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30.1 Introduction

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Seismic computations of long-span structures have long been an issue of great concern. Such

computations are usually executed numerically using schemes in the time domain (i.e., time history). For

short-span bridges, all supports can be assumed to move uniformly and the response-spectrum method

(RSM) is a suitable computation tool. For long-span bridges, however, various spatial effects such as the

wave-passage effect, the incoherence effect, the local site effect, and so on, may be important. Such spatial

effects cannot be dealt with directly by the conventional RSM. Instead, the time-history method (THM)

is the most widely used method for these systems. The time-history scheme requires solving the dynamic

equations for a number of seismic acceleration samples. The results are then processed statistically to

produce the quantities required by the designs. This process is rather complex and requires a considerable

computational effort. As a result, more efficient and effective methods are under investigation.

Seismic motions are random (stochastic) in nature (Housner, 1947). Spatial effects of long-span

bridges can be analyzed using the random-vibration approach (see also Chapter 5). In the last two

decades, many scholars and experts (Lee and Penzien, 1983; Dumanoglu and Severn, 1990; Lin et al.,

1990; Berrah and Kausel, 1992; Kiureghian and Neuenhofer, 1992; Ernesto and Vanmarcke, 1994) have

made great progress in promoting the seismic random analysis of long-span structures and its

engineering applications. Although available computational methods still need further improvement

both in precision and efficiency, the random-vibration approach, as a theoretically advanced tool, has

been gradually accepted by the earthquake engineering community. For example, it has been adopted by

the European Bridge Code (European Committee for Standardization, 1995).

Developed recently (Lin, 1992; Lin et al., 1994a, 1995a, 1995b, 1997a, 1997b; Lin and Zhan, 2003), the

pseudoexcitation method (PEM) is an accurate and highly efficient approach to the stationary and

nonstationary random seismic analysis of long-span structures. For typical three-dimensional finite

element models of long-span bridges with thousands of degrees of freedom (DoF) and dozens of

supports, when using 100 to 300 modes for mode-superposition analysis, the seismic responses can be

implemented quickly and accurately on a standard personal computer. Numerical results show that the

wave-passage effect is of particular importance for the seismic analysis of long-span bridges, and the

incoherence effect is of comparatively less importance. The details will be given in this chapter. The PEM

has been successfully applied to some practical engineering analyses (Wang et al., 1999; Liu and Liu, 2000;

Xue et al., 2000; Fan et al., 2001), and has been proven to be quite effective.

30.1.1 Basic Concepts of Random Vibration

30.1.1.1 Stationary Random Process

The probabilistic properties of stationary random processes are independent of time. A random process

is said to be strictly stationary (or strongly stationary) if its probability density function does not change

with time. However, such a condition is very difficult to satisfy in practical engineering problems.

Therefore, a wide-sense stationary (or weakly stationary) process is defined for which only the mean value

and autocorrelation function of the process are not permitted to vary with time.

A random variable x is said to be Gaussian-distributed if its probability density can be written in the

form

pðxÞ ¼

ffiffiffiffi

p2p exp 2 ðx 2 􀀊xÞ2

2s2

􀁻 !

ð30:1Þ

in which s is the standard deviation of x; and the variance is given by

s2 ¼

ð1

21 ðx 2 x􀀊Þ2pðxÞdx ð30:2Þ

30-2 Vibration and Shock Handbook

Probability density functions of typical Gaussian random processes are shown in Figure 30.1, in which

x􀀊 is the mean value given by the abscissa (horizontal coordinate) of the peak value. A smaller s

corresponds to a narrower and higher peak.

For a random process xðtÞ; if the joint probability density function of its values xðt1Þ; xðt2Þ; …; xðtnÞ at n

arbitrary time instants is Gaussian, then xðtÞ is said to be a Gaussian random process. Since the joint

probability density function depends only on the mean values and covariances of the n values, a weakly

stationary Gaussian random process is also strongly stationary.

