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26.06.16
1
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
26.06.16
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Response spectrum analysis – peak floor accelerations Christoph Adam©
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)
Response spectrum analysis – peak floor accelerations Christoph Adam©
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
" #$ %$i=1
N
∑ = φφiΓi!!di(rel)
i=1
N
∑!!u(rel)
" #$ %$+ φφiΓi!!ug
i=1
N
∑e!!ug
" #$ %$= φφiΓi
!!dii=1
N
∑
uk(rel)
ukN
k
1
ugTotal accelerations
Modal decomposition of u(rel)
26.06.16
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Response spectrum analysis – peak floor accelerations Christoph Adam©
Modal response history analysis Total accelerations
Page 5
!!u = φφiΓi
!!dii=1
N
∑ = φφiΓi!!di(rel)
i=1
N
∑ + φφiΓi!!ugi=1
N
∑ = φφiΓi!!di(rel)
i=1
N
∑ + e!!ug
!!u = φφiΓi!!di
i=1
n
∑ + φφiΓi!!di(rel)
i=n+1
N
∑ + φφiΓi!!ugi=n+1
N
∑ = φφiΓi!!di
i=1
n
∑
u(n)" #$ %$
+ 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
Response spectrum analysis – peak floor accelerations Christoph Adam©
Modal response history analysis Total accelerations
Page 6
!!u = φφiΓi
!!dii=1
N
∑ = φφiΓi!!di(rel)
i=1
N
∑ + φφiΓi!!ugi=1
N
∑ = φφiΓi!!di(rel)
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
26.06.16
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Response spectrum analysis – peak floor accelerations Christoph Adam©
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
Response spectrum analysis – peak floor accelerations Christoph Adam©
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
Proposed response spectrum method
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
26.06.16
5
Response spectrum analysis – peak floor accelerations Christoph Adam©
=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
Proposed response spectrum method
Page 9
correlation coefficients
E max !!Uk⎡⎣
⎤⎦ ≡ mPFAk
= Var !!Uk⎡⎣ ⎤⎦pk =
peak factors
Response spectrum analysis – peak floor accelerations Christoph Adam©
Proposed response spectrum method
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
26.06.16
6
Response spectrum analysis – peak floor accelerations Christoph Adam©
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
Response spectrum analysis – peak floor accelerations Christoph Adam©
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
26.06.16
7
Response spectrum analysis – peak floor accelerations Christoph Adam©
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
Response spectrum analysis – peak floor accelerations Christoph Adam©
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)
26.06.16
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Response spectrum analysis – peak floor accelerations Christoph Adam©
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
Response spectrum analysis – peak floor accelerations Christoph Adam©
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
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
12-story SMRF 24-story SMRF
26.06.16
9
Response spectrum analysis – peak floor accelerations Christoph Adam©
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
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
12-story SMRF
with peak factors no peak factors
Response spectrum analysis – peak floor accelerations Christoph Adam©
Application
Page 18
Spatial frame 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 = 7.20 rad / s ω1 = 4.14 rad / s ω1 = 2.38 rad / s
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
26.06.16
10
Response spectrum analysis – peak floor accelerations Christoph Adam©
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
ω1 = 7.55 rad / sω2 = 7.55 rad / sω3 = 31.0 rad / s
12-story SMRF
12-story WALL
Response spectrum analysis – peak floor accelerations Christoph Adam©
0.0 1.0 2.0 3.0 4.0 5.0mPFAk / g
hre
l
1 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-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
1 to 95% m.p. mode included
Application
Page 20
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
with peak factors
no peak factors
26.06.16
11
Response spectrum analysis – peak floor accelerations Christoph Adam©
ω1 = 7.55 rad / sω2 = 7.55 rad / sω3 = 31.0 rad / s
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
1 to 95% m.p. mode included
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
1 to 95% m.p. mode included
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
Response spectrum analysis – peak floor accelerations Christoph Adam©
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