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CompwfersStructures,ol. 2, pp.855-874. +rpamonma 1972. rintedn Chat Bri tai n

MONTE CARLO SOLUTION OFDYNAMICSt

MASANOBU SHINOZUKA

STR~CTU~L

Department of Civil Engineering and Engineering Mechanics, Columbia University,New York, N.Y. 10027, U.S.A.Abstract-The recent advent of high speed digital computers has made it not only possible but also highlypractical to apply the Monte Carlo techniques to a large variety of engineering problems. In this paper atechnique of digital simulation of multivariate and/or multidimensional Gaussian random processes (homo-geneous or nonhomogeneous) which can represent physical processes germane to structural engineeringis presented. The paper also describes a method of digital simulation of envelope functions. Such simula-tions are a~omplished in terms of a sum of cosine functions with random phase angles and used as thebasic tool in a general Monte Carlo method of solution of a wide cIass of problems in structural engineering.Most important problems for which the method is found extremely useful includes (a) numerical analysis ofdynamic response of nonlinear structures to random excitations, (b) time domain analysis of linear structuresunder random excitations performed for the purpose of obtaining a kind of information, such as firstexcursion probability and time history of a sample function, that is not obtainable from the standard fre-quency domain analysis, (c) numerical solution of structural problems involving randomly nonhomogeneousmaterial property such as wave propagation through random medium, and (d) dynamic analysis of extremelycomplex systems such as those involving st~tur~fluid interaction. Numerical examplea of some of theseproblems are presented.

I. INTRODUCTIONIN THE last two decades, much research effort has been devoted to the application of thestochastic process theory in the general area of en~nee~ng mechanics and structuralengineering for the purpose of predicting the dynamic structural performance with a betteraccuracy and of assessing the overall structural safety with a better reliability by consideringmore realistic analytical models of load-structure systems.Naming only a few, possible applications of this stochastic approach include (a)analysis of panel vibrations of aircraft and submarines induced by boundary layer turbu-lence, (b) analysis of ship oscillations caused by ocean waves, p~icul~ly during a storm,(c) analysis of aircraft response to gust vertical velocity, (d) response analysis of off-shorestructures to wave and wind forces, (e) statistical strength analysis of engineering materials,particularly of modern composite materials, with randomly distributed thermo-mechanicalproperties, (f) analysis of the effect of randomness in geometrical configuration of andmechanical constraint on a structural component due, for example, to fabrication errors onthe vibration and buckling eigenvalues and, (g) study of random surface roughness of bridgepavement and airport run-way for the purposes of analyzing the vehicle and aircraft

t This work was supported by the National Science Foundation under Grant GK 24925.t Presented as an Invited Paper at the National Symposium on Computerixed Structnral Analysis andDesign, at the School of Engineering and Applied Science, George W~hin~on University, W~on,D.C., 27-29 March, (1972).

855

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856 MASAN~KI SAINOZLJKAvibration caused by the roughness and of stress analysis of pavement systems under theaction of vehicles and aircraft.In spite of the recent remarkable advance in this area of study, however, the presentstate of art still leaves a number of difficulties unsolved that must be overcome before theapproach becomes more useful. Such problem areas include(1) random response analysis of highly nonlinear structures,(2) failure analysis of structures under random loading,(3) analysis of extremely complex systems and,(4) random eigenvalue problems.

The recent advent of high speed digital computers, however, has made it not onlypossible but also highly practical to apply the Monte Carlo techniques to a large variety ofengineering problems. The present paper presents a technique of digital simulation ofmultivariate and/or multidimensional Gaussian random processes (homogeneous or non-homogeneous) which can represent physical processes germane to structural engineering.The paper also describes a method of digital simulation of envelope functions. Suchsimulations are accomplished in terms of a sum of cosine functions with random phaseangles and used as the basic tool in a genera1 Monte Carlo method of solution to a wide classof problems in structural engineering, particularly those mentioned above.

II. SJMULATION OF A RANDOM PROCESS: A BACKGROUNDA basic representation of a homogeneous Gaussian (one-dimensional and one-variate)

random processf,(x) with mean zero and spectral density S,,(w) in the form of the sum of thecosine functions has existed for some time [I];

where @, are random angles distributed uniformly between 0 and 27~and independent ofcP,(k Z-9, and

Ak= [S:(w,)Aw]*, ok = (Ii - +,Ao (2)with

S;(o) =2&(o) (3)being the one-sided spectra1 density function (see Fig. I). For digital simulation of a simplefunctionf(x) off(x) and therefore off,(x), equation (1) is used with CD,being replaced bqtheir realized values (Pi;

-N.fW =& ,z,A/xcos(o,x- (ok).With the aid of equations (1) and (4), it is easy to show that equation (1) is ergodic at leastup to the second moment.Speaking of structure-related applications, however, until Borgman [2] published apaper simulating ocean surface elevation as a multidimensional process essentially in thesame form, little attention had been paid to this representation in spite of its substantial

