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  • The Islamic University Gaza Postgraduate Deanship Faulty of Engineering Structural Engineering Department

    NEW STRAIN-BASED

    TRIANGULAR AND RECTANGULAR FINITE ELEMENTS

    FOR PLANE ELASTICITY PROBLEMS

    SUBMITTED BY

    ENG. SALAH M. TAYEH

    SUPERVISED BY

    DR. ATTIA I. MOUSA Associate Professor, Dean of Faculty of Engineering

    Islamic University Gaza

    A Thesis submitted in partial fulfillment of the requirements for the Degree of Master of Science in Civil / Structural Engineering

    JULY 2003

  • -

  • Abstract

    I

    ABSTRACT

    In this thesis, the Finite Element Method of structural analysis is used to

    investigate and compare the performance of several strain-based elements. Two

    new strain-based triangular and rectangular finite elements in Cartesian

    coordinates system, for two dimensional elasticity problems are developed.

    Each of these elements has three degrees of freedom per node. The Strain

    Based Approach is used to develop and formulate these two dimensional finite

    elements. In this approach, finite elements are formulated based on assumed

    polynomial strains rather than displacements.

    Two main computer programs are developed to analyze the new finite

    elements. To test the performance of these elements, they are used to solve two

    common plane elasticity problems. The problems considered included are: the

    problem of a plane deep cantilever beam fixed at one end and loaded by a point

    load at the free end; and the problem of a simply supported beam loaded at the

    mid-span by a point load.

    The finite element solutions obtained for these problems are compared with the

    analytical values given by the elasticity solutions.

    Results obtained using the new triangular and rectangular elements are also

    compared to those of the well-known constant strain triangular element (CST)

    and the bilinear rectangular element (BRE) respectively.

    In all cases, convergence curves for deflection at specific points within each

    problem are plotted to show that acceptable levels of accuracy. Furthermore,

    convergence curves for bending stress at points on the upper surface and shear

    stress at points on the neutral axis are plotted; again convergence is ensured.

  • II

    TO MY PARENTS

    AND

    MY WIFE

    I DEDICATE THIS WORK

  • Acknowledgments

    III

    ACKNOWLEDGMENTS

    The work presented in this thesis was carried out in the Department of Civil

    and Structural Engineering at the Islamic University of Gaza, under the

    supervision of Dr. Attia Mousa, Dean of Faculty of Civil Engineering.

    The researcher wishes to take this opportunity to express his acknowledgments

    and gratitude to Dr. Mousa for his continued guidance and encouragement

    throughout the course of this work.

    My thanks extend to the lecturers at the Civil Engineering Department for their

    effort and their role in launching the M. Sc. program in the field of Structural

    Engineering.

    I am particularly grateful to my parents, whom, without their support I would

    have not been able to complete this work.

    Finally, I present my thanks for all people who helped me in completing this

    work..

  • List of Abbreviations (Notations)

    IV

    LIST OF ABBREVIATIONS (NOTATIONS)

    Symbol Notation

    x, y Cartesian coordinates

    T Thickness of the element

    E Youngs Modulus of Elasticity

    Poissons ratio

    U Displacement in the x direction

    V Displacement in the y direction

    In-plane rotation

    x, y Direct strains in the x and y directions, respectively

    xy Shear strain

    x, y Normal stresses in the x and y directions, respectively

    xy Shear stress

    [ ]C Displacement transformation matrix [ ]B Strain matrix [ ]D Rigidity matrix

    [ ]eK Element stiffness matrix [ ]Q Strain energy matrix

    { }e Element nodal displacement vector Total Potential Energy

    DoF Degree of Freedom

    CST Constant Strain Triangle

    BRE Bilinear Rectangular Element

    SBTREIR Strain-Based Triangular Element with In-plane Rotation

    SBREIR Strain-Based Rectangular Element with In-plane Rotation

  • Table of Contents

    V

    TABLE OF CONTENTS

    ABSTRACT............................................................................................................... I

    ACKNOWLEDGMENTS...................................................................................... III

    LIST OF ABBREVIATIONS (NOTATIONS)...................................................... IV

    LIST OF FIGURES............................................................................................. VIII

    LIST OF TABLES ...................................................................................................X

    1. CHAPTER ONE: INTRODUCTION ................................................................. 1 1.1 INTRODUCTION .......................................................................................... 1

    1.2 HISTORY OF FINITE ELEMENT METHOD ............................................... 3 1.3 PROCEDURE OF THE FINITE ELEMENT METHOD................................. 4

    1.3.1 Idealization of the Structure ........................................................................... 5 1.3.2 Formulation of Element Stiffness Matrix ....................................................... 6 1.3.3 General Procedure for Derivation of Element Stiffness Matrix ....................... 6 1.3.4 Assembly of the Overall Structural Matrix and the Solution Routine.............. 9 1.3.5 Convergence Criteria ..................................................................................... 9

    A- Rigid-Body Modes.......................................................................10 B- Constant Strain ............................................................................10 C- Inter-element Compatibi1ity ........................................................11

    1.3.6 Mesh Size Design ........................................................................................ 11 A- Number of Degree of Freedom.....................................................12 B- Aspect Ratio ................................................................................12 C- Element Distortions (Skewing) ....................................................12

    1.4 FINITE ELEMENT APPLICATIONS OF TWO-DIMENSIONAL STRUCTURES..............................................................................................13 1.4.1 Constant Strain Triangular Element (CST)................................................... 14 1.4.2 Liner Strain Triangular Element (LST) ........................................................ 14 1.4.3 Bilinear Rectangular Element (BRE) ........................................................... 15 1.4.4 Quadratic Isoparametric Rectangle Element................................................. 16

    1.5 SCOPE OF THE CURRENT THESIS ...........................................................16

    2. CHAPTER TWO: STRAIN BASED APPROACH...........................................17 2.1 INTRODUCTION .........................................................................................17

    2.2 REASONS FOR DEVELOPMENT OF STRAIN BASED ELEMENTS........18 2.3 HISTORY OF STRAIN BASED APPROACH..............................................18

    2.3.1 Strain Based Curved Elements ..................................................................... 18 2.3.2 Strain Based Shell Elements ........................................................................ 19 2.3.3 Strain Based Two Dimensional Elements..................................................... 21

    2.4 GENERAL PROCEDURE FOR DERIVING DISPLACEMENT FIELDS FOR STRAIN-BASED ELEMENTS.............................................................23

  • Table of Contents

    VI

    3. CHAPTER THREE: COMPUTER PROGRAMS............................................26 3.1 USED COMPUTER SOFTWARE.................................................................26

    3.1.1 MathCAD Software..................................................................................... 26 3.1.2 MatLab Software......................................................................................... 27

    3.2 IMPLEMENTATION OF THE FEM ANALYSIS.........................................27 3.2.1 Outline of the Major Program Steps ............................................................. 28 3.2.2 Programs Flow Chart................................................................................... 29

    4. CHAPTER FOUR: DEVELOPMENT OF NEW STRAIN-BASED TRIANGULAR ELEMENT............................................................................31

    4.1 INTRODUCTION .........................................................................................31

    4.2 DERIVATION OF DISPLACEMENT FIELDS OF NEW TRIANGULAR ELEMENT WITH IN-PLANE ROTATION ..................................................31

    4.3 PROBLEMS CONSIDERD ...........................................................................36 4.4 DEEP CANTILEVER BEAM PROBLEM ....................................................36

    4.4.1 Used Mesh Size........................................................................................... 37 4.4.2 Analytical Solution...................................................................................... 38 4.4.3 Convergence Results ...................................................................................

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