be usee solely ,to desc~ibe the separate", limiting performance requirements. The terms, levell, 2 and 3 will be discussed in detail in chapten 5, 6, 7,8 and 11. The fundame~~aJ distinctions between levels 2 and 3 cannot ealiilr be understood at,this stage umil the neces- ' sery background in probability theory has been co'ered (Chllptt!: 2). However, the three levels ,,~; of safety checking should be seen as a hierachy of methods in whi~hJev~1 2 methods are all "'lJ''f!: proximation to level 3 methods and in wi'Jch level 1 methods are a discretisation of level 2 methods (i,e. ~'in2 identical desiiTls 10 le'e! 2 method~ ior only a iew discrett! sets of values of the) structural design parametersl. t· ".1 for pra::tic:i.l purposes - for ex:amph:, Ic~ Citect use in d(>si~n or for el'3Juating level 1 partial fatton · it is necessary to haVE: :. method of rtliabilily analysis which is computationally {~t alld'~~ ;~, ,~.
- 23. BIBLIOGRAPHY 11 efiicient ano which produces results with tht- desired degree of acC'.lr.lcy. The only methods which currently sathi), these requirements are the level 2 method;;. although analysis by Montf.-· Cilflo simulation is sometimes feasible. In this book emphasis is pJ.acec. on tn.? theory and appli· cation of le'el 2 methods and their use in the design of le'ell structurAl codes. BIBLIOGRAPHY (1.11 CIRIA: Rationalisation of Safety and Seruiceability Factors.in Structural Codes. Can· struction Industry Research and Information Association, Report. No. 63, 1977. (1.2J Cornell, C. A.: Bayesian Statistical Decision Theory and RelUlbility.Based Desig/!. Inter· national Conference on Struct~iaI Safety and Reliability, Wa;hington, 1969 .. (pub. Pergamon 1972):_. 11.3] Ditlevsen, 0.: Uncertaint:r /I~od,!lillg. Mcqraw Hill,19B1. (1.4J - Freudenthal. A. M . .-Garretts, J. M. and Shinoz.uka, M.: The Analysis of Structural Safety. Jownal of the Structural Division, ASCE, Vol. 9~, ~o. STi, Feb. 1966. 11.5)- Joint Committee on Structural Safety, CEB.- CECM • CIB· FIP . L0U3SE· lASS· RILEI"II: First Order Reliability Concepts for Design Codes. CEB Bulletin No. 112, July 1976. 11.6J Joint Committee on Structural Safety, CEB· CECM· CIB· FIP· lA-BSE • lASS . RILEM: Interlltltiolltll System of Unified Standard Codes for Structures. Volume I: Common Uni· fied Rules for Different Types of Construction and Material. ~EB/~IP 1978. 11.7] Joint Committee on Structural Safety, CEB • CECt-,·j • CIB· FIP - L-.BSE . lASS· RILEM: Genera! Principles on Reliabilit)' for Structural Design. Intemational.Association for Bridge .and Structural Engjneering·,·1981. [1.8J Leporati, L: The Assessment of Structural Safety. Researeh StUdies Press, 1979. 11.9J Nordic Committee on Building Regulations: Recommendation for Loading and Safety Regulatiolls for StructurtJI Desigll ... NKB.Report No. 36, No .... 1S7S.
- 24. ,
- 25. Chapter 2 FUNDAMENTALS OF PROBABILITY THEOR:>: : ~ . ","." 2.1 INTRODUCTION .~ ., ' / ... , ' " ,'X$ e~pbsiied in"~'liiip~er J."mOdtrn structural reEability anaiysis is ~sed on a pro'babilistic: point ~f v'jew', It is il;ereC~re imp6rt~nt to get a profound knowledge obt least some part of probll~ility theory. It is beyond the scope oC thii book to give a thorough presentation oC pr9b~bili~y ~e~~y o~· 'a "r.iOfOUs' axiomatic: bash;" 'bat is needed to understand .lhe fol lowing Chllpters is so~e knowied'ge 0'( the (undame:1ta1 assumptions of modem probability theorr. c~~bined with a n~~be'r O:(derfnitions:me theorems'; It will be assumed that !-he reader is f3.miliar with the terminology :md the al~eora of.simple set theory. The purpose oi this ~haPter is th~r~(O~~ to gh:~c a ~lf-contain~d presenta.~ion cif p"rob'lbility thcor)-' with emphasis on concepts of importanCe :or structur::a reliabilit!-' analysis. 13 A SUlndard :,ay of delermin;n~ the yield stress of 3 material such as steel is to-perform a nu'~be~ O(~iT:ple. U!~~i~~. tes~~ with specimens mace" from·the material in ·question. By each test a v~lu~ r~)I: t~e .y!el.d str~iS is d~termined but tbis value will probably be different from test to test. Therefore. tn this connection, the yie'le stress must be taken as an uncertain quan· tity and it is in accordance with this point oi view said to be a random quantity. The set of 311 possible outcomes of such tests is called the sam'pI~.5pact! and eReh indi~.~ual outcome is a sample point. The sample space for the' yield stress. is the .open intel'!'al J 0 ;",,[ • tha~ is the sct ..... oi all positive real ·numbers. :SOte th!lt this sample space has an infinite number of sample ·· points. [t is an example of a cOlltlmlOu.! sample !pace. A sample space can also be discr~te . . namely when the sample points are discret; and count~ble·~~tities. Example 2.1~ Consider a simply.supp/?ned beam .• -B with t;.vo co.ncen~rated forces PI and P2 liS shown-on fi¥Ure 2.1. funher.let the possible vlllues of Pl and P:! be 4. 5. 6. and 3. -t. respectively. In this ex::.mple all values are in kN. The sample space ior the lOi1din~ will then be tl":e set . (2.11
- 26. 2. f'UNDA.IE?o.'TALS or PROBABILITY THEORY '-" - " ' - r ! P2 , , B, ,. ~ ' :3" . '7777. , :>I' , ' . , A Figure 2.1 This sample space is discrete. Further, it ha,s a finite number oC sample points. There· fore, it is caUed a finite sample space. A s:unpl~ space with the countable infinite num· ber of sample poinu is called .an infinite 5:Imple spacp. ?ot.e that the sample spaces for the loads P1 and P2 are 0 1 - ·{4·,.S,·6} ,"and:n 2 ~ . ~ '{3, 4}, respedivel)'. Also note that n ~.nl X: fi2,' ~~o~ as.ll~ e~~.rc,isf. ~!'Iat.~,~~mple space for the reacuon RA in point A.is ~A .. P]/~~ 121.3. 13/3, 1~/3, ' 1513; 16la}. > . , . A subset of a sample sp3ce is called an· ~~en/. ~~yent is therefore a se~ or sample'points. If It contains no sample paints. it is called an l~po~ble e~~nt, A cert(Ji~' e'uenf co'nt:ains ali the sample points in the sampJe ~pace • that is, a c~~ eye~t is equal t'o'the sample space itself. ~ . . - ' Example 2.2. Consider again the be:lm in ligure 2.1. The sample space fo ... ·the reaction RA is Sl A - {lIla, 12/3.· -13/3. 14/3, 1513.: 16/3). The. subset {IS/3, ,16/3) is the eventlh'at R . .a.' is· equal to 1513 or 1613. Let £1 and £.2 be tWO events. The Imion of El and E2 ls an event d,e~,o~ ~~ ' ~1? £2 and it Is the_subset. of sample points,~a~ ~Iong to E1 ,and/or E2, Tne interseclion of El and E:? Is an event denoted by £l' f'I £2 and,is ~he sl;lbse1..oi ~ple"po~U::~I~nglng to b'ot.1l E} 'arid E2. The tWO events ~1 and £2 are said t~ be mUluall." ex'?-lusilJe if the)~ ace disjoint. ihlit'is if they ha"e no sample poinu in common. In this c~ £1 f". E2 '" C!'-,: where 0 'fi titi Impossible event (an-emplY set},' .• -- ..... r .. ';. , .. , - ~. .. •.. .. . Let P. -bf-nample-'.ipace and E an event, The event com.aining all the sample'poini,s in n that . "an- o;'t in E is called the camp/eruen tary ·ellent and is denoled by E. Obviously. E u E • n an·dE()E-0. ' . • . .. , Il is easy to show thai the 'i~ter$ectlon and union operations obey the following commutative, associative. and distribuUye·hiws .. " ~ . El n (E2 n E31 '" (E1 n E2) n E3 } } (2.2) 12,3) (2.-11 i ~ "
- 27. 2.3 AXIOMS ASD THEOREMS 0: PROBABrL1n' TkEORY 15 Due to these Jaws it mtlkes sensE' to t'onsicier the intersection or the union of tOI!' events E1, E2 •...• En' These new l''ents are denoted and n• E] - El '" E::!. n ... . "1 En i-I U• Ej - El U E2 U ... 'J En i-I Exercise 2.1. Prove thl: so-called De Morgan's laws ElnE2c~ 2.3 A."{IOMS AND THEOREMS OF PROBABILITY THEORY (2.5/ 12.6) (2."} ) (2.8) In this section is shown how a probability measure can be assigned to any e·ent. Such a proba. billty measure is a set function because an event is a subset of the sample space. Further. the prob:lbility of the certain e'eot (the sample space itself) is unity. finally, it is reasonable to as· sump that the prohability o~ the union of mutually exclusive events is equal 10 the sum of the probability of the individual events. These assumptions .re given a mathematical, precise formu· lation by ·,he following funciament31 axioms of probability theory. Axiom 1 For any event E 0< P(E) < 1 (2.9) where lhe IUnction P is the probabilit), measure. peE) is the probabillty of the event E. Axiom 2 Let the sample space be Po. Then P(O) - ' 12 .10) Axiom 3 U £1' E2~ ...• En are mutually exclusive events then • • P( U Eil"l~p(Ei) i-I j_ ) 12.11 )
- 28. 16 _. ~'l!:-;OA;I£NTALS Of· PROBABILITY THEORY Exercise 2.2. Prove the following theorem~ PIE)' 1-PIE) (2.12) '2.131 CU-II Example 2.3. Consider the statically determinate structural system with 1 elements shown in figure 2.2. Lel the event that elemen~ Il il> fnils be denoted by Fi and tet the probability oi failure or element ~hl he P(F;). Further aui.l,me that failuf(!S oi the ir.dividual members are ~tatistically independent. thnt is P(Flrl F) • PIF1}' PfFj ) for an:: pair of (i. j). The failure of any member will result in system lailure (or this natic3lly determinate stmcture', Thus.' .. ,~': .. ' P(failure of structure) - PI Fl U ... U F-; I - P( U Fj ) ',- -. ,, ' ., . I-I. -l-PI U· F;l-i-PI ii ·F,) (:!.15J i-I i-I accor-ding to De ~Iorgan's law (2.81. Becaus~ of statistical {ndependen'ce.12,15) C31l be . " }yriuen P/f.:ll)ure of stnlct.ure) .. 1'- P(F1) , P(F~) , , . P(~) Let PIFl ) " PIF3J.·P(Fs'" P{F;}" 0.02, P(F2)" P{Fs). "".(),~l.:.3:"d ?W,) .'" 0.03. Then P(Cailure o( structure) " 1 - 0.98~· 0.991 • 0.97" 1 - 0',876'9 ';:0,1231 ; ' , I .... . . " . . .f; : FiJ;Ure 2.2 In many prnctk:1.! applications the probahility of occutrence of 1!'ent.E1 conditionai upon the ·tCcurrf>nce of en'nl E:!-i,; (.)( ~rt'al · inll~r~t. This prohahili.ty called the cOllditioIlDI:~N?babmty is d.molt'd PI E: .. E:!., :llld is defineu by
- 29. ir p! F.:!, > O. The conditional probnbility ;s not defined (or PfE2) ~ O. !::'ent E is said to be statistically independent of event E2 ir that is. H the occurrence of E2 d.oes not affect the p~~b~bilitY of El . from eqv:ltion 12 .16) the prob..,bility of the e'ent E1 n"Ez-is -givim' by If El nnd E2 are statisticnlly independent 12.181 hecomes 17 (2.1;1 ~2.1S i 12.191 TIll' rule 12.19) is calloo the multiplicacior. rule lind has alrelld~' heen used in example 2.3. Exercise 2.3. Show 1hat El and £2 are ,;tatistic3l1y independent, when ~l and I:::! are mltisticaJly independent. Exercise 2..1. Show that (2.20, . . , . Eumple 2.4. Consider ngain the structure in figure 2.2. It is now assum.ed (or the sake of simplicity. however. that only element 2 and 6 can fa.il. Therefore. P{Cailure or structure) • P(F:! U Fs) "" P(F:1:) + P(Fs ' - Plf:!':' Fs) (2.211 Ie f~ and FIS :lre statistically independent as in example 2.3 and if P(F:) ,. P(F'l):> 0.01 then P(failure of muctureJ Q 0.01 ... 0.01 - 0.01 • o.oi ,;, O.Ot9~ .~ , But if F:! and Ffi ine not independent then knowledge of P(FzIFIj) is required. If the two elements are fabricated ffom the same steel bar it is reasonable to expect them to h.l~·e the same strength. Funher. they have th'e same loading. :ind'then!{oTe' iii this ip!Ci.31 casc. one can expect P(F2 ! F6 ) to be close to 1. With PIP::!! Fa) • t one gets ,from 12,211 '<; ' . ·i •. : .' _ , , ":. . -" ,:: _. :-. -; .. ... . ::':: • PI {lliJure oi structure I '" 0.01 .;. 0.01 - ·1 . 0.01 .. 0.0100 . '. ': :;." , "":.:) "" Fin:lii;·. t~u/so"~:il1ed' &~'es' theorem ',vill be ceri"W. Let ttie sample space- n ':be divided into :'l mutually cxclu$i'e e'ents E 1 , Ez, ... ·. E::: 'fS~ fi¥Ure·2.3. ~here' n • 4); Le't· .. Che on event tn the same ~ample space. Then
- 30. 18 2: : FUNi:iA}.i·E~TALS OPPROBABILin' THEOR'l' f .. ;' FiiUre 2.3 '" peA lEI )P(E1) + peA IE2)~(E2) + ... + P(A IEn)P(En ) .. IPIA!E1lP(Ej ) i"l from the dt-finition (2.16) follow.1 so that or by usinG' i2.22j PI.'IEjIP(Ej , P(E11..t.. n ~P(AtEi)P(Ej) . j-I Thls .i~ t~'e important Bay.;5 ~ , t.~~o~em .. ,', (2.22) , '; ' - (2.23) 12.24) ~ . f . c, ;,::. ' 0' Example 2.5. Assume that a steel girder has to pass a given test before application. Fur- '~J ther, assump from elt.perience that 955'( of all girders are found to pass the test. b!Jt the :'':' test is assumed only 90% reliable. Therefore, z. eonclwion based on such ~ lest has a proba-:.t bility of 0.1 of being erroneous. The problem is now the following: What is the peoba. ."~ bility ~hat a perieC't girder will pass th~ lest? Let E be the. el~n.t th.at the girder is perfect ;.. .~ : .• a~d I~t A .bl? lhe> event In:n .il pas5~~ th~ lest. . . : .. &? G PIEiAj- 0.90 and ~~:,:
- 31. 2..& RANDOM VARIABLES 19 so that P(EIA):Ii: 1-0.90 = 0.10 Flam experience P(M" 0.95. The problem is lo find PtA IE). The events A and A are mutually exclusive, so that. according to (2.22) P(E) " P(EIA)P(A) + P(EIA)P(A) .. 0.90· 0.95+ 0.10' 0.05 '" 0.860 Finally, P(AIE) _ PiElA)' PlA) .. ~ c a 99~' PIE} 0.86· Example 2.6. ConSlder a number of tensile specimens ~esj~ed to su'pport a load of 2 kN. The problem is now to estimate the probability that a specimen can suppon a load of 2.5 kN. Based on previous experiments It Is estimated that thore is a proba. bility of 0.80 that a specimen can carry 2.5 kN. Further, it is known that 50% of those not able to support 2.5 kN fail at loads less than 2.3 kN. The probability of 0.80 mentioned above can now be' updated if the following test is successful. A single specimen is loaded to 2.3 kN. Let E be the event that the specimen can support 2.5 kN and A the event that the- test is successful (the specimens can support 2.3 kN). Then P(AIE) = 0.5, and P{E) " 0.80. Further P(AIE) " 1.0 so that Bayes' theorem gives PIElA) '" PlAIEPlE) .. 1.0· 0.80 .. 0 89 . P(AJE)P{E) + P(AIE)P{E) 1.0'0.80+ 0.5'0.20 . The previous value of 0.80 lor the probability that 8 specimen can carry 2.5 k!' is in this way updated to 0.89. 2.4 RANDOM VARIABLES , The outcome of experiments will in most cases be numerical values. But this will not always be true. If, lot example, one wants to check whether a given structure_can carry a given load the outcome may be yes or no. However. in such a case it-is possible,to'8ssign a numerical value to the outcome, ror example the number 1 to the event that the st;uc:ture can CaIY)' the !oad, and the number 0 to the event that the structure.cannot c~ ~,~.1oad. Note that the numbf!ll> 0 and 1 are artificially u signed numerical values and therefore. other 'alues could have been associated with the events in qUt:iti~n. !:. ~h;c W;':l.' H is possi_ble ~o identify p-o:sslble outcomes or a random phenomenon by numerical values. In most cases .thes,e ':'altJ..,:. · .... m ~imo!v be the outcomes of the phenomenon but as mentioned it may be necessary to assign the numerical values artificiaUy. In this wayan outcome or e'ent can be identified through the value of a function call£od a ron· dom LlQriable. A random variable is a function which maps. events in the sample spaCe!! into the real line R. Usually a random variable is denoted hy a capital letter such as X. To empha- &ize the domain of X the random variable is often writlen X: n~R_ The concept of a ~onti :l'.!o:J$ random vmable is illustrated in figure 2.4. The event E1 C n. where n Ii a continuous :;ampi~
- 32. 20 '. FUNDAMENTALS OF PROBAB!UTY THEORY sample space n __ ---_~"'.n.dom varillble X ---~~~~L-______ _ , R • b Figure 2.4 space, is mapped by the [unction X on to the interval (a ; bJ c R. If the. sample space i. discrete, the random variable is ca11ed a discrete random variable. In section 2.3 the probability of an event E is introduced by the probability measure P. In this section. it is shown how a numerical value is associated with any event by the random variable. This permits a convenient analytical nnd graphical description of events and associated probabilities. Usually the argument to! in X(w) is omitted. Similarly, the abbreviation P(X C;; x) is used Cor P({w :X(w) < x}). First consider a diM!rete random uariahfe X. This is a function that takes on ont'l ::J. f~te or countably infinite number of discrete values. For such a ~ndom variable the probabilicy mass function Px is defmed by px(x)" P(X = x) 12.25). where X is the nndom variable, and x ... Xl' X2 ' •••• Xli.' and where n can be finite or infmite. Note that difCermt symbols are used for the random variable and itl·values, namely X and x, . respectively. It is a direct consequence.of the axioms (2.9) ~ i (~.: l1) that .. ~tpX{x)-l j-l Pfa<;X<b}- ZPxlx,l- ZPxIX,) lI"j"b 1[1"" ~3:~1' ~.~:, ~:.~ ,~ ;i. ":'i' .I(~ ,; 'iI: . 1:,.- ~~pr,o bability distribution function Px ; Rf""""'R is related to Px ~.i. '. '. Px(x)::: PtX ~ xl '" .l'Px(xj ) 'I1<1[ 12.26) 12.27) 12.28) 12.29) Sy the de(inition (2.29) the value Px(xI is the probability of the event that the random vari· able X tak~s on values equal to or less than x. .
- 33. 2.4 RANDOM VARIABLES Example 2.7. Consider again example 2.1 and let P(PI .. 4) - 0.3, P(PI .5)" 0.5 and P(PI .. 6) .. 0.2. The probability mass function PPl and the p.robability distribution .. Cunction PP 1 Cor the random variable P 1 are shown on figure 2.5. Note that the circled points are not included in PPI (s). fpp, (x) 1. 1.0 F'---~", O. 0.5 ' ' ...->: x ! ! I I , , I • I I X 0 5 0 5 Figure 2.5 21 Next consider a continuous random mritlbte X. Thill is a Cunction which can take on any value within one or several intervals. For such a random variable the probability for it to assume a specific value is zero. Therefore. the ~babllitymass function deCined in (2.25) is of no in· terest. However, the probability distribution /Unction Fx : R~R can still be defined by FX (x) ., P(X <: xl xER (2.30) It is often useful to use the derivative probability function. This function is called the pro· bability density function fx :Rr"R aDd is defined by (2.31) assuming of course that th~ derivativeoists. ~ote that the symboJ Px (x) is used for .the. prob. ability mass function and the symbol !xiX') for the probability density (unction. Example 2.8. Figure 2.6 sho~s .. f:he probability density function fx and the probabUity distribution function Fx for a conti.D.uous random variable X. IF,,(x) 1.0+------------- --- ~~--------------------x ";': ....... , ,.
- 34. "2. FUNDAMENTALS OF PROBAmLlTY THEORY ";" I~ follows directly from the axioms (2.9) • (2.11) that for any probability distribution function (2) FX is non-decreasing Inversion of the equation (2.31J gives .. FX(x) e f,,(tidt (2.32) '"-- for II continuous random variable. From (?,.32), it follows that r fX (tjdt - Fx '-)-l (2.33) ."-- It is sometimes useCul to use a mixed continuoUl-<iilCTete random uarilJbl~. i.e. " continuous tandom variable admitting a countable number of discrete values with a non-uro probability as shown in figure 2.7. In this case the area under the curve in figure 2.7 15 equal to 1 - 0.2 - 0.1 • n.7. oX Figure 2 .• ~.5 MOMENTS In this section a'number of important concepts will be introduced. Let. X be a continuous ran· dam vari2ble. Then its probabilistic characteristic! are described by the distn'bution function Fx' Ho~er. in manr applications the form of Fx is not known in all details. It 1& therefore useful to have an approximate dc.!~rlp.tion of a random variable stressing its mast important fearures. When F X (or!X) is completely known, however, it is also of interest to have lome very simple way of der.cribing the probabillstic characteti;tie.~. For this puIpOlot the so-called momtmt$ are introduced here. When X is a random variable. Y '" Xk, where k is a positive integer, is also a random variable because- PI {w : XII ...: y:·1 exists for every y. ln the following it is assumed that an random van· 2:bles are ;:ontinuous random variables. if not. otherwise stated. The ~%pcctf!!d volue of X is de· ~:ned as
- 35. i·· .. 2.5~ MOMENTs . (2.34) The expected value is also called the ensemble average. mean or the first moment of X and the symbollJx is often used for it. By analogy with this the n'th moment of X is called Elxn 1 and is defined as E(Xn ) -C xn fx(x)dx (2.35) '-- For discrete random variables the integrals in (2.34) and (2.35) must be replaced by summa· tions. Note that the flISt moment of X defmed by equation (2.34) Is analogous to the location of the centroid of a unit mass. Likewise, the second moment ean be compared with the mas~ moment of inertia. : Example 2.9. Consi"-er the discrete random variable X defmed in example 2.7. The : discrete venion of (2.34) rives E[XI- 4·0.H 5·0.5 + 6,0.2,4.9 The most probable value is called the mode and is in this ease equal to 5,0 (see figure 2.5). Further E(Xl J ,. 16· 0.3 + 25· 0.5 + 36· 0.2 '" 24.5 Above. a new random vuiable Y • X· wu considered. Tnis is a spe.cial case oC a random vari· able which is a function of another random variable whose distribution function is known, Let Y • l(X), where f is a function with at most a finite number of discontinuities. Then it is possible to Ihow tbat Y .is a ~dom variable according to the definition of a random variable. If the {unction f is monotonic the distribution function Fy II given by Fy(Y) -P(Y" y)-P(X" f-' (YII-FXW' (y)) (2.36) and the density function fy by (2.37) or simply !y(Y)- !x(X)I~i (2.38)
- 36. !!.. FUNDA.lENTALS OF PROBABILITY THSORY Example 2.10. Let Y II aX + b. Then X - tV - b)/a:md I IY).r IY -b). Illl Y X a a I IL is important to note that the expected value oC Y .. [(X) can be computed in the following way without determining { '( ' .- 12.39) Exercise 2.5. Show that . " " E{ I fj{X)J .. I Elfj(X)J 12.40) i-I ;-1 SO that the operations of expectation and summation can commutate. Returning to the momenu of a random variable X. the nthcentroi moment oCX is defined by EI (X - JI. X)n I. where JI. X • E{ XI: ~ote that the first central moment of X is always equal to zero. The second central moment of X is c3.lled the t'ar;ance of X and Is denoted,by a~ or ~X) . The positive square root aC t~e v~iance. aX' i~'caJled the ~t~ndard. ~~.U~tiO~ oC X. Exercise 2.6. Show that : ". 12.U) 12.42) The standard deviation aX is a measure of how closely the values of the random variable X are . con~ntrated around the expected. value EIXI. It is difficult only by knowledge?C aX to decide whether the dispersion should be considered small or large heause this "ill depend on the ex· . pecLed value. However; the coe{ficient of variationNx ' defined by .--.- -·-vx .. .. ;; 12.43) , ~ives better information rc"ardim; the dispersion.
- 37. r ," ;1' -:. J. ' . 2.6 UNIV.RI.-TE DISTRIBUTIONS Example 2.11. Consider the same discrete random v3'rintlle X,as in example 2.9. where E(XI '" 4.9 and E(X1l = 24.5, The variance. there(~re. is''-' -, - '. I. VariXI - 24.5 - 4.9' · 0.49 and the standard deviation is ax' -yQ.'49 .. 0.7 , ",; Thus the coerficient of variation is Vx . ::: ~:~::: 0.14 The third central moment is a measure of the asymmetry or s/~ewness of the distribution of a random variabie.:- F,~r a continuous random vOlriOlble it. is defined b~ (2.44) ..•. 2.6 UNIVARIATE DISTRIBUTIONS In ,~h.is , ~tio~ ~!lle of most .. widely used probOlbilit~ distributions are introduced. Perhaps t~e most importa~t distr'ibution is the ~o~m~1 diitribuliori aJs'o called the Ga'ussian, distriqution. It .• . '" -r ' , ' , f ' !"~' ". "'. . is a two-parameter distribution defined' by' the densft-y func'fion ,",i. ;,; I, :, . . ,: /.. '-";"!.' .,,1:'1',,"· (2.45) where II. and a are par.ameters equal to.ux and ax - This normal distribution will be denoted N(Il,a). ',: 'The distribution functic:m c::on:espondin,~ ,t.o (2,t5),.:is ~.~en. by (2.46) This integral cannot be evaluated o~ a closed (o'nn, 'By the'substitution "'- ,.,': s.t:;p ,dt - O'ds (2.47) the equation 1_2,46) becomes,_ " (2.48) . where '1>:< is the standard normal distr(/mtiol1 (ullction defined by :" 25
- 38. 26 2. FUNDAM£NTALS C?F PROBAIUUTY THEORY 41 .(x)· ' ' r1n : eXPl-,?" ldt X v~:: - '-00 . ' .. (2.49) The corresponding $tand"rd normal density functioll ill (2.50) Due to the important relation (2.4S) only a standud normal table is neces~·. The functions ';x and 4lx are ",own in figure 2.8. f.;x tx, A -3-2-1 123 , 1.0 ---. .:::-:;-~- x Figure 2.8. .: , ; , ' r . . Let the random variable Y .. tnX De normally distributed N{~y. 11~)' Then the"l'lndom variable X isS3.id to follow a Joprjthmic normal distribution with th~ paiimeters ~rE R and Oy > O. The " IOjl:.normal density function is . . .; -. .,. 1 1 1 inx-,uy 2 'x(X.)~ ay$ xexP[-I<--,,-.-) J (2.51) where x;:' O. : Exezcili<! 2.'7. Derive the lo&,-normal densit)' function'·{2.51),by the USE' of equatioD (2.38). Let X ~ ]og·non:uii1ly distributee. with the parameten .u y' and ~y: Note that lAy and 0y are not equal to.ux :md "x:ll can be shown that (2.52) EsetcisC! 2.8. Lei X be log·normally distributed with the parameters Ily and ay. Show !~t . . lnx-lJy Fx (xl· P(X .;; x)· 4>(--,,-.- ) (2.54) wh"ft' .]. is the 5c:mciard no~ttl cii3tribution function.
- 39. !! .6 UNIVARIATE DlSTR1BUTlONS ! fXj)n 0.0 1i "0.41 T '~1 (2".1) 0.2 '-- 0.0 I 1.0 2.0 3.0 Figure 2.9 x ~7 i The log.normal density {"'nctions with the parameters (J.'y, (ly)- (0, 1) and (1/2, 1) are illu· strated in figure 2.9. , Example 2.12. Let the compressive strength X lorconcret.e be Jog·normally distributed with the parameters (j.lX' "y) " (3 MPa, 0.2 MPa). Then -"x - exp(3 + t . 0.04} - 20.49 MPa .-- ok .. 20.49'(1.0408-1) - 17.14 (MPa)~ Ox z 4.14 MPa and P(X," 10 MPa) at 0II'«lnl0 - 3)(0.2) '" 4>(- 3.467) - 204 • 10~ An important distribution Is the so-called Weibull distribution with 3 parameters tI. (and k. The density function ex is defined by (2.55) where).:;;' r andtI > 1. k > E. If r" 0 equation (2.55) is ~~(.X): .. t (~)JI:.~ exp{- (I)JI) , x;> 0 (2.56) The density function (2.56) is called a two-parameter Weibull density function and is shown iii i1i:; ... ~ : • t1 Tf F '" 0 and ~ - 2 in (2.S5)lhe density function is identical with the so-called Rayleigh density (unction 12.5'j)
- 40. 28 Figure 2.10 2. FU!'lDA.'tENT.~LS OF PROBABILITY THEORY ----- (k.~)· (I. ') I (k.~) · (1. 2) (k.~)=(2.2) -""-:.- 2.7 RANDOM VECTORS Until now, the concept of a mndom variable has been u~ed only in a one.(;jimensional sense. In section 2.4 a random variable is detined as a real·valued function X :n ......... R mappin~ the sample space n into the real line R. This definition can easily ~ extended to 3 vector·valued random variable X :nARn called a mndom vector (random n.t"ple), where Rn '" R X R X ." X R. An n-dimensional random 'ector X:n'""'R" can be considered an ordered set X '" . (Xt , X2 •...• Xfl) of one-dimensional random v31iables XI ;n,.-.,R. i = 1 •..• 1 n. Note that Xl' X:!: •.. . • X" oue defined on the same sample space n. Let Xl an? Xz be .. two random variables. The range of the random vector X'" (Xl' Xzi is then a subset of R~ as shown in figure 2.11. Likewise. the range of an n-dimen~ional random vector 15 a sub,se t o( Etft. ____ C-___________ " a'igure 2.11
- 41. :!.' RANDm,l VECTORS .- c ,:. Consider llgain two random "ariables X1 and X~ and the corre~ponding distribution rune· tions Fx and Fx 2 • It Is clear that the latter give no information regarding the)oint beha"jour or::<l and x 2. To describe the joint behaviour of Xl and ,X'!. th~ft?i"t probabifi. ty distribution. {unction FXI . x 2: Rlf'""' R is introduced and defined by . (2.58) It is often convenient to use the notation FX for FX1,xZ' where X .. (Xl' X2). The defini· . tion 12.58) can be generalized to the n-dimensional case , FR(x)::: p( n (Xi C;; "i» (2.59) ial where X .. IX1,·, .• X"n)and X - (Xl"" .xn). 29 A random 'ector can be discrete or continuous. but the ~rp.sentation here will be confined to continuous distributions. Only two-dimensional random 'ectors will be treated because genera- . lization to n-dimensional rando~ vectors is straightforward. The joint probability density {lInction (or the random vector X .. (Xl' X~) is dermed us (2.60) The inverse of {2.60) is (2.61) The distribution functions FXl and F:<2 for the single random variables Xt ~nd :<2 can be ob· tained from (2.61) . 3nd similarly (or EX.,. By dirre~nti3tion of (2.62) tx, (xt ) ,. )~~. !X:(x1• "2)dx'Z ~.nd correspondingly for (x r 2 fX:!(:t:':!' "'" fX(X.l,x~)dxl The density functions fXI and fX2 llre called marginal den.sity fUnctions. (2.62) (2.63) .' . 2.64)
- 42. 30 o F".OAMeNTALS OF fROBAlL/TY THEORY E.umplc 2.13. Consider again e;!l:3mple 2.1 and let a 2-dimensional discr~e random vec-tor X" (X~. :;.21 be defined on r.! by .. . ,;" P'(j, 3) ';' O~l ~ P( 4'~ ~i) - 0.1' P(5, 3}- 0.3 P(5, 4) ""0..2 P(6, 3}' 0.2 pte. 4) 0:0.1 . .; ' • ".f ..... 'The mass [unaJon Pi is illuStrated in fil.Il'e 2.12, and the mar&inal mass functions PX 1 and PX:l in figure 2.13, .... Note that PR(x1 , "2}" PX1 (Xl) . px,(x2)· Figure 2.12 r>x,lSl) 0.5 j I ( . ~x 4 5 6 1 Figure 2.13
- 43. ~ . 6 CONDITIONAL DISTRIBl.'TIONS 2,8 CONDITIONAL DlSTRIBUTION~ In equation (2,16) the probability of occurrence of foVt'ul 1.:1 ,·,lllditional upon the occurrenct' of event Ez was d~Iin~d by (2.16) In accord:tnce with this derinition the conditional prCllNlbi'.,y mass (unction for LWO jointly dis. tribui.ed discrete random variables Xl and X2 is ddinl!tJ U~ , . Px x (x"x2) . I' - '-'!L! .""''-;,:'-;...:- PX1IX:(xl x 2) - PX~(X2) (2.65) A natural extension to the continuous case is the followinv. lh'firiition of the conditional probabill!)' density function (2.66) where fx:(x2) > 0 and where fX:l is defined by (2.64). N"",. that PX 1 IX: is a mass function in (2.65) and fXll X2 a density function in (2.66). The two random variables Xl ~d X2 are said to be im.lt:p.'"fJl'~·t if . ~2.6i) v .. hieh im~lies (2.68) By integrating (2.66) with respe.ct to xl one gets the c<mdJlhmo' ·distribution function rl .. (;~1':~/-"., 7·2)dx~ .. FXIIX:(xllx2) - fXt(x2) (2.69) "ext by integrating with respect to x2 the so-called IOhJl/"'lllUbillty theorem is shown (2.70) Example 2.14. ·Consider again the two jointly dislritJlJt.t:d discrete random variables Xl and X2 from example 2.13. Note that but for e);ample
- 44. 32 _. FT.:NOAMENTALS OF PROBABILITY THEORY Therefore. Xl and X2 are not independent. Exerci~e 2.9. Consider two jointly distributed discrete random variables Xl and X2 with the probability mass functions PX t and PX2 given in figure 2.13 and assume that Xl and X2 are i~dependent. Determine the joint probability mass function Px for the ran· dom vector X • (Xl' X2)· 2.9 FUNCfIONS OF RANDOM VARIABLES In chapter 2.5 a random variable Y,. which is a function ((X) of anothe~ random variable X. was ' treated and it was shown how the density function fy could be deter:mi~ed on the basis of the densit~ function! x' namely by equation (2.38) (2.38) where X" C-l (y). This will now be generalized to random vectors, where the random vector y .. (VI- y2.·· · . Yn) isa function1 - (f1,· ·., (n)or the rando,1Tt ve~to~ X " (X1.X2'·· ~ ,Xn ), ~at~ .. . " (2 .71 ) where i .. 1, 2, ' . . , n. It is assumed that the functions fj,l .. 1, 2 •.•. , n are one-to-one (unctions so that inverle relations exist (2.72) It can then be shown that (2.73) (2.74) 'x_ 13V - I is the Jacobian determinant. Let·the random variable Y be a function (of the random vector X - IXI •. _ . . Xn). that is f2.75)
- 45. :. - 2.9 FUNCTIONS OF RAl'Dm.t VARiABLES 33 It can be shown that " (2.16) where i ., (xl' ...• xn) and f xCi) is the probability density function for the random vector x. Exercise 2.10. Show that " " E(l";(Xill - IE!f;(X;lI (2.77) i-I i-1 so that the'op'eration of expectations and summations can commut~'~~i£ompare with exercise 2.5. Exercise 2.11. Show that EliZ" ';(X;J1~ II" E!f;(X;)] (2.78) i-I i-I when Xl' , , , • X:1 are independent random variables, _ ~et Xl and X2 be two random -ariables with the expected values E[X1 ] ., Jlx 1 and E[X2, :a IlX2 , The mixed central moment defined by , (2.79) -: '~ ,"!to ...... <':" .. - •• "'-"" '. is called the cOlJariancl! of Xl and X2. The ratio, Cov(Xt , X.,] PXl X~ " aX t aX 2 - r, 'r; (2.80) : •. " where aX t and aX 2 ~ the standard deviations'of the ran'dom variables Xi and X2'.' is called the correlation coeffici.~nt. It can be used as a measure of mutual linear dependence between a pair of random variables. It can be shown that - ~,;;; .oX1.X2·-';;; 1. Two random variables Xl and X2 are said to be uncorrelated if .oX!::<2 .. O. It follows from equation (2.76) that . (2.81) Therefore. for uncorrelatec random variables Xl and X2 wt' ha'e E(X,·X,] = PX, JE(X,J (2.82) !
- 46. I 2, FUNDAMENTACs OF' PROBABILITY THEORY I ! , ! 1: is imponant to note th:n indcpentlem random variables 3n~ uncorrelat!d. but uncorrelated 'anables art' not ir. i!eneml independem . . :Note that (2.83) Therefore. the mutual correlation between random variables Xl' X! •...• Xn can be expressed by the so-called COt'Griancc mcurix C defined by .;', . .. .. . . • . . . . , c. ·..',Co,jX,.X,:.) 1 ..• " • • ,.: .: • ... ~ CovIX2 •Xnl . . I,· •. , ·Var!X21 .. ....... ':. V..{X,I J .... Exercise 2.1.2. Let the random 'ariable Y be defined by where Xl ' X:: are random variables and ai' 32 constants. Show that Varl '1 .. ai VarlXl J + 3i Var!X2 J' + 2a132 C~'[Xl' x:!] 2.10 MULTIVARl.-TE DISTRIBUTIONS (2.84) (2.85) (2,86) The most imponan~ joint density function of two continuous random variables Xl and X2 is the biLoariate norln~! d.~ns~t~.fu'}.rr.tion , j!:i'en ,br (2.87) ,· .. here - ... " ~1 " ... - ... ..:;; x2 .:;: -. and,.: ';:2 are {h.e-means, 0l:~ 02 · ~he standard deviations and p the correlation coefficient of Xl <lnd X2 , .. .... . .; . "1"," ' .:, Exercise !U3, Show that the mo.r~inal density functio~li ~Xl (Xl) for (2,8?1 are 1 lx,-~~ f X (Xl) c -=- exp! - - (----.l) J (2.88) • I , / 2:: a l 2 0 1
- 47. BIBLIOCRAPHY .. ' . ,;. The multiloariale normal ciensilY {uncllon i;; defined aJ; ," .. ' " ',." ;~ BIBLIOGRAPHY 12.11 12.21 12.31 " .. 12.41 ' (2.5] 12.61 12.7J An~, A. H·S, & W. H. Tan(/:: Probabili~y Concepts in Enginacring Planning and Design. Vol.l, Wiley, N. Y.,197S. Benja~in. {R: & C. A. Corn'ell: Probability, Statistics and Decision for Civil Engi. neers. Mcci~a~.Hii1: N.Y' .. 1970. Bolotin. V. V.: Statistical Methods in Structural Alecharlics: Holden·Day. San Fran· cisco,1969. Ditlevsen, 0.: Uncertainty Modeling. McGra ..... ·HiII. N.V .. 1981. Lin, Y. K.:'Probabilistic Tneory of Structural Dynamics. McGraw-HilL N.Y., 1967. Feller, W.: A~ Jntrod~ction to Probabilit)· Theory and ils Applications. Wiley, N.Y., Vol. I, 1950, '01. 1l,1966, Larson, H. J. & 6. O. Shubert: Probabilistic Models in Engjllacring Sciencc •. T. Wiley & Sam. N.)' .. Vol. I 6: fl. 1979. 35
- 48. .,: n· ;s -,:-1::1':: .... . ."" ! • • -
- 49. ·I.;·' .. ,:i. .;.,, : ~ ' -f " ·,·'t :. . ... 'I' :," ' :.' Chapter 3 I::' :: •. t-. PRoilAiifLIsTiCMODELs FOR LO,IiS AND RESISTANCE VARIABLES .' :.:I.Ji : .'~" • ,;,;:. '::}:1' ',.::, ", ·: t· · - .TJ."· , ~. :.' ',;,:' ·-,m:" .. l ... 1; ..... 1 ~.o..: . ": .. .. ... , !, .... : . . :.! .. , . ~r ,r. In .thls c;hapt.e~,.tpe a.i!1l)~;,~9. .~~.~.~jn.e ~he, w~y}n I~h~ch .~.~~bl~. p:ro~~l~~}i,c;: ~,~~~s:.,~,'r. ~e . developed t.o. represeq.t ~h,e .. ~l!~~r.tai"t!es. that.~xist,in typ.i.~, ~~je ~b~~5: ,}~~~.?aIlJi,9~ ; consider the problem of modelling physical ':uiability ,an.c;t IH~~n,turp t.~.t~~: ~~~s.~~C?~ . ~~ I~~ corporating statistical uncertainty, Load and ~islance parameters clearly require different treatment. ii."lCe loads are generally :ime·varying. A5 .di.s.cl!s,s~ in !<~,,:p~rs 9 :md ~9 .. ti~e-v:a.I1:! .ng . loads ~ best m~71~ed as stochastic processes, but,th}s i~: ~~t .3 c~~v,enient te~res,enta~ion COt use with the methods of reo ::ability analysis being presented here (chapters 5 and 6). L'utead. it is appropriate to usc the i istribution of the'extremelvah.ie 'or tile !dad :n'the iefe~e:1ce" period :or which the reliability 37 !3 required; or, where there are two or more:!i.me-'arying .roads :lctin~.,n a structu~e together, :he distribution of the extreme combined load or load effect. The particular problems associated with the analys~ or combined loading are discussed in chapter 10. The selection of probabilistic models (or basic random variables can he -:ti;ded into two parts • the ehol~e'of 5uitabh~" prob3bility Clistnbutions:,vith which to cha.rac.t.:!rize ~~~~hysical uncertainty in 'each c:ase and the C'hoice of-appropriate 'alues for the parameters of those distributions. For most practical problems neither task is easy since there may be a number of distributions which appear to fit the available data equally well. As mentioned above. loads and resistance variables require different treatment and .will be discussed separately. Hovevet. it is first necessary to introduce the i~po~ant subj~~:-~f ~h~ ~tatistic~i tbeo~y~ ~'{extTem~s '~hi'~ld~ of rele· .... ance to both load and strength variables. This topic is disc':1s~~ ,in .tb~i next :~;'.~ ~.tio~. 3.2 STATISTICAL THEORY OF EXTRE~IES In the modelling of loads and in the reliability analysis of SlrucrunJ systems it is necessar:y 1.0 deal with the theory of extreme values. For example. with tlme·v~ing loads. the analyst 1& interested in the likely value of the greatest load during the life' oC the., stru:~iu're. To be more ,necise. he wishes to know the probability ci:5tribution oi th~ £-reate!t. 'road. This may be inter· "reted physicaUy as the distrihut!on th~t. would-b.e obiain~ iJ. the ::o_a.xi~il!!l.Hre~ime I.oa~ were :neasured in an 'iniinite set of nomin:llly icie:::ical structures.
- 50. 38 3. PROB.ABILlSTICMqp~!-S FOR~OADS AND RESISTANCE VARIABLES , , In an analogous way, if the strength of a structure depenqs on the strength of the weakest 'Jf a number of elements· ioc example, a statically dett:rmin~te truss· one is concerned with the probability distribution of the minimum strength. In g~J.l~ral. one car estimate, fr(lm, test r~sults, ~~ refo,~s ~he,~~meter5 of ~?~ ~t~i~l~~ion of the instantaneous 'aIues of load or of the strength of individual components, and from this information the aim is to derive the distribution for the smallest or largest values. 3.2.1 Derivation of the cumulative distribution of the ith smallest value of n identically distributed independent random variables Xi .wume the existence oi a random variable X (e.g. the maximum mean.hourly wind speed in consecutive yearly periods) having a cumulative distribution function .'~x ~~ a ~orresponding probability density function fx ' This is often referred'to as the paren-t ~istribution: Taking a Sample'size of '(((e.g. h'years'records and n values oftheiniiXimum niean-hoilrlywind speed) lE!t t~e c'liinUiatlve districlIiti'on 'function of the ith sm::iUest'i.lalue X!l in the sample be F X" and _ ',' " I , I its correspoiuiing density function be fx~' - .. , ", " ",' .: ,J Then f~~ (x)dX '" co'nstant X probability that (i -1) values of X fall below :It ~- I ',1: ,", ,;- : _00',' _ -, ",:, ' , "-, " ~,probabi1ity that. (n -'i) values of X fall above i x. pro,bability that 1 valU!? of _X }ie~ in the range_l.: to, (x T d,~) , 0; cFr1 cx)(l-:- Fx(x»n-i,fx (x)dx ,lr>! _ (3.1) where the numb~r of ways otch,?osing,,{i -:-:-J),val~j~ l~~ :~han x, together with (n - i) values greater than x, (3.2) , , , -. • 1 ", - -', ThUs --FX~(Y) '" r:f~n(xjdii,,; Y ~Ftl (x){1 '-'F ~(x))n"";i f>i(x)dx 1 I "0 - L ',. 0 . . ',' (33) ThiS can be-sho~n to be equal t~ c· Figure 3.1
- 51. l .!! STATISTICAL THEORY OF £XTRE~IES [ (F".(Y)Ji In-i 1Fxty))i+l I'n-i) -;-- ~ 1 ; (i'" 1) + , 2 x (F ( }}i1'2 n " X Y _ (n_il(n-i)(fx ()')) ! (i+2) .•. +( 1) n-i n J (3.4) Exercise 3.1, Show that equation (3.4) can be derived from equat.ion (3.3) by expanding (1 - FX (x»n - i an.d integrating by parts. Equation (3.4) gives the probability distribution function for the jth smallest value of n values sampled at random from a vari£ble X with a probability distribution F x. Two special cases will now be considered in the following examples. Example 3.1. For i = n equation (3.4) simplifies to: FXn (x) .. (Fx(x))n • (3.5) This is the distribution function for the muimum value in a sample size n. Example 3.2. For i" 1 equation (3.4) simplifies to: (3.6) This is the distribution function for the minimum "alue in a sample size n. It. should be noted that F X",(x) may also be interpreted as tht!. probahilit)' of the non·occur· renee co! the event (X > x) in any ofn independent triah.$O that equation (3.5) follows imme· diatel)' from the multiplication rule for probabilities. Equation (3.6) mty be interpreted in an analogous manner. See also chapter 7 .. 3.2.2 Normal extremes . If a random variable is nonnally distributed with mean IlX with standasd deviation Ox the vari· able has a distribution function Fx (see (2.46» F (x.) - -- -exp(--(:.....!:.X.» dt x 1 1 1 t-II- 2 x • _oo..;z; Ox 2 Ox (3.7) If we are interested in the distribution of the maximum 'alue of n identically distributed normal random variables with paramete:-s Px and Ox this has a distribution function Fx"'(x) '" ~ - el:p<-.,(--X) )dt ( " 11 1 t - •• , ' • II • _ .. y'..!:J: Ox - Ox J (3.6) It S~OLL;: i"~- ;;~l ~~<:l tOOlt Fx: is not normtJ~l)' distriO!.lICC.
- 52. 3. PROBABILISTIC ;IODELS FOR LOADS A~D RESISTA:-:CE VARIABLES 1 1 . , r)t"(S) ·l.5 Figura 3.2, The probability density function fX." Z I ~ (Fx.") is shown in figure 3.2 (or various 4Iu~ of n and with X distributed N(O. 1). 3.3 ASY~fPTOTIC EXTRE~IE-VALl"E DISTRIBUTIONS It is fortunate that for.:l very wide class of parent distributions. the distribution functions of the maximum or minimum values of large random samples taken from the parent distribution tend tO~~lIds certain limitinl;l: distributions as tbe sample becomes large. These are called rJsYI'!!.ototic extreme-I:a{ue discrfbutions and are of three main types. 1. II and lIt. For eXa!),ple. if the particular .variable of interest is the mLximum of many similar but independent events (e.g. the annual maximum mean·hourly wind speed at 3 particular site) there are generally good theoretical grounds for expeding the variable to have a distribution function which is very close to one of the asymptotic extreme value distributions. For detailed iniormation on this subject the reader should refer to a specialist text. e.g. Gumbel [3.8J or Mann. Schafer and Slngpurwalla [3.111. Only the most frequently used extreme·value distributions will be referred to here. 3.3.1 Type ( extreme~value distributions (Gumbel di5tribitt~ons) Type {asymptotic distribution of the largest extreme: If the upper tail of the parent distribution falls off·in an exponential manner. i.e. (3.91 where g Is an incre3sing {unc~ian of x. then the distribution function F~· of the'la~est 'a!ue Y. from a large sample selected at random from the parent population. will be of the for~ Fy(Yi - expl-expt-o:(y -ullJ -"'''y''. formally. F y will asymptojic311r 2pproach the dist:-ibution given by the right' hand side of ~qu.:ltion !3.10J as n - "".
- 53. 3.3 ASYMPTOTIC EXTRE:'.IE·VALUE DISTRIBUTIONS fl' . Figur.3.3 The parameters u and Q: are respectively ::1easures of location and dispersion. u is the mode of the asymptotic extrem'e.valuedistribU'tic:l (see' figure '3.3). The me~n and standard deviation of the :ype I ma:dma distribution (3.10) are related to the parameters u and 0 as {oUows (3.11) 'nd a .-'- Y .J6 fl (3.12) _ or' : . . . .. -- ~" . . . . . ' .: :: .. , ... ,:, .. .~ ,. . . where "1 is Euler's constant. This distribution is positively skew as shown in flgUfe 3.3. . A useful property oC the type I maxima distribution is that the distribution Cunction Fyn for the largest extreme in any s3mple of size n is also type I maxima distributed. Furthermo~e, the standard devi~tton '~emai'ns constant (is c:dependent'of n), i.e. .. . ,;.;..~ , " . . ~ :. . ~ . ' ... ' . -. '. ' : '':'! : ~ ~" (3.13) This property is 'Of help in the anal~-sis :-o: load combinations when diCferent-num6e"rs of repe· titions of loads'nj need to-be considered ' see 'chapter 10). In this connee'tion. t(is uSeCul"lo be . 'able to calculate the parameters oTthe -eitreme ·vari.i!.ble y~ from a kri"owledge 'of the para~ . meters of Y. IC Y is type I mtlxima distributed with u:s;:ribution (unction Fy given by equation f 3.10) and with p~rameters Q: and u. then the e~me;::~ distribution 01 ma..'<.i:na genei~ted i'n n "i~d;epimdent trials has it distribution function FyA, ty)" ,exPI-=-: 1.1. expt.~, ~f.Y - .uHI .. ' ,
- 54. 42 3. PRODAB.ILlSTIC MODELS FOIl LOADS AND RESISTANCE VARIABLES 1 with mean given by (3.15) Type I asymptotic distribution of the smallest extreme: This is of rather similar form to the, Type I maxima distribution. but will not be discussed here. The reader should refer to one of the standard texts·5ee (3.81.13.111 or 13.51. ' 3.3.2 Type U extRme-value distributions As with the type I e:.;:trem~va1ue distribullons, the type II distributions 'are of two types. Oldy tbe type II distribution of the largest extreme will b.,e conside~ed here. Its di~tributionfu~ction Fy is given by Fy(Y) s::: &p(- (u/y)") y;o. O. u > 0, k > 0 (3.16) where the ~eterS u and k are related tO,the mean and ~~~~dde~iat~on by .. #J.~ =ur(i~l/k) "',1. (3.17.) . , 0y - u{f{l- 2/k) - r 1 (1 - l/k)]2 with k> 2 (3.18) . where r is the gamma function defined by ,- -11 11:-1 r(k): e Il. du . ·0 (3.19) It should be noted that for k '" 2, the standard deviation Oy is not defined. It is also of i~terest that if Y is" type II maximi'distributed, then Z;;. .l!ny is type I ma.xima·di·~tributed~ . Elo:etcise 3.2. Let Y be type II maxima d.istribuied with ~istribu~~.o~,~uncti~~l'Y and '. coefficient o!nriation ay/Jl.y' Show that the variable representing the largest extreme with distribution function (Fy(y»n has the same coefficient of variation. The type II_~~ d~tribu~ion is freqlJ.~ntly,used in modelling extreme.hY,drological and meterologica, l, events. ~F ~~.as the limiting distri,bution of the largest valLIe .of manY.independent ident.ically distribute~ .~.9P~_var~ables,_ whe~. the parent distribution_is limited to ,values greater than zero and bas an infmite tail to the right of the form 3.3.3 Type III exbeme--'alue distl'ibutions In this case only the t)'pe 1IJ asymptotic distribution of the smallest extreme will be considered. It arises when the parent distribution 15 of the form:
- 55. 3.3 ASYMPTOTIC EXTREME·VALUE DISTRlBUTIONS 43 with x;' £ (3.21 ) i.e. the parent distribution is limited to the left at a value x .. '" E. In many practical cases f may be zero (i.e. representing a physical limitation on, say, slrengthj. The distribution of the minimum Y of n independent and identically distributed variables Xi asymptotically approaches the form ","'llh y;;;' E, P > 0, k > E;;;' ° (3.22) as n .... ""'. The mean and standard deviation of Y are: (3.23) and (3.24) The type III minima distribution (3.22) is often known as the 3·parameter Vi'ejbufl distribution and has fr.equently been used for the treatment of fatigue and fr~cture· problems: For the special case f C 0, the distribution simplifies to the so·called 2·pa-rameter Weibull distribution (3.25) -10- _ 10 -t. . _10...,0; . : I -10~ I . -- -- I _10-.11 rormal .. k1Y' ~_-'.:- 'i . .-..p. type II maxima i I , -10~ I i: 'k...:... ... -; - I '-+ type 1 maxim. . i I 1 I )'1 U"ol·nonn~ 1 1 -10-1 II I i I ! I d •• I I ! I I I I I I 10-' ,., I , I 1 1O-1 ·j lAY I ! I I ! I iii/! I I i I 10-.1 I .,' . , I I , , 10""" / 1 , I 10<10'" 7 i , I : "I : , I , I I . I 0.' I . - - . . -0.::. 1.0 1., 2.0 3.0 4.0 Fi&ure 3.~. Cumulative distributions or djrrer~nt di$tribution [unclio:l'; (/oy ·1, "y • 0.2).
- 56. 3. PROBABILISTIC ~IODELS FOR LOADS A~D RESIST.-:CE VARIABLES with 13.26) 'nd (3.27) Comparisons of the type I maxima and type II maxima distributions with the normal and lognormal distributions are shown in figure 3.4. The random variables in each case have the same mean and standard de;ation. namely 1.0 and 0.2. 3.4 ~IODELLING OF RESIST_-"'~CE VARIABLES - MODEL SELECTION 3.4.1 General remarks In this section some general guidelines are given for the selection of probability distributions to represent the physical ,uncertainty in variables which affect,the.strength of structural components and complete structures - for example, dimensions, geometrical imperfections and material properties. Since each material and mechanical property is different. each requires individual attention. Nevertheless. a number of general rules apply. Attention will be restricted here to the modelling of continuously distributed as opposed to discrete quantities. The easiest starting point is to consider the probability density function fX of a random variable X as the limiting case of a histogram of sample observations as the number of sample elements is increased and the class interval reduced. However. for small sample sizes, the shape of the histogram varies somewhat from 'sample to sample. as a result of the random nature of the variable. Figure 3.5 shows two sets of 100 observations of the thickness T of reinforced concrete slabs having a nominal thickness of 150 m!", which illustrates this point. These data wer.e not. in fact, obtained by measurements in rea] structures but were randomly sampled from a logarithmic normal distriJution with a mean JlT"=' 150 mm and a coefficient of variation VT ". 0.15 (see appendix A). The corresponding density function fT is also shown.in figure 3.S. For comparison. figure 3.6 shows data obtained from'a real construction sit~. A clear disti!lction mus~ be made. however~ betwe-en a histo~am or a relative freguency diagram on the one hand and a probability density function on the other. Whereas the former is Simply a record of obsen-ations. the latter is intended for predicting the occurrence of future events· e.g. a thickness less than 100 mm. If the probability den,;ity function fx of a random variable X is interp_ret~d as the limiting case of a histogram or re!ati'e frequency dia~m as the sample. size i~[Ids tojnii~ity. the probaoiiity P given by ,x:! P=>P(:<'l<X':;X:!:I= ':x(xldx • x! I 13.231
- 57. Ui 0,03' 10 0.02 r ,I , 1 l--+---<,2.1:J,JLL14LL14LLC:;S.~_ :fmml 90 110 130 150 : Fi;:ure 3.5 ;.;; ;':urnher " t· SMtaono"d a.r.dI "d,e . 'intion .153.1:mm 12.;mm No. of rc'.dinijs 272' 14 10. , 1;0 190 210 1 1 nominal v.!u~ ! 1 1 1 1 :>,. 160 L • _ - .- Fi~ur~ 3.6~ Ilistu~~:m~.or.5Iab ~hic~n~5.'i measurements.. " t:' .. ~, •.•• ;c .. ,.' .".' .- .-. - ; .. :;. 45 " 'f'! .,.; -, --. ~ .t(mml
- 58. 46 : / .~, 3. PROlU.BrLISTIC MOD~ts;FOR LOAi$s~ND R~s'i~A~~~ V~~IABLES I clearly has a relative'frequency interpreUltion: ~ :;. if ~ very large sample of varilible X is obtained at random, the proportion of 'alues within the s!.:npie fo.JIin{!. in the range»1 < X " x~ is likely te. · to be ver), close to P. Ho~·ever. thi>; interpretatic:: may not in practice be too helpfuL All that ;;' can be said is that jf a variable X does in fact b ~xe a known probability density' (unction lx-and if it is sampled at random an infinite numbe: of tim~·~. the proportion in the range 1 );:1_ x!! [ wlll be P. . The problem of modelling is completely di!ferer.:. In gene!.afthe engineer is likely to have only a relatively small sample of actual observations of X. along with some prior information' obtained from a different source. The problem then is ho~ best"io use·aU'.this Information, Before this question can be answered it is necessary to deCin~ ex.actly what the variable X represents. This Is best explained by means of an elWDple. ( .. ( ..... . ". . . Example 3.S. Consider the mE:Chanical pro~rtie5 of a single nominal size of continuously· ~, . cast hot·rolled reinforc~ sWeI. Let. us rest.&.!t our attention to a single property, the dy· ~ namic yield stress, 0yd,delennmed at a controlled strain rate oC 300 micro·strain per minute ~ and deCined as the aVer.!ge height oC the stre.s-strain curve between strains of 0.003 and 0 .005.: i.e. (3.29). where 0y(e) is the d}'namic yield stress at s~-in e . Let us assume that tM property can be me3:-.Jred with negligible ex.perimental error and tha: all the reinforcine: hars from a single cast of ;;eel are cut into test specimens 0.6 m long and then tested. If 0vd is ploned against Z. the pjsition in the bar, the outcome will be of the form r' shown in figure '2.7. This is an example of a i:ep..wise continuous·state/continuous-time Sto· r· chastic process X(tJ in which the parameter: alii:!! be interpreted as the distance Z'along the - reinforcing bar. (See chapter 9 Cor further dE.:ails of stochastic processes). ,:'. The process is interrupted approximately e'r:.,-y 600 m because the continuously cast steel is cut into ingots and these are fe-heated and tc.Ued separately. The fluctuations in yield stress within each 600 m Jengtb are typically very S!D.all, i.e. in the order of 1 - 2 N/mm'. For each 600 m length f, the spatial average yield stresi a d is defined as 1 ~ ::: :~ . y 0yd ~2L o),ddE "_. :- (3.30) The variations in ~ from one roUe?: le~·gth. ~o'another: ~~ tYpi.cally I~~r, tp.an th~ within· le~gth . variations and are ciused mainly by d:.:ierences iil 'the terop.erature of the ingot at the start of-rolling and by a number of other factors.. Some typical data giving values of Oyd for consecutive lengths of 20 mm diameter hot·roU~ high-yield bins froin the same cast of steel are shown in figure 3.8 (along ,..;th values for the sl-Atic yield stress). These can be can· sidered as a continuoll5·state/discrete-time stochastic process. It can be seen that there is a fairly strong posith-e corrclatioo between aye for adjacent lengths, as might be expected. If £c is the totalle11Jlh of reinforcement proe'.1eed Irom a single cast of steel then the average yield stress lor the cast can be d~Cim·c as (3.31) ;.. . ; ).., . .. J :'1I
- 59. 3A ~10DELLING or RESISTANCE VARIABLES· MODEL SEl.ECTIO!' t °yd (NJmm 1 ) soo I :::Lj ~~~~ ~ .40 Zlm) ·'·+I----r---~--~----+_--_+----~I--~~--- 0. 500 1000 1500. 2000 2500 3000 3GOO . Fi,un 3.7 VUbtioJU in dynamic yiel4 stress alone a 20 rum diameter bot·rolled reinCorc:ine bar. soo .eo ... 10 " 20 • , ,. .. " 30 3S 40 50 bar number Fi,ure 3.b. Withil~as var;llialU in the yield streu of. 20 .mm aliLI"~ •• _ ""I·rolled reinfarcine bu. 47
- 60. -48 Provided that the '3riattons in yield sttesS alohg ~a~h 600 m cl~~n~~h oC co~tin~~u~I}: rolled b:tr car'! be assumed to be small in comparison with 'ariations in Oyd. the average yield stress for the cast may DC expressed ~ ~ -! . ~ . . uyd - n .::... q~'d(l) (3.32) j"l . :"'. " ), .;:; where ayd(i) Is the yield stress oC the jth bar and n is the number eCbars [olled"from' the cast. ," .,. It we are interested in the statistical dfs,tribution of the yield stress of reinforcing bars supplied to a construction site. accourt mu.s~ also !>e taken 9,' ~h~. Y~riati~n.s in ~d that occur from cast to cast. If the steel is to be supplied by a single manufacturer and very cloSe can· trol is exercised over the chemical compoi!;'ition of each cast, variaUonS ln Uyd will lM!' very. ~mnll; but if the chemistry is not well controiled significant difference's betveencw can 'Jccur. If bars are supplied by a number of ,diCterent manufacturers. systematic-differences hetween manufacturers will be evident even for nominally identical products (e.g. 20.mm diameter bars) because of differences in rolling procedures. r final effect which must tle t:1ken into account is Uie·systematic 'change in mean yield :it,ress with bar diameter as illustrated in figure 3.9. This phenomenon is quite inarkeci"iind is rarely taken Inw acc,?unt in structl,lr.al design~ " ,- "" !~ !I : Yield mc" (X!mm:) ,,0 ~ ,00JI _____ -; ______ -; ______ -;cOc·c,Cdc;·cmc'+',c,_icmC'n~' -. I I '00 1'1 10 20 30 '10 :..: Bako!r t.nd 1ckh2m (1979) o nak~fj. 19. 0 ) • ~anniuer (l!!GS) ','
- 61. 3.01 ~IODELLI~G OF RESISTA:-."CE VARIABLES - "IQOEL SELECTION ··· T· . . . !, . ,t; From the preceding example it is clear that there are mony sources of physical variability which contribute to the ovenil uncertainty in the yield $treS$ oC '" grade oC rei.nf~m:;ing steel. Le~ us now define the quantity X os the random variable representing the yield str_~ssp{a particular grade of-reinforcing steel irrespective oC source and where ~yie!d ~tressll is defill~.n a precise way. We now wish to es:uiblish a suitable probability density.function.Cor X to use'in further calciila~~o!ls .. ~i"iS.clear ;:hat 'the mathe~'atical Corm oC fll. will depend on the p~i~C~,lar subset of X,e.g.: Let ~I be t~~ event [b.. a rs are suppti~a by manufacturer iJ 81 !1e th~ even.t (bars .~re o'i~~~t~r jJ C be the event' (t~ars aie (r.~~ a sin ale cast of steel) The.ll"i.p ge~eral.the.de1"'~i~~'i~jl~tiOri's .~X' IXIA!' fxrS;. fxtAln 81.'. fxrA~ "'.Bj''"'I.C et·c. will all be different: not only·thee parameter.s. but'also then- shapes .. It is also clear that the probability density functioitti rep:esenti~g" all b~, iire;~tive of size or manufacturer. will not be oC a simple or standard fom je~,. normal."i6gJ1o,,;a1. etc.J.lt will take the (orm . . -. --. -- ~ . . . . .. " . ---- . __ . (3.33) (3.34) qj being the probability ,hat the bar is of diameter j. Equation 13.33) represents what is known as a mixed distribution model, ;..,,; H snould be-noted thatbeca~se .oc th; systematic decreaJIe in reinforcing bar yield' ~t~ess with incr:asi~g dia~eter, equation-(3.34) gives rise to a density runcd.on fXI A - which ~.:n.~tter and I has less pr~mounced tails I platykurtic) than any of the compon~,:,:t distributions fXIAI n BI' Furthermore~it,i~_~enerally Cound that .the dens:ity""functl?_n~~"iB~ representing bars of a particular .si~e considered.Q·er:;l.1I manufac.turels is highly positi;,eIY's~ew; The 're<lSon lor this is - .·c·,-: , ,,' " ' • .. . ,- .. ' I discussed in example 3..t.· be~o.w. Example 3.4. The );eld stre~:'~/hot-r?l1eci ~teel plates of a:siitgfe.'J.o{J1inal thiclmes5 and grade oC steel. supplied b ....· :1singfe mllnufact"un!r."ca·n" be ~nown to be closely represented by a log·normal proOabUity distribution 1~· e.9~~tiOn (2.51)). as illustrated by the cumulative- frequency diagr:uiis in figure' 3:10: IC'-howe·ve.r, data frQ!l1 .~ number of manufacturers lire combined. the'distribution becomes hi~ly ske:w. This is b~cause manufacturers with high pr~~ct ,ariabilit.y have to aim for hi~.,er rne~n "properties than-manufacturers whose products can be cio;cly controlied to achie';e the same specified yield stre.;:;;. tor'" Kiven probability of rejec::on. See figure 3.11. I:: ~hould be not~d. t~at the scales _chosen in -fjgures 3.10 and 3.11 Olre 5:,,:' that a logarithmic ::ormal distribution p~ots Ol~ a straiiht,line.
