111

Stochastic Structural Dynamics

Lecture-32

Dr C S ManoharDepartment of Civil Engineering

Professor of Structural EngineeringIndian Institute of ScienceBangalore 560 012 India

manohar@civil.iisc.ernet.in

Probabilistic methods in earthquake engineering-1

2

We have developed methods to study systems governed by

,

where 0 & 0 are specified,and is a vectorrandom process.The analysis has included characterization of responsemomen

MX CX KX F X X G t

X X G t

Recall

ts, pdf-s of system states, and reliability measures.

The equations governing the behavior of structures subjected to earthquake support motions have formsimilar to the above equation.Therefore, what are the issues that we need to considerafresh?

3

Focus: uncertainties in vibratory response of structures during an earthquake

• Stochastic models for ground motions• Response spectrum, PSD and time histories• Modal combination rules• Seismic risk analysis• Performance based structural design (PBSD)

Aim:• To introduce the basic ideas and facilitate future self‐study

4

When, where, and how earthquakes occurEarthquakes The details of ground motion

The effect of ground motion on structure

Uncertainties in earthquake engineering problems

Dynamic responseDamage and loss

555

Duhamel's integral

Alternatives for earthquake load specification

Time histories

Power spectral density Response spectra

Monte Carlo simulations

Spectralestimation

Spectrumcompatibleaccelerograms

Extreme value theory

Spectrum compatibe PSDusing extreme value theory

Not suited forreliability analysis

6

References

• R W Clough and J P Penzien, 1993, Dynamics of structures, McGraw Hill, NY

• N C Nigam and S Narayanan, 1994, Applications of random vibrations, Narosa, New Delhi

7

Response spectra

8

Effect based model for earthquake ground acceleration

Equation governing total displacement( ) ( ) 0

Relative displacement ( ) ( ) ( )

2

t t tg g

tg

g

mv c v v k v v

v t v t v t

mv cv kv mv

v

Response spectrum

2 ( )gv v v t

9

0

0

0

0

1( ) exp[ ( )]sin[ ( )][ ( )]

1 exp[ ( )]sin[ ( )] ( )

0.1

1( ) exp[ ( )]sin[ ( )] ( )

( ) ( ) exp[ ( )]cos[ ( )]

( )

t

d gd

t

d gd

dt

n n gn

t

g n

g

v t t t mv dm

t t v d

v t t t v d

v t v t t d

v

0

exp[ ( )]sin[ ( )]t

nt t d

10

2

2

0

0

( ) ( ) ( ) 2

( ) (2 1) ( ) exp[ ( )]sin[ ( )]

2 ( )exp[ ( )]cos[ ( )]

Systems with the same and respond identically

t t tg g n n

tt

n g n n

t

n g n n

n

c kv t v v v v v vm m

v t v t t d

v t t d

Remark

to the same ground motion.

11

Response quantities of interest

( ) :Maximum relative displacementForce in the spring (column) and hence the stresses

in the column are proportional to ( )

( ) :Of interest in the study of secondary

t

v t

v t

v t

systemsPounding of adjacent buildings

12

2

2

( ) [ ( ) ( )] ( )

( )( )( )

( ) base shear

( ): has units of acceleration (pseudo-acceleration)

seismic coefficient

Weight of the building seismic

ts g

n

s

n

f t k v t v t kv t

m v tA tmA t Wg

f t

A t v t

A tg

Force in the spring

coefficient= base shearBase shear height of the building= base moment

13

2 2 2

2 20

1 1( ) ( )2 2

1 [ ( )] ( )2 2

strain energy stored: has units of velocity

pseudo-velocity

n

n

n

U kv t m v t

mm v t V t

Uv t

14

Let a family of sdof systems with damping {η} and natural frequencies {ωn} be subjected to a given ground acceleration.

Let us determine the peak responsesover time as a function of {η} and {ωn}.

