On the accuracy and convergence of the finite element method for truss and frame structures

Abstract

During applied mechanics courses for undergraduates, the words `finite element' will be mentioned if a structural problem is so complex that it cannot be solved analytically or simply because it would require too much time to solve it by hand. The lecturer refers to a software package and explains how to work with it. Unfortunately, during these courses, no time is spend on the inner workings of this software. How does the program know, after virtually assembling the structural members and specifying the load, how the structure mechanically behaves? Another term that is mentioned in conjunction with the finite element method is `approximation'; the method is not an exact method, only approximated values are obtained. The accuracy depends on the amount of elements and the type of element. In this thesis the following question will be answered: what is the influence of the mesh size and the element type used in a finite element protocol for trusses and frames, on the accuracy of the displacements and forces and how does it affect the computation time? The goal of this thesis is also to get a better comprehension of the inner workings of a finite element program with a structural application. All this can be achieved by developing, analyzing and using a finite element program created in the MATLAB environment. To do this, prior literary study is required. Use is made of [1] and sub-paragraph 5.1.1 of [6]. A total of four MATLAB programs are written, one for trusses in 2D space, one for trusses in 3D space and two for frames in 2D space. Frames can namely be described by so-called linear and quadratic elements. What that means will be explained in paragraph 2.2. The structural models will be discretized and solved by MATLAB. The created finite element programs will be described and explained in the second chapter. After reading that part, it will be clear how the programs work and how to use them. In the chapter that follows, three elementary frame structures will be analyzed by modelling them with a varying number of elements and different element types. A comparison will be made with the analytical solution. Other aspects of the programs and how they compare to commercial finite element software is the focus of chapter 4. A geometrically complex structure, like an arch, will be modelled by beam elements and analyzed in the fifth chapter. The findings will be summarized and the central question will be answered in chapter 6.