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Monte Carlo Solution of Structural Dynamics 857

i(x) = fi $ Akcos(ukk - #+)where A, = Jyiqi=

uk = (k-@ho

- NI&LJ% -FIO. 1. One-sided pectral density.

advantage over the standard method in which a random process was digitally generated asoutput of an appropriate (analytical) filter subjected to a simulated white process. The useof 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 arandom process.Practical digital simulation of a multidimensional process has been made possible, asmentioned above, by Borgman [2] in principle through the preceding expression consistingof a sum of cosines and by Shinozuka [3] whose method reduces, in case of one-variate one-dimensional situations to the use of the following expression;

where VpL re as previously defined and ok are realized values of random frequenciesdistributed according to the density function ~(o)=S,,(w)/a~ with

Borgman [2] and Shinozuka [3] also investigated the digital simulation of the multivariate(but one-dimensional) process\ the former making use of the filtering technique and thelatter in the more convenient form of the sum of the cosine functions.

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858 MASANOBLI SHINOZUKAReference [4] shows that the autocorrelation function R(r) off(x) converges to RJ;) of

f(x) in the form of l/N2 as N-co. The same trend in convergence can be shown [4] to existbetween the spectral element S(o)Aw off(x) and S,(w)Aw off,(x).

It is interesting to note thatf(x) in equation (1) can be interpreted as a canonical expan-sion of a Gaussian process&(x) with mean zero and spectral density S,(o). To see this,express f,(x) in the form of the spectral representation

foW= ms dZ(w) (7)-ccwhere Z(w), called spectral process, is orthogonal in the sense that the increments dZ(w,)and dZ(w,) are uncorrelated when wr #c+.

Employing the orthogonal condition of Z(w), the autocorrelation function off,(s) isfound to be

m(~)=E[.l,(x)f,(x+~)] =s i"cEldZ(w)lZ (S)-CCwhere E indicates the expected value.Assume that the spectral density function S,(o) exists. Then EldZ(w)l= S,(w)do, and

equation (8) is reduced to the well-known Wiener-Khintchine relationship. For the casewhenf,(x) is real, equation (7) becomes [4]

fo(x) =s

y [cosxdU(w) + sin oxd V(w)] (91where U(o) and V(o) for any 020 are two mutually orthogonal processes, both real andwith orthogonal increments, such that

It is pointed out that if one defines

dV(o,) = [2Sy(wk)Aw]* sin (I+=@Ak sin @, (10)where c&, Ao and t& are defined in equations (1) and (2), then all the conditions imposed onU(w) and V(o) are satisfied, and equation (1) is basically consistent with the spectralrepresentation.It is seen from equation (9) that a homogeneous process is additively built up by ortho-gonal oscillations with random amplitudes.A canonical expression of a real random processf,(x) can be written in the form [5];

f& d = kElkuk(x) (11)

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Monte Carlo Solution of Structural Dynamics 859where ak are uncorrelated random variables with mean zero and uk(x) are real (deter-ministic) fUnCtiOnS f x; ak are called the COeffiCientS Of he canonical expansion and a&)the coordinate functions.

By multiplying both sides of equation (11) by cc,, taking the expected values and usingthe fact that ai are uncorrelated, one can show that

(12)where Dj is the variance of ai;

Dj=E[af]. (13)At this point, writefO(x) in equation (9) in the following approximate form;

j-(x) = j, [cos okxdU(~& + sin o,xd V(o,)]

= JZ 2 (A, cos f&XCOSI$ + Ak sin WkX in ok)k=l

= J? 5 Ak cos(~,x -$.)ak=l (14)

This is an approximation to./,(x) since the integration is replaced by a summation involvinga finite number (N) of terms. The degree of the approximation depends on (a) whether thecut-off frequency o, (see Fig. 1) is large enough and (b) whether Aw in equation (2) is smallenough so that

kAcuA;= I S&u)dw is@,)Aw (15)(k- l)A.a,is valid. Being the sample function of f(x), j(x) possesses the same approximate naturewhen considered as the sample function off,(x).

Within the context of this approximation, equation (14) can be interpreted as a(truncated) canonical expansion of f,(x) in the form of equation (11) if a, and oi(x) aredefined as

a2,-1 = JL~~cosr~,,tlzj= JFA,sinQj,

02j_i(X)=COSOjX (j= 1,2,. . , iv)ozi(x)=sino,x. (16)

In fact, the coordinate functions in equation (16) follow from equation (12) if the co-efficients ak in equation (16) andf(x) in the form of equation (14) are used therein.It can then be shown [!Yj again within the context of the approximation mentionedabove) that if these nk(x) defined in equation (16) are used, then for any given number of N

and for the particular selection of ak in equation (16), equation (14) gives the best approxi-mation to the random function&(x) in the sense that, among all possible series expansion

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860 MASANOBLI SHINOZUKAoff,(x) having the same number of terms, equation (14) minimizes the expectation of thesquare of the residual term at any value of x.

