Introduction to Earthquake Engineering - uni-kassel.de · From Clough, Penzien (3) ... Clough,...

Introduction to

Earthquake Engineering

Response Analysis

Prof. Dr.-Ing. Uwe E. Dorka

Stand: September 2013

Modeling of buildings

Prof. Dr.-Ing. Dorka | Introduction to Earthquake Engineering 2

masses lumped in floors

frames with plastic hinges

Soil-Structure Interaction


structure with soil in FE and BE

simplified model

boundary elements

From Clough, Penzien (3)

Modeling of bridges


superstructure, bearings and columns with FE

multiple base input

with D1=D2=5% yields w1=100.06 and w2=429.44

a = 0.520752958

b = 0.00325458

Rayleigh-damping:

C M Ka b w w

a w ww w

1 1 2 21 2 2 2

1 2

D D2

w w b

w w

1 2 1 2

2 2

1 2

2 D D

130

95

91

86

51

46

133

5

12

3

z20

y19133

z18

y17130

y16

x1595

y14

x1391

y12

x1186

z10

y951

z8

y746

y6

x512

y4

x35

y2

x13

DOFmaster

node

z20

y19133

z18

y17130

y16

x1595

y14

x1391

y12

x1186

z10

y951

z8

y746

y6

x512

y4

x35

y2

x13

DOFmaster

node

Model with 20 dynamic DOFs

Earthquake Loading


lumped MDOF-sytem with rigid

base translation (horizontal case)

From Clough, Penzien (3)

0tm v c v k v

where direction cosine, for buildings typicall "1"

t

gv v j v

j

eff tm v c v k v p

eff gt tp m j v

Re

fere

nc

e a

xis

vt

Description of non-linear behaviour


Hysteresis models (1D) – non-degrading models

From Wakabayashi (1)

Bilinear Model Trilinear Model

Ramberg-Osgood model


Hysteresis models (1D) – Degrading Models


Piecewise linear model

(Clough and Johnston)

Trilinear Takeda model

for rc-members




Hysteresis models (1D) – Slip-Type Models


Double bilinear model by

Tanabashi and Kaneta Slip-type model



Example for slip-type hysteresis model

(friction connection with multiple stops)



2D Hysteresis Model (2D Bouc-When)

From Pradlwarter, Schuëller, Dorka (7)


Time History Analysis

)(2

2

tpfKxxCxM rdtd

dt

d

2/1

;11

;2/1

1

1

n

n

n

N

N

N

Dynamic equilibrium:

with fr : vector of non-linear restoring

forces Shape functions for discretizing x in time:

Time Discretization

with:

tt /


Weighted Residual Formulation

d

NfNfNf

NuNuNuK

NuNuNuC

NuNuNuM

W

n

n

n

n

n

n

n

n

n

n

n

n

ndtdn

ndtdn

ndtdn

ndt

dn

ndt

dn

ndt

dn

1

1

**1

1

*

1

1

1

1

1

1

1

1

1

1

1

1

1

1

2

2

2

2

2

2

pff r *

;1

;

1

1

1

1

21

1

1

1

1

21

b

WddW

WddW

Weighting function W:

3-Point Recurrence Scheme


1

*

2

21

*

2

211

*

2

12

21

2

21

121

2

1

2212

n

nn

n

n

n

ft

ftft

uKttCM

uKttCM

KttCMu

b

bb

b

b

b

1

0

1 n

r

n fGuu

1

*

2

21

*

2

2112

12

21

2

21

12

0

2

1

2212

n

nn

n

n

ft

ftpt

uKttCM

uKttCM

KttCMu

b

bb

b

b

b 122 KttCMtG bb

with:

Stability and Accuracy



tYety )(

n

tttt

n yYeeYey

1

01

2212

2

21

2

21

22

kttcm

kttcm

kttcm

b

b

b

Solution for linear SDOF-

system:

and its recurrent form:

yields a characteristc equation:

1

1

exact solution:

stable solution with numerical

damping:


T/T

t/T

||

period elongation numerical damping

unconditionally stable Newmark

scheme without numerical damping 5.0 25.0b



Newton-Raphson Iteration

From Chopra (4)

1

0

1 n

r

n fGuu Since the response fr at time n+1 is not known, an

iteration within each time step is required to solve this

equation:


From Chopra (4)

Newton-Raphson Iteration

Examples of time history


Response of linear SDOF-system with T=0,5 s and z=0 to El Centro ground motion

From Chopra (4)

s

t

0

0

f elastic resisting force

u total acceleration

of the mass

f peak value of

elastic resisting force

u peak value of

total acceleration

w weight of mass


From Chopra (4)

y y

y

o o

f uf normalized yield strength

f u

m

y

uductility factor

u



From Chopra (4)

Response of elastoplastic system with T=0,5 s, z=0 and fy=0,125

to El Centro ground motion deformation

resisting force

expressed as

acceleration

time intervalls of

yielding

force-deformation

relation (hysteresis)

yield strength




From Chopra (4)

Deformation response and yielding of four systems due to El Centro

ground motion; T=0,5 s, z=5% and fy=1- 0,5 - 0,25 and 0,125

F

1m u d u k u m y t

m

Response Spectra


viscously damped SDOF oscillator

w w 2

Fu 2 u u y t

where: Eigenfrequency:

Damping ratio:

w k

m

w

d

2 m

-solving this equation for various w and z, but

only for one specific accelerogramm

-the maximum absolute acceleration of this

solution gives us the abzissa for the

following diagramm

From Petersen (2)

From Meskouris (5)

Response Spectra given by EC 8


The acceleration response spectra in EC 8 are

given with respect to the subsoil classes


From Meskouris (5)

Comparison of elastic and inelastic response spectra

Response Spectra given by EC 8

Modal Analysis


linear equations of motion for a

MDOF, homogeneous, undamped

case

0m u k u solving the eigenvalue

problem gives us the

modal matrix F

F F F F F 1 2 3 ..... n

with this modal matrix, the equations of motion

can be transform into modal coordinates F F F F

0

0

T Tm u k u

M K

or for a damped case with ground acceleration

F F F F F F F

T T T T

Fm u c u k u m j u t

M C K F t

now we have uncoupled equations of motion for n SDOF-

systems in modal coordinates


These n equations could be solved in

known ways, so we get n solutions in

modal coordinates in the time domain

1

2

j

n

t

t

tt

t

With the modal matrix we are able to

transform these solutions back into

local coordinates

F

1

2

j

n

u t

u t

u t tu t

u t

Modal Analysis

Modal Analysis using Response Spectra


n

j

j 1

S S Superposition of absolute values

The modal responses attain their peaks

at different time instants. How do we

have to combine these peak values?

n

2

j

j 1

S S

quare oot um of quare (SRSS)S R S S

n n

j jk k

j 1 k 1

jk

S S S

correlation coefficient

omplete uadratic ombination (CQC)C Q C

(1) Wakabayashi –

Design of Earthquake-Resistant Buildings

McGraw-Hill Book Company

(2) Petersen –

Dynamik der Baukonstruktionen

Vieweg

(3) Clough, Penzien –

Dynamics of Structures

McGraw-Hill

(4) Chopra –

Dynamics of Structures

Prentice Hall

(5) Meskouris–

Baudynamik

Ernst & Sohn

(6) Zienkiewicz–

The Finite Element Method

McGraw-Hill Book Company

(7) Pradlwarter, Schuëller, Dorka

Reliability of MDOF-systems with hysteretic

devices,

in Engineering Structures, Vol. 20, 1998

Elsevier

References

Introduction to Earthquake Engineering - uni-kassel.de · From Clough, Penzien (3) ... Clough,...

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