If a stationary random process has statistical properties that can be computed by taking the time

average of an arbitrary sample over a sufficiently long period, the process is said to be ergodic. A typical

seismic ground motion record is usually assumed to be a Gaussian and ergodic stationary random

process. Its expected value (i.e., mean value) can be computed by

E½xðtÞ􀀉 ¼ x􀀊ðtÞ ¼

ðþ1

xðtÞpðx; tÞdx ð30:3Þ

in which pðx; tÞ is the probability density function of xðtÞ:

In order to investigate the relation between the values of a random process xðtÞ at two different times,

the autocorrelation function of xðtÞ is defined as

Rxx ðtÞ ¼ E½xðtÞxðt þ tÞ􀀉 ¼ lim

T!1

ðT=2

2T=2

xðtÞxðt þ tÞdt ð30:4Þ

A stationary random process, denoted as xðtÞ; is not absolutely integrable in a region of t [ ð21; 1Þ:

Therefore, a subsidiary function xT ðtÞ is defined:

xT ðtÞ ¼

xðtÞ when 2 T=2 # t # T=2

0 elsewhere

(

ð30:5Þ

Obviously, xT ðtÞ is absolutely integrable within t [ ð21; 1Þ: Therefore, its Fourier transformation (see

Appendix 2A and Chapter 10) can be computed by

XT ðvÞ ¼

ð1

xT ðtÞexpðjvtÞdt ð30:6Þ

Let

Sxx ðvÞ ¼ lim

T!1

lXT ðvÞl2 ð30:7Þ

Equation 30.7 is the definition of the auto-PSD (power spectral density) function of xðtÞ:

p(x)

0.0

0.1

0.2

0.3

0.4

0.5

−4 −3 −2 −1 0 1 2 3 4 5 6 7 8

s = 0.5

s = 1.0

s = 2.0

FIGURE 30.1 Probability density functions of Gaussian random variables.

Seismic Random Vibration of Long-Span Structures 30-3

Note that the repeated subscripts in Rxx or Sxx can be represented by just one; that is, they can be

denoted as Rx or Sx : When xðtÞ is a zero-mean stationary random process, its variance is given by

x ¼

ð1

Sxx ðvÞdv ð30:8Þ

Figure 30.2 gives the auto-PSD curves of four typical stationary random processes, which show the

energy distribution with frequency for each kind of random process. The energy of a narrowband

random process is concentrated within a narrow frequency band (see Figure 30.2b) whereas the energy of

a wideband random process is distributed over a rather wide frequency range, as shown in Figure 30.2c.

The energy of a white noise process is distributed uniformly over an infinite region, v [ ð21; 1Þ; as

shown in Figure 30.2d. Using such a random process model results in mathematical convenience.

However, the white noise process does not physically exist. A single harmonic random wave has nonzero

values only at two isolate frequencies ^v0 (see Figure 30.2a), and its initial phase angle w is usually

regarded as uniformly distributed over ½0; 2pÞ:

Sxx(w)

−w0 w0

Wide-banded

White noise

x(t) = x0 sin(w0t+f)

x(t) Narrow-banded

x(t)

(a)

(b)

(c)

(d)

FIGURE 30.2 Auto-PSD curves of four typical stationary random processes.

30-4 Vibration and Shock Handbook

The auto-PSD Sxx ðvÞ of a stationary random process xðtÞ has the following properties:

1. Sxx ðvÞ is a nonnegative real number; that is

Sxx ðvÞ $ 0 ð30:9Þ

This can be judged from the definition of auto-PSD functions, that is, Equation 30.7.

2. Sxx ðvÞ is an even function; that is

Sxx ðvÞ ¼ Sxx ð2vÞ ð30:10Þ

This can also be deduced from Equation 30.7.

3. The auto-PSDs of the derivatives of xðtÞ can be computed from Sxx ðvÞ directly by

Sx_x_ ðvÞ ¼ v2SxxðvÞ; Sx€x€ ðvÞ ¼ v4Sxx ðvÞ ð30:11Þ

30.1.1.1.1 Wiener – Khintchine Theorem

Wiener and Khintchine proved that for an arbitrary stationary random process xðtÞ; its auto-PSD Sxx ðvÞ

and autocorrelation function Rxx ðtÞ are a Fourier transform pair; that is

Sxx ðvÞ ¼

ð1

Rxx ðtÞexpð2jvtÞdt ð30:12Þ

Rxx ðtÞ ¼

ð1

Sxx ðvÞexpðjvtÞdv ð30:13Þ

According to this theorem, if either of Sxx ðvÞ or Rxx ðtÞ has been found, the other can be directly

obtained.