There is another constraint to be imposed upon do. This constraint is due to theperiodicity of the sample function f(x). Obviously, the period of f(x) is T,=2n/Ao andtherefore, depending on the purpose of the simulation, Ao has to be so chosen that To islong enough for that purpose.A significant improvement in the efficiency of digital simulation has recently beensuggested by Yang [6] writing

j(x)=JzReF(x) (17)in which ReF(x) represents the real part of F(x) and

F(x) = kfi[2S~(wk)]feicpk}ei"~=is the finite complex Fourier transform ofwith wlr and (Pkdefined in equations (1) and (2). The advantage of equation (18) is such thatthe function F(x) can readily be computed by applying the fast Fourier transform (FFT)algorithm, hence avoiding the time-consuming computation of a large number of cosinefunctions.

In the preceding discussion, the spacing Aw in the frequency domain has been taken asconstant. This, however, does not necessarily have to be always observed. It is possible andin fact may even be advisable to use variable spacings depending on the extent of fluctuationof the spectral density to optimize the number of cosine terms in the summation in equation(1); finer spacings in those domains of frequency where the fluctuation of the spectraldensity is more rapid and coarser spacings elsewhere. It is also likely that such variablespacings will increase the period T, of the simulated process. If this is done, however, thesimulation through the FFT technique does not appear to be possible.

In the same reference [6], Yang also proposed to simulate the envelope process V,(x)of a random processf,(x) by following the definition of the envelope process [7]

v,(x)=Cm) +33x)1* (19)where fO(x) is the Hilbert transform off,(x) and, in the present case, can be written as

j&x)= r [sinordU(m)-cosWtdV(a)].JOIt then follows that fb(x) can be simulated as f(x);

3(x) =JZ f &sin(o,x - cpL).k=i (21)

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Monte Carlo Solution of Structural Dynamics 861

Hence, the envelope process V,,(x) can be simulated as V(X);V(x) = [f2(X) -t]Q)]f . (22)

As an example, consider the response process y,(t) of single degree of freedom systemto a Gaussian white noise excitation x.,(t) with the constant spectral density S,;

j;,(O+ 2boBo(O f&00>=x0(0* (23)It is well known that the spectral density of y,(t) is

s&d = so(02 - oo2)2 + 4~2f&L?and the standard deviation oY of y,(t) is

(24)

(25)With the aid of equations (l), (21) and (22), a segment of sample function J(t) of

simulated process r(t) for y,(t) and that of simulation V(t) of the envelope process V,(t)are computed and shown in Figs. 2 and 3. Figure 2 is for the case where the damping co-

---envelopeFIG. 2. Sample functions of a narrow-band random process and its envelope process.

efficient [=0.02 and hence the process y,(t) is narrow-band. This fact is well demonstratedby the smooth behavior of the sample envelope. When the local maxima of the sampleenvelope do not coincide with the local maxima (peaks) of the process, they reflect the localminima (troughs) of the simulated process. Figure 3 shows the sample functions of V(t) andy(t) for C=O.S. In this case, the process y,(t) is substantially wide-band and this fact isclearly seen from the much wilder fluctuation of both simulated envelope and simulated pro-cess, although the simulated envelope surprisingly well reflects peaks and troughs of thesimulated process even though the process y,(t) is wide-band. In terms of simulationefficiency, the computer time will be significantly reduced if one is interested in peak- and

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862 MASANOBUHINOZUKA

0-x envelopeFlo. 3. Sample functions of a wide-band random process and its envelope process.

trough-values of the process and if the process is narrow-band, since then the smoothnature of the envelope function makes it possible to use much larger interval betweensuccessive time instants at which the values of the simulated process is evaluated. The orderof magnitude of this interval can be that of the apparent period of the process, whichobviously is much too large for simulation of the process itself.

Again following Yang [6], write $(cr>asJ((t)=J&ImF(t) (26)

and hence V(i) can also be simulated through the FFT technique.It appears at this time that the method of simulation considered herein [equation (l)]

has a difficulty in achieving a reliable evaluation of the first passage time distribution whenthe threshold value is much larger than the standard deviation of the process. Lyons work[8] points to this fact, although this difficulty is by no means unique to the proposed method.It is suggested however, that a further investigation be performed on this point.

The Gaussian property of the simulated process [equation (l)] comes from the centrallimit theorem because it consists of a sum of a large number of inde~ndent functions oftime (see pp. 182-183 in Ref. [I]). Efficient simulation, or straight forward simulation ifnot efficient, of a non-Gaussian process appears to be an open problem at this time unlessthe process is restricted to a certain class of processes such as the filtered Poisson process.

