Uncertainty modelling and limit state reliability of

IJRET: International Journal of Research in Engineering and Technology ISSN: 2319-1163
__________________________________________________________________________________________
Volume: 01 Issue: 01 | Sep-2012, Available @ http://www.ijret.org 1
UNCERTAINTY MODELLING AND LIMIT STATE RELIABILITY OF
TUNNEL SUPPORTS UNDER SEISMIC EFFECTS
Mallika S1
, Srividya. A2
, Venkatachalam. G3
1
Research Scholar, Reliability Engineering Group, IIT Bombay, Powai, Mumbai, India, mallika@iitb.ac.in
2
Guest Professor, University College, Haugesund, Norway, asvidya88@gmail.com
3
Emeritus Fellow, Department of Civil Engineering, IIT Bombay, Powai, Mumbai, India, gvee@iitb.ac.in
Abstract
Underground openings and excavations are increasingly being used for civilian and strategic purposes all over the world. Recent
earthquakes and resulting damage have brought into focus and raised the awareness for aseismic design and construction. In
addition, underground tunnels, particularly, have distinct seismic behaviour due to their complete enclosure in soil or rock and their
significant length. Therefore, seismic response of tunnel support systems warrant closer attention. The geological settings in which
they are placed are often difficult to describe due to limited site investigation data and vast spatial variability. Therefore, the
parameters which govern the design are many and their variabilities cannot be ignored. A solution to this issue is reliability based
analysis and design. These real conditions of variability can only be addressed through a reliability based design. The problem
addressed here is one of reliability-based analysis of the support system of an underground tunnel in soil. Issues like the description of
the interaction between the tunnel lining and the surrounding medium, the type of limit state that would be appropriate, the non-
availability of a closed form performance function and the advantages of response surface method [RSM] are looked into. Both static
and seismic environment with random variability in the material properties are studied here. Support seismic response is studied in
terms of thrust, moment and shear forces in the lining. Interactive analysis using finite element method [FEM], combined with RSM
and Hasofer-Lind reliability concept to assess the performance of the tunnel support, has proven useful under real field situations.
Index Terms: Tunnel, Reliability, Random, Seismic
-----------------------------------------------------------------------***-----------------------------------------------------------------------
1. INTRODUCTION
The need today in civil engineering, more than ever before, is
toward providing economical designs commensurate with
safety. This requires a careful consideration of variabilities in
loads, geometry and material properties governing the behavior
of the given structure. Reliability analysis is best suited for this.
While considerable developments have taken place in
application of reliability concepts in structural engineering,
there is a need for more studies in geotechnical applications.
The importance of reliability analysis is best illustrated in the
evaluation of slope stability. Several researchers have
contributed to the understanding of slope reliability [1,2,3,4].
The measure normally adopted for expressing safety is the
factor of safety, which is evaluated from the relative
magnitudes of resistance (capacity) and load (demand) as FS =
Capaity (C) / Demand (D). Since capacity and demand cannot
be evaluated with certainty, assuming a random variation, a
ratio of expected values of capacity and demand called Central
factor of safety is preferred. However, when material properties
exhibit substantial variabilities, the minimum factor of safety
may not be the correct measure. A better measure of safety,
therefore, is the Reliability Index, β, which accounts for the
randomness of FS itself. It is defined as the number of standard
deviations by which the expected value of FS exceeds a critical
or acceptable value. This may be expressed as (Eq.1):
(1)
)(
1)(
FS
FSE

 
Assuming normal distribution of capacity and demand, another
parameter, Probability of failure, is expressed as (Eq.2):
(2))()(
2
1  
f
p
Where, φ (β) is the standard normal probability.
However, the variable values corresponding to the least value
of β would be the most critical [5]. The Hasofer-Lind reliability
index βHL is defined as the shortest distance from the origin of
the reduced variable space to the limit state function g = 0.
Consider a limit state function g(X1, X2…… Xn) with
uncorrelated Xi variables. The Hasofer-Lind reliability index is
determined by the following steps:
1. The reduced variables {Z1, Z2, . . ., Zn} are first determined
as (Eq.3):