15155/31/2012 15

Definitions

0

0

02

( , ) max | ( ) | Spectral relative displacement

( , ) max | ( ) | Spectral relative velocity

( , ) max | ( ) | Spectral absolute acceleration

( , ) ( , ) Spectral pseudo

d n t

v n tt

a n t

pa n n d n

S v t

S v t

S v t

S S

acceleration

( , ) ( , ) Spectral pseudo velocitypv n n d nS S

16165/31/2012 16

Definitions

Sd(ωn,η)Sv(ωn,η)Sa(ωn,η)Spa(ωn,η)Spv(ωn,η)

as a function of ωn with η as a parameter is called the responsespectrum for

Relative displacementRelative velocityAbsolute accelerationPseudo accelerationPseudo velocity

17

Connotes "frequency" on -axis."Frequency" often refers to the frequency parameter used in

defining Fourier transform. In this context we talk oftime and frequency domain representations of s

x

Spectrum

ignals.In the context of response spectrum "frequency" is not the

Fourier frequency but the natural frequencies of a familyof sdof systems.Often period is plotted on the -axis.x

18

2

2

2

0 0

0

0

2 ( )

lim ( )

lim max | ( ) | max | ( ) |

lim ( , ) max | ( ) |

max | ( ) |

ZPA: zero

n

n

n

n

n n g

n g

n gt t

pa n gt

gt

v v v v t

v v t

v t v t

S v t

v t ZPA or PGA

Limiting behavior of the spectra as

period accelerationPGA: peak ground acceleration

19

0

0 00

0

0

0 0

0

1( ) exp[ ( )]sin[ ( )] ( )

sin[ ( )]lim ( ) lim ( )

( ) ( ) ( )

lim ( ) ( )

lim max | ( ) |

n n

n

n

nt

d gd

td

gd

t

g g

g

t

v t t t v d

tv t v d

t v d v t

v t v t

v t

Limiting behavior of response spectra as

0

0 0

max | ( ) |

lim ( , ) max | ( ) |n

gt

d n gt

v t

S v t

20

00

00

00

0

0 (0, ) max ( )sin ( )

(0, ) max ( )cos ( )

(0, ) (0, ) except for very small .

0 (0, ) max ( )sin ( )

(0, ) max

t

pv n g nt

t

v n g nt

pv n v n n

t

a n n g nt

pa n

S v t d

S v t d

S S

S v t d

S

0

( )sin ( )

(0, ) (0, ) (0, )

0 & 0 0.2 (0, ) (0, )

t

n g nt

a n pa n pv n

a n pv n

v t d

S S S

S S

21

( , ) ( , )

1( , ) ( , )

( , ) ( , )

log ( , ) log ( , ) log

log ( , ) log ( , ) log

pv n n d n

d n pv nn

a n n pv n

d n pv n n

a n pv n n

S S

S S

S S

S S

S S

Tripartite plots( , ), ( , ) & ( , )d n pv n pa nS S S

22

Natural period s

log ,pvS T log ,dS T

log ,paS T

23235/31/2012 23

Why so many response spectra?• Each response spectrum provides a physically

meaningful quantity

Peak deformation

Peak strain energy

Peak force in the springBase shearBase moment

),( ndS ),( npvS

),( npaS

24

The shape of the response spectrum can be approximated more readily for design purposes with the aid of all three spectral quantities than any one of them taken alone.

Helps in understanding characteristics of response spectra.

Helps in constructing response spectra.

Helps in relating structural dynamics concepts to building codes.

25

Smooth design response spectra

Natural period s

,paS Tg

PGAZPA

1.0

Soil conditionDamping

PGA : arrived at based on seismic hazard analysis

26

Factors that influence response spectrum at a given site

• Source mechanism• Epicentral distance• Focal depth• Geological conditions• Richter’s magnitude• Soil condition• Damping and stiffness of the system

272727

Duhamel's integral

Alternatives for earthquake load specification

Time histories

Power spectral density Response spectra

Monte Carlo simulations

Spectralestimation

Spectrumcompatibleaccelerograms

Extreme value theory

Spectrum compatibe PSDusing extreme value theory

Not suited forreliability analysis

282828

2

0

2

zero mean, stationary, Gaussian random process;

~ 0,

max

exp

exp2

m

n n n g

g

g gg

m t T

X

x

x

x x x x

x t

x t N S

X x t

P T

How to generate a response spectrum compatible with a given PSD?

2

2

22

22 2

2

with &

x

x gg

x gg

H S d

H S d

292929

2

2

12

2

For a given probability , the corresponding is given by

exp exp2 2

22 ln ln

Let , be the given pseudo-acceleration response spectrum.

x

x x

xx

x

n n

p

p T

pT

R

12

2 2

We interpret , as the - th percentile point.

2, 2 ln ln (typically =84%)

n n

xn n n x

x

R p

R p pT

303030

2

22

2

zero mean, stationary, Gaussian random process;

~ 0,

To a first approximation we assume

n n n g

g

g gg

x g

x x x x

x t

x t N S

H S

How to generate a PSD compatible with a given response spectrum?

