APPLICATIONS OF DIGITAL SIMULATION OF GAUSSIAN RANDOM PROCESSES MASANOBU SHINOZUKA 1. INTRODUCTION. Columbia University New York, N.Y., U. S. A.

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1 APPLICATIONS OF DIGITAL SIMULATION OF GAUSSIAN RANDOM PROCESSES MASANOBU SHINOZUKA Columbia University New York, N.Y., U. S. A. 1. INTRODUCTION In the last two decades, much research effort has been devoted to the application of the stochastic process theory in the general area of engineering mechanics and structural engineering for the purpose of predicting the dynamic structural performance with a better accuracy and of assessing the over-all structural safety with a better reliability by considering more realistic analytical models of load-structure systems. Naming only a few, possible applications of this stochastic approach include (a) analysis of panel vibrations of H. Parkus (ed.), Random Excitation of Structures by Earthquakes and Atmospheric Turbulence Springer-Verlag Wien 1977

2 22 M. Shinozuka aircraft induced by boundary layer turbulence, (b) analysis of ship motions caused by ocean waves, particularly during a storm, (c) analysis of aircraft response to gust vertical velocity, (d) response analysis of off-shore structures to wave and wind forces,(e} statistical strength analysis ~f engineering materials with randomly distributed material properties, (f) analysis of the effect of the randomness in geometrical configuration and/or in mechanical constraint of a structural component (due, for example, to fabrication errors) on the vibration and buckling eigenvalues, (g) study of random surface roughness of bridge pavement and airport run-way for the purposesof analyzing the vehicle and aircraft vibration caused by the roughness and of stress analysis of pavement systems under the action of vehicles and aircraft, and of course, (h) response analysis of structures subject to earthquake acceleration or to atmospheric turbulence. In spite of the recent remarkable advance in these areas of study, however, the present state of the art still leaves a number of difficulties unsolved that must be overcome before the approach becomes more useful. Such problem areas include (1) random response analysis of highly nonlinear structures,

3 1. Introduction 23 (2) failure analysis of structures under random loading, (J) analysis of extreme values and, (4) property problems associated with random material. The recent advent of high speed digital computers, however, has made it not only possible but also highly practical to apply Monte Carlo techniques to a large variety of engineering problems including those mentioned above. This paper presents a technique of digital'simulation of Gaussian random processes and associated envelope processes. Such simulations are accomplished by summing a large number of cosine functions with w~ighted amplitudes and random phase angles and used as the basic tool in a general Monte Carlo method of solution to a wide class of practical problems in engineering, particularly those mentioned above. See, for example, References 1-1.

4 24 M. Shinozuka 2. SIMULATION OF A RANDOM PROCESS A basic represent~tion of a homogeneous Gaussian (onedimensional and one-variate) random process f (x) with mean zero and spectral density S (w) in the form of the sum of th. f t" h. t d f t 22 e cos1ne unc 1ons as ex1s e or some 1me (1) where ~k are random angles distributed uniformly between and 2TI and independent of ~.(k ~ j), and J Ak = [s~(wk)~wj/ 2, (2) with (J) being the one-sided spectral density function (see Fig.1). For digital simulation of a sample function f(x) of f(x) and therefore of f (x), Eq.1 is used with ~k being replaced by their realized values ~k; N f(x) = '{2 > Ak cos (wkx - k=1 cpk) (4) with the aid of Eqs.1 and 4 it is easy to show that Eq.1 is ergodic at least up to the second moment. Speaking of structure-related applications, however, until Borgman 2 3 published a paper simulating ocean surface