In the following sections, a method for digital simulation of multidimensional and/ormultiva~ate processes are briefly described. Wowever, for these cases, the rigorous discus-sions on the interpretation as canonical expansion, the use of the FFT technique in actualdigital computation, the envelope simulation, the problem of the first excursion time, thesimulation of non-Gaussian processes, the convergence of the auto-correlation functionand the spectral density of the simulated process to the respective target values, etc., aremostly subjects of future studies.

III. SIMULATION OF A MULTIDIMENSIONAL HOMOGENEOUS PROCESSThe autocorrelation function of an n-dimensional homogeneous real process f,(x)defined by

44s) = ~~(X~)~(X~)l

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Monte Carlo Solution of Structural Dynamics 863is even in 5 (symmetric with respect to the origin of the n-dimensional space)

R,(5)= R4- 5) (27)where x1 and x2 are space vectors and 5=x2 -x1 is the separation vector. Assume that then-fold Fourier transform of R,(c)xists. The spectral density function of f,(x) is thendefined as

S,(o)=1 Ccs2n) -03Ro(Qe-ird~ (28)where a is the frequency (wave number) vector and 0.5 is the inner product of o and 5,and, for simplicity

with n being the dimension of the vector 5. It follows from equation (27) thatsDR&)sin(co *&)de =0-C0and, therefore, from equation (28)

S&o) = S,( - 0). (29)Then

S,(o)= 1 msw --b: &,(Wos~~ 0% (30)and is real.It can be shown [9] that R&) is nonnegative definite and therefore it has a nonnegativen-fold Fourier transform;

S,(o) 2 0. (31)Based on these properties of S,,(w), a method of simulatingf,(x) is proposed in thefollowing:Consider an n-dimensional homogeneous process with mean zero and spectral density

function S,(a) which is of insignificant magnitude outside the region defined by

and denote the interval vector by(Awl, Aw2, . . . , Aw,)= Wlu-Wll w2u-w21 % - Q4ll

N, N2 I*** 1L, >(32)

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804 MASANUBU SHINOZUKAwhere usually w[= -co,,. Then the process can be simulated by the series

. cosiwA,.\1+(fJ?Ap2+ . %k,s--(14,h,,.L,,)where

@h !%> h,,=independent random phase uniformly distributed between 0 and 2rrwiki = wil + ( ci - $)AuJ, lii=l, 2,. . . , Ni i=i. 2.. . II.

(33)

As in the one-dimensional case, the digital simuiationJ(x) of.lIx) can be achieved byusing equation (33) with mklkl,, ,k, being replaced by their realized values qklkI, ,,k,,.

To avoid the lengthy expressions in the subsequent discussion, ,f(x) will be written inthe following compact form :

7(x)=$ f A(o,)cos(o, * x-q,) (34)IZ=where

N=N,N,. , . N,/t(o,) = [S,(o,jAo,AwZ . . . Ato,]* = [SJwJAw] i. (35)

It is noted that if the symmetric condition of S,(o) is used, N in equation (33) can be reducedby one-half. Furthermore, if the process is isotropic, N is reduced to N/2. Figure 4illustrates the significance of A(o,) for two-dimensional cases where, however, Aklk2iswritten for A(alk,, u&L

It can be shown [4] that the ensemble average of,f(xj is zero. and the autocorrelationfunction R(k) of,f(xj. becomes

RiS)= 2 A2i~,)c0do, . 5). (36)A=1Upon substituting A(~+)=S,(O~)AO, and taking the limit as N~CQ (in the sense thatN,, Nz, . . . . N,+ r*j simultaneousiy) one obtains

*LR(k)= ! S,(o)cos(o * &)do= K,(5)- V (37)where it is assumed S,(o)=0 for oo,.

This indicates that, when the ensemble average is considered, the simulated processf(x) possesses the target autocorrelation I?,,(%) and therefore the target spectral densityS,,(o).It can also be shown [4] that the temporal (or spatial) mean is zero and thetemporal autocorrelation R *(Q = becomes

R*(t)= f A2(o,)Cos(ok5,. (.7X)1. = I

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Monte Carlo Solution of Structural Dynamics 865

where A1 z 5 &,(w,~ , w2k I h, 41 2?k, = (k, -+ ,Au, , wzk2= (k2-$$

FIG. 4. Two-dimensional spectral density.As N+co, equation (38) becomes

R*(S)- Ws S,(o)cos(o - &do = R,(g). (39)-03From equations (36) and (38), it is seen that the processf(x) in equation (33) is ergodicregardless 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.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 densityKLZ I4 44so(09 Q(znz) * (I +c202)4/3 . +_&2+k2) (40)

where t and x represent the time and the distance respectively and, correspondingly, o and kthe frequency and the wave number. It is known that such processf,(t, X) is a satisfactorymodel of a fluctuating part of wind velocity along a straight line direction x. In the windstudy, however, it is customary to consider the Fourier transform &,(co, r) of the autocor-relation R,,(z, 5) =EL f.(t, x)f .(t +z, x+

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866 MASANOBU SHINOZUKAa familiar form for a fluctuating part of wind velocity at the reference altitude of 33 ft whereL=4000 t, K=surface drag coefficient, a=constant, c=L/(21rU,,) with U3, being themean wind velocity at the reference altitude. For U,, =40 mph, a=O.O2 ft/sec and K=0.03the sample functionsJ(t, 5) off,(t, r) are computed and shown in Fig. 5 at 5 =0, SOand 200ft. One can easily see in this example that the correlation almost disappears as the separa-tion t increases to 200 ft.

