The objective of this chapter is to discuss the concept of base functions and the basics of using some simple base functions to represent other functions. These base functions are needed in many commonly used analyses. An example of using base functions to approximate an almost arbitrary target function is the Taylor series expansion we have discussed. Here we say that an “almost arbitrary function” is not really arbitrary because, in theory, the target function must be differentiable an arbitrary number of times for the Taylor series expansion to be valid. The concept of linear independence of functions is important in understanding the selection of base functions.

Chapter 9 describes simulation or sampling methods for reliability assessment. The chapter begins by describing methods for generation of pseudorandom numbers for prescribed univariate or multivariate distributions. Next, the ordinary Monte Carlo simulation (MCS) method is described. It is shown that for small failure probabilities, which is the case in most structural reliability problems, the number of samples required by MCS for a given level of accuracy is inversely proportional to the failure probability. Thus, MCS is computationally demanding for structural reliability problems. Various methods to reduce the computational demand of MCS are introduced. These include the use of antithetic variates and importance sampling. For the latter, sampling around design points and sampling in half-space are presented, the latter for a special class of problems. Other efficient sampling methods described include directional sampling, orthogonal-plane sampling, and subset simulation. For each case, expressions are derived for a measure of accuracy of the estimated failure probability. Methods are also presented for computing parameter sensitivities by sampling. Finally, a method is presented for evaluating certain multifold integrals by sampling. This method is useful in Bayesian updating, as described in Chapter 10.