5 2. Simulation. of a Random Process 25 elevation as a multidimensional process essentially in the same form, little attention had been paid to this representation in spite of its substantial advantage over the standard method in which a random process was digitally generated as output of an appropriate ( analytical ) filter subjected to a simulated white process. The use of this filtering technique, although limited in its practical applications to a one-variate one-dimensional process has dominated a large number of papers involving simulations of a rar1dom process. Practical digital simulation of a multidimensional process has been made possible, as mentioned above, by Borgman 2 3 in principle through the preceding expression consisting of a sum of cosines and by Shinozuka 1 whose method reduces, in case of one-variate one-dimensional situationsto the use of the following expression; (5) where ~k are as previously defined and wk are realized values of random frequencies distributed according to the density function g(w) = S (w)/6 2 with f -oo s (w)dw (6) Borgman 2 3 and Shinozuka 1 ' 2 ' 11 also investigated the digital simulation of the multivariate (but one-dimensional) process,

6 26 M. Shinozuka the former making use of the filtering technique and the latter in the more convenient form of the sum of the cosine functions. Reference 2 shows that the autocorrelation function R(~) of f(x) given in Eq 1 converges to R (~) of f (x) in the form of 1/N 2 as N----7 The same trend in convergence can be shown2 to exist between the spectral element S(w)4w of f(x) given in Eq.1 and S (w) off (x), whereas the spectral element of f(x) given in Eq.5 converges much more slowly to the target spectral element 1 There is a constraint to be imposed upon ~w in Eq.2. This constraint is due to the periodicity of the sample function f(x). Obviously, the period of f(x) is T = 2n/4W and therefore, depending on the purpoee of the simulation, ~w has to be so chosen that T is long enough for that purpose. A significant improvement in the efficiency of digital simulation has been suggested by Yang3 in the following form: f(x) = y4w Re F(x) (7) in which Re F(x) represents the real part of F(x) and F(x) (8) is the finite complex Fourier transform of [2 S~(wk)] 1 / 2 e-i~k with wk and ~k defined in Eqs.1 and 2. The advantage of Eq.S is such that the function F(x) can readily be computed

7 2. Simulation of a Random Process 27 by applying the Fast Fourier Transform (FFT) algorithm, hence avoiding the time-consuming computation of a large number of cosine functions. In the preceding discussion, the spacing 4w in the frequency domain has been taken as constan~. This, however, does not necessarily have to be always observed. It is possible and in fact may even be advisable to use variable spacings depending on the extent of fluctuation of the spectral density to optimize the number of cosine terms in the summation in Eq.1 with finer spacing in those domains of frequency where the fluctuation of the spectral density is more rapid and coarser spacings elsewhere. It is also likely that such variable spacings will increase the period T of the simulat d process. If this is done, however, the simulation through the FFT technique does not appear to be possible. In the same reference 3 1 Yang also proposed to simulate the envelope process v (x) of a random process f (x) by folloo wing the definition of the envelope process. 24 (9) where f (x) is the Hilbert transform of f (x) and, in the present case, can be written as f (x) = J [sin wx du(w) - cos ll'x dv((-<!)] (1) where U(w) and V(w) are mutually orthogonal, real processes

8 28 M. Shinozuka wi'th orthogonal increments. A It then follows that f (x) can be simulated as f(x): A (11) Hence, the envelope process v (x) can be simulated as v(x): [ 2 -"2 J 1/2 v(x) = f (x) + f (x) (12) As an example, consider the response process y (t) of a single degree of freedom system to a Gaussian white noise excitation x (t) with constant spectral density S o o' y (t) + 2Cw y (t) + w2 y (t) = x (t) (13) It is well known that the spectral density of y (t) is s (lu) = y s (14) and the standard deviation 5 of y (t) is y ns 1/2 2C w3 6 = ( ) Y (15) Wi tl1 the aid of EqE:.1, 11, and 12, a segment of sample function y(t) of simulated process y(t) for y (t) and that of simulation v(t) of the envelope process v (t) are compuo ted and shown in Fig.2 and J. Figure 2 is for the case where the damping coefficient ~ =.2 and hence the process y (t)

