inns: 5 Date 0-7639 LIBRARY Michigan State University This is to certify that the thesis entitled RESPONSE OF SIMPLE BEAM TO SPATIALLY VARYING SEISMIC EXCITATION presented by WEIJUN WANG has been accepted towards fulfillment of the requirements for M. 8 degree in CB Major professor February 4, 1988 MS U is an Affirmative Action/Equal Opportunity Institution ‘bv1531aj RETURNING MATERIALS: Place in book drop to LlaRARJES remove this checkout from 4—3!!!- your record. FINES will _~, » be charged if book is returned after the date stamped below. RESPONSE OF SIMPLE BEAM TO SPAIIALLY‘VARYINC SEISMIC EXCIIAIION by Weijun Wang A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Civil and Environmental Engineering 1988 ABSTRACT RESPONSE OF SIMPLE BEAM T0 SPATIALLY VARYING SEISMIC EXCITATION By We 1 j un Wang 'The stochastic response of a simply supported beam is analyzed by using a space-time earthquake ground motion model. The effect of the spatial variation of ground motion is considered and results for several types of support motion correlations are compared. For practical use, an approximate method which can be used to quickly estimate the beam response is proposed. ACKNOWLEDGEMENTS The research described in this report was supported by the Nation- al Science Foundation under Grant No. ECE 8605184 and a grant from the Division of Engineering Research, Michigan State University. The author is grateful for this support. The author wishes to thank Professor R. Harichandran for his continuous encouragement and direction during this work. The author is also grateful to Professors R. Wen and T. Wolff for reviewing the manuscript and suggesting improvements. TABLE OF CONTENTS Egg LIST OF FIGURES .................................................... iii LIST OF SYMBOLS .................................................... v 1. INTRODUCTION .................................................... 1 2. ANALYSIS OF SIMPLY SUPPORTED BEAM ............................... 3 2.1 Ground Motion Model ........................................ 3 2.2 Random Vibration Analysis of Response ...................... 4 2.3 Numerical Results .......................................... 10 2.3.1 General Excitation ................................... 10 2.3.2 Some Special Cases ................................... 15 2.4 Approximate Methods for Evaluating the Response Variance ... 22 3. CONCLUSIONS ..................................................... 33 LIST OF REFERENCES ii Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 2-1 2-2 2-3 2-4 2-5 2-6 2-7 2-8 2-9 2-10: 2-11: 2-12: 2-13: 2-14: 2-15: 2-16: 2-17: 2-18: 2-19: 2-20: 2-21: LIST OF FIGURES iii _P_agg : Simple Beam With Support Excitation ................... A : Distribution of 03" for Different fig and f8 ........... ll : 03" for Exact Solution and First Mode Approximation ... 11 : Effect of Fundament Frequency of Beam on 02" .......... 13 : Effect of Beam Length on oi" .......................... 13 : Effect of fig on 0:" ................................... 14 : Effect of Ground Motion Frequency on oi" .............. 14 : Response Ratio for Cases 1 to 4 - L - 20 m ............ 17 : Response Ratio for Cases 1 to 4 - L - 50 m ............ 17 Response Ratio for Cases 1 to a - L - 100 m ........... 18 Response Ratio for Cases 1 to 4 - L - 200 m ........... 18 Response Ratio for Cases 1 to 4 - fi - 0.6, f - 4 Hz 20 Response Ratio for Cases 1 to a - fig - 0.35, fg - 2 Hz 20 Response Ratio for Cases 1 to 4 - fig - 0.35, fg - 4 Hz 21 Typical Response Ratio for Case 5 ..................... 21 Errors of Approximation to oi" for Different Beam Lengths ............................................... 2a Errors of Approximation to 03" for Different fig and fg 24 Plot of Q11(w) IH(w)|2 - L - 100 m, f1 - 1 Hz ......... 25 Plot of Q11(w) IH(w)I2 - L - 100 m, f1 - 5 Hz ......... 25 Plot of e11(w) and IH(w)|2 - L - 100 m, £1 - 1 Hz ..... 26 Plot of a11(w) and IH(w)I2 - L - 100 m, £1 - 5 Hz ..... 26 Figure Figure Figure Figure Figure Figure Figure 2-22: 2-23: 2-24: 2-25: 2-26: 2-27: 2-28: Plot of 011(w) for Different Beam Lengths : ............ Errors of Corrected Approx. Lengths ................... Errors of Corrected Approx. - 53-035, f - 2 Hz Variation of 0 Along the u" -p -o.35,£ -4Hz... g 5 iv to 03" for Various Beam to 03" for Various Len th - - 0.6, f = 2 Hz 3 fig g Len th - - 0.6, f = 4 Hz 8 fig g Length OOOOOOOOOOOOOOOOOOOOOOOOOOOO 28 29 29 31 31 32 LIST OF SYMBOLS The following symbols are used in this paper: C f(t) Hj(w) H (w,x) VJ Hud(w.X.