Vibration and Shock Handbook 30

Vibration and Shock Handbook 30 Every so often, a reference book appears that stands apart from all others, destined to become the definitive work in its field. The Vibration and Shock Handbook is just such a reference. From its ambitious scope to its impressive list of contributors, this handbook delivers all of the techniques, tools, instrumentation, and data needed to model, analyze, monitor, modify, and control vibration, shock, noise, and acoustics. Providing convenient, thorough, up-to-date, and authoritative coverage, the editor summarizes important and complex concepts and results into “snapshot” windows to make quick access to this critical information even easier. The Handbook’s nine sections encompass: fundamentals and analytical techniques; computer techniques, tools, and signal analysis; shock and vibration methodologies; instrumentation and testing; vibration suppression, damping, and control; monitoring and diagnosis; seismic vibration and related regulatory issues; system design, application, and control implementation; and acoustics and noise suppression. The book also features an extensive glossary and convenient cross-referencing, plus references at the end of each chapter. Brimming with illustrations, equations, examples, and case studies, the Vibration and Shock Handbook is the most extensive, practical, and comprehensive reference in the field. It is a must-have for anyone, beginner or expert, who is serious about investigating and controlling vibration and acoustics.

30 Seismic Random Vibration of Long-Span Structures 30.1 Introduction 30-2 Basic Concepts of Random Vibration Structural Seismic Analysis † Three Methods for 30.2 Seismic Random-Excitation Fields 30-11 Power Spectral Density of Spatially Varying Ground Acceleration † Several Coherence Models † Generation of Ground Acceleration Power Spectral Density Curves from Acceleration Response Spectrum Curves † Seismic Equations of Motion of Long-Span Structures † Seismic Waves and Their Geometrical Expressions 30.3 Pseudoexcitation Method for Structural Random Vibration Analysis 30-16 Structures Subjected to Stationary Random Excitations Structures Subjected to Nonstationary Random Vibration † Precise Integration Method † 30.4 Long-Span Structures Subjected to Stationary Random Ground Excitations 30-27 The Solution of Equations of Motion Using the Pseudoexcitation Method † Numerical Comparisons with Other Methods 30.5 Long-Span Structures Subjected to Nonstationary Random Ground Excitations 30-34 Jiahao Lin Dalian University of Technology Yahui Zhang Dalian University of Technology Modulation Functions † The Formulas for Nonstationary Multiexcitation Analysis † Expected Extreme Values of Nonstationary Random Processes † Numerical Comparisons with the Corresponding Stationary Analysis 30.6 Conclusions 30-39 Summary Particular considerations must be made during the design of long-span bridges with regard to safety during earthquakes These include: the wave-passage effect caused by the different times at which seismic waves arrive at different supports; the incoherence effect due to loss of coherency of the motion caused by either reflections and refractions of the waves in the inhomogeneous ground medium or the difference in the manner of superposition of waves from an extended source arriving at various supports; and the local effect because of the differences in soil conditions at different supports and the manner in which these influence the amplitude and frequency content of the bedrock motion This chapter deals with the random vibration approach to analyzing these structures, which is based on a statistical characterization of the set of motions at the supports This approach is particularly suitable for dealing with the above spatially varying input motions The computational problems may be largely overcome by the pseudo excitation method This approach is presented here Numerical comparisons are given to show the 30-1 © 2005 by Taylor & Francis Group, LLC 30-2 Vibration and Shock Handbook accuracy of the method and its capability of dealing with the spatial effects and nonstationary effects Further topics related to this chapter are discussed in Chapter 30.1 Introduction 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 ! ðx xị2 pxị ẳ p exp 30:1ị 2s s 2p in which s is the standard deviation of x; and the variance is given by ð1 s2 ¼ x xị2 pxịdx 21 â 2005 by Taylor & Francis Group, LLC ð30:2Þ Seismic Random Vibration of Long-Span Structures 30-3 p(x) 0.5 s = 0.5 0.4 0.3 0.2 s = 1.0 s = 2.0 0.1 −4 −3 0.0 −1 −2 FIGURE 30.1 x