9 2. Simu1ation of a Random Process 29 is narrow-band. This fact is we11 demonstrated by the smooth behavior of the samp1e enve1ope. When the 1oca1 maxima of the samp1e enve1ope do not coincide with the 1oca1 maxima (peaks) of the process, they ref1ect the 1oca1 minima (troughs) of the simu1ated process. Fig.3 shows the samp1e functions of v(t) and y(t) for C =.5. In this case, tb.e process y (t) is substantia11y wide-band and this fact is c1ear1y seen from the much wi1der f1uctuation of both- simu- 1ated enve1ope and simu1ated process, a1though the simu1ated enve1ope surprising1y we11 ref1ects peaks and troughs of the simu1ated process even though the process y (t) is wide-band. In terms of sim~1ation efficiency, the computer time wi11 be significant1y reduced if one is interested in peak- and trough - va1ues of the process and if the process is narrow-band, since then the smooth nature of the envelope function makes it possib1e to use much 1arger interva1 between successive time instants at which the va1ues of the simu1ated process is eva1uated. The order of magnitude of this interva1 can be that of the apparent period of the process, which obvious1y is much too 1arge for simulation of the process itself. Again, following Yang3, f(t) is written as f(t) = ~ Im F(t) (16) and hence v(t) can a1so be simulated through the FFT techniqu~

10 21 M. Shinozuka The Gaussian property of the simulated process (Eq.1) comes from the central limit theorem because it consists of a sum of a large number of independent functions of time. Obviously, the larger the value of N, the closer the simulated process approaches Gaussian. Recent paper by Yang25 indicates that if, for example, N = 5 is used in Eq.1, the simulated process differs approximately 1 %, 2 % and 5 % from the Gaussian process in terms of the first-order probability distribution at 3o, 46 and 5o level, respectively. Since the use of FFT makes it not only practical but also routine to use the value of N as large as 1, or more, the method of simulation considered here is the only realistic method of evaluating some of the hard-to-get quantities (e.g. first excursion probability). 3. SIMULATION OF A MULTIDIMENSIONAL HOMOGENEOUS PROCESS The autocorrelation function of an n-dimensional homogeneous real process f (x) defined by Ro (~) = E [fo (~1) fo (~2>] is even in ~ (symmetric with respect to the origin of the n-dimensional space).

11 3. Simulation of a Multidimensional 211 R (!;) = R (-!;) - - where ~ 1 and ~2 are space vectors and ~ = ~ 2 - ~ 1 is the (17) separation vector. Assume that the n-fold Fourier transform of R (~) exists. The spectral density function of f (x) is o~ o- then defined as s (w) =- 1 - o - (2n)n (18) where ~ is the frequency (wave number) vector and w ~ is the inner product of w and 1;, and for simplicity J ( ) dl; - J -oo -CD n-fold f ( ) d/;1 d/;2 dl;n -oo with n being the dimension of the vector 1;. It follows from Eq.17 that!;) d~ = and, therefore, from Eq.18 s (w) = s ( -w) - - (19) Then s (w) = o - (2n)n J -oo R (!;)cos (w. l;) dl; - (2) and is real.

12 212 M. Shinozuka 26 It can be shown that R (~) is non-negative de inite - and therefore it has a non-negative n- old Fourier transform; s (w) > - (21) Based on these properties of S (w), a method o simuo- lating f (x) is proposed in the ollowing: a- Consider an n-dimensional homogeneous process with mean zero and spectral density unction 5 (~) which is of insignificant magnitude outside the region de ined by and denote the interval vector by w -w nu n!) N n (22) where usually -x. w. -u by the series = - w Then the process can be simulated N n - L [s (w1k ' k =1 1 n w2k2'. 'wnkn)6.w1aw2..l\wn]2 4 where cos ~ - independent random phase uniformly distrik1k2.kn - buted between and 2TI X n

