response spectrum analysis for peak floor acceleration

26.06.16

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Response spectrum analysis for peak floor acceleration demands in

earthquake excited structures

Lukas Moschen, Christoph Adam Unit of Applied Mechanics

University of Innsbruck

Dimitrios Vamvatsikos School of Engineering

National Technical University of Athens

The 42nd Risk, Hazard & Uncertainty Workshop Hydra, Greece, June 23-24, 2016

Response spectrum analysis – peak floor accelerations Christoph Adam©

Motivation and objective

ü  Simplified approaches for estimating peak floor acceleration (PFA) demands

§  Lack usually generality in application

ü  Response spectrum methods with appropriate modal combination rules

(SRSS, CQC) more accurate

ü  SRSS and CQC rules originally developed for relative response quantities

(displacement, internal forces)

ü  PFA demand absolute response quantity

ü  CQC modal combination rule for PFA demands proposed

§  Closed form solution for peak factors and correlation coefficients

§  Based on concepts of normal stationary random vibration theory

§  Related studies: Taghavi and Miranda (2009), Pozzi and Der Kiureghian (2012,

2015)

Page 2

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Modal response history analysis Equations of motion

Page 3

M!!u(rel) + C !u(rel) +Ku(rel) = −Me!!ug

u(rel) = φφiΓidi

(rel)

i=1

N

∑

!!di(rel) + 2ζiωi

!di(rel) + ωi

2di(rel) = −!!ug

Γi =

φφiTMe

φφiTMφφi

uk(rel)

ukN

k

1

ug

Modal decomposition of u(rel)


Modal response history analysis Equations of motion

Page 4

M!!u(rel) + C !u(rel) +Ku(rel) = −Me!!ug

u(rel) = φφiΓidi

(rel)

i=1

N

∑

!!di(rel) + 2ζiωi

!di(rel) + ωi

2di(rel) = −!!ug

Γi =

φφiTMe

φφiTMφφi

!!u = !!u(rel) + e!!ug = φφiΓi!!di(rel) + !!ug( )!!di

" #$ %$i=1

N

∑ = φφiΓi!!di(rel)

i=1

N

∑!!u(rel)

" #$ %$+ φφiΓi!!ug

i=1

N

∑e!!ug

" #$ %$= φφiΓi

!!dii=1

N

∑

uk(rel)

ukN

k

1

ugTotal accelerations

Modal decomposition of u(rel)

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Modal response history analysis Total accelerations

Page 5

!!u = φφiΓi

!!dii=1

N


i=1

N

∑ + φφiΓi!!ugi=1

N


i=1

N

∑ + e!!ug

!!u = φφiΓi!!di

i=1

n

∑ + φφiΓi!!di(rel)

i=n+1

N

∑ + φφiΓi!!ugi=n+1

N

∑ = φφiΓi!!di

i=1

n

∑

u(n)" #$ %$

+ φφiΓi!!di(rel)

i=n+1

N

∑≈0

" #$$ %$$+ e − φφiΓi

i=1

n

∑⎛

⎝⎜

⎞

⎠⎟

r(n)" #$$ %$$

!!ug

Separation of modal series into 1,…, n modes and n+1,...,N modal contributions


Modal response history analysis Total accelerations

Page 6

!!u = φφiΓi

!!dii=1

N


i=1

N

∑ + φφiΓi!!ugi=1

N


i=1

N

∑ + e!!ug

!!u = φφiΓi!!di

i=1

n

∑ + φφiΓi!!di(rel)

i=n+1

N

∑ + φφiΓi!!ugi=n+1

N

∑ = φφiΓi!!di

i=1

n

∑

u(n)" #$ %$

+ φφiΓi!!di(rel)

i=n+1

N

∑≈0

" #$$ %$$+ e − φφiΓi

i=1

n

∑⎛

⎝⎜

⎞

⎠⎟

r(n)" #$$ %$$

!!ug

!!u ≈ φφiΓi!!di

i=1

n

∑!!u(n)

" #$ %$+ e − φφiΓi

i=1

n

∑⎛

⎝⎜

⎞

⎠⎟

r(n)" #$$ %$$

!!ug = !!u(n) + r(n)!!ug

residual vector -> high frequency contribution of ground acceleration considered

Approximation of total accelerations by the first n modes

Separation of modal series into 1,…, n modes and n+1,...,N modal contributions

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Total accelerations in terms of stationary random variables

