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Earthquake Analysis Earthquake Analysis :: Overview:: Overview
Earthquake Analysis Earthquake Analysis :: Overview:: Overview
Short Course on Nonlinear Seismic Analysis of Structuresby Durgesh C. Rai, C.V.R.Murty and Sudhir K. JainDepartment of Civil Engineering, IIT Kanpur
The material contained in this lecture handout is a property ofProfessors Durgesh C. Rai, C.V.R.Murty and Sudhir K. Jain of IIT Kanpur, and is for the sole and exclusive use of the participants enrolled in the Short Course on Nonlinear Seismic Analysis of Structuresconducted by them. It is not to be sold, reproduced or generally distributed.
Earthquake Analysis of Analysis of Structures
Simplified StructuresStructures
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Single mass SystemsSingle mass Systems
• Some structures have – Most of their mass lumped at a single location
• A single displacement unknowng p
5Walkway CorridorWalkway Corridor Elevated Water TankElevated Water TankFlyoverFlyover
• Most structures– Can be approximated for first-cut approximation
Single mass Systems…Single mass Systems…
6
Multi-Storey Building Multi-Storey Building Equivalent Simplified SystemEquivalent Simplified System
Basics ofDynamicsDynamics
Structural propertiesStructural properties
• Three independent properties– Mass m– Stiffness kStiffness k– Damping c
Roof(mass m)
Damping( ffi i t )
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Column(stiffness k)
(coefficient c)
3
Forces Forces and and equilibriumequilibrium
• Disturbance– External force f(t)– ResponseResponse
• Displacement • Velocity• Acceleration
u(t)f(t)
Roof
( )tu( )tu&( )tu&&
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f(t)
Column
• Internal forces– Inertia force– Damping force
Forces and equilibrium…Forces and equilibrium…
( ) ( )tuctfD &⋅=( ) ( )tumtfI &&⋅=
Damping force – Stiffness force
f(t)
Inertia force
Stiffness force Damping
force
External force
( ) ( )tuktfS ⋅=
( ) ( )fD
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• Dynamic equilibriumfI(t)+ fD(t) + fS(t) = f(t)
)t(fkuucum =++ &&&
Free Vibration ResponseFree Vibration Response
• Initial disturbance– Pull and release : Initial displacement– Impact : Initial velocityImpact : Initial velocity
• No external forceNeutral position
Extreme position
0kuucum =++ &&&
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position
• Subsequent motion
v0
Free Vibration Response…Free Vibration Response…
d0
0
u0
un
Tispl
acem
ent u
(t)
Exponential decay
Time t
12
∆t
TDi
4
• Dynamic characteristics– Natural frequency ω or Natural Period T
Free Vibration Response…Free Vibration Response…
k
– Damping
mk
=ω
13
• Critical Damping– Smallest value of damping at which
• No cyclic motion
Free Vibration Response…Free Vibration Response…
y• Gradual return to neutral position
men
t u(t)
Critically dampedCritically damped
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0
Dis
plac
em Time t
Under-dampedUnder-damped
Harmonic vibration ResponseHarmonic vibration Response
• Steady-state versus transient responses
ispl
acem
ent u
(t)
Time t0
Steady State Steady State
Transient ResponseTransient Response
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Di Steady-State
ResponseSteady-State Response
• Build-up of Transient AmplitudeHarmonic vibration Response…Harmonic vibration Response…
Dis
plac
