WAVELET-BASED EVOLUTIONARY RESPONSE OF MULTI-SPAN
1STRUCTURES INCLUDING WAVE-PASSAGE AND SITE-RESPONSE
2EFFECTS
3Van-Nguyen Dinh1, Biswajit Basu2 and Ronald B.J. Brinkgreve3 4
Abstract: Stochastic seismic wavelet-based evolutionary response of multi-span structures 5
including wave-passage and site-response effects is formulated in this paper. A procedure 6
to estimate site-compatible parameters of surface-to-bedrock frequency response function 7
by using finite element analysis of the supporting soil medium is proposed. The earthquake 8
energy content is represented by a composite power spectrum density function contributed 9
by the surface-to-bedrock frequency response function (FRF) and bedrock power spectra. 10
A long span multi-support structure is subjected to spatially-varying differential support 11
motions where the spatial-variability is represented by bedrock parametric coherency 12
models and time lags. In addition to the time lags due to wave passage effects, the site-13
response effects due to different soil conditions at the supports are characterized by 14
frequency-dependent time-lags. In an illustrative case study, a three-span two-dimensional 15
hangar frame is analyzed using the proposed formulations. The time-lags resulting from 16
site-response effects and computed by different FRFs show different variation in trend and 17
frequency content. The site-response effect is found to introduce additional frequency non-18
stationarity and leads to an increase in the frame responses but with slower attenuation in 19
time. 20
Subject headings: Seismic effects, amplitude-and frequency-nonstationary, spatial-21
variability, site-response, wavelets, parametric frequency response function.
22
1 Post-doctoral Research Fellow, Dept. of Civil, Structural and Environmental Engineering, School of
Engineering, Trinity College Dublin, Dublin 2, Ireland. E-mail: nguyendv@tcd.ie; nguyendhhh@yahoo.com
2
Professor, Dept. of Civil, Structural and Environmental Engineering, School of Engineering, Trinity College Dublin, Dublin 2, Ireland (corresponding author). E-mail: basub@tcd.ie
3
Associate Professor, Faculty of Civil Engineering & Geosciences, Delft University of Technology & Manager, PLAXIS bv, Delft, The Netherlands. E-mail: r.brinkgreve@plaxis.nl
1. Introduction 1
The ground motions at a site are random processes because of the complex 2
characteristics of source and paths of seismic waves. Moreover, they are apparently non-3
stationary in both amplitude and frequency due to frequency-proportional velocities of 4
seismic waves and due to travelling paths consisting of soil layers having different 5
properties. The structural responses under ground motions are consequently random and 6
temporally and spectrally non-stationary and they should therefore be stochastically 7
represented by time-varying statistical quantities such as the evolutionary power spectral 8
density function (PSDF). When statistical quantities representing ground motions (input) 9
are given for evaluating stochastic structural responses (output), the input-output relations 10
are needed. The random vibration theory is applicable only if the input is time-invariant. 11
For time-varying inputs, wavelet techniques are suitable tools as they can provide a joint 12
time-frequency representation simultaneously. Wavelet techniques have been used to 13
formulate the input-output relations of single degree of freedom systems (Basu and Gupta 14
1998, 2000) and multi degree of freedom systems (Basu and Gupta 1997, Tratskas and 15
Spanos 2003) where the input spatial-variation was excluded and proportional damping 16
was assumed. 17
The ground motions at several sites induced by an earthquake are spatially-varying 18
that consists of four distinct phenomena: incoherence, wave-passage, attenuation, and site-19
response effects (Der Kiureghian 1996). The spatial-variation of ground motions has 20
pronounced effects on structures. Stochastic input-output relations of multi-support 21
structures subjected to spatially-varying ground motions have been formulated by using 22
random vibrations (Hao 1994, Loh and Ku 1995) where incoherence and wave-passage 23
effects are considered and the ground motions are only non-stationary in amplitude. 24
Dumanoglu and Soyluk (2003) and Zhang et al. (2009) considered incoherence, wave-25
passage and site-response effects in their ground motion spatial-variability models and 1
