earthquake simplified analysis of structures

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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…

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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


k

– Damping

mk

=ω

13

• Critical Damping– Smallest value of damping at which

• No cyclic motion


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


( )tug

• Change of reference frame…

k)(

Seismic Response…Seismic Response…


Moving-base Structure 0kuuc)uu(m g =+++ &&&&&

Relative velocity/displacementRelative velocity/displacementAbsolute

accelerationAbsolute acceleration

Fixed base Fixed base

StructureStructure

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( )tumkuucum g&&&&& −=++

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• Duhamel’s IntegralSeismic Response…Seismic Response…


( ) ( ) ( ) ( ) 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…


( )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


– 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


– 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…


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



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



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



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


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++

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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…

earthquake simplified analysis of structures

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