emen
t u(t)
Time t0
Steady State Steady State
Transient AmplitudeTransient Amplitude
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Steady-State AmplitudeSteady-State Amplitude
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• Forced Vibration Response– Harmonic Excitation– Constant Amplitude
Harmonic vibration Response…Harmonic vibration Response…
Constant Amplitude
Normalised Displacement
umax/ust
UndampedUndamped
Under-dampedUnder-damped
17Frequency ω
0
Critically UndampedCritically Undamped
1
ωn
• Forced Vibration Response… – Resonance at natural frequency of structure– Critically dependant on damping
Harmonic vibration Response…Harmonic vibration Response…
Critically dependant on damping
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Seismic ResponseSeismic Response
• Change of reference frame– Rigid body motion causes no stiffness & damping forces
Mass m
( )tug&&
( )tum g&&−
19Fixed-base Structure
Fixed-base Structure
Moving-base Structure
Moving-base Structure
( )tug
• Change of reference frame…
k)(
Seismic Response…Seismic Response…
Moving-base Structure
Moving-base Structure 0kuuc)uu(m g =+++ &&&&&
Relative velocity/displacementRelative velocity/displacementAbsolute
accelerationAbsolute acceleration
Fixed base Fixed base
StructureStructure
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Fixed-base Structure
Fixed-base Structure
( )tumkuucum g&&&&& −=++
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• Duhamel’s IntegralSeismic Response…Seismic Response…
( )tumkuucum g&&&&& −=++
( ) ( ) ( ) ( ) τ⋅τ−ω⋅τω
= τ−ξω−∫ dtsinePm
1tu dt
t
0dn
Response of system at rest due to impulse P(t) Response of system at rest due to impulse P(t)
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( ) ( ) ( ) ( ) τ⋅τ−ω⋅τω
−= τ−ξω−∫ dtsineu1tu dt
t
0g
dn&&
Response of system at rest to random ground motion Response of system at rest to random ground motion ( )tug&&
• Response to random ground motionSeismic Response…Seismic Response…
( )tumkuucum g&&&&& −=++
( )t&& ( )tug&&
Time t0
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( )tu
Time t0
Dynamics of of SDOF Systemsy
• SDOF Model
Dynamic CharacteristicsDynamic Characteristics
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Inverted Pendulum ModelInverted Pendulum ModelOscillation of buildingOscillation of building
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• Two dynamic characteristics– Natural frequency
Dynamic Characteristics…Dynamic Characteristics…
k
– Dampingmk
=ω
k
m
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• The third dynamic characteristic– Deformed shape (or Mode Shape) of Vibration
• Can be obtained from static considerations
Dynamic Characteristics…Dynamic Characteristics…
fDepends on stiffness properties
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Shear TypeShear Type LinearLinear Flexure TypeFlexure Type
Equivalent Static ForceEquivalent Static Force
• Design EQ force
PSAmSDkukF max ⋅=⋅=⋅=
umax=SD
F
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F=VB
Methods of Earthquake Earthquake Analysis
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Established MethodsEstablished Methods
• Linear Analysis– Equivalent Static Analysis– Linear Dynamic AnalysisLinear Dynamic Analysis
• Time History Analysis• Response Spectrum Analysis
• Nonlinear AnalysisN li St ti A l i