carried out stochastic analyses of long-span bridges whereas the frequency non-stationarity 2
of excitations was neglected. The site-response effect was shown to contribute 3
considerably to the maximum response amplitudes. However, the influence of site-4
response effect on the frequency content of the responses of the bridge was unable to be 5
investigated. 6
A more general and realistic input-output relation has been proposed by Chakraborty 7
and Basu (2008) using wavelet-based framework where the ground motions are non-8
stationary in both amplitude and frequency, the excitation spatial-variation due to wave-9
passage effect is considered and the non-stationarity in both amplitude and frequency of 10
the output are evaluated. That work is extended in this paper to include the site-compatible 11
earthquake energy and site-response effect of supporting soil media beneath the supports. 12
In the literature, by including a term in the coherency phase, the site response effect is 13
considered in formulating complex coherency functions (Der Kiureghian 1996) and in 14
simulating spatially-varying non-stationary ground motion time histories (Zerva 2009, 15
Konakli and Der Kiureghian 2011). While this approach is suitable for ground motion 16
simulation, it faces a difficulty in spectral analysis of evolutionary responses. The direct 17
use of a complex function is not feasible for the second-order moment of the stochastic 18
responses as it is a real quantity. Instead, an alternative approach is formulated in this 19
paper to separately represent the lagged coherency and the phase. The site-response effect 20
is proposed to be characterized by frequency-dependent time-lags. Hence, the ground 21
motion spatial-variation modelled for stochastic response analysis in this paper 22
incorporates parametric lagged coherency, wave-passage and site-response effects. 23
For simulating spatially-varying non-stationary ground motions and analyzing 24
structural responses, the earthquake energy content has been generally characterised by 25
stationary two-sided Kanai-Tajimi (K-T) PSDF (Hao et al. 1989, Chakraborty and Basu 1
2008) and Clough-Penzien (C-P) PSDF (Hao 1994, Deodatis 1996, Dumanoglu and 2
Soyluk 2003, Zhang et al. 2009) amongst others. The K-T and C-P PSDFs are the most 3
commonly used in both parameterization and simulation of seismic ground motions (Zerva 4
2009). However, when using these spectra, the bedrock is assumed to be rigid and the 5
parameters for the sites where the structures are located are not estimated from the specific 6
geological profiles but are empirically assumed. As the dynamic properties of the soil 7
medium vary horizontally from site to site, such description of PSDF is unable to represent 8
the frequency and characteristic damping of the real soil conditions. Besides, the effect of 9
soil layer thickness is also not accounted for, whereas natural period of a soil layer is 10
proportional to its thickness (Kramer 1996). A thicker layer of soft soil may exhibit lower 11
frequencies close to the dominant ones of the seismic waves and cause more amplification 12
of the propagating waves than a thinner layer does. In order to account for the thickness of 13
soil deposit (i.e. the depth of the bedrock), the earthquake energy content at a site is 14
represented in this paper by a composite PSDF contributed by surface-to-bedrock 15
frequency response function (FRF) and the bedrock power spectrum. A procedure to 16
estimate site-compatible FRF parameters by finite element analysis (FEA) of the soil 17
profile under each individual support is proposed in this paper and applied in the case 18
study considered. This procedure overcomes some of the existing limitations by accounting 19
for (i) soil property horizontal variation and (ii) the effects of soil layer thickness in the 20
FEA. 21
Using the proposed representations of ground motion spatial-variation in Section 2 22
and earthquake energy in Section 3, the wavelet-based stochastic models of spatially-23
varying ground motions (input) and seismic evolutionary responses of multi-span 24
structures (output) are formulated in Sections 4 and 5.1, respectively. Section 5.2 reviews 25
an efficient wavelet basis function to be used in this paper. A flowchart explaining the 1
proposed methodology and the relationship among the equations presented in this paper is 2
provided in Fig. 1. 3
In the case study presented in Section 6, a three-span two-dimensional hangar frame 4