- 62. 50 ,. , ,-., , i. 3. PROBABILISTIC MOPELS FOP.. LOADS AND RESISTANCE VARIABLES • ; ., < 0 .• 0.7 0.' 0.6 0.' 0.' .. 0.2 O.l 0 .05 0.02 0.01 0.005 - . .. - .. _j -. 'I ' I . I I E I Y' i i I, . ; . .. I. I . . 1 i. 1 i ' I"~ "J. I I i 1 I·' . I 1/1/ .. d i A A" i/VI "" Mill i'i. , ;6, mm pi!lt,,:,: . . . ' fo' illure 3.10. Q.ainulativt! fr,quenc}' di~';;; (or yieid nrea of mild .t .... 1 plates. 0.998 . . 0.995 0.99 0.98 0.95 0.9 0 .• 0.7 0.' 0.' 00.4 , 0.3 . 0.2 .. O.l 0,05 0.02 . 1 . I I .. ' ! " . . 1 '. . I I 1/ I 1 '· 1 0- I I· I ,. I i . "". ·1· i h ... ',/. .', j.,," . .. i 1 1 ! " fl. I I: I I . 1 1 i · I ',t .1 , I I ! , 1 ' i I I I I, 1 i :i 220 24" 260 2.0 '00 320 , .. 'SO r::=+:=+t::i:===:+==t======+==::;:=t:~ I 0.01 0.005 Lf_-'-_--'-_--'-___ -'--_-'---'_--'-_'---'-__ ..L---'_-'-_ !t'lmm, l Filure 3.11. Combined cumulative frequency dia:f.m for 12 mm mUd steel platu (rom three mills.
- 63. 3.'; !l.IODELLl:-lC OF RESISTANCE VARIABLES· MODEL SELECTION 51 ,~ Ir" • . ,'. We now retu:n to the queslior. oi selectin!!: a suitable probability distribution to model the un· ce~~.i!lty in :r.e strength variable X. It should be clear from the preceding arg"Umt'nts tbat:l pfoc'edure 0: random sampling and testing of, say, reinforcing bars at a constructio~ site and attempts to fit a standard probability diitribution to th~ data will not lead to a sensible ou;· come. In partIcular, such a distribution will behave poorly as a predictor of the occurrenc(> of .'. values.of X.outside the range of the sample obtained. The only sensible approach is to synthpsise the probability distribution~'of X from a kno~le'dge of the component sources o(un;:::er· mintY. (as in:e'quatiori (,3.33». Admittedly thi!; ~pproach can be adopted only when such in· •• ,' .~,) .:. I, . ' , .. . formati?n is available. Expressing this problem in another way. it is important thal [he sta· tistical analysis of data should be restricted to samples which are homogeneous (O! more pre· eisely:. for which there is no e'idence of .~,on~hom~~e.~eitY). '!' ~urthe.r ~pect. o~: ~~em,,!g_ ?lust now be imrodue~. Models do not represent reality. they " ~oq~f .. approDmate it. As .is ~ell1:mown in other branches of engineering, anyone of a number of different empirical models may often be equally satisfactory for'some particular purpose., e.g. finite·element versus finite·dlffer1!nce approaches. The same is true of prob.3bilistic models. The question that must be asked is whether the model is suitable for the particular application where it is to be used. For most structural reliability calculations, the analyst is concerned with obtaining a good Cit in the lower tails of the strength distributions, but this may not always be important. for example, when the strength of a structural member is governed by the sum of the strengths of its components. This Is illustrated by the followin,e example . . Example 3.3. Consider an axially·loaded reinforced concrete column, a cross·section of which ii shown in figure 3.12, If, for the sake of simplicity. the Joad·carryin~ capacity of the column is assumed to be given exactly by: 12 R· re + ~' Rj (3.35) j"l where r~ is the load·carrying capacity of the concrete (assumed known) and Rj is the ran· dom load·cru:rying eapacil~' of the ith reinforcing bar at yie!d. Then, if the 'arious Rj are statistically independent, . 12 12 [IRI - EIre + ~., Rjl- re + ~ ElRjJ (3.361 j"l j"l and l!! VartRJ::. Vartrc + ~ Rjl :12 s = VarlRiJ (3.37) i·l i.l i.e . • • • • ~'''':'''.~r- • • • • Ficut. J.i~. Cr~a'WClion of felnfornd Ci;ltIe,ete column.
- 64. 52 J. PROBABILISTIC ).IODELS FOR LOADS A:-lD RESISTANCE VARIABLES (3.38) and 12 • • • (1:.' )' R j .. 1 R. . (3.39) Assuming Curther that the various Rj are also identically distributed normal variables, N(100.20) with unitsofkN,and thatrc""500kN.then . " ~R ::: 500 + 12 X 100 ::: 1700 kN' and (JR " 6'9.28 kN Since R is also normally distributed in this case, the value oC R which has. a 99.99% chance o( being exceeded is thus . , . , PR + 't>'l(O,QOOl)(JR -1700-3.719X 69.28 "'1442 k1'f This totalload·carrying capacity corresponds to an Dtieraglnoad..canying capicity oC ' (1-142 -:- 500)/12 • 78.~ kN Cor the individual reinCorcing bars, i.e. only 1.07 standard de· viations below the. mean .. For this type oC structural configuration (in fact. a parallel ductile structural"system in the reliabiiity sense· see chapter 7) in which the structural stren~h is governed ~y the.average strength oC the components, it can be anticipated from the above ~,·8.tthough it ,~lii not be Connally provoo ;,ere • that the reliability of the structure is not sensitive to the extreme lower tails of the strength distributions of the components. Hence the lac:::,k . ~f,.,!~.iJabmt.y oC statistical data on ex~remely, low strengths is not too import.~t, Cor such .c~es. ., '. Finally, jt,should be emphasised that these conch,.Isions are based on the assumption that _ .~~~ ,~;arious' ~I ar~ ,~ta.~ist.i~atly independen~. Exercise ·3.3. Given that; the column dfs'cussed'in examph~ 3.3 is subjected to an ~ialload of 1500 kN. calculate the probability that this load exceeds the load~~ng capacity. Re· calculate the probabilit.y under the assumption that the variou~ Rj ~ mutually Cully carre· lated (p '" + 1). 3.4.2 Choice of di5trib'~t'ioru; Cor resistan~e ~ariables It has already been mentioned' that unlessexpedmental data are obtained Crom an effectively homogeneous source, formal attempts to fit standard forms of probability distributio~ 'to the oata are hardlv worthwhile. When data from two or more sources are p~seni in a single sample; the overall sh;pe of the cumulative frequency d'istribution is likely to depe~d as much. i~' not more. on the relative number of 'obs'ervations {rom each source than on the actual. but unkno~n, probabiiitv distributiori"o[each·sii!i:popuhition. Extreme caution should therefore be exercised if the t}o,,; of probability distribution is to oe chosen on the basis of s~ple data ai<;m~ ... , A preferable approach is to make use oC physical reasoning about the nat1~~ ?_~ ~~ch: p~pc~l3.r: random "ariable to guide the choice of distribution. A number of limit,in,. cas~s ''fill .now be s~u. died.
- 65. 3A MODELLING OF RESISTANCE VARIABLES· ~IODEI.. SE.I..ECTI.ON ., . .:.··: :i ~,'.:_ " .!' ',-~ - . . . ' . - . ' . .' 53 Tire normal (Gaussian) distributio!l: As discussed in chapter: .. 2 ... this is one ot the'most important :. probabilitv distrib~~io~~·• . lt.·arl·s~s~~~ne·ve·~·~h;~~~do~ yariable of interes~ X is the slim of n ·ide~t.i~~ll~ distrib~ted inde~:~~~t ~~~~ :;~~".I~':~i.·.im~pe.ctive of,the probability distribution or Vi' provided the mean and variance or Yi are finite. Forma.lIy, if Yt , Y2 ,.::' Yn ~.ind.e~~der.tt .~den.. t. .ical~y! ,di~:~,lb~~!!~ !~~dom v~.~,l4!!~ .with. finite mean'py and firUte'variaiice a~, and if X .. Y~ +' Y2 + ... + Yn, then as n - 00 X-n~ Pta < ~ <: 0) -+ ,~(I'l) - 1)(0') Gym (3.40) ~or all tr,IJ(Ct < jJ), and.where·'" is the standard nonnal distn~uti?n r~n'cti'i:)~. T.his is known as .. the centraf'iimit theorem;' :..;·" ;V:-'· Provided a further set of conditions hold, the central limit theorem also appli~ to the sum oC in· dependent 'ariables which are not identically distributed, The rate: at ~~ich the sum tends to normality depends in practice on the presence of any domi'~ant n~n.~~rmai components. It is thererore clear that any structural member whose strength is a linear {unction of ~ number . ~ o(i!ld~pendent random variables may in generarb"e·con~idered. ·t~be g~vemed' bY' lh.e· ~?~'aJ law . • ; . ,.. ,, ' , ' !', I>~. ' ,",. -" , !-:-. ,. :I~ . " . . • • • ., ,, . r • . ,:: : Example 3.6, :Consider again the reinforced 'concrefe'columil discussed iri example 3.5. Since :, th.~ ~.tr!:.l)gt~. ~f,~hfJ!: concre~ is asrumed known and the strengths of the ielnfon:ing bars ha'e been asin~~~~ ~9 p~,ind~~ll:de_nt • .it}!lay' ~e cpnclyded.that.the load-currying capacity oC the coh.iinn R'is' normruly' distributed. (Whether this is. true in practh;e <;Iearly ·depends ona num· ' ber of other fa'c'tol:S arid' wnettlerthese'aSs&mpti~ns '!I.e :y~lId). . . " _.~' ... _ It is ,?~.e~mes ~gu~ .th.at,the normal distribution should not be used to model"resistance ''';ariables because it.8'!ves a finite probability . ~f negative strengths. However: this appa',ent Critidsm can be 3is~:n~ ,to··b~';ela~i~~i~ u~important iithe strength of a component can'be consldei'cd to be . the s~m ~{~ nU~'b~~-~tind~pendent tandom variables, thereby invoking the central limit theorem. The logarithmic normal distribution: The logarithmic nonnal (or log-normal) distribution is frequently used for mode~.~l~i~tanc.~ mia~.les an9 .. ~,'he. theoretical advantage of'precluding negative values. The mathematical form and parameters of the log.normal.d~s.t~ib1JJip'9, were rl:iscussed in chapter 2 (equation f2.51}). The log-Mnn,al distribution a.rises. naturally as a ,limiting distiibution when the riindo'm re'is~~~ X ~fthe p,~.u~t ~.f.~:~~~be' od~~~~:e~~!,.~.identi~aJlY distributed component variacle" i:e. . . 'X -Zl Z2': : .:Z~· ;, :nZ . (3.41) . ".,.. ",,'~": "" -, I-I ,i Clearly Y given b'y ' " ,,,. " Y <: I!nX::I QnZ1 -+- I1nZ:!.+- ... HnZn .. Il!nZi ,,:, '.:... .. j-I;· . tends to normality as n - -. follo~'ing the centtallimit' the6r~~:iieg~;d.ie~~·of·th~ probabili~y distribution oft!nZ .. The' probability' alstributio~ of x. th~~eio~~nd~' to~;;rds th'e log.n~~al. I .' .. , _,"' , ,. :. , as n increases.
- 66. I .' . . ... ,.. ~ ' " .: .... :: .' 3. PR08'BILISTId~IODEL5 FOR LOADS AND RESlSTAI-'CE VARIABLES -.-- ·,1-", ,Whether X may be regarded 'a, a Jog-normal random' ~'arial,ji~ 'in -~ny :pr;ca~;l situati'~I1" in which X is the .product of'3 number of randor:l va::1bles'at-~:nds' ~oh 'the dri:'ums~~'llc~'i;~_ Th'c l~g.norm:li distribution is .-howe'er:u$ed very ~.ja('ljj.jt .. 5yeuatiilit}.:Studi~. · · ~ . ' I ' I r . :. , ,;r": . " .. p . k"· (3.43) where k, /.l.and c.: are variables. It is therefoJ~ . to be e>:.pected tha~ suen.:,ath parameters,which are aCiecled;by friction, (e.g. the shear 'strength 'of cohesionieSl soils. cables, etc.) will ,tcnd to be log-normally dis· tributed, since spatial variations in the coefficient of friction p. wilt give rise to expres- " : sionsoilheiorm. -.: , :,- .!l . · h :ro:·~· . ~. , ,',';;.',1" - . . " :: .. :, ... ; :: •.. ~;. ":"'" 13.44) .: ,:;1' '.; ;1'' , .. ~,~:. , 1'.:':' ,: :-: !::'lc~' ~~-)< Th'e IVcibuU distribution:.This.distribution is used quite frequently to·model ·the distribution :'~(th~ si~ength or a 'struc~~~~comp~nent wnose stren~h is governed by size of its largest de· fect. I~ ~t is ~sumed that c~f~~~ compt;men.~. such as :weldedjointS,'contain' a large tiumber of s·mall·d·ete!!.~ a~d thaqhe,severity of these defects is distributed in'an appropnate··manner. the distribution.of the component Strength a,pproachesth;{Veibuli di;t'ii~~·tJ~~:· ~:di.scussed . in section' 3;3;:nt is one ofthe'so-<:alled asYn:ij)t'oti~ e:i~rem~'~~'ue di~trib~~ioru.' i~.density function is given in equatio:~s (2.55)~nd (2.56j' . .... ' . . . - . .otilcr, t!i~!rj~ution.s: A number of ocher common dismbuiions ·exist v.,hich may' o'n' occaSion~ be . _ useful for m.odelling the uncertainty.in -resis:ance 'anables'-" for·'eioimple.' itlf rectan'gUiir·,' beta. ". ,: ga~~a and t-distributions. for information of these distriljutions'tne :re~der'sho;ulcfcoflsult a .. ,.standard text, e.g.13.5!. ~ .. ' , .... ; . . ·' i' ,." ;:. ",,,,~ > ' ,;)'''' . . _ I"'." . -.1 ( " . 3.5.·MODELLING OF LOAD V .ARIABLE£~· ~tOD£L SELEc.T io!-t ·~· -,. ' .. .. . . . .. . ",. : "" . 3.5.1 .?~~~~ ,~~m~ks .. ,~ .. ,, _;,: ", : ~"'.;' ', ... co'," ••• ' ,_" . . ' ,._ . The·term load 1S genera1ly understood to mean those forces acting on a nructure. which arise from exterriai 'influences '. pnncipall;' th~ ~f!~ts of g~a'ity. 3~d aerO'~Y~~Il).i.~·~~;.~d hy~~odyna. mic effects, e.g. structural self·weight, superimposed loads, snow, wind and wave loads. The term action is now often used as a more general description to include both loads and imposed deformations. Examples of the latter are dimensional changes ~ris~~g iro~ tem~ra·ture. effects and differential settlement. Both loads and impo5ed deformations give rise ,to: ~t~ .of,action(' ffeels (often loosely referred to as load-eflect,S) within 3. structure. e.g. bending moments and ~he:lr forces. .: . Unlike resistance variables. most of which change 'ery lit.le during the life ot a structure. loads :md other actions art> typically time''arying c'.lantlties. The main exception of course is the. self· ~'ei~ht oi!~iinailen;·5t~c;~:r;I;·~d '~on,slru~':~~ c~~P~~·~nts. As me~~ioned earlier"time-. ' • ,_. '.-: - - ~; " .: : ,j" - -.. , -, " - • " • ':J. ~ying quantities are be~t modelled as stochastic proces~es , but discussion of this topic is post· ::: on~d to chapn'rs 9 anc 10.
- 67. 3.5 ~IODEl.Lll'G OF' LOAD 'ARIABLES· MODEL SELECTION i{ ~ ~ften helpfullO clas~ih' Io:"·.;~!~ ~nd olhef <lct.ions in aC::..Jrciance w'ith the following t!uel' attributes J3.9J. Each load or aClion can bt' described liS • permanent or 'ariabJ E- • fixed Of fre~ • static or dynamic These three Independent. attributes relate to the nature of t.he action ....· jth respect 1.0 • its variability in. magnitude with time . • its variability in position 'with time • the nature of the induced structural response Thus the 100ld imposed by vehicles on a lightly.damped long.span bridi:e could be described as being varja~lel free :md dynamiq. In general, -loads and actions cannot be sensibly classified ",ithout a knowledge of the structure on which they are acting. For any particular ~ctlon and structure, the attributes listed above also govern the nature of thertNctund .i:naJysis that must be.undertaken.:·;'"':: ., . i. ... '. r, . . To sO!'n~ aetre';"~e~I:ly' ~l.ioads· c'ould be ~onsider~d . to. be variable .. free and dynamic, but whethe:each' i~"~l~ifr~d '~';.uc'h depends on the r~sponse of the structure to·the loading. ··EU.mple 3.8: ·'Consider a steel bridge loaded solely Py a sequ~~e of paniaUy-laden 'ehicle:;. _tu far as the' imposed loads are concerned, the probability of failure of the bridge by a sim- . pie plastic collapse mechanism depends only on the weight of the heaviest" vehicle (assuming that only one vehicle can be on the bridge at anyone 'time). However, the prob'ability of failure by fatigue will also depend on (a) the weightS of the other vehicles and (b) whether the indivi.dual vehicles induce an}' ~ppreciable dynamic r~ponse. Clearly. there is only one iOun:e orloading. but the way in which it is classified and modelled is dictated by the fail-u. re ~od~ be'ing analysed. ' It should be noted that the preceding c1assirication appllt:s both to the ......... actions themselves . and to the mathemntical models thal are used to represent them. - A further disU'n'ction tha~ should be made is between' loading models used for the purposes ofn'ormal '(detern~inistjcJ desi5!n and t'hose: required for structural reliability analysis. To tak~ the' simplest case, ~t.nou8h a permanent fixed load i~ considered to be an action which does not 'a~'y i"it:·, ;.;r.~;:· '1r in position. it musi ge~erall): b~ cla&sed as a~ uncertain quantity.for the pUrPo:ses 'of rellab'jiit)' a~alysis;.: since ;,1 ;;~p.rlJl its mD.£"nitc;:!~;;·:!!,~,,!· ~c !:ncwn.-It must there' Core'be mode'ller.i~;~: r"a'rid~m 'variable. Hqwever, for determini~~ic design purposes it can be represented by ~'~ingl~ ~Pecified' constant. It will not have el=:caped the attention of the reader that t.he modelling of loads and actions requires a ceri;'iil d~grei'or s:~bjpctj,~ fudgement. The same is true for .rE:$istance 'ariables. This should not, however. be seen :l~ a limitation. since the aim is not to produce a perf:!ct im:lgt- of r£-ality (an impnssihlt.· task). but to dt!'elop a malhematlcaJ model of the phenomenon whit"h embodie$ its salient features and which can be u~t"d to make optim:e design decisions usin~ till' data <l.·ailabie.
- 68. 5';; :J, PROBABILISTIC ~10DELS FOR LOADS ANO RESISTANCE VARIABLES Finally, it should be noted that some *Ioads~ act in a resisting capacity for some failure modesCor example, a proportion of the self-weight of the structur·~ ~ most o'er-tuming problems, [n such cases , these IIloadSll are strictly resistance variables froni 3. reliabili'iy 'ie~vpoint. They are ge~erally easy to identify. '-"':' 3.5.2 Choice of,d(stributions for loads and other actions Ve now consider the process or defining appropriate random " anables and their associated pro· bability d.istrib.utlons to m~l single loads and other actloris, The modelling of combiraations o(ioads, is discu~ed in .chapter 10. As in the case of resistance variables, the procf!$s consists o,f thr~ distinct steps • • • precise definition or the random variables used to represent the uncenDinties 'in 'the loading selection of a suitable type of probability distribution for each rand~~ vari~bt~, and , . estimation of suitab.le d~tribution parameters irom available data ~d,any prior knowledge. l~ m~y respects the first, step is both the most important and the most difficult to deCide upon in,p~ctice . . Example 3.9. " Consld~r t'h/modelling of the asphalt suri~~ing ~~. ~ 'lo;,g.~pan ~teel, bridg~, Should the surfacing be treated as a permanent or a variable load? How should spatial varia· tions in this load be taken into account? Should variations in .density.as well as variations in ·thickness be modeUed? What is the probability that an additional lay.er of asphalt.will be placed on the bridge without removoiI of the origi~aI surfacing '~d 'how should this' b~ alA lowed for? These are typical of the questions that must be asked in any realistic load modelling prob· lem, They are also questions that can only be sensibly answered when the precise purpose ot the proposed Il'!liability :malysis is known. . . ,', ' The second step of selecting a suitable probability distribution for each random variable" can rarely ~e ~ade on the basis oi sample data alone and as in the case of resistance variables physical reasoning must be used to ~sist in this process. Some general,guidelines are,giveR .below. The third step of evaluating suitable distribution parameters is disc.ussed In section 3.6. Permonent loa.ds: The total permanent load that has~to' be 'Su'p·porter.i'by a struct~;e' i~ g~ne'ra1l~ the sum of the self.weights of many individual stru&ural elements and other·i;irts~· For 'this' rea-son (.see page 53) such loads are well represented by nonnal probability ciiStributions. Wbether the weigi'l,t.;s or individ)Jal structural elements can also be assumed to be normally distributed depend~. ~:m the nature of the processes controlling their manufacture, When the total permanent load acting on a structure is the sum of many independent components. It can easily be show~ that the coefficient o[ variation of the totallciad I~ g~~eralfy ~~ch less than those of itS'·components. Exercise 3.4. Given that the total load on a foundation is the sum of n independ,cnt but identically distributed permanent loads Pi' show , ~h~t t~e co~fIicier;t,t of va~ation . oq/:le .. ' . :otal load is only lIvn time.s that of the individual loads, . . ,'. i ' . , ' -,
- 69. 3.5 '-IODELLING OF LOAD VARIABLES · MODEL SEL~ON .1: " , f, . , ,',' ' ''. :'' ;1< ' . ;' .' -.• j,'. ... . ' . JL ' . • , .J '; i • VoriGble loads: For continuous U~e.-v~~i~g. lo.a~s .whi~h ca~ t ~ ~ ,~n.lquely, des c.!i,l?~~ ~Y a single quantity. X:(e;g. a magnitude),' one can defin'c a number of d.ilferent but related probability : . ,,' '. . ....... ~ '" . ~ ,., :. ;:,' '. ,. ' : ." :', '''' , .. ~ . . . distribliti'oh (uriction:s~ The most basic. Is the so~al1c d ot:bitrary.point,in.time or (iNlt-order dis- :" lribu'iio n : ~r X.·i . ,'. : .. ,< ,.",.: .. ' .. ,.; .. , . , :"","~,,v ,_. " . ; , '.: Let x.cn be the magnitude or a single tiRle:varying load X{t) at time t' ,.Fot,example. see figure 3,13',yhich 5h'~~ ~ c·~ntil~~-·~late/co·nti~'~~s,:timejstoch~~ic , prpce$5. _Then 'Fx: is the arbi· ,'" .' ' 1'1,. ,. " ,; ". :-- 'UI.'- ·' :" .: " , ;" ' ~ ~ - .' U I.' ... .... . • ... _. . . .• . • 'trary.poiL,t:rn-t1me'distribuUon of ,X{t) and is.,defined by _ .,' . j ,: - " , L I . • .. .:. . . . ,. , ' .. ; .. " .... , .. . ,: " (3,45) ' :'. / ' " 57 where t' is any randomly selected time. The corre,spon~i~.~ ~e,",~i~r (~':lct'?lI:. fx. is also iIl~trated . " , - ., ... ,', ," .:' . . '~', ,,! .. . . In figure 3 :13iTx may tUe'on a ':vide range of Corm anC:! 'depends on the nature.of X(~) - i.e. ~hether X(t.) is ~ deterministic or·s!ochasti~"ru.~~·ii~~ oit;~~.~·h~the~ the i~d'~~ ~e_both . '~~~ative and" positive"ViJiies~'e'f.c. "' ~~ .. ' ' .. '·~·,i. ,,"J'l ' I''''''~' ' , .. ,' 'j ' ' . < '., - . : : ' ::-O .:;~ " ."" ' . .;, Ex~'mph; 3.~O. ,Ir.th~.Joad)c.(t) htlS. ~ .detenninistic time-history given by ".J '. , ... , , : ,_c· " .. . ' x(t) 1:1 xsin(wt) t-.. .. . w",".'· .'. ,.,: -'.' , .' i.e: xn) is a si~usoidaUy-varying {orc;e of k.no'!¥n amplitude X, then i: can'be shown that , . ' .' '::., _. . - . . ,' . ..... ' ; .'7;.i; < :x "'-,X ;:~ " :-c:-:,)'; :--:~ : .. ~ , . ~ :::-- x which is a U-shaped distribution. ' . ;, ' ' . ',,-. -:',' - '" .:t· " xn) . :.' .i'·j, " T ~ ,: FlguR 3.1 3. nlustratian of continuoul tim~,vlr}'in f load. ",:; ..::', :.:.
- 70. 5$ . i .. _." . . .... "" '-' (· i,'l. · ... , ._ . • " , ", ' ". PltoI:lADILlSTldt.ioDELS FOP.' LOADS AND RESISTANCE VARIABLES '-j• . ' .. . ' ':' i . ," .;, I .; " • • __ ., .. . .... ! 'r: :':, ; '-; ~:.:" .. ,:, " ~anipfe 3.11", Variations du.e lolwaves in the surface ~le'lltion .<?f.lhe ~ea,X(t) .alt any ii.x"ed point remote fro'in;'the shor~ can be sh~;,·n·t·o .. ha.'.:e· ~)irs~.ord~r d~tribu,ti~.n ~X ~ .,whlch apprqximates very closely to the"normal distribution '(tor periods or,.time~n ~'~ich the sea-state can be assumed to be stationary) . . "". _':' . ,! " .,1. ,. , . j '" .". ,:. I ' • ~ Howe'er. wherr deiiling with" singie time-varying loadS and sQ..Qlled Cirst passage problems (i.e. _when !ailure:occurs if-and'onl)' if the loai{eJ{C?eedsi~;ome·t·n;e$h~I(i'v~~~i. the ror~ ~i the ar- . . " .' _''' . J " " .j .... .. ,; ,F~' J.~~' :'J ., . bitr&l)··point.in·time distribution is not of immediate relevance. The random variable which is oC importance is the magnitude of the largest extreme load that occurs during the referen~ ') - ::'- ' .. period T for which the reliabilit.y is to be determined. The latter might be the specified design liCe or any other perioci of time. ,'. . ',,'." ,.' : .';~ r •. :' .: .' ', ...... ', ... ;, j ,;:-;; ," ., ,!:':<) I ; If the loading process X(t) can be p..ssumedto Le ergodic,(see chapter 9)",the'distributiop of , ,' J.' . " ":':' ,"; ~:'l:I~':':'''' !,nl. I:'''') ',.c',,_"" 1 ,,' .. , ' .. ' '' ', .' the' largest extreme load can be thought of.1!lS ~etng generated, by. ~~p.l.mg the vaJ,u.es ,of xmu . ' ", ' " .. ".' "r;" .... : , " . ,; .• 0-.' --,~, - . ~. from'SicceSsive reference p.eriods T.lf the values ofxmu are , ~~pre~e!l~d. ~ytl?-e:random .yari-able Y, then ~y is the distribution function of the largest extreme load. The correspon.ding density function fy ·is illustrated in ,figure 3.l3·andean be' ~ompared ';ith;lh-J"ci~iUiYy-iunction of Ule arbitrary·poim-in·time distribution fx' Since. for a continuous loading process, the largest e:E:treme load that occurs during any reasonably long reference. period T corresponds to the largest of" fin:it"e num~i o'i p~~k ioads, . it can be seen from sections 3.2 and 3.3 that the pro~abi1ity dbtribution of the largest extreme is likeiy to be very closely approximated by one of the asymptotic extreme-value distributions. These distributions are frequently used for. representing the lD~irri~~'~r.time-varYing loads, It should be noted, however, that the precise form and parameters of the extreme-va1ue distribu· tion oe-pend very strongly on the aUlocorrelation':fuilciiori of the loadi~~ process X(t.). The can-cept of aut.Ocone-iation is discussed in chapter 9. ~;' , .. :: :::;, r:.".:- For the present purposes it is sufficient to state that the maxima of time·varying loads can in mOSt cases be represented by one of the asymptotic extreme·nlue distributions, with pa~ meten estimated by one of the techniques given in se-.::tion 3.6. 3.6 ESTIMATION OF DISTRIBUTION PARAMETERS It is assumed that the selection of the types of probability distribution for the Ianous load and resistance nriables has been made using the approaches and methods of re~~ning discussed -pre· viously _ The problem now is .0 estimate suitable numerical vaJUf!S for the parameters of. these distributions using availahle data. For single distribu~ons this req.~res just o~e set of data.' but [or the- more complex mixed distribution models such as &hewn in equation (3.34) various sets of d3ta are clearly required_ The overall process of parameter estimation consists of • initial inspection of the data • application of a suitable estima.tion procedur~ • nnal mod(!1 verific:Hion.
- 71. 3,6 ES11~lATlO!l: OF DlSTRIBt'TIO:-'; PARAMETERS It cannot 'be e~phasised.tou 5lrongly that the blind apPlic~tion of ~~atiSUCal p:ocedures can lead to ,,'ery misleading results amI that an initial inspection of the a'ailable da~a should always be undenaken before any formal calculations are made. Let I.lS consider the practical problem of estimating·the parameters ofa single disi'rib"ution func· tion fr.om a single sample of experimental·data. The first. step is to check the dau for obvious inconsistencies and errors. Manually recorded or copied data have a high probability of contain· ing at keast.some transcription errors. These should be eliminated if possible. The second step is to'plot the data in the form of a histogram to check for outliers and to confi.lin tbat its shape does o"?t devia.te markedly from.the shape of.the density function being fitted. If the histogram is" clearl}' bi·modal when a uni·modal distribut.ion is being fitted to the data or if the sample ap· peals'Co be truncated wnen the variable is ~sumed t~ be unbounded. checks 00 the data source . ~. arecleady'requii-edjri1consistenCies are often found to aris'e'when the'~~'t'of data h$ been ob· "', t~ned from.experimental test programmes in more than one iaboiator,:,i. Such lumping of data is often necessary when the sample sizes would otherwise be very small. but this should'be a~:: voided if possible. Checks on the consistency of the means and variances of the v~ous sub· . samples (see for exampl~ [3~5l): should generally be under~k~'n wh~n practic:abl~., .'" . The next step is to estimate the parameters of the selected distribut!on using one or more of the tecbruques described in section . -". 3.6.1 below. The baSic methods are :.. ,. • the method of moments • the'method of maximum likelihood • virious gra~hi~"ai procedures • use of order statistics. The last step is to check that the sample data are well modelled by the chosen distribution and parameters. ~lethods fordoing,this are briefly reviewed in section- 3.6.2. 3.6.1 Techniques for parame.ter estimation . This! i;;~~ larg~ subject in itself andonl)'. a brief description ,is p~~sible here. Readers unfamiliar w.i,th ~he :~,i~~S co.n~epts s~~~ld~lso study a specialist text (3.11J. 13.5],13.8]. It is assufliea in the ,f911o)'ing that the distribution function is known or has b-:en postulated .and that its parameters are now to be estimated, Depending on the distribution type. one; two. three or more parameters will be involved. The general procedure is to obtain estimates of the~ 'unknow~' parame~er~ in terms of appropriatE.' functions of the sample '8lues. The word estimale"is used in this contest advisedly. It should be clear that because of the random nature of the variable no sample. howe'er large. is completely representative of the source from which it deffi'e5;and indeed, small samples may be markedly unrepresentative. For example. in a ran· doni'sample of 10 independent Ob~~~'~r,j,~~~ drawn from _~.~ormal distrib~tioo. there is a prob· ability of approximately 1:1000 that by chance all obsen'ations will be grean:!' than the mean, Any attempt ~" p.:>timate the parameters (p, oj of the parem dis.tribUtion from this particular sample will result in cOll"idt'r3ble error. This difficulty cannOl: be eSl':l.ped. but the probability of Jargi> errors occurring decreases as llic s:'lmple size increases.
- 72. 60 3. PROBABILISTIC ~10DEtS FOR LOADS A;.IDRESISTANCE VARIABLES In essence there are two types of estimau~s for distribution p3Iame~ers that can be obtained . point estimates and in!ert.'al estimates: A point estim3.te is a single 'estimate 0"( the 'pa~meter whereas intetval.estimat.es allow certain additional confidence or prob~bility' s~temen'ri t~ be made. In this s:;:ction only-point estimates will be discussed. The different techniques of parameter estimation su~m:irised below correspond to the use of dif!erent functions of the sample data and give rise to different estimator~ 'ior'th~ para~~ters . .- number of desirable properties which characterize IIg00dll estimatorS·areiunbi~sedness. efficien. cy and consisten.cy. (For a precise definition of these terms, see .f or exa~ple. r3.ill). No ~sti· .." . ,' . ~ mator. however, has all th~e properties and in practice the choice' or estimator is iov~~,n:~ .by the particular requirements of the problem, or expediency . . Jethod of moments: Let'the variable of interest X have a probability de~~it~ iun~tion..rX' with parameters 81,92, ::". 6k . From equation (2.35) the jth ro?~ent o~ X is given by 'ij .. E[Xi] ... r xiixex)dx Since fx is a'functlon of the k parameters 8'1' 6~ ....• Ok' the ~ght han~ side of eql:1at!9n (3046) is also a function o~ the same k parameters and tj may he ~xpressed.as " ..•. . " (3.47) . Usin~ equation (3...16) to generate the first k momentS t. we obtaln k equations in the k un- J , ..• • " . .. ' . '. known distribution parameters 8j . If we now -consider ~ 'iandom sample of' th~ ~ariable X of size n '~ith ,values (xl' -Xa t.· ••• xn) the equivalent 'sample mOrl:lc'nts ru-e given b~ : ; . . (3.48) Finally. the moment estimators 8J. i'" 1, .... k for thek unkna'~. d~ttibU~iO~ p~me~fS 8, ,' may be obtained by equating the momen~~'~,~ -:f. ~~d the s~ple,momen~ ro, ~:'r' , ".: j -1. .. : I k (3.49) Example 3.12. Let X be a normally' distributed random ·aria.b·I~: having parameters jJ. and (1. The ~e~sity iunctio:n,given by equation (2.45) is ~ I • • ' • ':"-. (3.50)" . . Assume that a random sample of n 'observations of X has b~{(obi3ined.iil' x2, ~" ,' X~ ')~ The moment estimators for Jl and a= are now determined ;;.:; follows ... , J' • :,: ;'.: ,.;.:.,,: p.51) .,,;
- 73. .' ;3".6 E.S'T I~fATION" O" F D.I.S ,T R.tB.UT, . I.O..N PARA;IETERS, '.: . - . ", . ' The 'equIWlcht'iampie m~ments :lre -n 1 r?; " tnl xi m =1- L"X ! 2, , ~ i-I' . 1. • ;.) ·(3.5:.n -. ,.:.' .. , . ", .,:.' (3.531 Hence by equati-:tg terms, the estimators fJ and 0: for the parameters IJ and 0: may ~' , obtained from ri' .,; ~.; b". iJ -~ Li(-: ... ·.. , i-I . , .1' and .... : . .- " .. ~, : .,,' '.:: .. .,,,:-.. ' . .. .. (a.5;) Alternatively al may be expr:ssed as ~: I 1!'l ~ aZ "'- I{x. _p)l n i-I '. '-,,' .... " ',' (3;5S} where ~ is the sample mean. However, the form gi'en in equation (3.57) is in_fad prefer-able [rom a computational point of view. . . . . ' . . jj and a.:. ~.en .9y ~qua.tions{3.5.1?} and (3.51) ~e thus .the moment estimators of /l ·:t.nd a~. respectii.:elY: It s~o~.IJ~; ~!i~·9,ted,JlC~wever. that th.e bel!t u,nbiased estimator of (11 is not 0 1 butS2 -.(nl(n-l))oi. Method o"~.axj;;~m,i;~~llh~:d;. ~:i'~'~~th~d'i: ~~~eralIY, m.~r~:~~(~c,Ult to ap~~~,than the~;;':.: me'h~ . .of ~~~~,~~.:~r~~I:<~~~~i~i~g:.i~T,~.t~~;~.',~~.~~!~ti,9.;s: ~,ut. ~~~iml!m likelihood, estimatvrs of ,distribution puam~ters can ~ .~bp~,::~)~ ~~~e,_,a, n~~~r ~C d~~rable ,prqp.ertles 13.111 ;~' :-::·) : " Let th~ variable: '9:£ interest, X _have. a ,probability densit):; function' {Xi with'unknovn ·parame~~ "0 -:" (°1 , 02' ' .•• Sk). that are_,to he determined. Assume. in addition'. that a·partlcul3.r randoIn' sample (Xl' ~, ...• 1:n) ot the random variable X has been obtainec. The likelihood furt~tion : of this sampl~ i3 defi.n.ed. ~, . .: ., .. " :., , ." .. ~ .,,' Ilf:d~i'!~)' i-l . ,. ,:.,;,) (3.59)' . "." ; "'-. L exp~~~'se; ~~e. re_la~i~~~liKeljh.~oc!.~(hf.;'ing obse~y~d)he saf!1~le .:lS a iunction or the p3l"3r.leters . , ' ' _, ,. ' •. ) .• , . w'" •.• ••• 1.>_ ". • • Ii, Refl!rrin~ to equation 1 2.68)it:c~!1 be see,n tha~ the dgh.t hand si~,~ of e.quation.l3.59) is the' joint density functionfx 1,; ~2 •. ""~;·~~-1~~ . ~,~~. '.'_. : : .~_~ ~i} ' ?i"a ~a~pj.~ w.~~h .n,,~.1,~m~~,~_,xl,:.f~' " ... x~ I:'t:ike" at r.mdom from' the ·n.;riabl~ X: In ·.,-h i; ~~$e. hO~'e'~r,:: ~~ .i~ .~~e.~inp}~)·~lu~~'.~l ~,r ~~. ,' .. • xn tha~ :ire knO:n'~~d; the par~irieters'J that lre trC!a((~t1 3S '.-ariables.
- 74. 62 I .. ., .. ... . :so pnqsABILlSTIC MODELS FOR LOADS AND RESISTANCE VARIABLES The maximum-1ilielilwvd est:malorsV of the parameters 1i are--eHme'd 3S. ti.'~ value" of 1i tna~ m~~imize L, O~. equi.valently and Jlorc conveniently I the logarithm on .... rh~ ~'.ltlua~on of j : ' . • . . ' ' . . •••• • J • •. ' .,_ J • •• thus requires the solution of the set of k equations . "/ ~. . " ~ a8. iog(Cx(XjIO») '" 0 i-I J j = 1,2 •... , k (3.60, taking due acco~nt of any constraints (e.g. 0 < 0 < ... (or the parame'ter"'a~ of a normal distri· bution);, " .-' Exercise 3.5, Derive the ma.ximum·Jjkelihood estimators P and a fo,,: .. the parameters $.I and 0 of a normal distribution. Show that for this distribution, these-~lrmaton are the 'same as those obtained by the method of moments. , Graphical procedu.res: For most simple probability distributions. it is possible to plot the cumu· tativ~;distribution function F X for different. values of the ,variabie' x 'as' a sLrai~l1i~e. si~plY by pre-select int an appropriate ploLtine scalI! or t.ype of probability paper. See, for example, figures 3.4 ~d 3.10, Example 3.13. Let the random variable X have a 2·parameter Weibull distribution with parameters ~ and k and distribution function (3.61) Then. 't c. (n(- 2n(1- Fx:{):,))) is a linear function of y "" ~m;. since ' ' ,,;' " , ' ,- ~ " (3.62) The varlabies x andy therefore plot'as a straignt line 'on-ri_aiun;.rSciIes~'Equi;aient scales .in th~ orig'_"l.al quantities 'Fx(x) and x can therefore 'be' conStrUbied: ~ , . , ' ;,,'l - ': ' !' If we now obtain a random mmple of si'te n [rom a known ~)'pe of d,in:ribu[io~ [unction FX' but. with, unknowri.p:uameters':Sl,: the' cumulative' freqliiini.:y. di5't~iI;utioil : fo~;i1i1; '~~mpi~ can be ex: pected to' plot as" a'strai~h't: line if· the 'ap'p'ropnai'ec'piot'iii{g:'sc'~'e is'w~: it: U !~s'JaJ' 'to 'aider the elements 'of the '5'W'lpJe '~xl ,:):l~r', :'. ;"xn Tto obiiJri the 's'eqi~nce'li:~ ;i; :··: '.":·,'-~r~": .. , x!', where ~~ ls, .the smalles;yalue and xr: is the ith largest value called the jth' order, statistic. It will be reo called that the probabilit)' distribution function for the random ·L.;able Xi was derived in ~ec. tion 3.2," One estimate o[ the cumulativ(: distribution function FX (x,) (i.e. the~partib·u'l~r value '~( FX f~r X '" x.j ) is thus i,n, but prererable estimates are i,(n + 1) or (i --:.1/2)/0, since [or most. distribu· tion types they can be shown to be less biased. The cuinulati~e;"treqU:en'cy diagra'm is therefore obtained by plotting the points (xi' if(n + 1)) using scwes appropriate to the type o[ dIstribu· tion fUnctioo',lt"3houJd be noted,;however;'tnat"some ra~d8tti'de'ti~ci~'fro~:a'$trlighi: li~e ' are to be e~pccte<l;-particularlr for poinl$ at each 'end >:J! the:' liile: :' ' , For one" and ·tl~~o·pamrriete~ 'pi"c;b~bilit" dist·~ibutioris. es~imat.es of the QiStrib~ti~n par~~~t~rs can then'be obtaihed by dra:ing th~ ))beSt» straight line't.hrough th~ PIQt~~d poi~ts elli,~r by - , ' ' "' ,
- 75. 3.7 fNCLUSIOS OF STATISTICAL UIC::nTAIXTY 63 e~'e 'or'usi~g'.l!o~al.leasl-~quates m~thod. In both case~. ii is the sum or the horizom~1 squar",cl . de~iatioT$ fr~tn tne'iine which shouid be minimiseU. not the vcrueal (a!Suming the ax~ are chosl'n itS 'shown i~ figUre 3.41 :·Flnally. ~~~ .e~timates of the distrihution·parameters arE' obUlined from the slope and position of the best straight linl:. . '! .. , .• 1"." .. :; , ' . , , ; . . - • . Use of order stalistics: The graphical method disc~ssed above, is in fact a simpl2 application of 'o~d~r-statistics: A'd~Lailed d~ussion of this subject is beyond the scope of this book. The general apprC?a~h h;"~~~l;' ~h~ p!1ra~~te~ qf distributions. of known type is to use sets of coefficients " or ~~ighting' (ac~o~ in 'co'njunCtion with the order statistics to ::.btain "estimates of the parameters. 'The '6o~fficients are ~h~5en to give unbiased and h,ighly e:fficient estimates for samples'of particular size. The approach was first used anc has subsequently ~~!l f':lrt~er developed by Lieblein 13.10) ~ . for e:..:ireme.'alu~ ·distribution·s. S~ also 13 .111. This approach should not be neglected. in any serious application of th~se ·methods. 3.6.2 Model verification The final stage in the process of distribution selection and parameter est' f!~tion should be model vedric'ation. For situations in which only one set of data and no other information is a'aiable, the approach is straiihtrorward~ The simplest method is to check whether the sample dala plot as a re~nable straight line on the appropriate probability paper. If the distribution parameters have been estimated er;aphically. this step will have been taken as part of the estimation procedure. The Sllmple data shown in figur~. 3.10 may be considered to be a Jlgoodlt stralght-line plot •. Alternatively, a formal goodfless.of~fit tcst, such as the x2 test or the Kolmogorov-SmirnO' test may be employed to ascertaln the le'el oC probability a: which it is possible to reject the null h~p:othesi$ . t~3t ,ltthe random variabJe-X has a p.art.icular distribution function with certain stated parameters .. , Such tesU are widely desc..r:ibed, i!.g. [3.51. and will not be given here, In many struCtural reliability problem;. however, the basic variables are best de~~ribed by mixed distribution models !~:- which the test.> dp.sr.ribed above are not applicable. In other cases, the analyst may prefulo use some presc.ri;,oo distribution type 10 Rlf).!~! 8 ~qC variable. e.g. a log-normal distributio~':io' ~od~I-i"i~sIs~~'e ~ariabie, e'e~ though over the limited range of a'ailable data some other distri'b~tioti.: type ma~' in' fact give a better fit. The formal use oC goodness-or-fit tests in the context of structural reliability theory is therefore often limited. 3.7 INCLUSION OF STATISTICAL liNCERTAINTY ., As. mentioned pre'iously, the analyst is often faced with the problem of having insufficient data (or one or all of the basic variables which affect the structural reliability. ~et us assume. however. that there are good a priori reasons for assumine that a p:micular basic random variable X IS governed b)' a particu.1ar type of probability distribution. The problem arises therefore of select· ing the values oC the parameters e for that distribution. One approach is to·use single point es::mates (or t!le parameters· for example, the ma.'(i~:"0l~1ik:p· Iihood _estimates - a..'ld to ignore thc·acdiJi ~nal slatisikal uncertainty that arises when thE-re are too few data. Tnis approach may not 0(' laO unl'ons~n·~tj·~ .. 'l'i nc~ any non·homogeneitr in the data will tend to artificiaily enhance the 'uriance. A betl:t approach is to includl: the statistical
- 76. uncertainty In the p:trameters within the distribution of X iuelf. in terms of vhnt is known as the predicth'e distribution or x, If the probability density function of the random v3nabJe X. tor known parameters ij is written as lx (x ri) then the predictive density hx for u~certain !j is ~iven by hX(x} '" L fX(XIOHi (iflzldO (3,63) B , •• 1 :~ ;: ,,{i~ "! where fi (0 Iz) is the posterior probability density for 0' given aset oC data z· (zl' z2.···. 1:z). r;. (i1lz) can be obt.ained from Bayes theorem (see equation (2.24)) which can be expressed as wh~re L(81z) r~:(or ~ is the likelihood of '0 given the observation Z. and is the prior density or 8. before ~b~inin8 '~!1~ d~b'. ~d .; ' .. . .. ' (3,64) ., ~~ " '. -' . Fodurthe~ ~nformati~n the reader is r~rerr~d to Aitchison,and DUn:>more {3.11. ; BIBLIOGRAPHY " ;': .," (3.1~ Aitchison. J. & p~nsmore, 1.. R.: Statistical prediction Anaiysj~. Cambridge University Press. 19i5. ., ." :'J : I·) •• • · [3,21 Baker, M •• 1.; Variability in the Strength of Structural Steels· A Study in Structura(Safcty. Part 1: Material Variability. ,CIRIA Technic~1 ~ote 44. September 1972. [3.31 Baker, i'l, J. & Wickham. A. H. S.: An Examination of the Within-Cast VariabilitY'~f " the Yield Strength of Hot·Rolled High·Yie{d Reinforcing Bars .. Imperial College. De· partment of Civil Engineering Report SRRG/3/80, September 198~ .• ' {3.41 Bannister. J. L.: Steel Reinf~rcemeltt and 'f~n'd;;~t "'~r'Siructural Cone;ete. Par,t f: Steel for Reinforced Concrete. Concrete. VoL 2. July 1968. . .-1 I {3.51 Benjamin. J. R. & Cornell. C. A.:Probability.' Statistics an~ Decis~~n for Ci~il Engi!,!,e~rs. 1-IcGraw.HilI. 1970: [3.61 Edlund. B'. & Leopoldson. U.:Scatter in Strengti! ~I Data of S~nt~tural, Steel. Publica,tion 5i2:4, Dept. 5truct'. Eng.! ~ha1mers Un,iversity of Teehnolo~.:, 13 .71 F~rry Bor~es. J, & C::l.St:mheta. M.: Structural Safety. Laboratorio Nacional de Engenharia 1;l.81 13.91 Civil. Lisbon, 19i1. 'J ~ ' . ' • Gumbel. E. J.: .Slatistics of Extremes. Columbia Uni versity PrtSs.lr.(ai. '; .Joint Committee on Structural Safety. CEB .. cfc~f. elB . :FIP ~~I.~BSE::- lASS . R[L~~: General Principles on P_",U .. iJllf(v {or Structurai Design. [nternational Associ~tion for Rr~o:I o;c anu Structural Engineering. Report:> oi t!le 'orkinSj: Commissions, Volume 35. 1981. !
- 77. 65 13.101 Ucblein. J.: Efficient .Iethods 0; EX/"t", .. V f • _ • " UP. ..I'·II/url I U Commerce, National Bureau of Stand:lrIJ~_ " oogy. .S. Dllpartment of . "":J'>rt • il:.>1 R j -1.602. Hli- )1ann. ~. R.t Schafer, R. E. & Smgpur'JIa!:" . : ; D _ ' ' Rcfiobifit)' and Lfft! Data. John Wiley r,t. :;:"rJ~, ;9;.1,,'''/IIOdS for Stahlllit:al Analysis of 13.111 [3.121 :fathieu, H.: Jlanuel.securite des SlrUt~I~,.. C "". C""ill~ ,·'u I Bull~Un d'information Nos 127 and 12k, J :1;1'1. • ro- nternalionai du Beton. )fayne. J. R.: The Estimation of Extr~m" WI , • ntJJ • • 1·"'rt1l1t £1 d Vol. 4. 1979. 0 n llsttial Aerodynamics. [3.131 ~fitehell, G. R. & Woodgate. R. Y.: , 8""" .. , f " stNction Industry. Research and InformlJlI ".J rf'''.r 'o• adSi'n Orfl' rc Buildings. Con- . on '"~f" · I ; lr.illn. Re ort ":- Sentler. L.: A Stochastic Model for I,iu.! f P ~.}. London. 1970. -'''!I/) IJIl ""'HJ~' B . . Technology, Lund Institute of Techn',I,,'I.'/ I' . In uddmgll. Div. or Building , . "'!Pl')rt riO, 19i5. Lund, Sweden. [3.141 13.151
- 78. .,, .•. . 1 ., 1.' '! ; . ,.; .... ., · ...' : .. ; :,,1 d.
- 79. 67 Chapter 4 FUNDAMENTALS OF STRUCTURAL RELIABILITY THEORY 4.1 INTRODUCTION Structurai reliability analysis differs in many imponant ways from reliability analysis as prac· tised In, for example. the electr6rucs ani aeto-s~ac~·jndustrie~. in spite of the fact that the underlying probabilistic nat~e of the problems are the same. A clear understandmg of these differences is helpful and the)' ",ill be discussed later. Both branches have developed from clas· sical reliability theory··a subject which evolved as a result of the increasing need for reliable electronic systems during the 1940's, initially 'or military applications. The principles of reliability analysis ha,·e been applied to a "ery large class of problems. f!lng· lng from the design of control systems for complex nuclear and chemical plant to the dt!sign of specific mechanical and: structural components; as well as 'more generally in the field of electronics and aero·space. Reliability analysis should nol, however, be thought of as an iso· lated ~iscipline as.i~ is closely related to the theory of statistics and 'probability and to such fields as operations research, systems engineering,"quality' control engineering and statistical acceptance testing. .;,;. , ... '" 4.2 ELEMENTS OF CLASSICAL RELaBILITY THEORY Classical '~eliability theory was developed to predict such quantities as the expeeted life, or the expected failUre rate, or the expected time bet. ..... een breakdowns of mechanical and electronic · systems, given so~e test or failUre data for the system and/or its components. These predictions · may then be used to answer such questions as: What is the probability that the actual life of a partK!uiar system will exceed the required or specified design life? Taking into account the can· sequences of system malfuilctioiu,"is it economic to in~rease the expected life by providing · more component redundancy? What is the optimal period oftime that should elapse before a system or component is inspected or replaced? There are many questions of this type and the nalurt v.; ~:,,~ ~:/~tem, tht! use to.which it will be put and the consequences of failure dictate the type of analy,s is that shoutd b~ .;:":.:.!'ri:aken, . ;,~' "
- 80. 68 4. FUNDA.IENTALS OF STRUCTURAL RELIABILITY THEORY Reliability {unction: Typically. howe'er, the probability of failure of a system or component is a runctio~ of,ol>enlting or exposure time: so thnt ,the reliability.may, ~e expressed in tenns of the distribution function FT of the 'aryable T. the rand,om timeit?Jailure: The reliabil!ty tunc.cion 4tr ~hich is t~'e probability th~t t~e sy~ will, ~t~!l_~ C?~.rational at tim~_t is gl,ve,n by : .;~ . . (4.H. If the density function (T of the time to failUre is Imo~n .~h~n ~T' ~~y')e e~pressed ~ llT(t) .. 1 - rt fT(,)dT .. r~ fT(T )d, . lo It · (4.~1 . In some circumstances there 'are good a priori'rew'ns Cor sel~ting a particular form r~r CT - . for ~xample. a' Weibuil distribution, whosc' distributiori lunction takes the f~rm ' . , ;'. ;)" .•.. ; . FT(t)'" 1 - exp[- (tiaJ ' .... . ' . ~ .,,; . . .. ' ~: ' . (4.31 Substituting in equation (4.1) glves ' . .; t" 0 (4.41 ' .'.', Expected.life; , ,A further property of a system or component which is 'orinterest is the ex·-· peeted life or the expected,time for:which the syStem or component ~an 00 expected to o!'er. ate satisfactorily, Referring to chapter 2. this is airen by- . EIT; .~-.. 7'fT(r)dr .. r 6tT(t)dt . 0 0 . (4.5) since, integrating by parts, roo 6t T (t)dt .. (tlRT(t)J" + roo ttr(t)~t .. E[TI Jo 0 Jo (4.6) provided lim [tlRT{t)] .. O. ,-- Failure rates and hazard functions: Also of Interest- is a knowledge or how the rate' of failure " changes with time for any particular fonn of reliability function. The probability of failure within any given time interval I t. t + oS tl is the probability that the actual life T lies' in the range t to t ... -} t and is given by ".,.,
- 81. 4.2 ELEMENTS OF CLASSICAL RELIABILITY THEORY 69 ::,; '.' -~._:~" ' J !.~.-;. , ~ The average rate 3t. which failure occurs in any time interval {t, t + 6 tl is defined as the failure rate'arid is th-epioba~)jHty per unIt. 'time'that fai'hire 'o~curs Wi{hiri- th~inte~al :I~"'en':t~at:it': : ,' , . , ., .-' " . '. ',' "~. / ;." •.. " ; ,~ - ' J; .' ~'~ r.l: hu"nol. olreadY:;o&uired poor to time t. minieiY" , "'. - .. '. -. '. , ':, : .. .,,: : ... ~ . _' r> .( •. 6lT{t) - 6lT{t -4- St) . at41T(t) " .. . ~ ~;- . !" ,~"!' ... . . ~; ! .. ' . ; .. ~ . Tpehazprci function is defined as the, instantaneous failure'l"l1te as'the hltei'Vanlt approaches! zero. It ls normally denoted h and is given by , ' " '.~,-; • ' ',:,:1..-- ('.9)· ; .J;'he.p,ro~a~i1ity that a system or component.,which lias already s/Jrviu~dfor a'period'6f time' t, : ~vill fail .i~:~~ next .~.aU interyal o( time dt is thus simply h(t)dt:-Note-thafthis'probability is no!JTC9dt ~ ; ' :.I,,: ~ ;;' c'.::':' : · :f . . I : . •. ) · ..• ~ .. ' , .(, The use ot the hiiiariHiincU.on' is~in: indicating whet.her a :~"ysie'm :or ·compo~eni.·~eC;;mes p~o. gressively more or less likely to fail per uriit'time'as time'progieSs~s:-If itbed'-m~sjro'gr~s~-ive; ly more like!y , ~o Cail ,.then clendy action.should'be·taken ·to replace·the component'or systeni' at ~.me stage. or: t.o minimise .the consequences, of failure • . " : ,::~ .". " . ": . , ._ . • . !Example,4.1. Assume that some deliQte instnlment is destroyed by homontal accelera~ tiors greate.r .~8r:t O.. ~5g, 8!:,d that.t.h~ tj"!~jl1terval between major earthqu.akes c;1n,be as· sumed to be e.'tponentlally distributed. What Is the Conn oUhe hazard function? ' j .. ' - " .. . ,,:.: . . .. ~, •. ,-: " ;.;. :. , • • '" , . . .. ~. The uponential density functi~n.i~ .!pven by ) I '" ~" and' ,vhe~ ; is a c~nstant. Hence ~T(t) -r r,.(T)dT - )~ ~e-;rdT a e~·.t and thus " ; (4.10) . (4.11) I., i,_.' i.; , h(t)--R~rft-)c> -, (4.12) ' , For thiS'particiJJarfor~;of di~trih~tion of ti~e' ~o Cailur~~'the h~ard C~:n~tion is a '~onstant; ·and t.hUll th'e probabilit.y oC 'rililure pet unit: tim:e- is indei>en(lent;'c' pre~. ing: even~ . . ' ','. -.. , . ._ ·iC. ' '. ,, " ,or: ,~ Choice of reliobifity function: In' many areas of eiectronic~ ~I~'t.rica'j'and' m~ch"miC;;U ~n~~~e;- . ing; data 'ueavailabie'(oi the tim-e.-to "£ailure· on(e·nii·5(.ci(;;S"r:l~iay~. beiuing~:'dri~~~h~(tS an'd ' many-othereomponentS! In f;i'c't'data'bari:ks lor this t.ype o;ini3iIri;ri~ri eic'istin'm~';:ihdu~ . limeS: Ttie'fttime t.o la.ilure,,;·may, hovever; be'i:ecorded in ~s'i)'r ~um~r:~i ~ci;r~r Dpe~~ t.iori~':numbero( mUes driven; etc" .. rother 'than in uni"u -riC lIct.J~iime: :"-:-: .• ;:. ', . ~~:'! . '11 ' ''; !.' " :'" ....... , ...
- 82. '.: ~ : .. ~ ;.' ~: 70 <I.. FUtOAMENTALS OF STRUCTURAL RELIABILITY THEORY i .. .. ;.'.., .",, : ~ ~" ... :." '~;c. ," :.1 ,,:.· ·;;:;: .: . .... ,. , . . ~ .. . 1l: .. . · · I. :<: ·.··f. : ':"_:-~ . . .. . With some .types of comppnent. large quantities of d~ta may .b~ available o.n ~i!'le to failure add ~"'dist;ibuti'~n :~~;, b~ seleJ,~ ~d which gi~es. t. 'h: -.~. : bes't fil. ' to the,.d'.a t' .a. C...ar..e. mu...s.t be taken •. ,' " .. however. to ensure that tne data are effectively homogeneous and do not include, for exam· pie, ~ystematic ch'ani;es with time. . .. '-. . ; : . ~. ; .. Frequently, however, the data may be limited in number, in which case it is 'necessary to select th~ .. type, ~~ qi~tribution .a priori,.,froman understanding of the'physicaJ nature of the:faililie mechanism and/or from previous experience. , , ' . . f, .: 1. '1.' Systems: In general, the aim of design.is to proviq~ an assembly of compOnents which when a~ting together will perform 'saJsfacto;Uy Cor some nominal desfgn life eit~ei- With or without maivtenance .. frequently,- time;to .failure data are available for the individu3l componenu;·but .are .not av.~~le ·for the cemplete sys~m. The reason, for this is that'complete systems a,n; 'more difficult and costly to test and secondly because it is often tht aim of the design tcfj:,rorlde:inuch hjg~er,~elia~,i~~it);'~(),r ~n~ c;oml?,~e,te. sys.tem t~a~ for the. indiyidu~ comp.o:nents. by providint a hJf~ ~l,~~~t!! l~f::~~~~n~n~,~~t!':'~:SYS~ r~duJldancy . "'" ''':1 ,)' ·,:i .. ·• · ".,: k '''''.'' .;'. , ,,''';0 '. III the~case.s.it isoecess.aty .to·compute;the reliability function·alld'hazard·:funct!Cirl for the system from a knowledse of the reliability functions of the components and a'knowledge·of how the components are inter-connected. In general. account should also be taken of the possi· bility oJ dete.~t.ing and replacing components which have failed .;It is;:howevert of the utmost imparlance to predict the existence o(an')' adverse interactlo'r's'~ety~err 'the ~c9'mponent5 which may exisi' jn' th'e'~'ystem butYlllleti"h'Bv{noi afi~led the 'fail~r~'ra~'~'i't~i individ'~~I 'compo-nents when tested in isolation or under different 'condidon;;:: .. ' These problems will not be discussed in detail since they are dilferent from the sy.Stems problems that are encountered in structural engineering. The latter are discussed in chapt~rs 7 and 8. 4.3 STRUCTURAL RELIABILITY ANALYSIS .J: 4.3.1 General ·1·, Attention will now be focused on the par~icular features of structural reliability analysis. but firs.t, . som..e. o.,f 't .h..e, "pr .in ',c.i'p "al didf"fe rences between ~)ecttonicimechanical .s)'Slem·~ and s~ctuntl _ ~ , :;: .. : ':. ' . ... . :. ;,;: ',," ,r, i . • .•• · ,,· ; " ,~ .. ! . s:.:~t~~~)~~~lr~!, e,x,~i~~~ : ~f~~t;:el~:~r~~al" ,'~!ectr~nj~ ~d, ~~c~~~!::a1; co~p~n~nts .~n~l ,sys· terns deteriorate 'during use as a rcsult of elevated operating temperatures, chemical changes. me~~.~~ic~l .. ,;;~.,c.a.:~~~~t~,~e.~?~~.!n~!:~~,d ,~~r a .nu,mper o~ ?ti!~.r rea;;~I);~:,lailur~ .. of a parti. , cu ~~ .C?_£?,~~.n~~f, m,~Y r~~':~ tu~X~0p.c~rl ~~. of)e; ~f, t~!e~~ :r:e~0l'!~,( ~~:,i.~ ~~Y",~Il,c~~se.~J':!~~t-: l~' ~ ~:f~~~~L~~~~h~~.d~t~1~~.~.~~~~:..:~C; 5~'!:Ie ?~.her par~ ~f~th~ s):s~.e'!l' It ~ .n-,:~~, tJ:l~rerore., to thi. n.~}~ ~r~~ .. ~~ t~.e}H~;H.r.e,l~.t::~:~1 ~~, ~~sh.anical5ys.~e.~5f a,~thpug~,L.he p.rec.ise time at,h:' which »failurell occurs may be: ~~r.f.ifultto cs~abJis~.s.l?c.e. ~he, defj~ition . o' failur~ may iJe fu;zy or ~omewhat arbitrary.