£) Yj(X) L m Ru (f). Ra (T) A R- a (r) uAuB damping per unit length; flexural rigidity of beam; linear frequency (Hz); fundamental linear frequency of soil; modal linear frequency; frequency at which the Kanai-Tajimi spectrum has its peak; load which is the function of time; modal frequency response function; modal displacement frequency response function; frequency response function (transfer function); modal displacement; length of beam; mass per unit length; autocorrelation functions of accelerations at A and B; cross correlation function of accelerations at A and B; constant intensity parameter; spectral density function of ground acceleration; cross spectral density of accelerations at A and B; horizontal transverse displacement of beam; aA. (130:) us(x,t) ud(x,t) X Superscripts iv accelerations at supports A and B of beam; pseudo-static displacement; dynamic displacement; length along beam; apparent seismic wave propagation velocity from A to B; ratio of critical damping of soil; Dirac delta function; mode shape; modal damping ratio; modal response cross spectral density function; separation distance; frequency-dependent spatial correlation function; variance of curvature response for case i; variance of dynamic displacement response; variance of curvature response; area corresponding to correction Aaz(x); correction to approximation of ain(x); circular frequency (rad/sec); fundamental circular frequency of soil; modal circular frequency; first partial derivative with respect to time; second partial derivative with respect to time; second partial derivative with respect to x; fourth partial derivative with respect to x. vi 1. INTRODUCTION Due to the complexity of the earth's structure (inhomogeneity, anisotropy, etc.), rigorous investigations of structural responses to earthquake ground motions should consider the effects of wave propa- gation and spatial correlation of ground motions. These effects would be expected to be especially significant for large structures such as bridges, lifelines, dams, etc. Studies of the space-time variation of strong ground motion and its effect on structures has only recently gained prominence (Lab, at al., 1982; Bolt, et a1., 1984; Harichandran & Vanmarcke, 1986; Harichandran, 1987). In this report, the simplest model of a bridge or lifeline - a simply supported beam - is studied, and attention is focused on the stochastic response under seismic support excitations. Consideration is given to both wave propagation effects and spatial correlation effects. A stochastic space-time ground motion model resulting from studies of the space-time variation of earthquake ground motion based on the data obtained from the SMART-1 seismograph array in Lotung, Taiwan (Harichan- dran & Vanmarcke 1986) is used to analyze the effect of spatial varia- tion of ground motion on structural responses. The bending response of the beam due to transverse base-excitation is analyzed. In the analysis, stationary random vibration is assumed. The maximum response of the structure for various types of support motions with different characteristic parameters are compared. Also, the variations of the maximum response with variations of the natural frequency and length of beam as well as the fundamental frequency and damping of the soil are determined. Finally, approximate methods for the quick evaluation of the beam responses are evaluated. 2. ANALYSIS OF SIMPLY SUPPORTED BEAM 2 G o Mot o e In order to study the response of structures to spatially varying earthquake ground motion, a suitable ground motion model is required. In this study a space-time ground motion model proposed by Harichandran and Vanmarcke (1986) is used. In this model the ground Aaccelerations are assumed to constitute a homogeneous random field. The point spec- tral density function (SDF) of the ground acceleration, Sa (w), is 8 therefore assumed to be the same at all spatial locations. The correla- tion between the accelerations at two different points is characterized by a coherency function p(u,f), and the phase due to the time delay caused by wave propagation is accounted for by an exponential function, exp (-iwv/V). The cross SDF between the accelerations at two locations A and B is then written as the product of the point SDF, coherency function and phase delay SGABn(w) - Sug(”) p where p(u,f) - A exp [ 37%)- (1 - A + aA)] + (1 - A) exp [ 9%?) (1 - A + aA)] (2-2 ) o - k [1 - ]'” <2—3 > A, a, k, f0 and b are empirical constants, whose values for the radial components of a specific earthquake in Taiwan (Event 20 recorded by the SMART-1 array) were: A - 0.736; a - 0.147; k - 5,210; fo - 1.09 and b - 2.78; (2-4 ) f - w/2n - linear frequency; v - separation between locations A and B - L for the simply supported beam; V - apparent wave propagation velocity in direction AB. Any reasonable model may be used for the point SDF, and for a single-span beam the Kanai-Tajimi spectrum is adequate. This spectrum is obtained by passing a white-noise bedrock acceleration, whose spec- trum is So, through a filter (transfer function) corresponding to a one degree-of—freedom dynamic system, and has the form {1 + 4fi2[w/w g] } Sago”) ‘1 - [co/«>812!2 + afi’Iw/wg 1’s (2-5 ) where “3 and fig are the fundamental frequency and damping of the ground, respectively, and So is an intensity parameter. The parameters w and fig are usually estimated by fitting equation (2-5) to the spectra of real acceleration records. 2 Ra V b t on es onse ,ru = Horizontal. Displacement _-_.-_.