__________________________________________________________________________________________
(3)/)( xixiii XZ 
2. The limit state function is then redefined in terms of these
reduced variables.
3. The reliability index is the shortest distance from the origin
in the n-dimensional space of reduced variables to the curve
corresponding to the limit state function described by g (Z
1
,
Z
2
, . . . , Z
n
) = 0.
The Hasofer-Lind reliability index βHL can be expressed in
matrix form as (Eq. 4):
(4)))(1)((min   XCTXHL
where x vector represents the n random variables; µ is the
vector of their mean values; and C is the corresponding
covariance matrix.
The minimization of Eq (4) is performed subject to the
constraint g(x) ≤ 0 where the limit state surface g(x) = 0
separates the n-dimensional domain of random variables into
two regions: a failure region F represented by g (x) ≤ 0 and a
safe region given by g(x) > 0. A schematic of the Hasofer-Lind
reliability estimation is given in Fig.1, where superscript N
represents normative values and θ represents an angle as
shown.
Fig-1: Schematic of Hasofer-Lind Reliability Index estimation
Reliability analysis has also been done for Piles subjected to
axial and lateral loads [6]. Empirical, analytical and numerical
methods are available for analysis and design of tunnels. The
empirical methods such as the geomechanical methods consider
rock mass characteristics, which are otherwise very difficult to
model. Authors of [7] have brought out the significance of
seismic analysis, though the analytical methods, such as those
employed by them, generally consider ideal simplified
situations. Numerical methods have come to be employed
widely during the past few decades due to their ability to model
complex geometry, loading, stress-strain relationships and
construction sequences [8, 9]. Apart from the rigourousness of
the method of analysis, consideration of material properties,
especially their variabilties, also merits close attention and the
present study is an attempt in that direction.
2. RESPONSE SURFACE METHODOLOGY
Following are some uncertainties surrounding the seismic
analysis of the underground openings.
• Uncertainties in characterizing the seismic event
• Uncertainties in structural and material modeling
• Uncertainties/errors in method of analysis and performance
criteria
In the current study, the response of the underground opening
under static and seismic loading is analyzed considering the
ultimate limit state (ULS) as the performance criterion and
material properties as random variables (see for example, Eq.
5). The response surface methodology (RSM) is adopted herein
in order to study the response of the tunnel lining considering
the effect of the randomness of the ground material parameters.
Based on the observed data from the system, an empirical
model, which is a polynomial function of the random variables,
is built using regression analysis of the selected deterministic
analysis results. This polynomial, called the response surface,
can serve as a basis for further simulations and hence, for a
better estimation of the probability of failure (Eq. 6). The true
response, y, of a system is given by [10]:
(5)),...,,( 21   kfy
Where i are the independent variables and ε is the modeling
error. The second-order polynomial approximation of the true
response function involving two factors is:
(6)
2
222
2
111211222110 xxxxxxy  
Where xi are called coded variables, which are transformed
values of the actual variables i, to the domain of [-1,1] and βij
are called regression coefficients. Fig. 2 shows the number of
analyses required and the values of the variables to be used in
the case of three factors (a, b and c).
Fig-2: Three level three factor experiment

