A NOVEL FINITE ELEMENT FOR MODELING A FASTENER IN A LAP JOINT ASSEMBLY A Dissertation by Brian D. Foster Master of Science, Wichita State University, 2007 Bachelor of Science, Iowa State University, 1994 Submitted to the Department of Aerospace Engineering and the faculty of the Graduate School of Wichita State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy December 2014 Copyright 2014 by Brian D. Foster All Rights Reserved A NOVEL FINITE ELEMENT FOR MODELING A FASTENER IN A LAP JOINT ASSEMBLY The following members of the faculty have examined the final copy of this dissertation for form and content, and recommend that it be accepted in partial fulfillment of the requirement for the degree of Doctor of Philosophy with a major in Aerospace Engineering. Charles Yang, Committee Chair Walter Horn, Committee Member Suresh Raju, Committee Member Hamid Lankarani, Committee Member Wilfredo Moscoso-Kingsley, Committee Member Accepted for the College of Engineering _________________________________________ Royce Bowden, Dean Accepted for the Graduate School _________________________________________ Abu Masud, Dean, Interim iii ACKNOWLEDGEMENTS Without the guidance, assistance, insight, and above all, patience, from my advisor, Dr. Charles Yang, this work would not have been possible. For that, I am grateful. iv ABSTRACT This work documents the development of a novel finite element. This element introduces parameters that allow more accurate modeling for the specific application of mechanically fastened lap joints. The research here bridges the gap between two traditional methods commonly used in industry today. The novel element is more accurate than the first traditional method, by using a single-beam element to join plate elements with a linear solution. It is also more computationally compact than the second traditional method, which consists of an assembly of solid, three-dimensional elements with a non-linear solution. The case studies used are specifically limited to mechanically fastened lap joints pulled in tension, also referred to as “secondary bending.” The plates joined together with these fasteners are loaded up to a fraction of a yielding load of the plate material in order to maintain linearity. Isotropic materials are used exclusively for both the plates and fasteners. Ultimately, the fastening element created in this study is intended strictly for a linear calculation. The calculation contains a one-dimensional beam element with two nodes that connects two, two-dimensional plate elements together. To match the problem more accurately, this research introduces the principle of finite element Hertzian contact mechanics, which is specifically applied to mechanically fastened lap joints. This addition to the existing beam finite element allows for a more accurate simulation while holding to the simplicity of a linear solution of a single element with two nodes in a plate/beam/plate element modeling scheme. Transverse deflections resulting from the applied loading in this new finite element are first compared to the two baseline finite element models that the new element is intended to bridge. The new model is also compared to the deflections calculated from empirically formulated stiffness values generated for the neutral line model. v TABLE OF CONTENTS Chapter Page 1. INTRODUCTION ............................................................................................................ 17 1.1 Overview ............................................................................................................... 17 1.2 Scope of Work ...................................................................................................... 25 2. LITERATURE AND THEORETICAL REVIEW ........................................................... 26 2.1 Literature Review.................................................................................................. 26 2.2 Common Configurations Currently Used in Industry ........................................... 37 2.3 Beam Theories and FEM Implementation ............................................................ 40 2.4 Euler-Bernoulli Beam Theory and Finite Element Method .................................. 42 2.5 Timoshenko Beam Theory .................................................................................... 46 2.6 Axial Deflection .................................................................................................... 53 2.7 Combining Different Theories .............................................................................. 53 2.8 Hertzian Contact Mechanics ................................................................................. 54 2.9 Winkler Foundation .............................................................................................. 58 2.10 Literature Review Conclusion .............................................................................. 60 3. FASTENER ELEMENT FORMULATION..................................................................... 61 3.1 Problem Description ............................................................................................. 61 3.2 Finite Element Development ................................................................................ 63 3.3 Typical Mid-Plane Processing .............................................................................. 70 3.4 Linearized Cylindrical Hertzian Contact .............................................................. 70 4. TEST CASE STUDY 1 .................................................................................................... 77 4.1 Single-Bolt Test Case Layout ............................................................................... 77 4.2 Results and Analysis ............................................................................................. 83 5. TEST CASE STUDY 2 .................................................................................................... 90 5.1 Three-Bolt Test Case Layout ................................................................................ 90 5.2 Results and Analysis ............................................................................................. 94 6. TEST CASE STUDIES 3 AND 4 ................................................................................... 102 6.1 Three-Bolt Test Case Layout .............................................................................. 102 6.2 Results and Analysis ........................................................................................... 104 vi TABLE OF CONTENTS (continued) Chapter Page 7. CONCLUSIONS, RECOMMENDATIONS, AND FURTHER STUDY ...................... 116 7.1 Conclusions ......................................................................................................... 116 7.2 Recommendations ............................................................................................... 117 7.3 Areas for Further Research ................................................................................. 118 BIBLIOGRAPHY ....................................................................................................................... 121 APPENDICES ............................................................................................................................ 132 APPENDIX A AN ALTERNATIVE DERIVATION ................................................... 133 APPENDIX B AVERAGING OF STIFFNESS TERMS .............................................. 136 APPENDIX C METHODS USED FOR PRELOAD IN SOLID ELEMENTS ............ 137 APPENDIX D HERTZIAN CONTACT OF PARALLEL CYLINDERS .................... 139 APPENDIX E NEUTRAL LINE MODEL ................................................................... 141 APPENDIX F MODELS WITH AND WITHOUT RIGID BODY ELEMENTS ........ 145 APPENDIX G PROGRAMS AND SUPPORTING DATA FILES .............................. 149 vii LIST OF FIGURES Figure Page 1.1 Break down an assembly into smaller parts. ......................................................................18 1.2 Typical line elements. ........................................................................................................19 1.3 Typical surface elements....................................................................................................19 1.4 Typical solid elements. ......................................................................................................20 1.5 Solid element calculation (CATIA/Elfini processor). .......................................................21 1.6 Bar/plate element calculation (NX NASTRAN processor). ..............................................21 1.7 Types of bending in a lap joint. .........................................................................................23 1.8 Finite element design field .................................................................................................24 2.1 Assembly of 1D elements (Rutman et al., 2000). ..............................................................26 2.2 Multiple elements simulating fastener (Paroissien et al., 2006). .......................................28 2.3 Compression cone (Shigley and Mischke, 1988; Wileman et al., 1991; Lee, 2010a). ............ 31 2.4 Bolt/member contact regions. ............................................................................................36 2.5 Beam element (CBAR) with spider elements (RBE). .......................................................37 2.6 Beam element (CBAR) with