Probability density functions of Gaussian random variables 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 xðt1 Þ; xðt2 Þ; …; xðtn Þ 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 ỵ1 Eẵxtị ẳ xtị ẳ xtịpx; tịdx 30:3ị 21 in which px; tị 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 dened as Rxx tị ẳ Eẵxtịxt ỵ tị ẳ lim T!1 T=2 xtịxt ỵ tịdt T 2T=2 30:4ị 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: ( xðtÞ when T=2 # t # T=2 xT tị ẳ 30:5ị elsewhere Obviously, xT ðtÞ is absolutely integrable within t [ ð21; 1Þ: Therefore, its Fourier transformation (see Appendix 2A and Chapter 10) can be computed by ð1 x tịexpjvtịdt 30:6ị XT vị ẳ 2p 21 T Let Sxx vị ẳ lim T!1 lX vịl2 T T Equation 30.7 is the definition of the auto-PSD (power spectral density) function of xtị: â 2005 by Taylor & Francis Group, LLC ð30:7Þ 30-4 Vibration and Shock Handbook 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 s2x ẳ 30:8ị Sxx ðvÞdv 21 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 ẵ0; 2pị: x(t) = x0 sin(w 0t+f) Sxx(w) t −w w0 ω (a) x(t) Narrow-banded Sxx(w) t ω (b) x(t) Wide-banded Sxx(w) ω t (c) x(t) White noise Sxx(w) t (d) FIGURE 30.2 Auto-PSD curves of four typical stationary random processes © 2005 by Taylor & Francis Group, LLC ω Seismic Random Vibration of Long-Span Structures 30-5 The auto-PSD Sxx ðvÞ of a stationary random process xðtÞ has the following properties: Sxx ðvÞ is a nonnegative real number; that is Sxx ðvÞ $ ð30:9Þ This can be judged from the definition of auto-PSD functions, that is, Equation 30.7 Sxx ðvÞ is an even function; that is Sxx ðvÞ ¼ Sxx ð2vÞ ð30:10Þ This can also be deduced from Equation 30.7 The auto-PSDs of the derivatives of xðtÞ can be computed from Sxx ðvÞ directly by Sx_ x_ vị ẳ v2 Sxx vị; Sx x vị ẳ v4 Sxx 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 1 Sxx vị ẳ 30:12ị R tịexp2jvtịdt 2p 21 xx Rxx tị ẳ 21 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^ tị: T=2 x^ tị^yt ỵ tịdt 30:14ị Rxy tị ẳ lim T!1 T 2T=2 Ryx tị ẳ lim T!1 T=2 y^ tị^xt ỵ tịdt T 2T=2 30:15ị Also, their cross-PSD functions can be defined by means of the Fourier transforms of the corresponding cross-correlation functions: ð1 Sxy vị ẳ 30:16ị R tịexp2ivtịdt 2p 21 xy Syx vị ẳ 1 R tịexp2ivtịdt 2p 21 yx 30:17ị 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 © 2005 by Taylor & Francis Group, LLC 30-6 Vibration and Shock Handbook 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 RiemannStieltjes integration as f tị ẳ Av; tÞexpðivtÞdaðvÞ ð30:18Þ 21 in which aðvÞ satisfies the relations xðtÞ ẳ 21 expivtịdavị 30:19ị E ẵdap v1 ịdav2 ị ¼ Sxx ðv1 Þdðv2 v1 Þdv1 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 tị is 1 Sff vịdv ẳ lAv; tịl2 Sxx vịdv s2f tị ẳ 30:21ị 21 21 The PSD of f ðtÞ as given by Sff ðv; tÞ ¼ lAðv; tÞl2 Sxx 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 ð1 ð30:23Þ f ðtÞ ¼ gðtÞexpðivtÞdZðvÞ ¼ gðtÞxðtÞ 21 x(t) (a) g(t) (b) f(t) = g(t)x(t) (c) FIGURE 30.3 A uniformly modulated evolutionary random excitation f tị: â 2005 by Taylor & Francis Group, LLC Seismic Random Vibration of Long-Span Structures 30-7 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 and Chapter 8) M€y ỵ C_y ỵ Ky ẳ 2Mexg tị 30:24ị 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: ytị ẳ Futị ẳ q X jẳ1 uj wj ð30:25Þ With proportional damping assumed (see Chapter and Chapter 19), Equation 30.24 can be decoupled into q single-DoF equations u j ỵ 26j vj u_ j ỵ v2j uj ẳ 2gj x g tị 30:26ị in which 6j is the jth damping ratio and gj is the jth modal participation factor gj ẳ wTj Me 30:27ị According to the response spectrum theory, the solution of Equation 30.26 is uj ẳ gj aj g=v2j 30:28ị 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 yj ð 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 : qffiffiffiffiffiffiffiffiffi yk ¼ yTk Rc yk 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 pffiffiffiffiffi zi zj ðzi ỵ rzj ịr3=2 30:30ị Rij ẳ r2 ị2 þ 4zi zj rð1 þ r2 Þ þ 4ðz2i þ z2j ịr in which r ẳ 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; © 2005 by Taylor & Francis Group, LLC 30-8 Vibration and Shock Handbook 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 Sa ðvÞ 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 vị ẳ q X q X jẳ1 kẳ1 gj gk wj wTk Hjp vịHk vịSa vị ð30:31Þ in which wj and gj are the jth mode and the jth modal participation factor, and Hj ¼ ðv2j v2 ỵ 2i6j vvj ị21 30:32ị 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 vị ẳ q X jẳ1 g2j wj wTj lHj vịl2 Sa vị 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 © 2005 by Taylor & Francis Group, LLC Seismic Random Vibration of Long-Span