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Under random excitation, the fatigue crack growth of a structure depends primarily on the statistics of rises and falls of the random response process. Based on the envelope statistics, a close-form solution, expressed in terms of the hypergeometric function, for approximating the rise and fall statistics of random processes is presented. The solution is very simple to compute and its accuracy is very satisfactory when the random response process is reasonably narrow band.

In this paper an approach is taken in the time domain analysis, which yields an upper bound of the maximum response. A rational design rule seems to be that the less information or data one has, the more conservative design one should take. Although a structure with a single input is considered here, the extension of the present technique to a structure with multi-excitations appears to be possible.

The definition of stationary random envelope proposed by Cramer and Leadbetter, is extended to the envelope of nonstationary random process possessing evolutionary power spectral densities. The density function, the joint density function, the moment function, and the crossing rate of a level of the nonstationary envelope process are derived. Based on the envelope statistics, approximate solutions to the first excursion probability of nonstationary random processes are obtained. In particular, applications of the first excursion probability to the earthquake engineering problems are demonstrated in detail.

The first-excursion probability of a non-stationary Gaussian process with zero mean has been studied. Within the framework of the point process approach, a variety of analytical approximations applicable to stationary random processes is extended herein to non-stationary random processes. The extension is possible owing to a recent definition of non-stationary envelope processes proposed by the author. With the aid of numerical examples, merits of each approximation are examined by comparing with the results of simulation. It is found that under non-stationary excitations with short duration, the Markov approximation is the best among all the approximations discussed in this paper.

Efficient methods of simulating stationary and nonstationary random processes and envelopes, by using a series of sine or cosine functions or by using the fast Fourier transform, have been proposed previously. Without applying the central limit theorem, it is shown in this paper that the simulated random processes are asymptotically Gaussian processes as the number of terms, N, of sine or cosine functions approaches infinity. The accuracy of the first-order probability densities of the simulated random processes is investigated by using the fast Fourier transform. Numerical results are computed with respect to the variation of the number of terms, N, of sine or cosine functions used for simulation. It is shown that within the practical range of N, such as 500, the accuracy is remarkably satisfactory even outside the region of 3 standard deviations.

Spacecraft structures weight optimization based on fracture mechanics and reliability cost constraints applied to pressure vessel design

Efficient and practical methods of simulating stationary and nonstationary random envelope processes are presented. The stationary envelope processes are simulated by using the fast Fourier transform while the nonstationary envelope processes are simulated as the square root of the sum of a series of cosine functions and a series of sine functions with random phase angles. Typical applications of the envelope simulation are the simulations of peaks and troughs which play an important role in the analyses of the first excursion probability, fatigue and crack propagation. In particular, applications to the crack propagation under random loadings are demonstrated in detail.

Method has been developed to improve the lower bound of the first excursion probability that can apply to the problem with either constant or time-dependent barriers. The method requires knowledge of the joint density function of the random process at two arbitrary instants.

Study establishes distribution function of peak response values, based on frequency interpretation. Excitations considered include impact loading on landing gears and aircraft gust loading. Because of relative severity of excitations, prediction of fatigue and maximum response characteristics is important part of task of structural analysis and design.

The stochastic structural dynamic analysis for complex structures demands an analytical treatment involving nonstationary stochastic processes, under nonstationary gusty winds or earthquake accelerations. It is emphasized that the frequency domain analysis of such nonstationary problems with the aid of the Fourier transform technique is of vital importance in evaluating the mean value and the covariance function of the response, which are closely related to structural safety. An efficient method of numerical single and double Fourier transform and inversion with the aid of the complex Fourier series technique coupled with the Cooley-Tukey algorithm for evaluation of the Fourier coefficients is presented. Numerical examples indicate that the numerical inversion can be performed within a reasonable time on an IBM 7094.

A well‐posed boundary‐value problem is obtained of the first‐passage probability for a linear single‐degree‐of‐freedom vibratory system with a linear viscous damping and subjected to either stationary or nonstationary white‐noise excitation.