- 83. 01,3 STRUCTURAL RELIABILITY ANALYSIS ]n'c'ontrast to electronic/mechanical systems. structural syi~ems tend not to deteriorate, ex· cept by the mechani.5;nS'()f corrosion and fatigue, an'd in some cases may even get stronger- 71 for example: the incrEia.se in' the strength of concrete with timt, and the increase in the strength of soils as a result of consolidation. Whilst basic dat.a are a-ai.lable for the time to failure of electronic and mechanical components, no such information is avaiJable for structura,l componentS, because in general they do not fail in service. The probl~in is of a different nature. Structures or structural componenLS IIfailn when they encounter an extreme load, or when a combination of loads caus~s" an 'extreme lo~d ef(~ct of sufficient magnitud~ for the structure to attain a .failure state»; this may be an ultimate or a servicea~iJjty co~ition. Pan. of the problem, therefore, is to predict the magnitude of these extreme events, The other part is to predict the strength or toad..()e~e<:tion charactepstics of eacry ~tr')Jctural component from the info·~ation available at the ·de.si~ ~tage: I~ ·other "words, it is n~c.~~ary to synthesise probabi· listie models for the two parts ~~r the p;obl~~~ ·i'~~l~din~, on the one· h~d, all the unce;tain. ties affecting the loading, and, on the other, all the uncertainties aff~cting the streneth .or resistance of the structures. As discussed in chapter 3, statistical data may be available to quantify some of the sources of uncertainty and this information ca~ be included in a suitable form. Other sources of uncertainty may have to be assessed subjectively, A further difference between electronic/mechanical systems and suuctural systems is that whereas most structural.systems aIe aone-offl., all but the IllOst complex electronic and mechanical systems are produced in considerable numbers and can be assumed to be nomina.lly identical, Th~ . exi5tence of this underlying population of nominally identical systems or ·compo" Mnts ~eans that it is possible t.o interpret failure .probabillties in terms of relative frequencies, This is less justified for civil engineering syiOtems, and the probabilities· that are determined should not be thoue;ht of in this way. They should be thought of as intermediate results in a decisia:n·making process and with liule or no absolute meaning of their own • .As discussed in chapter I, the calculated reliability or failure probability for s.particuiar structure is not a unique property of that structure but II function of the reliability analyst's bck of know-ledge of the properties of the structure and the uncertain Illlture of the loading to which it, will be /i.luj=1.<;:d i.1 .~he future. 4,3.2 The Fundamental Case For a simple structural member selected at random from a population with II known distribu' tion function FR of ultimate strength R in some specified mode of failure, the probability of fail~r~ Pr under t.he action of a single known load effect sis ::.. I , ;. ' P, c P(R,..., 0;;. 0) = FR (,) = P(R/'" 1) (4.13) If the load effect S is also a random variable, with distribution function F S' equation (4,13.) is replaced by IU4)
- 84. i2 ... fL:SD.uIE~TALS OF STRl'CTCR,~L R~LlAIUL1T' THEORY ',.,. under the condition that Rand S are :>t3.t.i5tically imJe-pcndent. Equation I·L1-1) is best under· scood by plotting the density i~.mctio·n·s of R "aria "8,'us shown in figure '4:1 : It shoull.! be noted that Ram..! S must necess3.rily have tl":e s::ame dimensions (e.g. l~~ds a.nd load'carrying capaci· ties. or'bending moments and rle:o:ura! strength). Equation (.1.14) gi'e's' the'tothl probability ot (aHure Pr as the product or thtt probabilities of two'-independent events. suffin-ted alo'er:iI1 pOSs!ble'o~currences; namely'che' p'robability PI that S lies in the fange x.x + dx and the probability P2 that R Is less than or ~qual to x.. It is clear that ( 4.15) . " and (4.16) Under these conditions the reliabilit::-- ·i~ is the probability that the structure will survive when the load is applied, :nd is given by .' .+- ,11 - 1 ~ Pr : ,1 ~ ~ __ .~!lIXJ.rS~:":I~.X (4.17) ! : From considerntions o( symmetry, it COlO be s~n that. the re'liability may 3150 be expressed as (4.18) , ," . , :-:Luad eifec:t • S ".if. u .. i1i.iill~ mum>:nl-- - - ---- RUl$aIlW' R >I.g. 11IIXIH.1 (lIp.city .; . --.: -. . ~ !
- 85. ~.:J STRUCTURAL RELIABILITY ANALYSIS 73 : ... Fi~ure -'.2. Since in general it is not meD.ningiul to have negative stren~hs, the lower limits of integration in equations (4.17) and (4.18) mD.y In practice be repla:c~d by zero'.' It is also of interest to note thal although equations (4.17) and (4.18) give identical numerical values for Pr they are in fact fundamentally different. Setting (4.19) and (4.20) '. . . _ r : . . we obtain, respectively, the probability density function .o.f the resistance R' of structures in which failure occurs (llld the prObability density functi'an of the load ~(t~cts S' which have caused failure. These are not the same, since r * s at failure, see figUre 4.2- It should be not~~Uh~f Pro l.s nOLg:i1~!.l. py. .the._~~~. ~J. ~~erlap of the twO density tunctions f-R- (_r)._~;-._)- '_in f i~ 4.3::-~!!.is a common misconcepti~;' ~' - -. . .. __ ._--_. For the' general case, closed-Corm solutions do not 'exiSt "for the integrals in equations (4.17) and (4.18). There are. however, 3. numoer of ;pediu c~es: Example 4.2. 1C Rand S are independent normally distributed variables, Pr may be expressed as Pc =- P(M <: 0) (4.21) ~I· R-S (4..22) Thus (4.23)
- 86. ·- __ 4JW'DAMEl'!.TALS OF STRUCTURAL RELIABILITY THEORY M<O Failure , o 'JIM m Figure 4.3. Illustration O{t?,~ rel,iabUity i~dex Il. : '! and L' " (4,24) giving , oM .. (o~ + CJ§i'2 (4,25) Since Rand S are normal, M. a linea:" function of Rand S, is also normally distributed . _:_and (M - /-1M )/°14 is unit standard normal, giving - 0 ' ',;',,;: C';;,: _,~ -,;~" 1(. .' :",:,'::- " Pr-<fl( :P.!~)cq,LJJ~~JJ~,]' -'",- M ,_,:,-ca~ + (J~)2J (4.26)"· where <fl' is the standard normal distribution funct.ion and I1s.I1R'. as 'ana oR are'respecti'vely the means and standard deviations of random variables S and R. The r:e1iabilit)' index ~. may now be define,d as ine ratio JJM/oM or the number of standard deviations by which 11;'" exceed~ ~er?~,S~~ figure 4.3. Hence • (4,27) For the more general case where R and S are jointly normally distributed with correlation coefficient p. e9uation (4.27) still holds but oM is given by (4.28) Example 4.3. If Rand S are !:loth log-normt>,lIy d:stributed. Pr may be expressed ,as Pr - P(M' '" 1) (4,29) where 11' - RIS (4,30)
- 87. -I.3 .l PROBl.£).1S REDUCiNG TO THE FUNDAMENTAL CASE 75 Taking logar.itl:ams to .Lhe , ~ase e and putting M ." tnM.' gh:es M:. 2nR - i!nS· A-B (4.3) . .'" where A " .l!nR ~d B "l!nS. ,Failure.occ.ur! ""hel) M.' <1. or ,when hi <.O. ,But. it R and.5 are lo,normally distributed, A and 8 are normally distribut.ed, so that. tvl is also normal.lY distributed. Thus ' . .. - .... .. . , P .4,(IJB -IJA ) . ~( JJ~DS-/1tDR '. r (a' .l. a:)t (a' + 0' )t) B' A ~nS ~nR (4.32) The properties of the lognonnal distribution are such that if Y is 10gnonnaUy distributed and X:. tnY, t.hen 0'* .. 2n(V~ + 11 (4.33) m- y " lJy exp(-21' oi) (4.34) and IlX .. .l!n(ri.) (4.35) .. 'here my is the median of Y and Vy is the coefficient. ~f variation. of Y = ~'i Ipy. '. 5~b"i1ut;ng(in ,qUi~Onj(4;~2~~VO)S .) • n~ _R_._ IJR Vs + 1 P - 4' f ,/l!n((V~ + 1)(,~ + 1)) " ' . C4.36) Com'enient analytical expressions for PI do exist for some other combinations of distributions of R and S. However, this 5imple repreFentation of Lhe problem i& rarely of much praetic:~ use in structural reliability analysis, so we shall not consider the,m here. In general, equation (4.1?J or (4.18) net!dt to be ~'al:Jat~~ oy I,umerical met.hods ;-·-·- "~-- 4.3.3 Problemro ReduClnl to the Fundamp.ntal.Case In some simple situation,. although R an~ S may each be function~ of a number of oth~t random variables, it may be possible b~uneans of appropriate. transformations to reduce the problem, io the ~imple form. This i~ best illustrated by means of an example. Example 4.4. A slender cylindrical column of diameter D is to, be designed to carry a time· varyin~ axial load P, the maximum value of which In any 50 year period may be assumed to be lognor~ally distributed with a mean of 250'kN and a coerndtn: of l'llriation Vp ::: 0.25 . The load-carrying capacity of ~he column mar be assumed to be gO'erned by the re, latio nShip
- 88. ' .. , 76 ~ . FUNOA.t£~TALS·OF STRUCTURAL RELIABILITY THEORY . ',:,! ~ " -,. ' out owing to the nature of the end restraiols .. ,there is consid<!rable uncertainty in itS 'effective 'length '~. ihis uncf~nt); m:ly be repr~ented by, me:deUini L as a lognorm.u ,,ari. able 'vfth a' rri'ean 'o{'4 m and 3. coeWciem' of variation VL · 0.15. The other quantities C. the model uncertainty, and E, Young's modulus, may also be assumed to be lognormally distributed with the following parameters ". ."" .•.• ,: /J1;" 0.9 ' ;" , #J.E ,- 20~ ~N/mm.~ VE ,. 0.025 . . "~_ , ., .', . Assuming'that 'there is no 'uncertainty associated '~vith the diameter D:'~~d th.e required value of D such that the column has a reliability of 0.9999 for a 50 yea~ reference period, This problem may be solved as follows. If the applied load is p. failure' occurs wlte.n Cn- 1 ED~ ,: p' :::Pc --~·", !;' or 're-~'ngin'g',' ,'vhen' ",._~ , ,r- '_" 1 ", 0 - (64 Ll p)'i , ._," C:rlE '" ',.' ~ Taking logarithms t.o the_ base e give,s -.- " • -' .. <,-, ~' , .. ' .. (.,', " , ,, , ~ .,' " . . .'" " .: (~ .. ' ,') 1 . ~nID) - '4 {£nP + I!n 6-1 + 21!nL - enc - 3.enrr -l!nE) neplacin~ 0, the actual diameter. bye, the variable diameter at'which failure will just occur. ;md using the rules of expectation and variance for linear functions gives 1 . . ." . . .. .. .. . . . . ~1~nel-:t(E[enPI '~ E[2n6·lj' + '2EC2nLJ- Elinel..;.. 3E[Qmfl- E(QnEI) , and - ,,: " ', , .. " '. ' : ri,:; i:.: : 'f;',-"" ' " '" , ,~; : ' " Var[2n9 J . ·11S'(Varl2nPJ ·+k1 VarlfnLf+ :VarUnC," + Vai{2~I)I - ':1:': :: . . :. ~ ,' .. . .1_ .. .:"-, But from equation ~4:_3~):: _.. ....; : -; ': ~", .:,', V:uihPJ .. tn{Vp ,~~ ~J __ ~:~.~O.251' 4- 11 :~ 0.0606 . Similari>', V'ar (inCI:I O.q0995, Var(2nEj" 0 ,001)62 and Ver{tnLJ "" 0,02225, Substituting these values yields Varlene]" 0,010013 and thus "~ll ~ :::0 0,10006. From e~uations (-I.3-1) :lOd I·L35) , . ... . . :,, ', ElenP] = ~nJ.lp -t (1~ = ~n1250i - 0.5:< (0.0606)"" 5A9115
- 89. . 1 .t., · -1.3,4 TREAnlENT OF A SINOLE TI:IE·VARYISG LO ..... O " Simil:1fiy. E(2,.CI" - 0.1103 , £(RnEl- 5.3227 and ElfnLI" 8.2829 . . Sub~titu~io,,,,. !,ields, E[f.ndl. ~_,!:~92.?-: .~~~,E(~ne J "" ,utllA .. Iln(m(.,) r~o~ equation (4.35). , Hence the media~_.valu~ of.~! ~e . "'. ,e~p(.t39~S) .. . 80.S~ ~~. 'fhu~ th~, .vnlue of ~he dia-meter which will'resuft in (ailure 50%-orthe'tune is 80'.53' rilm .. 1. - ". " However,' tHe req~ir'ed ';;u~bilii:y 'i~<i:),9999i.' The: ~~~e 'oi'th~dii;;;et~r~rj"- ~vhich ~vili pro-vide this reliability is !ound as' (ollows: ' ., ; '. '.• :, ,.-' ; ' . ~ ' ' ::' .:.: . ., .. ',: jJ" '1>-1 {O.9999)""" 3.72 (from tables of the standard normal distribution fUnction), and since 2ne is normally distributed " rn De, .. ~n(,J + 3.'2:a~n~ .. 4.7646 ,- giving 0- .. 0 '" exp( 4.7646) = 117.3 mm. The required diameter for a reliability of 0.9999 in a 50 year period is thus 117.3 mm. I' ," 4.3.4 Treatment' or a'S ingle Time-Yary~g io~ .-'- . The:situation disCussed in seCtio~ 4.3.2:NaS that of :mu_:certain load eUe_ct S applied onCf! to a siru'~t~re" o(~ncer-iain re~is~ce R~: The' ~~P'I~, given' ;'b~~e :sh~ws h~w Ii and S'~ ~n be mo· "delled by' 3 nu'mber of com~'on-ent ~aridbies::rt';iso'~'n~'~s" h~~ a' ~~gle time.varyi·ni I;;ad ~ay be treated, The correct approach is to' uSe' the probability distn6utiori (o~ the load W corresponding to the muimum Intensity in the reCerence period T Cor which the reliability is being determined. In the preceding ~~ampfe,' this was the 50 yew e~treme maximum distribution. These distributions were dlscU5Sed In chapter 3. Provided the correct "distribution is used, the time element may then be neglected in the reliability calculations. '""Vben a reliab'nity problem 'in":91~es mo~e thad ~~e ~e~V~~g.lC?ad the problem is more com· plex (see chapter 10). 4.3.5 The General Case Only in the simplest of structures ciln the reUabilitY be ex:pr~ed i~ terms of the two random variables R and So' This is mainly beca~se R and S are not known !Ji o.re not convenient mathematical e:1pressions , e.g. 'R .. functio~(materi'~ properties, dimensions) ~ . ~~ " •. ,- ,.~ : . ..: .. ' ' --'; ' " --,." -' , .' ~" .. (! ' ,~,;,~Junction(appl_i.edJo.~! ~en~,ities, dimensions) '(4,37) "', ., : '. '~ - ,,'W: .;_: ,;, ' -"r.·, to ' Indeed Rand S.may not be s;.atist.iClllly_lndependent ,~ for example, cross·sectional dimensions ~lf~C~:tiOth ~e~~iona.1 .s.t~~~gf.~ a~d d~ad';;~~d~. I~'th~ case, the be~t ,solutkm to the problem is to express each limit state eqtLa.tio"n or failure function in terms of the set oC n basic Ilat:iables X which affect the Structur:ll performance. such th:1t
- 90. 78 4. FUNDAMENTALS OF STRUCTURAL RELIABILITY THEORY M·!(X,.X2 •...• X,)< 0 (4.38) • corresponds.to failure. M is the safety margin sometimes re~elTed to as the failure indicator .. !n general. the function r can take any form, provided that M ~ 0 corresponds to a failure 'state and M > 0 to a safe state. For simple problems, there is clearly no difficulty in finding a suitable form for f, but care should be taken in situations when some of the loads act in a resist~ng capacity (e.g. with loads res,isting. as opposed to causing. overturning). In such cases, a check should be made that (4.39) where Xi is any resisting variable known to be active in the limit state under considerat.ion. Setting ( 4.40) defines an (0. - ~) dimensional hyper-surface in the n-dimensiC?!.'Ial basic ,,:ariable space •. Th!.s surface is commonly referred to as the failure surface for the limit state under consideration and divides all possible combinations of the variables X which cause failure from all possible combinat.ions which do not. It should be noted that this is an entirely deterministic concept. The reliability of the structure may then be expressed as 4l-1-PI"1-~~ ... ) fXt.X2 •.... Xn(xl'x2 •... 'xn)dxldx.2 ... dxn (4.41) f(X) 0; 0 where !Xl>X2, ...• Xn (Xl' XI(' .•.• xn) is the joint probability density function for the n variables Xl' Note that the inte}ral is over the failure region. denoted we (see chapter 5.2). If, and only if. the basic variables X are statistically independent. equation (4.41) may be replaced by II· 1 -:- PI - 1 -)) ... ~ CXt (X 1)fX2 (X 2) ... Cx " (xn)dX1 dX2 . . - dxn ( 4.42) reX) < 0 Two practical problems are immediately apparent. First. there Is almost never sufficient data to define the joint probability density function for the n basic variables. Usually, there is hardl~ ~nough information to be confident about the marginal distributions and the covariances. Secondly, even if the joint density (unction is known, or in the case of equation (4.42) the marginal densities. the multi-dimensional integration requited may be extrem'e..l y time-<:on- sumin,. Analytic.al solutions do not exist for the majority of practical problems and the ana-lyst must resort to numerical methods.
- 91. 4.3.6 MONTE-CARLO METHODS 79 These difficulties can be overcome in practice by using t.he Jevel2 methods described in chapters 5 and 6. The only other possibility is Mont.e-Carlo simulat.ion, bat this too has its limita· tions. However, for the sake of completeness, the technique will be briefly discribed. 4.3.6 Monte-Carlo Methods Let us assume for siq.plicit.y that the basic variables Xi in equation (4.38) are stat.istically in· dependent with known distribution functions. The Monte·Carlo approach is to use appropriate random number generators (see appendix A) to generate independent sample values xi for each of the basic variables and to determine the corresponding value of the safety margin M from (4.43) By repeating this process many times it is possible to simulate the probability distribution for M by progressively building up a larger sample. This sample, although generated numerically, may be treated in the same way as any other statistical sample. In general, the exact probabili· ty distribution for M will not be of any standard form, although it may be governed by the form a./. the prob~bility distribution of ~Il,e fI!ost dominant. basic variable. ';~, t • . ~. The failure probability may be estimated in at least two ways. Since M.;;; a corresponds to failure, Pt may be expressed as P, ·P(M" 0)' lim kin (4.44) n-- where n is the total number of trials. and k is the number of trials in which f(x l • x2 ' ..• , xn) .. o. However, the ratio kin is a statistical variable whose sampling distribution, and in particular variance, depends on the number of trials n. For low failure probabilities andlor small n, the estimate of p{ given by kin may be subject to considerable uncertainty. Practical rules can be established giving the necessary number of trials for any given magnitude of Pl' The second approach is to fit an appropriate probability distribution fM" to the trial values of M', using its sample moments. Some suitable distributions are given by Elderton and Johnson (4.6J. See also [4.2J. This approach is the only one possible when the number of trials is small. Then (4.45) As a general rule, however, Monte-Carlo methods should be avoided if at all possible.
- 92. 80 ~. fL':.!D,MENTALS OF STRUCTURoL RELIABILITY THEORY BIBLIOGRAPHY ":. f4::1) ' Ang, A.'H . ..s:-'and Amin. M.: $4;c,y Ftw.tors and.Probability in Structural J)ajgn. Journal of the sm;;rG"raJ rii~ision. ".-SCE. July '1969, V~I •. 95. ~ , ' ".:", .: . ; ... , ' [4.2)' 'Baker. M. J.: The Reliabilir~' of Reinforced Concrete Floor Slabs in Office Buildings - " ~ A ProbabUistic Study. em,... Report 57, March 1976. {4..31 ~~1o~~, ·~. 'E. ~d' Pr~sch;~, F.: :lIllthematical Theory"otRefiability. . <1ohn Wiley and Sons, 1965. ' ..... . ; . ~ ... : {4,""I ' Bolotin. V. V.: Statistical .kthod$ in Structural Mechanics. Holden·Day, Ll~69. (4.5( (4.61 ( 4.71. (4.91 Cornell, C, A.: Bavesian Stati:lticai Der;ision. Theory and Reliability.Based Design. Int. Conf. Struc. Safety and 'Reliability of E~~. Struct .. washi~Kt~~; 'h:C:::9',11' Aprli, 1969. Elderton. W. P. :and Johrn;on. N. ·L.: Systems of Fre-queney-Curues. Cambridge Univenity Press, 1969. !ferry Borges, J. and Castanhet:a •• I.jl.: Structural saf~t~. National ~.aboratory oC Civil En· gineering, Lisbon, Portugal. 2nd Ed. 1971. Freudenthal, A. ),1.: SafelY, Reilability.and Structural Design. Transactions, ASCE; 1962. Vol. 127 (Part II). Freudemhnl. A. ~1. and Gumbel. E. J.: Failure'and Surviual in Fatigue. Journal of Applied Physics. Vol. 25, ~o. 11, 1954. f 4.101 FreUdenthal. A. ),'1.. Garrctts. J. M. and Shinozuka. ~I.: The .4.nalysis of Structural Safety. Journal ot the Structural Division. ASeE. Vol. 92. :-lo. ST1. Feb. 1966. r 4.111 Haugen, E. 8.: Probabilistic Approaches to Design. John Wiley and Sons, 1968. [4.121 Kapur, K. C. llnd Lamberson, L. R.: Reliability Engineering Design. John Wiley and Sons, 1917. [4.131 Pugsley. A. G.: The Safety of Structures. Edward Arnold, 1966. ,
- 93. 81 Chapter 5 LEVEL 2 METHODS 5.1 INTRODUCfION . In chllpters 1 and ol an introduction to the fundamental concepts of structural reliability is givell and the so-c3Jled le,'el '2 methods are brielly' mentioned. In this chapter, levei 2 methods will be treated in a' more detailed way and a" nUmber of simple applications ~Ill be 'shown. In chaput 6 (unher"sppiications of level 2 methods with 'sPecial emph~is' c'n non-nomany di;. tributed basit variables and correlated baSic .. "'3riaOles ate treated. ," . ", . As mentioned in secti~m 1.3.2. methods: oC re!i.abil~ty a~alysis are classified o~ the basis Qf :~e types at calculations performed and oi the a~prosimations made. The most :tdvanced methods are the lellef 3 methods. They can be characterized as being probabilistic ~ethod5 of an31y~:'" based on knowledge of the (joint) distribution of all bas,ic variables. In level 2 methods a m.::nber of idealizations compared, ~ith the 1Ot!:<'~!.>o level 3 methods . ~t;: m.~~e ~ I~ pacti_~ula.r it is:;.s· sumed that the failure surface described in 5.2 below can be sensibly 3ppro:timated by " .ta!lgent hyperplane at the point on the· failure surface closest to the origin·, ~hen the surf~ce has be,;n mapped into a standard normal space. A level 2 method is therefore a method of design or wa· lysis which in its simplest form comprises a check at only a single ·point on the failure surface. as op~osed to level 3 methods w~er~ tbe.probability content of the entire failure region is .evaluated. Level 2 methods provide a powerful set of tools for tackling a wide range of practical probli!ms. The relatively simple structural examples gi'en in this chapter and in chapter 6 are for the purpose of explaining the methods. :Vlore complex and practical applications are discussed in crrap. lers 11 and 12. 5.2 BASIC VARIABLES AND FAILURE SCRFACES One of the first problems one has to solve before the· reliability of a given structure t':lO be evaluated is to decide upon which variables (quan:ities or parameters) are of relevance. These variables called basic IJariaoies· can be geometrical quanritie~ (e.g. the area of the cross-section of a heam). material strength (e .g. rupture or yield strenr:.hl and ex.ternalloads (e.g. tr3ffic loads. wave.:lt wind loads). For a given srrucure each variable Xi- i " 1.. ' .• n is considered 3. realization of a random 'Miable Xl ' i '" 1. .... nand therefop.!. the set of variables ¥,. ix1 . . ... xn) is a re.:..iization of the random vector X '" IX 1_ .... Xn I . 1n other words. the variable x is a point in an !-di· mensionai basic uariable iipace.
- 94. 82 5. LEVEL 2 METHODS j In a leve13 method knowledge of the joint probability density {unction fx is required. but in the level 2 metho,d presented in this chapter only the expectations i '" 1, ... , n (5.1J and the covariances i,j -1. 2 •... , n (5.2) J is equal to Cov[ XI' XI]. . I ,- , are used. Note that the variance Vatl XI J - Q j In t~e {ollo~'ing ltjs assumed that a set of basic v~a~le,~.X 'r. (Xl"'," Xn) is c;:hosep in, such a ~'ay ,that a failure surfa,ce .. (or I!':'1i{ state sp.r(ace} ,~n .be .definecJ in . th~, n1i~ensi(:mal bas~c variable;$ pa~ w ~ ,A fail':lr~ ~urfa~ .. i.s a. su~[ace _di~ding the bll$ic . vlI!i~a~le spac~ _i.~.~. two regions namely a failure region w(.and a $Ore .regio,! w,' The failure region contains all reali:z.ations of X that would result in failure, and the safe region contains all realizations of X that would not result in failure. It is convenient to' descri~ the failure surface by an equation of the form , ' : ,; ~, (5.S) in such a w~y that positive values o'r i indicate safe sets of basic variables (the sire re!!ion) and non:po'sitive values of r indi~te uniare sets of 'ariables {the failure' re~ion}, i.e. when x E w, when~. S WI A 2<iimensiomi.l case is illustrated in figUre 5.1:The function f: wl'""'t R is called the failure function. It is important to note th3t the same failure surface can be described by a number or equivalent failure functions. ---:----:1----'6~----", Firure 5.1.
- 95. ". . ..... . . ~,' c .~ ;·~ ."· .... ; . ::.· c·-;'. '~c~ --. .. -,;., : -' ~ "!'.:'.- 5.3 UNtAR FAILURE FUNCTIONS AND NORMAL 8ASI~ ~~jUABLES" .• ~ :. ' '-'. ) '.~. 83 ., ... Let f ~ a failure function . The random variable M '" r(X) is then called a safet)' margin. ,"];' .. . ' Eiampl~ ·5.1 : COnuder the fundamental case with only t~o basic,"ariables (a l~d variable S and a strength paruneter R) and a failure function, (1 :,~~tjR" ~here .. '. . " (5.5) .. . -- The failure 'surface. the'failure region and the sale region are shown in figure 5.2. The col1'esponding.safet,.· margin M I Is g; ~e~ by ' -" . . .,. t lof1 ·R·..;... ·S (5.6) An.eq~~al~nt fah~re functi~n is 'f2': Rl~R~ where .'. - .. ; ! ,' •. .• , : .. :~. : , . , • . ' ' :! . ,... . . f 2{r •. sr= 2n1-," 2nr~,.i.n5 ~ .,; .. . (5.7) ~~th the. safety margin 11.12 ~' in ~ • ~n R - 2n~ -. ' (5.8) failure n,ion ... re rc~lon FifU'~ 5 .2 5.3 RELIABILITY INDEX FOR LINEAR FAILURE FUNCTIONS AND NORMAL BASIC VARlABLES For the case of a line'ar safety margin M and normal basic variables, the reliabilit.y index IJ is deftned by (5.9) where $1M is the mean of M and aM is the standard deviation of M. This definition o[ the relia· billty index 13 was used by Cornell 15 ,11 as early as 1969.
- 96. 84 , 3. LE"£L 2!>IETHODS Example 5.2.. Consider d'.,,: fundamental case treated in example 5.1 and assume that Rand S 3le uncorrelated. With ~.f ::: R -:- S one gets and according to equation (2.lSQJ. Therefore .uR -.us ,. (5 .10) (5.11) (5 .12) Let the safety margin M be lin~" in the basic variables Xl' ...• Xn (5.13) It is then easy to calculate the tdia.bility index 11. :.u (5.14) and -.--" :. p',.a.a·O'la. , II I J . ) ,. , - ..... :,. ,';. where the last tenn accounts fut correlation between any. pair of basi<; variables. Pij is equal to the correlation coefficientpxL}(J defined in equation (2.80), . Example 5.3.,.CoI:1Sider thLl staticaqy i,ndeterminate.beam shown in figure. 5.31oade~ by a concentrated (orce p and assume that the beam fails when Iml ;;. my. w~e-:e mf' Is a critical liInit moment and III is the maximum moment in the beam. Funber assume that P. i and mF ate tealizatioilll of uncorrelated random variables P, L. and MF with lolL = 5 m ~~lr :::I 20 kNm , ... Flpre 0.3
- 97. · 5.3 LI~EAR F • .;1LURE FU~CTIONS A."I;:O NOR:I,AL BASIC VARIABLES 85 The ma ..... imum moment is Iml,,,:, pl!/2 and.the,tefore. the fC?l!owing failure fur."tion em be used ' f(p, 11, mF ):: mF -t pI!:: 0 , < 'I (5.16) Note that.! in this case can be considered a deterministiC parameter because I1t. - 0 m. Therefore (S.16) c.an be,~ewritten .. {(p, mF ) '" mF -i p ~ 0 The correspof,lding safety m,argin is (S.17) and 5 J.I:o.t ;; 20 -'2. 4:: 10 kJ.'fm '5 a~t .. 4 + -4 • 1 '" 10.25 (kNmJ! (5.18) Therefore 10 iJ = ";10.25 ;; 3.12 15.191 Note that in the presentation above, a safety margin linear in the basic variables =.a:sbeen assumed. If the safety margin M is non-linear in X:: (Xl' ... ,Xn) then appro:d~a:e values for J.I:o.t an'd aM can be obtained by using a linearized safety margin M. Let By expanding this relationship in a Taylor series about (Xl' •.. , Xn) ," (f,ll •.... '':'n) and retainmgonly tbe linear 'terms one gets'",'-',' ' .. ". "- where af/aXi isevaluafed aC(J.i l ;: .. //-I~). From (5.21) approximate values for :';~,1 and u~l are determined by ,- ,._- ,
- 98. 86 '5, LEVEL 2 METHODS I Exercise 5!1. Prn'(> equations (5.22) and (5.23) and derive the speCial fonn of '(5.23) wh~:: the basic variables are uncorrelatcd. . Clearly. for non-linear failure functions, calculation of the reliability index p c 11M /OM on the basis of a iinearization as (5.211 will depend on-the choice of linearization point. In (5.21) the so-called mean point (tll •...• Pn ) is chosen, but as shown later. a point on the failure surface would be more reasonable. Experience shows that an expansion based on t~e mean point should noi be used. The reliability index i1 as defined by equation (5.9) will change when different but'equlvalent non-linea: failure functions ate used. This can ~asily be demonstrated by considering the fundamental case't.reat.ed in example 5.2. In the fundamental case only two unconelated basic variables R and,S are involved. As shown in example 5.2 the safety margin M '" R - S results in the rt!liability index 8 • ~R -Jls taA + aJ)"! By using toe equivalent safety margin (5.8) M .. 2n(RIS) ~ I1n R. -l1n5 one gets 6'=~ t",nIRIS) (5.24) (5.25) Exercise 5.2. Show that the following approximate 'alue. of p' defined by (5.25) . ; e: QnIJR - 2nIJs J(?li J: + (~)2 IJR Jls is obained by linearization of the safety margin M :: in(RIS) about (IJR' $iS)' T~e va!ues 5 and ()' from {5.24J and (5.26) ·are··different. The ~liabiJity index, defi~ed .~~. equation {5.9i is thus not invariant with regard to the choice of failure function. It is of course un· fortunaLe that a reliability measure can give different valu.es for the same problem. In the next section the reliability index is redefined, so that this problem' ~ solved in a simple way. In general. it is not possible to relate the reliability. ~de% fj defined by (5.9) to the probabilit)o· of failure Pf given by Pr c fX-(x)dx (5.27) , - ' f ~nere f X i~ the joint probability densit~· function and wr is the failure-ie¢~n defined· earlier in this section. However, when the safety margin M is linear in the basic '2riables Xi" i .. 1, ... ... _ n. and these basic 'ariables are normally distributed S{J.!i' 0jl, then the foUo;ng relation· ship exists.
- 99. '~ , " 5:3 LlI'OEAR FAILURE FUNCTIONS AND NORMAL BASIC" ARIADLES Si (5.281 where tlI is the standardized normal distribution function . Here. (5.28) will only be shown in the 2-dimensional fundamental case with 2 independent basic variables Rand S. and with the safet.y margin M ;: R - S, where R is normaUy dist.ributed N(UR' oR land S is nonnally distri· buted N(ps. os). Then. M is normally distributed N{JlR -us' .JaR + a~ l. and the probability of failure is : " ,' o-eu -p I P, =P(R-S < 0) = P(M < 0) «>( • ." .. ) = "(-pj "oR + 0S (5.29) according to (5.24) (see also Section 4,3,2). E:rample 5.4. Consider again the beam in example 5,3. where the reliability index 13 - a.12. If the basic variables My and P are normally distributed the probability of failure c:an be calculated by (5.29) PI - <1>(-6) - (~(-3.12) - 0.0009 Consider again the 2-dimen,sional fundamental case with independent basic variables R and S and let the means be PR and Ps and the standard deviations OR and os' FurUier. let the s3.re~y margin be MeR - S. The reliability ind~x, 6, can then be given a simple geomet~cal interpre· tation in a normalized coordinate system 8.5 shown in figure 5.4.-where the coordinates r' and $' Ilre realizations oC random variables R' .. (R - #R lIaR ,nd S' :II (8 - J.ls l/as' With these 1 'variables, the failure surface is given by ~ . '~:, (5.30) Therefore. the shortest distance from the origin to this'line:o,r (3!lure 'surface i~ eq'ualto 'theTe. liability indexfJ 5see' niure 5.4 s.'nd (5.24». ,:, . . , ":' . , . . " ' :l " u,.
- 100. 90 5. LEVEL:1: METHODS "p 4 k! 'p -1 kN " "L . 5m 'L a Om "E - 2'lC'k~/m: 'E O.5· .~07kNJm: "I "'10 ..... m· " 0.2'10"'& m4 By inserting 2'" 5, minto (5.37, the failure function can be y,Titten ei - 78.12 . P" 0 (5,3S) The basic vanablt.i t. E and P are then normalized ZI ;. (I -/lIlioI• Z2 "" (E - PEl/aE and Z3'- (P - /lp)J0p' In ~he normalized eoordinate system the failure"surhce is given by '.:' 0' 0 .2 2.} + 0.2522 ... 0.05 %.122 - 0.0391%.3 + 0.8138 " 0 The reliability index' and the design point are then determined by the foUowing equations ; _~~~~~-~0~,B~4~3~8 ____ ~~~_ 0.20.1 ... l:.25 o.:;. - O:05~Q. l 0.2 O.039la3 0.1 =-t(o.~- O.05liQ 2 ) . " I (0'')- ..... 00- II' , Q 2 -- -k' . _<1 • ~"o.l 0.3 =fo.C391 1. J , (5.39) where the same no:alion as in (5.35) is u~ed ~d where k is determined by putting 3 I al c a~ - ai -;. 0;'1 :: 1 i " l The fmt equation i."l (5.39) is written in a form suitable for iteration. The iteration Is now performed by choosing startinp: values for.81 o.l"IX:2' and O:s and calculating new values by (5.39.1 until only s:::::ali modifications are obtained. This is shoy;n i.n table 5.1. I Star. I Iteration No. ! 1 I 2 I 3 4 • I P 1-~55 II 3.~2 i 3.51 II 3.30 3.29 3.36 0, 1-0.;)3 ! -0.33 -0.23 -0.19 -O.lB 0, 1- 0.55 i -0.B2 I -0.91 i -0.9.5. -0.97 -0.97 0, , , ! i , 1 ..I. 0.5~ 0.22 0.23 0.19 0.18 0.17
- 101. .~' s.~ HASOFER AND Ul'O'S REUABILtTY I~UEX 91 " .' The reliability index is 5 • 3.29 and the ciesi!;l1 point A • 3.29(- 0.18,. - 0.9;. 0:17) '" (- 0.59. - 3.19. O.56).lote that tne correct ·siPl f~r Qi:'i - 1. 2. 3, was c:~ose!'1 f~r the 5brtin~ values in the table. This car. usually be done in the rolloVo·ing way . A positive sign can be recommended. when the conespondinj! basic variable is a .10adini variableI' (e .~. PI and a nega~iv~ sign 'A'hen it is a •. mr.ngth or geo~etrical variabJeM [e.g. E,and I) .. Exercise 5.3. Consider the elastic beam shown on figure 5.6·with a uniform· load p. lenltth 2 and critical limit moment mF' A5S!lme lhatp. (and mF . ar~ realizatio.ns oC uncorrelat.ed random variables P, L and MF with Pp - 2 Mp/m a" • 0.4 Mp/m The maximum bending moment is mmax. "' 1~ pel : Calculate the reliability index 11 for the following. failure ':'lode mmax ~ .mF·, 1. ~ ~ .,11 ,r Firurl! 5.6 In, eX~l!'pl.es.~ ;3 .and 5.~ and exercise 5.3. structures with give:,: loading, strength and djrnension~ have been analysed. Tne reliability index.is.calculated by an iteration method suitable for hand calcuiatio~s ·and c~mpu~er calculations .. 1t '~'pi. be s~~'w:n In example 5.6 that the same technique.!' can be used i~ ~fsigninl a s.tructure so .that ,it achieves .~ 8'iven reliability index (1. Example 5.6. Consider the elastic beam shown on fi~ure 5.i loaded wiLh a single load p . The maximum deflection is umu .. 1~2 ~. where C is the length of the beam. Assume that p. C. e and i are realizations of uncorrelat.ed random variables P. L. E and I. witi': tp ~. ___ .....I-! __ --:f:~ "J ~ -----~'!~'----"---~.~'---, Fij(lrt· 5 .•
- 102. 58 .. :'" , 5~ LEVEL 2.IETHODS 5.4. HASOFER A~D LIND'S R£LL-BILITY INDEX A serious objeetion to the reliability index'; as del1ned by equation 15.9) is its lack oC failure 'ruri~ti~~; invariance as 'aiscu~ed'in'ke~ti~~ 5':i ;iii tll"is se~tion .d~e ['eliabih~y' ~d~x p proposed 'by ~~~·f~r;.o.~·d.~i~1.in· 197" w,i11 ,i;e.introd~ced, an~ it ~~m.be shown that it is:1nvariant with re~pect t.o the choice of failure function. As bc!oce";the basic ';;ariablesare"cailed X z; (Xl' X:z •.. "., Xn) and the failure function f :wr"R, where 1.,1 is the n-dimensional basic variable space. The space is divided into a failure region we - {i:.f(i)..; O}~:and·:l. sale r.eeton .1.,1, ~. {i.: C(i) > O} by the Cailuresurface ijCJ '>3 (i:f{i) -.O} . . In this section it is"assumed for" the sake be simplicity 'th~t th~'b~i~' variabI1~ ~e uncorre14ted, i.e. Cov[Xp Xjl::. 0 for aU i and j. It will be sho~n in "the n~xt chap~er how to deal with safety problems when this assumption is rela..,,;ed." The first step in defining Hasofer and Lind',; reliahility index is to normalize the set of basic vari· abies. This new set Z· (Zl ' __ " Zn) is de·fi~ed. by " " "";,,' II (~": " i -1, 2, ...• n ': ., " (5.31) . ." "~" where J.I. XI and aXI are the mean and the standard deviatio~: oi" t"he r.md~m' variable Xl' Note that ;,IZI .. 0 and azl :, 1 , i = 1,2. _ . '. n (5.32) By the linear ma~ping defined by (5.31) the failure su~face"in the"~~o~rdina~e system is mapped. in· to a iailure surface in the z·coordinate system. The Cailure surface in the z-coordinate system divides the z'3pace into 3. failure region and a safe region in the same way as in the x,sPlce. Due to equation (5.32) the new z-coordinate system hG.$ an importunt characteristic. namely a rotational symmetry with le3peet to the standard deviations. Note that the origin 0 of the normalized z-cootdinate system will usually be within the safe region. A two-dimensional example is shown in figure 5.5. ·"HQ.sO;er~end Cind's reifrJbilit'): Inati t1 ;s' de!i;ted ~s 'the sHortest di$'ance" fr~m th-;"~"'ig;"ri "to the failure sur;ace in the normalized z.coord{na"i'€ s)'st'em: For"the two;dlrrieriSional caSe in'fig~re 5.5 Jl is e:,ual to the distance OA. The point A is calied the i:1e~igit point, By Usiii.~ ~ de!iniiio~" cir the reliability Index ;3. where 0 is related. to~ the" failure "suriace" and" not to the faii~~'function" a sa"Cety measure is obrajned which is failure function invariant because all equivalent failure functions result in the same' l'ailure"surface: €:Iilll!C ,.t!~lun Figure 5"5 ,. "1 ,7.2 I , I ! ! i ! j lilIiIC !illrll!!!! ..;
- 103. ','~, ~A HASOFER .... ~O tINO'S RELJ.-BIUTY I~OEX 89 The reliability index iJ as defined by equation (5.9) will coincide with the :elio.bility index) de. (ined above, when the failure surface is linear (a hyperplane), This -is ~hown (or the two-riimen. sional case at the end of last section. but can easily be proven in the gene:-J.1 n·dimensional COlle. Therefore. in this esse the important r~elatiori -(5.28) between {J and thl! probability of fuilure Pc can also be Cormulated. provided the basic varitibles Xi' i-I. 2: .... n are normally distributed (5,33) In the previous section it was shown how an approximate calculation of the reliahility index i1 defined by eq. (5.9) can be obtained when the safety margin ~ is non·lineu by expanding ~1 in a Taylor Series about the point ,XI' ': . " Xn) .. (~l ' _ .. , '~n J. It is no~ obious.crom lh~ above reIl)o~rks:that the twp"definitions of ~ will coincide if this expansion is mace 3bout t!le de~ign point. Th!s corresponds to approximating the non-linear failure surface b~' its tangent plane at the design point (A in figure 5.5). · . The definition of the reliability index IJ by Hasofer and Lind can be formt:.::lted in the following way 15.3-1) where aw is the (allure surface in' the 'z<:oordinate·s~~tem . The calculatio~ ot ~ can be unaer~ ken in a number of different ways (see also page 227). In the general c~~ where the failure , 1urf3ce is non·linear an iterative method must be u;ed. Here an iteration :ll!thod. with fast rate of convergence is given. The procedure will be illustrated by a number oi ii.rople examples where the failure lunction is assumed to be differentiable. In this case the diStance jj and the. unit vector Q '" (0.1' ..•• an) given by OA .. ~Q. ~here A is the design paine. can b~ determined by solving the fonowing n "+' 1 equations -iteratively . . ...., _. (f·,ll. (pa)2 fk k-l .. Qzc ',' . ' 1()1a 1• /10 2, • . •• po .... ) .- 0 where the failure sucface is give'n~b; , ,1'" 1. 2 •. , •• n (5.35) (5.36) Example 5.5. Consider th~ $;Im t' tie-am as in ",:xampl·c'"5.3 but now w: th the following de llec. tion _f~!~r!_critQrion ;JI ,:' . ~ i~ J.. . . umax -18 ti ;;:. 30 ~ ,.,),3. I where umax is the ma.ximum deflection. e the mOllulu ~ 01 i']a5,icity .:...,d i the reie".-ant mo· ment of inerti3. Further. let urn"'" ~ p. ~. e and i .~!.e reaiizations 01 un':·Jtrela.ted r:l.!:aom ·3ri· abies Urnu ' P. L. E.:lI1Q J with
- 104. 92 .'i. LEVEL Z ~IETHODS "p "" 4 kN 'p' 1 kN :JL ,. 6m 'L' Om "E ' 2-101 kNfm: 'E" 0.5-l0f kN,.il! The mean value i-lI for the moment of inertia is unkno .... -n. and the problem here is to determine i-ll so that;J '" 3.0, when the failure mode is uml<X ;;a. l~O - ~ and when o[ .. O_l-IlI' The fuil~re function is defined by 1 1 p~l 10oi-m-;i "' 0 0, 6ei -113 p z 0 ' .. ;' The basic variables P, E wd I ~e normalized. ZI"- (p.- ;.Ii-)IG,,, Z2 • (E-'IlE)/';~ Dnd Z3 ... (1 - iJ.I )/0[. In the normalize.~ . coordin~te Syst4!~ .t,he ~s.ilure functioa is given by The de~ign point is now yven' by ::>,,~;· ... ·"30;":The unk"n"~,W'n ~[' 0: 1, 1:12 nnd"3 ate thereiore d~"termi"nedJ)y the following system "of equations: _ 113(4 + 3QJ Pt .6.to:~2+ , ~.5a:2)(l+O;3Q3J" '" 1 " .. a 1 :cr ll3 " Q2 "=--~' 107 (3+ O.9(3)~1 a 3 '" -t· 10' (1.2 + 0.9 a~Jlll (5.40) ": ; J ..... here k is determined by the condition 0:; '+ 0:5 + a1 - 1. This "system can ~ ~i~ iteratively in the same ..... ay as the system 15.39) in example 5.5 was 501ved. , Iteration ~o ~ Start ! I 1 I 2 I 3 : 4 I "I 1 '10" I 116.10-7 I 158.10-7 16~'lO-J ! 167'10-7 , .. ., 40 , Ql "" 0.58 a:! ,- , I.' I 0.35 , 0.23 0.23 . 0.23 ' ,I -0.58 -0.90' -0:96 -,0.97 . -0.97 . 1 '." . , i . Q:J ~ " -0.58 -0.25 I ~O.14 -O.ll" ,,;1 ~O.ll I Tilble 5. :! The me:J.n" value for the moment of inertia corresponding: ,0 ~ '" 3 iS,:It .. 167.10-1 m~ .
- 105. B[BLIOCRAPHY 93 Exercise 5.4. Consider the same elastic beam as in exercise 5.3 . but with ~"I .. ~sumed unknown. Determine 1J.)ly so that 11 - 2.9 with the same failure f:.!nction and with O'''I~. "" O.OSPMF· BIBLIOGRAPHY [5.11 Cornell. C. A.:A Probability-Based Structural Code. ACI..Joum .• Vol. 66.1969, pp. 974-98~ . (5.2) Ditlevsen, 0.: Structural Reliability and the inuoriDnce Problem. Re5earch Report Mo. 22. Solid ~Iechanics Division, University of Waterloo, Canada. 1973. 15.31 Ditlevsen. 0.: Fundamentau of Second Moment Structurel ReliDbility Theory. Int. Re. search Seminar on Safety of Structures. Trondheim, Nom'ay, 1977. [5..11 Dyrhye. C. et a1.: Konstruktioner$ sikherhed (in Danish). Den Private Ingeniurfond ved Danmazks tekniske Hojskole, Kobenhavn. 1979. (5.51 Gravesen. S.: Level II Safety Alethods. Lectures on Struc:ural Reliability (ed. P. ThoftChristensen), Aalborg University Centre. AaJborg, Dennmk. 1980. pp. 29-38. [5.61 Hasofer. A. :'01. and N. C. Lind: An Exact and invariant F:rt;t Order Reliability Format. Proc. ASCE. J. Eng. Mech. Div., 197-1, pp.111-121. (5.71 Thoft-Christensen, P.: Some Experience from Applicatio.'1 of Optimization r,~r:hnique in Structural Reliability. Bygningsstatlske :'oleddelelser. Vol. 48.1977, pp. 31-H. .,,
- 106. , · 1. c. ! , 1, ., , . --~r_. ,:~,; ~x::. ~,-;.:---::
- 107. 95 I' Chapter 6 EXTENDED LEVEL 2 METHOD.S 6.1 INTRODUCTION In chapter 5 a detailed presentation of level 2 methods was given. It was shown that the relia· biJit;:, ind~~ (J of Haserer and Lind is failure fUnction 'inyaria~t' in the sen&e that. equivalent. failure functions result in the same re1iabHi~y. index. Calculation oC t.he reliability index has bf;!cn shown in chapter 5 by a number of examples, but only. uncorrefal~d basic lIaritibleihave 'b~~ tr~a~d .. ln _this chapter the treatmerif"'i11 be extended so that correlated basic lJorfubles can be induded. This exle.nsion is important. because in practical applications basic variables will oCten be correlat. ed and because correlation sometimes affects the "'alue oC fj appreci.abl~:. In sceli.on 5.4 it was shown that the probabilit.y of failure Pr can be related to the reliability in· dex 8. when the following two conditions nre fulfilled (a) the failure surface. is linear (b) the basic,.~~~,~~les Xi! ,i, - 1. 2, •... n are _ normall~' dist,ri~)Uted. " ' , ." . r: The one-one relation bet.ween Pr' ari'd ~ is given' by equation (5,33), (5,33) When the basic variables' are non-normally distributed t.his one·to-one relation is not vaIid:To retain this relation for'non·nonnaIly distributed basic variables also,lt is natural to approximnte a non-normally distributed variable with a norm~lly , distributed variable. In this chapter it will be ~ ' sho ..... n ho ..... this can be unde~E!.n_._
- 108. 96 6. EXTENDED LEVEL 2 METHODS 6.2 CONCEPT OF CORRELATION The concep.t of correlation has already been introdu~ in section 2.9. Let Xl and X2.be two random variables with the expected values ElXl , = JoIXl and E(Xzl = JoIX2 . Then the couoriance of Xl and X2 is denoted Cov{X11 Xz J and is defined by (see page 3:3,) The ratio (6.2) where 0X 1 and aX:! are the standard deviations of Xl and X2 is called the correlation coef· ficient. It can be shown that (6.3) The random variables are said to be unco"elated if Cov[X1, X2, ::I O. It follows from (6.1) that il a large and positive value of Cov[X1• Xzl occurs then the values of Xl and Xz tend to be simultaneously large or small relative to their means. If the value of Cov{Xl' Xz ) is numerically large but negative, then the values of Xl tend to be large when the the values of X2 are small relative to their means, and vice versa. When Cov{Xl' X2J is close to zero there is no linear relntionship between Xl and X2· Example 6.1. Consider the beam AB shown in figure 6.1 loaded with t.wo random loads PI and P2' The loads are assumed statistically independent. with means PPI ". 4 kN and IJP2 D 6 kN and standard deviations aPl = 0.4 kN and ap2 D 0.5 kN. The shear force QB and bending moment Ma at the support Bare 1 Q. - 27"(13 Pl + 23 P2) 6 M. - 27 (4 Pl + 5 P2 ) Qa and Me are random variables with the means 1 190 I-IQu - 27(13I-1P1 + 23J1p2)="27kN 6 92 IJMu -27(4/.1Pl + 5/.1P2)=gkNm and the variances .' __ 1_(131 0'2 + 232 0'2 } = 0.2185 QB 27' PI P2 . aMB -~(41a~1 + 52a~2}·0.4351 aQu = 0.467 kN aM» = 0.660 kNm The loads PI and P2 are statistically independent, but QB and MB are correlated. The covariance Cov[QBMaJ can be evaluated by (6.1), where
- 109. 6.2 CONCEPT OF CORRELA'I10N Ii t"' t ~ A • .1' ." ~. ," .1' ./3 r ~, V- !im ,r Figure 6.1. 1 6 EIQ.MBI·EI 27 (13P, + 23P2)27(4P, +5P2)1 = 2~' (52 EIPil + 157 E[P, P21 + 115 E[PiD In this equation E[P, P21- E[P,IE[P21 = 24(kN)' E[Pl} .. O~l + jJ~l :: 0.16 + 16'" 16.16 {kN)2 E[P~J c (J~z + 1J~2 :: 0.25 + 36'" 36.25 (kN)l Therefore EIQ.M.I- 2~,(52'16.16 + 157·24 + 115,36.25) -72.239 (kN)'m Dnd Finally, the correlation coefficient is 0.3051 PQaMB = 0.467'0.660 - 0.99 This value close to one indicates that Q8 and MB are strongly correlated. 97 Exercise 6.1. Consider the cantilever beam AB shown in figure 6.2. loaded with two sta· tistically independent random loads P 1 and P 2' w~ere .uP:: J..IPz 5 kN and apt = 0Pz • l ::I 1 kN. Let QS and MB be the shear Corce and bendmg moment at the support B. Show that lp, A t' .~ ,~ ./2 ./2 " ,'< Q .. 10 m j' Figure 6.2
- 110. ~QB"'10kN JJl..r . = 75 kNm O'Qs -lA142 kN "11.1803 kNm ':::mple 6.1 and exercise 6.1 random variables QB and M:s both depending on the same lbles PI and P2 are considered. Therefore, QB and MB a;:. c~rrelated although PI and P2 _' uncorrelated. ;ery often one needs to calculate the mean and variance fo~:!. random variable Y which is a :linear function of a number of random variables Xl' __ .• X:: Tne expected value ElY] is then given by • E[YI:::: ao ..,. I ajE[Xjl j"l l::ld the 'ariance Varl Yl by • • • VariYI" IafVar[Xj]+ I I ajajCov[Xj.Xjl j"l i * j V ... ere equation (G.5) is a generalization of equ~tion (2.86), SH- ~1so (5.15).' . :.et the random ·ariables Y1 and '12 be linear functions of Xl' X2 · ...• Xn , .. YI :: IajXj i-I • Y2 '" IbiXi j .. l :: '!::an then be shown that· , n .. " .,.' Il , n Cov[YI • Y21= IajbjVarIXi-] + I I ajbjCov(Xj.Xjl j"l i' j (6.3) (6.4) , (6.5) :'.'1 (6.6) (6.7) (6.B) Example 6.2. Consider again the probiem solved in examp:e 6.1. The covariance between QB and MB can now more easily be calculated by equatio:. !6.~!
- 111. . ··· ·r·· i , Consider aE!.:Iin :n~ ~J of n r3~do~ ~a~iables X ~ (Xl' .... An I with eXpfc;fld "aJue~ Ef XI i. i ::::I 1. 2 ..... n a.'ld :ariance and covariance collected in the CD"arianc.e matrix ex I.deiincf, ly equation 12.$'; 'I CO"I~I : Xn l I" (6.91 CovfX . X2J' ... : ....... n... , , . i .: j ! VariX.] . J Clearly. no correlation between any pair .oC random variables will exist if this matrix ex is a dia!lonai matrix.. It will now be shown how a (new) set oC random variables Y = (Y, .... , Ynl.. where Yj , j ". 1, 2 .... , n are linear functions of Xl' .... Xn. can be constructed so that the co~ponding CO"ariance matrix Cy is a diagonal matrix, that is 16.10) According to wel:-known theorems in linear algebra such a set of uncorrelated variables can be obtained by the transformation ~_AT~ ',, '1. (6.11) where A.is an o.nhogona matrix ,~ith '~~iu~n 'vectors equal to the orthonormal eil1en'vector.s. of eX' By this tran~for~aiio~ " . ' . .. . :. (6.121 (6.131 The dia~onal elements of Cy . i.e. Varl "il. i =: 1, ... , n are equal to the eigenvalue$. of eX. Example 6.3. Consider two correlated random ~riabJes Xl 2I1d X2 with the mean vector EIXI- (EIX,I. EIX, II ' (2. 3) and tnt> COl·.;:riance- matrix
- 112. 100 ex " r V:lrlX 11 Co,IXt · X'!1 I ~ COI/{X . .!. X, I Var(X~1 3 , ,:. : 1 . 3J ~ The characteristic equatioo or ex is with the roots;'1 .. 2 and;'~ .. 4. The conespomling orthonormal eigenvectors vi' and' v2 are determined by the equations , " Ii r-l 1; J vl~'r 01 ,nd ~2 '" '"r 0 1 i i, ~l 1 1_ 0 j L 1 -lJ LoJ resulti~~ in ; ':} 7f " T(l. -1) ::and ,:!:aT U . 1) The transtormation matr~x A is therciore gh'en by ~: .: '-:'f''r l 1'l - I ' L-1 IJ 111e PfleWIt unconelated random voriables Y - IY,. Y2J are therl!fore 0' f y l.i1rll il l 2 hJ 1 , - 1 -I : ; X ' I 1J IJ I Xo " - , ____ ! _.J:t'l!u~:..:Q~~cLyalu.es...ar.e _____ - - - - -·--- - ... ---- - - . - "- - •.•.. -- . . -.-- E[YII '" V; (2-3) =-iv'2 '7) - E[Y:!!c v 2- (2+ 3) "'I ,/2 .md tl~e cO:;rian~~ matrix c" .~ -; 01 • [2 ol , I 0 ~:! j LO ·1 J :
- 113. a.::! CO;':CEPT OF CORRELATlt" I - ~T - - ., , ·Cy ::.- C-x',A =~L I • ,~ I I -IJ [" ! i I 1 1 3 I' but Cy is of course ea~I"1 _, . .... IOI'rmmed by (6.111. 101 L-l Exercise 6.2. Let the '1,/ means E[Xj = [2,.,t, Si·;",:;;'nrrelate.d random ';ariables X .. (Xl' X.,. X3) have the LI~ covar1anl~e m:ur.:x - -1 -1 -1 i ex -[ 4 I -1 -2 Show that the random ""11 _ [:tr ""-j!:""'~'" are uncorrelated with tl." ""'.' ""I'ariance matrix Cy '" r3 0 I, I 0 6 I, Lo 0 " 6.3 CORRELATED BASIC V. I tl.BLES 1,- ~hi~ ~tiol1-it'i" "hown hO'1t I I.,. I lasofer I . . Structures with correlated h:'~Ir' am L:noreliaqilityindex p can be calculated for account of this correlation. JII "I:I,l"Iabl~. It Will be ~~own that i~ is o~ten important to take ing w3lo', First 3 set of Wlcorr,./ 11..>n;)..! th~ rt>iiability index;1 'was dermed in the follow· " . . '11".: hasic t·artabl. -:.a . . fallure surface IS expressed in IIw • >1 :, (.'( I •. ' .• .'( n) IS chosen. and the ubi.?::; Z '" ! 2 1 , .. , • Zn J is Obi .. ;, ':,rrespolldin!.: X·cr;ordinate system, Then a new set ofvari· fo!:owin!! way 'Po) hy norm;Ii':l;"J~ ti:e original set of oa.sic 'ariables X in the (6,15)
- 114. le'':' 6, EX~;DED LE'EL ~ METHODS W~;f!e /.I XI and aX; are the mean and the standard deviation of Xi' By :.:.~ linear mapping (6.15! t,;;. failure surface in th(' x-coordinate system i; mapped into a failuTe F..:rCace in the z·coordinate ~yi":~m. The reliability index (J is then defined as the shortest distance ::='om the origin to the f<.. .. :.m.' surface. in the normalized z-coordinate system. W:.-?n the basic variables X ... (Xl • ...• Xn) are correlated the first ste;: is to obtain a set. of w:.:orrclaled uariablc~ r r: (Y1, . _ .• 1'>1) as shown in the last section. 7ne next step is:then to ncmalize the un correlated variables and thereby obtain a set of nor~ized and uncorrelated l',;:-..::rbles Z '" (Z] • ...• Zn)' Fina1I~·. the reliability index can be define:' in the z-coordinate s;,',o:em as shown above. This procedure is illustrated in example 6.4. Example 6.4. Consider the simply supponed beam shown on flg-..:::-e 6.3. The beam is loaded with LWO concentrated loads Xl and X2 with the means EIX l J ,. E!X2} ,. ~ and the covariance matrix ex;:: I Var[XlJ LCov/X"X,/ wherc-l < p <1. Uncorrelated variables Y land 1'z are acc~r~ing to equation (6.1:, determined by Y=ATX-where the transformation matrix i~,""determined b:-' the eigenvectc:-; for Cx< The eigenvalues A1. AZ and eigenvectors vI' v2 for- ex are easily determined in tn:! simple case V}, --1 (1,1) "2 =,q. (1, -1) The transformation A is thererore gil-en by F='7111J II -1 and. according to the equations (6.11) - (6.13), _ ~T"'{. Y" = 4 (X,. + X,) Y=A X-Y .>1%. (X -X) ,2 2 1 2
- 115. 6.3 CORREL,'TED BASIC ,~RIABLES 1" r r/7n A • A ...y ~y "v J' Figure -6.a Uncorrelated and normalized variables Zl and Z2 are Cinally determined by Y,-EIY,I ,f2. . Zl· " Y, --2 (X1 +X2 -2}J)I",,'f+P " From the matrix equations (6 .11) . (6.13) it is seen that 0' XOte that in the z-coordinate system we have , , zTi - ex - EI~))TA(ATCxA)-"2 (ATCxA'- "2jT (x - EJXI) - (x - EIXIITA(ATCxA)" ATeX - [IXI) . (x- EIXI)T~(i - ~XI) -103 (6.16) (S.17) (S.16) (6.19, ··' TnedeC{nItfOricifthe rel,i6iliij'inde): -p-b)'-Hasolennu' Ln_"_" --_--..:..-.... _--'-...1 ... ______ . __ . , , P- min (tTZ)!'" min «X-EIXI,r~~1 (x-EIXlHz (S .20) i,;;wl i'iw" where 1.':1: is the failure surface in the z-coordinate system and Wx the failure surface in the xcoordinate system. The formal deCinition 16.20) is convenient if a computer is available, because calculation of .3 ilt by (6_20) formulated as a constrained optimization problem. The constrained Fle.(;her· Powell technique can be used for this purpose. For problems with a small number of basic 'ariables the same iterative method as used in the e. .. ampJes in chaptE:..-:5;c3n...be used. A small cie$k computer or pocket computet is suit'lble Cor this purpose_
- 116. [(is'easv to S!1! thatth~ reliability 'index' j d'en~eci. by' ~'u:ni~j, 'i6.:W) f.:;~:beeY:l.I'u:l.ted on ' th~ 'b~i~'~'f iJ,' :>ai~ty margYo ~ w'h~n t'h~ f~lur~ (u~ctio~ is.ii~~a~ ,inhe 'b'~si~ ·~~i;ble. Ld the ra.ilure function r :Rn ,,", R be given by (6.21) and'the corresponding: safety margin J:~e ,t'e~i~bjlitf. inqex ~ , is then ~!mpl}' ~~~ , to ~ :I" O!lx (6.22) i.e. the same v:l.lue as determined on the oasis of the relitlbility index fJ as defined by equation (5.91. An illustration of this is given in example 6.5 for the 2-dimension'aJ case. Example 6.5. Let Xl and X2 be r3ndorn variables with means E{X11 '" E{X 2J IC IJ. :md the covariance m:1trix exactly as in example 6.4. Further. let the safety margin. MX ~ d~f~ned by MX '" 3 0 ';- a1X 1 + :l2X2 (6.23) The purpose of this exam~le 'is t~ '~h~~~' th:~t '(6.20) 3nd~r6.22) }:ield the s .. in~ value {or the reli3bility index J. , ~r" Uncorrelated normalized.variables Zl and Z2 can be calculated (rom equations lS.lS) and is.I'). The inverse equations are r.r . Xl -jJ.'" T (a~Zl T ayT="'P Z 2) X2 =j J. + i1'1-. (a,/I"'+P Zl - a..;r::ti z:!) There(ore. th<!: s"lety margin :'o.tz can be written ( • :'o.lz ·"o"",I.I(a ~ ~ £J.. =-- l .... :1:!, ... 2- avl+pfal' +a2i.Zl T " 2~a"1-p(1l1-a2JZ2 (6.2-11
- 117. 6.3' CORRELATED BASIC VARIABLES .; . .... 0, . The reliability inde~ of Hasofer and Lind defined by 620) is equal to the distancc't""u the origin to the straight line (6.25). i.e. bO ~. v' bi' + b:'2 where bo ... 30 + ",tal + 0.2) and bi + b; ., a' (3~ + a; ~' 2pa132 l. so that :10 + ,.1(:1., + .:1.,) {3. , • ?- avai + a2 + _pal;J,~ From (6.23) - · .t .• .... . and Therefore. fS.221 git'es the same result as (6.2.). Example 6,6. To illustrate the importance of taking the correlation into account crill~ ,tll' example 6.;' with 0.0 : .0 and a1 - 112 '" l , Le ..... ith the saiety margin r . :WIX .=.Xl.;'"X2. Then. from (6.27) it follows that -~ 6:(1 + 01 ';" ' .. : .. whereip6 corresponds to 'no' correlation: between Xt and X2 (p .. 0). This relationshijl h". tween p and ~/~· is shown in figure I) .... I I I I I I .I .... I Figure 6A '.' ::, ..