Structures 30-9 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 ðvU ð1 li ¼ vi Sjj ðvÞdv < vi Sjj ðvÞdv ð30:34Þ vL The PSD values at the negative frequencies not have any intuitive physical significance and so the single-sided PSD Gxx ðvÞ is defined for applications in many engineering elds: ( 2Sxx vị v $ Gxx vị ẳ 30:35ị v,0 Thus, Equation 30.34 becomes li ẳ vi Gjj ðvÞdv < ð vU vL vi Gjj vịdv 30:36ị For general multiple input (xtị ẳ {x1 tị; x2 ðtÞ; …; xn ðtÞ}T ) and multiple output (yðtÞ ¼ {y1 ðtÞ; y2 ðtÞ; …; ym ðtÞ}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 vị: Syy vị ẳ Hp Sxx vịHT 30:37ị in which H is the frequency-response function matrix Also, the cross-PSD matrices between the excitations and responses can be computed from Sxy vị ẳ Sxx vịHT 30:38ị Syx vị ẳ Hp Sxx 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 ẵvL ; vU ẳ 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 © 2005 by Taylor & Francis Group, LLC 30-10 Vibration and Shock Handbook expressed as h ¼ ye =sy ; pffiffiffiffiffiffiffi n ¼ l2 =l0 p ð30:40Þ Based on these assumptions, the probability distribution of h can be derived as Phị ẳ exp ẵ2nTs exp2h2 =2ị ; h.0 30:41ị The expected value of h; known as the peak factor, is approximately given by Ehị < ln nTs ị1=2 ỵ g ln nTs Þ1=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 Eẵye ẳ Eẵh sy 30:44ị 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 " # !# " pffiffiffiffiffi exp p=2q1:2 h h2 PðhÞ ¼ exp exp 2nTs ð30:45Þ expðh2 =2Þ 2 in which n and Ts have the same meanings as the above, while the shape factor for the response PSD is d0 ¼ l21 =ðl0 l2 Þ 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: g Ehị ẳ ln ne Ts ị1=2 ỵ ð30:47Þ ð2 ln ne Ts Þ1=2 1:2 5:4 > < ne Ts 2:1 1=2 13 ỵ ln ne Ts Þ3:2 n T Þ ð2 ln ð30:48Þ sh ¼ e s > : 0:65 ne Ts # 2:1 in which ( ne ẳ 1:63q0:45 0:38ịn0 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 © 2005 by Taylor & Francis Group, LLC Seismic Random Vibration of Long-Span Structures 30-27 in which r1 and r2 are time-invariant vectors Substituting Equation 30.159 into Equation 30.139 enables the particular solution to be obtained (Lin et al., 1995a, 1995b; Zhong, 2004) as vp tị ẳ expatịv1 sin vt ỵ v2 cos vtị 30:160ị in which v ẳ aI H2 ị ỵ v2 Iị21 aI Hịr1 ỵ vr2 ị v ẳ aI H2 ị ỵ v2 IÞ21 ððaI HÞr2 vr1 Þ ð30:161Þ Thus, substituting Equation 30.161 into Equation 30.143 gives the general solution of Equation 30.139, that is, the HPD-E direct integration formula vtkỵ1 ị ẳ Ttịvtk ị expatk ịv1 sin vtk ỵ v cos vtk ịị ỵ expatkỵ1 ịv sin vtkỵ1 ỵ v2 cos vtkỵ1 ị 30:162ị The time interval is t ẳ tkỵ1 tk : 30.4 Long-Span Structures Subjected to Stationary Random Ground Excitations 30.4.1 The Solution of Equations of Motion Using the Pseudoexcitation Method In Equation 30.93, the PSD matrix of u€ b ; that is, SðivÞ in Equation 30.55, is an N-dimensional Hermitian matrix, while R is an N-dimensional real symmetric matrix Both matrices are usually positive definite or semipositive definite If the rank of R is rðr # NÞ; then by using Equation 30.111 it can be readily decomposed into the product of an N £ r matrix Q and its transposition; that is R ¼ QQT ð30:163Þ SðivÞ ¼ Bp JQQT JB ¼ Pp PT 30:164ị P ẳ BJQ 30:165ị Thus, Equation 30.55 can be written as in which To solve Equation 30.93, the right-hand side u€ b can be replaced by the pseudoground acceleration ~ ẳ P expivtị U b 30:166ị Thus, Equation 30.93 becomes the following sinusoidal equations of motion: Ms Y€~ r ỵ Cs Y_~ r ỵ Ks Y~ r ẳ Ms K21 s Ksb EmN P expðivtÞ ð30:167Þ The stable solution of Equation 30.167 is the pseudorelative displacement vector Y~ r ; whilst the pseudostatic displacement vector Y~ s can be computed by K E P expðivtÞ Y~ s ¼ K21 v s sb mN ð30:168Þ Thus, the pseudoabsolute displacement vector is ~ s ẳ Y~ r ỵ Y~ s X ð30:169Þ ~ s can be computed from X ~ s by means of a quasiIf necessary, any arbitrary pseudointernal force vector N static analysis Then, the corresponding PSD matrix is ~ ps N ~ Ts SNs Ns vị ẳ N â 2005 by Taylor & Francis Group, LLC ð30:170Þ 30-28 Vibration and Shock Handbook If it is assumed that SX€ ¼ SX€ ¼ · · · ¼ SX€ N ; denoted as Sa ; then Equation 30.165 becomes p P ẳ Sa BQ 30:171ị If only the wave passage effect is considered, that is, all lrij l ¼ in Equation 30.58, then matrix Q will reduce to a vector q0 with all its elements unity; that is Q ¼ q0 ¼ {1; 1; …; 1}T 30:172ị p P ẳ Sa e 30:173ị e0 ¼ {expð2ivt1 Þ; expð2ivt2 Þ; …; expð2ivtN Þ}T ð30:174Þ Thus, Equation 30.171 reduces to in which e0 is a complex vector Therefore, when only the wave passage effect is considered, Equation 30.167 reduces to p