supported on a horizontally-varying property soil layer and a thick elastic bedrock layer is 5
analyzed using the proposed formulations. The parameters of the C-P and K-T FRFs 6
compatible to the site beneath each individual support are estimated. The stationary PSD at 7
each support is calculated by using the parametric K-T and C-P FRFs. The stochastic 8
processes corresponding to these PSDs are used as orthogonal processes at different 9
supports for based modelling of spatially-varying ground motions and for wavelet-10
based evolutionary response analyses of the frame. The time lags computed by using K-T 11
parametric FRF vary by a moderate amount around a higher frequency whereas the time 12
lags computed by using C-P parametric FRF vary dramatically around a lower frequency 13
possibly due to the additional lower frequency filter. Comparing to the results with the case 14
where only wave-passage effect is considered, the site-response effect leads to an increase 15
in wavelet-based root-mean-squares of the frame relative displacements with a slower 16
attenuation in time. The frequency content of such responses is more non-stationary and 17
their instantaneous PSD peaks are higher. 18
2. Representation of Parametric Coherency Model for Ground Motions 19
The representation of ground motion spatial variation by a proposed introduction of 20
frequency dependent time lags for site response effects and accounting for other effects 21
such as incoherency and wave passage is developed in this section. 22
The spatial variability of ground motions at supports r and l is characterized in the 23
frequency domain by the coherency function rl() written in a complex form as
24
( )
exp ) ( ) (
rl rl i rl (1)where the real term, rl() , 0rl() 1, is the lagged coherency characterizing the 1
variation in space. In the literature, rl() has been represented by common functions 2
such as the ones given by Harichandran and Vanmarcke (1986), Luco and Wong (1986), 3
and Hao et al. (1989), for example. In this paper, the parametric coherency models are 4
estimated by using finite element-based seismic analysis of a geological soil medium 5
model including the bedrock. The coherency phase rl() represents the difference in 6
phase of the excitations at the two supports. When the wave passage and site response 7
effects denoted by the superscripts ‘wp’ and ‘site’ respectively are considered, rl() is 8 expressed as 9
site rl wp rl site rl wp rl rl( ) t (2)The use of representation in Eq. (1) for spectral analysis of evolutionary excitations 10
and structural responses faces a difficulty that the complex coherency is not feasible to be 11
directly introduced into any real-valued second-order moment quantity. To overcome this 12
difficulty, the coherency phase rl() is transformed into frequency-dependent time lags 13 as 14
site rl wp rl rl rl( ) .t t t (3)Consider a single soil layer under two surface sites r and l as shown in Fig. 2. The 15
properties of the soil layer are horizontally-varying. The propagation of seismic shear 16
waves from the bottom to the top can be characterized by 1-D wave propagation. The time 17
lag due to wave-passage effect in Eqs. (2) and (3) is computed from the separation distance 18
rl and the wave propagation velocities Vr and Vl beneath supports r and l respectively is
19 given as 20 21 l r rl wp rl V V t 2 (4)
1
The site-response effect between sites r and l is attributed to the difference in phases r and
2
l at the two sites as (Der Kiureghian, 1996)
3
1 ** Re Im tan ) ( ) ( ) ( r l r l r l site rl H H H H (5)When the behaviour of the soil column is dominated by its first mode or when the high-4
frequency components of the ground motion do not have significant contribution to the 5
structural responses, the functional form of the frequency response function (FRF) at a site 6
Hs(), s = r, l in Eq. (5) can be represented by the Kanai-Tajimi (K-T) filter function in Eq.
7
(6) or the Clough-Penzien (C-P) filter function in Eq. (7) (Clough and Penzien 2003), 8
s
s ss
s
T K s i i H 2 1 2 1 ) ( 2 (6)
f f
f f T K s P C s i H H 2 1 ) ( ) ( 2 (7)In Eqs. (6) and (7), s and s are the soil characteristic frequency and damping ratio, 9
respectively. The frequency f and damping ratio f used in C-P FRF, Eq. (7), greatly
10
attenuates the very low frequency components. The frequency-dependent time-lag due to 11
site-response effect in Eq. (3) is given as 12
1 () site rl site rl t (8)The time-lags trlwp and trlsite
