Chapter2 Dynamic Response Analysis

8/8/2019 Chapter2 Dynamic Response Analysis

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Chapter 2

Dynamic Response Analysis of

Bridge Structures

Tokyo Institute of Technology

Kazuhiko Kawashima

2005 Course

Uniqueness of Bridges for Structural Analysis

long in the longitudinal d irection, and s tructural

properties and soil condition are not the same along theaxial axis

consists of many structural types (deck br idges, arch

bridges, cable stayed bridges, suspension bridges, .)

have various shapes (straight, skewed, curved,

separation into several segments, and combination of

those types)

have various heights (short bridges and high bridges)

are made of various materials (RC, PC, steel,

composites)

Bridges are;

Types of Analysis

DNLDLDynamic

SNL

(pushover

analys is)

SLStatic

NonlinearLinear

Idealization of a Bridge

Idealization of a superstructure & substructures

Idealization of foundations

Idealization of soil response

Multiple excitation effec t (out of coherent GM)


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Analytical Model of Superstructures Analytical Model of Substructures

Idealization of Substructures

Soil springs

P5

P6

P7

P8

P9

P10P11

P12

P13

Deck

Column

3D FEM Idealization


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Plastic Deformation of RC Column Idealization of Bridges

Stiffness Idealization

i

ikK1

i

tt kK1

Mass Idealization

i

imM1

Element stiffness matrix

Total stiffness matrix

Total mass matrix

Element mass matrix

Time dependent stiffness

Idealization of Bridges (continued)

Damping Idealization

There are various sources which contribute to energy

dissipation. It is common to idealize the energy dissipation

in terms of the viscous damping.

i icC 1Since valuation of element damping matrix is generally

Difficult, the system damping ratio is often assumed by

he Rayleigh damping as

KMC (2.5)

KMC TTT

1

.

.

10

01

20..0

0..

...

.20

0..02

22

11

nn

2

22

21

0..0

0..

...

.0

0...

Orthogonarity

condition


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22 iii

i

ii

2

1

i

j

i

j

and can be determined by assigning for arbitrarytwo modes as

KMC

The above assumption sometimes results in problem,

because

Solution becomes unstable s ometimes

Does not capture the fact that inelast ic response of

structural members dissipate energy which results in an

increase of damping ratio

kmm

nTkm

kmmm

Tkmkm

k

k

k

1

1

n

kmmTkm

mkmmTkmkm

k

1

1

m

m

Strain Energy Proportional Damping Ratio

Kinematic Energy Proportional Damping Ratio

(2.6)

(2.7)

Strain energy propor tional damping ratio may be better in a

system in w hich hysteretic energy dissipation is predominant

Equations of Motion

)( gmF cku g

gmkucm

g

: relative displacement

: absolute displacement

: ground displacement

Single-degree-of-freedom oscillator


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i

i

g

Equations of Motion (continued)

Multi-degree-of-freedom system

gMBKuCM

T

uuuu ,........,, 21

001

100

011

000

001

ZYX bbbB

gz

gy

gx

g

u

u

u

u

Equations of Motion (continued)

Multiple Excitation

b

tct

uu

free nodal points

non-zero support displacements

b

tct

u

ufree nodal points

non-zero support displacements

ccsbb

s

suu

u

u

u

quasi-static displacements

dynamic displacements

csu

c

From definition, 0

(2.11)

The equation of motions can be written by enlarging

the mass, damping and stiffness matrices as well as the

dynamic load vector to account for the nb support

displacements

b

t

bbTbtt

b

t

bbTb

b

u

u

CC

CC

u

u

MM

MM

)()(

)(

)(

)( tR

tR

u

u

KK

KKbb

t

bbTbtt

The equations of motion ass ociated with n free nodal point

displacement become

R(t)u

uKK

u

uCC

u

uMM b

tbttb

tbttb

bb

(2.12)

(2.13)


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Substitution of Eq. (2.11) into Eq. (2.13) yields

ss

sbttb

s

sbtt

u

uCC

u

uMMR(t)uKuCuM

bb

s

s

bbTbtt

bb

s

s

bbTb

b

u

u

u

u

CC

CC

u

u

u

u

MM

MM

)()(

)(

)(

)( tR

tR

u

u

u

u

KK

KKbbb

s

s

bbTbtt

By def inition of the quasi-static displacement

0 ss u

KuK

(2.14)

(2.15)