If stationary random processes xðtÞ and yðtÞ are both ergodic, then their cross-correlation function can

be computed in terms of their sample functions x^ðtÞ and y^ðtÞ:

Rxy ðtÞ ¼ lim

T!1

ðT=2

2T=2

x^ðtÞy^ðt þtÞdt ð30:14Þ

Ryx ðtÞ ¼ lim

T!1

ðT=2

2T=2

y^ðtÞx^ðt þtÞdt ð30:15Þ

Also, their cross-PSD functions can be defined by means of the Fourier transforms of the corresponding

cross-correlation functions:

Sxy ðvÞ ¼

ð1

Rxy ðtÞexpð2ivtÞdt ð30:16Þ

Syx ðvÞ ¼

ð1

Ryx ðtÞexpð2ivtÞdt ð30:17Þ

For more details, see Lin (1967).

30.1.1.2 Nonstationary Random Process

Nonstationary random processes are generally short in duration. Their basic characteristic is that the

statistical properties vary significantly with time. An example is the process of a typical earthquake

record, for which the medium flat segment is often regarded as a stationary random process in order to

simplify the structural analysis. However, such simplification sometimes causes significant errors. For

instance, some long-span bridges are very flexible, with fundamental periods of approximately 15 to

20 sec. The period of the strong earthquake portion of a typical earthquake record is only approximately

20 to 30 sec. For such slender long-span bridges, the seismic excitations exhibit clear nonstationary

characteristics. In order to avoid computational complexities in the structural analyses, such excitations

are usually assumed to be stationary random processes. This chapter shows that the analysis of such

nonstationary random responses is made very simple by using PEM.

Seismic Random Vibration of Long-Span Structures 30-5

Nonstationary random processes are not ergodic because their statistical properties vary with time. In

earthquake engineering, the evolutionary random process defined by Priestly (1967) has been

investigated extensively. It is expressed in terms of the Riemann – Stieltjes integration as

f ðtÞ ¼

ð1

Aðv; tÞexpðivtÞdaðvÞ ð30:18Þ

in which aðvÞ satisfies the relations

xðtÞ ¼

ð1

expðivtÞdaðvÞ ð30:19Þ

E ½dapðv1Þdaðv2Þ􀀉 ¼ Sxx ðv1Þdðv2 2 v1Þdv1 dv2 ð30:20Þ

Here, xðtÞ is a zero-mean stationary random process, with auto-PSD Sxx ðvÞ; Aðv; tÞ is a deterministic

slowly varying nonuniform modulation function, and d is a Dirac delta function. The variance of f ðtÞ is

s2f

ðtÞ ¼

ð1

Sff ðvÞdv ¼

ð1

lAðv; tÞl2Sxx ðvÞdv ð30:21Þ

The PSD of f ðtÞ as given by

Sff ðv; tÞ ¼ lAðv; tÞl2Sxx ðvÞ ð30:22Þ

is known as an evolutionary power spectral density function.

Responses of structures subjected to nonstationary random excitations expressed by Equation 30.18

are not easy to compute. Therefore, the nonuniform modulation assumption is often replaced by a

uniform modulation assumption; that is, the nonuniform modulation function Aðv; tÞ is replaced by a

uniform modulation function gðtÞ: Thus, Equation 30.18 reduces to

f ðtÞ ¼

ð1

gðtÞexpðivtÞdZðvÞ ¼ gðtÞxðtÞ ð30:23Þ

x(t)

f(t) = g(t)x(t)

g(t)

(a)

(b)

(c)

FIGURE 30.3 A uniformly modulated evolutionary random excitation f ðtÞ:

30-6 Vibration and Shock Handbook

Equation 30.18 and Equation 30.23 are known as the nonuniformly modulated and uniformly

modulated evolutionary random processes, respectively.

Figure 30.3 shows a stationary random process xðtÞ and the corresponding uniformly modulated

evolutionary random excitation f ðtÞ with a given modulation function gðtÞ:

30.1.2 Three Methods for Structural Seismic Analysis

30.1.2.1 Response Spectrum Method

The equations of motion of a linear multi-DoF structure subjected to a ground acceleration excitation

x€g ðtÞ can be written as (see Chapter 3 and Chapter 8)