1 60.

t 60;; 40%2 20 0

-20-40-60

ok--7k-1~- 4b 510 1OsecFIG. 5. Simulation of wind velocity at different points.

IV. SIMULATION OF MULTIVARIATE MULTIDIMENSIONALHOMOGENEOUS PROCESSESConsider a set of m homogeneous Gaussian n-dimensional processes f:(x) (j= 1,2, . . .

m) with mean zero and with the cross-spectral density matrix so(w) defined by

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Monte Carlo Solution of Structural Dynamics 867where S;,(w) is the n-fold Fourier transform of the cross correlation Xq&).

Due to the fact that R~,(S)=R~j(-5), one obtainssy,w%i4 (43)

where the bar indicates the complex conjugate.The matrix p(o) is therefore Hermitian. As in the case of a one-dimensional multi-variate process [lo], it can be shown [4bthat the matrix S,(w) is also nonnegative definite.

Suppose one can find a matrix H(o) which possesses n-dimensional Fourier transformand satisfies the equation

so(o) =H(o)R(o)T (44)where S,,(o)is the specified target cross-spectral matrix and T indicates the transpose. Thenfj(X) (j= 1, 2,. . . , m) can be simulated by the following filtering technique [2, 111;

111 x.fjCx)= C f hjdx- 9hd5M5k=l -cc (45)

where hl,(x)is the n-dimensional Fourier transform of Hjk(ce);h,(x)= ms j,(o)e -iUxdw-m

and tfk(x) is an independent n-dimensional normalized white noise component such that

withS(x,-x*)=6(x,,-x,,)6(x,,-x,,) * * * ml,--XZ).

It can be easily verified that the fj(x) (j= 1, . . . m), as simulated by equation (45),satisfy equation (44) and thus represent the target processes.To find the matrix H(o) in an efficient way, one can assume that H(o) is a lowertriangular matrix;

Zjl l(N 0 0 0. .H(o) = HA4 H,,(o) . . . . 0. . . . . . . .

Substituting above into equation (44), solutions are obtained (see Ref. [12] for similarderivation) as

Dk(@ .H&>= -[ 1-t@) k=l, 2,. . . m (46)

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868 MASANOBU SHINOZLJKAwhere Dk(a) is the kth principal minor of Se(w) with D, being defined as unity, and

so 1,2,.. .) li-1,jHjk(a) = HAk(W) 1.2,. . . , k-l, k k= 1, 2, . . . 111J%CN j=k+ I , . . . rn

(47)

where

so 1, 2,. . . k-l. j,,2 ,.,. k_l,k = . . . . . . . . ..*................%,,I sL.2 *. . %,,-I SkO-1.k

sq, sy, . . * sg, k-1 sq,is the determinant of a submatrix obtained by deleting all elements except the (1, 2, . . .k- 1,j)th row and (1, 2, . . . k- 1, k)th column of SO(o).It is noted that the above decomposition is valid only when the matrix SO(,) is hermi-tian and positive definite as can be seen from equation (46).

Because the cross-spectral density matrix SO(,) is known to be only nonnegativedefinite, special consideration is needed in those cases where SO(o) has a zero principalminor. For the discussion on this point, the reader is referred to Ref. [4].

Since the real and the imaginary parts of cross-spectral density functions are respectivelyeven and odd in o, it can be shown from successive substitutions using equations (46) and(47) that

Re Hj,(O) = Re Hjk( - O)

for j>k , andImHjdw)= -ImHjk(-w)

Hjj(W)=Hjj(-U)>O

(48)

from which it follows that hjk(x) is real.If Hjk(w) is written in polar form;

Hj,(w) = I H j,(w)i eiejk (49)then, due to equation (48), the argument 8j,(w) is anti-symmetric in w, that is

with 8,,(w)=O.ejk(w)= -ejk(-w) (50)

Once Z-Z(w)s computed using equations (46) and (47) then instead of passing a whitenoise vector through filters, the process fj(x) can be simulated in terms of the followingseries

.fAx)= i i IHjAW;)lJGCOSIW, *X+Ojm(Wl)+@,nl],,I = I I = 1

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Monte Carlo Solution of Structural Dynamics 869

where oi, Ao, N and @,,,rare essentially the same as defined previously for n-dimensionalprocesses and

It can be shown [4] that the processesfj(x) (j= 1, . . . , m), as simulated by equation (5 l),possess the target crosseorrelation functions and hence the target cross-spectral density,with respect to an ensemble average.