__.e,,§_o_.x B on u A(t) UB(C) M We... 7 ‘ug Horizontal transverse acceleration of supports Figure 2-1 : Simple Beam With Support Excitation Consider a simply supported beam excited by horizontal transverse accelerations, uA(t) and fiB(t), at supports A and B, respectively, as shown in Figure 2-1 above. For a constant flexural rigidity EI, mass, m, per unit length and coefficient of viscous damping, c, per unit length, the equation of motion for the beam is 2 a mu,cas+mu_o (2_6) 2 at 2 at 6x where u is the horizontal transverse displacement of the beam. Decomposing the displacement into a pseudo-static part, us, and a dynamic part, ud, we obtain: u(x,t) - us(x,t) + ud(x,t) (2-7 ) where us(x,t) - uA(t) + [uB(t) - uA(t)] x/L x - distance from support A; L - length of the beam. Substituting (2-7) into (2-6) . iv . . mud + cud + EI ud - - mus - cus (2-8 ) For light damping cus is small compared to mas, and the equation of motion can be written as .. . V u + 9 u + £1 1 - d m d m “d 8(xvt) (2'9 ) where g(x,t) e - mus - - aAu - 15‘) - (13% (2-10) and the boundary conditions are ud(0,t) - ud(L,t) - 0. Using modal synthesis, we let ud(x,t) - 2 4’3 (X)Yj(t) (2-11) .1 It is well known that for a simply supported beam with length L, the normalized mode shape , ¢j(x), is a . $10!) - [12:] sin Lit—x (2-12) and the natural frequency w is J A - [51]“ [:32 Now consider a point load at x - {z- g(x,t) - 6(x-£)f(t). Substi- tuting (2-11), (2-12) and (2-13) into (2-9), mutiplying by wk and in- tegrating with respect to x over the length of beam, we obtain ~ g . 2 f3 ¢1 f(t) sec-s) dx Yj+in+ijj- - L - $3“) f(t) (2-14) since L J0 ¢j¢k dx - 0 j v k and L for the harmonic input f(t) - eiwt, the response is Y (we) - H (w;£) em (2-15) 3 YJ Substituting (2-15) into (2-14), we obtain C , 1;, arms) - [(032 2 15(5) J - w + 1w- - h) ¢ 2'16 H 1 where j(m) - 2 w - w + iwg j m In Hj(w), i is the modal bandwidth and is constant. Therefore, mej’ the ratio of critical damping for mode j, {j - -£—— varies inversely with natural frequency, and we can rewrite Hj(w) - 2 1 (2-17) m - + 12 J “’ ‘3‘”1” Using (2-15) and (2-16), equation (2-11) becomes ud(X.t;€) - 2 ¢J(X) Yi(t;€) J - § ¢J(X) Hj(w) ¢j<€) f(t) or ud(X.t;€) - Hud(X.w;€) f(t) (2-18) where Hud(x,w;€) - § ¢j(x) Hj(w) ¢j(€) is the frequency response function (or transfer function). For stationary support excitations, the autocorrelation function, Rg(x1,x2,r), of the input g(x,t) is Rg(x1,x2,r) - E [g(x1,t) g(x2,t+r)] an 1‘2] a“ - l - 1 - R (r) + R- (r) [ L L aA L2 uB x x X X +[1'L1]E2Raa(’)+il[1'fz] Rfifih) (2-19) A B B A where R (r) and R (r) are the autocorrelation functions of u and u , (1A OB A B respectively, and R (r) and R - (r) are the cross correlation func- a a a u A B B A ~ tions of a and a . (Note that R (r) - R (-r).) A B GARE OAOB Taking the Fourier transform of R8(x1,x2,r), we can obtain the spectral density function of g(x,t) x x x x 1 ,_2 l 2 ,, -.-][.-]s.. . ) Sg(xl x2 w) [ L L uA(w) L2 uB(w X]X x[ X2] _ + 1 - "l ‘2 S. - (w) + “l l - -— S" (w) (2 20) [ L L uAuB L L uBuA l where S- (w) and S (w) are the autospectra of a and u , respectively, uA OB A B and SBA93(w) and SfiBfiA(w) are the cross-spectra of 6A and 68. The spectral density function of the response is (Crandall, 1979) L L Sud(x1.x2.w) - Jo Jo Hud(x1.-w;€1) Hud(x2.-w;€2) Sg(€1.€2.w) dél déz - E Z We) ¢k(x2) HJ(-w) aka») ejkw) (2-21) 3-1 k-l where éjk(w), the modal response cross spectral density function, is L L éjk("9) "' Jo Jo ¢j(€1) ¢k(€2) 88(619529‘0) d€1 (152 (2'22) According to the definition the variance of response is the value of autocorrelation function Rg(x1,x2,r) when x1- x2 and r - 0. Since the spectral density function Sg(xl,x2,w) and autocorrelation function Rg(x1,x2,r) are Fourier transform pair and the spectral density function gives a decomposition of the variance (power) with frequency, so the variance of the dynamic displacement at any locations along the beam is the integration of spectral density function at the all frequency range with xl-xz-x 2 aud(x) - If” Sud(x,x,w) dw .. Z z 430:) Ifikhi) foo Hj(-w) Hk(w) ij(w) dw (2‘23) j-l k-l For well-separated modes, the contributions from the cross-terms (for which j i k) are very small compared to the terms for which j - k. Thus, as an approximation, we can write O Q aide) z X I¢j12 r IHJ.(w)|2 ejjw) dw (2-21.) -1 In order to evaluate Ojj(w), we use the ground motion model spe- cified by equations (2-1) and (2-2). Ojj(w) can then be obtained in closed form as ejj(e) - ——L—— [1 - ( 1)jp(L, f) cos (wL/V)] sfi (w) (2-25) (Jr)2 . us In engineering design, we are often more interested in stress levels, and the stress due to bending is propotional to the curvature u" (the second derivative of displacement u with respect to distance x. For a simply supported single span beam u" - ua + u; - ua 32_ a2_[ or u" - u - 2 ¢ (x)Y (t)] 6x2 d 6x2 j J J 1': - 2 2 ¢j(x) Yj(t) - Z ¢j(X) Yj(t) (2-26) J Repeating the same procedure as before (the only difference being the use of ¢3(x) instead of ¢J(x)), the variance of the curvature is a i.