Ms y~ r ỵ Cs y_~ r ỵ Ks y~ r ẳ Ms K21 s Ksb EmN e0 Sa expðivtÞ ð30:175Þ Furthermore, if the structure is subjected to a uniform ground motion, then the vector e0 in Equation 30.174 should be replaced by q0 ; and so Equation 30.175 can be further reduced to pffiffiffi Ms y~ r ỵ Cs y_~ r ỵ Ks y~ r ẳ Ms K21 30:176ị s Ksb EmN q0 Sa expðivtÞ 30.4.2 Numerical Comparisons with Other Methods 30.4.2.1 Song-Hua-Jiang Suspension Bridge The Song-Hua-Jiang suspension bridge (see Figure 30.10) is located in Jilin Province of China Its overall length is 450 m, with a main span of 240 m and a width of 28 m The finite element model had 2076 DoF, 445 nodes (including 12 supports) and 574 elements The static equilibrium position of the bridge included the effects of the initial tensions of the cables The earthquake action was determined based on the Chinese National Standard (Code for Seismic Design of Buildings GB 50011-2001), which FIGURE 30.10 Song-Hua-Jiang suspension bridge directly gives the ground ARS curve for the bridge The PSD curve was obtained in terms of the Kaul method from which the samples of the ground acceleration time-history can be produced (Kaul, 1978) For the analyses associated with the SH and SV waves, 100 modes were used for mode-superposition, whereas for the P waves, only 30 modes were used The apparent wave speeds used were km/s for P waves and km/s for SH or SV waves 30.4.2.1.1 All Supports Move Uniformly Figure 30.11(a) gives the axial force distribution of this bridge along the deck due to the seismic P waves, which travel along the longitudinal direction of the deck All supports of the bridges are assumed to move uniformly The following four computational models were used: Response spectrum method Pseudoexcitation method Time-history method using three samples Time-history method using ten samples © 2005 by Taylor & Francis Group, LLC Seismic Random Vibration of Long-Span Structures RSM kN PEM 30-29 THM (3 samples) THM (10 samples) 4.0E+2 3.0E+2 2.0E+2 1.0E+2 0.0E+0 (a) 50 100 RSM kN 150 PEM 200 250 300 THM (3 samples) 350 400 m THM (10 samples) 4.0E+3 3.0E+3 2.0E+3 1.0E+3 0.0E+0 (b) 50 100 RSM kN 150 PEM 200 250 300 THM (3 samples) 350 400 m THM (10 samples) 5.0E+2 4.0E+2 3.0E+2 2.0E+2 1.0E+2 0.0E+0 (c) 50 100 150 200 250 300 350 400 m FIGURE 30.11 Deck force distribution of Song-Hua-Jiang bridge due to uniform ground motion: (a) axial force distribution along the deck under P waves; (b) transverse shear force distribution along the deck under SH waves; (c) vertical shear force distribution along the deck under SV waves © 2005 by Taylor & Francis Group, LLC 30-30 Vibration and Shock Handbook TABLE 30.1 Central Processing Unit (CPU) Times Required by Different Methods for Stationary Analysis Method Used RSM (sec) PEM (sec) THM (for One Sample) (sec) Uniform ground motion Wave passage effect Wave passage effect and incoherence effect 80.6 15.1 24.2 209.1 29.9 36.7 Note: the CPU time for mode extraction is not included; extracting 100 modes needs 180.9 sec Figure 30.11(b) gives the transverse shear force distribution along the deck due to the seismic SH waves traveling along the deck All supports move uniformly The above four computational models were used Figure 30.11(c) gives the vertical shear force distribution along the deck due to the seismic SV waves traveling along the deck All supports move uniformly The above four computational models were also used here All computations were executed on a P3-750 personal computer The computation times for different methods are listed in Table 30.1 Figure 30.11 and Table 30.1 show that, when ground motion is assumed uniform, that is, the earthquake spatial effects are not taken into account, the RSM, PEM, and THM (using ten samples) give very close results if the excitations are properly produced The RSM is the most popular method, but the newly developed PEM may be the more efficient one The THM needs to be executed for a number of ground acceleration samples and so was inefficient 30.4.2.1.2 Wave Passage Effect Is Taken into Account Figure 30.12(a) gives the axial force distribution of the bridge along the deck due to the seismic P waves, which travel along the longitudinal direction of the deck All supports of the bridges are assumed to move with certain time lags; that is, the wave passage effect is taken into account The apparent P wave speed is km/sec The following four computational models were used: RSM (uniform ground motion is assumed for comparison only) PEM (wave passage effect is considered) THM (wave passage effect is considered using three ground-acceleration samples) THM (wave