contributing to the coherency phase rl() will be 13used separately from the lagged coherency rl() in the following sections. 14
3. Site-Compatible PSD of Ground Motions in a Soil Medium on Elastic Bedrock 15
In this paper, the geological profile consists a soil layer on a very thick elastic 16
bedrock layer. The PSDF of ground motions at a support r is related to that of bedrock 17
motions by (Der Kiureghian 1996) 18
2 bedrock
S H
Srr r (9)
where Hr() is the FRF of the soil layer beneath support r and Sbedrock() is the PSD of
1
bedrock. What follows later in the section is a proposed technique based on finite element 2
modelling of the soil medium to estimate the parameters of this FRF represented by a 3
parametric form. The PSD of bedrock is expressed as 4
2 bedrock bedrock bedrock 1 ) ( F T S (10)where Tbedrock is the stationary duration of the bedrock excitation stochastic process 5
contributed by the earthquake source and the source-to-bedrock path (Trifunac and Brady 6
1975). The estimation of soil FRF parameters and the formulation of Fourier amplitude 7
spectrum of motions at the top level of the bedrock (hereby in short called the bedrock), 8
Fbedrock(), are presented in the following sections. 9
3.1. Proposed Procedure to Estimate Site-Compatible FRF parameters 10
A procedure to estimate the FRF parameters of a single-layered soil column beneath 11
a support r by using Finite Element Analysis has been proposed. The soil column is 12
modelled and analysed using finite elements (PLAXIS 2D, Brinkgreve et al. 2008). The 13
excitation is a white-noise acceleration uniformly applied at the bottom of soil layer. The 14
unsmoothed absolute values of FRF of accelerations at the surface points with respect to 15
the source (bottom) are computed from the Fourier amplitude ratio as: 16
bed points , surface , , ; 1, , ; 1, , ) ( ) ( ~ N i N k F F H k i r k i r k i r (11)where N and Npoints are the number of discrete frequency intervals necessary and number 17
of surface points considered around support r, respectively. The smoothed absolute values 18
of FRF, Hr,i
k , k = 1, …, N, are obtained by averaging over Npoints number of surface 19points considered around support r. The parameters of the FRF are estimated by fitting 20
such smoothed absolute values to the functional form of the FRF, for example the K-T 1
spectrum in Eq. (6) or the C-P spectrum in Eq. (7). The parametric fitting (nonlinear least 2
square) in MATLAB is used for this purpose. Model parameters (s, s for K-T FRF or s,
3
s, f , f for C-P FRF) of the soil layer are obtained. After comparing the statistical
4
parameters (R-Square, RMSE), a final set of estimated model parameters and a parametric 5
form Hr
of soil column model at support r is obtained for each FRF functional form. 6This procedure is repeated for each individual support having different local soil 7
conditions. 8
3.2. Fourier Amplitude Spectrum of Earthquake Bedrock Motions 9
The Fourier amplitude spectrum of earthquake motions at bedrock is represented 10
by using the stochastic seismic spectrum (Boore 2003): 11
C E
M
GR P R A DFbedrock s , , (12)
where the scaling factor and the source spectrum are respectively expressed as 12 3 0 0 4 s s e e V F V R C (13a)
2 0 2 2 1 1 1 , M M M M E b a s (13b)in which, Re, Ve, Fs, Vs0 and are respectively the radiation pattern, partition of total shear
13
wave energy into horizontal components, constraint factor, and the shear wave velocity and 14
density of the source rock. The lower corner a frequency relates to the source duration.
15
The term b is the higher corner frequency at which the spectrum attains half of the high
16
frequency amplitude level, and is the weighting parameter. The moment magnitude M is
17
mapped from the seismic moment M0 (dyne-cm). The geometrical spreading function 18
RG is characterized by an empirical formulas well supported by data of distance range 19
from 10 to 1000 km with R R02he2 , R0 the epi-central distance and he the source
1
depth. The term P ,
R is the path-dependent attenuation factor and is dependent on 2propagation velocity. The diminution factor D(ω) accounts for the path-independent 3
attenuation of high-frequency waveforms and can be represented by the Kappa-filter. The 4
amplification factor A
is approximated by the source-to-site impedance ratio in a 5numerical scheme using the quarter-wavelength approximation method. 6
4. Wavelet-Based Modelling of Spatially-Varying Ground Motions Including Wave-7