My€ þ Cy_ þ Ky ¼ 2Mex€g ðtÞ ð30:24Þ

in which M, C, and K are the n £ n mass, damping, and stiffness matrices of the structure, and e is the

index vector of inertia forces. For short-span structures, all supports can be assumed to move uniformly

with the same ground acceleration x€g ðtÞ: If the structure under consideration has a very large number

of DoF, Equation 30.24 can be solved by using the mode-superposition scheme (see Chapter 3). First, the

lowest q natural angular frequencies vj ð j ¼ 1; 2; …; q; q ,, nÞ and the corresponding n £ q mass

normalized mode matrix F should be extracted. Then, yðtÞ can be decomposed in terms of these modes:

yðtÞ ¼ FuðtÞ ¼

j¼1

ujwj ð30:25Þ

With proportional damping assumed (see Chapter 3 and Chapter 19), Equation 30.24 can be decoupled

into q single-DoF equations

u€ j þ 26jvju_ j þv2j

uj ¼ 2gjx€g ðtÞ ð30:26Þ

in which 6j is the jth damping ratio and gj is the jth modal participation factor

gj ¼ wT

j Me ð30:27Þ

According to the response spectrum theory, the solution of Equation 30.26 is

uj ¼ gjajg=v2j ð30:28Þ

in which g is the gravity acceleration and aj is the value of the ground acceleration response spectrum

(ARS) at frequency vj: If the kth element of y, denoted as yk; is required, then the kth elements of all

yjð j ¼ 1; 2; …; qÞ are taken to compose a vector yk; which is then used in the computation of the response

(or demand) yk:

yk ¼

ffiffiffiffiffiffiffiffiffi

k Rcyk

ð30:29Þ

Here, Rc is the correlation matrix representing the degree of correlation between all participating modes.

Based on random-vibration theory, Wilson and Kiureghian (1981) have derived the expression for its

elements as

Rij ¼

ffiffiffiffiffi

zizj

ðzi þ rzjÞr3=2

ð1 2 r2Þ2 þ 4zizjrð1 þ r2Þ þ 4ðz2i

þ z2j

Þr2 ð30:30Þ

in which r ¼ vj=vi: This is the widely used Complete Quadratic Combination (CQC) algorithm in the

RSM. If the correlation coefficients between all modes are neglected, that is, if Rij ¼ dij (Dirac delta

function), then Rc becomes a unity (identity) matrix and Equation 30.29 reduces to the square root of

the sum of squares (SRSS) algorithm.

The RSM, as outlined above, is very popular in the seismic analysis of short-span structures. Some

extensions have been published (Lee and Penzien, 1983; Dumanoglu and Severn, 1990; Lin et al., 1990;

Seismic Random Vibration of Long-Span Structures 30-7

Berrah and Kausel, 1992; Kiureghian and Neuenhofer, 1992; Ernesto and Vanmarcke, 1994) in order to

deal with the seismic analysis of long-span structures. However, the efficiency and accuracy still need

further improvement before they can be widely accepted in engineering practice.

30.1.2.2 Time-History Method

Assume that all supports move uniformly with the same acceleration x€g ðtÞ; which is now given in a

discrete numerical form. Equation 30.24 can now be solved using the Newmark method, the Wilson-u

method (Clough and Penzien, 1993), or the precise integration method (Zhong and Williams, 1995). In

these THMs, the structural parameters can be modified at any time. Therefore, this method is good for

nonlinear problems for which structural parameters often vary with time, for example in seismic elastoplastic

analysis. A major disadvantage of THMs is that the computational results rely heavily on the

selected ground acceleration records. In general, a number of records must be selected for structural

analyses, and statistical results are then used in the designs. In order to reduce the computational effort,

usually only about three to ten records are used for statistical purposes.

When the wave passage effect needs to be taken into account, the same ground acceleration record is

applied to different supports with time lags and this generates x€b on the right-hand side of Equation

30.78. If the incoherence effect between the supports must also be considered, then the process for

generating x€b becomes rather complicated (Deodatis, 1990). In fact, real records of this type are difficult

to find.

30.1.2.3 Random Vibration Method

The random vibration approach is appealing for seismic random analysis of long-span structures.

Previously, because of its high complexity and low efficiency, it was not accepted as a method of analysis

by practicing engineers. However, this situation has changed considerably in recent years. Let us still

begin with Equation 30.24, which we can also apply to structures subjected to uniform stationary

random ground excitations. Now x€g ðtÞ is a zero-mean Gaussian stationary random process with a known

auto-PSD SaðvÞ representing acceleration excitations uniformly applied to all supports of the structure.