(X) - 02 u. (X) - E: E: ¢3(X) ¢" (X) I” Hj (- w) H k(w) éj k dw j-l k-l “ E: [¢3(X)]2 In IHj (w)l2 j(w) dw (2-27) where ¢3(X) - iii ¢j(X) - - [%]1/2 [%£]2 sin iflfi (2-28) 10 ume a esults 2 3 Ge c tat 0 Equation (2-27) represents the variance of curvature, 03", for the general case, i.e., considering the correlation between different sup- ports excitations as well as the phase shift (or the time delay for the seismic waves to travel from one location to another). If some parame- ters (such as the length L and fundamental frequency f1 of the beam, the fundamental frequency f8 and damping fig of the ground) change, ai"(x) will also change. Some numerical results are presented here which investigate the effect of these parameters on the beam response. A computer program was written by the author to perform the necessary calculations. Due to the very small contribution of higher modes to ain(x), and the symmetry in ai"(x) along the beam, only the first four modal responses were accounted for and only the left half of beam was examined. Figure 2-2 depicts the distribution of the normalized variance of curvature, 0:,(x), along the beam. The normalization is performed by dividing the values of 0:,(x) by afi"(L/2). It can be seen that aiu(x) - 0 at the supports (because the moment is zero) and the maximum value of a§"(x) is at midlength. Since we are mostly interested in the maximum variance, we will mainly focus our attention on ai"(x) at the midspan, i.e., at x - L/2. It should be pointed out, however, that the maximum value of 0:" is not always at the midspan, and may appear at some other location for specific cases. Figure 2-3 shows an example of this. For the case of f8 - 4 Hz, f1 - 1 Hz and fig - 0.35, the maximum values of can are at approximately x - 0.4L and x - 0.6L. This occurs due to the fact that when fundamental frequency of the structure, f is equal to 1 Hz, the 1’ 2n"( [‘/2) .(Xl/U 2 u - ....t(L/2) 2 u .(Xl/U 2 u I" ll 1-0 I ' l ' l r l ’ ,. 1;. tg-a Hz, 58- 0.6 I ’7’ _ 0.5.. -— — — 18-2 Hz. 55‘0'35 / _ ----- ‘3" Hz. leg-0.35 // / / -4 L-lOO m. v-iooo m/a ,’/ 0 6-4 (131 Hz / _. . C// / l .. x/ / . a / _ o 4 I / ,’/ ~ / 0.2-4 ,’/ z / // .. // a I!“ 0.0 I I I ' l l ‘ 01) 0.1 (l2 013 0u4 (l5 X/ L Figure 2-2 : Distribution of 03" for Different fig and fg 1-2 . 1 . l r r l , EXACT sownou , —- —- FIRST MODE ONLY ..l 1'0 L-SOO m. V-lOOO m/I / . [Ba-0.35. {8-4 Hz / / . 31-1 Hz 0.5- / _ , r 0:4 (15 ue X/L 2 -3 : on" for Exact Solution and First Mode Approximation 12 second mode frequency is 4 Hz and is equal to the ground motion ‘frequency. Hence for this case the second mode response becomes significant, and the maximum values of 03" for the second mode response are at x - L/h and x - 3L/4. The combination of the first four modal responses makes the maximum values of 03" shift slightly to the two points x - 0.4L and x - 0.6L. However, the difference between ai"(L/2) and au",max is small, usually less than 10%, and therefore this rare situation can perhaps be neglected. Figures 2-4 to 2-7 show how the parameters f1, L, fig and f8 affect 02 . Figure 2-4 and 2-5 indicate that can is very sensitive to changes u" in f1 and L. As the natural frequency and length of the structure increase, 0:" decreases very rapidly. It should be noted, however, that the variance of the 'stress response is equal to (EI)203", and E1 is expected to be quite different for beams of different fundamental fre- quencies and lengths. Figure 2-6 shows that 0:" does not change signif- icantly with fi8 over its normal range. When the damping ratio of the 2 ground, fig, decreases below that of the structure, {1 - 0.05, au" increases sharply, but when fi8 is larger than {1, 03" decreases only slightly. (It should be noted that fig is almost never less than {1.) As expected, Figure 2-7 shows that the maximum value of 03" occurs when the ground motion frequency, f8, is near the fundamental natural fre- quency of the structure (resonance). When fg tends to zero (i.e., when the base excitation becomes static) the dynamic curvature response vanishes, as it should. When fg is higher than and well separated from the structure fundamental frequency, f1, 03" is approximately constant. 2 u (L/Z) (x10”) 2 ll 0.. ,, (I./2) (x 10“) U 13 15+08~ I I I I I I I I I' E IE+O7E E IE+06; L== 100 m. v= 1000 m 1 E fg= 2 Hz. pg: 0.6 E IE+O5E E 1E+O4E E 1000é E IOOE E 10; E '3 i 1 I I I I I r I I I 0 1 2 3 4 5 6 7 8 9 10 NATURAL FREQUENCY :1 (Hz) ‘ Eigg£g_2;fl : Effect of Fundament Frequency of Beam on 02" u IE+094 J E a "L J 1“ ‘ 08 3 v=1000 m/s. flat Hz 33, 13+06 3 3‘ J IE+05E . 1E+04j 3 10004 J 1004 J i 3 104 E 3 1 1 l 0 100 200 300 400 500 LENGTH L (m) Fi u e -5 : Effect of Beam Length on 0:" .. (IA/2) (x10°) l4 L-Ioo m. v-1000 m/s . tau-2 Hz. 11-1 Hz 4 "o v-i 3.5 6? \, :3 I 91': 1... 4 4 l 1 -I 0 r I I r T i f I I I I T I I f T T l I 01) CHI (12 (13 (14 (15 (16 (L7 (18 (19 L0 THE DAMPING RATIO OF SOIL fig e -6 : Effect of fi on 02" g u 180 J r 7 I I U I —V I I l f I I I l I I 160‘ L-100 1n. v-1000 31/: j .. '1 120- _ 100- I 80- ‘ 60- .J a: 404 ‘ . I 20- ‘ d J o fif T T' j fir I T r r r j ' T I T i I r 0 1 2 3 4 5 6 7 8 9 10 GROUND MOTION FREQUENCY fg (Hz) . 