Analytic General Solution Employed to Calculate the Geometrical Characteristics in Structural Problems

Abstract

The automatic calculus in structural problems was a constant concern of the authors, several models of the phenomena being based on algorithmic generalizations of the analytical methods, where the solutions were either ‘exact’ or numerical. The calculus of the geometrical characteristics is important, being closely related to the calculus of the stresses, of the displacements, of the bucking and in other structural problems. This is why an algorithm designed to compute the geometrical characteristics, must take into account all these aspects. A first general and original solution was to create a Boole algebra based on simple shapes for whom there are direct calculus relations and to create complex geometries of the composite cross-sections using the addition and subtracting operations. This method offered an important advantage regarding the position of a very small set of points belonging to the section where the maximum stresses could be found. The weakness of the method consists of the very few ‘simple’ shapes which could be included in the library of the software application. A first direction to generalize the method would be the definition of new simple geometrical shapes, to be included in the general algorithm and in the library of the software application. Other shapes may be also included, such as the rolled steel girders: I-shaped beams, C-shaped beams and others. However, more general methods may be conceived, expressed as algorithm and implemented. The paper presents an original method to calculate the integrals, based on a mathematical generalization of the, so to-say, ‘classic’, theory. The method has some strong points, such as: it is very flexible regarding the definition of the domain; the effective calculus may be done either in a direct way or using the numerical methods; it can be easily adapted to employ approximative solutions in order to check the accuracy of the results using alternate approaches; the computer program may be conceived in several ways, fact which offers another aid in the generalisation of the method. The method was tested and the results are accurate. The next direction of generalisation regards the creation of an algorithm which identifies the most relevant locations of the points where the maximum stresses may be found and the areas where sudden variation of the stresses may occur. In this way, there may be calculated only small amount of relevant ad important data regarding the strength of the structure.