=0

small compared to inertia force

0 ss uKuKFrom

bst

bs

btts uBuKKu

1

KKB 1 represents a matrix of quas i-staticinfluence coefficients resulting from the nb non-zero

support displacements. If the system is linear,

all coef ficients in are invariant with time.B

(2.16)

ss

sbttb

s

sbtt

u

uCC

u

uMMR(t)uKuCuM

suMBtR )( (2.17)

n x n n x nb nb x 1

Multiple Support Excitation

(t)uu gs

(t)uMBR(t)uKuCuM gttt

suM

BtRuKuCuM )( (2.17)

(2.18)

(2.19)

Rigid Support Excitation

gZ

gY

gXrg

u

u

u

u

(t)uMBR(t)uKuCuM gtt

ZYX bbbB

n x n n x 3 3 x 1


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Linear Analysis

a) Natural Mode Shapes and Natural Frequencies

(t)MBR(t)KuCM g

CC KK

suMBtRuKuCuM )(

0 KuM

ieu

iii MK2

Linear Analysis (continued)

2KT

IMT

where,

)( 22 idiag

q

q

q

(t)uMBR(t)uKuCuM gtt

(t)uMBR(t)qKqCqM g

)( (t)uMBR(t)qKqCqM gTTTT

(t)Rqqq

* 2

)2( iidiag Only assumption

(t)uMBR(t)(t)R gT*

where,

(2.28)

Solving equation of motion for a SDOF system

(t)Rqqq* 2

(t)uMBR(t)(t)R gT*

Time History Analysis

Direct integration method

q

q

q

Determine iq iq iq

Determine forc e, stress and strain

iiiiiii Rqqq*22 ),......,2,1( ri


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),(),(),()(max iiDZiiDYiiDX

Tii hTShTShTStq MB

Response Spectral Method

(t)Rqqq* 2

(t)uMBR(t)(t)R gT* iiiiiii Rqqq *22 ),......,2,1( ri

maxmax )()( tqtu iii

(2.31)

(2.32)

2/1

1

2max

)()(

r

iiSRSS

tutu (2.33)

Nonlinear Dynamic Response Analysis

Equations of motion in the incremental form

(t)MBR(t)u(t)K(t)C(t)M gtt

(t)t)(t(t) (t)t)(t(t)

(t)t)(tu(t) )()()( tRtTRtR

(2.47)

(t)t)(t(t) ggg

Nonlinear Dynamic Response Analysis

(continued)Newmarks generalized acceleration method

0 1)()( Cduu

0 21

)()( CCduu 1)0( C

tt Cdutuu 0 1)()(

2)0( C

tt CtCdutuu 0 21)()(

)(t

)( tt

t tt t ttt tt

)(t

)( tt

)(t

)( tt

Newmarks generalized acceleration method (continued)

Constant Acceleration Method

2)()()( tttu

)(t

)( tt

t tt t ttt tt

)(t

)( tt

)(t

)( tt

)(2

)( tttt uuuu

)(2

tttttt uut

uu

)(4

)(2

ttttt uuuuu

)(4

2

ttttttt uut

tuuu


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Linear Acceleration Method

)()( tttt uut

uu

)(

)(

)(

)(

)(

)(

)(2

)(2

ttttt uut

uuu

)(2

2

ttttttt uut

tuuu

)(62

)(32

tttt

tt uuu

uuu

)(62

32

tttt

tttt uuttu

tuuu


t)(t(t))((t)t)(t 1

t)(t(t))/((t)t(t)t)(t 21

)(t

)( tt

t tt t ttt tt

)(t

)( tt

)(t

)( tt

2

1

4

1

6

1

2

1

cons tant acceleration method

linear acceleration method

(2.50)

(t)C(t)Cu(t)C(t) 31

(t)C(t)Cu(t)C(t) 52


constant acceleration

method

linear acceleration

method

21 /4 tC tC /22

tC /4324 C

05 C

21 /6 tC tC /32

tC /6334 C

2/5 tC

(2.51)(t)MBR(t)u(t)K(t)C(t)M gtt


(t)C(t)Cu(t)C(t) 31

(t)C(t)Cu(t)C(t) 52

(t)Ru(t)K~~

KCCMCK 21~

(t)uCCMC(t)uMBR(t)(t)R tg 43~

(t)CCMC 5

where,

(2.52)

(2.53)

(2.54)

(2.47)

(2.51)


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Accuracy of Computed Responses

)(tR

)(tR

)( ttR

)( tt

)( tt )( ttR

(t)Ru(t)K~~

KCCMCK 21~

Overshooting

Accuracy of Computed Responses (continued)

)(tR

)(tR

)( ttR

)( tt

)( tt )( ttR

Computed restoring forceSFuCuMttR )(

Exact restoring force = )( ttR

Unbalance force RR

SFuCuM


PSttt

ttP

RRR

R

aaa TE R

)max( ,ittR

.

.

.

