Nonlinear Static Analysis (Pushover)

Safety Assessment and Retrofitting of Existing Structures and Infrastructures
PUSHOVER ANALYSIS F. Di Trapani
Nonlinear Static Analysis (Pushover)

It is an engineering technique for assessing structural capacity against seismic actions in nonlinear field
Capacity should not be strictly synonymous with resistance but, more in general, we intend displacement
capacity or structural ductility with respect to a request from the earthquake (Limit State)
It is a technique based on some simplified assumptions and approximations therefore it can be used in
specficic cases
It can be a very useful tool to make a judgment on the vulnerability of structures or the effectiveness of
retrofitting interventions
Non-Linear Static Analysis - PUSHOVER

Non-Linear Static Analysis - PUSHOVER
What does it consist of?
Base shear
Top displacement
V (Base Shear)
δδδδ
Capacity curve
δδδδ (Top displacement)
δδδδu
Vmax
Application of a profile of forces or displacements monotonically increasing
Evaluation at every step of base shear and the corresponding top displacement and definition of a V-δ curve called a
capacity curve
Conversion of the capacity curve of the MDOF system into an equivalent SDOF capacity curve and check in terms of
displacement (or ductility) compared to the demand from inelastic spectra
Forcesprofile
1
2
3

Model Nonlinearity
σσσσ
The model has to evolve during the analysis in order to capture the damage associated
with the nonlinear response of materials
MODELING OF MECHANICAL NONLINEARITY
(in frame structures)
NON-LINEARITY OF MATERIAL
εεεε
εεεε
σσσσ
εεεε
σσσσ
CONCRETE Steel
Elastic stage Inelastic stage
Plastic Hinge models (concentrated plasticity)
Fiber section models (distributed plasticity)

Sample of nonlinear step-by-step response
(example with concentrated plasticity)
NL MODELLING OF RC FRAME STRUCTURES F. Di Trapani
(1) (2)
Myb
Myb
V1 V2
(3)
Myb
V3
Myb
Myc Myc
F1 F2 F3
(4)
V4=V3
Myc Myc
F4=F3
F1 δδδδ1 δδδδ2 δδδδ3 δδδδ4
Myb
Myb
OVERALL RESPONSEOVERALL RESPONSE
Base Shear
δδδδ
1
2
3 4
V2
V3=V4
V1
δδδδ3 δδδδ4δδδδ2δδδδ1
Top Displacement
M
Myb (Beams)
Myc (Columns)
δδδδ
ΘΘΘΘ
V
Sequence of formation
of plastic hinges during
the analysis and
associated deformed
shapes.
Sequence of formation
of plastic hinges during
the analysis and
associated deformed
shapes.

V
δδδδ Newton - Raphson Algorithm
R-F=U
Error
Iterations are carried out as long as the
error is less than the established tolerance
First iteration
Displacement to
first iteration
Determination of nonlinear response
Error on first
iteration )K(δK =
110 FδK = 11 UFR =−
221 FδK = 22 UFR =−
Forces within the
first iteration
)U() ∆δK(δF =
Load
increment
LINEAR Response
KδF =
NON LINEAR Response
equilibrium is rewritten every time as a
function to the evolution of K
The stiffness matrix updated during the analysis. The
determination of the nonlinear response is an iterative
procedure. In most cases Newton-Raphson algorithm is used.

Hypotheses for Uni-Modal Non-Adaptive (Pushover)
Lateral forces profile or displacements is monotonically increasing (the actual forces are cyclic)
The force profile is proportional to the first mode (first mode should have large participating mass ratio)
The lateral force profile has fixed shape and increases only in amplitude (the actual force profile varies with the
modification of the stiffness matrix)
1
2
3
Non-Linear Static Analysis
The capacity curve of the MDOF system should be converted into an equivalent SDOF capacity curve in order to do
safety checks using spectra (which are defined for SDOF systems)
4
Adaptive
(Hp. 3 is removed)
Pushover Analysis
Non-Adaptive
Multi-modal (Hp. 2 is removed)
Pushover Analysis
Uni-modal
General Framework for Pushover analysis

Evaluation of the Capacity Curve

m1
m2
m3
gx&&&&&& Mτ)xR(x,xCxM −=++
gx&&
The linear term Kx and replaced by a nonlinear term representing
inelastic restoring forces
x3(t)
x2(t)
x1(t)
Transforming the MDOF system equation into an equivalent SDOF system equation
Assumptions of load profile shape
R
x
Equation of the motion for a nonlinear MDOF system
4
2
Introduction of Hypotheses