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– Nonlinear Static Analysis• Pushover Analysis
– Nonlinear Dynamic Analysis• Time History Analysis
• Use in Design Practice– < 1960s
• Equivalent Linear Static Analysis
Established Methods…Established Methods…
q y– 1960s, 1970s
• Linear Dynamic Analysis– 1980s, 1990s
• Nonlinear Static Analysis– 2000s
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2000s• Nonlinear Dynamic Analysis
Equivalent Linear Linear Static Analysisy
Equivalent Lateral Force MethodEquivalent Lateral Force Method
• First mode analysis– Typical first mode shapes
Linear Parabolic
{ }⎪
⎪⎪⎪
⎬
⎫
⎪
⎪⎪⎪
⎨
⎧
⎟⎠⎞
⎜⎝⎛⋅ϕ=ϕ
M
M
Hhi
i011 { } ⎪⎪⎪
⎬
⎫
⎪⎪⎪
⎨
⎧
⎞⎜⎛⋅ϕ=ϕ
M
M
2i
i011h
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Low-to-Medium Period Buildings (T<1s)
Low-to-Medium Period Buildings (T<1s)
Long Period Buildings (T>2s)
Long Period Buildings (T>2s)
⎪⎪⎪
⎭⎪⎪⎪
⎩ M
M{ }
⎪⎪⎪
⎭
⎬
⎪⎪⎪
⎩
⎨⎠
⎜⎝
ϕϕ
M
M
i011 H
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• Base Shear VB using T1
Equivalent Lateral Force Method…Equivalent Lateral Force Method…
1B PSAMV ⋅=
• Distribution of force along height
Fi⋅= N
2ii
BihWVF
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∑=
N
1k
2kk
BihW
• Indian Seismic Design Code procedure– IS:1893 (Part1) – 2002
• Perform usual static elastic structural analysis
Equivalent Lateral Force Method…Equivalent Lateral Force Method…
ywith specified forces
No dynamic analysis doneDynamic actions considered through Response Spectrum used in computation of Base Shear VB
Fi
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Fi
VB
Linear Dynamic Dynamic Analysis
Two WaysTwo Ways
• Linear Dynamic Analysis– Time History Analysis
• Matrix Time History Analysisy y• Modal Time History Analysis
– Response Spectrum Analysis
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Matrix Time History AnalysisMatrix Time History Analysis
• Matrix equilibrium equation – MDOF Systems[ ] ( ){ } [ ] ( ){ } [ ] ( ){ } [ ]{ } ( )tx1MtxKtxCtxM &&&&& −=++
Ground shaking influence vectorDifferent for each direction of shaking
[ ] ( ){ } [ ] ( ){ } [ ] ( ){ } [ ]{ } ( )tx1MtxKtxCtxM g=++
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Linear actions
• Numerical IntegrationMatrix Time History Analysis…Matrix Time History Analysis…
( )( )[ ] ( )[ ] [ ][ ] ( ){ }∆ttxKβCM ∆tγ
∆t1
2 +⋅++
( )[ ]{ } ( )[ ]{ } ( )[ ]{ }[ ]( )( )[ ] ( )[ ][ ] ( ){ }
( ) [ ] ( )[ ][ ] ( ){ }( )( )[ ] ( )[ ][ ] { } ( )Mβ50Cβ2
txCβγM +
txCM
M∆ttuM∆ttuM∆ttuβ
∆t∆t1
∆tγ
∆t1
zgzygyxgx
2
&&
&⋅−+
⋅++
+++++−= 111
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– Obtain ( ){ }tx
( )( )[ ] ( )[ ][ ] { } (t)xMβ5.0Cβ2γ- + 2∆t &&⋅−+
Modal Time History AnalysisModal Time History Analysis
• Matrix equilibrium equation – MDOF Systems[ ] ( ){ } [ ] ( ){ } [ ] ( ){ } [ ]{ } ( )tx1MtxKtxCtxM &&&&& −=++
– Modal equilibrium equations for each mode k=[1,N]( ) ( ) ( ) ( )tuMtqKtqCtqM gxkkkkkkkkkkk &&&&& ⋅−=⋅+⋅+⋅ ∗∗∗∗
[ ] ( ){ } [ ] ( ){ } [ ] ( ){ } [ ]{ } ( )tx1MtxKtxCtxM g=++
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• Numerical integration– Solve for all k=[1,N] ( ){ }tqk
Modal Time History Analysis…Modal Time History Analysis…
( ) ( )( ) ( )∆ttqβKCM γ1 +⋅++( )( ) ( )( ) ( )
( )( )( )( ) ( )( ) ( )
( ) ( )( ) ( )( )( ) ( )( ) ( )ββ
tqCβγM +
tqCM
M∆ttuβ
∆ttqβKCM
∆t
kkk*kk*∆t1
kkk*∆tγ
kk*∆t1
kk*gx
kkk*kk*∆tkk*∆t
2
2
&⋅−+
⋅++
+−=
+⋅++
