Accurate and highly eЃcient algorithms for structural ...

Comput. Methods Appl. Mech. Engrg. 191 (2001) 103±111

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Accurate and highly ecient algorithms for structural stationary/non-stationary random responses Jiahao Lin *, Yan Zhao, Yahui Zhang Research Institute of Engineering Mechanics, Dalian University of Technology, Dalian 116023, China Received 7 November 2000; received in revised form 6 February 2001

Abstract An accurate and highly ecient algorithm series for structural stationary/non-stationary random response analysis, pseudo excitation method (PEM) is presented. The algorithm series, which have been developed since early 1980s, applies to the random vibration analyses of structures subjected to single/multiple, stationary/non-stationary random excitations, including the very attractive issue ± the solution of the response to non-uniformly modulated evolutionary random excitations. Each algorithm in this series is an accurate complete quadratic combination (CQC) method because the cross-correlation quadratic terms between the participant modes and between the excitations have both been included. Such algorithms are easy to use because structural stationary random response analysis is reduced to the analysis of structural harmonic response, while structural non-stationary random responses can be readily computed in terms of any step-by-step integration scheme (e.g., Newmark scheme or the precise integration method). Ó 2001 Elsevier Science B.V. All rights reserved.

1. Introduction Long-span structures are widely used in the design of many public facilities such as bridges, ¯yovers, dams, pipelines of nuclear power stations, and so on. Therefore, to ensure their safety under disastrous earthquake actions is of great importance. In the seismic analysis of long-span structures, many complex e€ects, e.g. wave passage e€ect, incoherence e€ect, local e€ect [1], and sometimes the non-stationary e€ect and/or non-uniform modulation e€ect have to be taken into account. These factors have caused great diculties in the structural seismic analysis. In the past 20 years, the research for such analysis methodologies has received special attention [1±10]. It has now been widely recognized that the most reasonable method for dealing with such multiple excitation problems is the random vibration approach. Among a great deal of research activities, the research work by Kiureghian and Neuenhofer (University of California at Berkeley) [1] and Ernesto and Vanmarcke (Princeton University) [2] are very representative. They all developed their research about the seismic analysis of long-span structures based on random vibration approach. However, when solving the high degree random di€erential equations, they all faced unacceptable computational e€orts. Finally they concluded: ``While the random vibration approach is appealing for its statistical nature, it is not yet accepted as a method of analysis by practising engineers'' [1], and ``The theoretical framework of a methodology for stochastic-response analysis to random-excitation ®elds is already available; however, its use by the earthquake-engineering community is viewed as impractical except for simple structures with a small number of degrees of freedom and

*

Corresponding author. Fax: 411-4708-400. E-mail address: [email protected] (J. Lin).

0045-7825/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 4 5 - 7 8 2 5 ( 0 1 ) 0 0 2 4 7 - X

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supports'' [2]. As a result, they presented di€erent response spectrum methods as the approximation approaches to the accurate random vibration method. Di€erent points of view over the problems of eciency and precision exist between the two approaches they suggested [3,4], and so far these methods have not been widely used. In addition, accurate computations of the non-stationary random seismic responses of such long-span structures have been considered even more impractical although they are also very important. An accurate and highly ecient algorithm series for the random vibration computation of complex engineering structures, known as the pseudo excitation method (PEM), has been developed since early 1980s [11±21]. By using this algorithm series, the above-mentioned diculties in the stationary/non-stationary random response computations of long-span structures have been satisfactorily resolved. Various programs based on this method have been developed in China for dealing with the seismic analyses of bridges, dams, o€shore-platforms, and so on. Complex engineering structures with thousands of degrees of freedom and dozens of supports can be analysed accurately and eciently on ordinary personal computers. The wave passage e€ect, incoherence e€ect, local e€ect, non-stationary and non-uniform modulation e€ects [22] can all be taken into account easily and accurately. In addition, the cross-correlation terms between all participant modes and between all random excitations have been included, therefore the PEM is a complete quadratic combination (CQC) method series. This paper is a summary of about 50 papers published by the ®rst author and his colleagues since 1985; therefore only the main conclusions are included here. 2. Structural responses due to a single stationary random excitation The PEM for stationary single excitation problem can be explained by means of Fig. 1: A linear system is subjected to a zero-mean-valued stationary random excitation p p whose power spectral density (PSD) Sxx …x† has been speci®ed. If y ˆ Sxx …x†H1 …x† eixt and z ˆ Sxx …x†H2 …x† eixt are two arbitrary stationary harmonic responses due to the pseudo harmonic excitation p x ˆ Sxx …x† eixt ; …1† then it can be readily veri®ed that p p y  y ˆ Sxx …x†H1 …x† e ixt
Sxx …x†H1 …x† eixt ˆ jH1 …x†j2 Sxx …x† ˆ Syy …x†; p p y  z ˆ Sxx …x†H1 …x† e ixt
Sxx …x†H2 …x† eixt ˆ H1 …x†Sxx …x†H2 …x† ˆ Syz …x†;

