FINITE ELEMENT METHOD FOR TWO-DIMENSIONAL ELASTICITY PROBLEM HANIMAH OTHMAN A dissertation submitted in partial fulfilment of the requirements for the award of the degree of Master of Science (Mathematics) Faculty of Science Universiti Teknologi Malaysia APRIL 2010
v ABSTRACT The purpose of this study is to develop a system which can be used to solve two-dimensional elasticity problem. A manual solution has been made to solve this two-dimensional elasticity problem based on the heat problem. In the first step, the main component is solved to get the stiffness matrix. Next, the initial strain and load vector, and the boundary vector are computed to complete the solution. The solution is computed and verified using MATLAB.
vi ABSTRAK Kajian ini bertujuan untuk membangunkan satu sistem yang boleh digunakan bagi menyelesaikan masalah kekenyalan dua dimensi. Satu penyelesaian secara manual untuk menyelesaikan masalah kekenyalan dua dimensi ini telah dibentuk dalam disertasi ini berdasarkan masalah haba. Dalam langkah pertama, komponen-komponen utama yang diselesaikan bagi mendapatkan matrik stiff. Seterusnya, nilai awal kekenyalan dan vektor limpahan serta vektor sempadan diselesaikan bagi menyempurnakan penyelesaian. Setiap penyelesaian dikira dan ditentusahkan menggunakan perisian MATLAB.
vii TABLE OF CONTENTS FINITE ELEMENT METHOD FOR TWO-DIMENSIONAL ELASTICITY PROBLEM i ACKNOWLEDGEMENT iv ABSTRACT v ABSTRAK vi TABLE OF CONTENTS vii LIST OF FIGURES x LIST OF SYMBOLS xi GLOSSARY OF TERMS xii LIST OF APPENDICES xiii 1 RESEARCH FRAMEWORK 1 1.1 Introduction 1 1.2 Background of the problem 2 1.3 Statement of the problem 2 1.4 Objectives of the study 3 1.5 Scope of the study 3 1.6 Outline of the research 4 2 LITERATURE REVIEW 5 2.1 Introduction 5 2.2 Elasticity 5 2.3 Types of Elasticity 7 2.4 Linear Elasticity 8 2.4.1 Initial strain 9
viii 2.4.2 Plane stress 10 2.4.3 Plane strain 13 2.5 Two-Dimensional Elasticity 14 2.5.1 Weak Form of Equilibrium of Two-Dimensional Elasticity 15 2.5.2 Formulation of Two-Dimensional Elasticity 17 2.6 Conclusion 19 3 COMPUTATIONAL MODELS 20 3.1 Introduction 20 3.2 Exact and Approximation Methods 20 3.2.1 Finite Element Method 21 3.2.2 Finite Difference Method 22 3.2.3 Boundary Element Method 22 3.3 Manual Calculation for One-Dimensional Problem 23 3.3.1 Example on One-Dimensional Elasticity 23 3.3.2 Strains 25 3.3.3 Stress-Strains Relationship 26 3.4 Manual Calculation for Torsion Problem 26 3.5 Some Method from the Journal 30 3.5.1 Introduction to Finite Element Techniques in Stress Analysis 30 3.5.2 What makes problems in theoretical stress analysis difficult? 31 3.5.3 The Finite Element Method 32 3.5.4 Example from the journal 33 3.6 Conclusion 37 4 CASE STUDY IN TWO-DIMENSIONAL PROBLEM 38 4.1 Introduction 38
ix 4.2 Case Study: Two-Dimensional Elasticity Problem 38 4.3 Finite Element Formulation of the problem 39 4.4 Manual calculation 45 4.4.1 Element 1 45 4.4.2 Element 2 51 4.4.3 Element 3 55 4.5 Conclusion 64 5 CONCLUSION AND RECOMMENDATION 65 5.1 Conclusion 65 5.2 Recommendation 65 REFERENCES 67 A WATERFALL MODEL 69 B GANTT CHART 70 C DETAILS IN CALCULATIONS 71 D MATLAB PROGRAMMING ALGORITHM 76
x LIST OF FIGURES FIGURE NO. TITLE PAGE Figure 2.1 (a) Direct or normal stress 6 Figure 2.1 (b) Tangential or shear stress 6 Figure 2.1 (c) Change in volume with no change in shape 6 Figure 3.1 Finite Element Discretisations of a machine component 21 Figure 3.2 Finite Difference Discretisations of a machine component 22 Figure 3.3 Boundary Element Discretisations of a machine component 23 Figure 3.4 The illustration 2-noded bar element 23 Figure 3.5 The iilustration for torsion problem 26 Figure 3.6 The illustration for torsion problem for element 1 27 Figure 3.7 Two-fold symmetry of geometry 34 Figure 3.8 Less mesh elements 35 Figure 3.9 More mesh elements 35 Figure 3.10 Mesh No 1 36 Figure 3.11 Mesh No 2 36 Figure 4.1 An illustration for case study 38 Figure 4.2 An element from the problem 39 Figure 4.3 Element 1 45 Figure 4.4 Element 2 51 Figure 4.5 Element 3 55
xi LIST OF SYMBOLS - Stiffness matrix - Matrix differential operator for shape function - Constitutive matrix - Boundary vector - Load vector - Initial strain vector - Shape function - Initial strain - Stress - Strain - Thermal expansion coefficient - Change in temperature - Temperature - Traction vector - Normal vector - Body force - Young s modulus - Poisson s ratio - Displacement vector
xii GLOSSARY OF TERMS FEM - Finite Element Method FDM - Finite Difference Method BEM - Boundary Element Method dof - Degree of freedom
xiii LIST OF APPENDICES APPENDIX TITLE PAGE A Waterfall Model 69 B Gantt Chart 70 C Details in calculations 71 D Matlab programming algorithm 76
CHAPTER 1 1 RESEARCH FRAMEWORK 1.1 Introduction Historically, Kevin and his friends such as Lame, Boussinesq, Cerruti and others are one of the first experimentally observed about the theory of elasticity. They mentioned that the rigorous to some boundary value problem of elasticity theory for the regions bounded by the surfaces prescribed by a single parameter have been obtained. However, Saint-Venant was reported his observation in the books of On torsion of prisms and On bending of prisms in which the semi-inverse method and Saint-Venant s principle were suggested. It should be proclaimed as the origin of elasticity theory as an applied discipline. Thus the elasticity problems represent, not a flux and temperature, but the traction vector and the displacement vector need to be found. The problem of finding a flux and temperature at certain nodes in heat problem is such a way in giving ideas about elasticity problem. Some problems in elasticity field such as torsion need detail calculations and a system must be used if it comes to a big number. It contains some crucial parts and need to spend more times when doing manually. The system developed here is just for certain problem using certain techniques in finite element method (FEM). By using this finite element method, all the calculations contain such as the differentiation and integration parts in most calculation and solving a matrix system