Transforming the MDOF system equation into an equivalent SDOF system equation4




















=










3
2
1
333231
232221
131211
3
2
1
y
y
y
x
x
x
ΦΦΦ
ΦΦΦ
ΦΦΦ
Φyx =
Eigenvector first mode1Φ












=












=
111n
1121
1n
21
11
1
/ΦΦ
...
/ΦΦ
1
Φ
...
Φ
Φ
Φ It is chosen for convenience of normalizing
it with respect to the top displacement
g11111111 xyyyy &&&&&& Mτ)Φ,R(ΦCΦMΦ −=++
11 yΦx =

Introducing the Load Profile Shape (Modal Profile)2
Vector of restoring forces
Vector of equivalent static
forces=
F3
F1
F2
In addition, since:
11 yKΦKxF ==
11
2
111
2
1 yy MΦMΦF ωω ==
It is assumed that the non-linear relationship between restoring forces and displacements is determined
through the application of a force profile F of the same shape as what you would have in the linear field
1
2
11 MΦKΦ ω=












=
1nn
212
111
Φm
...
Φm
Φm
λF
111 Φm
212 Φm
313 Φm
×λ
R F
F3
F2
F1
Force profile final form

g
T
11
2
11
T
111
T
111
T
1 xyyy &&&&& MτΦMΦΦCΦΦMΦΦ −=++ ω
The expression of R is replaced and both members multiply by T
1Φ
It is the equation of a non-linear SDOF system
1
T
1
T
1
1
MΦΦ
MτΦ
=Γ
1
1
1
y
D
Γ
=
MτΦT
1
*
m =
g1
2
1111 xDD2D &&&&& −=++ ωξω
§C 7.3.5 (Circ. 2019)
§C 7.3.4.2 (Circ. 2019)
Conversion of the MDOF system into the equivalent SDOF system in the case of modal force profile2-4
The modal
participation factor is:
Assuming the variable
D1 as:
The equivalent SDOF system is obtained by the scaling factor Γ1
(first modal participation factor with the eigenvector normalized at the top)
The mass of the equivalent SDOF is:

1
b* F
F
Γ
=
1Γ
cd
bF
*
d
1
c* d
d
Γ
=
Fb
dc
§C 7.3.4.2 (Circ. 2009)MDOF SYSTEM CAPACITY CURVEMDOF SYSTEM CAPACITY CURVE
EQUIVALENT SDOF CAPACITY CURVEEQUIVALENT SDOF CAPACITY CURVE
*
F
Conversion of the MDOF system into the equivalent SDOF system in the case of modal force profile2-4
PUSHOVER ANALYSISPUSHOVER ANALYSIS
Equivalent SDOF responseMDOF Response

Uniform profile
In order to consider a different possible modification of lateral forces as a function of the actual damage (e.g. damage
localized at the lower floors) another analysis with is typically performed using a force profile that is typically proportional to
floor accelerations and then to the floor masses. This is typically called uniform profile.
m3
m1
m2
Amplifies demand to lower floors
dc
Fb
Uniform
Possible real response
Modal
d*
F*
Uniform
Modal
MDOF
SDOF

Evaluation of the Demand and safety assessment
(N2 Method) – Fajfar 1996

Evaluation of the demand and safety assessment: Identification of the bilinear equivalent SDOF curve
*
F
*
d
*
buF
*
yF
*
buF6.0
*
yd *
ud
*
buF85.0≥
1
bu*
bu
F
F
Γ
=
1
u* d
d
Γ
=
SDOF
BILINEAR
EQUIVALENT
CURVE
The bilinear intersects the capacity curve at 0.6Fbu*
dy* is the yielding displacement associated with Fy*
Safety assessment is carried out by using elastic and inelastic spectra. To determined the demand it is necessary to characterize
the SDOF period T*, Stiffness K* and, reduction factor q*. This can be done by defining a bilinear equivalent curve.
Fy* is determined in such a way
that you get equivalence of the
underlying areas

*
F
*
d
*
buF
*
yF
*
buF6.0
*
yd *
ud
SDOF
BILINEAR EQUIVALENT
*
k
*
y
*
y*
d
F
k = MτΦT
1
*
m = *
*
*
k
m
2T π=
Stiffness Mass
Period
§C 7.3.6 (Circ. 2019)
Evaluation of the demand and safety assessment : Determination of K* and T*

==
y
E
F
F
q
The force required to the indefinitely elastic system can be obtained through the elastic spectrum
**
e
*
EE m)T(SFF ==
T
)T(Se
)T(S *
e
*
T
*
y
**
e
*
y
*
E*
F
m)T(S
F
F
q ==
*
yy FF =
The displacement demand will depend on q* and T*
FORCE REQUIRED TO THE ELASTIC SYSTEM
YIELDING FORCE
§C 7.3.4.2(Circ. 2019)
Evaluation of the demand and safety assessment : Determination of the reduction factor q*
Reduction factor
The yielding force is already known

