Finite-Element Analysis Of Two-Dimensional Frame Structures

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Parish, Benjamin
Applied Mathematics
thesis / dissertation description
The majority of the problems involving analysis of deformable solids are not tractable by direct methods. Exact solutions are at best, difficult to obtain and frequently unavailable. Because of the difficulty of getting exact solutions, many different approximation methods have been developed, one of which is the finite-element method. In order to explain how this method generates useful results, I will present the basic theory behind the finite-element method as well as discuss some general applications for using it. Most of this explanation, however, will focus on the application of the finite-element method to beams and frame structures. The basis of the finite element method is actually quite simple. In order to find the displacements and rotations of a structure under a physical load, a system of differential equations needs to be solved. The function u, defining displacements and deformations, is necessary for this solution. The amount of work when a structure is subject to arbitrary, kinematically consistent, virtual displacements and deformations must be equal to zero. Because these differential equations can almost never be solved exactly, an approximation is necessary. The Rayleigh-Ritz-Galerkin method provides a useful way to make this kind of approximation. Finite-element analysis requires choosing a finite number of trial functions and combining them in such a way that an optimum solution can be found (Strang 1). Two ways of applying this method are either to increase the number of subdivisions or elements in the structure, applying relatively simple trial functions to each sub-member, or to add more and more complex trial functions for each element. With these trial functions we model the actual deformations with estimated forms. The model I will be using applies the first of the above methods, therefore increasing the accuracy of analysis for deformable solids. Not only does this method lend itself to easier calculations, but it can be automated with a computer program.