- 118. 106 6. EXTENDED LEVEL 2 METHODS Exercise 6.3. Consider the beam shown in figure 6.5 loaded with two concentrated loads ~1 and Pz a .. the same point o~ the beam. Tne maximum deflection is; 1 ~ emax - 9.J3 ei where e is the modulus of elasticity. i the relevant moment of inertia and p ,. PI + P2' Further. let p, c,.I!. e and i be realizations of random variables P ~ PI + P:!' C, L. E and I with E!P1l = E[P21- 8 kN "p, -tip: ... 0.32 kN E:CI 2m "c Om E:LI 4m "L Om E:EI - 4·10·!m .... "E ., O.3·10-J m4 E;t) ,. 4'lO'MN/m: ", lO·MK/m2 All random 'ariables except PI and P2 are assumed uncorreiated :"The con;;lation between PI and P2 is given by ,'" Calculate the re1iab~~tr index 11 when.t~e fo.I,~o_wing failure criterion is u~ec , In examples 6.5 ·6.6 alld exercise 6.3 the correlation between random 'ari~bles could be treated in a simple way because these random variable, only appeared in a l,inear connection. The important equation (6.5) could therefore be u'ed in these examples. It was not necessary to construct new uncorrelated variables in the way presenU!d on page 9~"This i:: Dot the case in the next example. , .. ·here the failure runction is non-linear in ti:le ~wo basic ra::.dom variabies which are ah>:> correlated.
- 119. : 6.3 CORR,ELATED BASIC VARIABLES 107 ,..:' Example 6.;. Cons~der aeain thp beam shown in figure 6.J, but now only c and t are can· siderea reali:tations of ra~'dom variables. All the oth'er' variables are assumed La be deter· ministic.-The maximum deflection can thcrefo're be 'ritten ""here k is a constant. Further assume that c and r are realizations of random variables with EILl = 4 m, EI Cl .. 2 m. 0L = 0c • 0.25 m and CaviL. Cj .. 1/32 m1 • The covariance matrix is then . , so that the results from example 6.4 can ·be used directly. Uncorrelated random variables '1 and Y2 are given by y,.'1-(L+C) .y .il (L- C) 2 2 with E[ Y 11 .. 3J2 m, EI Y 21 .. ,/'2 m and the covariance matrix Cy · 1~.r1.5 0 J :.. 0 0.5 From (6.29) the random variables Land C can be expressed by Yl and 1'2 L . i2l ()"1 +y,) Let k '" 6.415·10~. ~~d let t~e failure cria!rion be ..... " 1 umu: :> iOO c .• : . . ,.,. 0' 1 - 6.415· 1O-J c2 <; 0 From (6.31) it follows that LC .1.(.' - V') 2 1 2 so that the failure criterion (6.32) can be written (6.29) (6.30) (S.31) (S.32) (6.33) where),; a.nd Y:l are realiz...1tiunr. or uncorr·~laied random variables Y 1 and Y:!. The reliability indr!l1. J can now be calculated b~' the iteratiVt'> meLhod shown in Ch3.?er 5 .
- 120. lOS Ij, EXTI::~DEO LEVEL:! ~I£1'HODS I .• "I Exercise SA. Determine the reliubilit)' index p for the .iar~[y problem ionnuiateu in (>xam· pie 0.7. 6.4 NON ·-SORMAL" BASIC VARIABLES : Until now only second order information hIlS been taken into uccount when ev.duating the r~lia· bility of·n: mucture. Let the failure .iurfa'ce be lin~ar ~~d let ih~ b~fc' ~:U:iahles be normally distributed. Then the' following rel3tio"n hetween the prcibabil"riy 'or" fail~;~ P; :lnd the reliability .,. . -.' . . .... -, ": ,; . ' -,".: . , .;~ -"', (6.34' : ; ~! .: .. ;;'1 . " ,~. where .,. is the stanclard!z,ed .nperna! distri~ution fllnc~ion~ _,', ' ~; ;.' .:1 .. :" It is -im'~rtlnt to r'~~ember that no information regardio,," the prob:lhilitV cii [llilure can he (lh· tain~ when the distributions of the basic.variables are .unknown. IUs 'reasonable in some situations t~' ~X~t that a gi~e~' ~~d~m variabl~ c~ b!! considered nonnaJiy distributed as a gaol! appro':<imatj:o'n. but oiten'such at'~sum'ption is q~ite unre3Sonable. Consider the yield stre5S oi a steel bar. For such a random 'ariable negative values cannot occur. It is therefore more reoJistic to assume lag-normal distribution in this case. To overcome the problem that the definition space ior the set of basic -ariables Xl' ...• Xn is not the whale Rn $pace. one can apply one·to-one transformations to the relevant variables. For the yield stress mentioned above this transformation should map R_ on R and be continuous. In this chapter the transformation ¢ven by Rackwitz and Fiessler [6.-11 will be usect. This transformation is chosen in such a way that the values of the original density functions fXI and the original distribution function FX, (or the r.mdom variahles Xi are equal to the corresponding values of the density function and the distribution (unction for a normally distributed variable at the design ]:lOint A (see figure 5.5). i.e. 16.35) 16.361 wherethed~3ignpoinl.-" (xi,'.' .xj,. _ .. x~) and where,ux; andux, are the tunknownJ mean and :itandard devi:tth:m of the approximate normal distributiCln. Solving (6.35' and 16.36) wilh re!!:ard:o Il~t and a XI we havE' .:d .-Llf.lxtij) {'Xi:: : - I'xr"j i 16.37) i6.381
- 121. 6." NON·:-lOR~IAL BASIC VARIABLES 109 Clearly the iterative method presemed in chapter 5 for calculoting the reliability index JJ for a gi~'en structure must he modified a little when the lr:msCorm3.tion shown abelve is used. On each step o( the ilerlllien new values (or 0Xand SJx, must be calculated (or those variables I where such 3. transfonnation has been used. This is iIIusuated in example 6.S. Example 6.B. Consider again the beam analysed in example 5.5. In example 5.5 the reliability index 11 was calculated solely on the basis of second order moments (or the relevant basic vilriables. namely the load p. the modwus of elasticity E and the moment of inertia l. It will now be assumed that the load P is Gumbel distributed with the distribution fun~:tion uee (3.10)) Fpfp) - exp(-exPt-a:IP - u))) and the density function fplp) "" e:<p(- expt- a:(p - u)J - alp - uJ!o The (,o parameters a and u can be calculated from the following exp~essions for the mean SJp and the standard deviation op (see 13.11) and (3.12)) J.l.p = U + 0.57722,'0: )'.!.' if l ' op ~ v'6' Q when IJp and O"p are known. With the same mean and standard deviation {or P. E and 1- as in . example 5.5 one gets Q:::> 1.2825 (k~,-' and U" 3.5499 kN. The Gumbel distribution is now transformed into a normal distribution with·the mean,llp llild the standard deviadon I1p given by the equations (6.37) Md 16.38). In the normalized coordinate system the failure surface is then'(compru:e with example 5.5) (10"""' + 0.2·10-4 Xl )(2'10' + 0.5,10' x!)- 78.12(pp + oj. x3) = 0 i6.39) The reliability index;1 can now. be calculated by an iterative technique analogous with the :' method used in example 5.5. The only diHerence is that the new values o( ,II 'p and 0;' must : be calculated after each step' in the iteration in the {ollowing way J.lp" xi -·tJ-1 (Fp(xi))a; 0";" - ~($.I IFp{xilJJ/fp{x;) where xl = Fp~ (<P(jlQ:3)). · , ", ; 0.'2 I, ", i "p i ;.Ip Table G.1 ~.' . Scart 3 3.53 -0.58 -0.52 1-0.58 l-o:;S , 0.58 I 0.35 ! 1..08! 1A.25 I' i 3.05-1) 3,-179 ., .. ' ,. . .. ' Iteration No:" 2 3 4 3.50 . 3AO . 3.34 -0.35 -0.2' -0,20 -0,88 I -0.94 1-0,95 0.31 i 0.26 [ 0.22 1.337! 1.237 I 1.176 3.5S0 I 3.678 I I 3.728 (6AO) (SAl) I 51 · 6 I , 3:33 '! 3,32 I -0,18 i -0.13 -0,96 ! -0.97 1 0.20 i 0.20 I 1.150 1 1:1'1'°11 3.7-1.6 ! 3.752
- 122. 110 f ',,(Pl. n"iP} i O.S T , ! I o'r 0.31 I, O. 2~ i 6. txTESDED LEVEL 2 METHODS o,t 0.0 .:--""'--L--+--f---t----<-..::::::~-_+--+_~ p 3 • , , Fis::urc 6.6 The reliability index is now IJ • 3.32 which is a small increase from enmple 5.5, where ~. 3.29. The Gumbel densil:,- function ip and the corresponding normal density function op are shown in figure.6.S. " .<ccording to the ec;.:.:ations (6.35) and (6.36) np and the corresponding distribution (unction ~p are connected with ip ·and Fp in the following,way npCx; J ,""fp(~;;' ):p(x;) :: Fp(~~ I where x'; i.i the. thirc coordinate of the design point (xi. x;. x;) - ~,al . Q:!, a3 ) := (- 0.59, - 3.22, 0.66) (6.42) (6.43) Exer~ 6.5, Consider again the elastic beam neated in example 5.6, but ~ume now :.hat tite load P is Gumbel distributed with the mean /-Ip • -4 kN and the standard devia· tion Dp - 1 kN. The other data.used in example 5 ,6 arc unchanged . . ~b, thi$ Gumbel db~bution on· a no~al distribution as ShO,,"D abo~ and determine the ex~c:ted value IJI, so that IJ ~ 3 (sol,u.tion IJI <:; l~l·lO·' m'). ; ': BIBLIOGRAPHY fti.ll Fl.it1e-·~n, 0.: F;mdam·ental$ o{Se~ond "!!,ment St~uctun:ll Rf!iiabilit)' Theory·. Int. Research Sem!:-,.!: on s.:afety of Structure;;. Trondheim. NOl"'"ay. 1977. (6.21 Ditle'Sen. 0.: i..'-:('crra inl :-· Modeling. McGraw·HiII.1961.
- 123. BIBLIOGRAPHY 111 16.3) Hasofer, A. M. and N. Lind:An Exact and Int!ariant First Order Reliability Format. Proc. ASCE, J. Ent!". Mcch. Dh·,. 1974, pp. III ·12l. 16.4) Rackwitt. R. and B. Ficssler: An Algorithm for Calculation of Structural Reliabilit,· under Combined Loading. Berichtf: ZUf Sifi:h~rhei~th~ril! der:Bauwerke, Lab. r. Ken· 5tr. Ingb., Munchen, 1977. 16.51" Rdliorui/~sati~n of Saf~ty !I~d S~rv~ceabiii"t)' FQcto~$ in Structural Codes. Report 63, , Constru,c tion Ind' ustr.y Resea. rch a.n d Info, r.m ation:Association. London. England. 1977. 16.6) The Nordic Committee on Building Re(rulations (NKB): The Loading and Safer,. Group. Recommendations for Loading and Safety Reg'i:l./ations for Structural Dcdgn. NKB·Rc· port No. 36, November 1978. (6.71 Theft·Christensen, P.: Introduction to Reliability of Offs/u>re, Structures. Lectures on O(fsho~ Engineering (cds. W. 'J, Graff and P~'Thott:'ciiri5tensen), Aalborg University C~ntre, Aa1.)org. Denmark. 19,78, pp, 53 • 72 .. ' .. ., 16,8) Thort-Christensen, p,: Some E:cperienc'c from Appl!cotion of Qptimfzation Technique in Structural Re/i11bility. By,nin1!'sstatis"ke Meddelelser, Vol. '48: 1977, pp. 31 • 44. " .. :~,
- 124. - ! ,":i .;.... , .. .' "
- 125. Chap'ter 7 ," ) " RELIABILITY OF. STRUCTURAL SYSTEMS ,7.1· INTRODUCTION , In' the -precedl~g ch~pt~h: rheconcept of reliabilIiy'liai m.Jnly been conce~ed with single :it'ructural eiemen'is such'~a beam or acoiumn. In" the fundament~l case the loading is described by a single random variable 5 and the st~~~gth by a sing;' random"~'ariable R (see 'chapter 41. The probability of failure Pris then defined as Pc " P(S;· R) assuming that the failure condition is R - S .;;; O. Pr can be calculated from Pf "') fR,s (~. 5) drds ~, (7.1) (7.2) 113 where fR,s is the joint probability density function and Wr the failure region ':(r, 5)Jr-5" 0;_ Vhen fR.S is known, the probability off~ure PI' can be calculated relatively easily from (7.2) by a suita'Lle numerical technique or by sim'..llation. When the loading of a single Structural element is determined by a number of random ~·ari· abies R = (R1" .. ,Rn) and likewise the strength by a number of random variables S;; !SI' ..•. Sm) equation 11.2) can be generalised and the probability of failure calculated if the corresponding joint probability density function is known. In the situations described above only one single structural member with a single failure mode is treated. The reliability of a real structure is usually much more difficult to evaluate since more than one element I member'J can fail and because there is possibility of more than-one failure mode for the system. To handle probiems of this kind it is sometimes useful to can· sider,from a systems poim of view. In this the real Structure is modelled by an equivalent sys· tern ~ mch a way that all relevant failure modes can be treated.
- 126. , 114 7. RELIABILITY OF STRUCTURAL SYSTE:'IS In this chapter the fundamenL3I systems· series systems and parallel systems· "ill be defined and it will be shown how such systems can be analysed from a probabilisttc po~.t I?,! iew. bOth when no correlation between the elements exists and when all elements are equiulY' correlated. In Chapter 8 some important reliabilli~;- bound~ io'r structur~i'~ys~ms ~ili be ~ti~.d,·.'~d for systems with unequally correlated elements some approximate methods of estimating the failure probability will be presented. 7.2 PERFECTLY BRITILE AND PERFECTLY DUCTILE ELEMENTS It is of ~eat importance for a struct.ural system whether its elements can be considered perfectly briWe or perfectly ductlle. A structural element is called perfectly brittle, if it becomes ineffective after failure, i.e. if it loses its load-bearing capacity completely by failure. The worth JlpedecUy brittlclt should be understood in a broad sense. If a tensile bar made of a brittle material fails due to a tensile force then such an element can reasonably be considered perfectly britUe. hecause its loading capacity is completely exhausted. But the'cbaracteriza· ticn »perfectly brittleJi ~n also ~ .. u~ed i~_~ ltuations, where no real fracture ~rs. , ~.g_. when the element is deflected so much ~h~t) t is ineffective in relation ~ a gi~ Ioa~g. -': typica1 oad-deDection cwve [or a brittle element is shown in figure 7.1. If an element maintains its load level after failure it is called perfectly ductile. A typical example of perfectly ductile behaviour is shown in figure 7.2, where it is assumed that ~he load p can be maintained d~ an increasing displacement.. To distinguish these two types of element behaviour the symbols shown in figure 7.3 can be used. llo:.d -!'-__________ dilplaccment Fi~rc!7.1 ., . oispllc:emcnt Fi!;ure 7.2
- 127. 7.3 fl!XDA~IENTAL SYST£:-"IS 115 brllllc ~l(!menl ductile clement Figure 7.3 7.3 F'UNDAMENTAL SYSTEMS As mentioned in the introductioll there are two fundatnental types of systems, namely series systems and parallel systems. A system of single clements is a series system if it is in a state of failure whenever any of its elements fails. Such a syslA!m is also called a weakest·link system. A typical example of a series system is a stat.ically determinate structure as shown in figure 7.4. Obviously, failure in any member of such a structure will result in failure of the total system. A series system with n eJements is generally symbolised as shown in figure 7.5. All elements in rigure 7.5 arc brittle elements but for a series s~'st.em the distinction between brittle and ductile elements is irrelevant bee:lUse the total system fails as soon a5 one element fails whether it is brittle or ductile. 1t is important to note that the idealisation of a structure by 8 series system as in figure 7.5 is only related to the failure interaction. Therefore, one must not interpret figure 7.5 as one in which all dements have the same load, although a load · S is shown applied to each end of the series system. Usually, an externaJ load on a series system Fi,Ufe 7.'" , " riJUre 7.5
- 128. ." Uo '. RElI.~BILI'fY ,OF ~-rRl!e!;t:ftAL SYSTEMS . will re~ult in differen~ loa~s .(st~sses) .~':' ~h~ ~in;gl~ ~~em.ents, ri~fe ';' ,5 is usefu].}n ~~lc~.liatini the diStribution function FR for the strength R of the series ~ystem. Let FR be .the qistribution . , . . . . . . .. . _" .. _ . .. , ... .. .", I . . function for the strength Rj of element i; then FR(r)<=P(RC r) - 1-p(R>rJ - 1-P(R1 > [1 ()R:> rzr"l· ·.· nRn.,?'rn) . . .- ' '', ' : ..' . "::;; ' :' " " ','- .:. . ".. ' . .. . • _ . _ :... • '. ... .-, : : ~ ,. n · .", • ,1,- (1 -: E.'RI.cr~p~l - FR:! (rzV : ' . (1 ,- , FR~ (rn» ~, l-::- !1 (1- FR/rj )) (7,3) ;' . .. I . ~ ... ,', , :~. . • :I,~ l , where it is ~med ~h:at - th;e stre~gths oOhc eleme:ntsareinciependent. ;,' .' .(':. --:" '. , ; .':: ;, ~ , ,', ' £:tample 7~1.' Consider 'a"simple'structure ciins:isting of two tchsiii b-~ as shown in figure i-.6. Let the strength --R"~ oc ieac'f· ten:siie bar b-e ~ ~d6'm(:;ariO:bhi witti the density function fRe,:showri"in' !the 'Sa~-e"{rgure, Lefthe'siiilctiir{bEi ib'aded"by n '~ingie tensile force S '" 1.1 kN, It is reasonable to model this structure as a ~rie5 system with two elements. The distribution function FR for the strength R of the system can then be de· rived by equation (7.3) if the strength of tW9 elements can be assumed i~dependent, One obtains wher.e -. ~. { 0 for, ,< 1 FR, If):a t,-t -'';'; for 1.,!, 1<,3 ~ , ,:' 1 (0' ,,. 3 ." 1 Ill. ' (r.l 1.0 0.1~, , .._ ~ ~~,kN 3 FiGure j ,6
- 129. ".3 FU~ DAME~.n AL SYSTEMS ,_ .~r ~.ubstitutlng FR .. i~ t,he .ex,pression for Fa one obtains ; . .' E1" ' :-. forr<l FR(r) • +tr~t forl';;;r<3 .. - , (or r;;' 3 .... . '.: . .:. I~ is then easy, to calculate the D,lean PR' and the variance tJ~ . (or.the -strength. R oC the ., :' sY~~~~. '?~~ ~ets.. ' . .' _. ,, ' ." '," '.. :. ' ., ;, :, The reli.abilitr, ~d:x ,tIS }~r the system, is the~ef9re t:-_ ... "' _, . ..~ _". ~ .•. . . . :,:!.: : - '. : ' . ,-- ~R'~ ~;S . . ' .tJs"'..Jo~+ ' o: . ' .. R~_ . ,s G 5(3 - 1.1 .. 1.24 ../279 while"thtl'reliabiiity index 11. (or a~ elem~nt is '-;; . . :-lote that Ps < Pe as expected. Also note that better values can be calculateci on the basis of f7 A) 3nd 16.341. When the distribution function FR for the strength R oC the series system is detarmined. the p'robabilily of failure Pr can be calculated as for a single element, by . !,-",l (l-FRt(rj))fs(f)dr (7.4) where Cs is the density function for the load S all the series system. , Example 7.2. Consider again the structure shown in figure 7.6 loaded with a deter· ministlc iorce S '" 1.1 kN. In this case the probability oC failure is simply Pr - FR (1.1) - -i' 1.21 + 1.5 . 1.1 -1.25;:: 0.0975 while the probability of failure for ~ ~ingle element is
- 130. '. RELIABILlTI' OF STRUCTURAL. SYSTDIS Exerci~ 7.1. To illustrate that lh!: probability of (ailure Pf of a series system increases with the number of eiements n. consider 3. system with n elements with the same distribu· tion function FR. as used in example 7.1 and the same forc.e S "1.1 kK. Calculate PI for no: 2. 5and 10. !'Oow tum to pilral/el S)o'stems. Failure in a sin¢e element in a structural system will not. always r~ suit in failure of the total system, because the remaining elements may be able to sustain the ex· ternalloads by redistribution of the loads. This situation is a characteristic of statically indeterminate structures. Frulure of such structures will always require that more than one ele-ment. fails. From a strength (failure) point of view such a set of elements is called a parallel system and the associa.ted fru:we state is called a failure mode. A real statically indetenninate structure will u'suaily have a great nu~b~r of fail~e modes, where each failure mode is modelled by a pa.rallel system . .and these parallel systems are then again combined as a serles sy5lem. Therefore, such a structural system will fail when the weakest mode (parallel system) fllih. In otber words. a parallel system will only fail when all elements in that system £ail. Therefore, the behaviour of such a system depends to a higb degree on whether the elements are perfectly ductile or perfectly brittle. A ponzl1el system wi! h n perr~cHy ductile elements is ~ho .... ;n in figure 7.7_ Because of the as· sumption of ductile elements, the strength R of this system is simply determined by (7 .51 where the strength of element i is given by the random variable Rj _ Note that when the random variables Ri , j .. 1, 2, ..• , n. are independent and are normally distributed N(lJj' tl'1)' then R is also normally dismbut"!!d Nfl.:, a), with EtR) - 1J c iJJj (7.6) i-I VarlR) " OR: = -f 0.,1 i .. 1 . (7.7) 2 ••• j • .. n . ~ ;,
- 131. i.a FUNDAMENTAL SYSTEMS 119 . I··· , " , . According to the Central Limit'Theorem it is r~ona~l.e to as~u~e R to be norma.!ly d,istri. buted if the number of elements is not too small. e~en .. i_n cases_wher~ .the distributi?ns ?f Ri . i = 1. 2, ..• , n. are non·normal. Example 7.3: Consider the system shown in figure 7.8 with 3 elements with the strengths R1, R2 an,d Ra' Let Rl and R2 be. normally distributed with E[ R1! = EI R2J c 5 k~ and O"R t .. (JR 2 '"' , k1l"._Further, lpt. R3 have a uniform distribution overt.he interval 18 kN : 12 kNJ. The random variables are assumed stat.istically independent. Consider first the subsystem consisting of the twoeleinents 1 and 2: According to' (7.5) the strength R12 of this subsystem is normally distributed with E[R12 J "" 10 kN and 0" R12 - .J2 kN. The total system can 1,10w ~ cons!dered a series sytems ,with two ele· .• ~e~ts; nl"l.f"71y in element ",ith, the st~en~~12 normally distributE!d_and an element with th~ ,~~re~gth ~a u~i~?r~I~~ist~l?utel~. The dtstri.butiqf!function F R3 for Ra is given by ,for. r< 8 for 8< [<12 for 12" r The distribution function FR of the total system can now be calculated from (7.3) r-l0 FR('1=1-(1-"(.,J2 1)(1-FR,(,1I r-l0 r-10 =,.(.,J2 I+FR,<'I-"C .,J2 IFR,C'I where '1' is the standard normal distribution function. 3 Fillurc 7.8 Exercise i .2. Consider the structural system shown in figure 7.9 consisting of 3 tensile bars. Assume that this structural system can be modelled by the system shown in figure 'J.8 and let its strength be as calculated in example 'j .3. Determine the probability of failure Pr for til!:. I:~'st~m when it is loaded with a sing:le force S '" 7.5 k:-, Wilat is Pi when S .. 10 kN'?'
- 132. 120 '. REtl.-BILITY OF STRUCTURAL 5'ST£.I$ ., ,.' . " Filun 1.9 P:uallel ductile systems are relatively easy to deal }Yith because of the simple relation (7.5) between the strength R of the system and the ~trencths R;. i '" 1.2. , ..• n of the individual elemenJ,s . • - parallel system with perfectly brittle elements is shown in figure 'j .10, IC:m element in such a system tails. iu load.bearing capacity is compl.etely .. e~~a~.5te~ .~he ,~ther 'elements mayor may not be able to prevent the g)'stem from failing'cO'mpletely .• ~y_ redistribution o[ the loads. In real structures' with a low degree of statical indete~;;;inacy, the . brittl;. iail~re· of one element will usually result in the subsequent i.lilure of other eiemet:!'r;; because of this.redistribution. If this is the case, the system behaviour is like a series system. ,However, Cor structures with a high degree of statical indeterminacy and a relatively high safety factor the system in figure 7.10 may be a reasonable model because in ;;,uch situations there will often be enough reserve strength capacity to c3.ITy the load after a brittle element (ailu(e. Let r t' r2, ... , I", where r < r~ < .. , < rn, be the strength of the n elements shown in figure 7.10. The strength r o( the system is then given by-r" max(nr1, (n -1)r2, .... 2r,,_I' rll ) (7.8) It can be shown that under cenain conditions r is a realisation of a random variable R which for large n approaches a normal distribution. .." - ' Fi~re] . ln ----
- 133. ;.3 FC~OA:-'IEST_l,L SY5TE:-'IS 121 ., s Failure mode 1 As mentioned above. iailure of a statically indeterminate structure can 30metimes be evaluated on the basis of a. number of f;illure modes where each (ailwe mode is modelled by a par.ill.el system. ~lodelling of the complete struCture will then be a series system of ;larallel subsystems as shown in figure 7.11. )i'Ote that a given elp.ment may appear i~ several ftilure modes . • ~Iso note that correlation in such a system can appear at least in two fo:ms. namely by corre.: lation between single elements and correlation between failUre modes. 5ys.tems with correlated elements or correlated subsystems will be treated in some detail in chapter S. In the next section. iundam~ntal sys~~ms with equally conel.ned elements will be disct!SSe:~ : . 'Example 1.4. Consider the statlca.lly iOcteterminate'~ with 3 panels shown in figure 7.12. Assume that' only the diagonals 1:2 •...• Ii can fail. This structure can thEm be modelled by the system shown in figure 7.13. ~ 1 - 3 • 777? f'jpre 7 .l~ , Fi1ure ~,'3 3 , 6 ,, , r-
- 134. 122 '. RELIABILITY OF STRUCTURAL SYSTE.IS 7.4 SYSTEMS WITH EQUALLY CORRELATED ELEMENTS Exact calculation of the probability of frulu!", for a I!"jven system with correlated strcnl!ttl..~ is usually not· possible. Howc'er, bounds for tn", probability of failure ca~. o!.~en be determined. This will be shown in cnapter S. However. in some Important cases it is.possible,to calculate the exact probability of failure [or the fundamental systems introducedjnsef.t.~~n 7.3. This has been done by Grigoriu & T':1tkstra for series systems and for parallel systems with ductile elements, on the assumption tliat.the strength pf the el~rn,ents can be modelled .. bY.no.rmaily distributed random variables, Ri . i ;:; I, 2, . : .. n, which are equally correlated with a common correlation coefficier:..t p-: Furt.her. it is assume-.:l that the loads are deterministic and constant in time and all elemen~ are designed in such a way that they have the same re.!i~bi1ity index Pe. Let the strength Rj of element i be N(/1 i' OJ) alld Sj the load effect. Theon 0' /10-80 p =-'--' e 0i . (7.9) If the coefficient of varia~on Vi "" Gi//1j is inse:ted into (7.9) this equation can ~e rewritten (7.10) The assumption of euqaliy coneJated elemen ... is relevant for sl~;m'e"stru'ct~res:: but for a great number of structures such an assumption can.'1ot be justified. However. it is of great advantage to use this assumption because the exact probability c~'th~n be calculated.,: B~aring this in mi~d it seems worthwhile investigating the. po~ibility of using a kind of »average)) coefficient of correlation in the general case where the co:relation is unequal. Such an investigation has been performed for parallel syS'"..ems Yiith ductile elements and for series systems. and it has resulted in two new methods for calculating-appro=?mate valu~s for the pr~ba1?i1ity of failure for such systems (see cha!lter 8). In this sectia:J"a brief !lresentation of the work by Grigoriu & Turkstra will be given. For a series system with n elements it has been shown by Stuart that the probability of failure on the assumptions mentioneo abo'e is given by (7.11) where 4' and <; denote the distributio~ and'demity function for the'standar_~_~_~!-l3s;~m random variable~ The variati,q~. of ;he probability of failure Pr with p is shown in figure 7,14 far n = 1. 2, 5, and 10 and::,:= 3.0.
- 135. ~A SYSTEMS WITH EQt.::ALLY CORRELATED ELJIE::"'TS . ! {I.012 I i !;:. -3.0 i --- I,~ ! I ,I ~ i I I '-.... I 'F=-C ~."". I .., ~. OlO {I.OOb (1.006 I '.'-l ~ 0.004 0.002 (r.000 0.0 . 'Ffrun! 7.14 ' '1 . o. o Ficurt ~ .15 0.2 0.' 0 .. ' . .. 3.0 ~ •• 2.11 0.' 0.6 0.' n-1'/ ! , 0.6 0.' 1.0 P 1.0 '.'1": ~ote thal. as expected. the probability of tallure Pr decreases with the correlation coefficient p and increa~es ~dth the number oC clemel]ts n. A formal reliabiiity index Ps for the aeries $~'stem can now be calculated by (7.12) The dependence of PSIP t on the correlation coeificientP for ~t - . 3.0 and Pe.- 2.0 and for n -1. 2. 5. and lOis shown in figure i.15 (lake~ from the paper by Grigoriu & Turkstra). ". . ' ., . I· · " i _
- 136. 12-' i. :tELlA8ILlTY OF STRUCTURAL SYST£,lS It is seen (rom fi gures 7.1-1 and 7,15 that the reliaoilit}' oi a :reries syst~~ is .incre~d i(the dependence betwi!en the strength of the elements is increase~. This result could be used in some design situ;l[ions • ior e:<ample. by stipulating that all members of a steel structure are irom the .iame balCh. Next consider a parallel system W{cll /I ductile ,demenrs having normally distributed strengths Rj identically distributed N(/J. IJ J and ~a[isfying the'~ame assumptions as above. According to equation (7.5), the strength R of the p~allel system is equal ~ ~ Ri • with f.I E[RI = f E[R,I-n" 17.131 i_I and .: n .L'· ' n . ,). . ,," . VarlRJ-- IVar/Rjl:+ p I tVar{RjIVar(RjJ),~. :::>na~+ n(n-l)p.a! ;-1 . i.j"1 ,,(7.H) . . ~ . where p is the common correlation coeifi<:ient, Let the load. on the system be S and the com· man element reliability index ~ ... Then. according to equation i7 .9) s - IS" ; =n.u-nPt!O' ;"1 ~=EIRI-S= . , and the reliability inde:.. i1S {or the system is , " (Var(RIl2 n/J -(n/J -n~ea ) _11 l n 1 - ~" 1 - pin Ino! -to nln-Uo!p)l 11 j' f'f{l'j' O.0015(j*;----,----,----,------;----;:n"~.rl-__, O.001001-..:...'c-~+~-_,..,.j~_:_--;----+_;;"'"'_!_-1 1I.000:!5,----------'-=,...<::..--77"'--------' O}.OOOOIl'_--=======--:...=""::::::=:-------::---:---::~~ 0.0 O.:! '1.-1 'J."; .-'., 1.0 JJ'. f'igur. i . Hi 17.15) (7,16)
- 137. 7.-1 SYSTD1S YolTH EQUALLY CORRELATEO nE:IE.'iTS 125 , .. ; :,,, : Figure 7.17 The variation of the probability of failure Pc :: '1I1-I1S) is .;hoJwn in figure. j .16 for n .'" 1. 2', S. and 10 and 3, .. 3.0. Note that p( increases with the correb:ion coe(ficient p ano de<:reases with [he number_of elements .tn >: 1). The dependence betwee-n :~e ratio Pe/:lS and p i.ishown in 11~re 'i.1'i"' fa'r n" I, 2, 5, 10, 100, and - . ~!i e:q)~~t~~d' i~-_~~ increases whe.ii" p :5 decreased or no is incre3.S«i. Therefore, the probability of failure ior n..:h a parollel system with conel:ned elements is underestimated if independence is assumed. This is in contrast to a series system. : .. Example 7.5. Consider again the structure in iiglue •. 12 modelled,by the system. shown in iigl.4re 7.1S. With the same assumptions as above and the further a.s..Sumptioru that all diago· nals ar~ equally loaded and behave in a ductile ·manner. the probab'i1ity of f3.ilure for the system can be ca1c:ulated. . The reliabili~Y' i~de,~ IIp fo~ 'a single pMel with' two d.i:l..gonals is, in accordance with equa· ti~n I? .It;~, given by . . 17.17) The · p~obability'. of failure P, for the system can now ~ calculated tram (7.1l) with n - 3, :Lnd;'e replaced by:s • and p reptaced by Pp' i.e. the ~orrelation coefficient.between the ~trength of the panefs. - , The correlation coeCCicient,op can be 'determined in the following ..... ay. Let Rt:md R2, be the strenJths of t~e diag~nals in a panel with the strength Rp := Rl + R2. Then Et Rp J '"' 2;.1. ':uiRpl .. 2a" + 2po" :. 2.(1 + PJo 1 and. :'O" .Rl' R2 ' - EIRt ~21- EIR11 EIR;!' '-'. . ' I J f7.1B) L~t R~ and R~ he the strenKths of twO panel~, Th!!l (7.19J
- 138. ,. .. ',. ..,, ' ,. ... 126 i. R!::LIABILln· OF STRUCTURAL SYSTEMS Finally p_.E IRP1Rfl2 1-EIRp_F _4(._oa: _-11 '1-4u: .. ..=2t;J. ... p VarlRpl 2(1+p)a:< . . 1+.0-" (7.20) The probability of failure Pf for th@system is therefore .;._ · .. ap+.JP;t] I P,' I - I"{ ,fI=P.. )j o{t)dt .__ I-pp (7.21) where Pp is given by (7.17, and Pp by (7.20). It is obvious from example 7.5 that the only calculation problem in estimating the:'probabm. ty of failure for structural systems satisfying the assumptions mentioned above is tlJ;e integral in equation (7.11). For De - 3.00 values of Pr are tabulated in Table i.1 for n :: 1~' 2 •.... 10 and.o = [0 ; 1.001. These values have been calculated by numerical integration using Simpson's formula and 200 intervals. 1 2 3 4 · " 7 8 , 9 I 10 0.00 13.50 26.98 40.44 53.89 67.31180.72 94.11 107.4 120.8 134.2 0.05 13.50 26.97 40.41 53.81 67.19 80.54 93.86 107.1. 120.4 133.6 0.10 13.50 26.95 40.35 53.iO 67.01, 80.27 93.48 106.6 119.8 132.8 0.15 13.50 26.92 40.27 53.74 66.75 79.88 92.93 105.9 118.8 131.7 0.20 13.50 26.88 40.16 53.32 66.38 79.34 92.19 .104.9, 117.6 130~ . 0.25 l3.S() ' 26:83' 40.00. "53.02 65.88 78.61 91.20 -103.6 116.0 ·128.2· 0.30 13.50 26.76 39:80 52.'6'2 65.24 77.68 89.93 102:0 113.9 125.7 0.35 13.50 26.67 39.53 52.10 64.42 76.49 88.34 99,96 111.4 122.6 0.40 13.50 26.54 39.18 51.45 63.39 75.03 86.39 97.49 108'.3 119.0 0.45 13.50 26.38 38.74 50.64 62.13 1 73.26 84.05 94.55 104.8 114.7 0.50 13.50 26.18 38.19 49.64 60.61 71.14 81.13 91.12 100.6 109.9 0.55 13.50 25.92 37.52 4S.H 58.79 68.66 78.11 87.19 95.93 104.4 0.60 13.50 25.60 36,69 46.99 56.66 65.78 74.45 82.72 90.65 98.26 0.65 13.50 25.20 35.69 45.28 54.16 62A7 70.30 77 •. 71 84.76 91.50 0.70 13.50 24.70 34.48 43.26 51.28 S8.69 65.61 i2.11 78.25 84.09 0.75 13.50 24.07 33.01 40.87 ' 47.94 54.39 60,34 65.89 71.10 76.01 0.80 13.50 23.28 3}.24 38.05 44.07 49.49 54.44 59.00 63.47 67.23 0.85 13.50 22.26 29.05 34.69 39.56 43.89 4i .i7 51.32 54.59 57.63 0.90 13.50 20.89 26.27 30.58 34.19 37.33 40.11 42.62 44.90 "47.00 10.95 13.50 18.91 22.49 2: ':oI2:.~0 29.~6 30.87 32.29 33.56 3'4.72 11.00 13.50 13.50 13.50 1 .... .,0 , ]" . .,0, 13 . .,0 13.50 13.50 13.50 13.50 T3blei.1. PrY-lOa after(i.ll)whh;ie " 3.00
- 139. BIBLIOGRAPHY Example 7.6. Consider t.he same stnlcture as in exa.mples ";.4 and i.5 and lei p '" O.S nnd D, = 2.85. By equation (~;17) ihereliabilit.~· index .ap ·for the panels is . ~ • . ., '_ . .;'; ; . '. and tne correlation caeCficient Pp between the strength of the panels is pp"2p/{1:+:pl. ~ 1.611.~". 0 .. 889 .. ;. The probability'of !ail~re'P( 'for the system is then luse table i.1 with n = 3) Pc:::> 25.88 . 10'" (7.22.1 Exercise 7.3. Consider the statically determinate structure shown in figul'e 7.18. Assume that this structure can be modelled by a series system with 7 elements satisfying all the as· 's~mptions mentioned above. Let the reliability index 0" (or the eleme~ts be O~ • 3.00 and the correlation coefCicient between the elementli p - 0.85. Calculate the probability of failure Pf (or this syst.em and compare Pr ~ith the probability of failure for a single element· . . " . , .. : Figure7.l8 BIBLIOGRAPHY Ii .11 Daniels. H. E.: Tile Statistical Tileo')' of the Strength of Bundles of Threads. Royal Slat. Soc .. Series~. yol. 133.1945. p. 405. [7.21 Grigariu. M. & C ~ TUr~strit : Safety of StructU'D/ Systems with Corn/Dted Resistances. Applied Math. ~iodelli~~,' Vol. 3. 1919, pp. 130-136. 17.31 Stuart. A. 'J.: Equally Correlated Variates and the Multinormallntcgrol. J. Royal Stat. Soc .. Series B, Vol. 20 .• 1958, pp. 313.~78 . . 17 .41 'fhoft-C~r;!>tense". P.: Fundamentals of Structural Reliability. Lectures 011 Structural Reliability led. P. Tho(t:Christimsen). Aalba~1 University Centre. Aalborg. Denmark. 1980. pp. 1~28. (7 .51 Thof ... C.h.risten5e~ . P. & J. D. Sore~~en: I.!eli~iI.it:·_ of Structural S.'5tems with Corre· lated Elcment~ Applied ~l3thematical :Iodclling. Vcil :- 6. i982.
- 141. 129 Chapter 8 RELIABILITY BOUNDS FOR STRUCTURAL. SYSTEMS 8.1 INTRODUCTION In chapter 7, the concept of modelling of structural systems by series and parallel systems was introduced. It was shown that in general the exact determination of the probability of failure of such systems is not possible and that a numerical calculation is often rather time-consuming. Howe~;r. upper and lower bounds for the exact probability of failure can often be formulated, ~lJt the pradical value of such bounds depends on how narrow they are. In this chapter two· ~:~ts of bounds will be derived, namely the so-called simple boun.ds and DiUeusen bounds: For systems with equally correlated elements, the probability of failure can be calculated exact· ly for a series system with n elements. This can have some important practical applications. It is shown in section 7 A that. under these assumptions, the probability of failure is given by (7.11) . where t1e is the common reliability index for all n elements and p the common correlation coefficient between any pair of elements. (n the same section 7 A, the following relation between the reliability index Os for a parallel system and the common reliability index t3e; for the n ductile elelfolents was given as _ / n t3s - (Je:' 1 +- pen 1) (7.16) The assumptions under which (7.16) is valid are specified in section 7.4. For a system with unequal correlation coefficients it is now natural to investigate the approxi· mation achieved by using the average correlation coefficient p in connection with equations (7.11) and (7.16). This has been investigated by Thoft-Christensen & S0rensen. They also suggest a very simple method to get approximate values for the probability of failure for a series system based on an equivalent correlation coefficient p. These approximate methods are presented in sections 8.4 and 8.5.
- 142. 130 E. RELI .... BILITY BOl":-:DS FOR STRl:CTI.'RAL SYSTE.'IS B.2 SI~tPLE BOUNDS H a system has a I!reat. number 01 ejem~nt.s it is convenient to use Boolean variables I~ rl!~::::...tthe state of the ele'!1en~. It is beyond the scope of this chapter to give a:d~tailed int~~uction to Boolean algebra. but all that is needed here is to assume that each element can exist in oniy one of t.wo slates, namely lI[ailufeJI or JOnon.failure", and to associate with element i. j ': 1. 2, .• . . • n a so-called Boolean variable (indicator function) Sj defined by 5,. ' { 10 if the element is in a non-failure state if the element is in a failure state Further. it. is useful t9 define a set of Boolean nriables Fj by F. -1-5. _ {O , ' . 1 if the element is in a non-failure state if the element is in a failure state (B.1) (8.2) Consider a system With n elements. By associating the state of each element by a variable Si' j "" 1. 2 •. , ", • n, the state of the ~ystem is determined by a vector (8.3) and can be described by a so-called s:,'stem (u.nction S~(Sl. The system r.unct.i~~ ',«;self is a Boolean '. ,'arlab!e de.fined ·by ' . . " .. ' . " if the system is in a non-failure state if the system is in a failure state for a series s,vslcm one gets by analo~' ",-jth "quation (7.3) that . Ss(S) - 5182 ... Sn .. i151 . j-t or (b~' definition) Sst>')" l-S " s (S) - 1-ll(l-F,) Note that . {O . Ss(F) - . 1 j-: if the sys~i!m is in a non-failure state if tke system is in a failure state For a parallel system tke system functior. is given by Ss(Sj-1-fl(1-Sj) i-I (8.4) ' .",. ·(8.5) (S.S) (S.7) (S.B)
- 143. S.2 SI~IPLE BOUNDS 131 or " SsIF)-II F, . ;-1 (S.9) Return now to a single element i. From the definition of S; and F; it follows lha: the eXre<"ted 'alues EfSjl and ElF;] are associated with thr. probability of failure P(F; ·1) [or element i. in the (oliowing way (8.10) and (8.11) The probability of {ajlure Pr for the system is ~en by analogy (8.12) As mentioned earlier. determination of the probability of failure PI for a system on the basis of equation (8.12) implies the calculation of I"l-dimensional multi.inte~a ls, However, very simple bounds for Pr can be derived lor a series s~'stcm with positive correlation between the stren~h of the elements. Assume that 1 ... 1 ; p(n Sj -l» PISj.q :::l)p(n Sj " l J (S.131 ;-1 .j" l for aliI" i "" n -1. The condition (8.13) expresses that the probability of non·!.allure for the system consistin:;! of the elements j .. 1. 2 •.... i..l...1 is greater than or equal to the product o~ the probability of non·failure for the systr-m l·{'nsisting or the elements j .. 1. 2 ...•• i and the probability of non-f(iilure for element j+llsee figure S.l), j-l "n1 ... . S-1) ,-, ,
- 144. d. RELIABILITY BOllNDS fOR STIiUl..'Tt;RAL SYSTE:l.IS From (S.13) follows that " " p,' P( U F, = 1) = I - P{ n S, • 1)' j-l i-I " 1 - P(St = l)Pt S:! - 1) .•. P(Sn .. 1) .. 1 - 11 (1 - PIF; .. 1)) Ci~arly .. Pr > max P(F," 1) i"1. n . ~ . . ;-1 .. : .. The upper and lower bounds Cor series systems (8.14) and f8.1~) can be combined " " mtL'< PIF; = l)"';; Pr " 1- fl(l-P(Fj -1)) i-I, n (8.14) (8.15) (S.16) Example 8.1. Consider the series system with two elements s!:lown in figu~e 8.2 :md assume that the reli:iliilit)'" index iJe .. 3.00. Funher assume thnt the equation 17 .11) can be used (or estimating the probability of failure. The probability of failure for each element is Bounds for the probab!lity of f:iilure p( for the system are according to (s:Hi) .0.00135';; PC" 1 .-11 - 0.00135)~ 0,00135 ';;; Pc < 0.00270 and-are in good a~meilt with table 7~1. ri~ure Ii.:!
- 145. 8.3 DITLEVSE:>: BOUNDS 13:3 The lower'bound in (8.16) is equal to the exact value for Pc if there is perfect dependence . between all elements (the correlation coefficient between any' pillr of elements is equal to 1). The upper bound in (8.16) corresponds to no dependence between any pair of elements (see equation (7.3». For a parallel system", simple lower bound :lIld a simple upper bound can be constructed by 3CgUing that perfect correlation between all elements now corresponds to the upper bound and no correlation between any pair of elements corresponds to the lower bound. Therefore. . : " llP(t'; - l),,' Pr-< " min P(Fj -1) (8.18) j-1 i-I, a Example 8.2. Consider one of the panels from the structural system shown in fig. 7.12. This panel is modelled in ligure 7.13 by a parallel system with only two elements. In exam· ple 7.S and 7.6 the reliability indu iJ? lor such a panel is cakulated assuming that the reo liability index for a single element is oJ", .. ~.85 and that the correlation coe(ficient between the elements is p .. 0.8. One gets,Jp = 3.0.0. The probability of failure for each element i, Therefore, by (8.18) ' " I ',' O.Oo.219~ " Pc" 0..00219 0' 4.8· 10~ " Pc" 0.219· 10.-1 . "(8.19) for 0 " p " 1. In the p~icular Cas<! w,ifh p '" 0.8 ,~~e gets 8.3 DlTLEVSEN BOUNDS The simple ~,unds prcs,ented in chapter 8.2 are usually rather wide because theY,correspond. respectively, to perfect. dependence between all elements and no dependence between any pair of elements. Several other bounds have been suggested in the literature: In this chapter a brief derivation wiU be given for the bounds ,by Ditle·sp.n for series systems. These bounds are ve~' narrow, especially for correlation coefficients below 0.6. It follows from equation 18.61 that 5lF) = 1 - 5 t 52 .!. , ~n "1-SlS2··· Sni l +,"IS:? " Sn_1Fa , 18.20)
- 146. 13' s. RELIABILm' BOUto:DS FOR STR1;CTURAL SYSTE"IS Hence. in accordance with (8.121. the probability of failure Pr i~ It is easy to see that (8.22) for i" 1.2,3, ...• n -1. B~' inserting (8.22) into (8.21) and bearing in mind thatal) probab· i1ities ace ~on·negative one gets the following upper and lower bound~ {or a series system " " Pr ~ l"PfFj ... 1) - I max P«Fi .. 1) " (Fj -I)} (8.23) i-I j-2 j<i .. and n . • 1-1 . Pf > NFl'" 1) + ":max[~(~i " i)- ,EP«Fj "1)(' (Fj - l)),.oJ j-2 j-l (8.24) The numbering or the elements may influence the bounds (8.23) and (8.24.1. Therefore. to get the best bounds one has to choose from the different possible numberings of the elements. In section 8.5, the bounds (S.23) and (8.24) will be used to detcnninc approximate methods to estimate the !>robability of failure of series systems. ~.4 PARALLEL SYSTE~IS WITH UNEQUALLY CORRELATED ELEMENTS In a real structure modelled by a parallel s!o.·stem with n e~f!1ents. the correlation coefficient between the elements ..... iIl csuaJl~: not be equal. Very simple bounds for the probability of failure (or !uch .. sySlem an,; presented in section B.2. equation (8.18). valid when the corre· lation between atly pair'of elements is positive. Unfortunately. these bounds are rather wide becau!l' thP:: CO:Tt'$pond to equal correlation bet-ween any pair o~ elements: na".lelY.D ~. 1 for the up~r bound and p = 0 [or the lower bound. . In chapter· 7 A parallel systerns'with'n ductile elements ..,,;ere inveitigated uivier the .roli~wing . assumptions 11) (2) " , . ', ., the loading is determ·j·nist.ic and constant in time . lnestrenih it:, j ~ 1. 2 ..... n of fue members is identically normalh' distributed I ,.. .. .' :1.IJ. 0 i (31 a:: elemenH are desi~~rH~d to have a common reliability index 0. (4) common correlation coefficient p between any pair oC elements.
- 147. I 8.4 PARALLEL SYSTEMS WI~~~ UN£!QCALL Y CORRELATED El,E;MESTS 135 .I. ~;·lder these assumptions it. was shf ..... n that the reliability index;3$ for the parallel !'),slem is related LO the reliability index~" (or the elements by (i .16) I) (i .16) Now the assumption (4) above will be relaxed. The correlation coefficient bet""een element i and j .. ill be denoted Pij and the con:esponding correlation matrix.C is defmed by P12' , .. 1 Pn2' :..' ... 1 (8.25) The reliability index 135 !o~ such a parallel system can now be calculated in a similar way as used in deriving (i .16) in section 7.4. One geLs P EIRI-S . s " 1 - (n/J - (n/J - nfleo»)tno· + a: IVulRJi" where {", . _ 1 P- ntn - 1) ,. " I " Pij ij"l j.j (8.271 ~~Y comparing (7.16) and (8.26) it is ,seen that rOt systems wit~ n,on-equal correlation caeffi· 'cients the reliability index ~s can be calculated by t.he simple expression (7.16_1 simply by in· serting .for p the auerage correlation c,?cfficiellt p defined by (8.27), P is the a'erage of all plJ'l f j. This result is interesting because calculation of the probability of failure f~r such a system can easily now be made for any correlation matrix (8.25), One anI:.' needs to make a set of curves, as shown in figure 7.16, once. E:tllD'lple 8,3. Consider a parallel system, with 5 elements. as shown in fi!!Ure 8.3 . • • ,. .. .. <= , 3 ,.
- 148. 136 S. RELIABILITY BOU~DS fOR STRUCTl:RAL SYSTE~IS Assume Ihat the a~sllmptions Ill: (2) and r 31 abo ... e ure fulfilled, with Pe '" 3.50 and let the .:orrelation matrix be l 0.5 0.2 0.1 ;,] 0.5 1 0.5 0.2 .'. ~o~ 0.5 0.5 0.2 0.2 0.5 1 '0.5 :.- 0.1 0.2 0.5 1 The average .:orreiation coeificient is The reliability index for the system is therefore ". '·c.. - I 5' j:'-. Ps .· 3.50.y 1 + 0.28 '; 4 - .;:).38 fS.2S) Exercise 8.1. Compare the exact result (S.2S1 with the upproximate results obtained by . :lSSllmiiig'nocorrelation Pjj '" 0 and p~riect correlation /.Iij · 1. S.5 SERl.ES SYSTE~,IS W.ITH UNEQUALLY CORRELATED ELE~lENTS In this section series sys~~mswit.h unequal correlation between their elements are considered. An exact determination of the probability of failure Pc for such a system will, as mentioned earlier, imply extensive nu~erical calculations of multi-integrals. In section 8.2 the following very simple bounds were'derived for a series system with n elements " " max PIFj '" 1)';;; Pf ';; 1-111l-P(Fj '" IJ) (8.16) iOo1. n i'" where P(Fj '", IJ is the probability of failure for element i. ~luch ~e"iter upper'and lower bounds - a~e .the Ditlevsen bounds 18.23) and (8.24). In some special cases. the exact probability of failure Pr can be calculated relati'ely easily. This has been shown in section 7.-1 under the following assumptions: f 1) the loading is deterministic and constant in time d:) . the strength R,. i · t, 2, ... ,n of the members IS id~nticallY ncirrilaJly disrnbU[ed ~(p., (J) 13) all eleml:!nt5 :.lfe desig-ned to have a common fl:!liability index:3~ I..}) common correlation coefficient p hetwt:>cn any 'pair of elements. t"nder Ihl'st,l assumpt'lons the probability of failurt:> P:. Li~i'enby !i.ll) r'" S, • ..:... ,/fj t n d Pf - l-J 1'/'1. II :10 t __ vl-p (i_Il)
- 149. :J.a SERIES SYSTE:IS WITH USEQU.~LL'i CORRELATEO ELE:IE.'ITS 137 Assu~pti6;' (4)'· will now be relaxed~ tn secd~n 8:4 "It is shown tha't use of the average · ~~~lat·ici"~l coem~ient p, deCio'&! by (8.27·j; gives the-co·n'eet v':JJue oC p( Cor:l pM-..tUci system with d.~ctile:·eiem~nts. ·it is 'therefore·natu"ra.I to·in ... estigate't~e ·Use of(j :IS an" equivalent correlation coefficlent and then use equation 17.11). Note that p "" P when the assuin'p"tlon (4) is valid! The use oi p has been tested with·a series system, with the· following simple correlation matrix c-r 1 Pl2 0.2 0.2 (8.29) PI:! 0.2 0.2 0.2 0.2 1 0.2 l02 0.2 0.2 1" For this example approximate values for the probability of failure con be calculated from !7 .11) with P equ:l1 to the a'erage correlation coeificient P defined by (8.27). The result is shown in figure 8,4 for n - 2, 3. 5, and 10. and J3 - 3.00 as the curves ® . t The upper and lower.-oounds t8.23) and (8.24) by DltJevsen have been calculated for some values of P12 and are snOvn' i~ figure'8_~ as in~e~aI5 @) _ The·Cu~es CD are upper bounds corresponding to in~ependeIlt elements (equation (8.16)). The·curve$ ® are approximations based on the so-called equivalent.~o~!ation coefficient P defined Inter in this section. P'i. O.Ol~ ,r= ==="=-='=O==========:::;;;;;;::;;;;;;~~~::::;.: :::;r: :==/,-(@j)- 0.012 T -----l... _ @~ 0 .010 T , O.OOB t n'"5 /'!> ~-------""---------------~ .. ~~~========~~ 0.006 - ---'1:3Q o.oo~ t).002 0.000 ~-----C":-:':'~----~o-------~----........o e==~----~C~lj ~~ -- -----"":-",--...,,------------------------"'-----------c5' "':"-'-.-'--@1: -Z"; 1 0.0 0.2 0' I 0.' 1.0 Pt2
- 150. 138 6. RELIABILITY BOUNDS FOR STRUCTURAL SYSTEMS It. appears from f~. ;. re SA that the DiLlevsen bounds (S.23) and 18.24) are '"ery narro ..... for n '", 3 and n .. 5 ~ F;r n".; 10 "so.me gap is disclosed. As P, f~r P1 ') ". 0.2 is" caJcuiated e~actly using p';' P', the lo'''' .. e~ bo:~nd (8.2."'1 is apparen.t1y ,:er1' c'lo~ to ~~!! exact 'alu,e for Pc for P12 in t~e. neighbouri'.ood of 0.2, .. - - .. .. .. .. .." .. .. .. For PI2 > 0.7 it is clear from figure 8.4 t.hat u~ing D-as an equivalent. correlation coef(icient is on the safe side. The curves ® do not decrease as fast ll$ the correct. values of Pr for P12 approaching 1.0. To obtain a better agreement a modification of P(Dl must be used. A mu~h better ~ement can be obtained by using.an equivalent correlation coefficient pin· directly defined br 18.30) where Pr.2(Pmu) is the probability of failure calculated (rom equati.on (7.11) with n" 2 and P ., Pm:u: t where ~ p 2.00 I 2.50 3.00 I 3.50 4.00 4.50 5.00 X10-3 I ;"}o .... x 10"" hno-s X1O" X1O-' X1O-' .0.00 44.98 1:3.81 26.98· 46~52 . 63.34 . , 67.95 ai.33 0.05 44.82: 1~3.63 . 26.97 ' . 46.52 63:34 67:95 57.33 0.10 44.63 1~3.39 23.95 46.51 63.34 67.95 57.33 0.15 44.40 1:3.08 26.92 46.49 63.33 67.95 57.33 0 20 44.13 1:2.68 26.88 46,47 63.32 67.95 57.33 1 0.25 . 43.82 1:2.18 26.83 46.43 63.30 67.94 57.33 1 0.30 43.46 1Z~.57 26.76 46.37 63.27 67.93 51.32 10.35 43.05 120.82 26.66 46.29 63.23 67.91 57.32 10 42.58 l1e.90. 26.54 . 46.18· 63.15· 67.88 57.31 .40 0.45 42.05 I!S.Sl 26.38 46.01 63.03 67.82 57.29 1 0.50 4.1.45 U"7.50 26.18 45.79 62.86 67.72 57.25 10.55 40.77 IJo.95 25.92 45.48 6~.59 67.56 5i.17 , 0.60 40.00 11~.10 25.60 45.07 62.22 67.30 57.05 0.65 39.13 111.92. . 25.20 44.53 61.67. 6.6.90 56.83 0.70 38.14 1~;34 ' 24.70 43.81 60.90 66.29 56.48 0.75 37;00 105.25 24.07 42.85 59.82 65.37 55.88 1 0 .80 35.68 ~ ~ 102.51 · -23.28 41.58 . " 58.27 63.96 . 54.91 10.85 34.10 g-;.91 22.26 39.86 56.07 61.80 53.31 . ! 0.90 32.14 P.~.Ol 20.89 37045 52,80 58041 50.61 ( 0.95 29.48 &3.73 1S.91 33.77 · 47 .. 5.4 52.61 45.68 I i 1.00 22.75 6:.10 13.50 23.26 31.67 33.98 28.67 Table 8.1. Pr after!"j .11/ 'ith n = 2. - -' ..
- 151. I 8.:; SERIES SYSTL1S lTH t:~EQt:ALLY CORRELATED ELE~1ENTS PmlX .. rna.: .:oj; i,j"l.11 - j.j 139 IS.31, Note that Prtc) gives the correct value when all Pij' i:t j, are equal. Prep) is shown on ligure SA. as the CUNes @.:; and it can be concluded that in this example the values of Prep} ar", clOSfl to the (lowerl bounds, but a little on the unsafe side. It. II very convenient to us@ approximate valul!S lor Pr based on Prep) or Prep). because one only needs: a table giving Pr as a function of nand p defined by (i .11). Then, for any correia· lion matrix C, approximate values cs.n easily be calculalecl by hand. ' To radiik'ie the use of equation (8.30) Pr.:! is tabulated in table 8.1. Example S.4. Consider a series system with the same 5 elements as in example 8.3. Let De • 3.00 but let the correlation coefficients be unchanged from example 8.3. The average correlation coefflcient is ,.; P -0.28 An approximate value Pi for the probability of failure'Pr is ~~en by e~l,:,ation (7.11) (use table i.1 or figure 8.4) Pc - 65.5 • 10""- I ' · .;:W ... ; ' • ! . ~, ,, • It can beshOWll that the Ditlevsen bounds are 63.8' 10'" < Pr < 64.2 • 10~ so that using p is on the safe side in this use. The same conclusion is true when p is used. One @:eLsby(8.30Ithattuse table 7.1 or table 8.1) ,:. : ... , ... ,; '''! .~i .. Pr(D) - (Pr,2(P J - Pr:2(Pmax JJ .. Pr(0.28) ~ (P~.~(~·:~.8) ~ P~,2(~;·5» - 65.5· 10'" - (26.8' 10....1 - 26.2' 10-4, .. 64.9 • 10 .... Enmpie 8.5. Consider a series system with 10 ductile elem,ents. Assume that the assGri1p':"-' tions (1),(2; and caJ' on page 136 are fulCilled, with Ii. - 3.00 and let p::: o.S and Pmu • 0.9. from equation fl.ll) (use table 7.1) the (ollo ..... ing appro:<imate value Pi Cor the probability o( (ailure can be determined Pc - 67.23' 10 .... The appro~.imate value P; based on the equivalent correlation coefficient p is then fuse table 8.1) .. .. 67.2' 10'" - (23.':: . 10-4 _. 20.9' 10'" , ... 6-1.$' 10""-
- 152. 140 $ . ItHIAHILITY BOl':>;OS FOR ::'''TRl"CTURAL SYSTE:IS The :iimple bounds for this I!xamplt! can be calculated from lS.16) ~th P:(Fj ::1) -13.5·10"'"' for i :: 1. 2 ....• 10. One "ets 13,5' 10"" , Pr '" 1 - (l- 13.0 • 10 ..... )10 13.5 • 10'" G;; Pc";: 13 .. 1.2 . 10-1 The approximations PrC;;) llnd PrIP) have been evaluated by Thoft·Christensen & Sorensen by ex· tensive simulations. A great number of correlation matrices C have been generated randomly as- 5uming the single elements of C to be unifl)rmly distributed in the inter'alIO: 11. ~on-positive definite m~lrices· ~ere of course rej@cted in this testing. Th~ results-of these simulations have been compared ~ilh the Ditl~vsen bo~nds a~d· it can be conCl~·ded· ~~a~, using the equivalent correlation coefficient gi'l!S approximate 'alues close to these bounds. Cnfortunately it is not possible to l'onclud~ that the use of Prtp) nnd Pr(PJ is on the safe side. A finlll iIIusu-.1tion o( the different methods available to evaluate the probability of failure of .. series s.v scems is maue for a system with the [oHowing corr-e,i:,l:t-io. n matr.i x c - r~. p p' p" 1 I p "n-l ,j C p 1 ~n~~j .' ... .-, ., pO~l po:":! . : 1 I J 18.32) The results 'for;Ie '. 3.00 are shown in figure 8:5 for n • 3, 5 and 10. The' u'~per bounds PrtO) are the curves CD . PrIP) afe the curves ® and PctP) the curves ® . The u'pper and lower bounds 1;i), ·Ditlevsen·are shown as intervals' 0 -. In this case the curves-ior Pc';; I and PclP) are rather close and lhey differ only slightly from the=upper bounds. Note the great im!'rovement obtained by using PrtP) or PrIP) instead of pc! 0), where P(IOl corresponds to assuming independent dement:!. With reK::ud to the Ditlevsen bounLis. it .:an be concluded from figure 8.5 that the)' may be rather wide. especi.uly for correlation coefficients greater than 0.7 and when n increases, For small correlation coefficients the bounds are narrow and are therefore of great imponance in evaluolting the probability o( failure (or series systems. For 3. st3tic:illy indeterminate :ltructural system. a failure mode C:ln often be rnodelled by a parallel system. Llsu:llly sever.11 f!lilure moO~s exist !Inci the total modelling·will be 3. series ot' p.::lr.llle) S)'sl.:>ms as shown in I'g. '; .11. In usinJ: the uppcoximate methods presented above for series systems. knowledgj:!: .OLlh..: rel1a1jir[~y-i·~c..I~x) for ~:lch p:uallel system is required and. :·unh ..... lbC'-l'orrela'tfQ';l -between the par.:lil~1 systems must be known. Xote that the methods aho·t> art> :):lsn:i on tht' !1SSumpt ion !h:n :.III p:.lrJII~1 syslt'ms ila'e the same reliability index, This wili USU:lily 11<.)1 he the l':lse. To t,)'t'fCome the difficult}' it may lJe useful to use an equiL'o/enr reliabiliry illd.:.-r. ~ det'int'J by
- 153. o.!> SERIES SYSTEMS 1TH l.'SEQl.'ALL'{ CORRELATED n£:L!c;STS 18 .33) j ... when .. 'I> is the distribution !uhction for the standard norm ill distribulion ilnd ':i lht rdil.bili~y index Cor element.!. Xott:" ihallj ,,:orfltsp(;ond; LO assuming thal ~he parallel syslems ure mutu:!l· Iy independent I .. ee 17.3»). Tt-{ere(ore: using)) is on-the sa ie side.