2

22

12 22

&2

g

n n n gg n n n gg n

n n

xn

x

d

H S S

313131

2 43

2

2, 2 ln ln2

To a first approximation we thus get

,.

1ln ln

gg nn n n

nn n

n n ngg n

nn

SR p

T

RS

pT

3232

2

2 2

22

22 2

(1) Set iteration N=1(2) Start with the initial guess on the PSD given by

,

1ln ln

(3)Evaluate and using

&

n n nNn

nn

x x

Nx gg

Nx gg

RS

pT

H S d

H S d

Steps

3333

12

2 2

21

2(4)Evaluate , 2 ln ln .

(5)Obtain an improved estimate of PSD using

( )( ) ( )( )

(6)Stop iterations if the PSD function has converged; if not go to

N xn n n x

x

N NN

R pT

RS SR

step 3.

34

0 100 200 300 400 500 600 7000

2

4

6

8

10

12

14

16

Frequency rad/s

Acc

eler

atio

n m

/s2

Target

Derived

Pseudo acceleration response spectrum

35

0 100 200 300 400 500 600 7000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Frequency rad/s

PS

D (

m2 /s

4 )/(r

ad/s

)Derived

Used

36

0 2 4 6 8 10 12 14 16 18 20-5

0

5

0 2 4 6 8 10 12 14 16 18 20-5

0

5

0 2 4 6 8 10 12 14 16 18 20-5

0

5

Acc

eler

atio

n m

/s2

0 2 4 6 8 10 12 14 16 18 20-5

0

5

0 2 4 6 8 10 12 14 16 18 20-5

0

5

Time s

37

0 2 4 6 8 10 12 14 16 18 20-0.2

0

0.2

0 2 4 6 8 10 12 14 16 18 20-0.2

0

0.2

0 2 4 6 8 10 12 14 16 18 20-0.2

0

0.2

Vel

ocity

m/s

0 2 4 6 8 10 12 14 16 18 20-0.2

0

0.2

0 2 4 6 8 10 12 14 16 18 20-0.2

0

0.2

Time s

38

0 2 4 6 8 10 12 14 16 18 20

-0.020

0.02

0 2 4 6 8 10 12 14 16 18 20

-0.020

0.02

0 2 4 6 8 10 12 14 16 18 20

-0.020

0.02

Dis

plac

emen

t m

0 2 4 6 8 10 12 14 16 18 20

-0.020

0.02

0 2 4 6 8 10 12 14 16 18 20

-0.020

0.02

Time s

39

Variance(σ2) = 1.0893 m2/s4

Std. deviation (σ ) = 1.0437 m/s2

3σ = 3.1311 m/s2

Time history no. Maximum ( m/s2)

Minimum ( m/s2)

1. 4.5063 -3.8202

2. 3.4130 -3.4098

3. 3.3419 -3.5512

4. 3.7498 -3.5094

5. 3.4756 -3.1706

Zero Peak Acceleration (ZPA) =3.2530 m/s2

40

-4 -3 -2 -1 0 1 2 3 4 50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Acceleration m/s2

Pro

babi

lity

Normal

Simulation

PDF of the simulated samples of ground acceleration

41

0 100 200 300 400 500 600 7000

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Frequency rad/s

PS

D (

m2 /s

4 )/(r

ad/s

)Target

Simulation

PSD function from 100 samples of ground acceleration

42

0 1 2 3 4 5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Acceleration m/s2

Pro

babl

ity

Simulation

Gumbel

43

Response spectrum method providesmaximum response of sdof systems asa function of natural frequencies anddamping ratios.How to deal with mdof systems?

MDOF systems can be decomposed into a set of Hope :

uncoupled sdof oscillators.How can this be taken advantage of?