…2† …3†

in which the asterisk * represents complex conjugate. That means, the auto- and cross-PSD functions of the two random responses y…t† and z…t† can be computed using the corresponding pseudo harmonic responses y and z. If the pseudo responses are two arbitrary harmonic response vectors, fy…t†g and fz…t†g, it can be similarly veri®ed that the corresponding PSD matrices would be 

T

…4†



T

…5†

fyg fyg ˆ ‰Sxx …x†Š; fyg fzg ˆ ‰Syz …x†Š: Now, consider a structure subjected to a single random excitation, its equation of motion is ‰MŠfy g ‡ ‰CŠfy_ g ‡ ‰KŠfyg ˆ fpgx…t†;

Fig. 1. Pseudo excitation and responses for an SIMO system.

…6†

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105

in which, x…t† is a stationary random process ofp which the PSD Sxx …x† is known. fpg is a given constant vector. In order to solve this equation, let x…t† ˆ Sxx …x† eixt , then Eq. (6) becomes the following harmonic equation: p …7† ‰MŠfy g ‡ ‰CŠfy_ g ‡ ‰KŠfyg ˆ fpg Sxx …x† eixt : Its stationary solution is fy…t†g ˆ fY …x†g eixt :

…8†

Using its ®rst q modes for mode-superposition, then fY …x†g ˆ  Hj ˆ x2j

q X jˆ1

cj Hj f/j g

p Sxx …x†;

x2 ‡ 2i1j xxj



1

…9†

 T cj ˆ /j fpg;

;

…10†

in which xj ; f/j g; 1j ; cj are the jth natural angular frequency, mode, damping ratio and mode-participation factor, respectively. According to the PEM, the PSD matrix of fyg would be ‰Syy …x†Š ˆ fyg fygT ˆ fY …x†g fY …x†gT :

…11†

Substituting Eq. (9) into Eq. (11) and expanding it gives ‰Syy …x†Š ˆ

q X q X jˆ1

kˆ1

T

cj ck Hj Hk f/j gf/k g Sxx …x†:

…12†

This is the well-known CQC formula [13,23]. It involves the cross-correlation terms between all participant modes, therefore it usually requires considerable computational e€orts. In order to reduce the e€orts, the cross-correlation terms are usually neglected, that leads to the following approximate square root of the sum of squares (SRSS) formula ‰Syy …x†Š ˆ

q X jˆ1

2

T

c2j jH jj f/j gf/j g Sxx …x†:

…13†

The main e€orts for Eq. (12) are q2 vector multiplication operations, while Eq. (13) contains only q such vector multiplication operations. However, the PEM algorithm, i.e., Eqs. (9) and (11) needs only one such vector multiplication operation to achieve the same precision as Eq. (12) in computing ‰Syy …x†Š. For example, if taking 20 participant modes, the difference in the computational efforts would be approximately 400:1. Therefore the PEM is also known as the fast-CQC method [13]. The SRSS formula is approximately usable only for lightly damped structures with sparsely spaces participant frequencies. For complex engineering structures, particularly if 3D structural models are used, their natural frequencies are always closely spaced, therefore the SRSS method does not apply. The conventional CQC method, i.e., Eq. (12) needs enormous efforts. This explains the dif®culty in the application of random vibration approach to practical engineering problems, except for simple structures with a small number of degrees of freedom and excitations. The PEM has resolved this dif®culty, and so very complex problems can now be computed on personal computers accurately and very quickly. 3. Structural responses due to multiple stationary random excitations Consider a linear structure subjected to a number of stationary random excitations. Its equation of motion is ‰MŠfy g ‡ ‰CŠfy_ g ‡ ‰KŠfyg ˆ ‰RŠfx…t†g;

…14†

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in which ‰RŠ is a speci®ed constant matrix, and fx…t†g is an m degree stationary random process vector, its PSD matrix ‰Sxx …x†Š is known. Since such PSD matrix is Hermitian, it can be decomposed into ‰Sxx …x†Š ˆ

r X jˆ1

dj f/j g f/j gT

…r 6 m†;