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– Superpose and obtain ( ){ } [ ] ( ){ }tqtx Φ=
( )( ) ( )( ) (t) qMβ5.0Cβ2γ- + kk*kk*2∆t &&⋅−+
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Response Spectrum AnalysisResponse Spectrum Analysis
• Matrix equilibrium equation – MDOF Systems[ ] ( ){ } [ ] ( ){ } [ ] ( ){ } [ ]{ } ( )tx1MtxKtxCtxM &&&&& −=++
– Modal equilibrium equations for each mode k=[1,N]( ) ( ) ( ) ( )tuMtqKtqCtqM gxkkkkkkkkkkk &&&&& ⋅−=⋅+⋅+⋅ ∗∗∗∗
[ ] ( ){ } [ ] ( ){ } [ ] ( ){ } [ ]{ } ( )tx1MtxKtxCtxM g=++
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• Concept– Superposition
Response Spectrum Analysis…Response Spectrum Analysis…
= +
ϕ21 ϕ22
ϕ11 ϕ12q1 q2
x2
x1
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• Estimate maximum response of each mode from Design Response Spectrum given– Solve for all k=[1,N] ( )maxk tq
Response Spectrum Analysis…Response Spectrum Analysis…
f [ , ]
– Obtain maximum modal response of structure
St ti ti ll ti t i t
( )tmaxkq
{ } { }kmaxkmax
ktt
)t(q)t(x φ=
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– Statistically estimate maximum net response
( ){ }tmaxtx
∑∑= =
⎟⎠⎞⎜
⎝⎛ ⋅⋅≅
N
1k
N
1smax
is
maxi
kksmaxi
ttt)t(x)t(x)t(x ρ
• Dynamic Characteristicsm2
m1
k2
Example: 2 DOF SystemExample: 2 DOF System
PropertyProperty Mode 1Mode 1 Mode 2Mode 2
m1k1
Equivalent SDOFs K1
M1
K2
M2
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q 1 2
Natural Frequency1
11 M
K=ω
2
22 M
K=ω
Natural Period 11 /2T ωπ= 22 /2T ωπ=
12
• Lateral Forcem2
m1
k2
Example: 2 DOF System…Example: 2 DOF System…
PropertyProperty Mode 1Mode 1 Mode 2Mode 2
m1k1
PSA (g) PSA (g)
PSA2
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PSA
T
PSA1
TT1 T2
SD 21
11
PSASDω
= 22
22
PSASDω
=
• Lateral Force…m2
m1
k2
Example: 2 DOF System…Example: 2 DOF System…
PropertyProperty Mode 1Mode 1 Mode 2Mode 2
m1k1
Mode Participation Factor
{ } [ ]{ }1
T1
1 M1mϕ
=Γ{ } [ ]{ }
2
T2
2 M1mϕ
=Γ
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Lateral Displacement
Factor 1
{ } { }
⎭⎬⎫
⎩⎨⎧
=
Γϕ=
22
212222
uu
SDu{ } { }
⎭⎬⎫
⎩⎨⎧
=
Γϕ=
12
111111
uu
SDu
• Lateral Force…m2
m1
k2 F11
F12
F21
F22
Example: 2 DOF System…Example: 2 DOF System…
PropertyProperty Mode 1Mode 1 Mode 2Mode 2
m1k1
L t l F { } [ ]{ } ⎬⎫
⎨⎧ 11F
F k { } [ ]{ } ⎬⎫
⎨⎧ 21F
F k
11 F21
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Lateral Force { } [ ]{ }⎭⎬
⎩⎨==
1211 F
uF k { } [ ]{ }⎭⎬
⎩⎨==
2222 F
uF k
Base Shear ∑=
=2
1ii11B FV ∑
==
2
1ii22B FV
• Lateral Forcem2
m1
k2 F11
F12
F21
F22
Example: 2 DOF System…Example: 2 DOF System…
PropertyProperty Mode 1Mode 1 Mode 2Mode 2
m1k1
11 F21
22
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Resultant Base Shear ( ) ( )22B2
1BB VVV +=
⎟⎟⎠
⎞⎜⎜⎝
⎛>⎟⎟
⎠
⎞⎜⎜⎝
⎛
B
2B
B
1BVV
VV
Usually, for regular buildings
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• Equivalent Static Force– If translation mode 1 dominant
FF
Example: 2 DOF System…Example: 2 DOF System…
F11
F12
F11
F12
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BuildingBuilding Mode 1Mode 1
1BB VV ≈
VB VB1
Nonlinear Static Static Analysis
Real StructuresReal Structures
• Stiffness characteristic– Actually Nonlinear
– Assumed Linear
[ ] ( ){ } [ ] ( ){ } ( ){ }( ){ } [ ]{ } ( )tx1MtxptxCtxM g&&&&& −=++
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Assumed Linear
[ ] ( ){ } [ ] ( ){ } [ ] ( ){ } [ ]{ } ( )tx1MtxKtxCtxM g&&&&& −=++
• Pushover analysis– Literally…!!