- 154. S. RELIABILITY BOUNDS FOR STRUCTURAL SYSTEMS Example 8.S. Coniljer a series system with 5 elements satisfying the assumptions (1) and (21 on pa~e 136. be: omitting assumptions (3) and (4). Let the reliability indices for the five elements be .e1 '" '2.50, P2 ·3.00. P3 .. 3.20. 114 - 3.40. ~5 '" 3.50 From (S.33) the C(!·.:.h:alent reliability index ~ can be calculated using a table lor •. One g." <}(- PI ., 1 - E - 62.10'10'" ) (1 - 13.50'10'" )(1 - 6.87'10'" )(1 - 3.3'7-10'" ) (1 - 2.33·10 .... ! - 0.991202 and p:: - 4>-1 (0.991202) " 2.37 Further. assume tha: the correlation matrix is the lOame as in example 8.4. The average correlation coefCicie:tt is then P : 0.28 and by numerical integration an approximate value Pi for the prooobility of failure can be calculated. p' .. 1 - "' 1.:.(2.37 + v"O":28 t)Js op(t)<:lt ... 43, 10-3 r _ .___ ... /1-0.28 . . Example 8.7. Consi:er a series system of two parallel systems with n1 and n2 elements. respectively. as sho~ in figure 8.6. Further assume that the IOtren~hs of the elements are identicall~' distribult-:: with the com~on , mean Il and standard deviation 0 and that the correlation coefficie=, between any pair of elements is p. Finally. assume that m elements are common to the t"-'o par:allel systems. In this case the coefficient of correlation Ps be· tween the two paral-=lsystems can be calculated in the following waS. Let Rl and R2 be the strengtns of the ~wo parallel systems. Then PR 1 :c n1 Il, PR:1 -- n2J' and rillure 8.6
- 155. BIBLIOGRAPHY 143 Therefore (8.34 .1 ~,:,~rcise 8.2. Det.ennine t.he coef[jcie~t. of corre,laUcn Ps betwe~n two pa,rallei systems with n1 ::I: n2 ~ 4 elements and m. 2 common elements, Assume that the elements have laentically distributed strength and common correlation coefficient p ::·O.B. BIBLIOGRAPHY (8.1J Cornell, C. A.: Boima.s on the Reliability of Structural S)'stems. J. Struct. Oil'.~ ASCE. VoL 93, 196i, pp. 1il ·200. 1F.,2) Ditlevsen, 0.: NarTOfL' Reliabflity Bounds for Structural Systems. J. Struct. Mech" Vol. 7,1979, pp. 435 - 451. (S.3) Hohenbichie:o.).L 8; R. Rackwitz: Non-normal Dependent Feetors in Structural 5otery. J. Eng. Mech., ASCE. Vol. 107. 1?81. (8.4) Rackwilz. R.: Close Bounds for the Reliability of Structural Systf:ms, Berichte tur Zu-o ve.~l~.ssigkeitst.heorie der Bauwetke, SFB 96, H,eft 29/19;8, LKI. Technische Unh'crsitat ~Unchen_ .(8.51 Thoft-Christen~n. P. & J. D. S0rensen: Reliabilit:o'. of Structural Systems with Com!tated elements. Applied Mathematical Modelling, Vol. 6.,1982. " . ',. - --:. :.---- .~ . -._--
- 156. "' ;, ""J ," "'; ' " • ' " ' ; "" " ;' "r:, "
- 157. us Chapter 9 INTRODUCTION TO STOCHASTIC PROCESS THEORY AND ITS USES 9.1 INTRODUCTION , In the preceding chapters, loads' and strengths have mainly been modelled by random vari· abies with associated distribution [unctions. However. a load 5 on:a given structure ;"m usual· Iy be time-varying S(U. The function SUI is stochastic lrandom) in the sen;e that the value of :; :at a pven time t is an outcome at a random variable. In this way, by modelling the time history and the r::mdnmness at a physical quantity by an (infiniteJ numher of random variables . .3. socalled $(ochu"ric process is obtained. In section 9,2 a more totmal definition of this concept wil1lJ~ I;l:iven. but it is not possible to give a detailed treatment of .he theory of stochastic pro· cesses here, Only the most rundament~1 n!=Hions will be introduced and only one special t~-pe of stoChastic processes will be described in more detail. A v~ry impo~t problem in re'iation to a stochastic process is the barrier crossing problem • . Consider. for example. the response of a struct~re e:<pressed by the rime·history of a given " stress, Whe~ moci'eUing the'time.ilistory of the stress by a stochastic process it might be'of interest to· evaluate th~'probaHility that the pro'cess stays within spe'cified bounds during the expected lifetime of the structure, This problem will also be briefly examined. 9.2 STOCHASTIC PROCESSES .-S mentioned above a stochastic process is an indexed set {X(tJ, t';; T; of random ,'ariables X/t). where all XCt) are defined on the same sample space n. ~ote thilt two different kinds of ':uiables are involved. namely the s,tOChaslic variables XU) and the varioble t, here caUed the inde", The i/lde:., st-l T is t~~pically 3 ' time.interval, but can'be -any kind o(finite ~et. a couritably infinite se, or.3. subset of R. For the sake of'simplicity t vill be assumed in the following to be the 'ari· able time. Th~ probabilistic struclure 0[':1 stochastic process can be pescribed in a way similar to random 'l'ctOTS. If the index set is a finite set then the stochastic process forms a rancom ·ector. The fnct that a stlXhastic procl'SS is a set 0; random variables make~ it natural to describe its ~robabil· I . istic ~tructure in a way similar to r:andbm vectors, but in this ClSe the index set is infinite. , I .. ,
- 158. 146 ,r",r-~-~' '- vq'iv(--VVv!.:. X(l,J-;-- ---- I I I 1 I I [. 1 :] . , ',',' ~ '. I ,I I tl I t~ I I I ,I ~ --- 1 l 1 I I .t T Fillure 9.1 xC'. 'i • I • • > realizatlon~ , • • • • • • • • • • Figure 9.2 For a fixed sample space n the outcome of the set of stochastic variables form an o,rdinary fu.nctioncalled a realization ,(~ee firure 9.1). A realization might be thought of as the outcome - '- -, . of an experiment. If th,e experiment is repeated, the new realisations will not be the same as in figure 9.1, but its p;obabilistic contents will b~ the same. In figure 9.2 values of X(t1l are , shown.for a number of realizations and also ~he associated de~s~ty functi~n fix} (x; h). Given two instants 'of time tl and;'2 some correlation ,between x(t1) and ::dt,2),wlli usually. exist, especially when the time.cJifference ltl - 121 is small. This is taken into consideration through the joint distribution function F {x :.rx} " x2 : L1" t 2) defined (see. (2.58)) by (9.1) This joint distribution function for arbjtrary.(t1., t2 J E Tl .is called the joint d~bution (unction of.order 2,The_ corresponding joint density f~!lction of order 2 is given by (9,2)
- 159. Tne definitions (9.L and {9.21 c:m easily be ~('nc!";lized 10 probability funct.ions of any oruer n. n == a. 4 •... In descrjbin~ a stochastic process the followin~ functIons lof time) are of ,:!rent interest. Tne mean value function Ilx (t.) is defined as the expected value of X(t) 11 X (t.) =EIX(tli c r .. xf{x :{x:t)cLx (9.3) The autocorrelation function Rxx (ll' t z ) is equal to t.he following joint moment of t.he ra.n · dom va.riables X(l)) and XCt!!) RXX Ct) :-t2 .1 - EIX(t1 lX(tz l} ., J ~_ ~ ~ _ x) xzf {:q(xt • "2 ; t} • t2 )d"l dX2 (9.4) The autocol)tJriancc function Cxx ftl • t 2 ) is the covariance of t.he random variables X(tl ) and X(t!!) Cxx (t1, l2) - E[ (X(t) ) -Ilx fll »(X(t2) -Ilx (t2 )1 c RXX(tl' 1,:) - IlX(L) )Ilx H:) (9.5) By setting t) = t2 • t in (9.5). the loar;ance il/nctiol! a~ ttl of tne random variables xm is obtained (9.6) Finally the autocorrelation coefficient PXXltl' t:) is defined in a similar manner to (2.80, by (9.7) For an important. J!TOUP of stochastic p;ocesses all fi nite dimensional distributions are invari:mt. to a linear translation of the index set. Tnis can also bt> expressed by the statement that. all di~· tributions are invariant to a t.ranslation of the time origin, Such processes are called strictiy ho· mogeneous or when the index parameter is time, $lrict1~, $latiollOry. When this invariant assumption only holds for distributions of order one and two the process is called weokly homogeneous or weolll~' slationar)'. In the follo'"ing, the wc:'rd stationary will be used in the last-mentioned meaning. An-'mpo:t.-'lnt consequence of the assumption of stationarity is that f{x }(x ; t) and F{x}(x : t) be· come$ independent of t so th.;.t t'.'~ cap_~_~it reference to t. Further, the second-order distributions (9.ll will only depend on the djfrere~ce -of ~I,t: ~:~<!~x_p.~!~meter r '" tl - t 2, The same is trut for all tht> other statistics mentioned above. _. _-
- 160. 148 9. INTRODUCTION TO.STOCHASTIC PROCESS THEORY AND ITS USES In practical applicatiol,ls, the modelling of a physical quantity by a stochas~ic process must often be based on a single realisation of a stationary Pf'?Cess. If only one realisation is a~. hMd it. is natural to estimate the mean value in the following way 1 rT Jl 2 if J x(r)dr o (9.8) If this time average approaches Jix for T ..... - the process is said to be ergodic in the mean value. In the same manner a process is ergodic in correlation if R(T):: T ~ T ~T-1'" x(t + r)x(t)dt o (9.9) approaches Rxx (1) for T - .... If this property holds Cor all moments, the process is called ergodic. Note that stationarity is an assumption behind the definition of an ergodic process so that any ergodic process is stationary but not. vice versa. 9.3 GAUSSIAN PROCESSES . In this section so-called Gaussian processes u.re treated. It has been stated several times that a linear transformation of a set of Gaussian (normal) random variables result in a new set 0'[ Gaussian random variables. This important property of Gaussian random variables is the molln reason why they ore used for modelling whenever it can be juseified. In a similar manner, it can be shown that linear operations on a Gaussian process results in another Gaussian process. A process (X(t), t E T) is Gaussian it the random variables X(t1l, X(t2l •... ,X(tnl nre jointly normal for any n, t 1, t 2 , .•. ,tn' The probability density function for the corresponding n·dj· mensional nth order distribution is then given by (see (2.89)) f{x){Xl'····Xnit.l·····tn) · (9.10) where C is the autocovariance mat.rix CXX(',",) ......... CXX:".'O)] CXX(tn , t.2) ........... CXX(tn• tn) (9.11)
- 161. 9.3 GAUSSIAN PROCESSES .I. .... and Mij Is the i. jth element in C - I • It is clear from the definition (9.10) that a G:mssian process is completely determined by the mean value [unction Jlx (t) and the autocovariance function Cxx (tl • t2). Therefore. a stationary Gaussian process is always strictly stationary. An important property of a Gaussian process (X(t)} is" that its derivative process (X(t)) is also a Gaussian process. Let x(t) be a realization of (X(t)} and let x• (t ) "(fdt x(t) (9.12) be meaningful. The derivative process {X(t.)} is then determined by the realizations x(t) when almost all realizations :t(t) or (X(t)} are considered. Ex.ample 9.1. Consider two independent nonnal random variables Xl and Xz with Jlx = JlX2 "" 0 and ail · a?cz = a1 • Let a stochastic process {XCtl} be given by ) X(t) "" Xl cos(wt) + X2sin(wt.) (9.13) where w is a constant. The random variables X(~). ti E T are clearly jointly normal and their ~tatistics are dc:ermined by the mean and autocorrelation of the process {X(t)}. By (9.13) "x(t) - E(X(t)] = 0 and by the definition (9.4). RX1 Xz (e l , t2) ;; E[(X1 coswtl + X2sinwtl )(X1 cosw t2 + X2sinw t2)} "" E[Xllcoswtl coswt2 + E[X2]sinw tl sinwt2 :: 0 1 COSW (tl - t2) since E(X1 X2) • O. From (9.15) oi (t) .. Rxx (t, t) - Jli (t) = 0 1 (9.14) (9.15) (9.16) The process {X(t)} is therefore a stationary Gaussian process with zero mean and variance cr'. Example 9.2. Consider the same process {X(t)} as in example 9.1. The autocorrelation coefficient is _ !lx. (t!. t.) _ _ PXX(tl,t2) 0x(t 1 )a X (t 2 ) -cosw(t1 t 2) so that the joint distribution density function is given by :x:~ - 2x1 x 2cos...,,. + x; 2a'Cl cos' ... ,) (9.17) (9.18)
- 162. ;;: .. ' . . ;: 150 · 9. INTRODUCTION TO STOCHASTIC PROCESS THEORY AND ITS USES An important property oC the autocomlation Cunctlon Rxx(r). ofa .. ~4-tionary stochastic process (X(t)) is the following. If Rxx(1") has 2 second deliva.live RXx(r) which is contin.u(u~at ~ a ,O then the derivative process O{(t»). defmcd by its rcalizations by . (9,12)~ . ,is also 8 s~~i.onary .'. . . stochastic process. An~. it ~an be sry.own that (9.19), and (9.20) so that thcre is no con-elation between :X(t)} and {X-(t)}. Further EIXI" ~ E{X). Example 9.3. Let (X(t)} be a stationary Gaussian process with zero mean. It follows then from the rem.,rks abOVe that the joint density function f {x} {xl is (9.21) 9.4 BARRIER C~OSSING PROBLEM In this section it will bt' shown for a stoch.astic process (X'~)} how the number of crossings of a given barrier (threshold I in a given time·i.,terval can be e5timat~, The p~esent.at.lon:liere is in accordance with the book by Lin. Figure 9.3 shows a realization x(t) in the interval It, ; to) 1 of a st~chastic process {X(t)} and a constan~ barrier x(t) ... ~. The number of upcrosslngs or ~his barri~r in the'time intetvaJ [t.; '; t2 J is foU:. In the following an upcrossing will be called a posi. Hve ptu;sage and a dOl"ncrossing a negative ptu;sage, , x{t) 1--111o"-----::-+-+---++-%(t). t
- 163. 9,4 BARR~E'.t C~pSS.H,jq P~q~.~EM 151 , To so:~~~ th~,~~~bl,e~ .?~ ~sl~mali~~ .t~~. ~F,=te.d number .of pos,~?'e passages?: ~ given barrier the so<:alled Heauiside step function H is a useful too. Heaviside's step (unction H is defined bi':(~ee ri~~e: 9.4) I :r ' J1 for x<O H(x) .. for x • 0 (9.22) l ~ for x>O ' :' .. By lot{Dai diUerenliation of the function H one gets the so-called Dirac della function b(x}. oS (x) is not an ordinary function in the sense that a definite value ,can be assigned to every x, For o1lr purposes, it can be defined by oS(x) _ lim. __1_ e-z",· . ,_o..J"2n£ (9.23) What is required here, is only the property that integration o( 0 (xliivc' Hix), For a stochastic process {X(t)) and a eh'en baerler x(tJ = ~, it is then convenient to define a - . :._. ';1'. '! ,.-; . ' . • ' new stochastic process {Y(t» by or Yet) c H(x(i)'" If ._ ' , . y(t) ~ H for x(t) < t for x(t) .. t II for x(t) > ~ l ~t_H_"_) ---'' 1' ! ;igUTt' 9..1 .. . , .... . :. , (9.24) : " .-;!:, (9.25) .. ,. '- '
- 164. 152 9. I;.lTRODl.'CTIO~ TO STOCHASTIC PROCESS THEORY AND ITS USES · i'" YU) 1- I I .;" Flture 9.5 ~y formal differentiation of (9,24) the derivative process {~} can be determined by (9.26) where the existence of X(t) is assumed. For a realization X(t) of the process {X(t)} the carre· sponding realizations y(tl and yet) of the processes {yet)} and ('l(t)} are shown in figure 9.5. Note that the realization },(t) consists of a number of unit Impulses. A positive unit impulse cor· responds .to a positive passage of the barrier and a negative'unit impulse corresponds to a neea· tive pass~ge of the barrier. These impulses are unit impulses because integration of y(tl over one impulse must yield + 1 or -1. By counting the number of such unit impu.l.;es in the time interval I tl ; t2 J the total number n of crossings of the barrier XI t) .. ! is obtained. This can also be formulated in the following way .1.. .1., ,n(t; " 1.t2)= 'Iy(tlldt- -IX(t)I'6(Xlt)-~)dt 't1 " · t l ·· ·· " . ;;, . (9.27) -----_._--
- 165. SA BARRIER CROSSING PROBC.E~I 153 From (9.27), the number n(tttl,,,Z) of crossings or passages of a ~~~~ barriercan be calculated for any realization :c(t) of the stochilstic process {X(t)} :'Such a set 'of numbers can be considered the outcome of a random variable N(t, t 1, t2). The e.xpected number of crossings con . now be determined • .. , ... E(NI!. ',. ',)/ : , ' E(IX(.)I . . , (X('),-Hid. - ," ; .' . ~ 11 . . . • L,. ... - lill.i(x-~)f:d.(x.xtt)dxd.idt:::: • 11 ~ . -;- .-.. • t~ ." • . - l.ilfxx(t. X; t)didt ~tl ," - .. (9.28) where fxx .. f{XXx} is the joint density function for {X(tJ: and ~X(t)j. It is convenient to consider the rate of crossings per.unit time "r instead of. the number of eros· sings ~ in the time interval considered. N' and N are related in the (oHowing way N'~, t 1 , t 2) - t2 N'(~, t)dt (9.29) . " EquatJon (9.28) can then be written ~n the more simple form (9.30) Now assume that the stochastic process (X(t)} 'is stationary so 'that ex..x is independent or the time t. Then (9.31) ~o th.a the expected rate of crossing per unit time E[N'(t, tJJ is independent of time, but of course dependent on the banier ~. The expected total number ot cto~ings in the time interval Ill; t21 is therefore (see (9.29)) (9.32) L'ling equations (9.31) and '9.32). the number of crossings of the barrier ~ is determined, i.e. upcrossings r positive crossings} ;is well tiS downcrossingst negath'e . croS:5~gs}. But for a stotion: l...~' stochastic process it is re;i$onnblt! to assume :hat any positive crossing is followed by a. negl!.- t;','e crossing. Thereiore j ,9.33)
- 166. 1~4 9, I!'TRODUCTION TO STOCHASTIC PROCESS THEORY AtJD ITS USES where N~ (t) is the rate of positive ttossings of the barrier ( and !'~ U, is tne pte of negative crossings of t.h,e barr,i~r ~. Note that. for p6~it.i'e crossings:i: > 0 so t~~t ~rom (~ .30) (9.34) and similarlY for negative crossings. Hence, for stationary processes. the joint,density function fxx is an even [unction in the variable i., The fundamental formula (9.34.> is ealled Rice', for. milia, Example 9.4. Let {X(t)} be a stationary Gaussian process with zero mean. The joint density function fxX is then given by (9.21). From Rice's formula (9.34) (9.35) For ~ - D.the expected rate of positive zero crossings is E[N' (0)).1-''2>. .,. 2:; Ox (9.36) Example 9.5. Consider a statiOnalY non·Gaussian'process {X(t)} with the (ollowing joint. densit.y function fXX (x, x)" , Fit:1m: 9.6 r 1. (1 ~ x) 1I :2( 1-X) for (x, x) E r~ 1: 0] XI-I: 11 for (x, xl E [0: 1) X [-1; 11 (9.37) othervdse .....
- 167. 9.01 BARklER CROSSING PROBLEM ! 155 The expected Tate of positive crouings of the barrier x(tl" ~ is given b,· (9.3ol) I . ; E:~~(oI'I>t(!=Odx.{t::~:: ::: ~~~'~:o o otherwise (9.38) Example 9.6. Consider the same stochastic pro"cess {X(t}} :~ in enmple 9.5. but in this case the joint density funcLion (9.37) is approximated by a 2-dimensional nennal distribution in such a way that t.he two marginal density functions ha'e the .. same means and variances. >~~~:~~~,a~od:;:I:~a!Unctions for the distribution (9.37) are shown in 'figure 9.7. It ~x""x"O ai -i The approximate normal distribution is therefore fXX (X,X).,3f! e-3)J1~-1 .5x l (9.39) (9.40) The expected rate of positive zero crossings for the corresponding stationary Gaussian process is E[N~ (0)1' 2 1 , :;;. 0.2251 (9.41) -'----+-I- --1.-> 1..··_ _ x' . ./f.. . . I ~ -1 . Fia;ure 9.; ElI:etcise 9.1. Approximate the joint density, function (9.3i·) by a 2-dimensional normal distribution in such a war that t!1e Tate of positive crossings of the ba'rriers to::O and ~ .. 1/2 is equal for the corresponding stochastic processes. (Answer fxxtx. xl a 0.56 e- 2.;;)JI~ -1.12 i
- 168. 156 9. IXTROOl."CTION TO STOCHASTIC PROCESS THEORY ~:-;D ITS USES 9.5 PEAK DISTRIBUTION The results derived in section 9.4 can be used to investigate the statistics of the peak distribu· tion of::l stQCh:l.Stic process {X(t}}, because peaks or troughs (e:urema in (X(t))) occur when the stochastic process {X(t)} has a zero crossing. The number of zero crossings of {Xft)} is equal to the number oC extrema in {X(t)}. The formulas derived in section 9.4 can there· fore be used when ~X(t)} and iX:(t)} :lre substituted for {X(t)}and {XC;)}. ;. When the process {X(t)} IS 'a narrow.wnd Gaussian process the distribution of the peaks can be chherm'i~ed in a.very si'mpie way.'A r~liz;'ti6~ ~'( a' ~:mow.banc!- ,~~oc.~ss· is' sh~.vn in figure 9.8. It. is similar to a sinusoid, but. the amplitude and phase are slowly' varying. The stationary response of a lightly damped' linear syste'rri wiil'often'be narrow.banded. when the input procesS is a broad-banded G;sU:I$ian process,"such as an eaithquake excitation. In this case the expected number o[ pe3ks abo;e the level U~ > oj per unit time is. with good approximation, equal to the expected rate of crossings of the'blirrier t, Le. equal to E[X~ (tl]. SimiJar.ly. •. the expected total number oi peaks' per unit time is equal to the expected rate of zero crossings E[N ~.(O)/. Therefor~, (he expected relative number of peaks above ~ per unit time is E(N~{nl - ~1~:x EIN~ lOll' , whete the formulas (9.35) :lnd (9.36) h·.il.ve been used. , , The distribution function F:! (n for the 'peak ma~itude (t > 0) is then given by t' -2"x F,CO- 1-, nod the density function fxCt) by t' f:;:(~) ""!r- e -2~~ 0<, ~ < .. x This distribution is the so-called Rayleigh distribution_ 19 .. 2) 19.43) 19.-14)
- 169. ~ :~ PEAK DlST~IBUTIO~ 157 Example 9.7. Let {X(t)} be 3. narrow·band Gaussian process with zero mean and let ' O'x ,. 1. The density function Cor the'peak ma.,anitude is then given by (9~44) _t~l . f:z:(n" t e (9045) LU :to .~. - , .. : ri, ... ,,,, 9.9. Density (unc,ion (9,45). Example 9.B. From an experimental investigation of the variation in bending moment with time in a ~ven section of a beam it is concluded that the moment can be modelled by a stationary narrow-band Gaussinn ,Process {M(t}} with ~M ::: 12 MNm O"~I" 2)'-[Nm P'~I - 0 MNm/see The rate of positive crossings of the barrier ~ ,: 18 MNrn is then (lS-12e .... ErN' (18)1 .. J:... 4.01 . 10-1 e- 2'4 _ 3.5-1 . 10-1 sec-I . +, 2:r 2 , The density function for the peak magnitude is given by _...L u - a )· f:!(~) .. t(~-12)e 2'" and the prob:ability of gettin~ peak magnitudes greater than ~ = 18 MNm is - -L'18 - ' 121' P{~ > 18) .. 1 - Fz (l8) " e 2'4 .. 0.011 (9.46) (9,47) (9.48) In the derivations above. only narrow'D:anded processes are considered, i.e. processes where the r:aLio ,0' 'exoectetl numher of zeto cros;;inl!s (l .. expected number of peaKs 19...19) , is approximately equal to 1 1 see iiPire- 9.8}. It" cin"be'shown that the ratio a lies between zero
- 170. 158 9. INT!,O.Q.U.~ON TO STOCHASTIC PROCESS THEORY AND ITS USES and one and that. in general the density function f;::(n for the peak magnitude is ~ven by (t t' 2C1~(l Q')+ ~(1+erft;L(5-_2)-t)le-2aX 2aX ax a where the error function erf i$ defined by erf(x) _ ~ ex e-t' dt /11 ~o Exercise 9.2. Show that (9.50) for 0: .. 1 is equal to (9.44). (9.50) (9.51) When 0: is very small (o: "" 0) correspond~g to a large number of peaks in rel~tion to the number of zero crossings thldormula (9.5'0) can be approximated by a normal distribution f ... (t) ot:- __, _ e- 2",, ~ - .,j""'E Ox (9.52) Example 9.8. Consider ~n: ergodic Gaussian process {X(t)} with ~X =- 5 and ax" 2. By analysing a realization of this process it is concluded that 'the ratio Ct of the expected num· ber of zero crossings to expected number of peaks can be set equal to O.S. Further, the expected number of crossings of the barrier ~ '" 9 is equal to 104 • The probability of getting peak magnitudes greater than 9 Is then .. , P(P 9) - 1 - '.(l)dl (9.53) '-- where H -5)' fxm - 0.lSO·e.- O.195(t -5}' + 0.075(t - 5)f1 + erf(O.265t»)e --.- (9.54) By numerical integration PU > 9) c;m' then be calculated from (9.53). Lower and upper bounds for P(~ > 9) can be calculated by considering the cases Q ,. 0 and 0: ::: l. For Q .. 0, one gets from (9,52) PU ~ ~) '" 1 ~ ~(9 ~ 5) _ 0.O~2i5 · andfora- c jrrOm(9.44) . _.1 (9'- 5)' P(~ > 9) :: e . 8 - 0.135 The ~ta.."ldard deviation ax for the derh'ative process can be calculated by setting' E(N~ (9» equal to t . 10-3 • One getS ax • 0.0';6. Thl! expected rale or positive crossings of any bar· rier can then be calculated from (9.35). ' .. - Exercise 9.3. Consider an ergodic narrow-banded GauS$ian process {X(t)J . By .analysing a reali:tation of this procl!5S il. is concluded that the expl!cted rat.es of positive crossings of the barriers t '" 0, 5. and 10 are 10-: . 10-' , and 10-5 • respecth·ely. Determine thl! mean and the variance for {X(t):' and ror {Xitf·. Sket!:h the de!"lsit}' {unction ror the peak ma!;nitude
- 171. I mBLIOGRAPHY 159 I and calculate the probability of obtaining peak values g-,'eater than 5. (Answer: P(~ > 5) c: 0.075)., BmLIOGRAPHY 19.11 .. [9.21 " ," "" ' ;" - -, ",:' .. , Crandall. S. H. & W. D. Mark: Random Vibration in Mechanical S~·stem5. Academic Press. N. y" 1973 . Krenk, S.: First-Passage_ Times_and Ext,:,em.es of Stochastic Processes .. Lectures on Strnc· . tura} Reliability (e~LP. ThoCt-Chri,sJensenl, Aalborg University Centre. Aalborg. Den· mark, 1980. pp. 75-98. [9.3] . Lin, Y. K.:Prob~.~ilistic Theory.of StructuratDynamics. McGraw.Hill, N.Y. 1967. {9A] Nielsen, S. K.: Strength- and Load Processes. Lectures on Structural Reliability (ed, . P. Thoft·Christensen), Aalborg University Centre. Aalborg, Denmark, 1980;pp. 39-73. [9.51 Papoulis, A.: Probability. Random Variables and Stochastic Processes. McGraw.HiIl, N. " .• 1965.
- 172. · ;, ,.
- 173. 161 Chapter 10 LOAD COMBINATIONS 10.1 INTRODUCTION The modelling of load variables Is treated briefly in section 3.0,. It is stressed there tha~ load variables and ather actions are typically time-varying quantities which are ~st modelled as stochastic processes. In section 3.5, it is al$o 5hown that when deaJing with a single fime-I.'o,)'ing load in connection with barrier crossing problems (see section 9.4) the detailed time variation is not ai relevance. This ls due to the fact that in such cases the distribution of the maximum value. of the loading prOcess in a given reference pcriode can be derived from the arbitrary-point-in-time distribution (see figure 3.13 on page 57). When the loading pro-cess is continuous then the probability distribution of the maximum value (largest extreme) is likely to be very closely approximated by one oi the asymptotic extreme value distributions. treated in section 3.3. In this way ioste.ad of modelling a single load variable as a stochastic process {X{t)} it is modelled by a stochastic variable,say Y (see'also section 9.5). Therefore. in reliability analysis, single load varlables imply no special difficulties. A number of examples in chapters 5 and 6 oC analysis and design of slml'.Je structures loaded by-single loads illustrate .' this fact. ..:'.-~'- When ~ore than one time·varying load vanableacts in'c~mbinGtion on a structure then the abo~e simplification cannot be used becaUse detenniitation of the' di'stribution of the combined load effect requires knowledge of the detailed variation with time of the individualloadine pro- " " ., :., ' ". cesses. This is· illustrated in figure 10.1, where realisations,p}.lt) and P2 (9 of two loading proces- ,ses {PI (t), 0.; t <; T} and {P:!:(t), 0 <: t <: T} are shown together with the sum PI (t) + P2{t), ,;, It is clear Crom figure 10.1 th.u the ma,.-rimum values of Pl (Ii, pitt) and Pl (t) + p~(tl during the:refert!llce Period need not appear at the same installt of time. For the specific realisation '''shown hete: the instants oftime tl ,'t~ '~d t3 for maximum of Pl (t), P2(t) and Pi (t) + P2 It) are all dirterent. Also note thic maximu~ i1ue of PI (t) + P2(t) is considerably snl'aller than the sum of the maxil"',lum values of PI (t) 'and P2(t). It is obvious from these observations that knowledge of the detailed time variation or' the' two ·Ioaciing 'variables in the reference period T is reo quired to determine the probability distribution of the sum onhe two load variables. Ther('fore. knowledge of the distribution of only the ma.:dmuJ;Il values of the individual loading processes gives insuificient information to evaluate the comaned dfect ~xactly_ I.
- 174. 162 o ~ T ' ,' "1(1I+ 1121U I I I I I I I I I I I .1 ... t, T Figurt 10.1 , ' ~ , .,' The intention of chapter 10 is to gi'e some Information on problems connected with load com, binations. Howeycr. a. thorou~h presentation of these problems is beyond tne sCope or this book. The reader is referred to the referenc;:es at the end of the chapter. The main intention is to give the necessary background for understandini the ideas behind an approximate method for dealing with load combinations. This method is very suitab~e fC?f use in conne~tion with the le~el 2 me~ods presented in chapters 5 amt~ . . 10.2 THE LOAD COMBINATION PROBLEM One of the fundamental problems in dealing "'ith time.varying loads modelled by stochastic processes is connected w,ith estimatio~ ,of the probability that the stochastic proce~~ defined as , the sum of the individual, processes cr.osses a given harrier (thrahold) during the .. Jel~n(.e period ~ T. More specificallY,let two IOllds.(or load c(!~cts) be modelled b~' sta"Uonary ~~·d ind~pendent . st~~astic ~"oc~es {X~ (t). 0" t'" T) and {X2(tl. 0< 1.< T} . . ln the following a1~ st9chastic processes.will have the same index set. so that l.he shorter notations {Xl (t)}. {X2(1.}}, etc. and {Xl.} ' {X2 1. etc .. c?". ~, used •. The com,bina~.ion problem can then be formulated .i,n l.he {oHowing way. What is the probability .t.hat the process . , . . (X(t)) • (X, (tl + X,(t)j (10.11 has a value larger than x(t) '" ~ during the reference period 0 C;;; t C;;; T? This probability
- 175. ·" I~!!· ~~ '~~A~'~~;1~,'~~:;t~~;; ~B~i.~~,'· :" .. ': .;..~ I ~ I I PrmoxXrtl>l. t~IO,Tlr= - P(X{O) > ~ I + P(one or more upcrossings of ~ I X(O) < ~) (10.21 where P{X(O) > 0 is the probability that the process {X(t)} h:ls a value greater than ~ lit t., a and the last. term in (10.2) is approximately equal to .-- Plane or more upc::ro,sing~ of t)· L P (n upcrossings of~) n"'1 . (10.3) The expected number of upcrossings (positive crossings) of a level ~ per unit. time ''for a sta1ionary ... process Is denoted E[N; (Hl in sect.ion 9.4. It is convenient to use a. shorter notation ~'x(n here. When £·X eu - EI N; en) is known, then the expected number of upcrossings in the time interval 10j TJ is equal to "x(U·T. i.e. ""- E!number of upcrossincsj'" "",(O'T" I n-P(n upcr05sings of tl (l0.4) .-1 It. follo's from (10.2). (l0.3) and (10.4) that P(maxX(t) > ~. t E [0: Tll" P(X(O) > ~) + l'XCO·T (10.5) In general P(XIO) > H « I'X (~). T and for most practical reliabllit~' problems "x(~ I' T <c 1. In such cases "x(~)' T is a gooO approxi~alion of P(max X(t) > ~, tEl 0: T]l. i.e. P(max (X(t) > L t E 10: Til'" t'x(U'T (10.61 The left hand side of l10.6) i~ equal to 1 - F :;:(t). where F~ It:- the dist.ribution function of the ma.'<imum value of the stochastic process {X(t)} in the time interval (0: TI. Therefore. (10.7) where "'x<t I' T < 1. By (10.;.1 the problem of calculating the distribution function F 1: for the maximum value oC {X(t)} .. -:X 10-) + X2tt») is reduced to that. of determining the rate of up· crOSSings (the expected number of positive crossings) "X(U - EIN~ em for {X(t.)). Unfortunately, exact expressions for "x (~l are only kno'n for some special kind' of processes, An obvious way of calculating "X(£) is to use Rice's fonnula (9.34) (10.8) where 'xx i~ the joint demity (unction for the prOCe5$ {Xft)} and its derh'ative process {Xn)}. The joint density (un.:t:on ex * can be derived by the l'G-caJled convolution integral
- 176. 164 10, LOAD CmIBINATlO:-'S (10.9) where f ..... x' and fv .; are the joint density functions for X" X, ana x~, X, ,'respectively, •. ... 1 1 •.. .. r .... ::! - N~te that,equa~ion (~0.9) is a generalization of the well·known convolution lnte~ in ele· .mentary probability theory. Also note that the first step in' calculating fx X is' to calculate (XIX I and fX:lX::I' This is in general difficult, but it has been:donefor some special stochastic processes, By inserting (10.9) in Rice's formula (l0.8) one gets VX(~)=~ xC ~. ··.f:<IXl(xl,Xl)fx':lX2(~-x~,i-i1.)dxldxldi • x-O • %1 --... %1 --... (10.10) (10.10) can be written in a more conve.~.ient form by .th~ .substitution i .. xl. ~ i2; (10.11) where the domain w in the Xl x:!.p)ane is shown in figure 10.2. In conclusion the procedure for evaluating the distribution function F% for the maximum value oi the stochastic process {X(t)} C {Xl etl"': X2;(t)} in the time interval [0; TJ is ~ .. - - .-~-.-..... _-- --------~~,--------.-"
- 177. 10.2 THE LOAD CmlBINATION PROBLEM (1) Calculate fXtx" and fX2X2 .f~r the two processes {Xl} and {X2} (2) Find "x(~):: E(N: ml by evaluating the integrals in (10.11) (3) Find an approximate expression for Fz from (10.7) 165 Step (2) above can only be performed exactly for special density functions. Usually numeriC1l integration must be used. However, upper and lower bounds for "X(~) can be derived by cha..'1g· ing the domain of integration (w) in {IO.l1)in an appropriate way. The upper bound is espe.:ial· Iy useful so its derivation will be shown here. The upper bound is obtained by changing tbe' domain oc'integration in the first integral on t.i.e right hand side from w to wI and the domain of integration in the second integral on the ripe hand side irom [,; to "'2' where'w1 and "'2' are shown ir.:·figure 10.3: Clearly, an upper bound of "xlH is then obtain~d (10.12i , ~vhere "Xl (x) and "x 2 (~ -~) ZIO-.Of upcrossings for the processes (Xl} and (X2j. The :nte· . grals in nO.12) are_m~chmo~ ~onvenient than the integrals in (10.11) because they only im'olve ~t~s of upcrossing of the processes {Xl} an~_ {X2} and the corresponding density fUnction3. It has been shown in the literature (~ee ,the,r~!e~nces at the end oi.chapter 10) that the upper ~ound . (10.1.?) is ~~ry.~lpse to the exact result. so that it ,can be used as an approximation for "xW. "x(~)""Jr_- _ ;x. 1 (xlfX 2 a. ~'x;dx-+JJ_-_ ~x 2(~'. ..".x)fX1 (X.)..~ x (ID.13) Fillllrl' 10.3
- 178. 166 10. LOAD COMBINATIONS Only :ne Stlm of tWO independent processes has been treated above. However. the same proce' dure ean be used for sums of three or more processes. For example. for the sum {X} of thre~ inde~ncient processes {Xl)' {X2l'and ' {Xl} one gets :' , . / .~ . ~ :x(t1O::,- I"x (x)fx X (i-X)+I'X {~)rx ... x (~ '-X)+Vx (x)Cx +x (~-X)J~~ • "'_ .. 1 2"'" 3 2 ." . ~ '., • ., I 2 . >- (10.14) , where the density runct!~ns f>;+ Xi are determ~ed as ~sua1 by the convolution integral r~x,-+ XliX) ::: ~: .. ,eX, (t.Hx,(X - t)d-~ (IO.IS) ~ (10.14' can cIISil).: .pe ge~eralize.d to sums of, mor~. than .three independent proce~s. ;', 10.3 TIlE FERRY BORGES ·CASTANHETA LOAD MODEL In this3e(tion a simple load model suggested by Ferry Borges and Castanheta will be presented. In thl5 model real loading processes a.re greatly simplified in such a way that the mathematical, probleI:1s connected with estimating the distribution function of the mnimum value oC 8 surq of load:ng processes are avoided.- Further. the Ferry Borges· Castanheta load model is very luitablt in connection with the level 2 methods presented in chapters 5 and 6. For ea~:' load process {Xi} it is assumed that. the load changes after equal so·called elementary: jlltcrlJ(J;'; of time 'ri ' This is illustrated in figure lOA. where the reference period T (e.g. 1 year) ,!S dh'ideci into hi intervals of equal length Tj - 'r/n j • nj is caBed the repel ilion number. Further It' is assun:~d that the load is constant in each elementa",· interval. The loads in the elementary i~Z ter ... als are icientically distributed and mutually independent random variabl~ with a density :0 lunctio:. (point·in'·time distributio~) fx~. This density function is shown as ~ c6,~tinuous ; : density function 'in 'figure lOA but itcan 'also be a density function of the mixed type (see page '22). Thi5 'is convenient if, foi exa'mple; iris desirable to have' the load value 0 with a finite proti. ability. Let the point.in·time distribution,for load process {Xi} be C~ and the corresponding < _ distribt::ion function F Xi then the distrib~tion of the maximum value in the reference period T is (FA; '"I. i.e, (see (3.5)) 1" o T F"il!'urt 1{1.~
- 179. I". 10.3 TliEFERRY BORGES·CASTANHETA LOAD MODEL "'''1' ' . f m,u X,(Xjl = (Fxt,,;))n, 167 110.16) "Thereiore: ~~r t)lis so-c"aJled reelangular PUlse:pr;ccss it is a simple task to e~"culate the dis· tribution of the " ~"a.~i!1U~ valu~ ,irt: tbe reference period T ."' "·.1" " When combinations oC"load processes {Xi};" {X:!l •.•.• {Xr} arc considered it is assimlE~d in the Ferry Borges ". Castanheta"load model that the loads arc"s"tochlUitically' jndeperi'd"cnt with integer repetition numbers nj'; where (10.17) "and where (Z+ is the set of po~itivc natural numbers) n1/ni_ 1 E Z. for i E {2. 3 •...• r} (10.18) .The ~onditions (10.17) and (10.18) are iliu~t~~ted i~ figurc "lO.5 ..... h~re r - 3 and n1 ,. 2. n2 '" 6 and Tl3 .. 12. Although the Ferr), Borges· Castanheta load model presented above is a gross simplification of """the 'real loading situation, experiences seem to verify that the model is capable oC reflecting the most important characteristics of load combinations. o ~ I I I I I I o T Flrur~ 10.5
- 180. 168 10. LOAD COMBINATIONS 10,4 CO.IBINATION RULES ' '' . r _ - .. :, " , • '; ' • • ' . ' '. ' • .1:> :- " ~', ,'" " . ;~! It ilas ~en empil:lSized earlier that two loading processes willusuilliy not reach their maximum value in a gi~'en rcierence period T at th~ ,same instant of time. It Is therefore too conservative to replace ffi3.1: {Xl (t) + .. . + Xr(tj} by max (Xl (tl)+ ... + max (X2(t.)}. On the other hand • . . ' .,. ' T, " . . "_ :'. T, ,. ".' _: ,_~ T. "il." i" , " , "~,,~ ' m~ {Xl (1) + .' . . + ~(t)} is :l ve'ry corri~~.ica~d:sio e hasti~ v:U"iabl~ t~ ~~~ in practice, so some kind of 3pproximation must be made. Using Turkstnzs ru.le, m.r' (X1(t) + .. ' + Xr(t)} Heplaced bY ','stochastic variables, namely .,;, (10.19) where t- i5 an arbitrary point in time. By this'rule the reliability ?f a structure is only checked at those point5 in time where the individual load pr~esses reach their maximum value. There, fore, the reliability of 11 stNCture will be oyerestim~ted. However, it has been shown that this overestimation is usually very small. :. more refined Nle has been formulated ,in connection with the Ferry Borges, Castanheta load model presented in section 10.3, tn this model the loading processes {Xl}' {X2}. :', . , {Xr} are rectaniU1ar load processes with "n{, °2, .. , • nr repetitions in the 'r~ference period T. where 01 C; n2 ';;; ••• < nt • FO,r r" 2, the rule gives the following 2 combinations for the loads: .. .. ,! " ' .. Combinau(]n I ;:':0. of repelitions vi loud Xo. I 1 2 1 I n, n2101 2 1 'n2 : ' ~: Table 1~>.1 For r - 3, the rule gives the following -l combinations: • ': !:! ' !':.I,:': .. r"
- 181. 10.~ COMBINATION RULES 169 I Combination ' No; or repetitions of load ._. - ! 'No. 1 2 3 r 1 "1 112/n1 ·, ." n3/n2 2 1 ". n3/n2 3 "1 1·_·· n3/nl 4 . 1 .1 ", Table 10.2 In general with r actions 2,":"1 different combinations of load have to be considered. Example 10.1.. Let the number of rectangular'puise processes be r · 3 and !~t the number of repetitio~s be n1 '" 3, ~2 z; 6 an,d :n3 :',';3.0, in the .referen~e pe~~d ~. }.ccording to table 10.2 the following combinations have to ~ checked: Combination 1: ::-'Jax13 rept. of XI) + Max{2 rept. of X2) + ~Ia.x(~ rept. of X3) Combination 2: (1 rept. of Xl) + Max(6 rept. of X2) + Max(5 rept, of XJJ ." Com.~~.atlon~: .M~.(~ .~~t .. o~~(~ )':~"'(~ ·[~p~.: ?f X2) "!' Ma~(l.~·~ep"<of. X3) : .;; .: ;~·pombin~tiqn 4: (1 rept. of Xl) + (~rept. of X2) + Max(30rept. of X3) Examples 10,2 and 10.3 show how. the reliability Inde;x p can ~.e calculated for a nructure loaded by r '" 2 time-varying loads ~adelled by Gaussian,rectan'gular pulse proceSJes, Example 10.2. Consider the indeterminate beam shown in figure 10.6 with two time-dependent loads PI (tl and P2(t). Let p(t)- PI (t) :'" P2(t). Let PI (tl be a real43.tion of a stochastic process {P1 (t)) and P1(t) of a stochastic process {P2(t)}. Funher,let {Pi (t)} be a Gaussian pulse process with J.l Pl (t) - 3 kN and op (t) - 0.3 kN and with nl • 1 repetitions in the design life (reference period) T - 1 year. '(ikewise,let iP2(t)} bea'Gaussian pulse process with ;.IP:dt) .. 2 kN 3lld O'P21t1 - 0.2 kN but with n2 = 12 repetitions in the design life 1 year. Realisations of the pulse processes are shown in r~ 10-.7, ' ~~_._...".,"",....., ______ .". _ ';"' _ '.J.t II(t) " Pl(t) - . P~(tl ~2.5m " Fillurf {1).1i
- 182. 170 10. LOAD COMBINATIONS ,. , 1/2 yen. 1 year 1 r'zel} . /' ~ 1/2 yelr 1 year The maximum value P2.fI'IAX of th~ load process {PaCt)) is then P2 max - max (P2iJ (10.20) ~ : • i~1. . . . . 12 _ ' . . ~ ; ~here P2i is the,load, l,eveJ i,n p~l~ i! Due to the ind'ependence o[ the _ pu~ and their identi. e:a1 Gaussian cH~tributions. the ~,!~tributi.on,·~tion ~X2 ~or X2 - ~2.~lU is given by . ~-~. ..... F (i.,) '" 41 12( ___') (10.21) X, • Op • 2 .~. ., " • :';ote that, X2 is not GalSsian.distributed •. fflere(ore. in connection with level 2 reliability analysis dr' design a transformation must be per-fonned., for example as shown in section 6A. By this transformation the distribution of X2 - P2.ml.X is replaced b~; 'a normal distribution with mctln ;IX, and standard deviation ax, where (see (6.37) and (6.38)) 2 ~(4;-'- 1 (4i 12(:i=!)i)' .. '. ax, = . ' ,' 0.2 x9-2 x9_2 "O:20 12'$11(-'--)c;(~) 0,2 0.2 , (10.22) (10.23) .... , -. . x; is the x::oCoordinate for the design point. ' .; , Example 10.3: Consider -the same beam as in example 10.2 and with the same loads. Further. lei the safety margin ~-i be given by (10.24) where the critical limit moment MF is 8 nonnally distributed r.mdom variable with J.lM '" ~ 20 ~N~ and 0r.:" '" 2 kNm; Introduce the random variable Xi - ~fF·-·~· Pl' Xl is normal,;' 1;:.- dlstnbutM With .. ; -... ~Xl =20-i.5""12.5kNm (10.25) aX, "' / 2:.a. (~ . 0.3,: .. 2.14 kNm (10.26) Tne safety <.largin can then be reformulated
- 183. 10.4 CO~tBINATIO=' Rt:LES 171 (10.27) ..... In the normalisecj coorcinat(' system t~e failure surface is th~n given by .r : • ,. .., (12.;•) -. -,,.'1 4 'xl' • --.,....( ~X,: !; + "o X;:'':.: ! 'J'oO (10,28) :. t" :.: ' - .' : The reliability index . ~ can ~ow be calcula~d by the same Iterative technique as used in '. ex~mple 6.iL With the usual notAtion' ' .. 2.511;" -12.5 ' . . . (J::I 2. • , (10.29) . 2.14 a1·· . . 2.tJ0x2. Q2 Q'] E -i-' 2.14 ° ,I 2.' 2 ....... k' ·;)(lx!! where (lX2 <md IJ'X2 nre given by (10.22) and (10.23) wlih xi -2 0.2 :.. · 1 stirt ·f-I-,--..,._I,,"'_"'_t_;o_n_N-,-°r·""·_'-:--I , , ....... :: , - 1 -'I 2 '1 3" 3.00 ! 3.87 r' 3.11 "'rs.li- -I .:-, -0.717;!I-o.989 ·,I-O.9g~:. ~~.~.9/· '. '0.717 i ",,o:'48! 0.141 0,138 ,-,' .." 2.151 : 0_573 0.439 .~:~:9 1 '0281 0:121 0.1 is 0.1191 2.31 j 2.31 1 2.32 2.32 L " (x: -2) I --o.z Table 10.3. The reliabi1ity index G/3 = 3.11: : - : ' (10.30) (10.31) (10.32) -, Exercise 10.2. Show that the reliability index for the structure in example 10.3 is is = 3.19 Ir the number of repetitiom"1 is equal to 6 (and not 12) but with aU other data unchanged. Examp1e 10.4. Consider the same structure as in examples 10.2 and 10.a: nie variation or the reliability index e with the number of repetitions n" for the load process {P.,(t)j is shown in figure 10.8. ~ .. . r: " -- -...
- 184. 172 10. LOAD COMBINATIONS :J' 3.:1 3.2 3.J '.0 ", 0 , 12 J6 20 Figure 10.8 Example 10.5. Consider :1gain the structure analysed in examples 10.2 and 10.3, but now the structure h.ts to be designed so that it has a reliability index tJ = 4.00. Let the critical limit moment MI" be normally distributed with unknown mean value ~~IF and standard deviation aM~ • .. D.l·IIMp-· All other data are unchanged. . " . The safety mar~in with Xl ,. ~IF' Xz ., PI and X3 .. PZ.max is (10.33) where Xl is N(/-IMF' , 0.1 iJ~IF')' X2 isN(3. 0.3) and X3 is"" N{/-IX3' aX3)' The formulae for the it~rative rro~css are . 7.5'+ 3 a2 + 2.5/-1Xa + lo"aXa a 3 /-IMF' '" 1 + 0.4a1 (10.34) 1 a1 = -"j(/-IMF' 0.1 (10.35) 1 0:2 =j{0.75 (10.36) 013 =t 2.5 UXa (10.37) and the iteration scheme is: Iteration No. St:1rt 1 2 3 /-IMp- 20.5 23.0 23.1 I .J -0.577 -0.928 -0.944 -0.944 ·2 0.577 0.340 0.307 0.307 1 " 0.577 0.152 0.124 0.123 ! 0.1:14 j I i aXa 0.122 0.120 0.120 I I I ! IlX3 2.30 i 2.31 , 2.32 2.32 i Table IDA. With P),fp- ., 23.1 kNm the reliability index ~ - 4.00.
- 185. 10.~ CO:>'IBI!'OATIO:-l RULES li3 - In the fin.al.example 10.6 iUs shown how the reliability index p can b~ ~:llculated fora struc. ture with 3 time· ... arying loads modelled as Gaussi.:m pulse processe'S. ~xantp'le ,10.6. Consider the simply supported ~am s ~own in figure 10,9. The beam is 'l()aded by 3 uniionnly distributed,time-dependent loads PI,(t), P2<'tl ,and P3(t.l. The Ferry Borges - Caslanheta load modelli~i j's used for the corresponding lOOld processes {PI ttl}, (P2(t)} and (PaCt)}. The mode1ii~g data are sJ;l0wn in t~ble ~9.3.:, The safety margin is M" "IF -t· 25(Pl + max (P2il + m3:~ 1.PSi O. ,. (10.3S) j·1, .... 6 j"I, : .. ,180 . where the critical limit moment )"IF is assume~ to be N"(l.2.50 kNm. 1.25 kNm). Note that in the last term in (10.38) the number of elemen.~a~ il!tervals:5 only 180 due to the fact that the load process {P2(tJ} Is onlYllSsume~ to be acth'e for 1 '2 year with n., :: 6 elementary intervals. " . - Equation (10.38) can be rewrit~n . " 25 M:: Xl - S max (X2i ~ rna.,," (X3i ll (10.391 j .. 1. ... ,6 1-1. .... 30 where. . 25. ' XI "';IF -SPI 15 ~(10.~375,1.397?) XZi is N(-0.20,0040) is N(- 2.00, 1.00) . I Load ~~~e~. No. of repetitions ~PI(t)' kN .a ~ I [~);_'_k .N.. _ _ ! I {Pl(t)} n1el/year .'· 0.50 0.20 , i ' (p,(t)i -- I~; -~6t!;m -0.20 ~:~~. c I {~3(t)} ' na - 3~~/Y,E!3r . -2.00 ,.' Table 10.5 :.:=" 11111 ! 111'1111111'11111111111111111 ,.;(1) . ITIIIIIIIIIIIIIUIIIIIIIIIIIIIII i '::;';. II II II 111I1I11 11III I1111II III 11II ,~t" )6~ . ~ ..... '" :- ,. p£.:'.: :~ , ". 5 m " .1 Dis!:ibution Gaussian Gaussian I GOlussian ..
- 186. 174 ..... ---~-.-, ... -.--.-,- - 10, LOAD COMBL~ATlONS • '., f Let X .. = rna..... [X<.i be approxima~ed by a normal distribution Xu,:X ,ax) and let: i"':.. .... 30 •. 3 -3 '~ X_ = X.,..o:-' X •. X5' wil: then be normally distributed' N(- 0.20 + P.x . .J0.40:'';' 'C X J2 J: ;) - , _ ,.,' ,'., ~,.C ' .:> -J "3 7 Finally, Ie:. Xa - max [X5i l be approximated by a norml'l1 'i:::. ... ibo.iuon N(pX ,eX ). , i~l_,: __ .. ~Q , .' , ,. . , .' . ,'" '" b b: .~1 appi~~~~iori~.a,i~ m~de a.i·t~,~;1~si~::p?~n~,pX;':+:,Xj,.~,~/:1~:tri~·~?,rmalised coordin~te system thef~lure_s,~rfacels theri.~~,el]-'-I:>Y , C" .~ " -' , " " .. 95," '.' -I' 'j .'- (10.93~+ 1.3975~·1)'--8 (PX b + 0xbXa)"'O and the iur.ition formUlas are . ",' 25. f.; .: "rr= . "8 pX~;~1~.935 . i.:a975"1 ~.~ G~~Ct'6 --..1.13975 ttl k 1 ' 1 25 , 0:6 =k 1 SOXb _ 1 "2 -~ 0.40 '1 • 0:'=k 2 0X3 13 __ P.x5 + 0.20 - -;-.3 .;. ,G0:60X ~ ";0.--10: - tax )~ _"._. . . 3 _ .- Tile iteration sc~eme ~ ~ follows: , !: , . , Iteration No. t Star- 1 2 "3. c 4 -. °X3 L. 1.000 . C.503 0.513_ 0:7Ci1 .. . 0:783 1, PX3 j-2.000 -0.005 -0.008 -0.233 -0.435 °x~ ~ 1.07'i'. 0.358 0.541 0.698 0.777 t-2.200 .- I I .. "'X~ 0.557 0.442 0'.302 0.121 I , J p. j 3.000 00450 2.951 3;386 3;662 - } , } ~ , .' . 1 • :" 5- I, 0.811 -0.521 I ) 0.802 , ! j 0.045 '3.738 .. -, ", J-0.707 :-0.781 ':7Q..€!~L .=.o .. :§39 _ ,=Q.',~Ht9 " -:Q.:487 ., ", 1 0.707' 0.625 0.771 0~84.2 0.8S7 o.8ul I I . I ., 0.707 I 0.322 0.615 0.496 0.455 0.442 , j , ! ". 0.707 0:83 0.7$9 0.868 0.891 I 0.897 I , I , i I , ! 3.00 , 0.: 7 4.45 3.86 3.77 I 3.76 , I , , Tatt!e 10.6 (10AO) (10041) (10.42) (10,43) (10.44)
- 187. ",' BIBLIOGRAPHY 175 "!' It is imponant to note that the values for J.l.PjHl and 0Pj(t) in table 10.5 are values adjusted in such a W3y that the ap?roxima~ed normal distributions iUI the m:..ximum dlslrihuLioni ""0: ~ ..'.< .'t!Ptable. BIBLIOGRAPHY 110.11 110.21 110.31 IIOA} /10.51 110.61 110.71 110.81 110.91 Ferry Borges, J. & Castanheta. M.: Structural Safety. 2nd edition. Laboratorio Nacional de Engenharia Civil. Lisbon. 1972. Madsen, H. 0.: Load Models and Load Combinations. T.~esis. Techn • .:al University of Denmark. Lyngby, February,1979. - Madsen, H. 0.: Some Experience with the Rackwitz·Fiessler AIeorithm for the CalculaUo'n 'of Structural Relio.bility 'under Combined Loading. DIALOG 77. Danish Engi. ,n eering .Aca.de m'):, Lyn.g by, 'pp; '73.98. .-' Madsen, H., KUcup, R. & Cornell. C. A.: Meon Upcf08sing Rates for Sums of Pulse· Type Stochastic Load Processes. Proc. Specialty Cont on Probabilistic Mech. and Struct. Reliability, ASCE, Tucson. Arizona, January 1979. Nordic Committee on Building Regulations: Recommendation for Loading and Safety Regulations for Structural Design. NKB·Report No. 36. Nov. 1978~ Rackwitz, R. & Fiessler, B.: Two Applications of Fir It Ord~r" Reliability Theory for Time-varianl Loading.s. LKI, Heft 17. Technische~ Unb: ~rsitiit Munchen, 1977. Turkstra, C.·J. &.: Madsen. H. 0.: LoadCombinatio;lS in Codified Structural Design. J. Struct. Di'., A5CE. Vol. 106, No. St. 12. Decem.ber i9Ba. Turkstra. C. J.: Application of Bayesian Decision Theory. StUdy No.3: Structural Re· liability and Codified Design. Solid Mechanics Division, University of Waterloo, Wawr- 100, Canada, 1970. Wen, Yi-Kwei: Statistical Combination of Extreme Loads. J. Struct. Oit· .. ~-SCE, Vol. 103, No. ST5. May 1977, pp.1079 ·1093.