2 using Gauss elimination system. These entire things are the main parts in solving this type of problem. Elasticity is just one of the applications in finite element method. The methods were in action when some forces are given to a solid that make changes in their shape. When this occurs, the shape for that solid are not permanently in their new shape but still have a chance to back normal but it depends on the force that given on it. 1.2 Background of the problem The elasticity problem is common problem that many engineers faced it. They give the problem to model and solve it numerically. This problem will be solved using finite element method for grid rectangle shape function. It depends on the value given such as the area, coordinate of the model and the flux at beginning. For instance, it does not give directly, but need to find based on the model graphically. However, the process are the same as heat flow problem except only for certain part such as the value of load vector, boundary vector and followed by initial strain vector. Each such solution has different shape function based on the number of element. Recent comparisons with every element show the different flux and temperature for every nodal point with surprising accuracy. The stress and strain value needs to be found by doing some other calculation before go through to the real problem. This success suggests that more complicated of three-dimensional problem might provide accurate physical models of more complex elasticity phenomena or even though with other shape function. 1.3 Statement of the problem Given a two-dimensional domain for an axially loaded elastic bar with a constant traction vector along the boundary by assuming a rectangular disk. The forces and the traction vector are given. In the same time, the mass density and the acceleration due to gravity are provided. The problem is to determine the finite
3 element formulation of elasticity for one element by getting the value of the boundary vector, the load vector and the initial strain vector. The result shows that one-fourth of this total force is distributed to each nodal point. The same results are obtained when considering the other elements and this implies that the uniform traction is equivalent to the nodal forces. 1.4 Objectives of the study The main objectives of this research are to: 1. Study the characteristics and problems of two-dimensional elasticity from previous. 2. Derive finite element formula for two-dimensional elasticity for grid rectangle problem. 3. Determine the solvability of the formulated finite element method. 4. Provide a finite element technique of two-dimensional elasticity for grid rectangle problem using software MATLAB. 1.5 Scope of the study This research will only consider the study on two-dimensional elasticity problem which is the extension from finite element method. A two-dimensional elasticity problem will be solved using finite element method and Gauss elimination is a technique to solve the matrix part. Apart from that, several requirements will only get from the problem itself or will be given directly. The solution of twodimensional elasticity problem will only focus on the problem which is using a grid rectangle as a method of solution in order to get their shape function.
4 1.6 Outline of the research This dissertation has five chapters including the preceding introductory part that have been discussed in the earlier sections. Chapter 2 will provide with some information on the literature studies related to this research. This chapter starts with the history of elasticity problem followed by some introductions on finite element method. Then, there will be a section that discusses on two-dimensional elasticity problem and finally some brief introduction on finite element method. In Chapter 3 will focus on the computational methods. This chapter starts with the process in order to complete the programming parts that have been used in some research by the other researchers. Some journal about using the elasticity or some parts in elasticity will be referred. All solutions are obtained by using MATLAB computer programming. Chapter 4 is mainly about the elasticity problems in details and some explanations about the problem. The manual solutions are used to compare the solutions using programming system. Most of the problems are nearly same as heat problem because it has a strong connection between both problems. There are conditions for elasticity to occur in heat problem such as for one-dimensional and two-dimensional problem. Manual solutions will be produced in this section. Finally, the last chapter will be the conclusion part that summarized the whole research. There are also some suggestions regarding related research area will be proposed in the last chapter, as well as the interesting applications of twodimensional elasticity problem in daily physical problems. Attachments have been provided in order to complete this research. Appendix A is a research methodology based on waterfall model. Appendix B is a Gantt chart that shows the timeline and milestones of the project that reflects with the Appendix A. While, the Appendix C shows the detail for calculations inside and the Appendix D shows the algorithm for the calculations.
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