Extreme of random field over rectangle with application to concrete rupture stresses

Probabilities of excursions of random processes and fields into critical domains are of fundamental interestin many civil engineering decision problems. Examples are reliability evaluations of structures subject torandom load processes, the influence of the size of a structural element on the carrying capacity of theelement when the local material properties vary as a random field over the element, evaluation of the riskof floods or draughts in connection with river and reservoir management, etc. Only for some very few specialcases of process and field models it is possible to obtain exact results for such probabilities. However, dueto the engineering importance of the problem, several approximate assessment methods have been suggestedin the past. The suitability and accuracy of each of these methods depends on the type of process or fieldunder consideration. Often recourse must be taken to time consuming simulation procedures. This paperrevives a conceptually simple approach that gives surprisingly good results in particular for wide band typesof random processes and fields. The closed form formulas obtained for smooth Gaussian fieldsover rectangles contain size effects both with respect to the area of the rectangle and the side lengths of therectangle. Published rupture stress data for plain concrete beams illustrate the applicability of the derivedclosed form extreme value distributions as models for distributions of rupture stresses related to weakest linkmechanisms.