passage effect is considered using ten ground-acceleration samples) Figure 30.12(b) gives the transverse shear force distribution along the deck due to the seismic SH waves traveling along the deck and Figure 30.12(c) gives the corresponding vertical shear-force distribution The above four computational models were used Figure 30.12 and Table 30.1 show that when the seismic wave-passage effect is taken into account, that is, the earthquake spatial effects are partly taken into account, the PEM and THM (using ten samples) give very close results The RSM, which does not consider the wave-passage effect, may give quite different results; these may appear larger or smaller than, or very close to, the results by the more reasonable PEM or THM analyses Therefore, such computations are necessary for evaluating the seismic spatial effects of long-span structures The PEM gives the most reliable results with the least computational effort and, therefore, this method is strongly recommended 30.4.2.1.3 Wave-Passage Effect and Incoherence Effect Are Jointly Taken into Account Figure 30.13(a) gives the axial force distribution of the bridge along the deck due to the seismic P waves, which travel along the longitudinal direction of the deck All supports of the bridge are assumed to move with certain time lags; that is, the wave passage effect is taken into account In addition, the incoherence effects are also taken into account Two coherence models, that is, the Loh –Yeh model © 2005 by Taylor & Francis Group, LLC Seismic Random Vibration of Long-Span Structures 30-31 FIGURE 30.12 Deck force distribution of Song-Hua-Jiang bridge due to wave passage effect: (a) axial force distribution along the deck under P waves; (b) transverse shear force distribution along the deck under SH waves; (c) vertical shear force distribution along the deck under SV waves © 2005 by Taylor & Francis Group, LLC 30-32 Vibration and Shock Handbook RSM (uniform) PEM (Loh model) kN PEM (uniform) PEM (QWW model) PEM (v=3km/s) 5.0E+2 4.0E+2 3.0E+2 2.0E+2 1.0E+2 0.0E+0 (a) 50 100 150 RSM (uniform) PEM (Loh model) kN 200 250 300 PEM (uniform) PEM (QWW model) 350 400 m PEM (v=2km/s) 4.0E+3 3.0E+3 2.0E+3 1.0E+3 0.0E+0 (b) 50 100 150 RSM (uniform) PEM (Loh model) kN 200 250 300 PEM (uniform) PEM (QWW model) 350 400 m PEM (v=2km/s) 5.0E+2 4.0E+2 3.0E+2 2.0E+2 1.0E+2 0.0E+0 50 100 150 200 250 300 350 400 m (c) FIGURE 30.13 Deck force distribution of Song-Hua-Jiang bridge due to incoherence effect: (a) axial force distribution along the deck under P waves; (b) transverse shear force distribution along the deck under SH waves; (c) vertical shear force distribution along the deck under SV waves © 2005 by Taylor & Francis Group, LLC Seismic Random Vibration of Long-Span Structures 30-33 and the QWW model were used for evaluating such effects The following five computational models were used: RSM (uniform ground motion is assumed) PEM (uniform ground motion is assumed) PEM (wave passage effect is considered) PEM (wave passage effect is considered also using the Loh –Yeh coherence model) PEM (wave passage effect is considered also using the QWW coherence model) Figure 30.13(b) gives the transverse shear force distribution along the deck due to the seismic SH waves traveling along the deck, and Figure 30.13(c) gives the corresponding vertical shear force distribution due to the seismic SV waves For both analyses the above five computational models were used The above computations as well were executed on a P3-750 personal computer, and the computation times required by different methods are listed in Table 30.1 From Figure 30.13 and Table 30.1, we can conclude that: When ground motion is assumed to be uniform, the RSM and PEM give very close internal force responses (i.e., demands), with the PEM being more efficient The wave passage effect is an important factor that affects the seismic responses of long-span structures To execute such seismic analyses, the PEM is not only theoretically quite reasonable, but also very efficient The incoherence effect appears to diverge when using different coherence models Herein, the influence caused by the QWW model is more evident than that caused by the Loh –Yeh model However, compared with the wave passage effect, the influence of the incoherence effect is of less importance 30.4.2.2 San Joaquin Concrete Bridge San Joaquin Bridge, located in California (see Figure 30.14), is a reinforced concrete bridge built in 2001 Its length is 36 þ 50 þ 50 þ 50 þ 36 ¼ 222 m, its width is 12 m, and the height of all piers is 16.76 m The finite element model had 367 nodes (including 10 ground z y nodes) and 366 elements Its basic natural period x is 0.811s Twenty modes were used in the modesuperposition analysis with all damping ratios being 0.05 The seismic analysis was carried out using the RSM and PEM, respectively The RSM 36m 50 m 50m 50 m 36m analysis was conducted according to the CALz TRANS Code (1999) with ARS ¼ 0.2 g, Type D x soil profile and magnitude Mw ¼ 7.0 The equivalent ground-acceleration power spectral density curve was produced by means of the FIGURE 30.14 Structural model of San Joaquin Kaul method All seismic waves were assumed to bridge travel along the longitudinal direction of the bridge The apparent P and S wave speeds were 3000 and 2000 m/s, respectively The internal forces in the deck (i.e., the axial forces due to P waves, the transverse shear forces due to SH waves, and the vertical shear forces due to SV waves) were all computed using the following computational models: RSM (uniform ground motion is assumed) PEM (uniform ground motion is assumed) © 2005 by Taylor & Francis Group, LLC 30-34 Vibration and Shock Handbook PEM (wave-passage effect is considered) PEM (wave-passage effect is considered also using the Loh –Yeh coherence model) PEM (wave-passage effect is considered also using the QWW coherence model) The computational results are shown in Figure 30.15(a) –(c) This bridge is not very long However, similar phenomena to those found for the bridge of Example 30.1 are still found Clearly, when the ground motion is assumed to be uniform, the RSM and PEM still give very close results If the wavepassage effect is taken into account, then the internal force distribution with the PEM will change considerably, particularly at the midpoint of the deck It is known that, for symmetric bridges, the antisymmetric modes will not participate in the symmetric motions under the assumption of uniform ground motion However, when the wave-passage effect is taken into account, this conclusion does not hold It is obvious that, even for this shorter bridge, the wave-passage effect seems to be quite significant The incoherence effect is comparatively not so important 30.5 Long-Span Structures Subjected to Nonstationary Random Ground Excitations A typical strong motion earthquake record consists of three stages In the first stage, the intensity of the ground motion increases, which mainly reflects the motion of P waves The intensity of the ground motion remains the strongest in the second stage, which mainly reflects the motion of S waves The ground motion will die down in the last stage Such a complete seismic motion is usually regarded as a nonstationary random process If the nonstationary property is assumed to takes place only for the intensity of the motion, then this random process is regarded as a uniformly modulated evolutionary random process However, if the shape of the ground motion PSD curve also varies with time (in other words, the intensity and the distribution with frequency of the ground motion energy both depend on time), then the ground motion is regarded as a nonuniformly modulated evolutionary random process It is usually accepted that when the intensity of the seismic motion in the second stage appears quite stationary while the time interval of this stage is much longer (e.g., three times or over) than the basic period of the structure under consideration, a simplified, stationary-based random analysis may be acceptable as a substitute of the nonstationary analysis In fact, the basic periods of many long-span bridges range from 10 to 20 sec, and the stationary portion of a typical strong earthquake is usually less than min, being only 20 to 30 sec in most cases Therefore, nonstationary analyses are appropriate for such problems Previously, such nonstationary random analyses have been considered very difficult However, by using the recently developed PEM, combined with the precise integration method, such analyses have become relatively easy 30.5.1 Modulation Functions Some popular uniform modulation functions are listed below: I ðt=t Þ2 # t # t1 > > < gðtÞ ¼ I0 t1 # t # t2 > > : I0 exp{cðt t2 Þ} t $ t2 ( gðtÞ ẳ t$0 t,0 gtị ẳ aẵexp2a1 tị exp2a2 tị ; gtị ẳ sin bt â 2005 by Taylor & Francis Group, LLC ð30:177Þ ð30:178Þ # a1 , a2 ð30:179Þ ð30:180Þ Seismic Random Vibration of Long-Span Structures 30-35 2 FIGURE 30.15 Deck force distribution of San Joaquin bridge due to incoherence effect: (a) axial force distribution along the deck under P waves; (b) transverse shear force distribution along the deck under SH waves; (c) vertical shear force distribution along the deck under SV waves © 