2 i u e -7 : Effect of Ground Motion Frequency on au" 15 2,3,2 Some Special Cases In order to examine the effect of correlation decay and phase shift between different support excitations on the response of the structure, we consider some special cases of support excitation. Case 1: Fully correlated and in-phase excitations at supports A and B. This corresponds to the current practice of considering identi- cal support excitations. For this case, p cos (wL/V) - 1, and from equation (2-27) it is clear that only odd modes contribute to the response. Case 2: Phase shift only. Here we assume that there is no loss in coherency between the support excitations at A and B, but that there is a time delay for seismic waves to propagate from A to B, i.e., p - 1. Case 3: The excitations at supports A and B are uncorrelated, i.e., coherency is zero. For this case p - 0. Case 4: No phase shift. If the waves propagate vertically, the excita- tions at every point would be in phase, i.e., there would be no time delay between support excitations at A and B and hence cos (wL/V) - 1. Case 5: Fully correlated but out-of-phase excitations at supports A and B. For this case p cos (wL/V) - -1 and only the even modes contribute to the response. Figures 2-8 to 2-15 show comparisons of the five special cases with the general case. In these figures, we denote the response vari- 2 i! 1,2,...5. Since we are interested in the maximum response, for cases 1 ance for the general case by 0:" and those for case 1 to 5 by a i = to 4 the response ratio, ai(L/2)/a§"(L/2), vs. the non-dimensionalized fundamental frequency of the beam, flL/V - wlL/(ZnV), is plotted; for 16 case 5, the maximum values of variance are at quarter-Span and three- quarter-span (the dominant mode being the second mode), and thus we consider the response ratio a§(L/4)/ai"(L/2). Figures 2-8 to 2-11 depict results for different lengths of the beam with V - 1000 m/s, fig - 0.6 and fg - 2 Hz. Figures 2-10 and 2-12 to 2-14 allow comparisons for various values of fig and f with L = 100 m and V - 1000 m/s. From these results we can conclude that: 1) 2) 3) The longer the structure, the larger the relative difference between oi and 03". The maximum ratio of ai/ai" increases from 1.5 to 4.5 as the length L increases from 20 m to 200 m. This clearly shows that the effect of spatial variation of ground 'motion on the re- sponse of the beam depends on the separation of the supports and the time delay for wave propagation. The extremes of the ratio ai/ain occur at approximately flL/V - k/2, k - l,2,3,..., for which cos (21rf1 L/V) - -1 and +1, i.e., when the excitations at frequency f1 are predominantly out-of-phase and in- phase, respectively. At the minimum points, cases 1, 3 and 4 (which do not contain any phase shift) give higher responses than the general case (except for very short beams). Case 2 has a lower value of 03" at the minimum points due to the assumption of fully out-of-phase excitations at all frequencies. The variance of curvature for the fully correlated case is always higher than that for the general case. This is because there is no coherency loss between the support excitations for the fully corre- lated case. In equation (2-27), p(L,f) represents the coherency between the two support excitations and it decreases very rapidly ..(l./2) 2 u a;(I./2)/a ..(L/2) 2 u (L/2)/a U 17 2.0 ; l ' L=20 tn. ValOOOm,-'s ———-— FULL CORR. . fig-OB, 18-2 Hz — — -— PHASE: SHIFT ONLY 4 ————— UNCORRELATED ------ NO PHASE SHIFT ~ h‘\"’""";7—:“: /._‘,\ \ .. / ~—________ 05«/ — 4 -I 0'0 r l— I 0.0 0.5 1.0 1.5 2.0 flL/V re - : Response Ratio for Cases 1 to 4 - L - 20 m 3.0 l l I a L-SO tn. V-lOOOrn/a FULL CORR. L 53-013. 11-2 Hz —- —- — PHASE sms'r ONLY 2'5“ ————— uncomursn ‘ - ------ N0 PHASE SHIFT 1 N )a—N——""--------—. \_—/ E .. 0.0 I I I 0.0 0.5 1.0 1.5 2.0 Figure 2-9 : Response Ratio for Cases 1 to 4 - L - 50 m a; ........_~./._ ._‘ // \\\ ,// ‘I— ‘ Qfifi"’ \" ‘ 1 . 0.0 | l I 0.0 0.5 1.0 1.5 2.0 f,L/V afi(L/2)/0§.(L/2) ‘ FULL cons. Jr 4'5? -— — -—PHAS£: sHIII'r ONLY 4.0 _ —-—-— UNCORRELATED .4 ------ NO PHASE SHIFT ~ 53-0-6. t‘-2 Hz L-ZOO In. V-IOOO tn/s 4 II j Eigure Z-ll: Response Ratio for Cases 1 to 4 - L - 200 m 4) 5) 6) 19 with increasing L, and is close to zero for high natural requencies. On the other hand, the dominant contribution to the response comes from the first mode and when p(L,f) is zero a will be equal to PHV 20:". This is why for longer beams (say, L > 200 m), oi z 20in for large £1. The results also show that no matter how long the beam and how high the fundamental frequency, over-conservative estimates of the response are always obtained if fully correlated and in-phase excitations are assumed. For case 5 (fully correlated but out-of—phase excitation), 02 is 5 always much less than can (see Figure 2-15). For this case, the first mode is not excited and hence the second mode is most dominant. Consequently the maximum value of a: is at f1 z fg/4,I i.e., f2 z f8. Note that the function IHj(w)|2 decreases very rapidly when ”j increases (wj - jzwl), and it is due