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## Finite element methods in probabilistic structural analysis: A selected review (1988)

Venue: | APPL. MECH. REW. |

Citations: | 30 - 0 self |

### BibTeX

@ARTICLE{Benaroya88finiteelement,

author = {H Benaroya and M Rehak},

title = {Finite element methods in probabilistic structural analysis: A selected review },

journal = {APPL. MECH. REW.},

year = {1988},

pages = {201--213}

}

### OpenURL

### Abstract

This review examines the field of structural analysis where finite element methods (FEMs) are used in a probabilistic setting. The finite element method is widely used, and its application in the field of structural analysis is universally accepted as an efficient numerical solution method. The analysis of structures, whether subjected to random or deterministic external loads, has been developed mainly under the assumption that the structure's parameters are deterministic quantities. For a significant number of circumstances, this assumption is not valid, and the probabilistic aspects of the structure need to be taken into account. We present a review of this emerging field: stochastic finite element methods. The terminology denotes the application of finite element methods with a probabilistic context. This broad definition includes two classes of methods: (i) first-and second-order second moment methods, and (ii) reliability methods. This paper addresses only the first category, leaving the second to specialists in that area. The contribution of this review is to illustrate the similarities and differences of the various methods falling in the first category. Also excluded from this review are simulation methods such as Monte Carlo and response surface, and methods that use FEM to solve deterministic equations (Fokker-Planck) governing probability densities. The essential conclusion is that the second moment methods are mathematically identical to the second order (except for the Neumann expansion). The essential distinction that can be made regarding stochastic FEM is the nature of the structure: It can be deterministic or random. By random structure is meant one with parameters that have associated uncertainties, and thus which must be modeled in a random form. Although the randomness in the structure can be of three types, random variable, random process in space, and random process in time, discussion will be limited to the first two categories. While keeping the emphasis on finite element methods, other techniques involving finite differences, which are useful in the study of multi-degree-of-freedom systems, are briefly mentioned. The present review covers only developments that are derived from the engineering literature, thus implying near-term applicability. CONTENTS DETERMINISTIC STRUCTURES UNDER TIME-DEPENDENT RANDOM LOADS The problem of deterministic structures excited by stationary time-dependent random loads is reviewed. The terminology " random vibration" analysis is often used to designate this particular category of problems. There exists an extensive body of work dealing with multi-degree-of-freedom systems excited by random loads; thus, discussion will be limited to a few representative papers dealing explicitly with finite element methods for the analysis of deterministic structures subjected to random loads. The justification for presenting the simpler and well-studied problem of deterministic systems lies with the fact that some of these methods are used in more complex and realistic cases where structural parameters are random. As will be seen in a subsequent section, the perturbation method, which is a primary tool for the analysis of random systems, reduces the mathematical model of a random structure to that of a deterministic one, thus making useful the simpler methods described below. Theoretical background Modal analysis The matrix equation resulting from the finite element discretization is The mode shapes $, are given by the solution of the governing equation without damping: ] are the modal (diagonal) damping and frequency matrices, and {g} = [®] T {F} is the vector of modal forces. The solution {^(0} i s given by where [/(/)] is the diagonal matrix of impulse response functions. Frequency domain analysis For linear stationary problems, it is advantageous to work in the frequency domain since convolutions in time become products. The Fourier transform of modal equation where Q and G are the Fourier transforms of q and g, respectively, and [H(o>)] is the diagonal matrix of transfer functions with elements Analogous to the single degree of freedom system, the spectral density of output {q} is related to that of input {g} by the linear relation: where * denotes the complex conjugate. An equivalent expression is obtained by using the relation where summations are over the number of degrees of freedom. Alternately, one could work with the equations of motion prior to their diagonalization, in which case the transfer function matrix is not diagonal: Existing finite element codes can be utilized for performing modal analysis in the context of deterministic vibration problems. These programs are easily amenable to frequency response analysis in random vibrations. It is sufficient to add post-processors to the eigenvalue analyzers in order to construct the matrices of cross spectral densities. Individual applications The basic elements of this method have been extensively applied to various types of problems. We have selected here a few representative papers to illustrate applications in structural analysis. A good book, among others, is by Augusti, Jones and Beadle (1970) consider the response of a cantilever plate modeled with rectangular elements under random excitation. Rus Sigbjornsson and Smith (1980) evaluate the wave-induced vibrations of offshore gravity structures. It is assumed, based on North Sea gravity platform data, that the ocean waves are stationary, homogeneous Gaussian. The structure is modeled using standard finite element techniques. The approach assumes that a stochastic velocity potential exists which satisfies the Laplace equation with appropriate linearized boundary conditions. Stochastic linearization is used to linearize the equations of motion. Weeks and Cost (1980) estimate the reliability of a composite structure, assumed to be elastic-viscoelastic, subjected to random loading. The use of finite elements is viewed as particularly justified in view of the abrupt changes in mass and stiffness in composites. A comparison of the finite element results with a closed from solution is used to validate the method. Pfaffinger (1981) disscusses the modification of the finite element code ADINA to estimate the extreme behavior of structures under multisupport excitation. The structure is assumed linear, and the dynamic excitation, stationary Gaussian. The outline of the theory and computational details of the implementation in a finite element code are given in a complete and useful way. Pires, Hwang, and Reich (1986) have used these methods in the analysis of nuclear containment structures subjected to seismic ground motion. Shinozuka, Kako, and Tsurui (1986) provide an extensive review of random vibration analysis using modal analysis. They also limit state probabilities. The use of the finite element method with modal analysis and reliability