Passage and Site-Response Effects 8
A wavelet based modelling of spatially varying ground motions including wave 9
passage and site response effects has been proposed in this section. In wavelet analysis, a 10
time series u(t) is represented as a composition of several time-localized shifted and scaled 11
wavelets (so-called the baby wavelets) a,b(t) of a basic wavelet (t), where
12
a b t a t b a , 1 (14)The parameter b is localizing the basis function at t = b and its neighbourhood, and the 13
parameter a controls the frequency content of the basis function by stretching or 14
compressing it. The discrete wavelet transform (DWT) has been used to simulate ground 15
motions (Iyama and Kuwamura 1999). Although the DWT is the most efficient and 16
compact, its power-of-two relationship in scale fixes its frequency resolution (Gurley and 17
Kareem 1999). Thus, the continuous wavelet transform (CWT) that allows more closely 18
spaced scaling than the 2i relationship is used in this paper. The CWT convolves the signal 19
u(t) with a set of baby wavelets as (Basu and Gupta 1997, 1998, 2000)
20
dt a b t t u a b a u W
1 * , (15)where (*) denotes the complex conjugate. Eq. (15) gives the localized frequency 1
information of u(t) around tb. W
maps a finite energy signal from the time domain 2to a finite energy two-dimensional distribution in the scale-translation domain. 3
A set of differential non-stationary ground motions ugr
t , r = 1, …, Ns, at Ns4
supports of a multi-span structure is considered. In practice, ugr
t is an evolutionary 5random process and can be expressed as (Priestley, 1981) 6
r t i r grt A t e dG u
, , (16)where Ar
t,
is a slowly varying time- and frequency-dependent modulation and dGr()7
is an orthogonal increment process associated with the rth support such that 8
dGrdGr*'
0, ' E (17)
dG
S
d E r 2 rr (18) In Eq. (18), Srr() is the two-sided PSDF of the stationary part of the random process and9
has been formulated in Eq. (9). The evolutionary random process in Eq. (16) can be 10
transformed in wavelet domain by using Eq. (16) and the wavelet coefficients at a 11
discretized scale aj, can be expressed as (Chakraborty and Basu 2008)
12
ugraj b Arjb ei bdGr W , ~ (19)where orthogonal increment process dG~r
satisfies (Spanos and Failla 2004) 13
dG~rdG~r*'
0, ' E (20a)
a
a S
d G d E ~r 22 j ˆ j 2 rr (20b)The function Arj
b represents the amplitude modulation for ugr
t, at a scale aj that can1
be the extended Shinuzoka-Sato amplitude modulation (Shinozuka and Sato, 1967) given 2 as 3
r t t j r j r j r j e e b A (21)in which rj, rj and rj are the parameters of the amplitude modulation for the ground 4
motion at the jth band of frequency and at the rth support. 5
Multiplying both sides of Eq. (19) by the complex conjugate corresponding to 6
another support l as being carried out by Chakraborty and Basu (2008) and considering the 7
frequency-dependent time lags due to site-response effects, the cross correlation of the 8
wavelet coefficients of seismic ground motions at two support r and l and at a scale aj is
9
ugraj,bW uglaj,b Arjb Ajrb trlwp trlsite EdG~r dG~l* W E (22)where trlwp and trlsite
are the time lag due to wave-passage effect and frequency-dependent 10time lag due to site-response effects, respectively, presented in Eqs. (4) and (8), and 11
dG~rdG~l*'
0, ' E (23a)
dG dG
a
a S
d E r l j j rl 2 * 2 ˆ ~ ~ (23b) The cross-spectral density function (CSDF) between the ground motions at two 12supports r and l is modelled as (Der Kiureghian 1996, Zerva and Zervas 2002) 13
rl rr llrl S S
S (24)
In this paper, the earthquake energy transmitted to each support r is completely 14
represented by its stationary PSDF Srr() in Eq. (9). Thus, the energy of the modulation
15
must be unit-normalised before convoluting with the power spectral densities in Eq. (19). 16
The energy content, IA, of a frequency-dependent modulation before normalizing A(t,) is 1 given by 2
N k N n n k T A At d dt At t I t 1 1 2 0 0 2 , , (25)The energy content is expressed in a band-dependent form as 3
nb t j N n j n j A N A t t I 1 1 2 (26)where Nj is the number of discrete frequencies in jth band and Nt is number of time
4
intervals. Hence, the unit-energy normalised amplitude modulation at a band j is given by 5
A n j n j I t A t A (27)5. Wavelet-Based Evolutionary Responses of Multi-Span Structures Subjected to 6
Differential Support Motions Including Wave-Passage and Site-Response Effects 7