If error > tolerance,

If error > tolerance,


re-compute using a smaller time increment of

numerical integration-This is always effective to

improve the stability and accuracy of solution.

add unbalanced forces to the incremental forces atthe next time step

use a numerical iteration


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Add unbalanced forces to the incremental

forces at the next time step

Unbalance Force at time t:

SFuCuMRR

R(t)u(t)K(t)C(t)M

Incremental Equations of Motion

Add the unbalance force to the incremental external force

u(t)K(t)C(t)M RR

SFuCuMRRR SFuCuMR

This method is effective when unloading and reloading

are important

Add unbalanced forces to the incremental

forces at the next time step (continued)

Numerical Iteration for the Equilibrium of

Equations of Motion

)(tR

)( ttR

)(tR

)(t

)(t)(tR

Equations of motion for Equilibrium

Collection

(i)(i)(i )(i)(i)RuKuCuM

SFuCuMRR

Unbalance force

RR )1(

(i))(i(i) uuu 1(i))(i(i)

uuu 1

(i))(i(i)uuu 1

S(i(i)(i)(i)FuCuMRR

Computer Soft-wares for Dynamic Response

Analysis

General Purpose Sof t-wares

ASKA

DYNA

ABAQUA

SAP

Multi-purposes

Well maintenance

Some kind of consensus for the results

Not so easy to modify

User routines may be included depending on programs


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Computer Soft-wares for Dynamic Response

Analysis (continued)

Hand-made

Easy to include problem-oriented routines

Difficult for maintenance for the use of other

groups and long-term maintenance

Few consensus for the results

Open-base forum for source code

Prepare well documented manual and example

problems

Structures of Computer Programs for Dynamic

Response Analysis

Input structural shape (coordinate) & properties

Input ground motions

At time t

Form time invariant structural properties such as mass

matrix, stiffness of elastic elements

Form time dependent properties such as stiffness of

nonlinear element

NLL KKK

Solve )(~

)(~

tt RuK (2.52)

Determine displacements, velocities, accelerations, force,

stress, and strain at time tt

Store the responses on a file

Check the accuracy by Eq. (2.62) or similar forms

If the accuracy is not enough, use small smaller time

increment, or add unbalance force to the next

incremental force, or iteration

ttt Repeat until the end of ground motion

Exercise of Chapter 2

Q. 2.1 Derive Eq. (2.17) from Eqs. (2.12) and (2.16)

Q. 2.2 Derive Eq. (2.28) from Eq. (2.23)

Q. 2.3 Derive Eqs. (2.50) and (2.51) for both the constant

and linear acceleration methods


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Q. 2.4 Develop a computer program for a SDOF oscillator

subjected to an arbitrary ground motion at its base using

the Newmarks direct integration scheme

a) Linear

b) Bilinear

Q. 2.5 Using the computer program developed in Q. 2.4,

compute responses of SDOF oscillators for the following

conditions

E

N

k

kr =0, 0.1, 1

T=0.5 & 1 s

a

b

Lateral Displacement


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Classification of Dynamic Response Analysis

Linear Analys is

Mode Superpos ition Method

Compute natural periods and mode shapes Compute response of SDOF systems

Compute response of a structure by mode

superposition

Time history analysis

Response spectrum analysis

n

ii tutu

1

2)()(

For example, square root of the sum of the squares

Required to compute natural periods and modeshapes

Standard procedure to compute time history andpeak responses of a structure

Restricted only to the linear analysis

Importance of the response spectrum analysis thatthe computer time is shorter is decreasing due to

progress of computers, how ever the importancethat the peak response can be directly computedbased on response spectrum is still existing.


Linear Analysis

Features of the Mode Superposition Method

Can be implemented to both linear & nonlinear

response analysis

All modes are considered in analysis

Only analytical method which is used for nonlinearstructures

Classification of Dynamic Response Anal

Direct integration analysis

Frequency dependent stiffness and damping can be

explicitly included in the analysis

Can be used only for linear analysis. However

material nonlinearity is sometimes included as an

equivalent analysis.

Well used for soil and soil-structure interaction

analys is.

SHAKE & LUSH families are often used.


Linear Analysis

Frequency Domain Analysis


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Idealization of Ground

Ground response is important in the evaluation of

structural response

Strong nonlinear behavior of ground Soil-struc ture interaction effect

Idealize both soil an structure

Idealize the constraint of soil by soil-springs

Differential ground motion, incoherency, spatial

variation of ground motion

Type of Ground Motions

Elastic response spectrum (Design response spectra) Ground accelerations

Ground surface accelerations

Bedrock accelerations

Accuracy of ground accelerations (measured

by an analog-type accelerograph)

Documents

Chapter2 Dynamic Response Analysis