For digital simulation of sample functions Offi( equation (51) is used with @,,,rbeingreplaced by their realized values.

V. SIMULATION OF MULTIDIMENSIONAL NONHOMOGENEOUSPROCESSES

Simulations of nonstationary processes have been studied dealing mostly with earth-quake ground motion. The common feature of these studies is that a nonstationary process issimulated by multiplying by an envelope function the stationary process generated either byfiltering a white noise [13, 141 or by a series of oscillations with random frequency andrandom phase [15-l 71.

The efficient method of simulation that has been proposed for multidimensional homo-geneous processes can be directly generalized to a non-homogeneous process characterizedby an evolutionary power spectrum as introduced by Priestley [18, 191.It was seen from equation (9) that a homogeneous process is additively built up byorthogonal oscillations with random amplitudes. This concept of orthogonal componentscan be extended to that of the evolutionary process f:(x) expressed as

f:(x) =Jr (x, o)[cos oxdU(o) + sin wxd k(o)] (53)where B(x, w) is a deterministic modulating function characterizing the nonhomogeneityof the process, and U(o) and V(o) are the same as defined in equation (10).Using the orthogonal conditions of U(w) and V(o), the mean square ofj!(x) is foundto be

E[_f%>] s mi.x, )S;(o)dw = m2(x, w)do0 f 0where S~(x,o)=B2(x,o) S;(o) is defined as the evolutionary power spectral density function.

For the detailed discussion and the estimation of the evolutionary power spectrum, thereaders are referred to Refs. [18 and 191.The direct generalization of the above discussion to n-dimensional process is obvious.Thus, if a real nonhomogeneous process has an evolutionary power spectral density func-tion, the process /z(x) can be simulated by

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870 MASANOBU SHINOZUKA

where B(x, o) is the n-dimensional modulating function and the remaining notations are thesame as in the case of a homogeneous process given by equation (33). It can be shown thatthe simulated process f,(x) possesses the target evolutionary power spectrum as N+co.

Note that in a particular case when B(x, o)=B(x), then f:(x) can be obtained bymultiplying a homogeneous process simulated from S,(o) by the spatial envelope functionb(x).A more detailed study with numerical examples on this method of simulation for non-homogeneous Gaussian process with an evolutionary power spectral density has been madeby Yang [6].

In the following section, the Gaussian nonhomogeneous process with an evolutionarypower spectral density is referred to as Gaussian evolutionary process for simplicity.

VI. MONTE CARLO SOLUTION OF STRUCTURAL DYNAMICSThe preceding method of digital generation of sample functions of a Gaussian processcan 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 functionsf(t, x) des-cribed in the preceding sections, one can generate sample functions of the simulated pro-cess J(t, x ) +m associated withf,(t, x) +m. Note that the mean value off,(t, x)+m is nolonger zero but it is equal to m.

(a) The method can be used in the response analysis of a nonlinear structure underrandom loading if such loading can be idealized as Gaussian homogeneous or Gaussianevolutionary process with constant mean values. In particular, if the modes pk(x) of thecorresponding linear structures are known, the solution y,(t, x) is in approximation ex-panded into a finite series.

(56)When equation (56) is substituted into the governing (nonlinear partial) differential equa-tion(s) of motion, one can usually get a set of simultaneous nonlinear but ordinary dif-ferential equations involving the generalized forces of the following form;

F,(t) = f ~dx)_L,c, xWD (57)wheref,(r, x) is the random excitation process and D indicates an appropriate domain ofintegration. Sample functions P,(r) can then be digitally generated from equation (57) withIb(t, x) replaced by (t ; 4;

F,W=s /4W3(4xW -DIt goes without saying that j(f(t,x) + m has to be used in place off(?, x) if the excitation pro-cess has a nonzero constant mean value since the simple superposition of solutions does notapply in this case because of nonlinearity.

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Monte Car10 Solution of Structural Dynamics 871The modes pL(x) often take the form of sinusoidal or hyperbolic functions or their

combinations. Therefore, the integration in equation (58) can usually be carried out inclosed form sincef(t, x) is given as a sum of cosine functions. This is one of the significantadvantages of the present method of simulation over other existing methods. In fact, ifthe domain of integration D represents a two or three dimensional space, the numericalintegration of equation (58) will usually become an insurmountable obstacle. Anotheradvantage is that the present method does not require the nonlinearity to be small ormoderate, a condition which has to be imposed for standard linearization or perturbationtechniques.