5.1 Formulation for Calculation of Evolutionary Responses 8
A formulation for calculating the evolutionary response including wave passage 9
and site response effects is derived in this section. Consider a structure having N degrees of 10
freedom (DOFs) and Ns supports subjected to spatially-varying excitation time histories
11
tugr , r = 1, …, Ns. The structure is modelled in a finite-element (FE) framework leading
12
to a discrete dynamical system model. Using the consistent mass matrix approach and an 13
assumption that the effect of entire velocity-damping coupling is negligible in comparison 14
to that of the inertia, the motion equations of the structure is given by (Clough and Penzien 15 2003) 16
g
ug Cu Ku ME M u M (28)where, u(t) represents the displacement vector relative to the support motions and M, C, 17
and K are the system NN mass, damping and stiffness matrices, respectively. In Eq. (28), 18
the NNs influence coefficient matrix E, whose kth column represents the displacements at
1
the unconstrained DOF when a support DOF is displaced by a unit amount while all other 2
support DOFs remain fixed, is expressed as EK-1Kg. The NNs matrices Mg and Kg
3
account for the coupling of the inertia and stiffness between structural DOFs and ground 4
motion DOFs. 5
Using modal transformation y = z where (i) y = <u u>T and the complex 2N2N 6
eigenvector for non-proportional damping case and (ii) y u and the real NN 7
eigenvector for proportional damping case, the uncoupled form of Eqs. (28) for the two 8
damping cases are, respectively (Chakraborty and Basu 2008) 9
t u z z Ns gr r r k k k k
1 , k = 1, …, 2N (29)
t u z z z Ns gr r r k k k k k k k
1 2 2 , k = 1, …, N (30)Transforming Eqs. (29) and (30) by a wavelet basis a,b(t) and using Eq. (19) and the
10
relations b
Wzk
aj,b
Wzk
aj,b and 2 b2
Wzk
aj,b
Wzk
aj,b gives 11
rj i b r N r r k j k k j ka b W z a b A b e dG z W b s ~ , , 1 (31a)
rj i b r N r r k j k k j k k k j k W z a b W z a b A b e dG b b a z W b s ~ , , 2 , 1 2 2 2 (31b)Solving Eqs. (31a) and (31b) that represent the equations of motion in wavelet domain, by 12
using Duhamel’s integral gives 13
rk i b r N r r k j ka b M be dG z W , s j , ~ 1
(32) where 14
b h
b A e
d M r i b j b k k rj
0 , (33)Using time-localization around t = b of the wavelet transform and the less oscillatory 1
nature of the band-dependent envelope function Arj
compared to the unit impulse 2response function hk
, Eq. (33) can be approximated as 3
b A b h b e
d A
bHMrkj rj b k i b rj
0, (34)
where Hk()is the conventional frequency-response function in the kth mode, which is
4
given for non-proportional damping and proportional damping cases as, respectively 5
k k i H 1 (35a)
i H k k k k 2 1 2 2 (35b)Taking wavelet transform of the modal transformation gives 6
Nd k pk k j j pa b W z a b u W 1 , , , (36)where Wup(aj, b) is the wavelet coefficient of relative displacement at pth DOFs and Nd is
7
the number of modes considered. Multiplying both sides of Eq. (36) by its complex 8
conjugate and applying expectation operator gives the second-order moment of the relative 9
displacement along pth DOF as 10
EW z a bW z a b b a u W E N k j m j k N m pk pm j p d d , , , * 1 1 , , 2 2 (37)Using the expressions of Wzk(aj, b) and Wzm(aj, b) in Eqs. (32) and (33) and the cross
11
correlation of the orthogonal incremental processes in Eq. (23b) while considering the 12
frequency-dependent time-lags due to site-response effects, Eq. (37) is simplified as 13
Wu a b
A
b f d E N jr j r N l l m r k N k N m pk pm j p s s d d
0 1 1 1 1 , , 2 2 , 2 (38) where 14
2
2 ˆ . 2 site k rl j rl wp rl l j j j a A b t t H S a f (39)The termsAjr
b and
site rl wp rl l j b t tA are the unit-energy normalised amplitude 1
modulations for ugr
t and u
t lg
at a scale j, respectively. By using the second-order 2
moments obtained from Eq. (38), the EPSD of the relative displacement along pth DOF 3
can be estimated (Spanos and Failla 2004). 4
Due to the time lags considered, for seismic waves already arrived at support r but 5
yet to arrive at support l, the modulation intensity at the latter support should be zero, i.e. 6
rlwp rlsite
0 l jb t t A if btrlwptrlsite
(40)The instantaneous mean-square value of a time-dependent process (Basu and Gupta 1998) 7
is used to compute that of the relative displacement along pth degree of freedom as 8
a i m j j i j p b t p a b a u W E K t u E 2 2 2 , (41)where ma is number of frequency bands and the term K is expressed as
9 C K 4 1 2 (42)