• Of course, analytically…
Nonlinear Static AnalysisNonlinear Static Analysis
, y y
52
14
• Outcome– Stiffness characteristics of structure
• Over the entire displacement loading range
NL Static Pushover Analysis…NL Static Pushover Analysis…
)x(pf =p g g
Useful in understanding relative performance
rce
rce Large damageLess
d
53Roof DisplacementRoof Displacement
Late
ral F
orLa
tera
l For damage
Nonlinear Dynamic Dynamic Analysis
Matrix NL Time History AnalysisMatrix NL Time History Analysis
• Matrix nonlinear equilibrium equation – MDOF Systems
[ ] ( ){ } [ ] ( ){ } ( ){ }( ){ } [ ]{ } ( )tx1MtxptxCtxM g&&&&& −=++
Ground shaking influence vectorDifferent for each direction of shaking
[ ] ( ){ } [ ] ( ){ } ( ){ }( ){ } [ ]{ } ( )tx1MtxptxCtxM g++
55
Linear actions
Nonlinear actions
• Numerical IntegrationMatrix Nonlinear Time History Analysis…Matrix Nonlinear Time History Analysis…
( )( )[ ] ( )[ ] [ ][ ] { }{ } { }[ ]
∆xKβCM jjT∆t
γ∆t
12 ⋅++
{ } ( )[ ]{ }[ ]( ){ } ( )( ){ }
( )( )[ ] ( )[ ][ ] ( ){ }
( )( )[ ] ( )[ ][ ] ( ){ }txCM
∆ttxCM
∆tt,∆ttxp
M∆ttuβf
∆tγ
∆t1
j∆tγ
∆t1
j
xgx0
2
2
&&
⋅++
+⋅+−
++−
+−=
β
β 1
56– Obtain ( ){ }tx
( )
( ) [ ] ( )[ ][ ] ( ){ }( )( )[ ] ( )[ ][ ] { } (t)xMβ5.0Cβ2γ- +
txCβγM +
2∆t
∆t1∆t
&&
&
⋅−+
⋅−+
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SummaryEstablished MethodsEstablished Methods
• Linear Analysis– Equivalent Static Analysis– Linear Dynamic AnalysisLinear Dynamic Analysis
• Time History AnalysisMatrix Analysis (THA)Modal Analysis (Modal THA)
• Response Spectrum Analysis
• Nonlinear Analysis
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y– Nonlinear Static Analysis
• Pushover Analysis– Nonlinear Dynamic Analysis
• Time History Analysis
Methods of Analysis…Methods of Analysis…
REQUIREMENTS AND LIMITATIONSMost Nonlinear Responses
Structural Configuration
Limited Higher Mode Responses
Approximately Proportional to Elastic response
Configuration be Regular
Higher Mode Effect
Linear Static Yes Yes YesLinear Dynamic Yes No NoNonlinear Static No No YesNonlinear Dynamic No No No
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Nonlinear Dynamic No No No
Thank you…
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