T*≥TCCASE 1
Equal displacement rule applies
*
F
*
k
*
yF Bilinear
Equivalent
*
max,e
*
max dd =
DISPLACEMENT
CAPACITY
*
d
*
ud
ELASTIC
DISPLACEMENT
DEMAND
INELASTIC
DISPLACEMENT
DEMAND
Evaluation of the demand and safety assessment : Safety checks





≥=
<+−=
)TT(q
)TT(1
T
T
)1q(
C
**
d
C
*
*
c*
d
µ
µ
dµ
*
yd
*
y
*
u
c
d
d
=µ
cd µµ ≤
(q* and T* are known)





≥=
<





+−=
)TT(dd
)TT(1
T
T
)1q(
q
d
d
C
**
max,e
*
max
C
*
*
c*
*
*
max,e*
max
*
y
*
max
d
d
d
=µ
Intermsofductility
*
u
*
max dd ≤
*
ud
Intermsofdisplacements
§C 7.3.4.2(Circ. 2019)
Capacity
Demand
Demand Capacity
)T(Sdd *
De
*
max,e
*
max ==

T*<TCCASE 2
Evaluation of the demand and safety assessment : Safety checks





≥=
<+−=
)TT(q
)TT(1
T
T
)1q(
C
**
d
C
*
*
c*
d
µ
µ
dµ *
y
*
u
c
d
d
=µ
cd µµ ≤
(q* and T* are known)





≥=
<





+−=
)TT(dd
)TT(1
T
T
)1q(
q
d
d
C
**
max,e
*
max
C
*
*
c*
*
*
max,e*
max
*
y
*
max
d
d
d
=µ
Intermsofductility
*
u
*
max dd ≤
*
ud
Intermsofdisplacements
§C 7.3.4.2(Circ. 2019)
Capacity
Demand
Demand Capacity
*
F
*
k
*
yF Bilinear
Equivalent
ELASTIC
DISPLACEMENT
DEMAND
*
max,ed
DISPLACEMENT
CAPACITY
*
d
*
ud
INELASTIC
DISPLACEMENT
DEMAND
*
maxd
*
maxd
)T(Sd *
De
*
max,e =
*
yd

Evaluation of the demand and safety assessment : Safety checks in the ADRS plan
Elastic spectrum (µ=1)
Inelastic constant ductility spectrum
Bilinear equivalent curve (capacity spectrum)
Elastic spectrum (µ=1)
Inelastic constant ductility spectrum
Bilinear equivalent curve (capacity spectrum)
(acceleration) (acceleration)
(displacement)(displacement)
ay
ae*
S
S
q =
*
*
e*
aeae
m
F
)T(SS ==
*
*
y
ay
m
F
S = 




≥=
<+−=
)TT(q
)TT(1
T
T
)1q(
C
**
d
C
*
*
c*
d
µ
µ
dµ
dµ
The safety check can be done graphically by
superimposing the normalized capacity curve
with the constant ductility spectrum for the
requested µd and is satisfied if the
performance point is exceeded.

Determination of Vulnerability indexes (TR=const. Hp.)
Sa
Sa
PGA
PGA
d,g
c,g
d
c
E
⋅
⋅
==ζ
T
)T(Sae
Reference Elastic SpectrumThe PGA demand is associated with the
reference elastic spectrum. It is the spectral
acceleration in correspondence of T=0
The PGA capacity is associated with the seismic
performance of the structure. This can be larger or
lower than the demand, and this means that the
earthquake exactly inducing the limit state is different
and has scaled (up or down) elastic spectrum
The vulnerability indexes, defined by the coefficient ζζζζE are conventionally evaluated by carrying out the ratio between
the PGA capacity and the PGA demand. This ratio can be larger or lower than 1 in the case that the system is satisfying or
not the safety check.
(S= Soil factor)
The PGA of the earthquake inducing the limit state
(LS) is found by imposing the capacity parameters of
the SDOF
Spectrum inducing LS
cPGA
dPGA

Determination of Vulnerability indexes (TR=const. Hp.)