Once the sample functions Fk(f) are evaluated from equation (58), then the samplefunctions qk(f) of qt(t) can be numerically evaluated from the (simultaneous) nonlinear butordinary differential equations mentioned above (replacing of course F&t) by F,(t) therein).The experience shows that this phase of numerical work is not a serious problem. Finally,the sample function jY(t,x) of the solution ~,(t, x) can be obtained from equation (56) withqn(t) replaced by gk(r). The temporal average of F*(t, x) over a sufficiently long period oftime produce the mean square response in the Monte Carlo sense if the processes involvedare crgodic. Otherwise, the ensemble average has to be considered.

Reference (201 represents a typical example of such analysis. A segment of a samplefunction of the tip asflection U (0, t) of a vertical pile of uniform cross-section in deepwater (Fig. 6) having nonlinear drag effect and subjected to unidirectional wind-inducedwaves is shown in Fig. 7, where a segment of a sample function of the response of the cor-

Fro. 6. Ocean-pile ystem.responding linear pile (without drag term) is also shown for comparison. In this study, theexcitation is due to waves under fully developed sea conditions with mean wind velocity Vfor which the Pierson-Moskowitz spectrum S:(m) for the ocean-surface elevation has beenused ;

where ol=8.1Ox 10-3, fl=O.74 and o,=g/Y with g=acceleration due to gravity and Y=mean wind velocity.

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872 MASANOBLI HINOZUKAThe application of this type of Monte Carlo approach has also been made to other non-

linear structural response analysis [3, 4, 21-231.(b) The method can be applied to the failure analysis of a structure with spatiallyrandom variation of strength and other material properties. In this case, sample structures

are generated by digitally generating such spatial variations of strength and other materialproperties. When correlated spatial variations are observed on more than one materialproperty (e.g. Youngs modulus and density), usually a multidimensional multivariate pro-cess has to be generated with the aid of the method described in Section IV.

FTC. 7. A section of sample response function at mean wind velocity 23.6 ft/sec

Applying to each of these sample structures a sample stress history of a random stressprocess, the fatigue life of a sample structure can be computed under the assumption of acertain fatigue failure mechanism. The statistical variation of the fatigue life thus computedestablishes its empirical distribution under the random stress process in the Monte Carlosense. This approach was successfully taken in Ref. [24]. A similar problem in which theempirical distribution of the static failure load is to be found for a concrete structure withspatial strength variation is treated in detail in Ref. [25].

(c) The method can be employed effectively when the structural system to be analyzedis complex even though it involves neither nonlinearity nor random variation of materialproperties. The mean square responses (displacement, shear force and bending moment)of a large floating plate to wind-induced random ocean-waves are computed in Ref. [26]taking the temporal averages of sample response functions as in Ref. [20]. The analysis isessentially numerical since sample functions of the wind-induced oceansurface elevation aredigitally generated and the corresponding response functions are numerically obtained.This was done because the ocean-structure system considered was too complex to solveanalytically. Another example of this kind is the study of the dynamic interaction betweenmoving vehicles and a bridge with random pavement surface roughness [27]. In this problemthe random pavement surface roughness is digitally simulated for numerical responseanalysis.In some problems of mechanics, a structure is considered complex when its materialproperties are spatially random. The wave propagation through a random medium is oneof these problems. In Ref. [28], the stress wave propagation through a finite cylinder with

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Monte Carlo Solutionof Structural Dynamics 873random material properties is treated under the condition that the one end of the cylinder isacted upon by an impact load and the other end is free. A set of one hundred samples ofcorrelated random material properties (Youngs modulus and density) are generated thusproducing one hundred sample cylinders. The finite element method is applied for the stressanalysis to compute maximum stress intensity in each of these cylinders due to the impact.An empirical distribution of the maximum stress intensity is then established in the MonteCarlo sense.(d) Finally, the method is often useful when the problem is to determine eigenvaluesfrequencies and buckling loads) of the structure with random material properties. As inthe case of the wave propagation problems, sample structures are generated and the stati-stical distribution of eigenvalues of these structures are treated as the empirical distributionof the eigenvalue of interest. An example of this problem is given in Ref. [29].

REFERENCES[1] S. 0. RICE, Mathematicalanalysis of random noise, in Selected Papers on Noise and Stochastic Pro-cesses (editedbv N. Wax). DD. 180-181. Dover. New York (1954).[21131[41