By means of the modal superposition scheme, that is, Equation 30.25 to Equation 30.27, the traditional

CQC method can be established (Clough and Penzien, 1993):

Syy ðvÞ ¼

j¼1

k¼1

gjgkwjwT

k Hp

j ðvÞHkðvÞSaðvÞ ð30:31Þ

in which wj and gj are the jth mode and the jth modal participation factor, and

Hj ¼ ðv2j

2 v2 þ 2i6jvvjÞ21 ð30:32Þ

is the jth frequency-response function. For a real long-span bridge, the number of structural DoF n

usually ranges from 103 to 104, and the numbers of v and q typically range from 102 to 103. Equation

30.31 includes all quadratic terms of the participating modes, and it must be repeatedly computed for

dozens or hundreds of frequencies. Although it is a simple form of excitation, the computational effort is

still considerable. Therefore, in engineering practice, the following SRSS method obtained by neglecting

all j – k terms in Equation 30.31, is generally used in place of the above CQC method:

Syy ðvÞ ¼

j¼1

g2j

wjwT

j lHjðvÞl2SaðvÞ ð30:33Þ

This is frequently recommended in academic literature. The SRSS formula is an approximation of

Equation 30.31 that neglects the cross-correlation terms between participating modes, thereby reducing

the computational effort to about 1/q of that required by Equation 30.31. However, this approximation

can be used only for lightly damped structures for which the participating frequencies must be

sparsely spaced. For most structures (in particular their three-dimensional structural models), some

participating frequencies are often closely spaced. Hence, the applicability of the SRSS approximation is

30-8 Vibration and Shock Handbook

somewhat questionable. It will be seen in the next section that PEM will produce results identical to those

from Equation 30.31 with much less computational effort.

The random-vibration analysis outlined above is executed in terms of power spectral densities in the

frequency domain and therefore it is also referred to as the power-spectrum method.

A diagonal element Sjj in the PSD matrix represents the auto-PSD of a random response j: Assume

that this response is significant only within the frequency domain v [ ½vL; vU􀀉: Thus, the ith spectral

moment of j can be computed by

li ¼ 2

ð1

viSjj ðvÞdv < 2

ðvU

viSjj ðvÞdv ð30:34Þ

The PSD values at the negative frequencies do not have any intuitive physical significance and so the

single-sided PSD Gxx ðvÞ is defined for applications in many engineering fields:

Gxx ðvÞ ¼

2Sxx ðvÞ v $ 0

0 v , 0

(

ð30:35Þ

Thus, Equation 30.34 becomes

li ¼

ð1

viGjj ðvÞdv <

ðvU

viGjj ðvÞdv ð30:36Þ

For general multiple input (xðtÞ ¼ {x1ðtÞ; x2ðtÞ; …; xnðtÞ}T) and multiple output

(yðtÞ ¼ {y1ðtÞ; y2ðtÞ; …; ymðtÞ}T) problems (or MIMO problems), the response (i.e., output) PSD matrix

SyyðvÞ can be computed using the excitation (i.e., input) PSD matrix Sxx ðvÞ:

Syy ðvÞ ¼ HpSxx ðvÞHT ð30:37Þ

in which H is the frequency-response function matrix. Also, the cross-PSD matrices between the

excitations and responses can be computed from

Sxy ðvÞ ¼ Sxx ðvÞHT ð30:38Þ

Syx ðvÞ ¼ HpSxx ðvÞ ð30:39Þ

Equation 30.37 to Equation 30.39 have simple forms and are comparatively convenient for engineering

applications. However, they must be executed for dozens or hundreds of discrete frequencies. For

complex structures, such matrix operations may require extensive effort. PEM, which will be introduced

in the next section, is a better alternative than these equations.

If the first N modes are used in the modal superposition analysis, numerical tests for seven bridges

show that taking ½vL; vU􀀉 ¼ ½0:7v1; 1:2vN 􀀉 seems to be a good choice for the integration interval, where

v1 and vN are the first and the Nth natural angular frequencies of the structure.

It is inconvenient for engineers to take such spectral moments for practical designs. However, some

approaches have been suggested to estimate structural responses (or demands) in terms of these spectral

moments. Two popular approaches are described next.