Proposed response spectrum method

Page 7

!!U ≈ φφiΓi

!!Dii=1

n

∑ + r(n)!!Ug = !!U(n) + r(n)!!Ug

Var !!U⎡⎣ ⎤⎦ = Var !!U(n)⎡

⎣⎢⎤⎦⎥ + r(n)( )2 Var !!Ug

⎡⎣

⎤⎦ + 2Cov !!U(n),r(n)!!Ug

⎡⎣⎢

⎤⎦⎥

Var !!U⎡⎣ ⎤⎦ = Cov !!Ui

(n), !!Uj(n)⎡

⎣⎢⎤⎦⎥

j=1

n

∑i=1

n

∑ + r(n)( )2 Var !!Ug⎡⎣

⎤⎦ + 2 Cov !!Ui

(n),r(n)!!Ug⎡⎣⎢

⎤⎦⎥

i=1

n

∑

Assumption: Ø  Set of ground motions represent Gaussian random process with zero mean

Ø  is Gaussian with zero mean

Ø 

Ø  Central peak floor acceleration (PFA) demand

!!U

E !!U2⎡⎣⎢

⎤⎦⎥ = Var !!U⎡⎣ ⎤⎦

Variance

E max !!Uk⎡⎣

⎤⎦ ≡ mPFAk

= Var !!Uk⎡⎣ ⎤⎦pk

peak factor

quantity to be determined

variance

uk

k

!!uk


Var !!U⎡⎣ ⎤⎦ = Cov !!Ui

(n), !!Uj(n)⎡

⎣⎢⎤⎦⎥

j=1

n

∑i=1

n

∑ + r(n)( )2 Var !!Ug⎡⎣

⎤⎦ + 2 Cov !!Ui

(n),r(n)!!Ug⎡⎣⎢

⎤⎦⎥

i=1

n

∑Modal decomposition of variance


Page 8

Var !!U⎡⎣ ⎤⎦ = φφiΓiφφjΓ jCov !!Di , !!Dj

⎡⎣⎢

⎤⎦⎥

j=1

n

∑i=1

n

∑ + r(n)( )2 Var !!Ug⎡⎣

⎤⎦ + 2r(n) φφiΓiCov !!Di , !!Ug

⎡⎣⎢

⎤⎦⎥

i=1

n

∑

Ø  Pearson’s cross correlation coefficient and central peak modal coordinate

ρi,j =Cov !!Di,

!!Dj⎡⎣

⎤⎦

Var !!Di⎡⎣ ⎤⎦Var !!Dj

⎡⎣

⎤⎦

Cov !!Di,

!!Dj⎡⎣

⎤⎦ = ρi,j Var !!Di

⎡⎣ ⎤⎦Var !!Dj⎡⎣

⎤⎦ = ρi,j

E max !!Di⎡⎣

⎤⎦

pi

E max !!Dj⎡⎣

⎤⎦

pj≈ ρi,j

Sa,i

pi

Sa,j

pj

E max !!Di⎡⎣

⎤⎦ ≈ Sa,i = Var !!Di

⎡⎣ ⎤⎦pi

Var !!U⎡⎣ ⎤⎦ = ρi,jφφiΓiφφjΓ jSa,i

pi

Sa,j

pjj=1

n

∑i=1

n

∑ + r(n)( )2 mPGA2

pg2

+ 2r(n) mPGApg

ρi,gφφiΓiSa,i

pii=1

n

∑

ρi,j

E max !!Di⎡⎣

⎤⎦

central spectral pseudo-acceleration at i-th period

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=pkpi

pkpj

φφi,kΓiSa,iφφj,kΓ jSa,jρi,jj=1

n

∑i=1

n

∑ +pkpg

⎛

⎝⎜⎜

⎞

⎠⎟⎟

2

mPGA2 rk

(n)( )2 + 2mPGArk(n) pk

pg

pkpii=1

n

∑ φφi,kΓ jSa,jρi,g

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

1/2

CQC mode superposition rule for central PFA demand of the k-th floor


Page 9

correlation coefficients

E max !!Uk⎡⎣

⎤⎦ ≡ mPFAk

= Var !!Uk⎡⎣ ⎤⎦pk =

peak factors



Page 10

Gg(KT) = G0

1 + 4ζg2(ν / νg)

2

1 − (ν / νg)2( )2 + 4ζg

2(ν / νg)2

ρi,j =λ0,ij

λ0,iiλ0,jj

ρi,g =λ0,ig

λ0,iiλ0,gg

λl,XY = νl

0

∞∫ GXY(ν)dν = νl

0

∞∫ Gg

(KT)(ν)HX(ν)HY*(ν)dν , l = 0,1,2,..