__________________________________________________________________________________________
3. STATEMENT OF PROBLEM
In order to evaluate the effect of uncertainties, a real tunnel and
the earthquake to which it was subjected or considered. The
case study chosen concerns a tunnel of 6 m diameter and
overburden depth between 21.0 and 25.3 m [11].
The center of the studied section of the tunnel is approximately
24m deep with a 21 m thick overburden and is embedded in the
Shongsan formation which comprises six alternating silty sand
and silty clay layers. A typical cross section of the formations,
the alignment of the tunnel and related information are given in
(Fig. 3). The material properties of the formation and that of the
tunnel lining are listed in Table 1 and Table 2 respectively.
Fig-3: Cross-section of the soil stratification at Shongsan
airport tunnel site (Gui and Chien, 2006)
Table-1: Material properties of the ground medium.
Ground properties Mean values
Bulk unit weight γ (kN/m³) 18
Modulus of elasticity Estat (MPa) 28
Poisson’s ratio  0.3
Cohesion c (kN/m2
) 30
Friction angle φ (degrees) 31
Modulus of elasticity Edyn (MPa) 253
Table-2: Tunnel lining parameters (a) a Steel (b) Concrete
Grade 60 Steel (16 Nos.)
Modulus of elasticity of steel (MPa) 210000
Poisson’s ratio 0.2
Tensile strength (MPa) 420
Shotcrete Thickness (m) 0.3
(a)
Shotcrete Lining
Modulus of shotcrete (MPa) 30500
Poisson’s ratio 0.2
Tensile strength (MPa) 0.36
Compressive strength (MPa) 42
(b)
4. METHODOLOGY
Interestingly, when material property variation is the main
consideration, reliability analysis requires only a set of
deterministic analyses using different combinations of material
properties. Hence, the methodology used consists essentially of
three steps. In the first step, a series of deterministic analyses is
carried out. Since there are three variables (factors),
considering mean and ± standard deviation, the number of
combinations works out to (3)3
= 27. Hence, 27 deterministic
analyses are carried out using PLAXIS and thrust, moment and
shear are calculated in the lining. Next, using these values and
RSM, an equation is built for each of these quantities in terms
of three variables. This serves as an equivalent alternative
function for the unknown performance function. In the third
step, the resulting equations are used to arrive at the least
reliability index.
Two cases as shown in Table 3 have been considered. The
objective of choosing the two cases is to identify the effect of
the random nature of the material parameters (cohesion c,
Elastic modulus, E and angle of internal friction φ) on the
lining performance, under ultimate and serviceability
conditions within static as well as seismic environment. To
perform the reliability analysis the material properties of the
ground are considered as normally distributed random variables
with the mean and Coefficient of Variation (COV) as shown in
Table 4. For the chosen COV, there was no likelihood of the
normal variables turning negative. However, for a higher COV,
say more than 30%, there is a possibility of normal variables
turning negative. In such cases, assumption of a lognormal
distribution would help. The effect of their randomness on the
performance of the tunnel support system is then studied using
RSM, where, the response surface built is evaluated to obtain
the least reliability index βHL through repeated iterations and
subsequent convergence.
Table-3: Cases studied
Case ULS SLS
1. Static Thrust, Moment, Shear
force
Deformation
2. Seismic Thrust, Moment, Shear
force
-
Table-4: Properties of chosen Normal Random variables.
Ground properties Mean
values
COV
(%)
Modulus of elasticity Estat (MPa) 28 13
Modulus of elasticity Edyn (MPa) 253 13
Cohesion c (kN/m2
) 30 30
Friction angle φ (degrees) 31 10

__________________________________________________________________________________________
4.1 Performance Functions using RSM
Based on the limit states defined for the two cases listed above,
the performance functions for the thrust, moment, shear force
and deformation responses are as follows:
•ULS of thrust: Nc/N -1 =0
•ULS of moment: Mc/M-1=0
•ULS of shear : Qc/Q-1=0
•SLS of deformation: yc-y=0
Here, the notations Nc, Qc, Mc correspond to the ultimate
capacity of thrust, moment and shear force calculated based on
ACI 318-05 [12] for the given lining material properties as
listed in Table 5. The serviceability limit state is defined by
limiting the displacement yc of the lining to 5mm.
Table-5: Structural capacity of the tunnel lining
Thrust capacity Nc (kN/m) 7993
Moment capacity Mc (kNm/m) 283
Shear capacity Qc (kN/m) 3888
For the seismic analysis, the accelerogram of the earthquake
that occurred in November 14, 1986 with a magnitude of 7.8 at
a distance of approximately 120km from the airport and at a
depth approximately 34m is used. The peak ground acceleration
PGA is 0.13g (Fig 4). Fig. 5 shows the FE model used for the
dynamic analysis. In order to apply RSM, 27 sample points
were taken initially as shown in Fig.1. The numbers of
experiments were chosen [13] such that the convergence of the
order 10-1
is attained within three to four iterations. The
selected sample points are (ci,φi,Ei), (ci+1.2c, φi,Ei), (ci-
0.3c,φi,Ei), (ci,φi+1.4φ,Ei), (ci, φi -0.3φ,Ei), (ci,φi,Ei+E) and
(ci,φi,Ei-E)
Fig-4: Accelerogram of Taiwan Earthquake, November14,
1986 [14] (National Geophysical Data Centre)
Fig-5: Finite Element model used for seismic analysis
5. RESULTS AND OBSERVATIONS
The results are analyzed and discussed in terms of thrusts,
moments, shear and displacements. The associated reliabilities
under appropriate limit states and corresponding critical
material property combinations are examined.
Table 6 illustrates the iterations conducted for the RSM used in
obtaining the reliability index for the seismic case. Table 7
illustrates the minimum reliability obtained for the
displacement response.
Table-6: Iterations performed to obtain minimum reliability
for (a) thrust, (b) moment and (c) shear response
Initial
values
Iteration # for thrust
response
1st
2nd
3rd
β 0 1.08 3.60 1.70
c (kPa) 30 23.63 49.59 48.85
Φ (deg) 31 33.60 32.95 35.20
E (MPa) 253 441.75 462.46 368.60
(a)
Initial
values
Iteration # for moment response
1st
2nd
3rd
4th
β 0 0.82 3.47 2.84 2.07
c (kPa) 30 25.20 13.17 2.75 1.42
Φ (deg) 31 32.95 23.43 25.89 24.53
E
(MPa)
253 425.07 485.84 493.88 569.20
(b)
Initial
values
Iteration # for shear response
1st
2nd
3rd
β 0 0.003 0.56 0.31
c (kPa) 30 30.00 33.67 33.54
Φ (deg) 31 31.00 30.04 30.35
E (MPa) 253 334.32 344.18 331.22
(c)