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L. E. BORGMA;, Ocean ;;a;k simulation for engineering design. J. Waterways Harbors Div., Proc.AXE WW4,556-583 (1969).M. SHINOZUKA, Simulation of multivariate and multidimensional random processes. J. acoust. Sot.Am. 49, 357-367 (1971).M . SHINOZUKA and C. -M. JAN, Simulation of multivariate and multidimensional random processes 11,NSF-GK 3858/24925, Technical Report No. 12, Department of Civil Engineering and EngineeringMechanics, Columbia University, April 1971, to be publsihed in J. Sound Vibration.V. S. PCJGACHEV,Theory ofRandom unctions, translation by 0. M. BLUNN, pp. 228-232. PergamonPress, Addison-Wesley, Reading, Mass. (1965).J. -N..YANG, Simulation of random envelope processes, private communication, to appear in J. SoundVibration.H. CRAMER nd M. F. LEADBETTER,tationary and Related Stochastic Processes, p. 249. John Wiley,New York (1967).R. H. LYON, Statistics of combined sine waves. J. acoust. Sot. Am. 48, 145-149 (1970).S. BOCHNER,Lectures on Fourier integrals (English translation by M. TENEBAUM nd H. POLLARD)pp. 325-328. Annals of Mathematic Studies No. 42, Princeton University Press, Princeton, N.J. (1959).H. CRAMER nd M. F. LEADBETTER,Stationary and Related Stochestic Pr ocesses p. 161 and p, 135.E. S. EBY, Synthesis of multivariate Gaussian random processes with a preassigned covariance. IEEETrans. In form. Theory, 773-776 November (1970).F. R. GANTMACHER,The Theory ofM atri ces, Vol. 1, p. 37. Chelsea, New York (1960).M. SHINOZUKA nd Y. SATO, Simulation of nonstationary random processes. Proc. ASCE 93, EMl,114 (1967).M. AMIN and A. H. -S. ANG, Nonstationary stochastic models of earthquake motions. EMD J. ASCE94, Em 2, 559-583 (1968).J. E. GOLDBERG, . L. B~GDANOFF nd D. R. SHARPE,The response of simple nonlinear structure to arandom disturbance of earthquake type. Bull . seis. Sot. Am. 54, 263-276 (1964).H. GOTO and K. TOKI, Structural response to nonstationary random excitation, 4th WCEE, Santiago(1969).M. SHINOZUKA nd P. W. BRANT,Application of evolutionary power spectrum in structural dynamics,Tech. Rpt. No. 3, NSF-GK 3858, Columbia University (1969).M. B. PRIESTLY,Evolutionary spectra and nonstationary processes. J. Ray. Stat. Sot. B27, 204-236(1965).M . B. PRIESTLEY, ower spectral analysis of nonstationary random processes. J. Sound Vibration. 6,86-97 (1967).

[20] M. SHINOZUKA nd Y. -K. WEN, Nonlinear dynamic analysis of offshore structures; a Monte Carloapproach, in Stochastic Hydraulics (Proceedings of the First International Symposium on StochasticHydraulics), University of Pittsburgh Press, pp. 507-521.[21] M. SHINOZUKA nd Y. -K. WEN, Monte Carlo solution of nonlinear vibration. AIAA J. 10, 39-40(1972).

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874P21

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MA~ANOBIJ HMOZUKAY. -K. WEN and M. SHINOZUKA,Monte Carlo solution of structural response to wind load, Proceedingsof the 3rd International Conference on Wind Effects on Buildings and Structures, Tokyo, Japan,6-11 September 1971, Part III, pp. III, 3-I-111, 3-8.R. VAICAITI~,C. -M. JAN and M. SHINOZUKA,Non-linear panel response and noise transmission froma turbulent boundary layer by a Monte Carlo approach, NSF GK 3858 and GK 24925, TechnicalReport No. 13, Columbia University, 1971, AIAA No. 72-199, AIAA 10th Aerospace SciencesMeeting, San Diego, California, 15-19 January (1972), to be published in AIAA J.H. ITAGAKIand M. SHINOZUKA,Applications of Monte Carlo technique to fatique failure analysisunder random loading, NSF GK 3858/24925, Technical Report No. 16, Department of Civil Engineer-ing and Engineering Mechanics, Columbia University, June 1971, to be published in a special technicalpublication of ASTM, SPT No. 511M. SHINOZUKA,Probabilistic formulation for analytical modeling of concrete structures, TechnicalReport CR 72.005, Naval Civil Engineering Laboratory, Port Hueneme, California, Sponsored byNaval Facilities Engineering Command, under Contract N62399-71-C-0022, November (1971).Y. -K. WEN and M. SHINOZUKA,Response of a large floating plate to ocean waves, Journnl of Water-w ays Har bors and Coastal Engi neeri ng Di v., Proc. ASCE, Vol. 98, No, WW2, 177-190, May (1972).ASCE.

[27] M. SHINOZUKA nd T. KOBORI,Fatigue analysis of highway bridges, NSF GK 3858 Technical ReportNo. 8, Department of Civil Engineering and Engineering Mechanics, Columbia University, January1971, presented at the ASCE National Meeting on Structural Engineering in Baltimore, 19-23 April(1971).[28] J. C. ATOLL, B. NOSSEIR nd M. SHINOZUKA, mpact loading on structures with random properties, tobe published in the J. St ruct . M ech. 1, No. 1 (1972).[29] M. SHINozuKA.and C. J. ASTILL, Random eigenvalue problems in structural mechanics, AIAA J. 10,456462 (1972).

(Received 1 February 1972)