≥=
<+−=
)TT(q~
)TT(1
T
T
)1(q~
C
*
c
*
C
*
c
*
c
*
µ
µ
*
ud *
y
*
u
c
d
d
=µ
Form the bilinear curve of the SDOF has the
following capacities:
Ultimate
displacement
capacity
Ductility
capacity
*
q~
1. Substituting the ductility
capacity into the q-µ-T
relationships one can found the
reduction factor associated with
the current SDOF and the
spectrum of the earthquake
inducing the limit state





≥=
<





+−=
)TT(dd
)TT(1
T
T
)1q(
q
d
d
C
**
max,e
*
max
C
*
*
c*
*
*
max,e*
max







≥=
<
+−
=
)TT(dd
~
)TT(
1
T
T
)1q~(
q~d
d
~
C
**
u
*
max,e
C
*
*
c*
**
u*
max,e
2. Substituting and the elastic
displacement associated with the
spectrum of the earthquake inducing the
limit state is found
*
q~ *
ud
*
max,ed
~
3. Given the proportionality
of spectral ordinated with
respect to PGA one can set:
E
d
c
*
ae
*
ae
*
De
*
De
*
max,e
*
max,e
PGA
PGA
)T(S
)T(S
~
)T(S
)T(S
~
d
d
~
ζ=∝==
T
)T(Sae
*
T
Reference Elastic Spectrum
Spectrum inducing LS
dPGA
dEc PGAPGA ζ=
)T(S
~ *
ae
cPGA )T(S *
ae
4. PGA capacity can be also
found.

Technical code definitions and prescriptions

NTC 2018 and Circular 2019: Definitions and Prescriptions
Nonlinear static analysis allows to determine the capacity curve of the structure, expressed by the relation Fb-dc, in
which Fb is the shear at the base and dc the displacement of a control point, which for buildings is generally
represented by the center of mass of the last floor. For each limit state considered, the comparison between the
capacity curve and the displacement demand allows to determine the level of performance achieved. To this
end, a structural system equivalent to a degree of freedom is usually associated with the real structural system.
At least two distributions of lateral forces must be
considered (one for each group).
Main distributions (Group 1)
Secondary distributions (Group 2)

Group 1 – Main Distributions
if the fundamental mode of vibration in the considered direction has a mass participation not less than 75% (60% for
masonry buildings), it applies one of the following two distributions:
- distribution proportional to the static forces referred to in § 7.3.3.2, using as a second distribution the (a) for
Group 2,
- distribution corresponding to a trend of accelerations proportional to the shape of the fundamental mode of
vibration in the direction considered;
in all cases a distribution corresponding to the trend of the horizontal floor forces
acting can be used.
These are calculated in a linear dynamic analysis (Response spectrum analysis), including
in the direction considered a number of modes necessary to achieve a total participating
mass of not less than 85%. The use of this distribution is mandatory if the period
fundamental of the structure is higher than 1.3 TC
F3
F1
F2

=
ii
ii
hi
hw
hw
FF
111 Φm
212 Φm
313 Φm
iii mF Φ=
F
F components are the
combination of modals
forces component according
to combination rules

a) distribution of forces, deduced from a uniform acceleration trend along the height of the building;
b) b) adaptive distribution, which changes as the displacement of the control point increases as a function of the
plasticization of the structure;
c) multimodal distribution, considering at least six significant modes
Group 2 – Secondary Distributions
ii mF =
m3
m2
m3

How many analyses to perform in total ?
X
Y CM
Group 1
Center of Mass CM translation +/-5%
for each direction with positive and
negative verse
+Y(+)
-Y(+)
+Y(-)
-Y(-)
+X(-)
+X(+)
-X(+)-X(-)
X
Y CM
Group 2
Center of Mass CM translation +/-5%
for each direction with positive and
negative verse
+Y(+)
-Y(+)
+Y(-)
-Y(-)
+X(-)
+X(+)
-X(+)-X(-)
8 Analysis 8 Analysis
TOTAL 16 ANALYSES

Applicative Example

CASE STUDY – GEOMETRIC DATA

Bond CLSBOND STEEL
fc=27 MPa
εcu=0,005fy=450 MPa
CASE STUDY – Mechanical data

MODEL

MODAL ANALYSIS
Mode 1 Mode 2 Mode 3
T=0,84 s T=0,25 s T=0,13 s

MODAL ANALYSIS












=
0.0282
0.0549
0.0755
0.0883
1ϕ












=
0.319
0.621
0.855
1
1 nϕ
Participating periods and masses
Eigenvectors first mode
2
kNs27.11 =Γ
Participation CoefficientMasses on the floors
m/kNs55.61m 2
=1
m/kNs55.61m 2
2 =
m/kNs55.61m 2
3 =
m/kNs55.61m 2
4 =