…15†

in which r is the rank of ‰Sxx …x†Š. Next, constitute r pseudo harmonic excitations p fxj …t†g ˆ dj f/j g eixt …j ˆ 1; 2; . . . ; r†:

…16†

By substituting Eq. (16) into the right-hand side of Eq. (14), the harmonic responses can be easily obtained fyj …t†g ˆ fYj …x†g eixt :

…17†

Other harmonic response vectors can also be obtained easily, denoted as fzj …t†g fzj …t†g ˆ fZj …x†g eixt :

…18†

The corresponding PSD matrices can be computed by means of the following formulas [14,20]: ‰Syy …x†Š ˆ

r X  T fYj …x†g fYj …x†g ;

…19†

jˆ1

‰Syz …x†Š ˆ

r X



T

fYj …x†g fZj …x†g :

…20†

jˆ1

For long-span structures subjected to di€erential ground motion, the equation of motion in the global coordinate system (assumed to be ®xed to the center of the earth) can be written in the partitioned form as [16,19,20]            xs Mss Msm Css Csm x_ s Kss Ksm xs 0 ‡ ‡ ˆ ; …21† xm x_ m Mms Mmm Cms Cmm Kms Kmm xm fm in which the subscript m represents the master degrees of freedom, i.e., the support displacements; while the subscript s represents the slave degrees of freedom. The absolute displacement vector fxs g can be decomposed into two parts fxs g ˆ fys g ‡ fyr g;

…22†

where fys g is the quasi-static displacement vector [23] which satis®es fys g ˆ

1

‰Kss Š ‰Ksm Šfxm g

…23†

and the dynamic displacement vector [23] fyr g satis®es the following equation: ‰Mss Šfyr g ‡ ‰Css Šfy_ r g ‡ ‰Kss Šfyr g ˆ

1

‰Mss Š fys g;

…24†

in which fys g ˆ

1

‰Kss Š ‰Ksm Šfxm g:

…25†

It should be pointed out that the relative displacement vector fys g derived from Eq. (21) cannot be reduced to the conventional solution when fxm g represents uniform ground displacements [16]. This is because Eq. (21) assumes the damping forces to be proportional to the absolute velocity vector f_xTs ; x_ Tm gT . In order to avoid this inconsistency, the damping forces should be assumed to be proportional to the relative velocity vectorfy_ rT ; 0gT that leads to Eq. (24). The displacements fys g thus obtained can then generate the conventional results due to the uniform ground excitations.

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When the ground acceleration PSD matrix is known, the corresponding pseudo acceleration vector fxm g is easy to generate according to Eq. (15). If instead, when the ground displacement PSD matrix or velocity PSD matrix is known, then the acceleration PSD matrix can be obtained by multiplying x4 or x2 on these PSD matrices. The other computations are exactly the same. 4. Structural responses due to single non-stationary random excitations 4.1. Uniformly modulated evolutionary random excitations Consider a linear structure subjected to an evolutionary random excitation f …t† ˆ g…t†x…t†, in which g…t† is a given slowly varying modulation function, while x…t† is a zero-mean-valued stationary random process of which the auto-PSD Sxx …x† is also known. In order to compute the PSD matrices of two arbitrary response vectors fy…t†g and fz…t†g, the following pseudo excitation should be used [16,17]: p f …t† ˆ g…t† Sxx …x† eixt : …26† Under the action of this deterministic loading, the equation of motion (6) becomes p ‰MŠfy g ‡ ‰CŠfy_ g ‡ ‰KŠfyg ˆ fpgg…t† Sxx …x† eixt :

…27†

For seismic problems, the structure is initially at rest, namely, fy0 g ˆ fy_ 0 g ˆ f0g when t ˆ 0. Thus the time-history of fy…x; t†g and fz…x; t†g can be computed in terms of the Newmark or Wilson-h schemes [23] with x regarded as a parameter. It has been veri®ed that [17] the response PSD matrices can be accurately computed by using the following equations: 

T

…28†



T

…29†

‰Syy …x†Š ˆ fy…x; t†g fy…x; t†g ; ‰Syz …x†Š ˆ fy…x; t†g fz…x; t†g :

It has also been veri®ed that if the precise integration method [20,24] is used in solving Eq. (27), the eciency would be many times higher than using the Newmark method. 4.2. Non-uniformly modulated evolutionary random excitations Provided the structure is subjected to a non-uniformly modulated evolutionary random excitation f …t† [22] Z 1 f …t† ˆ A…x; t† eixt da…x†; …30† 1

in which A…x; t† is the known non-uniform modulation function, and a satis®es the following equation: E‰da …x1 † da…x2 †Š ˆ Sxx …x1 †d…x2

x1 † dx1 dx2 ;