13 J. Simulation of a Multidimensional 213 k. = 1,2, N. l. l. i = 1,2,,n As in the one-dimensional case, the digital simulation f(x) of f(x) can be achieved by using Eq.23 with ~ k1k2..kn being replaced by their realized values ~k k k 1 2""" n To avoid the lengthy expressions in the subsequent discussion, f(x) will be written in the following compact form: where N f(x) = Y2 L A(!!,!k)cos (!!,!k. X + rpk) (24) k=1 N = N1N2 Nn It is noted that if the symmetric condition of S (w) is used, - N in Eq.23 can be reduced by one-half. Furthermore, if the process is isotropic, N is reduced to!_ Fig.4 illustrates 2n the significance of A(~k) for two-dimensional cases where, however, ~ k is written for A(w 1 k, w 2 k ) It can be shown2 that the ensemble average of f(x) is zero, and the autocorrelation function R(~) of f(x), becomes R(~) (26) 2 Upon substituting A (~k) = S (~k)6w, and taking the limit

14 214 M. Shinozuka as N -m (in the sense that N 1,N 2,,Nn---+ oo simultaneously), one obtains R(~) = j so(~)cos (~.Q dw = R (~) - -m (27) where it is assumed S (w) = - f'or w 4. wt and w > ~u. This indicates that, when the ensemble average is considered, the simulated process (~) possesses the target autocorrelation R (~) O- sity S (w). o- 2 It can also be shown and therefore the target spectral den that the temporal (or spatial) mean ~ f'(x)> is zero and the temporal autocorrelation Rl' (_Q = <::: f'(x) (~ + ~)> becomes Rllf(~) N 2 = L_ A (~k)cos (~k ~) k=1 (28) As N---+ oo, Eq. 28 becomes R (_Q = 1 S (w)cos (.!. ~) d.\q = R (~) -oo (29) From Eqs.26 and 28 it is seen that the process (~) in Eq.23 is ergodic regardless of' the size of' N. This makes the method directly applicable to a time domain analysis in which the ensemble average can be evaluated in terms of' the temporal average. Note that the simulated process is Gaussian by virtue of' the central limit theorem. It is important to note that the FFT technique can and

15 J. Simulation of a Multidimensional 215 should be used in the digital simulation of multidimensional Gaussian processes writing Eq.24 in a similar form to Eq.817. As an example of digital simulation of a multidimensional process, consider a two-dimensional homogeneous Gaussian process f (t,x) with mean zero and spectral density S (w,k) I w \ Q\Wi (JO) where t and x represent the time and the distance respectively and, correspondingly, w and k the frequency and the factory model of a fluctuating part of wind velocity along a straight line direction x. In the wind study, however, it is customary to consider the Fourier transform S (w,l;) of the respect to Eq.3 is wave number. It is known that such process f (t,x) is a satiso x+l; >] only with ~. In fact, the form of S (w,~) consistent with e -Qjw)\1; I (31) a familiar form for a fluctuating part of wind velocity at the reference altitude of 33 feet where L = 4 ft., K =surface drag coefficient, ~=constant, C = L/(2nu 33 ) with u 33 being the mean wind velocity at the reference altitude. For u 33 = 4 mph, oc =.2 ft. sec and K = O.OJ, the sample functions f(t,x) of f (t,x) are computed and shown

16 216 M. Shinozuka in Fig.5 at x =, 5 and 2 ft. One can easily see in this example that the correlation almost disappears as the separation ~ increases to 2 ft. 4. MONTE CARLO SOLUTION OF STRUCTURAL DYNAMICS The preceding method of digital generation of samle functions of a Gaussian process can be used for the Monte Carlo solution of the following structural problems. It is pointed out parenthetically that by adding a constant value m to the sample functions f( t,~) described in t.he preceding sections, one can generate sample functions of the simulated "process f(t,~) + m associated with f (t,x) + m. Note that the mean value of f (t,x) + m is no longer zero but is equal to m. - (a) The method can be used in the response analysis of a nonlinear structure under random loading if such loading can be idealized as Gaussian homogeneous or Gaussian evolutionary process with constant mean values. In particular, if the modes fk(x) of the corresponding linear structures are known. the solution y into a finite series: (t,~) is an approximation expanded