In Eq. (42), is a parameter for discrete representation of the scale aj = j and
10
d C 2 ˆ (43)5.2. Wavelet Basis Function 11
Although in theory the proposed stochastic seismic evolutionary response are 12
applicable for any wavelet basis function satisfying the admissibility criterion in Eq. (43), 13
the choice of the basic function is important for the efficiency and accuracy in 14
computation. Besides, the analysis resulted from CWT relies heavily on the scale 15
discretizations and the selected frequency range (Kijewski-Correa and Kareem 2006). 1
Several wavelet basis functions were shown advantageous in characterizing ground 2
motions such as the Mexican hat wavelets (Zhou and Adeli 2003) and Harmonic wavelets 3
Spanos et al. (2005). Tratskas and Spanos (2003) modeled the nonstationary base-4
excitations and estimate the stochastic evolutionary responses by using harmonic wavelets. 5
Harmonic wavelets were also used by Spanos and Kougioumtzoglou (2012) to compute 6
statistically-linearized evolutionary responses of nonlinear oscillators subject to stochastic 7
excitation. The modified LittleWood-Paley (MLP) basis function (Basu and Gupta 1998) is 8
used in this paper as it provides high accuracy in spectral analysis (Spanos and Failla 2004) 9
and advantages in numerical computation by enabling energy computation of any signal 10
with non-overlapping frequency bands. The MLP wavelet basis pair in time and frequency 11 domain is given by 12
t
t F t F F t 1 1 1 2 sin 2 sin . 1 2 1 (44)
1
F 4 1 | ˆ | 1 , 1 1 2 F F = 0 otherwise (45)whereF1 is the initial cut-off frequency of the mother wavelet. If F1 = 0.5 Hz, Eqs. (44-45) 13
are reduced to the original forms of MLP basis function (Basu and Gupta 1998) as 14
t t t t .sin sin 1 1 (46)
1 2 1 | ˆ | , = 0 otherwise (47)The scaled Fourier transform is 15
1 4 1 | ˆ | 1 F aj when j j a F a F 1 2 1 2 = 0 otherwise (48)The admissibility criterion coefficient,C , in Eq. (43) becomes 1
1 2 ln 1 1 2 1
d C (49)It is noted by Basu and Gupta (1998) that
21n, n 4 is found reasonable based 2on investigations on several ground motions recorded. However, since a small value of 3
leads to increased computational effort, a value of
214 has been chosen (Basu and 4Gupta 1998, 2000, Spanos and Failla 2004). A higher value of can also be chosen in case 5
of ground motion with relatively smooth Fourier spectra. 6
6. Numerical Example 7
An application of the proposed theory and derived formulations in this paper realting 8
to evolutionary response of structures with wave passage and site response effects is 9
presented in this section. In order to illustrate the wave-passage and the site-response 10
effects on the stochastic evolutionary responses, multi-span structures that exhibit 11
considerable vertical and horizontal responses should be examined. A three-span two-12
dimensional frame of a hangar shown in Fig. 3 is therefore considered. The cross sectional 13
area and moment of inertia of the columns are Ac = 1.2 m2and Ic = 0.144 m4, respectively
14
and those of the beams are Ab = 2.0 m2 and Ib = 0.667 m4, respectively. The frame material
15
parameters are: Elastic modulus Eb = 2.01011 N/m2, Poisson ratiob= 0.29, mass density
16
b = 7860 kg/m3, and the modal damping ratios for the first two modes 1 = 2 = 0.02. The
17
first five natural frequencies of the frame are 6.72, 10.02, 12.07, 15.67 and 33.96 rad/s. 18
Geological profile beneath the supports is shown in Fig. 3 and is given in Table 1. 19
Using the procedure presented in Section 3, the estimated parameters of Clough-20
Penzien (C-P) and Kanai-Tajimi (K-T) FRFs compatible to the sites beneath the supports 1
are shown in Tables 2 and 3. In these tables, R-Square stands for coefficient of 2
determination and RMSE for root mean squared error (standard error). The estimation of 3
parameters for Clough-Penzien (C-P) spectrum is better as its RMSE values are smaller 4
and R-Square values are larger and greater than 0.5. 5
The stationary PSDs at the supports calculated by Eq. (9) using parametric K-T FRF 6
and C-P FRF are shown in Figs. 4(a) and 4(b), respectively. The stochastic processes 7
corresponding to these PSDs are used as orthogonal processes at different supports for 8
wavelet-based modelling of spatially-varying ground motions employing Eq. (22) and for 9
wavelet-based evolutionary response analyses of the frame employing Eq. (38). The 10
variation of frequency-dependent time lags between the left support and other supports due 11