…31†

where Sxx …x1 † is the PSD of a zero-mean-valued stationary random process x…t†. The Riemann±Stieltjes integration has caused essential diculty in the conventional response computation. It has also been proved [15] that by constituting the following pseudo excitation: p f …t† ˆ A…x; t† Sxx …x† eixt : …32† Eq. (27) becomes ‰MŠfy g ‡ ‰CŠfy_ g ‡ ‰KŠfyg ˆ fpgA…x; t†

p ixt Sxx …x† e :

…33†

Then the time-history curves of two arbitrary response vectors, denoted as fy…x; t†g and fz…x; t†g, can be computed in terms of Eq. (33), and the required PSD matrices would then be [15] ‰Syy …x†Š ˆ fy…x; t†g fy…x; t†gT 

‰Syz …x†Š ˆ fy…x; t†g fz…x; t†g

T

…auto-PSD matrix†;

…34†

…cross-PSD matrix†:

…35†

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5. Structural responses due to multiple non-stationary random excitations For the case of uniformly modulated evolution random excitations applying on a structure, the equation of motion (14) can be modi®ed as ‰MŠfy g ‡ ‰CŠfy_ g ‡ ‰KŠfyg ˆ ‰RŠff …t†g;

…36†

in which ‰RŠ is a speci®ed constant matrix, while ff …t†g is a m degree evolutionary random process vector, it can be written as 9 8 g1 …t†x1 …t† > > > > > = < g2 …t†x2 …t† > ; …37† ff …t†g ˆ .. > > . > > > > ; : gm …t†xm …t† in which gi …t† is the modulation function of the ith excitation …i ˆ 1; 2; . . . ; m†, and the PSD matrix ‰Sxx …x†Š of the zero-mean-valued stationary random process vector fx…t†g: 9 8 x1 …t† > > > > > = < x2 …t† > …38† fx…t†g ˆ .. > . > > > > > ; : xm …t† is also known. Decompose the PSD matrix ‰Sxx …x†Š ‰Sxx …x†Š ˆ

r X jˆ1



dj f/j g f/j g

T

…r 6 m†;

in which r is the rank of ‰Sxx …x†Š. Then constitute r pseudo excitations p ffj …t†g ˆ dj gi …t†f/j g eixt …j ˆ 1; 2; . . . ; r†:

…39†

…40†

By substituting Eq. (40) into the right-hand side of Eq. (36), the zero-mean-valued transient solution fyj …x; t†g and fzj …x; t†g can be easily obtained by using any step-by-step integration scheme. It has been proved that [16,17,20] the corresponding time-dependent PSD matrices can be accurately computed by means of the following formulas: ‰Syy …x†Š ˆ

r X  T fyj …x; t†g fyj …x; t†g ;

…41†

jˆ1

‰Syz …x†Š ˆ

r X



T

fyj …x; t†g fzj …x; t†tg :

…42†

jˆ1

This algorithm can be extended to the case for multiple non-uniformly modulated evolutionary random excitations [20]. The details are not described here. 6. Example: seismic analysis of Hong Kong Tsing-Ma bridge The ®nite element model of Hong Kong Tsing-Ma suspension bridge is shown in Fig. 2. Its deck and towers are modeled by 3D beam elements, and the main and suspension cables are both modeled by cable elements. It has 769 nodes (including 17 supports), 1010 elements, 2257 degrees of freedom. In order to compare the precision, the ®rst 60, 100, 150, 180 and 200 modes were used in the mode-superposition analysis, respectively. Six cases were computed, i.e., (1) uniform stationary ground excitations; (2) stationary ground excitations with wave passage e€ect considered; (3) stationary ground excitations with wave

J. Lin et al. / Comput. Methods Appl. Mech. Engrg. 191 (2001) 103±111

109

Fig. 2. Seismic analysis of Hong Kong Tsing-Ma bridge.

passage e€ect and incoherence e€ect between supports taken into considered; (4) uniform non-stationary ground excitations; (5) non-stationary ground excitations with wave passage e€ect considered; (6) nonstationary ground excitations with wave passage e€ect and incoherence e€ect between supports taken into account. The variances of selected 1800 displacements and 300 internal forces were computed for each case. In order to compute such variances, 81 frequency points were used in the region x 2 ‰0:1; 5:4Š s 1 . For nonstationary response analyses, the time region used was [0,16] s, and the time-step was 0.1 s. The seismic P waves travel along the longitudinal direction of the bridge. For cases 1±3, the variances of a typical support shear force are shown in Fig. 3. For this example, clearly, at least 150 modes are required for the mode superposition analysis. The algorithms described above have all been coded into the program system DDJ/W developed in Dalian University of Technology [25]. A Pentium-2 personal computer (main frequency 350 MHz) was used for this example. The CPU time used for each case is listed in Table 1. It can be seen that with q ˆ 150

Fig. 3. Variations of a support shear force variance with participant modes.