44

Building frame under earthquake support motion

45

Equation of motion for total displacement

00

)()(

0

0

0

0

000000 11

3

2

1

33

3322

221

3

2

1

33

3322

221

3

2

1

3

2

1 txctxk

xxx

kkkkkk

kkk

xxx

cccccc

ccc

xxx

mm

m gg

Relative displacement ggg xxyxxyxxy 332211

Equation of motion for relative displacement

g

g

g

xmxmxm

yyy

kkkkkk

kkk

yyy

cccccc

ccc

yyy

mm

m

3

2

1

3

2

1

33

3322

221

3

2

1

33

3322

221

3

2

1

3

2

1

0

0

0

0

000000

Tg txMKYYCYM

111

)(

46

2

2

0

( )

1 1 1

2

2 1, 2, ,

( ) exp( )[ cos sin ]

(

g

T

T T Tn n n

T T T Tg

n n n n n n n g

T

n n n n dn n dnt

n g

MY CY KY M x t

Y Z

M I K Diag C Diag

M Z C Z K Z M x

z z z x n n

M

z t t A t B t

x

) ( )nh t d

47

1

1 0

2

01

1

Displacement: ( ) ( )

{exp( )[ cos sin ] ( ) ( ) }

Elastic forces:

Base shear:

Overturing moment:

N

k kn nn

tN

kn n n n dn n dn n g nn

s

N

snn

N

o n snn

y t z t

t A t B t x h t d

F KY K Z M Z

V F

M x F

48

2

{1} ( )

{1} ( ) ( )

{1}

2 ( )

modal participation factor or modal excitation factor

g

t t

tn n n n n n g n g

tn

nn n n n n n g

n

n

MV CV KV M v t

V Z M M K KM z C z K z M v t v t

M

z z z v tM

49

2

0

1 1

1

0

2 ( )

1( ) exp[ ( )]sin[ ( )] ( )

( ) ( )

( ) ( ) ( ) ( )

Put ( ) ( )

max ( ) ( , ) m

n n g

t

n n gn

nn n

np p

nk kn n k kn n

n n np

nkn kn k kn n

nn

kn n kn D n nt T

v v v v t

v t t t v d

z t v tM

V Z V t z t V t v tM

V t v tM

v t S

0ax ( ) ?kt TV t

50

0

2

2

2

Spring force( ) ( )

Recall

( ) [ ]

( ) [ ] { ( )}

( ) { ( )}

max ( ) ?

s

n n n

s n

ns n n

n

ns n

st T

n

F t KV t K Z

K M

F t M Z

F t M v tM

F t M a tM

F t

51

1

11

1

22

0

21 2

2

1

( ) 1 ( )

1 ( )

Recall {1} 1

( )

( )( )

( )

( ); ma x ?

N

B si si

nB n

n

t t

t nB n N

n

NN

N

N

Bt T

nB n

n n

V F t F t

V M a tM

M M

a tM

a tMV a t

M

a tM

V t t tM

Va

Base shear

52

2

2

1

The quantity has units of mass and is called

the effective modal mass.

Prove that total mass.

Total mass: 1 1

1 1

1

1

1 1

n

n

Nn

n n

T

t tn n

n nn n n n

n n

T

M

MM M

Z M M Z

M M Z M Z

M Z ZM M

M M

Note :

2

1 21

1 n

n

Nn n

T Nnn n

MM

M QEDM M

53

0(0, ) ; (0, ) 0; ( , ) 0; ( , ) 0

, ,

(0, ) 0; (0, ) 0; ( , ) 0; ( , ) 0

iv

g

g

ivg

EIy my cyy t x t y t EIy L t EIy L t

y x t z x t x t

EIz mz cz mx t

z t z t EIz L t EIz L t

Modal combination rules : what is the basic problem?

gx t

54

12

0

0

Eigenfunction expansion

,

with 2 ; 1, 2, ,

What we know based on response spectrum based analysis?

We know max ; 1, 2, , .

What we wish to know?

max ,

n nn

n n n n n n n g

nt T

t T

z x t a t x

a a a x t n

a t n

z x t

0 1max n nt T n

a t x

55

0 100 200 300 400 500 600 7000

2

4

6

8

10

12

14

16

Frequency rad/s

Acc

eler

atio

n m

/s2

Target

Derived

56

0 01 1max max

The extrema of for =1,2, , are likely tooccur at different times and they may have different signs.Response spectra do not contain infor

n n n nt T t Tn n

n

a t x x a t

a t n

Difficulty

Remarks

0 1

mation on times atwhich extrema occur nor do they store the signs of the extrema.

max can occur at a time instant * at which none

of ; =1,2, , need to attain their respective extr

n nt T n

n

a t x t

a t n

emum

values.

57

0 1

0

How best to obtain max in terms of

max , and the modal characteristics of the vibrating

system?

Modal characteristics: natural frequencies, mode s

n nt T n

nt T

a t x

a t

Problem of modal combination rules

hapes,modal damping ratios, and participation factors.

These rules are formulated based on methods of random vibration analysis.