2005 by Taylor & Francis Group, LLC 30-36 Vibration and Shock Handbook Nonuniform modulation models have rarely been investigated Lin et al (1997a, 1997b) suggested the following nonuniform modulation model: Av; tị ẳ bv; tịgtị ẳ exp 2h0 vt gtị v a ta ð30:181Þ in which gðtÞ is an amplitude modulation function; bðv; tÞ is a frequency modulation function; and va and ta are the reference frequency and time, which are introduced to transform v and t into dimensionless parameters In principle, va and ta can be arbitrarily selected Once they have been selected, the factor h0 ðh0 0Þ can be adjusted accordingly to make the high-frequency components of the nonstationary random process decay more quickly than the low-frequency components and, thus, simulate the seismic motion more accurately When h0 ẳ 0; that is, bv; tị ẳ 1; Av; tÞ reduces to the uniform modulation function gðtÞ: 30.5.2 The Formulas for Nonstationary Multiexcitation Analysis For nonuniformly modulated multiexcitation problems, the pseudoexcitation for the corresponding stationary problems, that is, Equation 30.166, is extended to ~ v; tị ẳ Av; tịPexpivtị U b ð30:182Þ in which the kth diagonal element of the N £ N diagonal matrix Aðv; tÞ is the modulation function Ak ðv; tÞ of the excitation which is applied to the kth support of the structure In the case of uniformly modulated excitations, it is only necessary to replace all the nonuniform modulation functions Ak ðv; tÞ by the uniform modulation functions gk ðtÞ: Other formulae remain entirely unchanged The N £ r matrix P can be generated by means of Equation 30.165 to Equation 30.174 Each column of €~ ðv; tÞ can be regarded as a deterministic acceleration excitation vector By substituting it into the U b right-hand side of Equation 30.93 and solving the equations of motion, a column of the matrix Y~ r ðv; tÞ can be produced Because Aj ðv; tÞ is a time-dependent and slowly varying function, the pseudoground displacement matrix can be computed approximately from ~€ ðv; tÞ ~ b ðv; tÞ ¼ U U v2 b ð30:183Þ The pseudoquasi-static displacement matrix Y~ s ðv; tÞ can then be computed from Equation 30.92 Then, the PSD matrix of the absolute displacement vector Xs ðv; tÞ is SXs Xs ðv; tÞ ẳ Y~ r v; tị ỵ Y~ s v; tịịp Y~ r v; tị ỵ Y~ s v; tịịT 30:184ị ~ e ; has been computed, then the PSD matrix of the If a group of pseudointernal forces, denoted as N corresponding internal forces Ne can be computed from ~ pe N ~ Te SNe Ne v; tị ẳ N ð30:185Þ When the ground acceleration PSD matrix is known, the corresponding pseudoacceleration vector u€~ b is easy to generate according to Equation 30.163 to Equation 30.166 If instead, the ground displacement PSD matrix or velocity PSD matrix is known, then the acceleration PSD matrix can be obtained by multiplying the displacement or velocity PSD matrices by v4 or v2 ; respectively © 2005 by Taylor & Francis Group, LLC Seismic Random Vibration of Long-Span Structures 30.5.3 30-37 Expected Extreme Values of Nonstationary Random Processes The evaluation of the peak amplitude responses of structures subjected to nonstationary seismic excitations has also received much attention (Shrikhande and Gupta, 1997; Zhao and Liu, 2001) Previously, only very simple structures could be computed However, by using the PEM, complicated structures can be analyzed, as is briefly described below To evaluate the expected extreme value responses of a structure subjected to nonstationary Gaussian excitations, the duration of which the intensity of the excitation peaks exceeds 50% of the maximum peak intensity denoted by ½t0 ; t0 ỵ t is taken as the equivalent stationary duration in order to use Equation 30.40 to Equation 30.49 to evaluate the desired expected extreme values Provided that the timedependent PSD of any arbitrary response yðtÞ; that is Syy ðv; tÞ, has been computed over that equivalent duration using the PEM, then the equivalent stationary PSD over that duration is S0yy vị ẳ t0 ỵt Syy v; tịdt t t0 ð30:186Þ To compute the extreme value responses based on Equation 30.177, the parameters t0 and t are chosen as t0 ẳ t1 p 2; p t ẳ t2 ỵ ln 2=c t1 = ð30:187Þ Thus, the equivalent stationary random responses are obtained and the subsequent processing can still use Equation 30.40 to Equation 30.49 30.5.4 Numerical Comparisons with the Corresponding Stationary Analysis The example of the Song-Hua-Jiang suspension bridge of the last section is used here for the seismic nonstationary random vibration analysis The results are compared with those from the corresponding stationary random-vibration analyses with the ground assumed to move uniformly (i.e., at an apparent wave speed vapp ¼ 1), or to move at a limited apparent wave speed vapp (with the wave-passage effect is taken into account), which is km/sec for P waves and km/sec for S waves The nonstationary random excitation model ztị ẳ gtịxtị was used in which the auto-PSD of xðtÞ is assumed to be identical to that used for the stationary excitation in the preceding section The frequencydomain parameters also remained the same The modulation function had the form of Equation 30.177 with t1 ¼ 8:0; t2 ¼ 20:0; and c ¼ 0:20: The duration of the earthquake was t [ ½0; 25 ; and the time stepsize was Dt ¼ 0:5: The nonstationary analysis results are shown in Figure 30.16(a) to (c), and are compared with the results of the corresponding stationary random vibration analyses Clearly, for such a long-span bridge, the wave passage effect is quite significant in its seismic analysis, as seen in Figure 30.11 to Figure 30.13 In addition, whether for uniform ground motion or for differential ground motion (i.e., the wave-passage effect is considered), the nonstationary responses are always smaller than the corresponding stationary responses The maximum difference between their corresponding peak values may reach up to 23.1% for the present problem, as shown in Table 30.2 For very slender bridges, this nonstationary property will be even stronger By means of the PEM combined with the precise integration method (its HPD-E form for the modulation function used in this example), such modification can be fulfilled quickly and conveniently The computational effort required by the nonstationary analysis is only about 25 (see Table 30.3) © 2005 by Taylor & Francis Group, LLC 30-38 Vibration and Shock Handbook kN Uniform-Nonstationary v=3km/s-Nonstatinary Uniform-Stationary v=3km/s-Stationary 5.0E+2 4.0E+2 3.0E+2 2.0E+2 1.0E+2 0.0E+0 (a) 50 kN 100 150 200 250 300 Uniform-Nonstationary v=2km/s-Nonstatinary 350 m 400 Uniform-Stationary v=2km/s-Stationary 4.0E+3 3.0E+3 2.0E+3 1.0E+3 0.0E+0 (b) 50 kN 100 150 200 250 300 Uniform-Nonstationary v=2km/s-Nonstatinary 350 400 m Uniform-Stationary v=2km/s-Stationary 5.0E+2 4.0E+2 3.0E+2 2.0E+2 1.0E+2 0.0E+0 50 100 150 200 250 300 350 400 m (c) FIGURE 30.16 Deck-force distribution of Song-Hua-Jiang bridge due to uniform and differential, and stationary and nonstationary random ground motion: (a) axial force distribution along the deck under P waves; (b) transverse shear force distribution along the deck under SH waves; (c) vertical shear force distribution along the deck under SV waves © 2005 by Taylor & Francis Group, LLC Seismic Random Vibration of Long-Span Structures TABLE 30.2 30-39 Comparisons of the Expected Extreme Values of Deck Internal Forces Ground Motion Internal Forces in the Deck (kN) Uniform ground motion Axial force N due to P waves Transverse shear force Fy due to SH waves Vertical shear force Fz due to SV waves Wave-passage effect Axial force N due to P waves Transverse shear force Fy due to SH waves Vertical shear force Fz due to SV waves Distance from the Left End (m) Nonstationary Responses Stationary Responses Increases (%) 72 258 274.95 712.19 146.00 337.60 832.47 165.81 22.8 16.9 13.6 420 270 246 264.51 1292.17 134.00 325.61 1515.75 152.89 23.1 17.3 14.1 TABLE 30.3 CPU Times Required by PEM for Nonstationary Analyses (Units: Seconds) 30.6 Modes Used 30 100 Uniform ground motion Wave passage effect Extracting modes 450 543 143 1142 1479 181 Conclusions For short-span structures, the ground spatial effects are negligible, and the seismic analyses using RSM, PEM, or THM (with a sufficient number of samples) are relatively close to one another provided that the ground accelerations have been produced properly, and so are almost equivalent Although they have almost the same accuracy level, their efficiencies are quite different Of the three methods, if the structural models are rather complex (e.g., the FEM models have thousands or more DoF and need dozens or hundreds of modes for mode superposition), then the PEM will have the highest computational efficiency For long-span structures, the wave passage effect is an important factor for structural seismic responses The influence may produce more conservative, or more dangerous, designs, which is difficult to predict by intuitive experience Thus, computer-based analysis is a preferable choice The PEM 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Group, LLC 30: 80Þ Seismic Random Vibration of Long-Span Structures 30- 15 Its second half gives Ksb eb ẳ 2Ks es 30: 81ị Substituting Equation 30. 79 into Equation 30. 78 and using Equation 30. 81 gives