- 188. · " .. .,. ','
- 189. 177 Chapter: il ; . , : .~., • . _'. · -.- ..... '1 •.• • APPLICATIONS.T9 STRUCT)JR.AL CODES .. . -" . . ",', ; 11.1 INTRODUcrlON Structural codes are documents which .. by their very natut,e. a,re subjec.t~o ,p;(! riodic;:revision and amendment; but the decade. 19.70 . 80 was-a time of marked activity in, code development. " This is!still continuing .. The main teatutes have be,en . .. . _ ..... " . • the 'replace~ent ~C many simple design rules by more scientffic'ally:based ca.lculations de' rive'dfrom 'expCrimimtal and theoreiicairesearch, . • ' ""the 'mov~ tow'ards limit 'state aesign ; whereby the designer and/or code writer specifies • • • • the relevant performance requirements (limit states) for each structure esplicitly; and where separa,te sets of calculations are required ,to c,heck that the struc~re will not attain each .. umit state' (at a given level 'of probability). ' .. . . . . : " " " , . -"- .. . . : .: --- . " ' ;., ' . - . . ' .. the replacement of single safety factors or load factors b{sets of partial coefficients, the improvement of rules for the treat~~nt~r ~ombinati~ns of'i~;ds an(i~ther actions, • .,: .. '.:. ' ': I ~ ; . 1 • " . " , . , , '.,, ' . ' ,' the use of structural reliability theory in determhling rational sets of.partial" coefficients, and'!"" ,", , '",. . , ,- . ' .. " . . .. ';''' . .. -': the 'prepaiatici~ o'r '~~ei" -Co~es {11.7J for different iy~'s~cif structural materials and forms of construction: and steps towards international code h'armonisation; particularly within the European Economic Community (EEC), . ,."" ' ,.:' '~ " ' :'lt'~' .:'.) " ." ' " ' " ~I It should not be thought that an these developments have bee.f! fully c().ordinaq!d. or that an ,the changes to practic:al .cod.e,s that have, ~aken place are n~essarily. of _gre~t, benefit. Indeed many recen-t changes in 5tructural ; co.~e."h~ve not been met wi~h e~th~si,asm by practising en· gineers. often 'for' goo(1 reason. Nevertheless. each of the features mention~ above is of rele- Y~ce t~ f~~'re' ~~d'~ de~~io~~~~i :' '. , • . - . -' ,r' . -, ,,' , . . .y- . '-, . •.•.• ;. ,.. In comparison with the idealised models used for calculation purpose~, the actual behaviour of most structun!S is extremely comtliex and there is a tendency. as man! research is undertaken . ' ' ,.'"._ .. :, ,: . ' "~'" ' .." '. ' . ,. , ., . -, ' , ,,' .. . . and more becomes kno:-vn;'for' the design procedures set out. In structural codes to become in- " c'te~i~gl~';~~;h;:~~d (i~~otv~': S~~h- ~'bange~generally inir';~e design :costi 'and increase the risk oi major errors being'madi:'They cannot be classed 38 improvements unless the new proceaures resultiri iinpro,ved itahdards DC safetY'anc:i/or reduced costs of construction and main- ~ •. tenarlce: ,,'" i :.,:', :" .: _: . .r -; " I .:, .;' ·W.''':'·'' .,;, '-.~ -. -
- 190. 178 I I 11, APPLICATIONS TO STRUcrt"RALCODES ' - -,--,. , :I , j It is therefore cleJ!" that the ))beSb codes are not necessarily those with the most scientifically , j advanced desi{!n clauses, As will be Ciscussed later. there may often be adv,ant.ages i~ using sim.t.~ plified design rules. The efiect of thu will be to make the overall C:ons'Nc~i~n· ~lightIY jess eco;'. ornie and the reli;!.bili*' or those s~ri:cl~'m desisn~d to'UHi"code margfnaily moni''ariable, for any specified standard of reliability, In previous chaplets. various aspects of structural reliability theory have been discussed, together with the problems of modelling load and resistance variables. In this chapter we can· sider how these techniques can be u.c:ed in the development of conventional structural codes. 11.2 STRUCTURAL SAFETY AKD LEVEL 1 CODES 'I; ; , "A$ ,lmentioned in chapter I, leuell design methods were'described audesign,methods in w~ich appropnate~ of structural reliabilitY' are provided 'on a structural element basis (occasio~ally on a Stmct.unU basis) by the use 'of a number of' partial safety fadors (partial 'coefficients) ; ,related to,pre-defined characteristic or "n~~inal values of tbe major structural and loading van-, 9bl~, Alevd 1 c~e is' ther~fore ,a ,~~" nv~~ti~na1 -d. ~t~rmi~&ti"~ c, o.. de'-' ~", ~, ~'i ~,h '' the nominal i' "strength,S:.:o(.lbe structural members designed to that ,code are gO,vemed by a nU,mber of partial I coefficients or by equivalent means,,' .. ". ," , " , , :) Th~ ~fety ~d' sen,icea.bUity oc'pracdcai'struc'tures ire achieved by the u'se of suitable partial coefficients in de~gn. together with a~~ro~ri~te ~~ntr~{"m~s'u;es, Both ~e ~Sential and it is h~ipfl,d"'t~ 'distin'guish" U,eir ind~~iduil:iol:~~': ! !, ',' ,r> l." ,~, ",' '; " ' . 11 " . ,' l' ~. _' " Let us first eDmme the role of partial coefficients. Consider a structure subjected to a random " ", - , , ' . , ':',', ,, ', " ','. .,. ,~'" , ':' .c . ' time-varying Joad Q having a specified nominal magnitude Qsp' The stiuctui'~',~~; proportioned ;. , ~o: carY)' a,desiin IO,a?8d "7 "Q'4p, :whe~e ')'Q is a Ji>,arti;~; ~~,~f~icient"~~, ~,i~~" I~~~, The effects of :increasing'1Q by{ssy,-20% will in Ii!,..,neral be ':":, ';'","1-',, ;. • • : ',' an increase in tneactiJ:il capacity of the struCture to'support the load Q, ' • an iricrease in ~e ' 5i:zes of the'S:lUcturci!members;and the self~weight' of the structure, " • ~ 'iflcrease in the"eoih 'of the smicturah~;stem, : • • 'so~~' i~'3se in: ~he actual c3PsChy oC the structure to'resist'ahy other load' Q', and ~ , " ' ,, ~, ', ' ~, an increase in the safety of the structure as charitCterised ' b~' a'reduction in the probabilit~: , that it will fail in any given reference period T: ' "--,, ~ ' : . '.-, ,,:" , ' ,Ln,,' "I' "I;n, , ...... " . , ,'; ' , '.,{,' '", ~ '! l! t?~. d~si~ ~ep~I ,~~ ,~ mat~rial is ~ve,n b!' ~d ~, e_I~!"f.~,,~, ;n~."e:~e~:IP. ~:~~~~,~~;~i~~d material ;' strength and 1., is a partial coe~fici('nt, an incr,~ase in 1'm will in general have the same effects , ,, ' ., .. , "" ,. ' ; ' • • , ,'.,' -'J" • !, ~, ~inC[e~~, lQ "::' .. ' }", ";',' , <:!'_H';' ", There are some clrcumstances,:howe.er, when in,c~~5.in. 1,Q o.~.i~ 1 m .. J?ay not gj'e rise to : these effects. For example, the acrullioad-carrying capacity of a structural ~e~ber, as oppos~ to its nominal capaclt>·, may decrea..::.c 0: may not significantly increase if, for example, any change in lQ or ":n: results in the cit"li~er using larger diameter reinforcing bars which. in spit~~ of having the samt specified yield saeu as the bars they are replacing, may ha'e a 10ft..::: ~ean; . -" :",~
- 191. 11.2. STRUCTURAL SAFETY AND LEVEL 1 CODES 179 yi~ld stress (se,. figure 3.9). Simil~ly. sr:::.all changes in 1'Q or 1m may sometimes have no effect on either the dimensions or the salet)' of some structural member!. This is because of the dis· crete nature of many structural components (e.g. rolled steel beo.ms) and the nlled to round up to the next section size above when designing. In such cases the !ictual strength. and hence the reliability. is not a continuous function or the partial coefficients. WE' now consider the reasons for usine; partial coefficients as opposed to single safet.y factors or load factors. The main reason is that only by using partial coefficients can reasonably con· sisumt standards of reliability be achieved over a range of di £rerent designs within anyone code. As will be discussed in section 11.4. the most consistent. standards can be achieved by assoclatinr a partial coefficient or some other safety element with each major source of uncertainty (i.e. with each basic variable). Partial coefficients are also essential for the rational treatment of load combinations. and in panicular !or situations in which the t-otalload effect in part of a struC' ture is the difference of two load effects of approximately similar magnitude but originating !rom different load sources - e.g. the effects of gravity loads and wind loads in the up· wind columns of a tall building. We no' return to the question of control measures. The safety and serviceability of a struct.ure are influenced as much,lt not more. by the nature of the control measures that are in operation as by the magnitude of the partial coefficients that are used in design. Control takes two main forms • quality control of materials and fabrication. and • controls to avoid the occurrence 0: major or gross errors in the design and construction processes. COl1teol of the first type Is aimed at reducing 'ariability in the mechanical properties of struc· tural materials and maintaining appropriate mean properties. For example, the vari ~bility in the yield stress of steel can be reduced by improved control all chemical composition and rolling conditions. Such control will. in general. reduce the probability of structural failure and lhus in· crease safety. Both the form and the parameter.; of the probabiiistic models for r('sist"l).ce variables discussed In chapter 3 are dependent on the standards of quality control and Inspection thal are in operation. Control of the second type is clearly more difficult to achie'e since the sources of possible errms are almost unlimited. This is the subject of chapter 13. We continue here with the problem of devising a suitable procedure for evaluating partial coef· ficients or olher safety elements for a le'ell code. The term safety element is uSl!d as a generic term for partial coefficients nnd additive safety elements (sel! section 11.3.3). A logical sequence of steps is as follows • ioCt l~mits on the range of structures and materials (or which the code will be applica~le, • specify the detenainistic functional relationships to be used as the basis for each design ciause, • select the general form of into probabilistic modcl ~ tor Uh, ... uri"~s luad u:u! resistant£< vari· abies and model uncertainties.
- 192. 180 • .; specify appr9pri3te quali~y COf).tr9J.r~e.asures and aeeeptance:critcria,Coc the manufacture and fabrication of basic materials and components. ' "",1". • determine the parameters of the relevant models from loading data and from materials data obtained under tr.e specified standards of quality control and inspection, • select a suitable safety format· the number of partial coefficients and their position in the design equations (i.e. the variables associated with partial coefficients), etc .. • select appropriate representative values of all basic random variables (e.g. nominal. charac· teristic or mean values I to be used as flXeci"detenninistic quantities in the code, • determine the magnitude of the partial coefficients to ,be used in conjunction with the above representative values to achifl"e the required standards oC reliability. , Procedures such as this h!l'e already been used in the application-ofstructural -reliabUit.y theory ,to practical level 1 codes, e.g,lll.6], (11.101. [11.121. Some of these st.eps have'already been con. sidered in some detail. e.g .• he modelling oC load and resistance variables, and others. e.g. quality . cont.rol·procedures, are: be~'ond the scope ol this book: In~ the ~~~~d'~j:;~c=th~'~hapte~ we shall concen'trate 'on the questlon'oC choosin'g'suitable saeeiy '(annals [or 5truetu~ co~e5 and on the 01 ; ';~, ealculation of panial coeff!cients. 11.3 RECOMMENDED SAFETY FORMATS FOR,LEVEL 1 CODES The safe~y format of a code is defined as the way in which the various clauses of the code regulate the_ degree of safety, .or more.gene~y' the reliability, of structures desig~ed.to the code. In p~ic'-:llar, it. concems:.the numbeq)[,partial coefficienls or other safety element.s to be used. UJeir positions in the de~l gr. equatio~ •. an~ rulesJor load combinations., . The following recommenClations for leyel l ' codes are based' on the work of the Interriational Joint Committee on Structural Safety [11.71. [11:81. and ate likely'to fonn tile bastii'of a new international standard to replace ISO 2394: General principles for ·the· verification of the safety of strudures. 11.3.1 Limit state functioQS and cbeckiDg equations As discussed In chapters 4 and 5,. the genetaI conditions for a limit state' not to be'exceeded may be expressed as ':' (lLl) where X are the' n basic random 'ariables whieh influence the'limit state:~'~d is the limit stat~ func~:~~ (t'ail~re·t~n~ti~;n-)_ · ,,_ '. ' ....... . , . .. , . . _ :.,1"' '.::.. The varia~les X ~ay be su~i~ded i.nto 'ar~ibl: loads and .actions. Q. p':fO"!~ent !.?~~~~. 9. rna· teriaJ properties E. geomel::cal parameters D . .md model uncertainties X (see equation (1.1 l), . . ~!:. ' ,':. ' : '. . . . m ' _.'" ,,',' . In addition. each limit stiltt function is likely to involve one or more constants c, EQl.;Iation (11.1) may thereiore be re",'ritten as
- 193. u:, ... :.r :.; ." :!! :':, :J, '. ~:C':T : .. ;.:','! .... 11.3. RECOMMENDEDSAF&TY FOR~IATS FOR LEVEL 1 CODES. lSI ", .1 ' . : 'l.:!,""::f.:..l7. ;;'" :_-:._ • " , _ f(Q,.G,E,D,Xm,c» .O . .;, '.: ";" ! l~ b c(). ;: ; .: .-:. : '.1,:.·· .: ' .• ·.t , ., (11.2) For the purp'oses of a level! code, the equivalent deterministic cri:~erion it?r safety checking (i.e. checking the sufficiency or a structure'or stni"cfuriilmember whose de5igri'pro'perties are given) 'll; ~ '" 11(~ ' : .".LJ:~·.· ,.: .. , .. '"!',.,- ~il~' .(11.3) ' .' where , f is the same limit state function as above, involving n quint-lties id and-m c;:onstants C, and <lei is the deterministic design value of the random variable Q, etc.' If the aim is to design, as o ppose.~, ~? : c~~?k ",.~.: gart!cp!~ ~t,.~I~J,~l ,~e~~~~J~ ~~y or~en be possible to invert equation (1!.3) to give the minimum design ... a1ue ol some convenient' resis tance variable· for example. a dimensit;~ :D'·~'r-'a';.;Jlion ii~~I;rs.~i,~:·' ·} ,'~;.:-:, .• :, ,tt , , .~ ,, '.' -,' .. 'Hence, ttie'piOcess o:(de~'i"~ing"a! i;tructural member i~i,.olves L. <>': " .;; . . .... - ;~. ' -. "': :. :;: .. c:; :u .':' 1'.. '. ,',' "'," '; ~ , ; :); .. . r.'. .ellA) .;,1: :-. .dete~ination q!. lh.e ,d~i~J.o~d.s .~d " 'i' ., .... ,. '. . ":, ,.: • '. ~l~tio~ ,oynate.t:iaJs ,~,d .ct~~ rmina.t.io~. of the d~iil1 vaJ~es of their relevan.t. mechanical .· ,Rrop'e,~I,e.~,~d' · ";'." : . ' .. ," . '.. . .") . ~., . • . ~~!~c~ion of pri~ary" d~~?!1;~,i,?~;~ ~.d ,,~?, sa~j~~ t~,e particular engin~g and architectural requ¥,emen,ts, and ", _ . ,', __ , " '" .~. , • ' determi'nation of the remaining unknown dd to satisfy eq,;,ation (11.4). " !~. '!Iany .c.~es it m~y .. ~ot ~: ,I?:0_~s~~!~ ~r co~~~~.ient ~o !i!~p~!~qu ~tion (~L.4) in ,e~pli~it fonn . in which case the design process will involve a number of tria1·and~rror_ca1culations to find the . , ; .. ~~u~ ~~l~e~~r,d;·~ha~:~~i~{i~~tn~ :i~~qua~~~y_.~li.~). rh~-:~i ~:~~_~.~U;~ normal ' .. ' approach t~ dl!$ign. .. ." .. . .. . :nJ' 'J: ,1:·~ ·:1 .... ~ : ··n.-" ;.,. Let us now re.-examine equation (11 .2), For many structUrH it is possible to re-write this as . ..l .. , .. ''', ':,;',', .(11.5) . '" . ' .;, _ . :j' .. , •. !".: .',J- .. ; f~~~~~n:~~~!,~~;~~~~~!~~~~:~~~~,.~~~'~:~_:~~~):.· _~ . ,~~~.: .·:~~ ... ::1~:::::'.·.;~'::~~~- ~ s · represents a road efrect or"actlon eHeel'luneuoR and 5 • 5('), ,
- 194. . . -- "~ l ··'·"r"·"-~' f ~: 182 11 , APPLICATIONS TO STRUCTURAL CODES "'~ ; ;' ~. ': ., . X. is. model uncenain", "".dated with 'he pmi,ulu f.'m .f the .'esiot"''' function, . . - :~.' .:" Xs is a model uncertainty associated with the particular form of the load eflect 01 action effect (unction. ". , "C, ~ · .,', . ~d wh!r~:PR and Ds ,are sets of different. dimensions. In equation {ll.5} the resist.ance function r and the load effect function s are shown as uncoupled; and because they share no common "ariables the two terms are also statistically independent. 1£ such uncoupling is possible, then the deterministic checking equation cone· sponding to equation (1l.3) rna!,' be el.:pressed as ",' where ~.R is "a Paiti~Jcoefficie~'t-on the computed ruiStan~' " ' . , ' "'. : .. '11!: ", ', . . , rs is a partial coefficie~t_ pn ~e.,co~pu~d, l~d .eCfec~ and where the subscript d denotes the design IJalue of the variable. (11.6) " : The design process generally involves iterative or trial.ind-erro'r caiculitiollS to find a set of dimensions dRd which in conjunttion with the design values of the load and streng.t? variables satisfies the checking equation. Equation T11.6) is the mo.st geni!ral form of the checking ~quation for' a'structure in which Rand S can be uncoupled. In th4 case, the safet}' or. sen'iceability·of a structure (the prob. ability that the limit state der~ed by the par~i~~ f~ra: ~f'th~ fu~ctions r and s ~ill ~ot be leache~) can clearly be increa!ed or decrease(fby a!djusting a~'y or 011 of'the';(ri'-l) inde- .'c' 'pendent desiill values Xci ' (e.g. ~ or ed) and the hvo"partl8J c:o~fficien~'R and lOs" Substi· tuting these values into equation (11.6) gives the, required value of the rem:aini;;g'quantity . generalJy:~ dimension. Because there lS'a~::infi'nite "ufmber'of set.'s·,~,(n~"-l) valu~ xd which will give the same design, the problem facing the code writer is to select,l.he ~besb'set of . ' . . : . ,)" . ~ - :' , " . , ' ... ' .. - , values xd" This is discussed in section 11.4 • . :) ' It should be 'n"oted'that in' pr..ctice the qU'anUties'"R aTid: S ma:y: orten be c"or'rt'la'led'hencltse .. ,. t· df c"omm'on 'parameters~ For example, the ·self; ..... eight of a'reinforc'ed con~rete beam and hence ttie" inid.~pan beiiding ~:o'inent S will be:weakly Correlated lV'ith the b~'ani·s ·inoinent..Carrying capacity R, as both are functions of beam depth. ,'r., "': . " .' 11.3.2 Cbaracteristic values of basic variables The term characteristic value was introduced in the'late' 1950'5 at the time when'probabilistic concepts were first. beinE introduced into structural codes; and when it was recognised that few basic variables ha"e clearly deflned upper or lower limits that can sensibly be used in design. Characteristic values of actions an~ mat~,ri.a1. propert!!!S qas.ed, ~:m.a p~escriJ:l,eq probability p of not being exceeded ...... ere considered to ~ m~re ~tio~d th~' ~i~~~ ~iect.ed values, " ~~ ~ ' . ' ~ ". " .::' :'~' , ~ . ' .' . . " , .
- 195. ,11.3. RECOMMENDtDSAFETY FORMATS FOR LEVEL 1 CODES 183 The characteristic value xk of a basic random variable X Is defined as the pth fractlle of X given by . . -. . "' ~, . (11.7) where Fx is the inyerse distribution function of X. and p is a probability which'depends on the type of variable being considered (i.e. a load or a strength). " The'selection of the probability p is to a large extent arbitrary but is influenced by the follow· ing considerations "';'.-. ", • characteris!:ic ~aiue~ C?f loads: and other actions are values which sh~uld rarely. b~ exceeded. • characteristic values of ~aterial strength properties should normally .be exceeded by actual properties, . . • the values of p should neither be so large nor so small that the values x ... are not occasionally encountered, • it is often sensible to use previously adopted nomina] values as specified chartlcteristic values" x'P. The distinction between characteristic vo/ue and specified ctuJracterisric ua/Ill! (specified value) should be made clear. The former is a fractlle of a random variable, vhereas the latter is some specified single value'of the same quantity ~ a constant. For practical reasons it is generally necessary for the user of a level 1 code to work ,with specified valuas of all the design variables rather than with actual characteristic values, some of which will not be known at. the design stage. For example" the actual characteristic value of the 28-day cube or cylinder strength of concrete is likely to depend on the particular supplier or contractor and is not known in advance_ In this case it is necessary for the quality control procedures specified by the code wrirers to be such that the actual characteristic.strength of the material exceeds the specified strength by an appropri.:1te margin or with a stated probability. Similarly, the user of a code should normally work with specified deterministic values of loads and other actions; it is the responsibility of the code writers to relate tnese values to the distributions of the actual loads and actions, ..... .:l t:- recommend associated partial coef!jcients or other safety elements. 11.3.3 Treatment at geometrical variables Geometrical varia~les ate of t:wo main types· structural, dimensions (e.g. the depth of a beam) and geometncal imperfections (e.g. the out·ot-straightness of a column) . . ' 'J. Structurc( dimensions: 'fhe uncertainties in most structural dimensions D are generally small ani::i for fhis rea'sort the mean value PD maY'b.e taken as"th"e characteristic value (i .e. d~ .. lAD) · Tolerance limits are specifica in codes for most struct.u ia,.i dimensions, and if these are of the ' . ' ~ . form I11.SI
- 196. 184 II,' APPLIC'TIO~S TO STRL'CTURAL CODES d •• fil/urc 1l.1 (a) . r I I I I I II I II I :! 't-- d FiJUto: ILl (b) '.". then the actual characteristic value dk and the specified nominal value dsp will generally be very close· see figure 11.1 (a). It should be noted, however, that unless the standard of inspection is high the probability that the dimension D will exceed the specified tolerance may not be negligible. See. Cor example. figure 3.S. Geometrical imperfections; The strength of ~any structural ~~mbers,' for example most plates. columns and shell structures. depends not only on cross-sectional and overall dimensions but v.-lso on the magnitude of relevant geometrical imperfections I. ,', '" For such structures it is normal to specify an upper limit ~ on the imperfection magnitude, Le. 111.9) In this case, t can be taken as the specified characteristic value of I. isp' The probabil.~~y, that i$p will be exceeded will g~nerallY be small and will depend on the standard of Inspection. The actual characreristic value of the imperfection ik can convenient.ly be chosen as the.. 95% (ractile of 1 and the acct!ptance criteria designed. so that isp exceeds i" by an appropriate ~argin (or with a stated probability) - see figure 11.1 (b). A histogram of some typical plate.panel imperfections (plate-panel out.oi.flatness) obtained (rom measurements on the steel box-girder bridge at Aust in the U.K. is sh?wn in, figure 11.2. The quantity Q) is the ratio of measured imperfection to the specified maximum imperiection t. r I 1~ . ... o 0.25 tUO 0.75 1.00 1.25 1.50 Fic:ur~ 11.2 PI:.tc plncl imperfections· AU$t Bri~l(e. ........
- 197. 11.3. RECOMMENDED SAFETY FORMATS FOR LEVEL 1 CODES 185 ':, i: 't· Design 'v'alues'of dimensions ar;ld_~mp_erfect.ions:.Typically •. ttt.e,~~nQar,d ~e~i~~i,'?n.s;!3f ~~~metrical variables 'are,independent of-nominal dimensi.o,~sjf!: .. g,fo,r gi"el),site c:<?ndi,t!pf!s,~p~ ~~dard de' viation,in'thethickness of;! 100·mm·slab.is likelY.,to be,_ab_o.!;l:t,~~~,,&aI;J;le,~_t,h,~~ __ pr~.~O,O,mm slab; giving a reduction in the coefficient of variation for increasing nominal thickJ;1.e~_).:For this :-eason the most uniform standards of reliability can be obtained over a range of different struc- ,d_,., -' . , . ·c.' [' .' "co",. " ',' "l'. ',.., ',' ". ~. _." ._.; _ ,-', _ .. . ..... ,_.' ~ures _by using design values'cd and id of the geometrical"variables reI3.ted'to the specified values, ~'follo~ ., ': <'," ,"';,,'" ;." ~~,;r;,:,: ,'.' :h' , -,--:: "'_'-', " <'r' .·ii'· (11.10) id = isp + 0i (11.11) , ').i: j .' '~:. j • "., ' ., 'where~ad'and 0i~are additiue safety elements. ·r ,," . ;!ir • ..'; For' tnany.stniCt;iJres.'hdwever'.:the'prob~bIIitY of failure is insensitive'to sniail-VariationS in ":hriI~!tuiaid,ifnerisions':Forrthe~e cases; ~d" mi.d: ,ij"shciuldbe set,tci zero arid' the uncertainties -'1~ 6 and :I"should be"aHowed for by modifications" to' Uie·partiiLI'coeffiCierits-'o'n the other de' i!~igJl'varl~bleL A: formafinethod for'doirig this-:is:discus'sed'in'section 1104: '~I: ,"'. '_~)l 11.3.4 Treatment of material properties -: Ye s,~~l re.stri,ct9l;1r a~~~1.1.tion _~o _the stren~h prope~t.i.~s:~~:st:Uf~~~:,~~~~r~aJ;~!,4e',l0~~ E. Fpr,~~c,~ v.ari~ble. the~.~ar~c,~~ri~tic:_~~,~u.e ek.s~r)Uld,be.s~:~~ i~~~~,!t.~.as ~.F,!!~~,~_~~~~ ~~~,~ prob- . ability, gJ~, 1 ~p}"of bein,g,exc~~9.e~?. in 3ll~ ~:in~l~ trial,r;r t~t:}'ypi~y:~.q.~ ;ta~e.~, ~o_be between 0.95 and 0,99, corresponding to the 5% and 1% fractiles of the, v~able E. However. as mentioned in section 11.3.2. the user of a level 1 code may·~ft~~'no~'k~~~ th~ ~ctual'charac~~ ristic values for his material properties in advance. and it is generally ne~essary to design using specified characteristic values, esp ' The acceptance criteria for a materiai'sh6uld hi! devised so ~~~t ek exceeds eap at a stated level of probability Pe' It should be no~ ~er~ ~.ha~ the u,ncer· . tainty associated with the event (elt > esp ) arises as a result of imperfect 'k'no'wledge ofthe ma. ~rial supplied and, t:he d~fficu,lties of obtaining sufficient_ samp~e data at th~ ,appropda~,'time. The ~robabiiit~ p '~;tnust be'~'I~Q.~I!l di1)tl~!iui:~h;d fr~~ ·t,he'p~obi~ility·p ni'~~'tion~d ~b~ve. ,e... " ;-"',""_' .. ;.:~. ,,' ~;';':'-,'-;_': .. r;. rh~:4e~iGn ~!71~~ ed oq~e ~~r,e:~_gth of a_~ate~al-, is o,b'~~~T;~ :f~~rr- th~e sp~ci,~e~ :~~,charaderistic 5~p.gtr, <!~ Xollo~s (11.12) .~,-.' '~here-1'~ isa partial coefficient on strength~ ; ... 1l.3.5TTeatment of loads and other actions - ,~' 'i" The classification and modelling of loads and other actions were discussed in chapt,h 3. Most loads differ from other basic.~arj._?bles in tha:~lth~yva.ry,:s!gr,tit1cantly ~i,th ~~me and are general· ly not amenable to effective ~on·t~ol. Ther~.are ~ofol:l~ ~~~_~.le e~ception-s to both thpse generali-sations. .1
- 198. I 186 11. APPLICATIONS TO STRUCTURAL dODES . ,.;. Because of the time-varying nature of most loads, the problem of assessing the combiriC:u ii' q-,' -"feet ;0£ a'number 'Of differeiiUoads acting on a structure has been seen so arise'.'This was dis· 'cussed'bl ch'ap'ter'lO in ttie'contextof reliability' analysis. As might-be-expected a.rather simi" lar'problem' arises'1il treatingctimbined loads within the framework of a deterministidevel . ",' '. i,· j • ::.,.'. ;r- ',' J.. , Character~tic.vQ_lu.es: The. uncertamt· in most permanent I~ads is small and for this reason it is custom~' 'to use th~ ~e~ or n~~.d ~a.iues of p'~~·i.u;~nt lo~ds; i~' m;ost design !!aJculations. For the same reason it is appropriate that the characteristic value gk of e~ch pec;nanent . load G is taken as its mean value JlG _ "'a may be considered to be the average permanent load taken over all nominally similar structtues and obtained by using mean"dimensions and mean . densities. Fat a time-varying load Q, the characteristic value qk is nonnally defined as that value which has a prescribed probability p of not being exceeded within a given reference:per.~od)t,is there- ,; I for~ the, p~~ fractile of .tJle extrem~ ~u~ distI:ibution qf. the: ~oad. correspqHf!ing'~o}hat referenceperiod., Up.to the present, date (l982) few .~at~?~~ loading ?ommitte~~j~~y.l!'.a::~~Pted ·to rationalise, their spe~if~~~J9.aqs,~ng t~~se _line~, but,progre,~ i~, bein.g m~de in .~his direction. Wind loading codes are perhaps the most a.9yanc,e~jn this respect .. e.g. [l1.~l,':Ihe nominal loads specified in most loading codes 'ary rather widely in terms of their probability of exceedance. - Single"time:liaryinglo~ds::'If a structure or structural 'component is'subjectea td only perma'~ e-~t"lo~ds 'G' ~a 'one·tirii~-v8rj.ing load Q~ the load~combin'ation problem does ri'o't arise', In this c~e";'til~'a1ues 8d .inci q~ to be usec' m the design: or sa"fety checking places's (ct; equa-. tion (11.6)T~eobtained from " "(11.13) " c'':,: j',,:-1) ');',,; "'!: . :0 '}~.~)JQ.~k ",' ,.,,-- '(11.14) ,. '~'-"-,,--":qC!: ,,;': " ': ,:.~, .',',. ,':, '," ' , •• ,.', " ,... ;,.. . " , I ''- 'wh~re ? !G ... ~d"l iQ ~f~~ar~~ coefficients and "gk and qk are characteristic ~a1ues of the ran-dom vai-hiblesG'and Q':'r~~peciively_ "', " ' c, .. For'fai'lure modes in which part'of the permentmt load acts in a stabilising'hr reslsiing:'sense and part in a de-stabUi.sing or loading sense, different values of "ICG should be used for:the two c~mponents; "1 fG '" 1 when the load is stabilising the structure and "1 fG ' l ~hen it is not. Combinations of time-varying loads: When a structure has to resist a number of stochastically independent time-varying loads, it is clear that the probability of two or more l?ads eltceedinr: their characteristic values simuluneously is small. If the total load effect in a member were to be determined from '; '., .-l , ')1:- (11.15)
- 199. ~~ 11.3. RECmlMENDED SAFETY FORMATS Fon LEVEL 1 CUl.J:r;.;:; ~1· where 1 WI and £kl are t~e values Ofl fG and qk {or the fjn;t of m permanent load~. qu is the characteristic value of the first of n time.varying loads Qj • 18i 7CQl is the partial coefficient associated with the load Q1 when this load is acting alone, and c il> the· load effect function, implying a linear or. where appropriate. a non-linear analy· sis of the struc~uro:: .,u).;Jer the action or tho:: bt:::::""d loads. the resulting load effect S would be extremely conservative. For this reason it is necessary to introduce a set of rediiction factors 1}1 Oi (1/1 Oi '" 1) to be applied to the time-varying loads Qi to ;;.; take account of the reduced probability of the design values of the loads being exceeded simultaneously. The total design load effect is therefore given by (cf. Turkstra's rule, p. 168), (11.16) In principle, if there are n time-varying loads, it is necessary to undertake n design checks (e· quation (11.6» on the structure, using a separate set of ¢o factors for each check and with . ¥Oii - 1 for't,~e~th ~~eci:k~ , . For the jth design check equation (11.16) may then be Ie-written as " Sd :0 C,(gdl' ... , gdi.· ... gdm' qdl'· .. ,qdi''''' qdj' .•. , ~n) .. s (id., qd' dSd ' c) (11.17) where The need for a number of design checks using di!!erent sets of ¥ 0 factors arises from the fact that throughout a structure the contribution of each separate load Qj t~ the maximum load· effect in any member. varies considerably from member to member. For exwnple, although snow loading may dominate the load effect in the roof beams 01 ... m'.llti-storey building, the same loads have only a small influence on the tota1load effects in the ground floor columns. In practice. with detailed knowledge of the strucuire being designed or checked, it is often pos· siblE' to reduce the number of safety chE'cks significantly. Equations (li.-:n and (11.16! are the most general form 0: checking equations that arE' eO'is·
- 200. 188 11 .• -PPLICATtONS TO STRUCTURAL CODES .. aged (or use in level 1 cocles. Some rather less general forms of checking equations have also ' been' suggested 1 11".7-J. 'In practical'codes'the desfgn requiiements ~~y 'b~ ~;d~ cgn~ide~a'blY simpler, '" . ,. . -i - ' , ,"'-". : 11.4 :o.IETHODS FOR THE EVALUATION OF PARTIAL COEFF1CIENTS Any reader who is unfamiliar with the theory of level! codes may be somewhat concerned . by the apparent comple:city of the safety t:hecking rules set out in section 11.3 and by the apparent arbitrariness of some of the steps. Because of the inherently probabilistic nature of most structural safety problems, it is clear that safety checking procedures which" :tre couched in deterministic terms will have some degree of arbitrariness. This cannot be avoided. The design clauses given in level 1 codes should be interpr~ted as a set of decision rules. the outcome of which can be modified by changes to a set of cO'ntrol pa;ame·te~. the partial coef· ficients. The process of selecting the set of partial coefficients to be used In ·a particular code should be seen ~ a proc~.s of op~imization such that the outcome of all designs underUlken to the code is in some sense optimal. This should not be c.Q.nfus~d wit~ the concept of optimizing individu;.5tructures. Whether or not a formal opeimization is ~~ert."1ken in pro.cUce. it is useful to HUnk of. the partial coefficient selection' process in this ';"ay: .It. is th'en clear that the possibility exists for using any simplified set of design clauses together with a modified set of partial coeCficients which on average will achieve the same degree of saiety as the more complex. set. The penalty to be paid for using the simp~ified design rules is some increase in rna_ terials'usage." . . . ': ~ ; .. ,. , . ~.'. :: ::..''': '"", : '' In the remainder of this section various formal procedures Cor the detennination of partial coefficients are discussed. 11.4.1 Relationship of partial coefficients to level 2 design poiqt It was shown in chapter 5 that for the reliability analysis of a particular structure. the level 2 method invol'fes the .mapping of the set of n basic random variables X to a set of independent standard normal "ariables Z. This results in the mapping of the limit. slate failure surface given by tn.1S) to a: failure surface in standard nonnal space 111.19) The reliability index 11 is defined in Z space as the shoite~t distance from 'the origin to' the failure surface ana is !iven by (see (5.34)) " .. 11 - min · !L:l.n2" i~. ~.~ i·t ....
- 201. 11.4. :-'1ETHODS FOR THE EVALUATION OF PARTIAL COEFFtCIE!>lTS 189 , J:h~ PC!!~,~P.~ t,~,~ ~~.i.l~r~ s]Jrfl'c.e:. .. . .. hlch is.closest to the origin is' referred to as the' design point (see figure 5 •. 5), :m{has c.o:.C!~l;Iinat~~ . WCJ:l' QCJ:2• '.' .• ~Qn) .. 'where (s3e (5.35))' ,'~ ': ' " , ..... . (11.21) with (11.22) andz·"'tJii. By using the inverse mapping .' . " i=1.2 •...• n (11.23) we obtain the set of values x- for th~ original basic "az:iables ~ co.rresponding to th.c. design pol~'t z· . tr·the variables X are all "~~m:illY di.~tributed.·then .th~ set of.valu~ x~ .are the values of the variables at which failu~:'i's most Iik'~ly to ~~ (U-this ev~nt were to happ~n). Le . . " i "' .. 1. 2 . ... : . .' n (11.24) 'j, " , ,;' ~here (.r~ Is t~efaii~~~ re~on .... .. :~: ... ~': . ... ,. .;.' .. " rr x,_~·n~~~:~·~.~~:~~:.'~~~~:~;;~·~~,~~~~~p &~V~ _~~I~,~I?~r~x!mate. . ~'.j • . It ~an now ~.:s.e~~' ~~~lir ,~~ft.'(a!u~~;~~.w~r,e;.to .. be used as the design values xd:in a de~ermiiiistic level ~: desi~;cai7.~,!~F?~"tJl~ :r;~!Hl1p_tn.!c,tu.re;would have. a .reliability;index IJ-and a relia:' b~iity, ~, . ", ~~ ,-.:~)(,-::, (1)'T~1-ls., i( ~ is.M .acsep,!able, reliability (or the structure.; a satisfactory , set of partial coeWcients is given by .. ... :, .. ;; .,. ... ":! . ' . '.... !":t ; w'here ~ '" is the sPeCified yalue.of the. re3i$tance variable XI' and by - 'I ,,. , .'PI ·;; ·. ' .' _ . . , . " . :oed " x-! :fx('~(zj-» ': r· ."'" ,. _ _ L _ .::L = ~,,--'- ' .' ,. . ... J X'PI xSPJ x~P~ ~., •. ;, .... ; .• ,: .. ; . .• .. : , _, . . " .: •. , .... : ~'":.:.h~." .: .. , . where x is the speci'fjed value oC the loadlng,varioble X·-: t ,· .~PI' :,: ' . . ,' .. ., .; ,,,.. ',' , . ... - .. " '. '"" ' - J • . .' ::':. '", ':')::l.i:"'! .;. ,] .. i .' . "'. ' :::. . . ;;.: . Ex~ple 11:.~; '!f ~j i~ ~_~?r!1]~~'r dj~tri_~.t~,d: l.oading ,'3riable. then Fit ('litz:)) J.lx + (lpiox ." - I '.' , . ! 1, x XJ PI "PI , "1. .• (l1.25] , (11.26) i 11r.27
- 202. 190 I 11. APPUCA'I'IONS TO STRUCTURAL 'f0DES '! ! ' Assuming that the parameters /.Ix and Ox of the variable X· are known or can be es~i. I oj I J, '". . .. I mated .• tha.t :tsp ' is.. given al}d; that the reliability index' tHs 'specified, tile c'a)uation'of the partial coe~Cicie!1l 11 requires only a kno"'ledge of tiie'5ensitiVjtj: ;{~Ctoi '~;~ Example 11.2. If Xi is a l08:no.fl!lally disuiiJute~. resistance· varia~le. ~en x x a SPi SP, 'Yj - F-' (4)(Z~))· .1, X, I IoIX,eXp(-tin(V.=+11+o.,,(2n(V1'+lW) where X '" Jlx, is the mean of Xi' and Vi is the coefficient of variation of XI' (11.28) .. "Again, if the' Pararne'ters ~x . and V'I'ire kn~~n. ~d x and p are given, then 7j ,Can be . ~. .1 . _. , , , .. l!Pj ,'. " . evaluated from a knowledge of the sensitivity factor "j' -, ", ' '-: .... -: .. -,' , ." '-' ,r - Equations (11.27) and (11.28) and similar relationships"for othe~ types of probability distribution are only of direct use, when the values Ii are known. 1n general, the val~eo[ t,JI'; 'depends not only on the p~eters of the random variable Xi' but on the v~,~es ,of ~?~, !'.a~~te~ oP?e other random variables, on the value of fJ lind on the nature of the limit $tate function. For a panicular structure and fallure mOdeS~e serisitivity'f:a~i~rs {may b~ ev~~atedfro~ " ,equation (1l.21}.,However,.the use 'of this equation unplies a teli~bilitl"an"aly'siS" o(ul~ st~cture .. '~d if this isolo be·.undertaken there is'-litUe poiiltin :following it 'with." a iei;el 'j:sat~fy check. F~~rmore "this approach. 'leads to a partial'coefficierlfon e~~ly: ti~ic'vaii.a.b le, wh.i ch is to.o - many for practical use b design. -. ~ •... A procedure is therefore required for the det.ermination of a limited nu.mber or p:artial coeffi· CientS or additive safety elements (cC n, where n is the numt.er of basic 'arlabJes) whict-: will be applicable over a range oC different failure modes and for a range of different structural types covered l:"y a code of practice/Sucn a procedure is di5Cu~ in' sectioI111':4':S. 'Before this. w~ shall consider IIJl approximate direct method for the eva.luatio·n of panial coefficients. 11,4.2 Approximate direct method for the evaluation of partial coefficients The difficulty with the approach sugCested 'a}iO"e 'was see~ to· Ii~·iri the:~vai~-ation' ofsui~ble sensitivity factors n. Experience shows that over fairly large ranges of design parameters the indhidual factors ~i oiten do not ch·ange·dram'atic:illy. FUlth"enoore, i:ie~~~sk' ""! ,~ ;,, it is always possible to choose a conservative set of sensitivity factors Cor use with equatic:'n , I -f I " I I I I .'
- 203. 11.'. MEniODS FOR THE EVALUATION OF PARTIAL COEFFICIENTS l~l (1l.26~. 0:1.:1t t 0:2 ,., ... . - ± O:n"": 1 is s.uch a set, '."'~en the sign of the factor is taken as posi· tive for l~dlng variables and nepative {or resisting variables: although in most pra.ctic~l cases . '~'~:o~')d b~·t~· cons~~~tiv~. - . , Assumine that the limit state function may be split into a resistance term R and a load effect term S, as in equation (11.6), it has been proposed (11.51 that the sensitivity !aetors should bE-ezpzessed as , , . - .,' '. ' ·; ' :··:t:r:~~·¥.·the sen~i~ivitY factor If?r the iih 'resistance ,variable, 0:5,1 -is the sensitivity factor for the jth loading variable, (1l.29) (11.30) . ';iR ' and as axe estimates of the sensitiv'lty fadlo~ for the-c~~posite variable~ Rand S in the limit state function R ..!. S '" 0, 1 . ' '' · ' OR I ~ a Iactor which d~pe~d$ on th~ ~elative, importance of the ith .re~istance van· " ' able; and· . as,1 is aJactor which.d~peri.d$ on the r~lative importance of the jth Joading·variable. Assuming that the:irue values a are kn'own (i.e. from' a lev~1 2 analysis) and tlte' v~bl~'X are ranked (taking due account oC sign) so that ',. . ; .... -140: crR,i '" 0 ,." : tH.al; " ,; wbere nR + ns' ''' n, the Lata) number of basic variables, the quantities R} and 51 may be termed the leading resisting and loading variables, respectively. ,' .,: For a wide range of structural members, the following empirically.based values can be shown to be satis[actory (11.33) i - I, 2, ... , nR (11.34) (11.35) HeDce, for the loading variables R) and 51' &R .1 -.Q;s 1 'c 1 giving a~ ,.~ . = - O.S and QS,} - 0.;. ~s apprc:;.·:h is viable only if the designer has prior knowledge of the reiative imoorta.'lce (sen' sitivity r.r.nking) o{ thf> various variables. This information can be gained'by ex:peri~nce and by the occasioDal)eveJ 2 analysis.
- 204. ", "" 192 11 .• ~PPLlCA nONS TO STRUCTURAL CODES Having estimated the sensitivity factors 0; from equations (11.29) to. (11.35). the patti,ai,coef. flcients 1; and 1j' or the design values of the variables xi and xr. may ~e obtained direc~ly from equations (11,25) and (11.26), This proc~ is iUustrated in the following simple example. Example 11.3. The encastre steel beam shown in figure 11.3 is to be designed against plastic collapse to resist a uniformly distributed superimposed 'load Q and a pennanent load G. Q, G, the yield stress of the steel Ey ' and the model uncertainty Xm affecting the piastic moment of resistance of the section are assumed to be nonnaJly distributed . random variables. with the parameters given in table 11.1. The yield stresses at the plastic hinge positions A. B ana C are assumed to be UU! same and the geometrical variables are assumed to have no uncertaintY. It is desired to evaluate' the partial coeWcients tQ' la, tEy and lX," for a reliability index J1 :0 4, and to determine the required plast~c mod~I1us zP' By consideration of the mean values and coefficients of variation of the variables and the nature of the limit state function it may be assumed that Thus. llQ :0 asaS•1 .. O.i X 1.0:0 0.7 Q Ev -«RaR.I ""-O.8X 1.0 - -0.8 !kXm "QRQR.2 ::::-0.8 X (../!-1) =-0.331 lla masas .2. = 0.7 X (.j'[ -1)" 0.290 Variable "x 'x Vx Q kN/m 40.1 6.015 1590 : G kN/m 30.0 1.5 5% Ey N/mm' 293.6 23.49 8% x" 1.0 0 . .06 6% Table 11.1 j I I I I ! ! ! ! I ! !G. Q ~., c ~ ------ ------ g -T- ., y. f11:U~ 11 .3 x'P 50.0 30.0 255.0 .~ .. 1;9;
- 205. 11.-1. ;IETHODS FOR THE EVALUATION OF PARTIAL COEFFICIENTS 193 '; ' .. "'~ . . . '.- and. the design values x*: . are,gi:,~1') by : .. ..-- . ;., c:. q~ ~, ~~.,+ ~.~~~~}}6:94 k~!~"" e; i:. IlE'" -t=- "E·:JldE,··::::i'218:4 ·N/mm1 ", "1 " "1.," ) ."1.,.. , x· :c Ilx + a Jla. '" 0.921 .m . " 1D: .. ~. :..;:; ... l_xm . :.:. ,, ' g* = IlG'+ tl"eJloc = 31.74 kN/m : ... :" J."; " '-,"1'_ ,-, ,- ,. , ''":1'" ~ ," . , ,.' . These values !1nd-lh~ ·partinl.c~~fficier:t~ found from equatio'~ (11.25) ~nd (11.26) are listed in table 11.2: '. . ,. .., . " ., ... ~ " .. ,.',-, Variable xsp -,.1 .. x . . ···1 C!', Q kN/rri '" 50.0 .' '; -56,9·' 1'.13 :'. G kN/~ 30.0 "'af:i 1.06 Ey N/mmJ 255.0 2.18.4 1.1,"I: , I' Xm . 1.0 ··· ·.0.92 1.09 j 'r, " .Table 11.2 By application of virtual work. the required plastic modulus zp may now be detennined from , ' / .' :-" : .. '4 ',' . .' £l ., ;~ " 10~ (1egs~ :- 1~,~sp}~ ,~;~4,( 1E"1 zp) ,1x,;;. (1l.3S) Substituting the appropriate values from table 11.2 lives zp :: ~89 X lOS, mm' .. Finally, i~ is ol-interest.to u~,the level .2 method to determine th-ereli nbiIiiy.o{this stnlc· .ture when the plastic modulus has the value found by the above method. The failure func· tio~ ca'n be 'written :is ,., ' , . ., . '-, . . .:.t;'l, ~ . , . ' . 'l~.! ' .• ' • > - , ' , ':' • ,-- ; ,;!"" :, ,1 (11.37) Using the' methOds of chapter 5 and the paramete~ Crom table. ~ 1.1 pves a reliability in· dex 11- -lAS. This is larger than the briginall~·' ~I~ted value of 4;0 showing that the approximate method of determining partia(c~iheii!!nt.s is safe, at least for the'structure and set of variables examined,. , .. i ' , ~ It must be stressed that great care must be taicen~ when using the sppmKimate method.' tor the evaluation of partial coeHici;~~~if ~h;·r~·~i~~'~.~i~~d~·'of th~ ~nsitivity C;ctDr.; '; ar~ not J~-' ~ . ..... .... ."._., .• ..- known. C:tre must.niso b~ l~_~~ .:whe~ ,ther~ is . @ppr~i,~J;?~~ .~~~is..!!~ ~nc~rtain~y i~ ~~l':arn· nU!ters of the probability di.st~~,~tiq!:l5 10r t~e ~asicf~ables ,be~ause of lack ,?f ~~ta:-:~.fethods of including statistical uncertmnt.v were introducec:i."in .'.s.~.t.i 'Ori 7" o .f c h"Jp.t"e r, 3 . S,e,e al-s,-o .{. 11.111I. . I •• -- ' ,r; " . "~" " , .,' ' :. ',,'"
- 206. 194 11. APPUCATlONS TO STRUCTURiL COD~~'" " 11.4.3 General method for the evaluation oC partial coefficieots 1 PracLil:a1 cooes should have the smallest number'of partinl coeCficients that, is cOMi:i~ern wilh reasonably uniConn standards of reliability; moreover, th: ~e .~anial coeffic,ients should be applicllble w a wide range of sr.ructural components. This means that they must be applicable over a fange of sensitivity factors without being unsafe or.unduly consen'aLive. A suitable gen· eral method for the evaluation of such a set of partial coefficientS is now presented. The first sta.,"e of this process is to decide upon im' a~prop~te standard oC reliiibility or target failure prob:lbi/ity (or the structures (or more generaU}', s.tn1c~ural c;:omponents, e.g. beams, columns, sl~s) that will be designed using the new code. This is aJso a pre-requisite Cor the pro~ edi1re' des~~bed in sec~iio~ 11.4.2. The"choicEifs'gen'eraUy'made by ~ process of probabilistic calibration to an existing code. e.g. see [-11.61. .... - . ... " Studies of the reliability of structural components designed to r.raditional codes typically show very wide r:mges of reliability. An appr~pri~~ choice;f!Jr the tUied:ii~ure probability Ph for a new code is the weighted average of the failure p'robabilities e~hibited by components designed to existing rodes, provided that the least ~iiable co~p~neDt exhibited satisf~ctot), performance in actual serrice. The latler is not always eas)' to verily because existing codes may not have been in use for a RlCficiently long period of time-and structures may h~ve been subjected to only a fraction of t.;eir design loads. Tbe weighting factors 'wi should be selected to represent the previous freque:lcy of usage of each structural compOlleni included in the calibration and should be, such that I Wi = 1.0. '. , .... . i" - ; :~." " ,' Use or the ,"eighted average failure probability rather than, say, the weighted average reliabllity in~ex mearu chat the target (:lilure probability Pn lends to ~ governed by the less reliable com· ponents in a:istine codes. This assumes a measu~ of ec~noal)' in the new cOde, but care has to be taken th2t'these-reliabilities are not· too low.,,: c . ' A more'dire=t appioacli'ti:i""Uie 'choice' Of target fiLilute probabilities has been recommended by the Nordic C'o~mihee on Buildihg RegUlations (NKB) (11:10). In this,-ihe 'target failure probability depencis on the consequences of failure and on the nature of the' fmlu"re mode, as shown in . table 11.3. , ; .::. ~: Failure .' ,'., , Failure"t yPe .' ." consequences ·1 11 Ul Not serious 10~ 10~ 10·J 3.09. :. 3.71, A ,26 Scrio~s 10~ 10-~ ~'. . 1O~. . ,,3.71 4.26 4.715 . V'ery·i eriol.i 10-1, '10" 10-'· 4 .~~ 4.75 5'.20 ',; -:;. Table 11 .3. Target railure probabilities and conesponding reliability indices (11 .10).
- 207. 11 .... METHODS FOR TIlE E'ALUATlON OF PARTIAL COEFFICIENTS 19& The target failure probabilities given in table 11.3 are for a reference period of 1 year. but . "should be treated as' operational or notional probabilities and not as relative frequencies. The failure types are defined as ductUe failure with reserve strength capacity resulting from strain hardening .-., " IL ductile failure' With no reserve capacity . »" • . ...•. 'm brittle'fallure and instability H,aving cho~en. ~ target failure probability. the problem of selecting a set of partie..l coefficients :; for a code, or part of a code, may now be reduced to the application of the following simple principle. Choose the set of partial coefficients:Y. so as to minimise the quantity S given by m S ~ I wj .:l(Pfi (7), P,,) j "' l Subject to the constraint m },-,; Wj Prl" '" Pit with and where m ,--Y," ' c.:,. '" 1.0 is an agreed function of the quantities Pn(lf and Ph' Cl1.38) (11.39) iii the failure pcobability of the jth structural 'c~~ponent desi:.,rned usini the set a! partial coefficients ;:. . is the tal"get failure pro.bability. W 1& (w11 ...• wm) is a set of weit/.hting factors. indicating the relative importance of each of the D1 structural components included in the partial factor evaluation: In general terms. the aim of this approach is to minimise the deviations of the probabilities Pn from the target probabUity of failure Pn . whilst maintaining the average probability. of failure at the target level. Experience has shown that the values of the partia! coefficients are general· ly very insensiti'e to the form of the objective function used (equation (11.38». Suitable func· lions are: ."" m S2 '" J: Wi(-Q>~I lPn (1)i -+ 4.1 (Prl})l j-l where ~ is tnt' rdiability index. (11.40) ellAl)
- 208. 196 Clcnriy. man)' other possibilities c:ocist~ Obt::aining the. solution to equations (11.38) and (11.39) is a problem of co~~ned mini"!is.ation for w~ich 11 number o( st3nd~rd t~hniques and computer programs are avaib.hte. Nevertheless. the-total amount of computational effect is considerable because all the probabilities Pfi need to be re-evaluated for each adjustment .~ th~ partial coefCicients :Y: The code writer is Cree to choose as many portial coefficients or additive safety~~emen!S as is considered appropriate for a given code. A practical n.umb:~~ is generally considerably Il)!ss than the number oC basic random variables . . reduction in the number of partial coefrlcients can,be achieved by constliUnirig 'the uftvented coefficients t9 be unity. Provi.qect~qua,tion fll.39} Is satisCied. the effect of these additional const~aints is '(0 'irlc~ease the deviations from the target failure probability Pu and to increase ~~e .av~:ag~_ ~,~.C?unt _t:?J Illa~rial used wh~n ~e~gning to the code. The penalty to be paid Cor irtcreased simplicity in- -ihe code-safety format is thereCore some increase in the initial cost oC construction. When applying this procedure over a number of codes rOr difCerent construction maierials, e.g. steel and concrete. a further constraint that should be considered is to ~&ke~the partial coef· ficients on loads and other actions the same in each code. irrespective of construction material. Such an approach has many practic3.l advantages. _ FinalJy, a note of caution. In chapter 5 it was mentioned that the reliability index as defined by equation (5.9) is not invariant with ~gard to the choice of failure function. A similar problem of hlCk of invariance arises when the partial coefficients used in a code are not directly associated with their corresponding sources of uncertainty. This occurs when the number of partial coefficients is constrained to some small number. [n such cases, the partial coefficients should be used only with the precise form of the design equations (failure functions) for which they ,were derived. 11.5 A..'l EXA..IPLE OF PROBABILISTIC $:ODE CALIBRATION The general method lor Ute evaluation or'~~ial coefficients which was described In the p~evious section involves a considerable a,J;l1ount of effort and computation and is not easily illll5tratcd by a simple example. For this r~~o"n., some results that were obtained during the'probabilistic cnli',mttion 11l.6l~rthe U.K. S~I Bridge Code BS 5400: Pan 3 (11.3) are included here as an illustration of the meth";;d".-'."---------,---·· 11.5.1 Aims of calibration BS 5400: p~ 3 is a level i ~od~ in whi~h the degree ~f structural reliability is contolled by a number of partial coefficients (partial CactorS. The code repiacM an earlier British Standard. BS 153 [11.11 and was developed mainly for the purposes of incorporating technical improvements in many oi the design clauses: but at the same time the opportunity was taken to ration· alise the safery ~!o ... isions and to chanoge from a permissible stfe~ to a limit state approach. In evaluating the partial coefficients, the a~eed policy was to achieve the same a/Jerage !elia.bility for components designed to the new code as the C1/Jerage inherent in deSigns to the pre· ·,iQUS code 9$ 153, but at the same time to reduce the scatter in the reliability of the 'arjous
- 209. 11.5. AN E... . A;IPL£ OF PROBABILISTIC CODE CALIBRATIO:.l 197 components. An ob'ious'liriiifutioifo('this ivork is that it' was restricted' to 'a'~t'u'dy bf'stru'~tural components rather ·than structutaI systems: " " -, .. ;.. . "'-,;.l:' ,. - ; ~,.- .-:, ),;. , _ ._ ,_" -' 1':~'-~/0 " "'1"- ,. " - , • - . • ~,. -"'a" ~". ' A , ~;ow~-i:"~var:. s~-:~;ing_~~~: ~~~~~"~~~~.~~ ;j~~;al:ulation pr~ed~re i~,give~. m,.~~~,,~~::: Obtain 'd~ta o~ . . "', lOad an~( sttength' parameten ". " '-' ... ... ,. c: '0'; .". ·1,".' .: .... . -cc ...... "" . DeCin'e set of,s_tructu11l1 compo.ne.nts am~ ~yeightil1g ";'fa ciorS w.' based -on' frequency of usaae: such that LWI;> 1.10 .< , '.~ ... -.... .':: ; ; ;~; ;;;, .... " " . ; "1'" . j " .. ~m;"'od0~,;'1;S,1~(;o!·~i,lc~O·1.~d-.. S ..i.. '.'-n'. J" ·.'' .'. ~ Determin!! failure probaQ,i1ities Pn 1&3 and strengths L-----------r-j-------'-----' I Determine Pet '!'.~w_d'tn53 }( 1 ,' ... , Choose approximate values of partial coefficients:; Fi~re 11.·1-. ,Pr90abilistic calig~~tion · oi 85 5400: Pa~ 3 to B5,153. ,i. 'i.
- 210. 19S 11.5.2 Results of calibration . . Figure 11.5 shows. l.he sC3ttl~r i~ the .c~mpu~ed failUre pr;o,baJ:?ilities ior the major itrpctural com· " ~-nent:s desi~ed to "the umiu of BS 153 which were)n,cluded in th~ caliocationcalc'ulations. The failure prqbabilities exhibit very wide sca.t~c:r vary'U)g. ~.Yt:~ man)' oniers,ofmagnitude. In addi~ion: th~i~ a;.~ $igniricM~' di!cerenc~s' i~ 'th~'av~~;g~"';;li-~biii~y ~'r &ii,,;;~~ tYPes of co~·po. nent. Neither of these facts is surprisinc: since the code was originally based on det.enninistic concepts with no regard for the relative magnitude of the various uncerta.b1ties. It should be noted that the modellint: did not allow for-the possibilitY '9f"gross elTGIS in deSIgn or ,construc· tion and for" this reason the' probabmiie's :shci~ld be';h:;ldq;;~ied as a ~~ of ~ia'tive safety • ' . "'" •. • ,"' ~. " I" " ' "" " " ' " and not as failure frequencies. The target failUre probabUitY~PfL" for the new code as 5400; Part 3 was determined as the "'eighted average of thdailure probabilitiedor components -designed to BS 15~ and was 0.63 X 10 .... In caJculating:UltS 'valu'e";"'s"tiffened compression flanges aDd unw~ plate panels were excluded, the former because they had not been"shown to behave" satis!!:ctorllY in service and the latter because the data o~ model uncertainty were coruidered ~gua&e':;": " :,"," " " . The partial coefficients for the new code were determined for use with checking equations of the form Ihm2 funct.ion(fyhml. other parameters) - ef~ects of t'YfOI GI • 'Y!G2G2' "lfQQ] > 0 (11.42) where fy iJ the yield-stress of the steet, Gl is structu;a(self w~ight. ' G2 is superimposed permanent load, Q is traffic loading. ".1Ql is a partial coefficient on yield stress which applies throughout the code, )"m2 is a partial coefficient on the~computed resistance which varies "itb type of component, and lfGl.-.rrC2 and 'trQ are pa.~ial ~~fficients o~ loads. ~aiues.of the partial coefficie_nts obta:!~ed by mi~imising~e_" quantity SdeCined by equation (11.38), subject to the constr~nt given by equatio;n (11.39), arc listed in column 1 of table 11.4. TIle other columnS in this table show the values of partial c~!(icients 'm2 when other constraints are introduced: For example, -colu;n~ 3 shows the effect of setting 'rm} ~ 1.0, lfQ • 1.5 and 'reG =~~.l~ _(given ~er_~. ~ a ~ei,h,~" average of 'rIGI ~nd ,7m2)' an~ t~us effec?vely eliminating "ml from the code. r.ne quantity S is given in :he ~ penultimate'row of iable "11.4 fof~ach" of the' SetS~ or partiiu coef· ficients calculated and car: be seen to increase as additional constraints are introduced. The, ql!a.. .. nity I WjBj given ir. :he 18st row of the Lable is the ratio of the ~mount of steel used when
- 211. 11.5. AN EXAMPLE,OF PROBABIUsnC CODE CALIBRATION 199 ., . Stiffened (om- l-----Bz:J:3::$I------:---:----------- prtnion'n.nlci r 1 InerlluinJ nfely We~ ~-:----------f3E!~B!3HE!----------~------~------- o ; 1f.?fl..r**lnOl inclu6ed iD dutrminltion oJ Ph Filu.e 11,5, Failu.e prol>abilil!es lor eompDncnu delilnllli to flS 153 and earl)' nan~" ruh,s (lrom 111.61), t PI~· (/.63 x 10-6 I Strul.l r-------B:rTl-mHfI--IfI~I!"lH!h---------- T~I NQwcOornp()siu· b".m (i.nles , " n:,:; ' ",;;;, I. ." ,. i ~I SptriefCwean(e!d n ca(onm"-" rL ---------j,,-!jTi-. ~·-____ IIi 1II. ~111E1 3I' mI: E3-------- Less ·steel I Mon, utel Webs ~-------------jll..,.~~:~~$~~:.~....:::::.::.:.::.--:..:.-= ,,,.,' I : .. , 110111: 11111 0.5 , o 'c=:J ranee of C.ilu~ ptob.bilitie$ P, cg (hanec in 51HI us:1,C /) 1.~ Fi~url' 11.6, FaiJu~e pr(lbab!lili ~ 5 rot eompunenl5 IiL-si&nltd I ... 85 5400 :ono the eH1IC1 on ,L'<! IIUIt; ffr~ml 11 1.611.
- 212. 200 11. APPLICATIONS TO STRUCTURAL CODES 1 2 3 4 5 6 7 Funy op· timised , .. coefls. Increasing constraints .- . 'Y.fG .. 1.16 1.13 1.13 1.13 1~13 1.13 1.13 ,., T'Q 1.4'7 1.50 1.50 1.50 ' LSD 1.50 . 1.50 .. 1ml 1.08 1.08 1.0 1.0 1.0 1.0 1.0 '-----~-~-- L_''':''' -- e-:-- --.' 1---- ! "(~2 for: I I I I Struts 0.98 , 0.98 1.03 1.05 1.0 I ! Beam nang'" 1.09 Loa 1.17 - 1.15 1.2 Stiffened com· .. pression Danges 1.27 1.28 1.37 1.35 1.4, l.3 1.35 Webs ' . 1.25 1.25 1.34 1.35" 1.3" :' :. Plate panels 1.08 LOB 1.14 1.15 1.1 Ties 1.09 1.09 1.18 1.20 1.2 ~..wiPnlX 10 .... 1 .632 .632 .632 .658 1.146 1.072 .0.288 S , .073 .086 .142 .225 .282 5.95 8.80 I'T" WiSI .936 .938 .939 .942 .933 1.00 1.04 Calculated !t.ounded or ~.rbitrary : coefficients coefficients Table 11.4. Partial coeCf~elents . ror various degre2s of constraint (fro':1 {1l.61).
- 213. BIBLIOGRAPHY 201 .,' ~; ' .' . designing ith the new'co'de (wi"th :the partial coefficients give"ri} to thetlmourii'used when de. signing wJt~,..original .cq~e B~ 1.5.3. U,se. of,any or. .th~ se.t.s .of pan!~l coefficie.n.t.s in columns ,1-t V!?~idJ?~~:~,f.f!~~~~'.~~~~~i:!~g,,?i~;.~.f:;8~:i~,~~~IY(,~~ i~ ;i~ei.~o'~~.~rnp~i~<~ ~~,T.p_~ed w"tth -as 153. Further constraint! on -the nU~~~,r.?f p~~t~~co~!!!ci~n.~_,~s~ !~U~~ ~N~ ,~~!p_~- In practice. tbe balance between the simplicity of the safety fonnat and the savings in material must be de-cided by'"the code.writing committee." ::.' . No code calibration study ;h6'~ld" be:conside~ed to be' co~plet~ until 'the 'err~cts of the change in safety format on the design of all components within the scope of the code have been examined. It is important to know the range of failure probabilities for each type of componl'nt when using the proposed set of partial coefficients. It is also of interest to know the changes in the quantities of materi3ls that will be used compared with enrlier codes. The latter is perhaps the most tnngible measure of change in safety levels. This is illustrated in figure 11.6. BIBLIOGRAPHY 111.11 111.21 111.31 111.41 111.51 111.61 111.71 11 .1.81 {lL91 British Standards Institution; Specificalfon for Steel Girder Bridges. BS 153, 1958. British Standards Institution: Basic Data for the Design of Buildings. Wind Load3. C,P. 3: Chapter V,Part 2. 1972. British Standards Institution;Steei. Concrete and Composite Bridges. Part 3: Code of Practice for Design of Steel ~ridges. BS 5400, Part 3,1982. CIRL: RationaJiSiltion of Safety and Serviceability Factors in Structural Codes. Construction Industry Research and Information Association, Report No. 63, 1977_ Euro·Code No.1; Common Unified Rula for Different Types of Constructions and Jlaterial. Appendix II: Guidelines for Determining Partial CoefficientS. Draft. February, 1981. Flint. A. R., Smith. B. W .• Baker. M. J. and Manners. W.: The Derivation of Safety Factors for Des{gn of Highway Bridges. Proc. Can!. on The New Code for the Design of Steel Bridges. Cardiff, ~Iarch 1980. Granada Publishing. 1981. Joint Committee on Structural Safety, CEB - CEC;·[ - CIS - FlP - IABSE· JASS RILE).- t: International System of l!nified Standard Codes for Struclure3. Volume 1: Common Unified Rules for DiflereJ?t Types oC Construction and Material. CEB/FIP. 1978. Joint Committee on StruCtural Safety. CEB - CEC~1 - CIS - FIP - L-BSE - {ASS RJLE) I : General PrinCiples on ReUabiUty {or Structllral Design. International Asscci:.tion for Bridge and Structural Engineering. 1981. Linci.~ , C.: RdiJJbility-Ba.sed Strucrunzl Codes - Optimisation Theory, International Research Seminar on Safety of .Stry.ct~res und~ .D,ynamic Loadins:. Vol. 1. Trond· heim. June, 1977.