to site-response effects calculated by using Eq. (8) are shown in Fig. 5. The time lags 12
computed by using K-T parametric FRF vary by a moderate amount around the frequency 13
of 4 rad/s whereas the time lags computed by using C-P parametric FRF vary dramatically 14
around a lower frequency of 1 rad/s. This fluctuation in the time lags in the C-P FRF case 15
around a lower frequency may have been resulted due to the additional lower frequency 16
filter (Eq. (7)) which in some cases may be a more realistic representation. The C-P FRF is 17
therefore used in the computation for structural responses in the following example even 18
though the K-T FRF could have been used in the computation with equal ease. 19
The influence of site-response effects on the amplitude and frequency non-20
stationarity of the frame relative displacement responses have been examined. Figs. 6(a) 21
and 6(b) show the root-mean-square (RMS) values of the relative vertical displacement at 22
the midpoints of the left span and mid-span calculated by using Eq. (41). When only wave-23
passage effect is considered the RMS values decrease and attenuate faster in time than 24
those when both wave-passage and site-response effects are considered. Similar influence 25
of site-response effects on the relative horizontal displacements at the top of the first and 1
second columns can be observed in Figs. 7(a) and 7(b) respectively. The site-response 2
effects are also shown to alter the amplitude non-stationarity of frame displacements. 3
The EPSDs of the relative vertical displacements at the midpoints of the left span and 4
the mid span using the parametric C-P FRF and Eq. (38) are shown in Figs. 8 and 9, 5
respectively. Figs. 10(a) and 10(b) show the corresponding PSDs at t = 5 sec. The effects 6
of site-response on the amplitude non-stationarity and on reducing the rate of decay of the 7
RMS response envelope values in time as seen in Fig. 6(b) are also observed in Figs. 8(b) 8
and 9(b). Figs. 10(a) and 10(b) show that compared to the case with only wave-passage 9
effect considered, the frequency content are more non-stationary and the peaks of the 10
instantaneous PSDs are higher in the case of combined wave-passage and site-response 11
effects. In addition, the attenuation of the response energy is slower in time. Similar trends 12
are observed in the EPSD of the column relative horizontal displacements in Fig. 11 and 13
the PSDs at t = 5 sec in Fig. 12. 14
7. Conclusions 15
A wavelet-based evolutionary response formulation of multi-span structures supported on a 16
soil medium and subjected to spatially-varying differential support motions including 17
wave-passage and site-response effects has been proposed in this paper. The spatial-18
variability of support motions is formulated by bedrock parametric coherency models, the 19
time lags due to wave-passage effects and a proposed alternate way to represent site 20
response effects by frequency-dependent time-lags. The earthquake energy content is 21
properly characterized by a composite PSDF constituting of the parametric surface-to-22
bedrock FRF and the bedrock power spectrum. The site-compatible parametric FRFs are 23
proposed to be characterized in this paper by carrying out a FEA of soil media beneath the 24
supports. In an illustrative case study, a three-span two-dimensional hangar frame is 25
analyzed using the proposed formulations. The time lags due to site-response effect and 1
computed from different FRFs show different variation in trend. The site-response effect 2
adds frequency non-stationarity to the frame responses and results in an increase of such 3
responses with slower attenuation in time. The formulation in this paper provides a more 4
accurate seismic analysis of long-span multi-support structures as it accounts for the non-5
stationaries in both amplitude and frequency of excitations, properties and variation of soil 6
media beneath the supports and a representation of realistic earthquake energy content. The 7
proposed formulations are generally applicable for any wavelet basis function and 8
structures. 9
Acknowledgements 10
This research is carried out partially funded under the EU FP7 funding for the Marie Curie 11
IAPP project NOTES (Grant No. PIAP-GA- 2008-230663). The authors are grateful for 12
the support. The authors thank the anonymous reviewers who have given valuable 13
comments. 14
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