Correlation coefficients

l-th cross-spectral moment

Hg(ν) = 1

zeroth cross-spectral moment between i-th and j-th random modal acceleration

zeroth cross-spectral moment between i-th random modal and ground acceleration

cross PSD

Kanai-Tajimi PSD

FRFs

FRF of i-th modal acceleration FRF of ground acceleration

Hi(ν) =

ωi2 + 2iζiωiν

ωi2 − ν2 + 2iζiωiν

Kanai-Tajimi PSD – characterization of ground excitation

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foa-CQC-pf modal combination rule Ø  Assumption: first order approximation of cross-spectral moments

Approximations

Page 11

ha-CQC-pf modal combination rule Ø  Assumption: hybrid approximation of cross-spectral moments

CQC-nopf modal combination rule Ø  Assumption: ratios of peak factors are unity

mPFAk

≈ φφi,kΓiSa,iφφj,kΓ jSa,jρi,jj=1

n

∑i=1

n

∑ +mPGA2 rk

(n)( )2 + 2mPGArk(n) φφi,kΓ jSa,jρi,g

i=1

n

∑⎡

⎣⎢⎢

⎤

⎦⎥⎥

1/2

pkpi

=pkpj

=pkpg

= 1

mPFAk=

pkpi

pkpj

φφi,kΓiSa,iφφj,kΓ jSa,jρi,jj=1

n

∑i=1

n

∑ +pkpg

⎛

⎝⎜⎜

⎞

⎠⎟⎟

2

mPGA2 rk

(n)( )2 + 2mPGArk(n) pk

pg

pkpii=1

n

∑ φφi,kΓ jSa,jρi,g

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

1/2


Approximations

Page 12

mPFAk≈

pkpi

φφi,kΓiSa,i

⎛

⎝⎜

⎞

⎠⎟

2

i=1

n

∑ + mPGArk(n)( )2

⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

1/2

mPFAk

≈ φφi,kΓiSa,i( )2i=1

n

∑ + µPGArk(n)( )2⎡

⎣⎢⎢

⎤

⎦⎥⎥

1/2

t-SRSS-pf modal combination rule Ø  Assumption: modal response uncorrelated; peak factors and truncated modes considered

t-SRSS-nopf modal combination rule Ø  Assumptions: modal response uncorrelated; peak factor ratios unity;

truncated modes considered

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SRSS-pf modal combination rule Ø  Assumptions: modal response uncorrelated; truncated modes neglected

Approximations

Page 13

Classical SRSS modal combination rule (SRSS-nopf) Ø  Assumptions: modal response uncorrelated; peak factor ratios unity;

truncated modes neglected

mPFAk≈

pkpi

φφi,kΓiSa,i

⎛

⎝⎜

⎞

⎠⎟

2

i=1

n

∑⎡

⎣

⎢⎢⎢

⎤

⎦

⎥⎥⎥

1/2

mPFAk

≈ φφi,kΓiSa,i( )2i=1

n

∑⎡

⎣⎢⎢

⎤

⎦⎥⎥

1/2


Ground motion modeling in frequency domain

Page 14

Reference solution: outcome of RHA Ø  Seismic hazard representative for Century City (Los Angeles)

Ø  92 site compatible ground motion records from PEER NGA database

Ø  Median and dispersion of record set match design spectrum and target spectrum

in period range 0.05s ≤ T ≤ 3.0s σt = 0.80

G0 = 0.18

Gg(KT) = G0

1 + 4ζg2(ν / νg)

2

1 − (ν / νg)2( )2 + 4ζg

2(ν / νg)2

Calibration of Kanai-Tajimi PSD

ζg = 0.78

νg / (2π) = 1.79Hz

0.0

1.0

2.0

3.0

4.0

5.0

0.1 1 10 100

Sa

/ g

f = ν /2π [Hz]

0.0

0.1

0.2

0.3

0.4

0.1 1 10

Gg

f = ν / 2π [Hz]

Gg(ground motions)

Gg(KT)

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0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

hre

l

6/6

2/6

4/6

5/6

3/6

1/6

mPFAk / g

RHACQC-pf mode 1-95%

CQC-pf mode 1

CQC-pf mode 1+2

CQC-pf all modes

6-story SMRF

Application

Page 15

Planar SMRF structures Ø  6-story ( ), 12-story ( ), and 24-story ( ) structures

Ø  Linear mode shape

Ø  Seismic active mass concentrated to BC connections. Roof only half mass.

