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Interactive Analysis of General Beam Configurations Using Finite Element Methods and Javascript by CHRISTOPHER HERNANDEZ Presen

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Interactive Analysis of General Beam Configurations Using Finite Element Methods and Javascript by CHRISTOPHER HERNANDEZ Presen Interactive Analysis of General Beam Configurations using Finite Element Methods and JavaScript By CHRISTOPHER HERNANDEZ Presented to the Faculty of the Graduate School of The University of Texas at Arlington in Partial Fulfillment of the Requirements for the Degree of MASTER OF SCIENCE IN AEROSPACE ENGINEERING THE UNIVERSITY OF TEXAS AT ARLINGTON December 2016 i Copyright © by Christopher Hernandez 2016 All Rights Reserved ii Acknowledgements I would like to thank my family for always being there for me and supporting me through all the years I’ve been in school. Without them I would not have been able to achieve all that I have up to this point. I would also like to express my gratitude to Dr. Lawrence, who gave me the opportunity to not only learn new skills but was also there to guide and help me make my work the best that it could be, and was instrumental in shaping this work into what it is today. Finally, I would like to thank my friends who I have been with me since my undergraduate studies. November 21, 2016 iii Abstract DEVELOPMENT OF FINITE ELEMENT BEAM ANALYSIS APPLICATION USING JAVASCRIPT Christopher Hernandez, MS The University of Texas at Arlington, 2016 Supervising Professor: Kent Lawrence Advancements in computer technology have contributed to the widespread practice of modelling and solving engineering problems through the use of specialized software. The wide use of engineering software comes with the disadvantage to the user of costs from the required purchase of software licenses. The creation of accurate, trusted, and freely available applications capable of conducting meaningful analysis of engineering problems is a way to mitigate to the costs associated with every-day engineering computations. Writing applications in the JavaScript programming language allows the applications to run within any computer browser, without the need to install specialized software, since all internet browsers are equipped with virtual machines (VM) that allow the browsers to execute JavaScript code. iv The objective of this work is the development of an application that performs the analysis of a completely general beam through use of the finite element method. The app is written in JavaScript and embedded in a web page so it can be downloaded and executed by a user with an internet connection. This application allows the user to analyze any uniform or non-uniform beam, with any combination of applied forces, moments, distributed loads, and boundary conditions. Outputs for this application include lists the beam deformations and slopes, as well as lateral and slope deformation graphs, bending stress distributions, and shear and a moment diagrams. To validate the methodology of the GBeam finite element app, its results are verified using the results from obtained from two other established finite element solvers for fifteen separate test cases. v Table of Contents Acknowledgements ...................................................................................................................................... iii Abstract ........................................................................................................................................................ iv List of Illustrations ..................................................................................................................................... viii List of Tables ................................................................................................................................................ x 1. Introduction ............................................................................................................................................... 1 2. Installed Software Licensing ..................................................................................................................... 1 2.1 SOFTWARE LICENSING BEFORE THE INTERNET .............................................................................. 1 2.2 Types of Licenses ............................................................................................................................... 2 3. JavaScript Programming Language .......................................................................................................... 5 3.1 JavaScript: A Brief Overview ............................................................................................................. 5 3.2 Developing Computational Applications with JavaScript .................................................................. 6 4. The Finite Element Method .................................................................................................................... 10 4.1 Overview of FEM: History, Advantages, and Concepts ................................................................... 10 4.2 Beam Models: Euler-Bernoulli Beam ............................................................................................... 14 4.3 Formulation of Beam Element Stiffness Matrix: Euler-Bernoulli .................................................... 16 4.4 Formulation of Beam Global Stiffness Matrix: Euler-Bernoulli ...................................................... 24 5. JavaScript Finite Element Beam Analysis .............................................................................................. 32 5.1 An Application in Three Parts .......................................................................................................... 32 5.1.1 Input Spreadsheet: The Main Menu ........................................................................................... 33 5.1.2 Solution Output Window ........................................................................................................... 41 5.1.3 Plot Output Window .................................................................................................................. 42 5.2 Sorting Data Into Arrays ................................................................................................................... 43 5.3 Creation of Element & Global Stiffness Matrices ............................................................................ 47 5.3.1 Stiffness Matrix Creation ........................................................................................................... 47 5.3.2 Effects of Boundary Conditions On Global Stiffness Matrix .................................................... 50 5.4 Spring Support .................................................................................................................................. 63 5.5 Applied Forces and Moments ........................................................................................................... 64 5.5.1 Application of Distributed Forces .............................................................................................. 65 5.6 Beam Refinement .............................................................................................................................. 70 6. Solution and Outputs ............................................................................................................................... 71 6.1 Displacements ................................................................................................................................... 72 vi 6.2 Displacement Output Window .......................................................................................................... 75 6.3 Plot Output Window ......................................................................................................................... 77 6.3.1 Lateral Displacement Diagram .................................................................................................. 78 6.3.2 Slope Diagram ........................................................................................................................... 79 6.3.3 Shear and Moment Diagram ...................................................................................................... 81 6.3.4 Bending Stress Diagram ............................................................................................................. 88 7 Output Comparison .................................................................................................................................. 89 8 Conclusions ............................................................................................................................................ 105 9 Future Work ........................................................................................................................................... 107 Appendix A- Additional Application Images ........................................................................................... 111 Appendix B- Beam Problem Outputs and Comparisons........................................................................... 113 vii List of Illustrations Figure 1 Circle Area Application Example 1 ................................................................................................ 6 Figure 2 Circle Area Application Example 1 Script ..................................................................................... 7 Figure 3 Circle Area Application Example 3 ................................................................................................ 7 Figure 4 Circle Area Application Example 2 Script ..................................................................................... 8 Figure 5 Client-Side Application [7] .........................................................................................................
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