== 2
1ii
1ii
1
T
1
T
1
1
m
m
φ
φ
Γ
MΦΦ
MτΦ

LOAD PROFILES












=












×
×
×
×
=
65.19
26.38
62.52
55.61
61.550.319
61.550.621
61.550.855
61.551
F
MODAL
Scaled by 61.55
1
0.86
0.62
0.32
UNIFORM 1
1
1
1
m/kNs55.61m 2
=1
m/kNs55.61m 2
2 =
m/kNs55.61m 2
3 =
m/kNs55.61m 2
4 =
Scaled by 61.55

0
50
100
150
200
250
300
350
400
450
500
0 0,1 0,2 0,3 0,4 0,5
MODALE
UNIFORME
Baseshear[kN]
Top Displacement [m]
UNIFORM
MODAL PROFILE
1° Yield strength 1° Yield strength CollapseCollapse
UNIFORM PROFILE
RESPONSES
Capacity Curves

0
50
100
150
200
250
300
350
400
0 0,1 0,2 0,3 0,4
Serie1
FXTRI BIL
0
50
100
150
200
250
300
350
400
450
500
0 0,1 0,2 0,3 0,4 0,5
MODALE
UNIFORME
BaseShear[kN]
CONVERSION TO THE EQUIVALENT SDOF
0
50
100
150
200
250
300
350
400
0 0,1 0,2 0,3 0,4
Serie1
Serie2
BaseShear[kN]
MDOF SDOF
BaseShear[kN]
UNIFORM
2
kNs27.11 =Γ
BaseShear[kN]
MODAL
m/kN6323*k
s99.0*T
m046.0*d
kN290*F
m/kNs4.158*m
y
y
2
=
=
=
=
=

DEFINITION OF THE DEMAND

g31.0*)T(Se
91.0*T
=
=
g28.0*)T(Se
99.0*T
=
=
MODALUNIFORM
T [s]
Se(T)
[g]

Determination of q*
49.1
*F
F
*q
kN328*F
kN04.4914.15881.931.0*m*)T(SeF
91.0*T
y
E
y
E
==
=
=××==
=
UNIFORM
32.1
*F
F
*q
kN328*F
kN4354.15881.928.0*m*)T(SeF
99.0*T
y
E
y
E
==
=
=××==
=
MODAL

m362.0d
m07.0
6323
435
6323
4.15881.928.0
*k
*m*)T(Se
*d
kNm6323*k
*
u
max
=
==
××
==
=
m361.0d
m065.0
7472
491
7472
4.15881.931.0
*k
*m*)T(Se
*d
kNm7472*k
*
u
max
=
==
××
==
=
Determination of the displacement demand and safety checks
MODAL
T*>TC
CASE 1
*
**
e*
De
*
max
k
m)T(S
)T(Sd ==
Verified!
UNIFORM
Verified!
*
u
*
max dd ≤

Nonlinear Static Analysis: Final Considerations
1. This is an approximate method for assessing the seismic performance of structures with respect to a specified
seismic demand.
2. It is a static method used to simulate the effect of a dynamic action
3. It provides good results and is applicable IF the structure is dominated by a fundamental mode
4. Load profile increases in amplitude but does not change shape even when structure enters nonlinear field
5. To take into account possible oversights arising from this hypothesis, verification is usually required for different
force profiles
6. To perform the checks an equivalent SDOF is identified
7. Checks are performed in terms of ductility or displacements.

Faijfar, P., Gaspersic, P., 1996. The N2 method for the seismic damage analysis of RC buildings, Earthquake Engineering and
Structural Dynamics,25, 31-46.
Peter Fajfar (2000) A Nonlinear Analysis Method for Performance-Based Seismic Design. Earthquake Spectra: August 2000, Vol. 16
Faijfar, P., 1999. Capacity spectrum method based on inelastic demand spectra, Earthquake Engineering and Structural Dynamics,
28, 979-993.
Anil K. Chopra and Rakesh K. Goel (2000). A modal pushover analysis procedure for estimating seismic demands for buildings.
Earthquake Engng Struct. Dyn. 2002; 31:561–582 (DOI: 10.1002/eqe.144)
D.M. 14/01/2008. Nuove Norme tecniche per le costruzioni.
Federal Emergency Management Agency. 2000. FEMA 356 “Prestandard and commentary
for the seismic rehabilitation of buildings”, Washington, D.C., Stati Uniti.
Federal Emergency Management Agency. 2001. FEMA 368 “NEHRP Recommended
provisions for seismic regulations for new buildings and other structures”, Washington,
D.C., Stati Uniti.
ESSENTIAL BIBLIOGRAPHY

Nonlinear Static Analysis (Pushover)

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