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Table 1 CPU time used for all cases (h:m:s) Modes used

Mode extraction time

Stationary ground excitations

Non-stationary ground excitations

Case 1

Case 2

Case 3

Case 4

Case 5

Case 6

60 100 150 180 200

0:01:58 0:03:18 0:06:05 0:07:10 0:07:15

0:00:8 0:00:12 0:00:17 0:00:20 0:00:23

0:00:10 0:00:14 0:00:19 0:00:23 0:00:25

0:00:38 0:00:56 0:01:19 0:01:33 0:01:44

0:00:58 0:01:35 0:02:19 0:02:46 0:03:05

0:03:49 0:06:10 0:08:12 0:10:50 0:12:01

0:15:35 0:24:51 0:36:46 0:43:49 0:48:48

modes be used, if the ground motion is assumed to be uniform and stationary (Case 1), only 17 CPU s were required by PEM for computing more than 2000 responses. If the conventional CQC method, i.e., Eq. (12), was instead used, then 17  q2 ˆ 382500 s or 203 h CPU time would be required. That is unacceptable by engineers. Case 2 includes the wave passage e€ect and Case 3 further includes the incoherence e€ect, both were considered very dicult [1±4]. However, by using the suggested PEM, only 19 and 79 s were used to complete all these computations accurately. Cases 4±6 are the equivalents of Cases 1±3 when the excitations become non-stationary. Such non-stationary analyses are considered much more dicult than the corresponding stationary ones. However they can still be completed within 3±40 min accurately. 7. Conclusions For any linear structure, the stationary random responses can be computed by means of harmonic vibration analyses, while the non-stationary random responses can be computed by means of an arbitrary direct integration scheme. Therefore, all these random response analyses have become very simple by using the PEM. This method is simple, accurate and very ecient, therefore is very useful. Acknowledgements This work has been supported by the China NSFC (No. 10072015) and NKBRSF (No. G1999032805). References [1] A.D. Kiureghian, A. Neuenhofer, Response spectrum method for multi-support seismic excitations, Earthquake Engrg. Struct. Dynamics 21 (1992) 713±740. [2] H.Z. Ernesto, E.H. Vanmarcke, Seismic random vibration analysis of multi-support structural systems, ASCE J. Eng. Mech. 120 (1994) 1107±1128. [3] H.Z. Ernesto, E.H. Vanmarcke, Closure on the discussion, ASCE J. Eng. Mech. 121 (1995) 1038. [4] A.D. Kiureghian, A. Neuenhofer, A discussion on Ref. [3], ASCE J. Eng. Mech. 121 (1995) 1037. [5] M.C. Lee, J. Penzien, Stochastic analysis of structures and piping systems subjected to stationary multiple support excitations, Earthquake Eng. Struct. Dynamics 11 (1983) 91±110. [6] A.M. Abdel-Gha€ar, I.I. Rubin, Suspension bridges response to multiple support excitations, ASCE J. Eng. Mech. 108 (1982) 419±434. [7] Y.K. Lin, R. Zhang, Y. Yong, Multiply supported pipeline under seismic wave excitations, J. Eng. Mech. 116 (1990) 1094±1108. [8] T. Qu, Q. Wang, Advancement of multiple input seismic response analysis research, World Earthquake Engrg. 11 (1993) 30±36 (in Chinese). [9] C.H. Loh, Y.T. Yeh, Spatial variation and stochastic modeling of seismic di€erential ground movement, Earthquake Engrg. Struct. Dynamics 16 (1988) 583±596. [10] R.S. Harichandran, E.H. Vanmarcke, Stochastic variation of earthquake ground motion in space and time, J. Engrg. Mech. 112 (1986) 154±175. [11] J.H. Lin, A deterministic algorithm of stochastic earthquake responses, Earthquake Engrg. Engrg. Vibration 5 (1985) 89±93 (in Chinese).

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