__________________________________________________________________________________________
Table-7: Iterations performed to obtain minimum reliability for
displacement Response
Reliabilityβ 0 0.26 4.24 0.60
c (kPa) 30 28.74 20.64 24.28
Φ (deg) 31 31.40 31.07 30.78
E (MPa) 28 27.35 12.64 12.55
The minimum reliability corresponds to the last iteration for
each response. The corresponding critical values are also listed
alongside. It is also worth mentioning that the successive
iterations do not result in perfect convergence for the given
degree of tolerance. This could be attributed to the sampling
points chosen and could be refined further to obtain proper
convergence. The following are the equations ( Eq. 7 to 10)
derived for the Thrust (N), Moment (M), Shear Force (Q) and
deformation of the tunnel lining:
(7))
2
0.0001E
2
0.0102
2
0.0014cE0.0009
0.0002cE0.008c0.137E1.3690.496c6.360(/Nc



N
(8))
2
0.0002E
2
2.26φE0.0838
0.826cE21c0.171E23.3143c(203/Mc



M
(9)
2
0406.0
2
1.19
2
19.514.1
206.043.6737.0561221c14400/cQ
EcE
cEcEQ




(10)
2
0.463E
2
0.00575-
2
0.00279c
E0.1960.377cE-0.0986c2.29E-1.3616.4c-282yc



y
A comparison of the static and seismic reliability levels of the
lining response is given in Table 9.
Table-9: Comparison of reliability of the static and seismic
case for the lining response
Thrust
β
E
(MPa)
cc
(kPa)
φc(deg)
Static 1.77 2.02 34.60 23.40
Seismic 1.70 368.56 48.80 35.20
(a)
Moment
β
E
(MPa)
cc
(kPa)
φc(deg)
Static 5.77 58.60 20.02 37.10
Seismic 2.07 569.20 1.42 24.53
(b)
Shear
β
E
(MPa)
cc
(kPa)
φc(deg)
Static 1.30 25.51 34.93 34.74
Seismic 0.31 331.22 33.54 30.35
(c)
5.1. Ultimate Limit State (ULS)
The reliability of the lining system is analysed considering the
thrust, moment and shear capacity of the lining under Ultimate
limit state condition.
From the RSM based reliability analysis results (Tables 6, 7),
the minimum reliability is obtained for the shear response
where =0.31 indicating a probability of failure (pf) of 37.83%
under seismic condition. However for the thrust and moment
response, where >1, there is less probability of failure
(pf<4%). This indicates that the chosen lining system is reliable
against the randomness in ground material properties for the
thrust and moment response. Table 9 also confirms that shear
response under seismic case is poorer compared to the static
case. This implies that strengthening against shear failure
would be advisable.
5.2 Serviceability Limit State (SLS)
The displacement response indicates  of 0.60 indicating a
probability of failure under serviceability condition as 27.42%.
CONCLUSIONS
The response surface methodology incorporating the
randomness in the material properties of the ground and the
subsequent reliability based analysis of the underground soil-
support interaction has been found to be useful. The study
shows:
1. For the chosen lining system, the reliability decreases
under seismic conditions considering the random nature
of the ground properties chosen.
2. The reliability of the lining is low for the shear response
under seismic case.
3. However considering the total response of the lining
including the thrust, moment and shear for the ULS, the
reliability is more than the SLS. Deformation and, hence,
SLS is found to be the governing criterion of estimating
the reliability of the support system as it provides the
least reliability index.
4. Thus, reliability based analysis gives an optimum
solution to the design of underground support system.
ACKNOWLEDGEMENTS
The computational and other support provided by IIT Bombay
is gratefully acknowledged.

__________________________________________________________________________________________
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[14] National Geophysical Data Centre., University of
California, Seismograph station, Berkeley, CA and
Institute of Earth Sciences of the Academia Sinica,
Taipei, Taiwan.

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