Probabilistic Methods in the Theory of Structures

The following sections are included:

• Dedication
• Preface to the Third Edition
• Preface to the Second Edition
• Preface to the First Edition
• Contents

For an adequate description of structural behavior, probabilistic methods must be resorted to. Properly speaking, an element of probability is embodied even in the deterministic approach, which claims to “simplify” the structure by eliminating all aspects of uncertainty. Under the deterministic approach, external loading and the properties of the structure are represented as though they were fully determined, and available (often highly sophisticated) tools yield, with sufficient accuracy, the strains and stresses in systems with complex configurations. At the same time, these stresses are compared with allowable ones obtained by dividing their ultimate levels by a “safety factor,” so as to yield a level below that of failure, a practice that recognizes the uncertain, and random, features of the stress distribution in the material. This is how a probabilistic consideration is admitted “via the back door”; indeed, the safety factor has often been referred to as the “ignorance factor”…

The following sections are included:

• RANDOM EVENT
• SAMPLE SPACE
• PROBABILITY AXIOMS
• EQUIPROBABLE EVENTS
• PROBABILITY AND RELATIVE FREQUENCY
• CONDITIONAL PROBABILITY
• INDEPENDENT EVENTS
• RELIABILITY OF STATICALLY DETERMINATE TRUSS
• OVERALL PROBABILITY AND BAYES’ FORMULA
• PROBLEMS
• CITED REFERENCES
• RECOMMENDED FURTHER READING

The following sections are included:

• RANDOM VARIABLE
• DISTRIBUTION FUNCTION
• PROPERTIES OF THE DISTRIBUTION FUNCTION
• MATHEMATICAL EXPECTATION
• MOMENTS OF RANDOM VARIABLE; VARIANCE
• CHARACTERISTIC FUNCTION
• CONDITIONAL PROBABILITY DISTRIBUTION AND DENSITY FUNCTIONS
• INEQUALITIES OF BIENAYMÉ AND TCHEBYCHEFF
• PROBLEMS
• CITED REFERENCES
• RECOMMENDED FURTHER READING

In this chapter we present some widely used discrete and continuous probability distribution and density functions.

The scope of the knowledge acquired by us so far suffices for reliability analysis (recalling that reliability is the probability of nonfailure performance) of the simplest structures, described by a single random variable. We are concerned with generally differing structures called upon to realize different functional assignments and will not consider nonfunctional ones. In other words, reliability is associated with a purpose—exploitation of the structure in accordance with defined goals. The acceptability criterion consists in the reliability exceeding some specified level.

We now proceed to concrete examples of structural reliability.

With the aid of the theory of a single random variable, we were able to calculate the reliability of a structure characterized by such a variable, for example that of a bar under a random tensile force, with strength treated as a deterministic quantity. We considered also the “reverse” example, where the force was assumed to be given and the strength considered as a random variable. In practice, however, both force and strength are random. Moreover, the beam may have to bear several concentrated loads rather than a single one, and so on. In these circumstances the theory must be extended to the case of multiple random variables…

Hitherto we have been able to consider only the reliability of structures described by a single random variable. Such an analysis was, however, confined to simple cases. Now we extend the discussion to structures described by two or more random variables, beginning with the two-dimensional case.

This chapter deals with some basic aspects of the random function, which is in essence a generalization of the random variable.

In this chapter we turn to random vibration problems with intensive reference to the results of Chapter 8. After a review of relevant deterministic results, we proceed to cases of single- and multidegree-of-freedom systems. Finally, we demonstrate the normal mode method and the dramatic effect of the interaction between different modes, usually overlooked in the literature.

Having so far confined ourselves to discrete structures, it is natural now to proceed to continuous structures in which excitation and response are generally random not only in time but also in space. The solution via the normal mode method will be derived, and the effect of cross correlation between different modes will be shown again to be of marked importance.

Up to now we have been concerned with problems capable of exact or effective approximate solution. These, however, constitute only a small fraction of the probabilistic problems encountered in the different branches of engineering. In cases where exact solution is cumbersome or impossible, the Monte Carlo method is a logical remedy, especially with the advent of high-speed digital computers. Even if it may seem a “last resort” at present, it is bound to become more and more attractive in the future.

The following sections are included:

• Appendix A. EVALUATION OF INTEGRALS (4.15) AND (4.22)
• Appendix B. TABLE OF THE ERROR FUNCTION
• Appendix C. CALCULATION OF THE MEAN SQUARE RESPONSE OF A CLASS OF LINEAR SYSTEMS
• Appendix D. SOME AUTOCORRELATION FUNCTIONS AND ASSOCIATED SPECTRAL DENSITIES
• Appendix E. GALERKIN METHOD
• Appendix F. FIRST ORDER — SECOND — MOMENT HASOFER-LIND METHOD

The following sections are included:

• Additional References (1984–1998)
• Additional References (1999–2016)
• Index