17 4. Monte Carlo Solution 217 {32) When Eq.32 is substituted into the governing {nonlinear partial) di erential equation{s) o motion, one can usually geta set o simultaneous nonlinear but ordinary di erential equations involving the generalized orces o the ollowing orm: {33) where {t,~) is the random excitation process and D indicates an appropriate domain o integration. Sample unctions Fk{t) can then be digitally generated rom Eq.33 with {t,x) replaced by f{t,x): It goes without saying that f{t,x) + m has to be used in place o f{t,x) i the excitation process has a non-zero constant mean value since the simple superposition o solutions does not apply in this case because o nonlinearity. The modes?k{x) o ten take the orm o sinusoidal or hyperbolic unctions or their combinations. There ore, the integration in Eq.34 can usually be carried out in closed orm since {t,x) is given as a sum o cosine unctions. This

18 218 M. Shinozuka is one of the significant advantages of the present method of simulation over other existing methods. In fact, if the domain of integration D represents a two or three dimensional space, the numerical integration of Eq.)4 will usually become an isurmountable obstacle. Another advantage is that the present method does not require the nonlinearity to be small or moderate, a condition which has to be imposed for standard linearization or perturbation techniques. Once the sampl~ functions Fk(t) are evaluated from Eq.)4 then the sample functions ~(t) of qk(t) can be numerically evaluated from the (simultaneous) nonlinear but ordinary differential equations mentioned above (replacing of course Fk(t) by Fk(t) therein). The experience shows that this phase of numerical work is not a serious problem. Finally, the sample function y(t,~) of the solution y (t,~) can be obtained from Eq.)2 with qk(t) replaced by qk(t). The temporal average of y 2 (t,~) over a sufficiently long period of time will produce the mean square response in the Monte Carlo sense if the processes involved are ergodic. Otherwise, the ensemble average has to be considered. Reference 6 represents a typical example dealing with random responses of nonlinear string and plate. The analysis for the nonlinear string is briefly presented here. It is pointed out, however, that the digital simulation of the random process for this example was performed

19 4. Monte Carlo Solution 219 using the multidimensional equivalent o Eqs.5 and 6. The di erential equation and the boundary conditions or transverse displacement u o a string under initial tension T are, in dimensionless orm, given by o 2 u + 2~ ou ot 2 at = = ~(.!!) i<o~> <IX 4 [ ] o 2 ii (.!;) ox 2T L u :fcx, t> u(o,t> = U:<1,t> = (J5) (J6) where X = X/L u == u/u t = t/2l(.f- = (x 1t)L T L = T =? = (x,t> = string length initial tension mass/unit length orce/unit length ~ = Y]L/~f To Y] = A = E = u = coe icient o linear viscous cross-sectional area o string elastic modulus o string re erence displacement damping Assuming an approximate solution consisting o irst M

20 22 M. Shinozuka terms of the modes of the (corresponding) linear string, M u(x,t) =2:::: n=1 b (t) n sin n11x (37) Substituting Eq.37 into Eq.35 multiplying by sin m11x, and integrating from to 1, one obtains db n dt ( 2nn) 2 b n 1 + ay J :f<x, t> sin nnx dx (38) n = 1,2,,M where = u y L AE a:..= ~ y2 The above equations can be solved numerically without difficulty if the associated generalized forces 1 Fn(t) = J f(x,t) sin nnx dx (39) are given numerically. This can be accomplished easily since f(t,x) can be

21 4. Monte Carlo Solution 221 readily simulated in a manner described above. Note that the integral in F (t) can be carried out analytically ben cause (t,x) is a sum of cosine functions. The result of the analysis will be shown in a diagram where the root mean square (r.m.s.) of the response is plotted against that of the excitation. Since, however, the statistical fluctuation in the r.m.s. of the simulated excitation process will undoubtedly reflect on the r.m.s. of the simulated response process, both are estimated by means of temporal average although the theoretical value of the r.m.s. for the excitation process is known. For this purpose, Eq.J7 is used for the response upon computing b (t) and equivalent n of Eq.5 for the excitation. As an example, a nondimensionalized pressure field, f(x,t) with a nondimensionalized generalized spectral density (4) is applied to the string. Eq.4 is obtained from the crossspectral density S (w 1,~ 2 ) of the following form; (41) with (42)

22 222 M. Shinozuka A result is obtained for a particular set of values of parameters involved (~ = 2TI/1, a = 4TI, c =.7 and ~ =.5) and shown in Fig.6 where r.m.s. value of the nondimensional displacement u = u/(.1l) at the midspan is plotted against that of the nondimensional excitation f = L f (x,t)/t The accuracy of the present approach is first checked against the exact solution obtained for the corresponding linear string. Open circles indicate the r.m.s. values based on simulated excitation processes consisting of 1 cosine terms (N = 1) and on the temporal average computed over T = 5 cycles of fundamental period 2LV~(T ) of the linear string. Open triangles, however, indicate the values based on N = 5 and T = 5. It is evident from the computation for the linear string that a use of N = 5 and T = 5 produces a reasonable approximai.ion. In fact, the r.m.s. value of the nonlinear string (dashed curve) is obtained by interpolating those points (solid triangles) computed with N = 5 and T = 5. In the present example, the displacement is dominated by the first mode, both in linear and nonlinear cases, occupying approximately 98 % of the total displacement at the midspan. Finally, it is pointed out that fewer number of cosine terms would be needed if Eq.23 was used instead of equivalent of Eq.5 for the simulation of the excitation process f(t,x). The application of this type of Monte Carlo approach

23 4. Monte Carlo Solution 223 has also been made to other nonlinear structural response analysis (b) The method can be applied to the failure analysis of a structure with spatially random variation of strength and other material properties. In this case, sample structures are generated by digitally generating such spatial variations of strength and other material properties. When correlated spatial variations are observed on more than one material property (e.g. Young's modulus and density), usually a multidimensional multivariate process has to be generated. Applying to each of these sample structures a sample stress history of a random stress process, the fatigue life of a sample structure can be computed under the assumption of a certain fatigue failure mechanism. The statistical variation of the fatigue life thus computed establishes its empirical distribution under the random stress process in the Monte Carlo sense. This approach was successfully taken in Ref.28. A similar problem in whjch the empirical distribution of the static failure load is to be found for a concrete structure with spatial strength variation is treated in detail in Refs.1J and 19. (c) The method can be employed effectively when the structural system to be analyzed is complex even though it involves neither nonlinearity nor random variation of material properties. The mean square response (displacement,

24 224 M. Shinozuka shear force and bending moment) of a large floating plate to wind-induced random ocean-waves are computed in Ref.29 taking the temporal averages of sample response functions as in Ref.6. The analysis is.essentially numerical since sample functions of the wind-induced ocean-surface elevation are digitally generated and the corresponding response functions are numerically obtained. This was done because the ocean-structure system considered was too complex to solve analytically. Another example of this kind is the study of the dynamic interaction between moving vehicles and a bridge 8 with random pavement surface roughness In this problem the random pavement surface roughness is digitally simulated for numerical response analysis. In some problems of mechanics, a structure is considered complex when its material properties are spatially random. The wave propagation through a random medium is one of these problems. In Ref.1q, the stress wave propagation through a finite cylinder with random material properties is treated under the condition that the one end of the cylinder is acted upon by an impact load and the other end is free. A set of one hundred samples of correlated random material properties (Young's modulus and density) are generated thus producing one hundred sample cylinders. The finite element method is applied for the stress analysis to compute maximum stress intensity in each of these cylinders

25 4. Monte Carlo Solution 225 due to the impact. An empirical distribution of the maximum stress intensity is then established in the Monte Carlo sense. (d) Finally, the method is often useful when the problem is to determine eigenvalues (frequencies and buckling loads) of the structure with random material properties. As in the case of the wave propagation problems, sample structures are generated and the statistical distribution of eigenvalues of these structures are treated as the empirical distribution of the eigenvalue of interest. An example of this problem is given in Ref.9.

26 226 M. Shinozuka REFERENCES [1] Shinozuka, M., Simulation o~ multivariate and multidimensional random processes, J. Ac. Soc. Am., 49, 1 (part 2), 357, January [2] Shinozuka, M.mrlJan C.-M., Digital simulation o~ random processes and its applications, J. Sound and Vibration, 25, 1,111, l3] Yang, J.-N., Simulation o~ random envelope processes, J. Sound and Vibration, 21, 73, [4] Shinozuka, M. and Sato, Y., Simulation o~ nonstationary random processes, Proc.ASCE, EMD Journal, 93, EM 1, 11, [?] Shinozuka, M. and Wen Y.-K., Nonlinear dynamic analysis o~ o~~shore structures; a Monte Carlo approach, Proc. o~ the Int. Symp. on Stoch. Hydraulics, Pittsburgh, 57, June 1971.

27 References 227 [6] Shinozuka, M. and Wen,Y.-K. Monte Carlo solution of' nonlinear vibrations, AIAA J. 1, J7, January [7] Wen, Y.-K. and Shinozuka, M., Monte Carlo solution of' structural response to wind load, Proc. of' the Jrd Int. Conf'. on Wind Effects on Buildings and Structures, Tokyo, Japan, Part III, J-1, Sept [8] Shinozuka, M. and Kobori T., Fatigue analysis of' highway bridges, Proc. of' Japan Soci ty ~_!_~ivil Engineers, 28, 137, Dec [9] Shinozuka, M. and Astill, J.C., Random eigenvalue problems in structural mechanics, AIAA J., 1, 4, 456, April [ 1 J Vaicai tis, R. and Shinozuka, M., Simulation of' flow induced vibration, ASCE --~~-.!:i.onal Structural_ -~~gineering Meeting, Cincinnati, Meeting Preprint 2218, Apr il [11] Shinozuka, M., Monte Carlo solution of structural dynamics, Int. J. of C~I!IP~:t;ei'~- -~_E:d Structures, 2, 855, 1972.

28 228 M. Shinozuka [12] Vaicaitis, R. and Shinozuka, M. and Takeno, M., Parametric study of wind loading on structures, Proc.ASCE, J.STD, 99, ST 3, 453, March 1973 ~3] Shinozuka, M., Probabilistic formulation for analytical modeling of concrete structures, Proc. ASCE, M. EMD, 98, EM 6, 1433, December [14] Astill, C.J. and Nosseir, S.B. and Shinozuka, M., Impact loading on structures with random properties, J. of Struct. Mech., 1, 1, 63, [15] Vaicaitis, R. and Shinozuka, M. and Takeno, M., Response analysis of tall buildings, Proc. ASCE, J.STD, 11, ST 3, 585, March [16] Shinozuka, M., Digital simulation of ground acceleration, 5-th WCEE, Rome, Italy, 2829, June [17l Shinozuka, M., Digital simulation of random processes in engineering mechanics with the aid of FFT technique in Stocha~tic Problems in Mechanics, Ariaratnam, S.T. and Leipholz, H.H.E.,University of Waterloo Press, 277, 1974.