- 214. . , ~,: 202 11. APPLICATIONS to STRUCTURAL CODES 111.101 Nordic Committee on Building Replations: Recommendation j,o r Loading an~ Saiety .Regulat.iolls ror Structural Design. NKB:,Rep9rt No, 36, Nov. 1~78. . _._ i 11'1:,11J 'RaclFY:'itz. R.: lrnplemell-t~hon' 'of ProbaoUlstJc S~fety ·Conc'f.!pts iii 'De~lgn. and Orgallisa. 111.12J iiOr:a, Codes. PrOc~edings of the 3rC! iritern~dOri.J Con!~rence on Sirii'cbhaI Safety and Reliability, T'r6ridheini, 19S1-.'Ersk~ier.1981~'~: . 'Ii," Parimi, S. R. and Lind, N. C.: Limit State Basis {or..Cold. Formed Steel DeSign. Proc. ASCE! J. SUllct. Div" Marc~ 1976, yolo ~O~J, No: ST3 ... ,
- 215. 203 Chapter 12 APPLICATIONS TO FIXED OFFSHORE STRUCTURES 12.1 INTRODUCTION In the previous chapter I structural reliability theory was found to be a valuable tool in the de· velopment of rath:lnal safety formats f~r design codes and for the evaluation of par~ial coefiicien~. This is a major a~ea of application. However, the main purpose of codes is to ens.~re that structures are designed and built in such a way that they can safety resist the (generally uncertain) loads to which they will be subjected; and it is therefore logical that reliability theory should be used directly as part of the design/analysis process, when appropriate. The main pote~ ti'al for ?irec:'t application is with s~ctures baving large failure conseque.nces 'or where the use gives immediate savings in construc,tion costs. Other areas of application are in the assessment of existing structures and in research. The examples given in earlier chapters are mainly of a simple nature and are included to illustrate the various calculation methods. The purpose of the present chapter is to discuss more complex structures where the direct application of the theory could be used in the process of safety assessment. Offshore structures have been ch'osen because of theirecon~mi·c importimce and because they are toe subject of much othe.r res"~arch. Consideration will be reo stricted, however, to steel jacket structures, on which a number of detailed reliability studies have been carried out, e.g. (12.21, [12.26J. Nevenheless. many of the principles are equally applicable to other types of installation. In the following, attention is focussed on the reliability ot offshore structures under the action6fextreme environmental loads (wind, waves and currents). Various aspects of the 11)0- "delliiig of:iacket stnictures are discussed. but not the broader features of oUshore safety such as bl~";"'.outs, fires, E'xplosions. collisions and other h~~ds. For information on these topics, see, for: example, 112.15]. 112.27]. 12.2 MODELLING THE RESPONSE OF JACKET STRUCTURES FOR RELIABILITY ANALYSIS Jacket structures are currently by iar the commonest type of fl . ....:edoffshore platform and are likely to continue to be used for oil and gas production in moderate water depths (up to 200
- 216. 204 11 . . -ppI.ICAnO~S TO fIXED OFfSHORE STRUCTURES :, .. -,.' (" (" Figurl! 12.1. Elentions or various 'jacket ~ttuctur .. 112.xxl • • ;> '" .. ' ) -,,' m, ~or the foresee~bte future.; .. ~ltt:tough in greater depths, structures ~uch as tension leg plat..forms rq .. p·s) may be fou?d. to be an ec(>nomic proposition. Existing jacket structures range in height from a few tens of metres. as found in shallow coastal_wate!3:' to over 300 m.-TheY also ~ary app~e'ciablY with regard to the number of main legs 3IId the ammgement of bracii'tg, .. ;th consequent effect on the degree of structural redundancy and the,;unount of local damage that can be sustained prior to collapse. ~levatlons of a number '~i di!Cerentjacket struc- , " rures (drawn to different scales) are shown in figure 12.1. Structures such as that shown in figure 12.1 (d), the Cognac platConn in the Gulf of Mexico, are immensely complex. employing thousands of tubular members and welded connections in the main jacket alone. For reasons of size. t~9 actual res~~nse of such structures to,;,vind and wave loads WIll be subject to consIderable uncertainty. but most of the pnncipal sources of uncertainty are clearly identifiable. A major difference, between the early shallow·water jackets and more recent offshore structures is ih ~heir response to wave loading. The smaller structures tend to' be relatively stiff and to c,a rry smaller deck loads and are thus not subject to sigmficant dyuamic response. In . addition. they tend to be situated in more,sheltered environments; ~todem detp·water struc-tures. on the other hand. arc likely to experience appreciable dynamic motions and the reliability of such st~cwres cannot be d~ter~ined- without a.full dynamic anaJysis. The reliability analysis of a large offshore jacket involves the same steps"that: musL'l>e undertaken i!1 a co~v~n,tiona! stochastic dynamic analysis, but in addition the main structural 'and loading variables and the parameters of the wave spectra are treated as r.uKiorn quantities. The sources of uncertainty affecting the tiehaviour oC dynamically sensitive jacket struct'ures • .. . 'r. ~ e' " can oe grouped under four main headings
- 217. 12.2 MODELLI:-lCi THE RESPONSE OF JACKET STRUCTtiRES FOR RELIABILITY ANALYSIS 205 .. . i'l ". '. : 1, .:'.' :,: .• ; 'J . " • .Those a((ecting the loadin'g; . • extreme wind s~d ~~tr.e~~ Cl:!':.~~.~.:p'~~ . th.~s'pec~r.al:form of th~. extreme sea·state the extent of marine growth ", ".,- hydrodynam.ic forees, given.t.hE! · w!l·~r particle velOCities and accelerations perrri;ri~~t ~n'd"serrii'~-er~~~~t d~k':;loads . '" vari~ble 'declt' loads .. ~. .._ ..• . " ,. " . ~. : ..... Those 'affecting the stru_ctu~!1l .respon.se:. the eCC(!ct of unc(!rt:1in scil propenies on natur .. 1 frequency .• '· ~ ': .o· · -: thedfect of 'arlable deck loads on natural frequency . stru~~i.i;~l and·hYdrOd;namic . d~~·Pir'g . the' pea'k resp(m~ in a '£lven refere~e p~od given the rpot mean square (r .m.s.).respense .. '.~ ::.:' : -, ~' ::"' •. '; ' : ~~, : .. ~ .; ... . ~; .• :. . ;~., .. q ;Ii;' - '. • Those affecting component' stren'gth: ' ,_ material properties • geometrical i~~rirc~ions r: . • ~ mode,l. un~ert~t.i~i i~.p~edi'?ted c:~mponent strength • Those a.flection system_be~;i~~f:" '" The purpo.~~,ofa : 5tructunJ.reliability analysis is to incorporate aU these sources or uncertainty within a S~l)gJ~.!e(of·~:atcui~ti(:ms which ,can 'then be used to predict the relative likelihood of each o.C a large nurrib"er ,oC possiblerfailure ;events. From this information and with knowledge of the consequ!~ces of the. v~C!ti~··~~.i1~!'e< d~~io~s can be made about the overall adequacy of the structure and the stren~h oC individual components . . .:;c·~: · :,.' .. .... . . ·-':- f; ,· .:' To be able to undertake the'se calculations it is first necessary tl)' develop appropriate probabil- " .•• ' . ,., '., ' . ' .. ' ," " ., . ',' '. ", .,. " _' " I .' , . • i' ... ... ~. . '.' .•• ':,." . ' istic'models for the basic load and r~sis~ce ~ariables. as discussed .in . ch;lp~e!"; ;l. 1n addition. further deterministi~ lor sto.chastic) m~de1s·.re" ~quired to'relate-the o,;iriU 'resp'onse of the structure. .t.o. t..h..e. a..pp..l.i -.e-d.. l.o. -ad,.i-ng" ,. ..a..n...d.. _ t. .o. define the behaviour of the indi;dual structural com- ·~ .. ·-··-I ponents in terms·.ol.the overall resp·onse .. The.word model is used here to emphasise the Cact that th~ actucil· bl!!hiJuiour of·all structUres is'extiemelY complex and that all calculation pro- ; ..... - .. .. i' ;, :. "·,1 I cedures ,! n.v()lv~.~ .~~~~.er of !~!!!!.aE~J:1~~. f.o"t .~~~mple, a typical jacket structure has a large (strictly infinite) numb4ttofdegrees oC freedom ror dynamic response, but with careful mo· d~lIing all the signific~t c~htributlons to the response can be obtained by using a relatively small number of lumped masses. Figure 12.2 shows the main models and calculation ueps in the level 2 reliability lUIaiysis of :1jncket; strUctur~' usin~ s~ec·trai ::i'na.r;;si~. TJie prO:b~ss i1lustr:1ted is ro~ ~~~· ,~·~t~rr?i.~~~i~n ~f . the reliability of component members of the structure. conditional on no other members having {ailed. and where !allure is oi theJ.i!:t-.passage t~:pe. For multi-ehment redundant
- 218. I . . . - . P.d.f. of extreme wine.! 'dodt~· or joint p.d.!. of wave height and period I P.d1. of drag and inertia eoeff-icients Current velocity profile and I p.d.!. of surface velocity J Spatial distribution of marin,e V growth and p.d.!. of thicknesS at m.sJ. Mode shapes and frequencies . {or mean mass and stiffness properties I .. P.d .Cof materi."l pwpenies'and I . I geometrical imperfections I :t .. ". - , - .'" •••. ' "ui,. -; u HXi't u .... SIIU". STH"CTC'.~, ' . SEA-STATE MODEL: I Spectral representation of waves for I extreme sea·Slate , I I , W.- VE MODEL: • Calculation of wilt>:!! partic1t! velocities and acc:elel'ations at different depths LOADINC MODEL: " .Calculatic.H.l of hydrodynamic forces and damping, taking into account re-lative movements between jacket and water and spatial separation oC piles f- , NATURAL FRE QUENCY MOI?E~: Modification of mode shapes IUld ha-turarfreQuencies to allow lor variable foundation compliance and variable deck loads ... RESPONSE MODEL: Calculation of variance of structural displacements . .. ~ EXTREME VAI.UE MODEl.: C31culation of peak displacements- STRUCTORA"L ANALYSIS MODEL: Calculation of member lorces and mo· menLS and 'hence the _totalJoad effect S in the member of interest MEIoIBER STRENGTH MODEL: Calculation of member load-carrying I capacity R I ." Calculate trial value of reliabilityln- .dex ~ and determine better let of val-ues ,of tlW .basic vari?bl~s i-i .. ., I p,P, I Figure 12:2~ Mod~1s' ~d calcul~tion 'steps in the 'level 2 reliability analysis '~! a jacket struc- .ure using spectra! analysis. " , !, , ! I I I , I I
- 219. structures this is only the first stage of determining the system reliability. but the results can be used with the methods of chapters 'i and 8 to calculate upper bounds on the system failur!? probability. Accurate lower bounds can be found only by examining the sequences of memo ber failures which result in complete collapse. This is the subJect of actiVE; research and is be· yond the~e"'ne of t h .. ;-resent book. h should be noted. however, that the failures of different structures members are in general strongly correlated because of common loading variables. This has the effect of reducing the separation between the bounds. Moreover, for structures which are dynamically sensitive, the change in fundamental natural frequency following the »brittle)) failure of just one major component (e.g. by fatigue or fracture) is likely to lead to a significant increase in dynamic response followed by rapid failure of other members within a short period of time. In the following sub·sections a brief description is given of various loading and'response mo· dels for dynamically sensitive offshore structures that can be used within a level 2 reliability analysis of such a structure, 12.2.1 Sea·state model Waves are generated by wind blowing over the surface of the sea and are the m.ajor source of . loading for most offshore structures. At any fixed position in the open sea, the level of the water surface varies randomly due to the passing waves and may be modelled as a stochastic process X(t), where the index t denotes time. As mentioned in chaprer 9, the propenies of X{t) which are of grel,ltest importance are the mean value function J.lx{t) defined by (see page 147) (12.1) and the autocorrelation function R~x(tl' t2) defined by (12.2) As previously discussed, when the firsi and second order'distributions of X(t) are independent of absolute time, the process is said to be weakly stationary. In this case, the autocorrelation function is dependent only on the time separation l' = tl - t2 ana may be written as RXX('f), It is clear that the level of the water surface at any position in the open sea is not a stationary p~oces.s, because of changing meteorological conditions which are to some extent both season. al and cyclic. However, for reasonably short periods of time (3·6 hours) the assumption of stationarity is no~,unreasonable. The reason for this can be seen by examining the frequency components of horizontal wind speed as shown in figure 12.3, In t.emperate latitudes major meteorologiCal u;;~'lrbances occur v.ith a typical frequency of about 1 every 4 days and the wave cC)Jlditions that are generated build '-!p :md die away at the same rate. with the worst conditions lasting about 6 hours. The process of wave generatloli i~ cOf"'piex, depending on the distance (fetch) over which the .... <lnd blows and many' other faciors, e.g, see [1~,.2.11, 1;:". will not be discussed here,
- 220. 208 12. APPLICATIOXS TO FIXED OFFSHORE STRLoCTt:RES 1 (mISJ~ I 5 t I ~ ..1. ,1 i ,1- i 11 O~~----+I:------+~--=::;==::::-r------~ ______ ~I~~~ 10 10 . 2 10 .1 10 100 1000 [rQ11Ueney (cych.'J.i10U,1 ~'i::ure 12.3. S~celtum or hori'.wntal .... intl Jl('('rilt B'ookk~~n 112.3. J Spectral representation: Ie the variations In the water surface elevation aboue mean still water lel-'ef are denoted by TILt), this quantity may be modelled as a zero-mean. continuous-space. continuous·tlme stochastic process, which. over short periods of time (say. 6 hours I mny be assumed to be stationary. The autocorrelation function of t~e process Is then (12.31 and the mean square .~pectral density tor spectral density) S~II(W) is defined as the complex Fourier transform [12.30J of R'1I1(T) and is given by 1 .- S~,,(w) · 2" Rq.,(T)e- i .... dT (12.4) '-- where w is circular frequency in radians p~r s~ond. (Note: Fa" equation 12.4 to apply, the mean value or the process p,,(t) must be zero I .. The associated inverse relationship or inverse Fourier rrun.sionn enables R'l'l(r) to be reco."~red from S~'l (w) and is given by R'l'lC'r) " "" S~'l(c..I)eil'Hdw (12;5) '-- Equations (12.4) ond (12.5) are known u "a Fouri~r transform pair. S~~(W). ,as defined b~~ equation rl2A ,. is a real. non·neE;:ath'e. !~'en (i.e. S~'l(~ I :z ~~'l (-w» function of Col. , ' . , . . , ' ,. ' ., '" ,- It should he noted thai the fact_a! li2Jl' ~'nich appean in ~a.ua~ion (12 . ..1) is pul instead in equa-tion f 12.5l by some auihc"~$~ .. ~'e~" important propert~· of the spectr3i density is obtained by ~ombin'ing equations (12.31 and r p.51 and setting r .. O. namely !
- 221. , ": , '.-:," . ::.: 12.::! "tODELLIXC; THE RESPONSE OF' J,CKET STRL'CTL"RES FOR RELIABILITY M':,-L i"51S 209 Le. the ,area under the spectral density cun'e is equal to the mean square of the process. But. using~qu~rio~ (2 .. il)·it· can easily be sho~'n thot and, since iJ , is zero. it follows that where a~ is the variance of the process. (12.7) 112,8) The {orm of the spectrum S~IJ(w I deflned in equation l12..l1 is known as a double-~ided spec· trum and im-ol~'es the .concept,or.-negative as well as posith'e frequencies. This fonn 'arises na· tur:dly·from the basic definitions. bu.t is noUhe most helpful far :engineering purposes. For the latter. the equ~"a!ent one-sided spectrum S~'1 (w 1.0 C; w " .... is generally.adopted and will be used here 'e_~_ fiiUt'e 12.3), Clearly. E!Jj: I tH" )~ .. ~~'1.(w)dw .. ~ :.~~'1 (~.Idw 112.9) . ·and thus S":'l '(W) - 2S~.,{w). If the circular frequ·ency ·w . me.uured in radians per second. is .. replaced by cytlic:frequency f. 'measured in Hertz rcycles per'second), then it is eas~.' to s:~ow that (12.10) [Note: Confusion between single and double-sided spectra ;).nd between spectra expressed in Hl!!rtz and those in radinns/second is 11 common source Qf error!. ":;j. .• .. ;'1 Practical measure, of sea-staletThe height H of a single waxe is nonnally deOned as the total range of 11' tl in the time interv,al To between two consecuth'e zero up:crossings bY·1Jlt). see figure 12.4. To is the corresponding wtHle period, and in general more than one ma:cimum and one minimum of nit) will occur during this time, ' .. -, The sea"state itselfi, characteriied ,ty the'distribution O(:W3'C heights and penods. Two of the most common prac'tical measures.o£,sea·stateare )lsigniCicMt w'ave heighOt ' H~' and _mean zeroc rossing period. Ti:' .H, is defined ninhe menn hei'iht of the highest one.third,;of·all the waves recorded during 11 period of obsen'ation T. 11na is sometime! denoted H1. j :(or-Hll.:3}). Tl. is the mean of the s.equence of times To bttween 5uccessi'e up-croisings ot" the mean still water le'el during the same period of observation T. Both H~ and T'l ine thus sample statistics relatin!;! to the obsen'eri sea·state. As might be e:o;pccted, H, and T, 3re not independent. the !ar~r values of HJ generally being associated with W1H'eS of longer period.
- 222. 210 12. APPLICATIONS Tb FIXED OFFSHORE STRUCTURES . ' " ,", I [' I J Fil:U1C 12.4 , Wjegt:l: 112~381 has obtained a regression "relationship between log Hs and log T:r. based on data ·Crom the~North Sea and the Gulf of Mexico: When H~' is measured in metres, Tz i,n seconds was found to be given by . .. :,'" (12.111 Howeyer, as shown by Draper and Squire 112.14J in data· obtained for a complete year in the North Atlantic, there is considerable scatter in the relationship between HI and Tz. Houmb , and Overvik (12.19) have.fittl!!d·a r,wo'p&J:ameter, Weipull distribution ~o3925. samples of data obtained from the oorthern North Sea to determiOl! a conditional distribution of T:z: given H., as follows . (12.12) where· .1,:. , ! ' . .. ' with h .. in metres. :Not surprisingly. (Wre are.discrepancies between the results obtained during different investigations, probably. ~ to .differing fetches and non-homogeneous wind Cields •. but:for heavy seas (large· H,) the various predictions or Ti are quite close and the conditional variance of T~ reasonably small. t: .....
- 223. 12,!! MODELLING THE RESPONSE Of' JACKET STRUCTURES FOR RELIABILITY ANALYSIS 211 Praclical spectral {orms (or water sur{ace elevation, TI(t) : Research by 'arious oceanol!raphers has led to a number oC semi-empirical expressions for the (orm of the spectra 5.,,, (w I of waler surface elevation TIlt), (i.!enerally called wa'e spectra). Two commonly used SpecLJ"d are the Pierson-Moskowitz (P·M) 112.311 and the JO~SWAP 112.171, pjerson and Moskowitz found that for fully developed seas (Le. those that occur when the wind blo ..... s uninterrupted over an effecth'e1y unlimited fetch) the wa'e spectra approximate to a single form which is dependent only on "dnd speed and two dimensionless constants Q and /1. In its original Corm, the P-M spectrum was given by (12.13) where w is frequency in radians/second Wo .. glu19.S g is acceleration due to gravity (9.81 m/s' ) u19.s is-wind speed in mls at a heigh~ oC 19.5 m above mean still water level Q .. 0.0081 (Phillip's empirical constant) ~ c 0.74 Making use of equation (12.10), the P·M spectrum may be expressed as a function of cyclic frequency f, as follows O"C<- (l2.14) The spectrum is shown in figure 12.5 tor ... mollS values at wind speed ul~. 5' This illustrates the increase in wave energy with increase in ,ind speed and the corresponding decrease in the Crequency of the spectral peak (increase In dominant wave perioJ). Th~ frequency of the peak fp is obtained by difterentiating SI)r, (C) 'with respect to r and equating to zerO, giving r = (O.8/1)O.~S -"P 211'u19•5 'Finally, combining equations (12.14) and (12.15) gives o<;C,,- . ;" (12.15) (12.16) The JONSWAP spectrum was developed from the results (Jf the,Joint North Sell Wa~c Project 112:17] and" ~pplie$ to wind genemted ..... aves under.conditions.oflimited fetch :md homogene· ous wind fields .. 1t is gh'en by
- 224. 212 t 2. APPLH~A nONS TO FLXED OFFSHORE STR UCTURES tS'l'l1fl ! £00 ! : U19•S • 30 mJs 4001 200. 0.05 0.10 FiiU~ 12.5. Pierson·M05kowitl' .peetrum Cor vMoUl";nd lpeedl. u19." . . . • f S'l'l (f) :::: (2:1: ri exp(-1.25 (Tl),a where a .. 0.076 i-O,2:! r I" 3,5...L i-O. 33 P ulO . a .. exP(-t ({ - fp}l Harp)l) '1 is the peak enhancement factor (typically in the range 3·5) and where the non-dimensional fetch i .. plUtO " . g is <l.cceleration due to gnvity (9.81 m/s:) • • ~ 00 :>:":: . x is the true fetch in metres u to ~s th~ wind . sp.:e~ in m/.s a~: a height of l~ ~ above mean ~tiU water level a .. cit = 0.09 , O.l!! (12.17) ·(12.18) (12.19) (12.20) "-' . The form 'ot equation 112.1i) is similar to equation 112.16) but with the addition of" peak enhancemen,t term 'l' . lbe exponent <1. gi'en by equation 112.20L h:lS a mlLximum I!alue of unity whichjoccurS when f .. Cp' The peak of the JO~SW..P spectrum is therefore l' times
- 225. 12.::! ~IODELLING THE RESPONSE OF JACKET STRt.:C·Tl·RES FOR RELIABILZTI- ANALYSIS 213 th~,'equival~nt' (Mpe'nk;. ~'howing' that m'ore en~rgy :5 'co~cent;ated in the;narrow band of fre~ que~~ie5 ~~~~ndth~ p;~k. It'~vill ~~o be .seen iha~ th~ p;U~meters a mc:i f~'are both dependent on the' n~n~i~en~ibrial fetc'li para'meter i. both dec,ea~lng as' the non-dimensional retch is reduced (e.g. if the real fetch l( is reduced while the 'wind speed' is:kept conStant);'; ,'. . .... "., "il.·· "'." .... ,.- ....... ' ,.--, '"' Fot::gi~'en values, of Ct andfp ' there i~ greater ene~gy associate.d, 'With the JONSW AP spectrum th~ ~th' the 'p.~t spebt~m 'since:~vith'tbe 'eni1~nced peak: the are~ under the JONSWAP .. spectrum. and hence o~ (t) (see equation (12.8», is larger. Hmvever, it should he noted that the two ~pe~tra are not directly comparable because one is for fully-developed seas and the other is for ~onsidions of limited fetch. A further difference is the neight' for "the reterence wind speed. The JONSWAP spectrum. for a fetch of GOO;"m and 'arjous wind speeds. is shown in figure' 12:6.:1' -' ,: The JONSWAP spectrum has fh'e independent parameters. x. U~'o.'''( ,"oa ~nd'~b' some of whi~h ~a~::~e ~reated as random variables, In the JONSWAP experiment, the shape parameters ,,(, 0a and 0h displayed considerable scatter but. unlike the parameters a and f p' showed no significant trend with i. Their mean values were 1.1 ~ 3.3.1.1 ~ 0.07.1.1- !l: 0.08. However. the most im- 'Y. ,'.'" ,,~ "." ':;1>' .,." . '.':," portant quantities are probably uIO ' oX and ,,(, all of which should be treated as random variables in the reliability analysis of a dynamically sensitive offshore structure. Because fetch is depend- , : ' ' ' . , ent on wind direction for any given offshore location. the information ideally required is the joint probability distribution of extreme wind spee~ and dir~~io.n. toge~her with a suitable probability distribution for "(, Use of parameterized JONSWAP·type spectra: In the preceding paragraphsthe.Pierson-Moskowitz and JONSWAP spectra have been discussed in :he contes.t of finding suitable models for ~elating e~t;em'e wi~d' ~p~ed't~ sea-state, for the purpose of u~dertaking 'iel~bilityanatys~s of dynamically sensitive offshore structures,'If the par:lme'ters oi the p'robability'(ijstributions'~f t S'I,,(C) m~/s 150G!- j 1001)0. U10 "·Him/s MS" Fi~ure 12,6, JONSWAP spectrum (or "anous wi~~ speeds, u: O'_ / 0,10 .Cetclt 600 km .,." 3.3 cr." 0.07' cr1>" 0.09 0.15
- 226. 2H I extreme wind speed are known. then thi~ appro;Jch can be adopted (see also section 12.3.1'. However •. in. .. s~~,e. ." a.!!~ s.uff~~i~~t st3.tist.i~al, ~~_ta o~, w~ve. . ~ei~h~~. ~~,d ~.~,';1~ ~a~:~.~7 ~~'~ilab!e to allow the. reliability anal'st lO fonnulate the problem d:rectl)' in .terms of the probabiljw " '. "' , "~,,, . . ' .. ""'~,., , ' " " '.,,, ,,,· ~ 'i": '-; "1<7 ••• -.,;: disuibu.tions. 9f,th~_,~~~:~t<~;t~ p':arame.te~",~.g~ Hi ,and Tz ' l~ , s.~~? "8:5es." i~,.~.s, ~~.c:s.~a;.~ ,~~, ~~.Iate the wave spectra t.o. thes~ sea,st8t~ . p~meters, ~ .. ,;.~! . ~ i-. ,': " : If fl( t) ,is assume~ to be a nati.onary zero·mean Gaussian pr.9ce,~s I(see,. chapter 9) it has been shown by Ri~~ . (12 ~3·2} (12,33j that the expected 'n~mber c'C up:~'rossi~~~ t;". '1(~J' th~~ugh tbe '.. . • . '. . . . . ,' , . ' .;-,.J-;.o::: •. .; .. ' . .' , le~'el zer~ pe~ .un~t; ~i.!"~. is giv~n by : ". , • _ . 1 "0 - E[~~ (0)1 ~ (m2/moji, where mo and m2 are r~spectlvely the z~r.oth and second m,o ments of the spectrum S (t);" ~, de£aned. f.or the g~neral case, by , . ,~ Hence, Tz:the mean iime '~tween 'zero up-crossings. is given by ,:- . ':. -. ::.. ,:. , The spectral bandwidth parameter f is defined by' , o!~ • mom.j -m~ '., · mom.j . (12.22)· · ~ . ' . _ .. (12.23) .(.1.. 2.24) and 'Ii'~s'i~ th~-~~nie a < ~;.~ 1. 'hen ~ is close t~ zero, the spectrum co~sis~ of a nanow band of f;~-~~~.n.'.~.ie s'~~d" ;he.- s~tru~ 'is termed tw,,:ow.band. -, ' , . Consider now the heights ~ of all the wave peaks (maxima in 7f{t» measured from the mean still· ..... ater level (7f(t) ;: 0) • see figure 12.4, Canwright and Longuet·Higgins have shown (see figure 4 of 12,10]) that f.or reasonably narrow·band spectra (t < 0.4) the mean height of the hignest one.third. of_all pe~~s, ~l/3' is gi'en by (12.25) Furthermore, it. can be shown that, for reasonably narrow·band spt!Ctra, the largest depth of a trough between zer.o cr.ossings is almOst equal to the height of the wave crest which preceeded it. so that. the wave h'eight H!:1 2~. Thus the significant wave height Hs may be determined from (12.26) Tht last qu.anlity .. that will be .d~5C.ribed h~'r~ is'Tp' the eX~l.ed time b.etween successive peaks (ma.'Cma) in the process '1(t). FO,t a stationary zero·mean Gal~ian proCi!SS, Tp can be shown
- 227. 12.2 MODELl.INC THE RESPONSE OF JACK.&T STRUCTURES FOR RELIABILITY ANALYSIS 215 112.91 to be related to m2 and m~ by (12.2" Hen<.:e .In: ra~io or the number oC peaks to the number zero crossings. which is a measure of tho; depee oC inegularily oC '1(t), is given by . , r - Tm':, C m2J(mOm~)2 (12.28) with 0 < r< 1. For narrow-band spectra r is close to u.nity, and the number of ma..ximo. is not significantly SreaLer than the number of zero up-crossings. Use may now be made oC equations (12.23) and (12.26) to relate the sea·state parameters Hs. and T, to the moments of the spectrum S"" (fl, and hence to obtain expressions for the spectral paramet,ers an terms of H& and T;. Houmb and Overvik: 112.19) have undert.:J.ken such a para· rpeterization of the JONSWAP spectrum (equation (12.17) to determine the parameters 0:, i and fm for different sets of values of H, and T,. This inConnation can. be used, together wiLh the Ilppropriate extreme joint. distribution function of Hs and T, for a particular offshore location whet. this is known, as the input for a structural reliability analysis. 12.2.2 Wal'e model Wave theory is a subject in its own right and cannot be discussed in detail in a book on s(fIJCtural reliability theor)~. Attention will be focussed therefore on two-dimensional smaIl-amplitude simple harmonic waves, generally known as Airy waves. They are two·dimensional in the sense that the wave creslS are assumed to be paraUel and to e~tend to infinity, and of small am· pliwde in comparison with .wave·length and water depth. Airy waves cannot really be considered to octer in practice (actual deep-water waves correspond more closely to such wave theOlies us Stokes' 5th order) but :lre imponant because of their central role in spectral analysis in which the sea state is represented by an infinite number of Fourier components: For a thorough presentation of Airy and other wave theories the reader should study a specialist text, e.g, Kinsman 112.21], The assumptions on which linear, or Airy, wave theory ar~ based are as follows: the flow is two-dimensional, Irrctational and incompressible, the pressure at the free surface is negligible, the non·linear terms in the equations of motion are negligible compared with the linear term:>, the water surract displacementS art~ smail so that th~ (velocilY), te"rms are lIegli Rible. Given these assumptions, it can be shown 112.21J that the water surfuce elevation 11 above still water lavel may be expressed as a function of time t and of horizontal distance x in the direction normal to the crest llX. U'" acos(kx - wt)
- 228. .;. 216 12. APPLJCAT[O:-<S TO FIXED OFfSHORE STRUCn' RES where .;"" , is the wav~ amplitude " k , is tbe waue number (i!1.radsJml . w is the wave frequency (in rads/s) Figure 12.7 shows a cross·section through a typical linear wave. It should be noted that this differs 'r~m figure 12.4 in which 1J is plotted for some fixed pOint as a function of time. Defining"the waue period To" 2rrlw, the UIQve length L:II 2./k"and""the woue 1I,!ighr Ii· 2a, e'quption (12.29) may be ;'e·writte~ as J1(X;'t) - !I-Cos2ir"{E.-..l..) : (12~301 , 2 _ ._ L .. To but, because of its greater simplicity, the (onn ,f'equatl'on (12~29) will be retained for rU:;tJ.er calculations. :" . Ii:. . . . . _. The loading on an of £Shore jacket structure depends, however. on the water particle velocities ;~p ~nd ~~celeratjons up' F.'0r"v~rtica1 m~mb~rs i,t is ne<:essary to know the horizoni:at ,.c~I:T;P()oo nen~ Up and lip as functions 0,£ dep~h : ~~king z as the vertical distance measured up~ards from still wat~r level (see figure 12,7). linear wave theory gives, the wate~ particl~yeIQc!~,~s and accelerations as • ' coshk(z+d)" up(z.x. t) -aw sinh kd cos(k:"<-wt) ... I) - ! cosh k (1. + d). (kx ) , . ~p l,~' x. , - aU) sinh kd SIO - w t L TIn, x) ,'-: :;1" , -.', (1.2.31) .:" (12.32) , t · I' I . , ~ .. up,' u;I' U;I fi!turt l2: ,j . Lihl!lfw~ve. ;
- 229. ->:." ",r',;;, t·r· 12,2 MODELLING THE RESPONSE Of J.-CKET STRl.'CTt;RES FOR REUABILITY A:>:ALYSIS 217 .. ::. Fu~he;~m.o~e, .i~5a.n .be, shmyn}hat w. k,~l!g.:~ .. a}:~'r~l'!te(;l. by·~heexpress.on ',' (12.33) -. .which allows the wave number k to be evaluated ,for knownwater'depth d'and any selected .frequency.component W'.' Equations (I2:31) and (l2,32)ate'strittIyvalid only:fot'-d < z <. O,butit is possible to apply:them_for distances z up·tothe'Cree water surfaCe~' .. , . . The effe~t.of current.: l!ndElr,~:-;~r~me storm conditions the predominant loading on offshore .~tructures.is.~that.~ue Y)"~a,,:~s,,blt cl,I.rrents are al!'io pre_s.~nt and-their effect may be signifi- . c!l.n.~. In r~rnqte,.,ofr~h9'5e l()ca,~i~,m~.,-~!dal c~n:ents are generalIYos;m~, but wind, generated cur· rents are likely to arise under storm conditions. It is normally assumed that the.current velocities at different depths follow a simple power law function of the current v,elocity uc:(O) at the;sea surface, giving , -dO:;;;; z", ° (12.341 where d is the total water depth and a is a constant. It is"kn'6wn that w~ve~~urrent interaction h~ ~n effect on hyd~~d:-Ol~mic I~~~li~g. but this is not easy to allow for, and is generally ignored. The structure is'-;:nerefore ari31ysed ~imply by using the total particle velocity at each level z, obtained bY"'ec:oriaI-addition of the components up(z) and uc:(z). Ideally. the infonnation required for a structural reliability analysis is,t~E!_jClint.pis_~~bution function of wind speed, wave direction. current speed and current direction: but this informa· tion is likely to_be,difCicultor, imp.ossible, to obtain for a ,given offshore location-and the ana· lyst may be forced to'adopt some simplifying procedures. e.g. anai:!o'Sing the st~cture first wit~ .zero current velocity and then again. assuming that the cunent and wave directions are the same. to see whether the effect of current is significant. 12.2.3 Loading model Fltlid loading; In section 12.2.1. the concept of a wa'e spectrum was introduc,eci and ways were descrih.ed for relating the parameters of-the spectrum either 1::0 '~,ind speed or to the sea state statistics Hs and Tz. In section 12.2.2. the equations of linear wave theory were given . . Elnabiing water: particle y.elocities and accclerations,to-be determinea-, for components'oia wave havi~g.a.mpHt~de a and fr,:!quency w. In this section.-'l'e examine the'forces acting on slender l1?-,e~qe~s imnwrsed in,~ fluid: which has an-in~tant~neous horizontal velocitY:I.,.-and accelera· tiot:t,~p..... ; i. The'most widely accepted approach to the calculation of the iia'e forces on'a 'ertica!'sE!nder rigid body extending from the sea bed to'above the wat!"r sU:rface is that dUe-to :Ioris'ori.'et al. [12.281. It is assumed that the Wa'e force can be expres.sed as the sum of
- 230. 218 .. ~ .~ 12. APP~ICATIJNS TO FlxtD OFFSHORt STRUCTURES --- --..• . .... .. _- -_ .. a drag Corce proporlionallo the sql,lare 0: the water fankle 'eIOCit)'~.w~ichmay ~.e repre· sented by a drag coeCCicient having subs"tantially the same value as for steady flow;and an inertia f!Jrce proportional to the horizontal component of the accelerative force exerted ·on the virtual maiS of the water displaced by the body. It.should be noted,thatthe virtual mass governing·the inertia force is ·itself composed of two .parts. the. mass or water displaced by the actual ·v.olume of the.slender-body and an (external. Iy) entrained mass or water which moves with the bodr. The latter is known as added mass and depends on the shape of the body and ·its surface roughness. ~ 'Marison's assuJ'!Iptians can be lal(en to be valid provide(! that t.he body does not significantly affect the wave·pattem. ·This is ·iNe if the width of the body D is less than about 20% of the relevant wavelength. The·total force at any instant w time't acting on an incrementallen·gth of the body dz is then given by , ; (12.3S) where p is the water density up . is the instantaneous W8!er particle vel~ity nonnal to the 10ng1tudinal.~is of the body . .. up is .the co.rre~pqnding water panic.1c ,,:cceleratioll A.: . is the cross·sectional area of the bod)' Cd is a drag c?,!fficient. em is an inertia coeffiCient . The total horizontal Corce on the vertical rigid body can be obtained Cram .:t .. 0 ,P(t) ". dP(t) .... - d .. ~ ! . (12.36) To distinguish between the co~tributions of the actual displaced mass per unit length pA and the added mass (em -l)pA. equation (12.35) may be re-written as. ! . (12.37) J:. However, most fixed offshore structures cannot be assumed to be rigid and;·indeed,·those in deep ..... ater may exhibit appreciable dynamic response. In these circumstances, thfdnduced' structural velocities Us and accelerations il• maybe·significant in comparison with up and U·p necessitating a modification to equation (12.35). Following the discussion by Chakrabanl .112.1).1 on,a pap~r by Malhotra and Penzien f12.25] it can be sho ..... n that the revised form . of. ~~o~on's eq'"!~tio~ to allow for structural motion is (12.38,
- 231. 12..2 ~toDELLING THE RESl'ONSE OF J.'CKET STRUCTURES FOR RELIABILITY ANALYSIS 219 Tht main difficulty with the applico.tion of Morison 's equation. either in its original or n:vis~t1 forms. lies In the proper choice of the coefficients Cd and em from a wide range of pubJisht'd data. Although these coefficients can be shown to vary sysl~m.aticaH};'with" other par3meters such as Reynolds' number, Keulegan-Carpenter number and relative roughness II 2.16 J th~re is still considerable residual uncenalnty. Cd and Cm may therefore ,be sensii:lly tteatl:d o.s ran· dam variables and must be included as such in a structural reliability analysts. (See section 12.3.2 for further discussion of Cd and em)' Other 1000ding: Ap~r:t from the effects of waves and currents discussed above, offshore struc· tures are subject to"wind loads, superimposed deck loads, drilling and operationallo~ds. together with both minor and major collision' loads, Each needs to be taken into account in a ~a~ric assessment of structural reliability; but relatively si~ple mO,dels can .be used Cor these actions when the overall response is insensitive to them, which is often the case. Major collision loads are of course an ex~eptioJ? and require special tleatment. 12.2.4. Natural frequency model "The 'naturnl frequencies of structures in their various lInormalp modes of viorntion depend on the spa'tiaJ distributions of mass' a~d component stiffness. Offshore jacket suuc:t.ures un'tike typical building structures contain very few non·structural components. at least below the level of the 'superstructure: Because of this and becaus"e'the u~certa"inty in Young's modulus E for'steel is 'cry small, the overall stiffness of such frames can be considered a deterministic function of E and the member dimensions, Furthermore, because the overall stiffness of the structure to horiz'ontal wave loading is a linear function of the stiffness properties of :a hlrge number oC individu· al Components whreh fo~ fabrication reasons can be assumed to be independent, the total un· certainty in the nifCnc.!o.' of the jacket itself can be neglected. The major remaining source of uncertainty is In the shan-term elastic properties of the soil foundations. A discussion of this is beyond the scope of the present text, but should not be overlooked in practice, Uncenainties in the mass of an offshore structure arise from the possibility of ~h3l"lges ip the superstructure loads during the operating life such as would result from the storage o~ additiunal equipment and struC'tUtal modific:ltions. These uncenainties should also be modelled and their effect- on natural frequency taken into account. 12:2.5 Evaluation of structural response Structural model for loading and spectral analysis: The reliability analysis of an offshore structure requires the n:f;;tit;on oC a spectral analysis a number oC times, depending on the number of ba. sic variables affecting the response, Fot this reason, the mathematical idealisation of the structure that has to be ma(i"e"r;;I"-.~h:l pur. poses of representing the loading. the dampinIJ: and the distributed masses should be as simple as possible and yet relain tht' importanl structural lind hydrodynamic characteristics of the ac· tual jacket. A jiiicket Slructure can be idelllist:d for the purposes' of wa'e loading by 3 number of 't'rtical
- 232. 12. APPLICATIONS TO FIXED OFFSHORE 5'TRGCTI:RES ,;. .,, ::' members extending' f~om the sea bed to, above the water surfnce and di~ided 'irao zo'n~s e~ch representing members at a panicular iOCOltion in the real structure. The dlame~e~ of each'sec. tion of the idealised model can be chosen such that the frontal area exposed' to the wa~es is equal to the sum oC the indh;dual tubular members in thnt zone. Using ~Is apP"?ach, th,e actual value of the drag coefficient Cd can be used, but it is necessary to adjust the inertia COe[ficie~t C~ by the factor -InllDl where D is the diameter of the equ~vai~~'~, tUb~I~ , member and Di are t'be actual diameters. For the purposes oC determining the dynamic response, a jacket structure may be idea1!sed suHiciently accurately by II. number of lumped masses, particularly Cor t~e (I~t mode of vibration which dominates the o"e~lI r~~ponse to wave loading. Spectral analysis: The preceding parts of section 12.2 have provided the b~i~' models for"ihe spectral analysis oC a J~cket structure under wave ~d'current loading. The input t~ ~~c~ ' cal. cul3tions ~· the sp~ctrum oC wat.er·surCace elevation S (w) and the vertical profile ~f cur- . . _ ,_ . "'I ; , l ;, .,. rent velocity. The output Crom the analysis is a spectrum oC structural displaet!:ments 5ss(,-,), or more precisely a spectrum oC modal amplitudes in each of the normal mod~~ of vibra'tion which are considered. The complex set of calculations which are required to obtain 5ss(w) from 5'1I1(w) arc summarised in Appendix B. :'vlaking use oC the ieneral relationship given in equation (12.8): the variance of the structural displacem.!nlS o~ May be -obtained {rom a~ "'~" 5SS(w)dw ,0 (12 .. 39) where SSS(w) is a single-sided spectrum. It should be emphasised that the final outcome from such a sPectral ~~ysiS is ' probabilistic in nature in that only the v3rlance (and the m,ean value) onti.e displacehl~n~ ~e determined. Further analysis is therefore requirid to obtain suitable peak stresses for use in:design or saCety checking, as described in secti0Q",12.2.6 below. However, it i3 clear that as itself as de. fined by equ3tion (12,39) is a deterministic function oC the set of b3Sic random variables X (e.g. wind speed) which form the patameten; of the input spectrum S'l'l(w), together with ' other basic random variables such as drag and inertia coefficients. We may therefore write (12.40) "This-provides tht: .b,,:,i5 ~or cr~~~n~ 'a. fai1u!e ~unction oC the form.~~scrib,ed in chapters 4. 5 and 6 suitable lor use in.a le'el2 reliability an~lysis. ._;'._:' . ':',,-';_-'-_ 12.2.6 E'aluation of peak re5p'on!e It was assumed in sections 12.~.13nd 12.2.2 that the water su-riace elevation :,)It) can 'be sen- 5ibly modelled for short penocis of time as a. stationary zero·mean Gaussian precess. However. hec.ause of the non.line3.r~ta! term fCd pOll lu 112) in ~lorison's equation. the final , p p
- 233. 12.2 MODELLING THE RESPONSE OF JACKET STRUCTURES FOR RELl .. "BILlTY ANALYSIS 221 structUf31 response will be non·Gaussian. As discussed in App~ndix B. an approximation to the true ~~sponse variance can be obtained by a met~od of equival:~tynearisation in which the root mean square error of the Iincariscd response is ~inimised. The response may therefore be interpreted as an equivalent GaWlsian process. with Sgg(w) being the spectral density of the result.ing displacements S(t) (e.g. thE!: ampl~tu.de of t.h: [iut mode of vibration). FrOID a knowledge of SSS(w) It is possible to determine the peak response of the sttuctui"1! during an exposure time T. Makinc use of equation (12.21). the e:c.pectc<fnumber of maxima in S(t)"occurring during timeT is T ~ N -"T'" T(m4 /m2 )2 p where (see equation (12.22» the spectral moments m2 and m" are ~efi~ed .by .- 1 1'- mn - f"Sss(f)dCa--" w"Sss(w)dw • 0 (2:r) ~o (12.41) . (12.42) It is then necessary to find .the distribution . ~nc:tion of the m;utimum S. of t~e N ir:edivi~u~ maxima of Set) that occur during time T. This.has been shown "':or Davenport (12.~.~ ) t.o b~ (12.43) ~here (12.44) : !. -- . ".' and. "0· (m2/mo)2 as.defined in equation (12.21). . - .. ~ . - ' ,;- ' . . . ! 1r •• • ~ I ;' : .. 1.5 .. 1.0 .- ,,;, '". . ;0.5 : ~ ;" -1 o ~~. --- 3 , , FilUl"e 12.8. z
- 234. I 222 12. APPLlCA1Jl0:'lS TO FIXED OFFSHORE STRUCTURES :,;.. . :. .~ ;.:.. .. :-.-.-.~~.:: -- _.-.. ..:. i·' , ' ,-. . .. ,,,. , _. The der-iit)' function Cz is ploned in fi~rl.! 12.8 forvarioJ5large values oC ~·T. The mean and sumdat: deviation oC Z are given b)' ! 10 .- ' ,, ' , . I .,... ' , •. , ~, '. ~ " -. ",: • ' 11' "; '·;: ,;-1 . , " °2 .. .,f6 ! (2 In ~o T):! ," .. , where . ~ - 0.57.72 (Euler's constant). FinaJly. the random variable S. representing the peak modal displacement, is obtained (rom , r (12.47) In deterministic' design procedures it'is cammon io'choose ,the .peak factor .. Z .·,JJz so that (12,48) Howe'e:: ln' a structuraheliability analysis it is possible to treat Se as 'an additional basic vm· able y,·;'j~h ~ontributeS'to 'the overall uncertainty.-· .~- 12.2.i Other models In sectio:u 12.2.1 to 12.2.6 various mathematical models have been given which provide ~he structunl and reliability analyst with a method o( computing the probability distributiori'"of .t!I.e peaj: moda1 displacement of a Steel jacket structUre in a given period of e~posure t.o a stationl.."Y (strictly ergodic) random sea. As discw.sed in section 12.2.4, the uncenainties in ,the stiffness pr.openies oi jacket structures can be wumed to be small in comparison ~i'th stfeng!b p~peril~ ~'nd up to 'the i'evels' ot load at whlcr. the first structural component fails it is reasonable to assume that the overall behavi. our of tile structure remwns linelU> For ilny particular 'a1ue of peak displacement, tbe forces and '~lO!!lents may be computed for any part of the StrUcture bY 'conventional methods of linear ac11ysis. This' information can then be used in conjunction'with appropriate deterministic models for the s4'ength 9[;individual structural components to assess whether or not they will iail. , Finall)'. vdth.knowledge of the probability distributions of the relevant basic variables and model t.mcertaintie~, it is possible to ~etermi,!l.e the failure probability for each structural component for any given sea siate and, by integrating over all possible sea slates, to evaluate the total failure probability f~r any co~ponent for the desiiIJ life of the structure or for any other period of time. Some results for ~a1culations oC this type are-given in section 12.5.
- 235. 12.3 PROBABILITY DISTRIBUTIONS FOR IMPORTANT LOADING VARIABLES 12..3 PROBABlLlTY DlSTRIBl1I'lONS FOR IMl'ORTANT LOADlNG VARIABLES Attention will be restricted here to the modelling of three quantities which are of panicular ililj:l ... ~'~h.:l' in th~ reliabiiity analysis of jacket str:uctures: wj~d sp~ed and ~Iorison's coeCficien~ .Cd ~~d -?m' 12.3.1' Wind sp,eed 223 For th~ dete~m:inLstic desi~ and analysis of structures it is c~mni~n p~tice to ';too a qua"1tity such as th~ 50-year return wind speed (ib; wind speed that has an assumed prob~bility of being exceeded of 0.02) as a fixed deterministic value; and co~~iderable ellon has been 'expended in determining ~hiS statistic for both on·shore and ·off·shore areas. However, this is not of direct relevance in a reliability assessment. . . In sec~on 12.2.1 it was seen that the input to a stochastic 'd~namic reliability analysis of~ jacket structure could be either the probability distribution of extreme wind speed or the . joi~~,~s.~~e .distri.bu~ti~.n of th.e parameters Hi and T:t -in both ~ the distrib~tio~~ correspon~ i to the ex.~eme co~ditions in the reference peri~d for whicp th~ l~eliabili~is to ';)e determined (e.g. 25 years)' WhIch approach is used In practice must be governed by the avail- ~~~~ . . . . . .. .. Relia~le wind records. ha'e been obtained at exposed coastal stations for a number of decades and there are often advantages to be gained i~ predicting the distributions o( extre,me seastates (rom these wi~d records. rather than trying to use. short-term :wave statisti~~ obt;ained from a particular offshore location. However, as more wave data are obtained the,b.':IJan~e may change. Only wind speed is considered here. As discussed b~' Davenport 112.131. a simple model of the atmospheric boundary layer which assumes that large scale turbulence is composed of uncorrelated Gaussian fluctuations of wind speed from orthogonal directions leads to a Rayleigh distribution (see page 27) for the mean· hourly wind spee~! a~ la particular location. The Rayleiih distribution is.a special ease of the 2- parameter Weib~lI dis~ribution and it .is fo~nd that both long and :hon.tenn contiil!-10US records of mean-hourly and m~an..daily wind speed arc acc~rately modelled by the latter. see figure 12.9. As staled in chapter 3. the 2·parameter Weibull distribution has a distnoution function (see page 43) y> 0 (12.49) . jf it ~ ~.::-t.!:'d that. the parent mean-hourly wind speed X follows th.is distribution. the maximum ot n independent samples taken fronl the parent ...... ill approach a type I asymptotic di,,· tribution of the l~rgeSt extfe'me. since ttie W~'i':'Ul1 ~ist.Cibu·tion ~ ~ ~pper wi which f~ off in a negative exponential manner (see page 40) . . As there are 8760 hours in a year, it is a reasonable assumption that th.e <J:Im.ual maximum mean· houri): .~ind spee~ ; Y is typt' I maxima (GUmbel) distrib~ted (see page 40), i.e. (12.50)
- 236. 224 12. APPLICATIONS TO FIXED OFFSHORE: STR{]CTL-'RES if,.!)!) I u.9999g2 0.99!t9 E 0.999 r 0.99 r 0 .90 r Wcibull 0.80 ~ sC31e 0;;0 0.60 0 .50 1)..;0 1).:).0 t 0.20 • O.t.from 1972·74 I • ruo , ! 1) .03 I'- --'---L-L•.C 'J ."...,L -'-__' ----'_L L LL.L.u.._~ , , , 5 7 8 9 10 2:0 :30"'0 5060 80 100 fh,:urc 12.9. ~tean·o:iaily wind s()Ced rrorn M.V. Famit. (57" 30·:-<. 3'El (US +-___________________ --. 1).14 0.12 0.10 .·mean·hourly parent }t . y (m/s) o+-----~----~~=-~--~--+_----4_-- 10 20 30 .. 0 50 filure 12.10. Distrib~tiQns or pan:n~ Inti annual eir.tr'cme winds (or Lcrwlek; .:':./:' ' . ~ Taklnc the useful life of an offshore platform as 25 years. the assumption that th~ 25' year maximum mean·hourly wind speed has this distribution is even more reaSonable, unless there are other influences ';.,·hich begin to dobunate at these low probability Ie!,·els. Such an innuence· eould be the presence oC a numbcriol dillerent storm generating meehan.isms (e.g. tr~pical cYclones in addition to fully developI{d pressure systems). However. lor areas such as the North Sea. the major stonn generating mechanisms can be assumed to be of a single type. I
- 237. . 12'. 3' PROBABILrr'V' OISTRIBUTlONS FOR LlPORTANT LOADING VARIABLES 225 "" Analysis of"wind dab '(~~iri 'a 'iargtfn"~'rhb~r'of m~tc6rological stations. e.g. [i'2".221. ~n~~s th~t nlthough the-m~~ii: nn~~a}'maXi~u;:n! meai1~hourt~ ~i~d s'~eed ~~yvarv fto~ ~~e l~;~~ti~'n ~to-: "' ' ". .' , . .. ' . " . .., , .. ,. . ., ...... ', ., ., " . ". r., . ~ ". ,," . : ,.". ''-C' .. " ' another. the coefficient of variation o( this quantity is sensibly constant for locations of similar exposure: Taking !.C~vick to be reas'oiia"biy rep~nt.aiive o('th'~ 'northem North' Sea~ a typf~1 valu'e for the coefficient ~f vU1aildn'ol annu;ij maxmmm mean.hourly extremes is ·12·~13%. The distribution of annual 'maximum v~iies Cor Lerwlck is shown in 'figure i2.10. If annual maXima vahi'es of meari~h'oJIly ..nnd speed are assumed' to be t),'fie' J 'maxima ;distri· . buted, then the -tirobability 'Ciistribution oCextreme values corresponding'to larg~r'periOds of time may be 'determined from equation (3.15). . " . .', , I ' ' . , . _ ' ;'. ..,; - -. ' " ' , ', . ~ . " • .': i .; ." " '" .. [,n .. the .p.re.c.e ding paragrtlphs reference is made to mean-hourly and mean-daily wind statistics . -;;, , . , " ., . , ~ , " , . .. : '. .. , ; , . . . .'. ';:. '.' ' . ·However. whereas mO,st wind dat,a are recorded,in,terms ot 3·second ~sts. or 8.$ mean-hourlv (ihne.averaged) ~al~~~':'~o~t ~~jor si~~ I~t fro~ 6.12 hou~. Conversio~" i;~tors ~e avaii. . ' . "'",," . ,'. :,""';'. ..'"'''' '," '.. .. . ; , !. . ,". ""'(' , ,,';>l" ". . able to convert from mean-hourly values to speeds averaged over a null1 ber of hours· e.g. [12.91 : but some unce~ta'iniy ·;.nust ~. atta'~hed t~ Utese v~lues' beca&~.t the -f~cto~s '~hat ar~ appropriate Cor m-od~rate ~xirem~ ~ay not ~ ~~~ct. r~r ~evere e~bemes. , ' . ' " . - . 12".3.2 Madso'n's coeicidients The Morison Coree coefficients Cd 'and C~ (see equations (12.35) 'and (12.36)) 'have been shoWn by experiment to exhibit considerable scatter. A critical evaluation of all the available in{onna· tion has been made by the British Ship Research Association (12,81. . ";.' , ". ", .. ';: ... :! ; Under uniform conditions wfth steady n~w. constant temperature. viscosity. rouRhness and geometry the drag coefficient Cd is (probably) a deterministic quantity and considerable experi- '. mental'work has been'undertaken to determine values; but because of ttie diCCiculties of scaling. experimental resulta'obtainedat low: Reynolds' numbers on'small modelS'cannot easily be ap., plied to Cull scale structures with flow at high Reynolds' numbers:') ,, 1 ' • Under typical o(fshore "conditioru;;howevet~ there is oscillating flow wit.h ·orbit.al water-par. :i ; ticle motion. Work by Sarpkaya.(12.35I, wing a large U-shaped vertical' water tunnel;';illowing the attainment of now 3.t high Reynolds' numbers has shown that. for smooth cylinders, the drag and inertia coefficients depend on both the Reynolds' number,Re and the Keulegan~Car· penter number NKC:' For rough cylinder.;. Cd and Cm have vnlues which become nea~ly independent- oC Reynolds' number for values of Re in excess of about 4 X 10'. but which depend: on Keulegan-Carpenter number and relative roughness kID. Although the values of Cd obtain~ by Sarpkaya Cor smooth.cylinders in. oscillating flow are about the same as the values·obtained· for smooth.cylinders In steady-flow ,(name.ly;'between 0.6 3.nd 0.7). the:values obtained -for '" rough cylinders.in oscillating flow range from 1'.2 to 1.8 dependin'g on' relative roughness: These values are very:high :lnd cast· some doubt on.'the-values oC about·O .. ; : which are normally .used· ' in design.-Howe'er: the:cbnditlons.under.whi.ch these experiments were'con4uded are'very '" dissimibr«to the waye,.condiUcins·experienced in~the open sea. -·'" ·c -' j'. "', " · ... i -:; , . " • D Ii T . -, Rt .; um" . P- Ind N KC ,; -1;- .... h~re u" 'i.' th', amplituda ~r hori~(;nt~1 oscill~to;)' veloelty. 0 iii{h~'e'y::' .; . - _.• , ', " .. ," " , ; ,;~" ... ~• .': "'" .. __ •• i_; " ,)._ .. ',' .: : ·"·Y· ._· . ;,.· ·. ;-,;r,·': linder diametn, T 11th . .... : .. e period. p is the density oC ",ater .in·d· ~ ii its viscosity.
- 238. I 226. 12. AP~t;~.AT~qNS TC? t:lXED OFFSHOR~ STR!)enJRES , ,j- - --. 1~ . 7~~~~ast to. ~he above are som~ ~~s~!~s .t?bttined".bY .~~~_~ f?;?~ .~p.ll.s~a1e sec,tion of a 325 mm diame~!' pile expo.::ed too ~'~ve loading, ~ reported b): ~ and Hibbard (12.20]. This I : ', ,_.:.' " .:.0 .' , ' .' . '. , •• 1 •••• :"" '" __ _ '1 _. ,... _" ~r","?ilte,,~ an.~.s~s,men~ o(.~~o~~on~s~,~o.t:({i.?~:~.ts, .~~~~pe,n~,~~~.?~ ~y wave .. lh,:O?:. The ~~an ~al~e, .O~taifled for Cd :n.'~ 0.61.W~lh_~,c3e,~fi~~~~t ~rV_~a?on of 0.24, wh~r~~ for em' the I cC?rre~IX?~di?s: ~ean and ~oefficienl ~f "v~!~~i~~;~!;~~ ~~9. ~ .0~2 .l'r!.?, si~ifi can,t ~jf(e,re.nces were found for se,as wi.th sign.ifieant w~ve heights ~~,~om 0.8 to 3.0 m. , :" Experi,mental, da~give no c.I~~, !n~~~.H~n o~ ,~h~t ,ar~ app~p~~e. val~.e.s ~o take,for Gd and Cm for off.sho:~e ~~P~,cati~:ms : Although t~se force coefticie~u may .weU be co~pl~x detenninislic fWlctions of basic parameters such as viscosity,.~ensitY,,';Uld Dow geome.U)·._these relationships are not k.n,?~~. What is m:or~~ ~ven if t.hey were ~~~wn for s~ple is"~~~ d . ~il.e.s " i?teraction ~!feeLS betwet:~, i~,d~~'~,:,a1 pile:. ~~ a jacke~.~h;, ,eff:~Ls o~ ~~, m~~ gr.~~t.~ and th~~fn~raH~ion of waves and 'currents would .render any simpl~ th~ory invalid when applied to real structur~s, The're is thetl!r~iel $~'rY :~o~si-derabl}' uncerLai~~;:'in' the c~~~~~ i~" is appropriate to c~~- . " .... , _ .. ,., 1 ' " .' " " '~'" .' " " . - ,'" .J." • 'J" ... ' . •• ' . sider Cd and' C;., "as random yariables wh'ere tlietolfl:l uncertai.rltY ~ fr.o~.the. cOmbined .ef-fec'i of a large n~'mbe; ~/f mainl)' unkn~wn ·,in~,:,:en~~s. M~~ vaiues' for Cd ,and C~. of 0.75' and 1.8. respectively. with coefficients of variation of tictweeD 2.5~ and 3;;;% provides a reasonable pr?babilistic model for these force coefficients for members of typical jacket structures, The coefficients can be sensibly assumed to be nonnally distributed. " .(.. . ' r, ... , :" ,·, . . , 12.4 METHODS OF RELIABILITY ANALYSIS .12.4.1 ~enen1 : , e'" ' ; _ i .~.' , The basic prir.~ipl~ and meth.o,ds,of §tructural re~ia~~litytheo~)' " have, been_discussed in.chapters 4, ;; and ~ •. bu; ~ .. nll.mbe:r of &C:lditionai topAcs,need to be_considered here in connection with-offshore structures. In ch~p~rs _ 4 and ,5 it was seen that the key s~p.in . any analysis is the identification of a suitable ma~h~_l!1:atical model which defines survival· in tenus-of a function· (12.51) where X is,a 'cctor of basic random variables (e.g. wind speed, material properties. dimensions, etc.J . ln most of the simple structural examples given in earlier chapters. the form oUhe failure func~io n f is es.plicit and is easily determined, as-for.example in the case of-the propped cantilever beatp. gi" en in, example 5.3. However. for complex structural systems this is 'not the case. For dynamically sensitive· offshore structures.the [unction_fincorporates ail the'models [or loading and response given in section 12.2', .. together with the.fallure criterion far the structural component under consideration., e.g. the buckling oC a stiff~n..ad. tlilbular member under the combined aClion of end loads and, moments. The only sensible.form of expression for f in this case is a camp1.!ter program. Reg3,f~less 91 tne . c~mp:icx ity of the.(uncti9n .~. ,~e , princip_IC$.o.f reliability analysis are 'the same. pro1d~d that the basic randam 'uiables do not have p~~et.e1S which themselves change with , 10- '"". , ,_ . .. ,,' ,"
- 239. 12..1 .!lI£THODS OF R£LlABlUTY ANALYSIS 22; time. As shown in equation (4.42) the reliability is given by ~=l-Pr=l-~r·.· .rj[ x- I"X 2 .. ·· ..X . . (X,.X-" ... . ,x •• dx,dx"- ... dx n "- (12.52) I(X) C; 0 where [Xl' X 2 ••..• x" (Xl' x!! ..... xn ) is the joint pro'bability density function for the n variables Xi' As discussed in chapter 4, various methods can be used to .determine it from equation (12.52 ) when the failure fu.nction {is of a simple form and when n ·is .small. Howe'er, for complex {ail· ure (unctions involving spectral analysis ani), level 2 methods are seriously worth considering . . Exactly the same theory can be followed as in chapters 4 and 5, but an algorithm more suitable for computer·based calculations is given below. This can be used when the failure criterion cannot be expressed explicitly in terms of the basic variables. 12.4.2 Level 2 method The level 2 method in its most general form may be interpreted as comprising the following steps: 1) specification oC a failure function f in terms of n basic random variables X 2) creation of a hyper-surface (known as the failure surCace) in the n·dimensional space of the basic nriables X (x-space) by setting f(X) = 0 3) defining the joint density function fx: for the n basic variables 4) mapping the failure surface in X·space to the space of n independent standSId normal variables Z (z-space) 5) finding Ihe shonest distance. iJ, from the origin to the failure surface. For linear failure functions and for sets of basic variabJes which are jointly·normal. it has ~een stated in chapter S Ihat the following exact relationship holds (12.53) In nearly all praclical problems, however, these conditions nr£, .,ot satisfied and the relationship is only approll:imate. Nevertheless, the errors are generally very small- see. for example, (12.3l. In chapter 6, it was shewn how correlation between basic variables can be taken into account and how the level 2 method can be modified to allow for uncorrelated non·normal basic ·ari· abies. In the f01l0;ng, it is suggested how the analysis can be undertaken if information is available on the complete joint densit)· function fs:: .. General method: ·),.$ pointed out by Hohenbichler and Rockwitz 112.181. it has been shown by Rosenblattl12.341lhat if X is a random vector With a continuous distribution function Fx then if
- 240. 225 1:!. APPLICATIONS TO fIXED OfFSHORE STRUCTURES Ul -F1(X1 ) U2 - F2 (x2 I x 1 ) (12.54) the variables U l' U 2' ..•• Un are uniformly and independently distrihuted: .. The functions Fj •• i - 1.2. '" • n are defined in (12.56) . Hence. if Zj and uj are related by I'" 1. 2 ..... n (12.55) the variables Zl' Z:!, ... , Zn are independent standard normal. The conditional distribution functions F1(xilx1. )(2' .... xI_I) in equation (12.54) are given by ' " .. fl(xt • x2 ... ·.Xj _ 1; sildsi ~~ .. (~(~1.' x2•· .. ,xi_I' $i)dsl where the partial joint density functions fj • i '" 1. 2 •... , n are defined by f j (:t1, x2 ...... xj ) - )~ ..... )~ .. fX(xl , x2 •• .. ·x j • $1+ ·1 ... · ,sn)dsi+ l':"~~' an~ where fx: i.s the joint density function of the original set of basic variables. : .(12.56) (12.57) The inverse relationships corresponding to equations (12.54) and (12.551 m gi~en.by Xl - Fi" «fl(Zl» , ;:(2 = Fi' ($(1:2 )IX1 ) (12.58) Finally. substituting equations (12.58) in equation (12.51) the safety margin M may be expressed '" (12.59) Equation (12.59) gives M as a !gene;ftilly non-linear) function of n independ~nt st.:mdard normal vari"bles for which the method of ~aJy.sis given in section 4 of chapter 5 is directly applicable. The reli3Dilily index iJ is given by
- 241. 12.~ :I&THODS Of RELIABILITY .-NALYSIS 229 <E " p= min zp'l 1 (12.60l t ... ~w i·l where QW is the failure surface in Z.space. The fundamental problem with the preceding formulatlon:is that the j~int density -Cunction fx: is ruely known and therefore this approach cannot be used. However. in many cases the components of the random vector X may be assumed to be stochastically independent in which c~e equations (12.58) become i - 1,2 .... ,n (12.611 and •. " (l2.62) ComputaUon.al procedure: It is assumed that a failure function'g OC the r6:;m'~'en b;' ~qua~ion (12.62).h3.5: been obtained for a particular struct.ural problem and that Z is 3 vector'of ninde- , pendent s':'tndard. normal variables. Let z· '" (Iii, t; ..... z:> be the coordinates of the closeST< point on the fai1uresu';fa~e t~ the origin. Then expanding g as a Taylor series about z· " g(i· + Si):e. g(i".) + X 8 ih i (12.6~) 1-1 where (12.64) But for smallliZ' (U.S5) and thus " J:ajhj A 0 (12.60) i-l However. 8. the distance from the origin to a random point z-on the fl1ilure-surface is given by " I (} :IXt;)~ i€ aw (12.6i) i-I and for IJ to be a minimum (or ma."(imuml
- 242. I ~:: . AI'Pl.IC,TIO:-;:S TO FIXED OfFSHORE STRUC1il'REi ! If li is a minimum. and assuming that only one minimu:o exists, .' 'givhi.g (12..70) , Comparison of eqatior.s (12.6B) and U2.iO) shows that for any arbitrary set of small values oz. a n.2cessary condition {or them both t:> be satisfied and for z- to be the closest Point is i == I , 2, ... , n (12.71) . where c is a non·~ constant.' For the purp05es.o! calculating ~, let i' be an estimate of the coordinates of the closest point and let (zo + ozl bi: a better estimate. Tiler •. hy expanding g as a Taylor"series about the point ZO we obtain. as bEiore " g(io + oi):l:: Blz o)..;. I e.t hi (12.72) i-l where (12.73) AssUming now that ~'i(o .;. 6 z) ~ z- and that af' ~ ai' i = 1, 2 . .. ,' n. it can be seen that the left hand &ide of equana:o (12.72) is approximately zero, giving • '.,g(io) + Laibl <lo 0 (12.74) j-l and from equation f12.il) io::l,2 .... , n '(l2.i5) Su~tituting equatiQ!l (12.75) in (12.14) (12.76)
- 243. I::! ... ~lET:HJDi:i OF RELlABlLlTY ANALYSIS and tim;; Finally. re.substitutingpfc in equation (l2.75) gives . I " a~ 2,f:r-:-.g(ZO) , ;&1 " ; ,Z-)( af) and from equation (12.69) " 1 ~. 1.1' I';)' i'. i-l i = I, 2, ... , ri 231 £12 .771 (12.78) . (12.79) Equations (12.73) and (12.78) together 'with equation (12.79) provide the basis for an efficient iterative method for calculating fj. The procedure is as follows: 1) Select a trial vector ZO , preferably in the region of z- . 2) Determine ar, i '" 1, 2, ... . ,n from equation (12.73). 3) Determine a bett~r estimate of z· from equation (12.7Si. 4) Repeat 2) and 3) to achieve convergence. 5) Evaluate (1 from equation (12.79). In most practical situations the form of the failure function g will be such that it is not ·explicit· ly differentiable. In this case, it is necessary to use numerical differentiation, but this rarely causes difficulties. It should be nOTed that ,he sensitiity factors 0, (discussed in chapters 5 and 6) are not directly calculated using the above procedure. but may be e .... aluated from Q. j :: z"fP i::z 1. 2 •... ,n (12.80) Finally, a caution about local minima. Depending on the form of the failure function g. there mayor may not be local minima (and local maxima) present. Convergence to one of these points would result in a dangerous o'er-estimation of the reliability index and hence ~. In case- of doubt, t.he analysis can be started using a range of different trial vectors 'i 0 and searches for the failure surface can be made in il number of predetermined directions. These problems tend not to arise in simple structural problems, but for complex structures more care has to be taken.
