Approximation method to compute domain related integrals in structural studies

Abstract

Various engineering calculi use integral calculus in theoretical models, i.e. analytical and numerical models. For usual problems, integrals have mathematical exact solutions. If the domain of integration is complicated, there may be used several methods to calculate the integral. The first idea is to divide the domain in smaller sub-domains for which there are direct calculus relations, i.e. in strength of materials the bending moment may be computed in some discrete points using the graphical integration of the shear force diagram, which usually has a simple shape. Another example is in mathematics, where the surface of a subgraph may be approximated by a set of rectangles or trapezoids used to calculate the definite integral. The goal of the work is to introduce our studies about the calculus of the integrals in the transverse section domains, computer aided solutions and a generalizing method. The aim of our research is to create general computer based methods to execute the calculi in structural studies. Thus, we define a Boolean algebra which operates with ‘simple’ shape domains. This algebraic standpoint uses addition and subtraction, conditioned by the sign of every ‘simple’ shape (-1 for the shapes to be subtracted). By ‘simple’ shape or ‘basic’ shape we define either shapes for which there are direct calculus relations, or domains for which their frontiers are approximated by known functions and the according calculus is carried out using an algorithm. The ‘basic’ shapes are linked to the calculus of the most significant stresses in the section, refined aspect which needs special attention. Starting from this idea, in the libraries of ‘basic’ shapes, there were included rectangles, ellipses and domains whose frontiers are approximated by spline functions. The domain triangularization methods suggested that another ‘basic’ shape to be considered is the triangle. The subsequent phase was to deduce the exact relations for the calculus of the integrals associated to the transverse section problems. Thus we use a virtual rectangle which is framing the triangle, being generated supplementary right angled triangles. The sign of rectangle and the signs of the supplementary triangles are conditioned by the sign of the initial triangle. In this way, a generally located triangle for which we have direct calculus relations may be used to generate the discretization of any domain in transverse section associated integrals. A significant consequence of the paper is the opportunity to create modern computer aided engineering applications for structural studies, which use: intelligent applied mathematics background, modern informatics technologies and advanced computing techniques, such as calculus parallelization.