29 Ref'erences 229 [18] Wilkins, D.J. and Wolf', R.V. and Shinozuka, M. and Cox, E.F., Realism in f'atigue testing, the ef'f'ect of' flight-by-flight thermal and random load histories on composite bonded joints, ASTM, STP 569, 37, 1975 [19] Shinozuka, M. and Lenoe, E., A probabilistic model f'or spatial distribution of' material properties, J. of' Eng. Fracture Mech., 8, 217, [2] Shinozuka, M. and Levy R., Generation of' wind velocity f'ield f'or design of' antenna ref'lectors, accepted f'or publication in Eng. Mech. Div. Proc.ASCE. [21] Shinozuka, M. and Vaicaitis, R. and Asada, H., Digital simulation of' random f'orces f'or large scale experiments, accepted f'or publication in AIAA~~ [22] Rice, S.O., Mathematical analysis of' random noise, in Selecte<!_Pape_~-~---~-~- ~~-~se an<i_ S_t_o~h~~~_!.~--~!'_o_~~~ses_ edited by N.Wax, Dover Publications, Inc., New York, 18, 1954.

30 23 M. Shinozuka [23] Borgman, L.E., Ocean wave simulation for engineering design, Proc.ASCE, J. of Waterway and Harbors Div., ~' WW 4, 556, Nov [24] Cramer, H. and Leadbetter, M.F., Stationary and Related Stochastic Processes, John Wiley, New York, 249, [25] Yang, J.-N., On the normality and accuracy of simulated random processes, J. Sound and Vibration, 26, 3, 417, 1973 [26] Bochner, S., Lectures on Fourier Integrals, English Translation by M. Tenebaum and Pollard H., Annals of Mathematic Studies, 42, Princeton University Press, 325, [27] Vaicaitis, R. and Jan C.-M. and Shinozuka, M., Nonlinear panel response and noise. transmission from a turbulent boundary layer by a Monte Carlo approach, AIAA J., 1, 7, 895, July [28] Itagaki, H. and Shinozuka, M., Application of Monte Carlo Technique to Fatigue Failure Analysis under Random Loading, ASTM, STP 511,168.

31 References 231 [29] Wen, Y.-K. and Shinozuka, M., Analysis of Floating Plate under Ocean Waves, Journal of Waterways, Harbors and Coastal Engineering Division, Proc. ASCE, 98, WW2, 177, May 1972.

32 232 M. Shinozuka s,(w) N f(x) =.J2 ~ Ak cos(wkx -1/\) k=l where Ak = Js7 ( wk) b.w wk kb.w Fig. 1 One-Sided Spectral Density

33 .. '= ~ a_ Q;., > c as -=- ~ I> Fig. 2 4,_.t '\. I \ I -y(t) envelope and its Envelope Process - sec "r1 cl<i.!:; (!> "' N w w "'o = 2 n; ~ =.2 '--=--- 1-, Sample Functions of a Narrow-Band R.andom Process.,

34 'Q s.,...,., ~ > <= o<s.:::.,., I>- 2-1 / ' ~nv~lop~ and its Envelope Process N VJ >!». ~ Ul... ::T l:j N ~ Ill s =.5 -y(t) W = 2 7r Sample Functions of a Wide-Band Random Process Fig. 3

35 Figures 235 Fig. 4 Two-dimensional Spectral Density

36 236 M. Shinozuka t 6...._ 4 "' 2 > at X= Oft t 6.. u.._ "' ~ ~ 4 2 > at X= 5ft t 6.. u "' 4-2 ::::: > at X= 2 fl SI.'C Fig. 5 Simulation of Wind Velocity at Different Points

37 Figures a: a: c:( c:( z z z Lo,J 'w - :::i z...j T = 5 I N = 1 T =5, N = 5 A.& LINEAR (:I ~ z Lo,J :1: w u c:(...j a.. "' i5...j c:( z v; z w :1: z z....n :1: cr R. M.S. OF NON DIMENSIONAL EXCITATION - Fig. 6 R.M.S. Response of Non-Dimensional Displacement (String)

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