to this that a: is only 2 to 15 percent of the value of 03". For other cases, the response variances are sometimes over-predicted and sometimes under-predicted depending on the length L, fundamental frequency f1 and wave velocity V. Here one point worth noting is that for case 4 (neglecting phase shift but considering coherency decay), the response is usually always higher than that for the general case. The difference, however, is relatively small ( aZ/ai" usually being less than 1.5) when flL/V is greater than 1. The values of fg and fi8 also have an effect on the ratio 03/02" u . From Figures 2-10 and 2-12 to 2-14 it is apparent that for the same 2 length and velocity, the relative difference between 0% and 0 NC 2 becomes larger when fg increases from 2 to 4 Hz. The ratio al/OU also changes as fig changes but no simple trend is apparent. Of the five special cases, cases 1, 2, 3 and 5 do not consider the 20 5.0 I I , 4 51 -——- FULL CORR. r ' q —— — —- PHASE SHIFT ONLY 1 4,04 —-—-— UNCORRELATED I. . ------ NO PHASE SHIFT L a 8-5- « - J E 3 0 L=100 m. v=1000 m/s 4 V - q o \. —. A cu \ ..I V N ‘ b 1f0 1.5 2.0 f1L/V F1gg;g_2;lg: Response Ratio for Cases 1 to 4 - fig - 0.6, fg - 4 Hz 3'0 I I J 7 """"-""'— FULL CORR. 1 — — "— PHASE SHIFT ONLY 2.5-1 «4 —'—"— UNCORREIATED ------ NO PHASE SHIFT H L==100 In. Vv-IOOO m/s “I fl‘-0.35. 18-2 Hz ..(L/Z) oi(L/2)/0 r I 011 (LS 1A0 1n5 21) flL/V Figure 2-13: Response Ratio for Cases 1 to 4 - fig - 0.35, fg - 2 Hz a§(L/2)/ag.,(L/2) Figuge 2-14: Response Ratio for Cases 1 to 4 - fig - 0.35, fg - 4 Hz ,(L/z) 2 u 053(1./4)/a 3.04 0.15 (105 0.00 21 I l I FULL CORR. _' —' —— UNCORRELATED """" NO PEUSE SHIFT -- -- —- PHASE SHIFT ONLY 7 L=100 In. V=1000 31/: 3380.35. {Ba-i» Hz flL/V I Ti 1 L-iOO m. V-lOOO 111/- '03. f -2 Hz 5: z 4 FULLY CORRELATED BUT OUT-OF-PHASE “ T ‘____,__, I I I T 00 05 L0 L5 f ‘L/ V Figuge 2-15: Typical Response Ratio for Case 5 2.0 22 correlation decay between different support excitations, so they are simpler than the general case; case 4 does not consider wave propagation but accounts for correlation decay. From the numerical results, it is apparent that the coherency affects the variance of curvature more than the time delay, and therefore in order to obtain more accurate response predictions, at least the correlation decay between difference support excitations should be taken into account. 2 4 A roximate Methods 0 Evaluatin the es onse Variance Equation (2-27) allows the evaluation of the response variance of the beam excited by random ground motion. However, for practical pur- poses, it would be desirable to find an approximate method which simpli- fies the numerical computations while yielding reasonably accurate results. Since the first mode response is most dominant, it should be easonable to consider only the first mode response in equation (2-27). i.e., a2 (x) z [¢"(x)12 In (w)|2 e (w) aw <2-29) u" 1 _m l 11 Assuming that .¢11(w) is 'slowly varying in the vicinity of the narrow spike, IH1(w)I2, we obtain the commonly used approximation: 2 fig o11(w) IH1(w)|2 dw z o11(w1) If” IHl(w)| dw n - ¢11(w1) 3 (2-30) zglwl where the well-known result I” lul(w)|2 dw - " 3 (2-31) -co Zflwl 23 has been used. Thus we can write 02.00 .... [$0012 <1» (w ) A (2-32) u 1 ll 1 2; w3 1 1 The above approximation was evaluated for different beam lengths, and fig and f8 values. Figures 2-16 and 2-17 give the percentage error of the approximation compared to exact value of 03" at midspan. The results indicate that when the fundamental frequency of the beam, fl, is small (say, lower then ground motion frequency f8), the approximation is quite acceptable - the error usually being less than 10%. But when f1 increases, the error also increases - up to -70% at f - 10 Hz. In 1 order to obtain better results, an improved approximation is necessary. In order to derive an improved approximation, the behavior of the functions in equations (2-27) and (2-32) were investigated. Figures 2-18 and 2-19 show comparisons of the actual and approximate shapes of the product ¢11(w)|H1(w)I2. It can be seen that for f - 1 Hz (i.e., f l l < fg)’ the actual curve and approximate curve almost coincide, but for f1 - 5 Hz (f1 > £8) there is a significant deviation between the two curves. This is the main source of error. To reduce the error we can perhaps add a correction term when f1 is large. From the separate plots of @11(w) and IH1(w)|2 plotted in Figures 2-20 and 2-21 it can be clearly seen that the maximum value of ¢11(w) is at a frequency slightly lower than fg’ after which ¢11(w) decays very quickly. For IH1(w)|2, the peak occurs approximately at f - f1 and the values of IH1(w)|2 become very small when f departs from f Note also 1’ that the magnitude of |H1(w)|2 decreases extremely rapidly at all fre- quencies when f1 increases. Thus, when f1 is larger than fg’ the pro- duct of ¢11(w) and |H1(w)|2 becomes significant at the lower frequency range. We should, therefore, add a correction when f is larger than 1 % ERROR OF APPROX. TO a:,(I./2) 24 30 I I r . I I ' I I f . T I I ' I . 20 L= 20 In -—-— — L= 50 In 10 — - — — L=100 m -------- 1.8200 m o ’__‘ {8-2 H2. 5830.6 ‘\ v=1ooo m/s I 3 O— MWWuNJJMMWT I l 4 5, 6 7 f1 (Hz) Eiguge 2-16: Errors of Approximation to can for Different Beam Lengths Figure 2-17: Errors of Approximation to 03" for Different fig and f8 30? I f I . I . I . I . I I I ' I ' I R q ., «I a? 20-3 L=100 In. V=1000 m,’s 18-2 Hz. 58' 0'6 '5 :1: 10-:: / \ -— —- - — fg‘z H2. flg=0.35 "5 b 03 ...... ~ -------- '3'4 Hz. 38:0.35 : o E \ \\ ...—— a \ 5 E_: -10.: \ \\\ / / \é‘ >3 3 \ \‘v’ a o: '20—: E C. 1 : D. -301 ‘1 <3 1 .. C: -40; -: I: -5o-‘ ~‘ 0 : \ 1 c: : \ : C: -50-: \ 1 a d I .— — — ~ \ : :1 \ / / " N -701 \ -" 1 3 1 -80 I I l l'i r fi ‘ fi I F r I I l l I I 0 1 2 3 4 5 8 7 8 9 10 1, (Hz) ‘f’,,(m)|H(w)l2 ’,.(w)lH(w)l" (MW) 25 8 ' I ' l 7 I I I o T --— ACTUAL CURVE l 7_ -. - - - APPROXIMATIVE CURVE q 1 6~ L=100m. V=1000m/s " . fig: 0.6, fg=2 Hz 1 5‘1 1,: 1 Hz a 4- l .. . l ‘ 3- q 2« J “ 1— I " / . O—Ir——— r r I V I ' l o 1 2 3 4 5 LINEAR FREQUENCY f= 02/21!“ (Hz) w: Plot of one») IH(w)I2 - L - 100 m, £1 - 1 Hz 7 . fi— . I t I r l . ' —— ACTUAL CURVE 6 - ' - - - APPROXIMATIVE CURVE L-lOOm. V=1000m/s fig= 0.6. {822 Hz ff 5 Hz - LINEAR FREQUENCY f= w/zn (Hz) Figure z-igz Plot of ¢11(w) IH(w)l2 - L - 100 m, £1 - 5 Hz 4’..(w). |H(w)lz ¢,,(w). |”(w)|2 26 120- i i 1004 f¥/\ , /" 80- H I l I I I I I 1 U I U I I ----- IH(w)l' (x2000) ("u(u) L=100rn. V=1000m/s figs 0.6, fg=2 Hz f,= 1 Hz .0 ll _ I. . 'I 60- 'I — H I. . 'I 40¢ 'l .. ‘I . I. y 'I 20" l I -" ' I ‘ I | a I \ \ 0 j" .37? T r r V f F I ‘r U— T I" 'r r T 1' 0 5 10 15 20 LINEAR FREQUENCY f= 02/er (Hz) Figure 2-20: Plot of 011(0) and IH(w)|2 - L - 100 m, £1 - 1 Hz r I I I I U I I I l I I I I j I I I I 12"“ ----- IH(w)l' (xw') - . .5 —— M“) w s . 0 _. 1 0 § , L=100rn. V=1000m/s ~ § :1 58?: H‘ ‘ ll Cl ll u \ l I 50‘ l I ‘I l I l l l ‘ : z - d l I d I I 204 ; l _ .° I \ . p5..- - 7 _ ' \\ _ __ __ .— I ‘ M O r r ; # r1r_ _I*' T r fl r—-F-‘T—fi hi fi 0 5 10 15 20 LINEAR FREQUENCY f= 00/211’ (Hz) Figure 2-21: Plot of 011(0) and IH(w)I2 - L - 100 m, f1 - 5 Hz 27 the frequency at which ¢11(w) attains its maximum value: To determine the frequency at which @11(w) attains its maximum value is difficult for all conditions, but for the fully correlated case the peak occurs at the frequency at which the Kanai-Tajimi spectrum has its peak, namely f (was p 2B3 fg (2-33) For small fig, fp z fg' Figure 2-22 depicts plots of Q11(w) for differ- ent beam lengths and it shows that the location of the peak of Q11(w) changes for the general case depending on the parameters used. The frequency V/(hL) plays an important role, since cos (2nfL/V) becomes negative as f increases beyond V/(hL), and therefore ¢11(w) begins to decrease (see equation (2-25)). If fp < V/(AL) then the peak of ¢11(w) occurs near fp’ otherwise it occurs near V/(AL). Hence it would seem reasonable to add the correction when either f1 is larger than fp or V/(AL). In equation (2-30) it was assumed that ¢11(w) was constant and equal to ¢11(w1). The shaded area of Q11(w) shown in Figure 2-21 was not accounted for. Assuming that in the frequency range from O to wl - 21rf1 we can approximate IH1(w)I2 by |H1(0)I2 - l/wa, the correction to equation (2-31) may be written as A02(x) - lz [¢i(x)]2 x (shaded area in Figure 2-21) m w 1 - i: [¢i(x)]2 [ Io ¢11(w)dw - w1¢11(w1) (2-3A) The integral in the above equation may be evaluated through a numerical quadrature rule such as Simpson's rule. The integrand is now 28 l 160 F r I I IV I T "—T I I I I I ‘ \ -——————- L= 50 H) a - — — L=100 In - ----- L=200 H1 — fig=0.6, fg=2 Hz. v=1ooo m/s' ‘13“(0) FREQUENCY (Hz) Eigure 2-22: Plot of ¢11(w) for Different Beam Lengths a smoothly varying function without the peak-like behavior of |H1(w)l2 and can therefore be evaluated easily without significant error. The corrected approximation may thus be summarized as follows: r n 2 _I_. 1! l¢1(X)] 011(w1)2fi1wi , when wl 5 mp and wl 5 2L 0:"(x) - I ' (2-35) n 2 2 [$1(X)] ¢11(w1)§§§;§ + Aau" , elsewhere L where w - fo . P P Figures 2-23 and 2-24 show the percentage errors between the corrected approximation and the exact solution of 03" at midspan. As expected, the errors are greatly reduced. The corrected approximation was usually accurate to within 15%, although for some cases it was in error by about 25%. In all cases the corrections substantially improved 29 10 ‘ I— I I I T I I I I I ' I I I ' I I l x: 20 m : C 50 m .I E 100 m j 2.. 200 :11 i < : L 100 m. V 1000 tn, 3 {‘32 Hz. 33' 0.6 . O 101 —- — 18-4 {-12.5 a 0.6 1 Q: 8 . a. . / \ — — — ‘3’2 Hz. £8=0.35 : 95 5; / \ ------ {8-4 Hz. ”2‘0-35 _: C: I 5:3 -—-—— -\ I 5‘ 63‘ ”’ “\ -1 c: i _: z N .5 -.. -.-"‘- :_—. "" O b / ‘~\: 0 o : E. Ed _ O I Q: I O _ a»: .. c: 2 E3 I 0 J R i I I t 8 9 10 f1 (Hz) Figure 2-24: Errors of Corrected Approx. to 03" for Various fig and f8 30 the variance estimates. Finally, the distribution along the length of the beam of the exact solution, the exact first mode reponse, and the corrected approxi- mation, are compared in figures 2-25 to 2-28. The variance ai"(x) is 2 normalized by au",exact (L/2) in these firgures, and it can be seen that the corrected approximation matches the exact first mode solution very well. As discussed earlier, for some cases there are errors arising from considering only the first mode, especially when f1 - fg/h (for which case the second mode response gives significant contribution). But as far as the maximum responses are concerned the approximation is still acceptable. 31 1.2 I I I I T I I r I . EXACT SOLUTION . -- — FIRST IIOOE ONLY 1'0“ ----- FIRST MODE APPROXIMATION , , r " I; ‘ L-100 m. V-lOOO m/a ’ . \ 0 8~ fig-Ofi. :8-2 H: d :3 “-1 Hz / g I // . Nb: 0.6“ '- :: . ii I J 04 // - I (124 A 01) ' l_ . r ' I ' I ' 0.0 0.1 0.2 0.3 0.4 0.5 X/L Figure 2-25: Variation of 0:" Along the Length - fig - 0.6, fg - 2 Hz 1.2 7 I _] U I fl I ' I q EXACT SOLUTION - ._ —_ FIRST MODE ONLY 1-0“ ————— FIRST MODE APPROXIMATION I; L-IOO In. V-1000 m/a \ 0 5'“ fll-o°6' f‘ml Hz J :2 ' 11-1 Hz Nb” 0.6'1 .1 > . ' :4 :g044 ‘ b 1 02- " 0.0 r I ' I T I O0 01 02 03 04 05 X/L Figure 2-26: Variation of 03" Along the Length - fig - 0.6, fg - 4 Hz 32 102 T— I I I I I I t T . EXACT SOLUTION . — —- FIRST IIOOE: ONLY 1'0‘ ----- FIRST MODE APPROXIMATION " It; L-100 m. V-1000 In/a E 0.8- fig-OJS. f‘-2 I-Iz N" 0.6“ i J ... E / , my 0.4- — 4 . (l2— _ OJ) . 1 ° I If I ' r I (10 0.1 (12 013 0u4 (15 X/ L Eiggge 2-27: Variation of 0:" Along the Length - fig - 0.35, fg - 2 Hz 1.2 l' I 7 l I I T I ‘ EXACT SOLUTION ‘ — -— FIRST MODE ONLY 1'0“ ----- FIRST MODE APPROXIMATION a L-IOO m. V-IOOO In/s } 0.8a ‘e'°°35- ‘z'4 “2 _ x: 11.1 Hz g 1 8’ 0.61 - "e \ d A x ‘1, 0.44 - N O O I ~1 O.O ' 0.1 I 0.2 0.3 0.4 0.5 X/L Figure 2-28: Variation of can Along the Length - fig - 0.35, fg - 4 Hz 3. CONCLUSIONS From the above study of the response of a simply supported beam to differential support excitations caused by the Spatial variation of ground motion, the following conclusions may be drawn:' 1) 2) 3) 4) 5) The spatial variation of ground motion will significantly affect the response of structures to the random ground motion, especially for long structures; The excitation condition (or input condition) is very important for estimating the response of structures to ground motion. For simply- supported beam-like structures, assuming identical support excitations will lead to conservative response estimates compared to the general case which includes correlation decay and phase delay. Assuming out-of-phase fully correlated support excitation is un- reasonable and will seriously under-predict the responses. The correlation between support excitations is more important than the time delay (or phase shift) in predicting the response for typical simply-supported bridge lengths and seismic wave velocities. The response of a beam-like structure to random ground motion is very sensitive to the length and the fundamental frequency of the structure, though the properties of the ground soil also affect the response. For a simply supported beam, estimates of the response to correlated support excitations obtained by considering only the fundamental 33 34 modal response yields sufficiently accurate results. For practical purposes, the proposed approximate method can be used to obtain very reasonable results with a minimum of computations. If the natural frequency of the structure is very low (lower than one quarter of the ground motion frequency), consideration of higher mode responses will give better results, since higher mode frequencies may lie within the dominant excitation frequency range. LIST OF REFERENCES LIST OF REFERENCES Bolt, B.A., Abrahamson, N., and Yeh, Y.T. (1984). "The Variation of Strong Ground Motion Over Short Distances," Proceedings of the Eighth World Conference on Earthquake Engineering, 183-189. Crandall, S.H. (1979). "Random Vibration of One- and Two-Dimensional Structures,” in Developments in Statistics, P.R. Krishnaiah (ed), Academic Press. Harichandran, R.S., and Vanmarcke, E. (1986). "Stochastic Variation of Earthquake Ground Motion in Space and Time," Journal of Engineer- ing Mechanics, ASCE, 112(2), ISA-174. Harichandran, R.S. (1987). "Stochastic Analysis of Rigid Foundation Filtering,” Earthquake Engineering and Structural Dynamics, 15(7), 889-899. Loh, C-H., Penzien, J., and Ysai, Y.B. (1982). "Engineering Analysis of SMART 1 Array Accelerograms,” Earthquake Engineering and Struc- tural Dynamics, 10, 575-591. Masri, S.F. (1976). "Response of Beams to Propagating Boundary Excitation,” Earthquake Engineering and Structural Dynamics, 4, 497- 507. Masri, S.F. and waadia, F.E. (1977). "Transient Response of a Shear Beam to Correlated Random Boundary Excitation," Journal of Applied Mechanics, Transactions of the ASME, 44(E3), 487-491. Scanlan, R.H. (1976). "Seismic Wave Effects on Soil-Structure Inter- action," Earthquake Engineering and Structural Dynamics, 4, 379-388. Vanmarcke, E. (1976). "Structural Response to Earthquakes," in Seis- mic Risk and Engineering Decisions, E. Rosenblueth and C. Lomnitz (editors), Elsevier, Amsterdam, 287-337. 35 N)! All“ R" m“ Ll 51