methods which determine the probability of failure of various modes is presented in a thorough manner. To (1984) investigates the effects of a wide class of nonstationary random excitations on the response of multi-degree-offreedom structures discretized by the finite element method. The solution method consists of decoupling the equations of motion and expressing the components of the response vector by Duhamel's integral. This leads to closed form expressions for the response variance and covariance. Nonstationarity is modeled using evolutionary spectral densities. The method is valid only for cases where proportional damping can be adopted. In another paper (1986), To proposes to use a central difference method to directly integrate the discretized equations of motion. This approach does not use frequency domain or modal analysis and is thus adequate for nonlinear and nonstationary problems. A study of the accuracy and numerical stability of the method is included. The advantage of this direct integration scheme is that it does not require explicit functions for nonlinearities in the structure and allows for a wide variety of nonstationary random excitations. Concluding comments This section has presented the application of FEM analysis of deterministic systems subjected to random loading in some relatively simple situations. This capability proves useful for considering random systems, since the subsequent section on the perturbation method, which reduces random systems to deterministic ones under random loading, will be based on these methods. Problems of greater complexity, such as the response to non-Gaussian excitation, have not been discussed here since the focus of this review resides on analysis of uncertain systems. It is expected that the approaches discussed in the subsequent sections would be applicable in such cases. Computer codes such as SAP IV and ADINA have been modified to compute matrices of spectral densities. Existing codes and algorithms provide means of computing the transfer function matrix. Post-processors are required to perform the required multiplications to obtain spectral densities. The case of nonlinear systems must be handled using finite difference methods since there is no linear relation between input and output. Before considering random systems, the application of FEM for a solution of one stochastic equation is reviewed. FINITE ELEMENTS FOR THE SOLUTION OF ONE STOCHASTIC EQUATION Theoretical background Since all of the following papers use the Galerkin method of weighted residuals as a numerical solution technique to solve a random differential equation, the essence of the method is recalled with the first paper reviewed below. In the Galerkin method, the weights are identical to the basis functions. A significant review of the method of weighted residuals has been given by As an example to illustrate the Galerkin method, the following approach by Wong (1986) is presented for the solution of the string equation excited by an external random forcing function, with random boundary conditions. The random functions are assumed to be uncorrelated, and rest initial conditions are assumed. Expressions for time histories of first and second order moments are compared first with those obtained by applying the finite difference scheme directly to the governing equations of motion, and then with the exact solution. Consider the wave equation excited by a random forcing function: where f(x, t) and g(x) are the random excitation and initial condition with given autocorrelation functions. Using a discretization procedure, the time history of the autocorrelation function of u is'sought. The spatial domain [O.w] is divided into N-l intervals connected by N nodal points, and a trial solution u a of the form is assumed, where prescribed basis functions 6f(x) are given and satisfy the boundary conditions. Then u a also satisfy the boundary conditions for all a t (t), which are random functions. The differential equation for the residual is a measure of the accuracy with which u a approximates u. As N increases, the residual should become smaller, the exact solution occurring for a zero residual. As an approximation to this ideal, the weighted integral of this residual is set to zero: w, are the weighting functions chosen to be equal to the basis function 6 k (x) in the Galerkin method. Upon integration by parts and substitution of the series expansion (11) into Eq (13), a set of ordinary differential equations in time is obtained. Using matrix notation, one has where and {a(/)} is the unknown vector of nodal responses. A solution can be obtained by using the central difference ap- A recursive relation governing the time-increment solution is obtained: where where In the direct finite difference formulation, both the spatial and time derivatives are approximated by their difference equivalents. Again, recursive relations for the mean and variance are derived. The finite element and finite difference solutions are compared to an exact solution. Individual applications Tasaka and Matsuoka (1982) and Tasaka (1985) for the nonstationary case, consider the one-dimensional heat diffusion equation with random initial conditions g(x) and random external excitation f(x, r): both from a finite element and finite difference point of view. The procedure and equations are very close to those of the preceding example. It is assumed that there is no statistical correlation between the two random functions, g(x) and f(x, t), and that the boundary conditions are zero. The solution to Eq (20) assumes the usual form of a series of linear space dependent interpolation functions weighted by time dependent functions. For the element k, «*(*,/) = !>,*(*)«?(')• This equation is similar to Eq (11), except it is written for a single element. The Galerkin scheme is applied to the governing equation to obtain a time-dependent equation governing one discrete element. A forward time difference is then used to obtain the solution to this equation in a recursive form. That is, the solution at a given time is defined in terms of the solution at previous time increments. Mean values and covariance matrices can thus be constructed in a similar recursive form. A finite difference scheme for the numerical solution of the heat equation is considered next by the authors, using a forward difference approximation in time and a central difference ap- For the finite element scheme, Eq (22) is used to derive relations governing the mean and the covariance matrix. Comparisons between finite element and finite difference approaches are discussed. Comparisons of finite element, finite difference, and exact solutions reveal that, of the two schemes examined, the finite difference is better from a computational and accuracy point of view. The authors of the above three papers all concluded that, for the specific problems considered and numerical schemes used, the finite difference approach was superior in accuracy and computational cost when compared to the finite element. However, no general conclusion is possible, and it is recognized that numerous factors could affect such comparisons. This is especially true for larger scale problems than those considered. Sun (1974) examines deterministic system equation (23) where random force f(x) is given white noise properties: