Using the Direct Determined Fully Probabilistic Method (DDFPM) for determination of failure

Abstract

DDFPM has been developed as an alternative for Monte Carlo in the assessment of structural reliability in probabilistic calculations (Marek et al. 1995). Input random quantities (such as the load, geometry, material properties, or imperfections) are expressed as histograms in the calculations. In the probabilistic calculations, all input random variables are combined with each other. The number of possible combinations is equal to the product of classes (intervals) of all input variables. With rather many input random variables, the number of combination is very high. Only a small portion of possible combinations results, typically, in failures. When DDFPM is used, the calculation takes too much time, because combinations are taken into account that does not contribute to the failure. Efforts to reduce the number of calculation operations have resulted into the development of algorithms that provide the numerical solution of the integral that defines formally the failure probability with rather many random variables: n n D f X X X X X X f p f d ,..., d , d ) ,...., , ( 2 1 2 1 ∫ = , where Df represents a failure area where g(X)≤0, f(X1, X2,...., Xn) for the function of the combined density of probabilities of random quantities. The algorithms are implemented into ProbCalc features of the software will be presented here. Parts of the calculation can be carried out simultaneously. The random input variables can be expressed by means of histograms with parametric and non-parametric distribution created from sets of random quantities that have been measured or observed. Figure 2. Structure resistance and load response – probability density curves The chart shows probability density curves for the structure load response fS(x) and structure resistance fR(x). The mutual location of the both curves, fS(x) and fR(x), characterize and specifies the area where a failure may appear and enables the failure probability to be calculated. That area is hatched separately for both the density fS(x) and fR(x). The failure occurs, when the following condition is fulfilled: Z smax (pf = 0), − might, but need not necessarily, occur if rmin ≤ s and s ≤ smax, where the failure probability for all possible s values is pf (0 ≤ pf ≤ 1) , − occurs always, if rmax < smin (pf = 1). Note: Theoretically, the points of intersection may exist in infinitude, this being for instance the case of a parametric normal distribution. The failure probability can be analytically calculated using the well-know formula (Kralik 2006; Teplý & Novak 1999): ∫ ∫ ∞