- 244. 232 12. APPLICA nONS TO FIXED OFFSHORE STRUCTURES 12.5 SOME RESULTS FROM THE STUDY OF A JACKET STRUCTURE .•.. / In the last part or this chapter which is based on [12.2J. some results are given from the analy· sis of a typical del!p'",ater jacket structure. The-purpose or this is to illl.i&tritiithe use of the various models and calculation procedures that :havc been described Ii. eartler sections. It is not intended as a critical review of the reliability of jacket structures w'hic~l a subject beyond the scope of the present text.' Figure 12.11 shows the elevations of 3 structure inte~ded lor barge launching at. a location in the North Sea with a mean water depth of 156 m. It was designed in aecordance ~ith the'rules' o! the American Petroleum Institute (12.11 with fuU allowance for dynamic response, and was used as a basis for a sensitivity study 112.231. U2:2). The. structure VJL'> subjected to a fullstruc· tura! analysis to obtain natural frequencies. mode shapes and influence. coefficients for forces and moments in the various members. Thi; detailed deterministic analysis was necessary to provide input for the simplified models' used in the reliability ~YSis an'd t~' i~i~rPret the response in terms of stresses, as discussed in section 12.2. • ".j ..... '-' '. ". The reliabillty analysis was undertaken to check the ~ety 'of the '~ctUre under the action of wind, wave and current loading and to assess the sensitivity of the dCSign'to th~ various ran· dom variable~ that affect itS behaviour. Use was made of the JONSWAP spectrum to relate e:o:· treme wind speed to sea·state. For present purposes, only results for a single tubular membe:- are discussed (indicated by the arrow in figure 12.11). This leg member acts as a strut carrYing a combination' of a"C:ialload and moment, and failure was deemed to occur 3t the collapse load predicted by the API design rules [12.1), but treating the dimensions and material properties as random variables and set· ting the permissible stress equal to the yield stress. " FiguTe l2.n.
- 245. 12.5 SmofE RESULTS FROM THE STUDY OF A JACKET STRUCTURE 233 " Variable Distribution ·0 "0 a Annual extreme €i-hourly wind Type I extreme 26.1 rnls 3.3 + 0.900 Yield stress Log·normal 380 N/mml IB.O -0.214 CUrrent speed Normal O.m/s 0.14 + 0.203 Cd . Normal 0.75 0.225 + 0.199 Marine gro'Gth at m.w.1. Normal 0.16 m 0.06 + 0.154 Strength model uncerbinty Norma1 0.061 '0.023 -0.099 '. Cm. Normal 1.80 0.27 ..:.. 0.063 DeCk load Normal 24000 t · 720 ~ 0.041 Thickness of leg wall . Normal 33.5 mm 0.34. .. ::-:.O,.94p ..•.. Leg diameter Normal 4191 mm 16.8. . '..",. -0.012 Damping rono Normal 0.03 0.01 .:::'0.001 Others - - - ,. (0.146) Table 12.1. Analysis of jacket structure. The panmeters of the probabilit.y distributions of the imporunt "ask variables are given in table 12.1 . Using this data, it was found that failure of the member oc;curs :with a probability o~ abo~t ~.q 'x 1~~1 .in .a reCere~.~.e. period .of 25, y.!~~ .• but .no ~cular significance should be attached t~ .this nu~be~. ~hat are ~f interest, ~~~vev.er. are,theparamer.ets OJ; which indicate the relative sensitivity of the failure probability to the various basic variables. These are ranked in 'order or'decre~ngi~~o~~~c~ i~ ~bl~ l2.:i.'The f~~~ th~t wind speed is ·dominant is: not surprising because it is the main loading variable and because its extreme value is subject to considerable uncertainty, The relative importance of Cd compared with em is also not unexpected • The contribution of the geometrical variables to the total uncertainty is negligible. An alternath .. e approach in determining sensitivity is to examine the effects of ,~I??e,~~ changes in the distribution parameters. This Is illustrated for changes in standard deviation aX in figure 1~_12a, ~d fo~~hanges in mean 101)( in figure 12.12b. These show 'that the failu're probability is e~t~rnel~· . se~ti,,!,e:to the varia.ryc~ of .the extreme wind speed and to the mean value of the ri,e~~ st~s_. The eCf.ect o.~ m~.~;~ ~r9~~~h is relati.v~ly unimpo~t. Som~ [ur~her ~Hs using a l1).ore sophisticated. shell analysis fodhe stiffened cylindrical leg me:nhers. in place of the,API failure crite~on. have been reported in [12.21.' This has enabled importa~t. ~p~c1usions ~ b.e dr;'wn al?~~t ~e re~a.*~ im~rtanc~ of..geo~etri~ jmper.f.e<;t!,ons in g~vemrrg' fai.lur~. An immediate field or application for the type ot an!l.lysis d~ribed · 3bi)'e ·is to:a..:;.sist with de· cision makin~ in areilS such :lS the selection of optim3.1 fabrication tolerances and ::1 the scheduling of cle:ming programmes for the remo'al of marine growth. The main future usc. how-
- 246. 1:' , .:t': :'IC",Tlo:-:' TU nXl:::u Un"SHOItE fTRUC'1'UR£~ , '- 234 ! 1'( I. lO~ ! " L- {h.ld nrC'SI "-, !' 5 _-----:--cm '[ 1.25 .,S Fi!lur" 12.12a. Fil!ur~ lZ.12b. -, .. 2 ever, will bein'the reliabilityaSses.sme:lt oN:"omp:ete structural systems. Th,e'devel:OPment:Of the close bounds'·disCussed in 'chapter S has brou§:!lt the goal of compl~te system reliabiHty ·:.analysis in sight, but it. would be irre#ionsibie to ?retend that. with the p}esent st~te of k~ow. lede-e. othe problems-of undenaking:a complete ar.:JJys,i.s: of a complex struct.ure invol~i~g many failure modes are not formidable. " ,: .,',-, BIBLIOGRAPHY , ,- ;1-- ', .... 112.1) . Arperican Peuoleum Institute: Recomr::ended Practice (or Planning. Designing and . Constructing Fixed. Offshore Plalfornu. 'API RP 2A Seventh Edlticin;'Jan: 1976. 112.21 Baker, M.J. and Rain"achandran, K. :"Reliability AnQ1Y'~'cn a T~' i~ the iJesign' of Fixed Offshore Platforms. The Integrity of Offshore 'Structures. Proc:2nd Inter- : :_ 'national Symposium; Glasgow, Jul~. . 19S1. Applied science' Publishers', i9Sl. ," ': '12.3 )" "- Baker. ~CJ.'and Wyatt~ T.: Methods oj' Rellab'jlii;, Analysis for jacl:el Pfatio;;ns. ProC. 2nd Intf:!rnational Coniercnce all :he Behaviour of Offshore Stru~tures, Lon· . don. Aug .. 1979.
- 247. .I:IIHLi'.Jui{Ar-!I) 112.41 112.51 112.61 112.7J 112.8J 112.9J Hattjes. J. A.: Probabilistic Aspect of Occan W/H·CS. Safety c~ Structures und~r l)y. namil' Lo:.tds (ed. Haland.1. et aLI Volume 1, Tapir. Trondheim. 19::. Borg~n.an. L: ~.: TIi~ .Spcclral Densiry iar Oceall lVavc Force". Cuastal Eng. Conf .. .-SCE.1965. BOfi.man" L. E.: A.Statis/ical Theor)' {or Hydrodynamir: Forccs on Objecfs. Hy· draulic Engineering Laboratory Wave Research Projects. ljz~:'ers ity of California, Berkeley, 19.65 .. Brebbia. C. A. and Walker. S.:Dyrnzmic Arwlysis of Offshore Structures. Newnes· Butterworth, London, 1979. The British Ship ReselU'ch Association: A Critical Ewliualion of the Dala on Voile Force Coefficients. Report No. W.2iS. Aug. 1976. British Standards Institution: Fixed Offshore Structures. Draft (or Development, 0055: 1978. (12.1 0) . Cart~·riiht,. D. E. ~~ .~~guet.Higgins, M. '5 .: The Stlltisticc! D~trjbulion of the . Maxima of a Random Function. Proc. Royal Society, A 23:.19:'6. 112.11) . Chakrabarti, S. K.: Discussion on 112.251. Journal Eng. Mecn. DI"., ASCE, EM3, June 1971. 112.12J. Davenport, A. G.: Note on the Distribution of the Largest ,'alue of a Random FUnc· tion with Application to Gust Loading. Proc. Institution of Civil Engineers, London, ,V ol. 28, 1964. . ' '. 11 .2.13J. Davenport. A. G.: The pependence of iYind Loads on ltlereorological Parameters . .. Proc.lnt. Conf. on Wind Effects on Buildings and Structum, Ottawa. Sept. 1967, ,.vol. I. 112.141 Draper, L. and Squire, E. 10.1.: Wallt's at Ocean Weather Ship Station India. Trans. Royallnsdtute oC Na'lai Architects, 196i. 112 .151 Fjeld,S., Andersen, T., Myklatun, B.:Rish Analysis of Offshore Production and Drilling Platforms. Proc. Offshore Tech. Conf. Paper 3152. Houston. 1978. 112.161 Hallam,:1. G., Hear. N.,J. and Wootton, L. R.: Dynamics oj Marine Structures. Re· port URS. CIRIA ynde~waler Engineering Group. 2nd edition, Aug. 1978. 112.1 i I Hasselmann, K. et at.: Measuremenls oj" Wind·wave Growth :;nd Swell Decay dUring the Joint North Sea Waue Project (JONSIYAP). Deutsches Hydrographisches Zeit· schrift, Hamb.urg, .Reihe A 18") Nr. 12. 112.181 Hohenbichler, M. and Rackwitz, R.:Non·Normal Vectors ir. Structural Reliability. Journal Eng. Mech. Div. ASCE, EM6. Vol. 109, Dec. i9Bl. 112.191 Houmb, O. G. and Overvik, T.: Parameterization af Walle Sp~clra and Long Term Joint Disrrlbu/ion of Watl~ Height and Period. Proc. 1st International Conference on the 8eha~jou.r of9ffshore Structures., Trondheim.19i6. 112.201 Kim. Y. Y. and Hibbard, H. C.: Anai:-'sis of Simultaneous IrC('C' Force alld Ifacer Particle Velocity Mcasuremcllls. Proc. Offshore T~ch. ConI. Pap('r 2192. Houston. I9iS.
- 248. 236 12. APPLICATIOl'S TO FIXED OFFSHORE STRUCTURES [12:211 Kinsman. B.: Wind Vaves. Their Generation and Propagation on the Ocean Surface. Prentice-Hall. Englewood Cliffs, :-;. J •• 1965,·. [12.22] Laboratorio Nacional de Engenharia Ci;!: Wind in Western Europe. Results of an Enquiry. Proc. 34/13{4047. LNEC. Lisbon. Feb. 1973. [12.23] L. E. A. Offshore Management: Permanently Located Offshore Structures • Jacket Sensitiuity Study. 1979 (unpublished). [12.24 J Malhotra. A. K. and Penzien, J.: Response of Offshore Structures to Random Waue Loads. Journal Str. Div. ASCE, STIO. Oct. 1970. [12.25] ~Ialhotra. A. K. an? Penzien. J.: Nondeterministic Analysis of Offsllore Structures. Journal Eng. _~I_ech};>.i~~ ASCE. EMS, Dec. 1970. [12.26] Marshall, P, W. ap.d ,Bea, ~, G.: Failure .Uodes of Offshore Platforms. Proc. lst Inter· national Conference on,the Behaviour of Offshore Structures. Vol. 2, Trondheim. 1976. [12.271 Moan. T. and Holand. I.: Ris/t Assessment or Fixed Offshore Structures· Experience and Principles. Proc. 3rd International Conference of Structural Safety and Relia· bility. Trondheim, 1981. (12.28J )'-Iorison, J. R .. O'Brien. ~L P .• Johnson. J. W. and Schaaf. S. A.: Tile Forces Exerted by Surface Waves on Piles. Petroleum Transactions, AThIE, Vol. 189, 1950; [12.29] Moses. F.: Reliability Analysis Format for Offshore Structure~. Proe. Offshore Tech. Conf. Paper 3046, Houston', 1978. [12.301 Newland. D. E.: An Introduction to Random Vi!:Jrations and Spectral Analysis. Long-man, London, 1975. [12.311 Pierson, W. J. and Moskowitz, L.: A Proposed Spectral Form for Fully.Develop~d Wind Seas based. on the Similarity Theory of S. A. Kitaigordshii. Journal or Geo; physical Researe~,.vol. 69, No. 24. pee. 1964. {12.321 Rice, S. q.: .i.~Jathematical Analysis. of Random Noise. Bell Systems Tech. Journal • l12.331 - [12-.34] (12.351 [12.361 . Vol. 23, 1944. Reprinte.d in Wax, N. (ed.):Selected Papers onNoise and Stochastic Processes .. Dov:er, N. Y .• 1954. Rice, S. O.: .. Hathematical Analysis of Random Noise. Bell Systems Tech. Journal; Vol. 44, 1945 (Reprinted with [12.32}) • . Ro:::enblatt. M.: Remark~ o:n.·a Mult(-Variate Transformation. Annals of MaHil!matical Statistics, Vol. 23, 1952. Sarpkaya, T.: The Hydrodynamic Resistance of Roughened Cylinders in Harmoliic Flow. RoyallNst. :--.iaval Architects. S!Jring ).[eeting,-1977, Paper No.4. 5tuueroep Problematii!k ,-,on Offshore Constnleties .Stu POC): Probabilistic Refia· 6ifity' Analysis for Offshore S}ructlfres. StuPOC·V-5·6. Netherlands Industrial Coun· <:il for Oceanolo~" English El~ition. 1980. ,I
- 249. 91BLlOGRAPHY 237 [12.37] Van der Hoven. I.:Pou;er S.occtrr<m of .1'orizontal Wind Speed in the Frequency Range from 0.0007 to 900 Cycff)s per .=iour. Journal oC :-'Ieteorology, Vol. 14. 1957. (12.38J Wiegel. R. L.: lVaVf)S and Wove Spectre ,;nd Design Estimates. Con!. Deep-Sea Oil Production Structures. University of Cilifomia. Berkeley. 1978.
- 251. 23f! Chapter 13 RELIABILITY THEORY AND QUALITY ASSURANCE 13,1 INTRODUCTION The £irst 12 chapters of this book are devoted to various aspects of Structural reliability theory and its application to design and safety chec~g. [-n undert.u..ing 8 reli~bility anaJ)'sis, the engi· neer should take account of all known sources of uncertainty and should use this information to control the probabilities' of structural failure and unserviceability within acceptable ranges. This ean be done either directly, by modifying some pan of the structure, or indirectly by modifying the partial coefficients. Allowance should be made for the possibility of the occurr· ence of all recognised fallure modes, e.g. shear, buckling, plastic collapse, etc" together with various modes or unserviceability. It is widely recognised, however, that most structural failures occur for unexpected reasons and in ways that have not previousb' been encouDtered l13.21. t 13.7 I. 113.111.113.1-1 j. (13.17). No .discussion of s~ructural reliability theory i.s t.herefore complete without some consideration being given to these additional causes of failure and their possible treatment. 13.2 CROSS ERRORS 13.2.1 General In'bhapters 3 and 12 '3 coruiderable amount of space is used to' discuss the probabilistic modelling of loads anc! resistance variables. These models are selected or devised in such a way as 'to embody those "features of the phYSical quantity tha1 are essential for the analysis of the practIcal problems being considered. It should"'not be' thought that the models are intended 'to be a perfect mirror image of reality. but rather as a lltoollt in a decision making process. Depending on the nature of the decision to be taken. the lttooht may need to be changed. For example, in modelling the yield stress of steel, a simple log-normal distribution may often be used (see chapter 3i. but em o~~e[ " ...... "ions a mixed distribu~on model (see equation (3,.3.~P ~o_ul.~ . ~. !nl?re appropriate. One important assumption that has been made is that the probabilistic models for loads and re· sistanc~ variables are representa~ive of events during a f'~icula.r period of time· the lire of the structure in the case of loads and other'actions, and the period of construction in the case of
- 252. 240 13. RELIA51LITY THEORY AND QUALITY ,SSURANCE materi.,,1 pro!>l!tties. In fact. the models are condition3.1 upon or pre-suppose certain stnndards of design checking, quality control. Inspection and maintenance. It. tor example. the st-a~'dards of qunHty control used in the manufacture ~f a structural material change. one would expect to see some change in the probabili.t>' d~tribution function of that v3riable. If the standards of quality control are significantly d~f!erent._between different manufacturers or suppliers (e.g. in the case of steel or concrete), it may ~ convenient to use a mixed distribution model to allow (or these differences. The problems that. need to be considered here. however. are or a diHerent nature. The vast majority of structurnl failures occur. beeause oi gross errors. A gross error is de~in~d as a major or fundamental mistake in some aspect of the processes of planning. design. analysis. construc· tion. use or maintenance oC a structure that ;"as the potential Cor causing failure. 'Gross errors occur because of inadequacies in the sfandards of quality assurance· the proceSs by which the various components of complete IIbuilding process~ (mentioned above) areco·ordinated with the aim of achieving the design. objective. It should be noted. however. that not aU gross errors make '. ' .' " . .. , . . a st,?cture weaker • th~Y can abo make it unnet~:;ary strong. Typical examples of grl?ss errors are mistakes in d~gn .c.~culatlons. use 0.( t)"te wrong siu o( celnCorci!,g ba!!, or grade.?r st:eel . misinterp~tations._of geot~hnical survey data. subjecting the stNctu~ to a .~las5 of loading (or whicJ:l it was not btended. etc. A gross error should·not therefore be considered as some extreme ~alue in the tail of.the prob· ability. distribution used to model a particular random variable, but a discrete event G which radically alt.ers the probability of Cailure by changing the models that are applicable. Example 13.1. Assume, Cor the sake of simplicity. that a structure has a resistanc:e R which is dependent on 'only on~ basic variable. the yield stress" of steel. and thai it is sub-jeeted to a single load e(fect S. For simttUcit.y. let Rand 5 be normally distributed and ~R = 380 N/mm1 oR " 25 N/mml ~S -230N/mmz )) · · ··oS "43N/mm1 , . Then. from equa~ions . (4.26) and (4.27).,3 = 3.02 and Pr ~ 1.3 X 10-1 • Assume now that, ll lower gtad~ of steel is used in place o( th4! ,correct grade, 50 ~hat R is dist~~uted with para~~ters ......... . .. ' giving,3' ., 1.0 and Pi .. 0.16. The values of l-lti and I-lit' iiven above are typicaJ Cur the yield stress or grade 50 and 43 weldable structural steels. and, it can therefore be seen that a gro~s-err()r- !~volv!ng the 3ubstitution 0-( gra'de 43 steel ror grade 50 in a critical part. of a structure 'is quite likely to cause failure· a chance oC.about 1 in 6 w,ith the aroitrary assumptions made here. , The precedin~ e:<ampie is somewhat ~implistic aJi little attention should be paid to the '~ctual numbers used. However. it illustrates the princip~ that the models which ::lre used under nor· mal conditions. without the presence of gross err~rs. are no longer applicable .when a gross
- 253. ,: 13.2 GROSS ERRORS 241 error ~urs. This does not mean t~at relia1?ii.ity thel?r:'. c~n~t be used u~der ~hese conditions .• it means that the models hnve to be amended. The problem is to know the various forms which gross errors can 'take. 13.2.2 Classification of gross errors .: ',. Table 13.1 gives a ge'~eral cii.ssi£i~tion of the nature and SOWCe5 ot'gnm' e~;or.(atong· with some examples. No such list can,·oC·c·ourse;·tiihomprehensive. indeed, by'-their very' nature, some potential gross errors or hazardous situat'ions must ex1st'wh'ich have not yet been reo cOiJ1tsed. rg'nora~~e of phe~omena' s~c'h is'iatigu;e~' britUe fracture and the deterioration of concrete made from high':aluminae~mei;t areyplc'al examples trom the'past of 'erTorS ;i~ de· sign concept. Early designers cann'ot he' cntic:.se'd for not knowing abO'u't such :efCects, just as there is no reason to suppose that'cuirentlir unrecognised failure modeS will n'ot: cause acci· dents in the Cuture. ., ~.- ' Those gross errors which c:ause st~c~u.ral .coI13p_se or 1:'nse~ce~bility can ;al.so be cl¥Sifi.ed according to the type .of Cailure which occurs. as shown in table 13.2. The word failure, here, should be interpreted in ~he general sense of fJ.ilure to comply with some performance requirement and not just collapse ILe. it includes unserviceability). Two tyPes are given: those in which the structure Cails ina,predictable manner by, one of a .number oC foreseen failure modes ;" here.called,type.Aj and those in which unforeseen failure modes occur· caUed type 8. Source Design con:cepi ::'1. ',--. ,: .... , Nature · ' ; I,. Poss'jbIJ;'failu'iijl m6d'e unrecogrused ::'JnCO~t nattli-e 'of ~~~-:iSshmed" '-u '.' ~ . · Omission or a load or load com· bination Design " Mismterpretation of geoteclinical . arid ' " :data ·.. " analysis " Computational error in analysis . • Misinterpretation of units , • Error in detailing " Misiriterpretatfcin of dravhlgs Construe· - Use of incorrect material tion . ·Inconect fabrication ·, ,. • Incorrect construction : Inspection I' Gross defect not dct~ted ; Use I · Accidental loading • Change o( use without structur.1l assessment · Need for specialist maintenance overlooked .. . . Example ' P .'" · ':'. 'neglect oClater.lt 'torsionai buckIhig .: . " room'used for'$tortt'gl'! of heavy equip. nient in office premises' · ~. erreci '~f'Wo~d~~~~~: ~;~~te over- -, looked!' " .... . . • ' .. soft stra.tum not deteCted :-: " '1" , . . error In computer program • •.•. kilogrammes interpreted:3.S l;lewtons .•. •.. ,20 .mm;bars-useQ.. instead o~ 40 mm .";'.: 100 mm'slab instead of 150 -mm' · .. grade 43 steel used instead of grade 50 , . ; omi~on of heat trea~.I1l e nt · .• error In position or reinforcement · .• crnck in .,-,'eld.· .~; ~ " -,·.f H , . '. seveni'impact Dr explosion I .. .'domestiC prernises'used'(or' public I . library •. . ' • . ', cathodic protection system becomes inoperative Table 13.1. General c1assifiCiltion or the nature and .sources of gross errors.
- 254. I 242 13, R[L1ABJLlTI' THEORY AND QUALlTI' ASSURANCe: . , ; Type of failur: ;; I Type A: I :~;::~:~::::~h I the structure '¥, Q~signed I ~ , .~. I T)'pe I?: , . ' I Failu~e i!1 a mode of b;h, aviow: ag!linst. ~'hi~~ the structu~~ was n!l~ designed . . ',' , .. -. ~ , Gross error, Errors afjecting: ioad<:arrying capacity , .,,' "', ,,1.1. ability to, remain servic~a~~~, . app1ie?l.~ad~s} :, I , , . ~rr~,~sJh~~: ~el~'~~, tp}I~~ fu~.dar:nent~~ ~t1~er- , standing ,of st[.~,ctu'll~ ~e~aucqur, .arj~~lIg f,,?m:' p~?r.~sion·,s ~gn;~ranc~. ~n.~~,~~'Jgno,~ance engineers' o,ersight Table1.3.2: Classi~cation 'of gro~s errors accorQing to' type of failure. :'< '"':, . ' Failures·Of type ~. ' arise because of'gross errors in the calculation of design loads and/or loadcarrying capacities, and/or. because of weaknesses'which,are introduced into the structure during construction. Most of the examples in table ' 13 ~1 are of this tYpe,l'Failures of type B occur mainl)' because of lack of knowledge, However, diulnction must be made between failures which ~~!'he result of ignorance within the slruc~~.ral e-ngineering pr'~fes-sion as a whole and those tlial occur becaUse -of ignorance- or negligenc'eby' atfindividuat or design team. In .th~·'~~;;~~~~~', th~ _~;gin~er or te~-i~ ~J~,iriy resPC?9~i:b(e; f;;~-:h~·iailute. In the for-mer, be or they.are just'unfonunate, unless It can be shown that currently accepted practice has been extrar~.~ted to en unreasonable extent. ,", The prevention of failures which arise from lack of knowledge within. the profession as a whole is clearly impossible and occasJonal failures of this type wiil continue to occur ~ The)' will then be researched and this will add to the general fund of engineering knowledge ~ Failures of type'-A';'and:faihiresoi type B resulting irom an engineer's ignorance or negl,igenc::~, are in thea!,)' pre'entable, but this requires an appropriate level'of expenditure,on edUcation, -training; d~sign c~~cking. qu'aiity control, inspection, mai~tenance, etc. The planning and co. ordination of these ·uious U!osks is the subject of qUaUl)' assurance, This. i.~ brieO)' referred to in section 13_;4 . . ' Tables13,Tand-13.2 show onlY'two of many possible ways of classifying gross errors. For instance;. elTOrs can be classified 'according to nat.ure .of the :,rror(table' 13,1) type of fa!Jure 3S$OC!ated with the error (table 13.2) ,c~nsequences of.lailur,e..at:ising from the error those responslblC' for causing the error those responsible for not. detecting the error . etc.
- 255. 13.3 INTERACTION OF RELIABILlTI' THEORY AND QUALITY ASS!;RANCE , !;. of tou,l 60 2. .+-~-L~~~~==~ . AI Ben 1-: F 0 H A. Lack of formal qua.liriudons B. Lack of education C. uck or uperien« D. Lack of ability to communicate E. L:l.ck of authority F. Incompetence G. Nelll.ence H. Sharp pracdc. FiCl.lrc 130.1 , hl~ly,b ofulldcrlyinll caU~1 of 120 Jlructu r.~I. f.ilun;:s in blildin~. [rom 11S.S. 243 COnsiderable su'ccess has beeri achieved in the analysis of structural fallure data using this type of classification 113.8J. 11S.11), 11S.17J. t13.18). An important conclusion is that many gross e~ors occur because'of lack of experience on the part of those un:dert.aking the work and because the fundamental behaviour of the structure is often not fully ·lriderstood. Figure 1S.1, taken from 11S.S}. illustrates this point. In a study of 120 building failures: it waS found that over 60% were due to lack of experience by the personnel concezned and that in about 50% of all cases the major cau~ of failure was a lack of appreCiation of the relevant design concepts • for example. ignorance of the need to design against lateral torsional buckling in unsupponed compression flange, . Such findings are of cons.iderable. value in the planning of quality '~ran7e scn~:nle~. 'but this will not be discussed here. 13.3 L'/TERACTION OF RELIABILITY THEORY AND QUALITY ASSURANCE lS.S.1 General The phrase . building process. has been used in the preceding Sections'oS;; general tenn to inClude' planning. design, analysis, construction, maintenance and use·of a-structure. It will also be used here, to cover all aspeCts and stages of a structural de'velopment~ but it should be emphasised that it is applicable to all structural projects, not just buildings. For a given structure and location, the building process can be dhi.ded. into two distinct stages: 1) preparing a precise ~ete~mlOl.i!'tic specificat~on for the structure, and 2) building the' structure and checking that. the speCiticaijc~ i~ ·met. ~ . ,;, The specification. im'olving documents 'and drawings, will typically contain ir,{ormation of the type the column shaIJ contain 16 40 mm diameter reinforcing bars the nominal thickness of the slab shaH be 200 mm . the structural steel shall have a n?minal yield stress of 350 Ximm1 .,"
- 256. 13. RELIABILiTY THEORY AND QL'.-LITY "SStiRA.~CE Th.is information. which is passed from the designer to the contractor (or builder), is nnd needs to be of a deterministic· nature, The second part of the building proccs-') im'olves the transformat ion of th~ 'specification into physica.1 reality - the structure. and checkIng that.i"t is satisfac· tory. For new structures. the role played by reliability theory is in the preparation 'of-thestructural specification, either directly, by subjecting the proposed design to a reliability analysis or, indirectly I by using a code in which the partial coefficients have been asses-sed probabilistically. As previously mentioned (for e"ample in chapter 3), the probabilistic models used in reliabilityanalysis are conditional upon the specified. standards of quality control an~...!!ccept:mce tests for the materials, and on the standards of inspection for the finished struct·ure. Stages 1) and 2) of t,he buildi~g process .~7 ~~~ should be) intima~ely relat~d: , ~h~l r~~~ .of. qu~ity assurance. in its broa?es.t · s.~nse. ~s to ~oo.rdinate. rationalise and m~:mitor t~ese two stag!~, Reliability theory also has a role to play in the assessment of existing structures. particularly when structural damage has occurred as 3 result of accidentalloeding, or when a structure is being' assessed for a r3dical change of usc. It should be noted that in all cases, ihe questions to be answered are ~(the type: Is the s;ructure strong enough? Should the nominal dimen· sions be increased or reduced, and if so by how much? 13.3 . .2 The effect of ~05S elTOn on' tbe choice of partial coeffi~ients - The problem that must now be considered is whether knowledge that gross ettors can "occur during the processes of design and construction should affect the rational choice of partial coefficients for use in level! codes. This is best explored by means of an example. Example 13.2 . . /taken from [l3.l)). Take a failure fu.nction of the fonn ((G, R, K, S)o and let the safety margin :1 be , M "'GR -KS (13.1) where R is a continuously distributed random strength variable, N(PR' aR) S is a continuously distributed random load eUed, N(/lS' as) K is a discrete·valued model uncertainty, and G is a gross error which modifies the strength parameter R, and assume that the qU3Jltities R. S. K and G ale 5tatistic3.lly indep'endent: This is:1 reasonable :!ssumption. as le3St as far as independence of G and the other 'wables is concernl!d. ,;ince it may be 3Ssum¥c. for example. that the p~obabllity of having an incorrect size of reo iniardn,! bus :s unrelated ,0 the 'leld stress of toe bars or to the loads that are subsequent-ly .:lFlplie~d to the structure. ' 1 .. ., ~
- 257. Let the initial cost of the structare be given by '; :. 245 (13.2) where a is a constane. and let the consequential cost of Cailure, should it occur, be (13.3) The probabiliey oC fallure, given a gross error of magnitude g and a model uncertainty of magnitude k. may be expressed as Pri g, k - P(gR- kS" 0) and the expected conditional total c;ost as ErCrlg,-kP' c1 ~ ·c; p~ig. k ." (13.4) . -.....:: :: (13.5) Assuming that the model uncen~nty K can take the following discrete· values with prob. ability mass p(ki ) k; 0.4 0.5 0.6 p(ki ) 0.2 0.6 0.2 Table 13.3 the expected total cost, given a ;ross error g. E( CT 19l. is 3 . ~ ' .. . E[CTIgJ 3 I (e, +" 'ciPrlgJP(k) i-I ·., (13.6) The expected total cost given that .there is a &!ioss ~rror. g~it.h a probability pIg). and no error (g . 1) with a probabilit.y 1 - p{g) is thus ' " " 3 3 ··· E[CT(g)] '" .E (c, + crPrlg)p(k)p(g) + X (el 4> crPr)p(kl(l- p(g» (13.7) i-I i - I Substituting for c} and cr from equations (13.21 and (~3,3) gives 3 3 EICT(g)]:cI !.,~ QJ.la(l + JP!ig)p(k)p(g) + SaPR (1 + ~Pr)p(k)(l-p(g» (13.8) i-I · i-I Variable • I ,I. R - I 0.07 S 100 I 0.10 (I" - 10 eost units ~ ::::: 20 I Table 13A
- 258. 246 !"iI."I>Ll~ I :30 I !20 . I I lIO I I ; "100 .. ~ . I I 90 - I I s. I I 70 I 0.1 Fi;ur ... 13.2 13. incf('uinll ~rou ~rror malni:udC! ----l I I (e)· 0.05 I I I I I l- I " I I P(ll:-O~-~ I '.5 .1 Ie 1.' ine-fuiin, cross errol.malnitud(l 1 EIC'rh:}I ,. •• 1 I Gross elTOI'S considlllJed in c:bv.icc __ or partial coe:rrleienu ____ 'Gros.,.;· error. ne~lccled In" eheM« of -----..I I I putial c:oef(icienu . ' . - F"~=U~~~~~~-~", 1500 I "J. ''.. ',ElCrIl:)"a .'8.6~1 1000 EICT~!1I~R'OP11..J' '" ,,~ cl=786.5t---------------------,,------------~~=o==~======:: 5 •• '.1 0.5 Fi~urc 13.3 (P(C)· 0.05). Initial <:oct ro: I'll. - 78.6;; I I I 1.0 ' .. "
- 259. BfBLlOGRAf'Hl' 241 Let us now undertake an unconstrained minimisation of EI CT tgl I with respect to the quantity P R and denote the minimum value of IJ.R by IJ.R,optU::l. This is shown in figur .. 13.2 for the set of parameters given in table 13.4. An important result for this particul:.: se~ of 4SSUmptions i5 that if the quality assurance process is.able to restrict the frequency of gTOSi errol"$ to less. than 25<., then ~R.opt is very insepsitive to. the occurrence of gross ettors of any magnitude. A more important result, however, is that even if the frequency of gross errors rises to say 5%, although. J.lR ,opt shows a marked increase {or gross erron of-moderate magnitude, the expected total coSt given by ." ..... '. . ~ ,. . , ~C:r(g)I.IJ.R,opll .. .-:..- ~'1 + crP,lg,IJ.R,oPt)p(k)p(g) .j-l 3 . + ;E(C, .+ crPrIPR,OPt }p(k){l- - p(g)) .1- 1 ..... •. . :, . (13.9) is inse"nsiti~e ·tO tll~' deci~i:on of whether or·not. to allow i~r "th~~~ssibility of gross errors in calculating J.lR.~pi (see figure 13.3). At the worst, the total expected cost diffe~ by only 15%. This example indicates that, at least far the set of models and parameters chosen. the possibility of the occurrence of gr~ss errors should not. influence the selection of partial coefficients for use in structural design. Some further results arc given in l13.11. The extent to which these results can be generaJised depends on circumstances. but it Is considered that under many can· ditions the optimisati~!il qf expenditure on t~e control of 8rOSS errOrs can be .un~ertaken ind£'o pendently of th.e choice of partial coefficients. Similar conclusions have been reached by others, e.g. 113.6). 13.4 QUAUTY ASSURANCE The respective vaJues of reliability analysis and quality assurance have been explored earlier in this chzpter and bave been shown to be entirely compatible. The analysis of many structural failures (see e.g. (13.2J) shows that the majority could not have been prevented by minor in· creases in panial coefficients. This is consistent. with the results obtained from example 13.2 and indicates that relatively more resources should be deployed on control. inspection and checking - i.e. quality assurance, A discussion of this large subject is beyond the scope of the present text but the reader is referred to (13.91 for further study. BIBLIOGRAPHY 113.11 113.2) Baker, M. J. and Wyatt, T. A.: Methods o{ Reliabilil)' Anal.vsis for Jackel Platforms. r:c=. ~Tld InternationaJ Conference on the Behaviour of Off·Shore Structures. Lon· don,1979. Blackley. D.l.: Anai)'sis of Structural Failures. Proc.Institution of Civil Engineers. Part 1, Vol. 62, feb. 1977.
- 260. 2.,l,8 [13.3[ [13.4 [ [13.5] [13.6[ [13,7[ [13.8[ (13.91 13. RELIABILITY THEORY AND QUALITY ASSUR.A~CE Blackley, D. I.: The Nature of Structural Design and Safety. Ellis Horwood. Chichester, 1980. Bosshard, W.:Structural Safety • .-I .. Matter 0; Decision and Control. IABSE Surveys S 9/19, LOBSE Periodica 2/1979. CIRL!.: Rationalisation of Safety and Serviceability Factors in Structural Codes. Construction IndustrY Research and Information Association. Report :-10 .. 63, 1977. Ditlevsen. 0.: Formal and Real Structural Safety. Influence of Gross Errors. Lectures on Structural Reliability (ed. P. Thaft·Christensen), Aalbarg University Centre, Aalborg, Denmark. 1980, pp. 121·147. Feld, J.: Lessons from Failures of Concrete Structures. ACI ~Ionograph No. I, 1964. Institution·of Structula!. Engineers:Structural Failures in Buildings. Symposium. London, April 1980. "' Joint Committee on Structural Safety, ·CEB • CEC:-l . CIB • FIP • IABSE • IASSRILEM: Gener~l Principles on Quality Assurance for Structures. IABSE. Reports of the Working Commissions. Vol. 35, 1981. [13.10] :Vlatousek, :'01.: Massnahmen gegen Fehler im Bauprozess. Dissertation );0. 6941. Eidgenossische Technische Hochschule (ETH), ZUrich. 1981. [13.11] i¥latousek. M. and Sch.r:te~der, J.: Untersuchungen zur Struktur desSicnerheitsPr:oblems bei Bauwerken. Sericht ~r. 59, Institut fUr Baustatik und Konstruktion.ETH, ZUrich, 1976. [13.121 Melchers, R. E.: Selection of Control LelJels for Maximum Utility of Structures. Proc. 3rd International Conference on the Application of Statistics and Probability in Soil and Structural Engineering. SydneY,1979. [13.131 :-Ioan, T. and Holand.l.: Risk Assessment of Offshore Structures· Experience and Principles. Proc. 3rd International Conference on Stru't;tural Safety and Reliability, Trondheim, 1981. [13.14] Pugsley. A. G.: The Safety of Structures. Arnold, London. 1966. [13.15] Schneider. J.: Organisation and J./anagement of Structural Safety during Design. Construction and Operation of Structures. Proc. 3rd International Conference on Structural Safety and Reliability. Trondheim. 1981. (13.161 SIA: Weisung fUr die Koordination des Normwerhs des SiA im Hinblick auf Sicher· heit und·Gt!orauchsfiilligkeit von Tragwerken. SIA 260. Schweiz. Ingenieur· und Architekten-Verein ISlA), ZUrich. 1981. [13 .. 17] Sibly. P. G. and WaL1.;:er .• . C.: Structural A.ccidenfs and their Causes. ?roc. Insti· tution of Ch'U Engi .... leers. Part 1. Vol. $2. :'ofay 19"ii. [ 13.1B I Tveit. O. J. and Eva.'ldt. 0.: EXperiences with Faiipres and Accidents of Olfshore Structures. Proc. 3rd International Conference onlStructural Safety and Reliability. Trondheim. 1981.
- 261. ·: '. 249 Appendix ~ RANDOM NUMBER GENERATORS 1. GENERAL For simulation studies and Monte-Carlo analysis use has to b~ made of long sequences of random numbers (generally"pseudo·random-numbeI5). Th~se are ~ost conve~i;n'tI~ 'g~~e~t~ using a digital computer. The increased use of "such studies in recent years has meant that reo liable lIbra.ry functi,?ns have been made available on most computer systems. It CM normally be assumed that these (unctions have been (ully tested for random behaviour. 2. UNIFORt.-l RANDOM ND-lBER GE~ERATORS ~iost digital random number generators are based 0':1 uniform pseudo·random number generators of the multiplicative congruence type. A uniform random number generator is 'one which 'generates successive independent realisations ui oi a random variable U having a rectangular density (unction. usually in the interval (0,11. Le. . giving elsewhere !-I. < 0 9<u<;1 u>1 The menn and st:mdnrd deviation of the random 'ariable U can be shown to be ." . (A.1) lA.2)
- 262. 2~O APPENDIX A. L"DOM NUMBER GE~ERATORS ! 3. ~1ULTIPLICAT1VE CONGRUENCE METHOD Tnis method produces a series of pseudo·random numbers ri trW. eventually rep~ats. bul, if correctly desigllt'd, oniy after a very lung cycle. The neXt numbe: in the pseudo.r~rid~rri' series is relaud to the previous number by the relationship' .'. ' . '. ;" .' ' . I' , " (A.3) where a and m are integer constants and are relatively prime. Starting with an integer »seedn io the firSL pseudo·random number r 1 in the interval (0, 1) is obtained !rom ~ a iQ-hm ~ m '" h + m '" jl ;.. m '"' h + r , (A.4) -~:~;~~.·.i£~~i! ~~~»,~:~~l p#<~(~e qUO~ie.~~ (~!~.»);~' a~~ jl 'is the in~g.~;Part. 11 ... {a.io_:~ hmi i..s:.the seed for the second random number. In general. the integer constants a and m are chosen to obtain the lopgest possible cycle. It can be shown that"i(, :, .nd ; .. , .-' ...... . ',. ""'. , " ' " . .. t , ·," " where band t are inte[!ers, then ~he length of the intl::ger sequence before repetition is" of the orderof2(b-2 J. EXample .j".l. Let a "" 5f .. (8 x 391 - 3) :: 3125 and m - 2u "" 61108864. Let the start· ing seed iO • 1234567 « 2"). '.". Then ~ . 3125 X 1234567 c· + ~ ". 57 ..: 32816627 . -- 48901 m 6i1QS864 Jl m 67108864 ~l. The first random number is the fractional part of aio/m '" 0.48901 - r1 and the new seed {or generating the second.number is i1 ". 32816627. The first few terms of this random sequence are' ~. ~i.,-__ '. o 1234567~- 1 3281~627 '- ·0.48901 2 96151S3~O.1432S 3 49781667"" 0.74185 0.27921
- 263. APPENDIX A. G£SEP..ATJOS OF RANDO~I DEVIATES 251 The lenith of this sequence before repetition is approximately ~26-2l .. 224 .. 16.77216. A commonly used random number generator on. for example the CDC 6000 Series com· puters is rn+1 c186277rll (modulo 2") .. , , ', This has ~pprox.imateJy 246~ (7.04 X lOll) random numbers before repetition. a sufficient· ly large number for most purposes. The pseudo;rahdom numbers r; generated by this method can be'assumed to be independent rea· lisations u1 of a random vimable U hl'ing a rectangular distribution with 0 <; U'" 1. 4.. GENERATION OF RA..~DOM DEVIATES WITH A SPEClfIED,PROBABlLITY DISTRI· BUTlON FUNCTION Fx , ,. A convenient aeneral method consists ,of generating a random number. r, as described above and then, by making use of equation (A.2), finding the corresponding random deviate x of tbe ran· dom variable X Crom , ':; ... (A.5) where Fx is the required distribution (unction. it is therefore nec~ss~y to il~d the inverse func-tion F'I , giving .. '. . . ', ' (A.S) This is valid for all roms of distribution function. but two'classes or variable' exist which reo quire different treatment. Class A: The distribution function FX bas an inverse FX which can be expressed in closed form In thiS cas':' ~~~ ::L..~dom de-ute .x can be generated simply· by obtaining successive values (A.?) Example A.2. Let X be type I extreme (maxima) distributl:d with distribution function 'Fx(x) .. exPl- exp(':" a(x - 'l))} (A.B) and paramel,ers,. and Q. •• Then (A.9)
- 264. 252 APPENDIX A. RANDOM :iUMBER GENERATORS Hence. a sequence o[ independent random deviateS xi or the random variable X may be obtained from , '. (A.I0) where [I are rectangularly distributed random numbers in the interval fO. 11. Class B: The distrihutioD {unction Fx has an inverse FiL which cannot he expres:!lcd in dosed . tonn In this case the general pro~ure is the same. but the inverse function has to be evaluated·' either graphically. by numerical integration, by table !ook,up and interpolation, or by, fitting " an ~ppropria'te polynomial. The last three methods are suitab~~ for¥:o';'puter applic.ation .. . 5. SPECIAL CASES: GENERATION OF RANDOM DEVIATES HAVING NORMAL AND LOG-NORMAL DISTRIBUTIONS The normal and log.nonnal distributions are two of the Clus B (unctions, but be<:ause of their irequent use they dHerve further at~ntion. In addit~.o~ :f;o t¥ general method described ". above, a number of special methods exist for normal variables., These me~hod~J~.ay . ~~,? be used . Cor generating.log4nonnaJly distributed random mimbers. by the 'use of an app~opri~~, transfor • . mation. Genen:ttion of random normal deujaies from the sum of n rectangularly distributed random deuiates. The fact that under very general conditions. the distribution 'function Cor the sum of a series oC independent random variables tends to a normal distribution as the number of variables in the sum increases (reCer to the central limit theorem in chapter 3) can be U5~~ generate random numbers having a distribution which approximates very cl~se,y. to nonnal. , Most -computer routines use the sum of 12 or more independent rect:Ulgularly distributed ran4 dom numbers ri' If the latter are generated in the interval to. a1. their sum ~~ can euily be shown to be approximately normally distributed with mean j.I~ given by ;":t .. an/2 and variance 0; by o~ .. al n/12 For the simple case when a::. 1 3lId n .. 12. r given by -~".. .. r,. -6 i-I (A.ll) (A.12) (A.13)
- 265. APPESDIX A. BIBLIOGRAPHY 253 is approximately normally distributed with zero mean and unit standard deviation. This approach gives excellent approximations to the normal distributi on for deviates within two or three standard deviations from the mean. but for extreme values the approx.imation becomes increasingly poor. unless n is large. For example. the random variable r defined by equation (A.13) cannot lie outside the the interval [-6, 61. Generation of random normal deuiates using method due to Box and .fuller Box. and Muller [A.l1 have shown that if r1 and r2 are Independent random variables from the same rectangular distribution in the interval lO,I), then Nt and N2 given by , Nl " (-2~nrl)~cos2I1'r2 , N2 '" (-2 Qn rllT sin211'r2 (A.H) are Independent random variables. normally distributed with zero mean and unit variance: The advantage oC this method is that it is accurate over the complete range and depends oniy on the randomness and independence of r1 31ld r2 • BIBLIOGRAPHY IA.ll Box, G. E. P. nnd Muller. )of. E.: A Xote on the Generation of Random Normal Del'iates. Annals of :-lath. StatistiCS, Vol. 29, 1958. {A.21 Hammersley I J. M. and Handscombe, D. C.: Monte Carlo Methods. Methuen. 1964. [A.3J Tocher, K. D.: The Art of Simulation. The English Universities Press,1963.
- 266. " ', · ,:: ,." ," ~, , " .. i. ; 'P ; " ; ' I •• , .. I',
- 267. 255 Appendix B SPECTRAL ANALYSIS OF WAVE FORCES .... ; -. : ' ,; .... 1. INTRODUCTION The purpOse of this appendix is: to d~ri"e a 'r~lationshiP "between t~e sp~trum of water surface elevation' 5 1)1) (w) aod the spectru'm oC structural displacements SSS( w) COt a typical multipile jacket structure in a given normal response mode. The theory is derived from the work of Borgman lB.II and Malhotra and P~nzien IB.3]. For a discussion·oC the general theory of ran" " dam vibr8:Uo~ and sP'~iia:1 ~alysiS See, for ex'amPle, (B.21 or IB.H. 2. GENERAL EQUATIONS OF MOTION ' .ossuming tha); ihe/sirucbire can be id~alised as a lumped mass system (see figure B.l). the equa. tions of motion 'may t>e expressed in the well-known form (B.l) . , : ! .dl1 " / " Fi~urc B. l .
- 268. 256 APPESDIX 8. SPECTRAL ANAL )"515 OF WAVE FORCES where m is a ciia~qn3t mass matrix Cs is a structural domping m;ltrix k is a structural stiffness matrix Us is a ... ·ector of displac!'!ments of the various lumped masses du d:u Us - CIt ::"dt" and where pIt) is a vector of wave loads Pi(t) acting on the lumped masses '(it," itt+1)/2 Pi(t):: dP(t) • (ZI + ~t-I )/2 Substituting for dP(t) from equation (12.37). equation f B.1) becomes where Up is a ... ·ector of water particle velocities ca is a matrix. involving parameters such as Cd' D. p (see chapter 12) Cb is a matrix involving parameters such as em' P. A (see chapter 12) A is a diagonal._m~~ri~. <?~_c~s.~tional areas. Letr '" up -cis' Equation"CS.3) may now be written in terms oci and ~ as, with - - - E = (c, - c) r + car Ii I (B.2) (B.3) (8.4) (8.5) The right hand side of equation fS.4) is completely defined tor a given ~~ve f~e~uen!=r, .~on. tainin~ terms im"olving only water panicle displacement. velocitY' and .u:celeration. The terms on tne left nand side Ofe of the st:mdard (arm ior 11 linear multi·degree .oi freedom system with . -- "1- viscous damping. except for the error term' it which involves a quadratif tenit in r"
- 269. APPENDIX B. SPECTRAL ANAL YSlS OF WAVE FORCES , J .~ In order to make equation (B'A) line~r.:~ i~r~inin;i~~d in'th~ '~ea~~~u~'~' s'~~~~ ~~t~'en dropped (see (B.3j). This leads to a damping matrix c with terms C ij ::: 0 , .. I t j } .. .... ; .. " (B.6) ".". ,,' where a· .. is the square root of the relative velocity variance at the level of mass i. Hence r,l ..,' '. __ _ , ... where Ce is a diagonal matrix of terms, referred to as ~U~? d~~ping co~fficients. ·On re'arrangement,equation (8.4) then becomes (B.7) (B.8) 25i The term (m + cb - P A) ~ the effective m~s matrix. c is a damping rna,tox i~ciuding com-' ponents of structural and fluid damping. Cell is the contribution to the loading due to drag. ~nd cb ~p is the contribution to the loading d~e to inertia effects. 3. MODAL ANALYSIS The matrix equation (B.8) represents a set of coupled differential equations. which can be uncoupled by usini the system's eigenvector m;trix (~'ode shape matrix), ;:'as a ~sfor~ation matrix. (;' is obtained by ignoring the damping term ~~s). ,Let- . (B.9) Pre-multiplication by ;'T and substitution in ,equation (B:8) ,gives, (B.10)' Mi + c ~ + Ki .. pet) (B.11l The generalised mass m.1trix M and stiffness m3trix K will be diagonaJ due to ortr.ogonality but the damping matrix C may be full. C may be diagonalised by error minjmiza:io~ ~ i~ ... . the drag term linearization. Howe·er. for highly damped systems. it. is sufficiently accurate to use the diagonal terms of C ana to ignore the off-diagonal terms. This approad: is followed below.
- 270. 258 APPENDIX B. SkCTRAL ANALYStS or WAVE F~RCES . .. .< l.. ..' '.' ., ,.. .. " ., ,- .•.. -. - . .. - I! EQuation (B.II) consists of a set of unco:;.pled modal equations which can be used to obtain the structural ~es.~nse. ~ ,~ach lIno~a~,. mode separatel~·. tfeating ~~~h.. ~ a one. degree of ftee· 'dom system: ' , .... -." .'. .... . , ... ,. "., i . .: '. , 4. SOLtJI'lON STRATEGY . -., (: . Consideration will be restricted to response in a single mode. Thus, equation (B.1.1) may be re.- written for mode j as (1.1 :::t Mjj• C - Cjj• K" Kjjl (8.12) or , . " .- : u ·, Mij + CXj + K:r:J • L:"'JiPj(t) j .. l ".' ' . .. ... . ,.-. . ' : . (B.13) where n is the nUlDber of lumped masse~ Il?d. Pi(t),. cr,il ~p,[ tFb.lI~pi' . '.' . ~ "' Usltlg complelt: number representation (I ... ,I-I), let • . ,-. 'I ." ': : " ~ . . ~r~'> " ': ' I. ', ., .. . Substitution {or' ~, and Xj in equ~tion (B,l.S) gives .. . . : " _" :':: : " ',J ' ... , ... . '.' , • ' . . . So , " (-w~M+ ICo)C+ K}:!j '-":L' -"'jjPj(t) 1-, H~I~') t ~'liPj(t) • 0 i"l . . , " where H(lw) is kno .... -n as the complex freq!ency response (unction. Thus . • . n , X·j(w)",wxj - c.wH("..I) .l'+jll(t) i"1 :-.' ;: . . ~ ,; " " ; ' ." ' ) ~ ' :'(B.14) (B.15) (B.17) ! -. (B.lS) (B.19) ancHhe ~'i~ctural velocity for each mass j r~po[,,#tig in mode·j is iiven·Crorn'equation·(B.9) as ' ,'. •• v' :r,1 '.. .' - , . ' . ~., '.';.', -:,. " .;."]
- 271. APPEt-:DlX B. SPECTRAL ANALYSIS Of' WAVE FORCES • UJ;,ij(W) '" ~'iJXjlW) c "'IJ,wH(,wl I "'jlittJ i-1 259 (B.20, It. can be seen from equations (B.3) 0 (B.B) that the structural displacement for any given component of wave frequency 1.,1 can be e'aluated only if the Iinearised damping coefficients are known. Unfortunately, the damping coeCCicients are dependent on the variance 01 the reo lative velocity between the structure and the water particles (see equation (B.6)) requiring an it~rati'e form of solution as described below. Particle velocity uarjance: For a given component frequency and depth Z, measured from the mean water level (positive z measured upwards), the panicle velocity is obtained ~from linear wave theory (equatiQns (12.29) and (12.31» as • ..... cosh k(z + ·dl Up(Z,71,W)=W sinhkd fl . The corresponding water particle velocitY spectrum Sup ~p (:z:; w) is gi'en by v{here U;(W) is the complex conjugate of up • Thus, _ 2 cosh2 k(z+dl S~pup(z.w)-w sinh~k.d S'I'I(w) and finally the particle velocity variance aJ (:r.) by • O"~ (ZJ "' {- Su: (z,w)dw p .0 p (B.2Ii (B.22) (B.23) (B.24) Force actin, on a single pile: The generalised force, Pj'"(tl: 'obtained b}; taking th~ ~e1. of terms correspllnding to mode j Crom the riSht hand side of equation (B.lO). is given by ., . ' I· ·· " pJ· et) - .c~.." ".IP,. (tJ" ~"" "',.I (c·"u p.1. + c• ,.I.I Up ,1.J i-l , i-I (B.25) Substituting for cr,ii and up,1 from equations (B.El) and (8.21) and noting that by combining equations (12.29) and (12.32) .. ', ... ( ) ~coshk(z+dl ·up Z.1f.W CIW' sinhkd II (B.26) 'equation (.8.25) becomes (8.27)
- 272. 260 APPENDIX B. SPECTRAL ASALYSLS OF WAVE FORCES The only term in the above which Is not known is 0i.i' the square root oC the relative velocity ·ariance. At the commencement of the iterative C3lculation. it may be assumed that Gr, i equals O'up,l' which is given by equQtion (13.24). Structural velocity variance: The structural velocity Us,ij(w) at the level of the jth mass is given by equation (B.20) as " u.s,ijew);; '''ij,wH(IW) Z"'JiPj(t) - "'ljtwH(Iw)Pj(w) j-1 where PjCw) is defined by equation (B.27). The structural velocity spectrum Sa,ij(w) is obtained frolU where u;.ij(w) is the complex conjugate of us,ij(w), Finally I the structuml velocity varillJ1ce may be obtained from oJ '" I'" S.I,(w)dw .,11 • 0 .1 (B.28) (B.29) (B.30) Structural tH!locity • particle velocity couariance: The structural velocity-particle velocity covariance u,~ ,,11 ,( '" Cov[ u. 'j' U ,J) is the area under the structural velocity.particle cross- "'~,'J p,l ,I p,l spectrum and is given by , (B.31) Structure-particle relariue uelocity variance: The relative velocity between the structure and the water particles at the level of mass I and for vibration mode j is , (B.32) (B.33) giving IB.34) IJi(i I Llbtuined frum lhe above may then b~ re-substituted in equation tB.o I!to ohtain a better ~atimate of the Iinearised fluid dampin~ terms. '
- 273. APPE~DIX B. SPECTRAL ANALYSIS OF WAVE FORCES 261 5. ~IULTlPLE PILES Tho preceding theory applies to a wave force acting on :1 singh! venical member or pile. or to a br3ced frame lying parallello the wave front. However, if the structure consists of two Of mote members or fnunes. the wave forces acting on the separate parts of the structure are dif· ferent, and depend in part on the ratio of wave length to the distance between the frames. The concept of a multiple pile tr3nsfer function IB.II can be used to modify the drag and in~rtia coe!ficients to allow for this effect. This approach t.nkes account of the relative positions of the members with respect to 'Wave length at each component of frequency. The modified coefficients c~ and ct, are gh'en by (B.3S, " . , cb DCb ( I I cosk(xn -xm))l IB.361 m .. t ft-1 where (:tn - 1m.) is the horizontal distance betwet!n the nth Qnd mth pile. k is the wa'e num· ber (see page 216), n' is the number uf piles or frames in the direction normal to the ..... ave front. and c. and cb c:m be obtained from the equations on page 256. In pt:lcticai applications, it ls clear that there may be considerable uncertainty in the total. force acting on the structure due to shielding effects. wave slam. the'effects ot' members in· clined to the vertical, but. at present these innuences cannot be allowed foe in a rigorous way. It is suggested, however, that some account of this is taken when m9delling ~Iorison's coer· ficients Cd and em' 6. COMPUTATIONAL PROCEDURE The calculations described above require considerable computational e£{on· particularly when they are us~d within an iterative level '2 reliability anat;sis. In particular, the areas under the various speetra n~ to be evaluated by numerical integration, and careful selection of frequency points is required to obtain sufficient accuracy without undue loss of comput.'ltionai eflicieocy_ BIBLIOGRAPHY {B.l] Borgman, L.E.: The Spectral Density {or Ocean Waue Forces. Coastal Eng. Can!.. ASCE.1965. lB.2J ~ewl3nd. D. E.: An Introduction to Random Vibrations and Spec.!ral Anaiyois. Longman. Lonaon. 1915. (B.31 ;"Ialhou3. A. ; ~. and Penden. J.: .'1on·Determinis tic Analysis of Offshore Structures. Joumo.l Eng •• lech. Div., ASCE. E~16. Dec. 197q: ' ~ !BAI Lin. Y. K.:Probabifistic Theory of Slructural-Dynamlcs. )1cGraw·!iill. :-l. Y .. 1967.
- 274. ! - ,1
- 275. INDEX Additive safety elements, 179, 185 Actions, choice,of distributions, 56 Airy waves, 215 ~ce,quality.247 Asymptotic extreme-value distributions, 40 Autocorrelation coefficient, 147 Autocorrelation function. 147 Autocovariance function, 147 A~erage correlation coefficient. 134 Barrier crossing: definition, 150 Rice's formula, 154 Basic variables: characteristic value, 182 correlated. 101 definition, 81 design values, 182 non-normal, 103 normal,88 specified characteristic value. 183 Bayesian reliability, 9 Bayes' theorem. 17 Bi'ariate nonnal density fUnction, 34 'Box and Muller method, 253 British Standard, BS 153, 196.198 BritUe elements, 114 Central moments, 24. 33 ChanM:teristic valUf5, 183, 186 Codes: actions, 185 calibration, 196 geometrical variables, 183 levell. 178 loads,185 material properties.18S safety formats, 180 Coefficient of variation, 24 Conditional distributions: defmition.31 density function, 31 distribution (unction, 31 mass function, 31 Control measures, liS Correlated basic variables, 101 Correlation coefficient: average, 135 definition, 33 equivalent, 137 Covariance: definition. 33 matrix, 34, 99 263