30.1.2.3.1 Davenport Approach

With the seismic excitations assumed to be zero-mean stationary Gaussian processes, an arbitrary linear

response of the structure subjected to such excitations, denoted yðtÞ; will also possess the same

probability characteristic. It is also assumed that if a given barrier (threshold) a is sufficiently high, the

peaks of yðtÞ above this barrier will appear independently. Let NðtÞ be the number of upcrossing of a

within the time interval ð0; t􀀉; then NðtÞ will be a Poisson process with a stationary increment

(Davenport, 1961). Denote the extreme value of yðtÞ; that is, the maximum value of all peaks by their

absolute values, within the earthquake duration ½0; Ts􀀉 as ye; and the standard deviation of yðtÞ as sy :

Define h as the dimensionless parameter of ye; and n as the mean zero-crossing rate, which can be

Seismic Random Vibration of Long-Span Structures 30-9

expressed as

h ¼ ye=sy ; n ¼

ffiffiffiffiffiffiffi

l2=l0

p 􀀋

p ð30:40Þ

Based on these assumptions, the probability distribution of h can be derived as

PðhÞ ¼ exp ½2nTs expð2h2=2Þ􀀉; h . 0 ð30:41Þ

The expected value of h; known as the peak factor, is approximately given by

EðhÞ < ð2 ln nTsÞ1=2 þ g

􀀋

ð2 ln nTsÞ1=2 ð30:42Þ

and its standard deviation is

sh < p

􀀋

12 ln nTs

􀀄 􀀅1=2 ð30:43Þ

in which g ¼ 0:5772 is the Euler constant, while the expected value of ye is approximately

E½ye􀀉 ¼ E½h􀀉sy ð30:44Þ

This quantity is the demand usually required by engineers.

30.1.2.3.2 Vanmarcke Approach

In the preceding paragraph, the barrier a was assumed to be sufficiently high. Therefore, the peaks of yðtÞ

above this barrier will appear independently, and NðtÞ can be regarded as a Poisson process. Vanmarcke

(1972) considered that the barrier a should not be very high. Therefore, the Poisson process assumption

should be replaced by the two-state Markov process assumption and the probability distribution of h

becomes

PðhÞ ¼ 1 2 exp 2

" 􀁻 !#

exp 2nTs

1 2 exp

􀀄

ffiffiffiffiffi

p=2 p q1:2h

􀀅

expðh2=2Þ 2 1

" #

ð30:45Þ

in which n and Ts have the same meanings as the above, while the shape factor for the response PSD is

d0 ¼

􀀄

1 2 l21

=ðl0l2Þ

􀀅1=2 ð30:46Þ

Here, d0 is a bandwidth parameter with values ranging from zero to one. For a narrowband process, d0 is

close to zero. Based on the probability distribution function shown in Equation 30.45, Kiureghian (1980)

proposes the following approximate expressions for the peak factor EðhÞ and standard deviation sh when

10 # nt # 1000 and 0:11 # q # 1; which are of interest in earthquake engineering:

EðhÞ ¼ ð2 ln neTsÞ1=2 þ

ð2 ln neTsÞ1=2 ð30:47Þ

sh ¼

1:2

ð2 ln neTsÞ1=2 2

5:4

13 þ ð2 ln neTsÞ3:2 neTs . 2:1

0:65 neTs # 2:1

8><

ð30:48Þ

in which

ne ¼ ð1:63q0:45 2 0:38Þn0 d0 , 0:69

n0 d0 $ 0:69

(

ð30:49Þ

Gupta and Trifunac (1998) made numerical experiments to compare the above two models using 1000

simulated time-history excitations. Their research shows that for most practical purposes in earthquake

engineering studies, the effect of the dependence among level crossings is not significant.

30.1.2.4 Comparisons of the Three Seismic-Analysis Methods

The RSM is the most popular method for the seismic analysis of short-span structures. Some extensions

have been made to allow this method to be used in the seismic analysis of long-span structures.

30-10 Vibration and Shock Handbook

However, the accuracy and efficiency still need further improvement for practical applications. The THM

can be used in the seismic analysis of long-span structures without theoretical difficulties. However, its

major disadvantages are that it requires a good selection of the ground acceleration record samples and its

computational cost is very high. The random-vibration method is appealing because of its statistical

nature. In the past, the random vibration computation of complex structures has been very costly. This

issue has received much attention in the last two decades. As will be described below, PEM has

remarkably improved this situation. Now, long-span structures with thousands of DoF and dozens of

supports can be computed far more quickly and accurately on a personal computer.

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