Ø  5% Rayleigh damping

ω1 = 7.33 rad / s ω1 = 4.22 rad / s ω1 = 2.42 rad / s

1.0 1.1 1.2 1.3 1.4 1.5 1.6pk

hre

l

pk(hrel=0)=pg

6/6

2/6

4/6

5/6

3/6

1/6

mode 1-3

mode 1

mode 1+2

all modes

6-story SMRF


0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5mPFAk / g

hre

l

RHA

CQC-pf mode 1-95%

CQC-pf mode 1

CQC-pf mode 1+2

CQC-pf all modes

12/12

4/12

8/12

10/12

6/12

2/12

12-story SMRF

0.0 0.5 1.0 1.5 2.0 2.5 3.0mPFAk / g

hre

l

RHA

CQC mode 1-95%

CQC mode 1

CQC mode 1+2

CQC all modes

24/24

8/24

16/24

20/24

12/24

4/24

24-story SMRF

Application

Page 16






12-story SMRF 24-story SMRF

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0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5mPFAk / g

hre

l

12/12

4/12

8/12

10/12

6/12

2/12

12-story SMRF

RHA

CQC-pf

ha-CQC-pf

foa-CQC-pf

t-SRSS-pf

SRSS-pf

1 to 95% m.p. mode included

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5mPFAk / g

hre

l1 to 95% m.p. mode included

12/12

4/12

8/12

10/12

6/12

2/12

RHA

12-story SMRF

CQC-nopf

ha-CQC-nopf

foa-CQC-nopf

t-SRSS-nopfSRSS-nopf

Application

Page 17






12-story SMRF

with peak factors no peak factors


Application

Page 18

Spatial frame structures Ø  6-story ( ), 12-story ( ), and 24-story ( ) structures





Spatial wall structures Ø  6-story ( ), 12-story ( ), and 24-story ( ) structures

Ø  Parabolic mode shape ω1 = 12.58 rad / s ω1 = 7.55 rad / s ω1 = 4.49 rad / s

A7.32 m

7.3

2 m

3.6

6 m

3.6

6 m

0.30meccentricmass

!!uAxk

!!uAx1

!!uAx2

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0.0 1.0 2.0 3.0 4.0 5.0mPFAk / g

hre

l

RHA

12/12

4/12

8/12

10/12

6/12

2/12

12-story WALL

CQC-pf mode 1-95%CQC-pfmode 1 CQC-pf all modes

SRSS-nopf all modes

SRSS-nopfmode 1

0.0 1.0 2.0 3.0 4.0 5.0mPFAk / g

hre

l

RHA

12/12

4/12

8/12

10/12

6/12

2/12

12-story SMRF

CQC-pf mode 1-95%

CQC-pf mode 1

CQC-pf all modes

SRSS-nopf all modes

Application

Page 19

3.6

6 m

3.6

6 m

!!uAx1

!!uAx2

ω1 = 4.14 rad / sω2 = 4.21 rad / sω3 = 4.59 rad / s


12-story SMRF

12-story WALL


0.0 1.0 2.0 3.0 4.0 5.0mPFAk / g

hre

l


12/12

4/12

8/12

10/12

6/12

2/12

RHA

12-story SMRF

CQC-nopf

ha-CQC-nopf

foa-CQC-nopf

t-SRSS-nopf

SRSS-nopf

0.0 1.0 2.0 3.0 4.0 5.0mPFAk / g

hre

l

RHA

12/12

4/12

8/12

10/12

6/12

2/12

12-story SMRF

CQC-pf

ha-CQC-pf

foa-CQC-pf

t-SRSS-pf

SRSS-pf


Application

Page 20

3.6

6 m

3.6

6 m

!!uAx1

!!uAx2


12-story SMRF

with peak factors

no peak factors

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0.0 1.0 2.0 3.0 4.0 5.0mPFAk / g

hre

l

RHA

12-story WALL

CQC-pf

ha-CQC-pf

foa-CQC-pf

t-SRSS-pf

SRSS-pf


12/12

4/12

8/12

10/12

6/12

2/12

0.0 1.0 2.0 3.0 4.0 5.0mPFAk / g

hre

l

RHA

12-story WALL

CQC-nopf

ha-CQC-nopf

foa-CQC-nopf

t-SRSS-nopf

SRSS-nopf


12/12

4/12

8/12

10/12

6/12

2/12

Application

Page 21

3.6

6 m

3.6

6 m

!!uAx1

!!uAx2

12-story WALL

with peak factors

no peak factors


Summary and conclusions ü  Robust CQC modal combination rule for central PFA demands proposed

ü  Analytical expressions and approximations for

§  Correlation coefficients

§  Peak factors

ü  Reliable assessment of the central PFA demand

ü  Peak factors

§  Significant for midrise to highrise structures

ü  Correlation coefficients

§  Insignificant for planar structures

§  Significant for spatial structures with closely spaced modes

ü  Truncated modes

§  Significant for lower stories

Page 22

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Thank you very much for your attention!

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response spectrum analysis for peak floor acceleration

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