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Ann Arbor, MI 48106 Thick Plate Theories, with Applications to Vibration DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By N. F. HANNA, RSir. M.Sc. * » * * * The Ohio State University 1990 Dissertation Committee: Approved by Prof. O. G. McGee Prof. D. A. Mendelsohn ProLA. W.LE1SSA Department of Engineering Mechanics DEDICATION To my Lord Jesus Christ ACKNOWLEDGEMENTS I would like to thank most my adviser, Professor A. W. Leissa for suggesting the topic and advising me throughout its development Special gratitude and sincere regards are extended to Professor O. G. McGee and Professor D. Mendelsohn for being on the reading committee. Finally, greatest sincere thanks and appreciation go to my family for their love, support, understanding and encouragement without which this study would never have been completed. VITA August 16, 1957 Bom, Cairo, EGYPT June 1979 B.S. in Civil Engineering Ain Shams University, EGYPT August 1985 M.S. in Civil Engineering The Ohio State University Columbus, Ohio, USA 1981 - 1983 Teacher Assistant, Civil Engineering, Helwan Univer­ sity, Cairo, EGYPT 1981 - 1983 Structural engineer, Sabbour Consulting firm, Cairo, EGYPT 1985 - Present Graduate Teacher Assistant, Math Department, The Ohio State University, Columbus, Ohio, USA FIELDS OF STUDY Major Field : Engineering Mechanics. Mechanics : Strength of Materials, Continuum Mechanics, Composite Materials, Vibrations of Continuous media. Mathematics : Linear Algebra, Tensors, Calculus of Variations, Calculus. Numerical Analysis : Numerical Integration, Numerical Methods of Linear Algebra, Ritz Method, Finite Element Method. TABLE OF CONTENTS PAGE D e d ica tio n ......................................................................................................................... ii ACKNOWLEDGEMENTS.............................................................................................. iii VITA ............................................................................................................................... iv LIST OF FIGURES......................................................................................................... vii LIST OF TABLES .................................................................................. ix CHAPTER I. INTRODUCTION .............................................................................................1 II. LITERATUREREVIEW ....................................................................................3 2.1 Introduction ....................................................................................3 2.2 Classical Plate Theory (Thin Plates):- ......................................... 4 2.3 First Order Shear Deformation Theories ....................................7 2.3.1 Reissner's Approach ...............................................................8 2.3.2 Mindlin's A pproach ................................................. 11 2.3.3 Ambartsumyan's Approach ............................................ 18 2.4 Higher-Order Shear Deformation T heories ........................... 24 2.4.1 Higher-Order Theories using Reissner's Approach .. 25 2.4.2 Higher-Order Theories using Mindlin's Approach .. 32 2.4.3 Higher-Order Theories using Ambartsumyan's Approach ................................................................. 44 2.5 Discrete I.ayer T heories .............................................. 47 2.6 Two Dimensional Analysis using Finite Elements ................. 50 2.7 Three Dimensional Analysis ..................................................... 54 2.7.1 Exact Solutions ............................................................. 54 - v - 2.7.2 Approximate Solutions ................................................... 56 2.8 Other Reviews of the L iterature ............................................. 57 2.9 Other Ref erences ...................................................................... 58 m . ENERGY FUNCTIONALS FOR FREE VIBRATIONS OF RECTANGULAR PLATES ......................................................... 59 3.1 Classical Plate T h e o ry .............................................................. 60 3.2 Mindlin's T h e o ry ...................................................................... 61 3.3 Modified Kant T h eo ry .............................................................. 63 TV. DIFFERENTIAL EQUATIONS AND BOUNDARY CONDITIONS FOR MODIFIED KANT TH EO RY ...........................................68 V. FREE VIBRATION ANALYSIS OF ISOTROPIC HOMOGENEOUS RECTANGULAR PLATES USING THE RITZ M ETH O D .............................................................. 76 5.1 Mindlin Theory : ...................................................................... 76 5.2 Modified Kant T h eory .............................................................. 79 VI. RESULTS FOR FREE VIBRATIONS OF COMPLETELY FREE RECTANGULAR THICK PL A T E S....................................... 82 6.1 Introduction ................ 82 6.2 Symmetry Consideration ........................................................ 83 6.3 Convergencc Study ................................................................ 86 6.4 Comparison with Classical Plate Theory .............................. 92 6.5 Thick Plate Deformations ........................................................ 98 6.6 Effect of Varying Poisson's Ratio upon the Frequency Param eter............................................................................ I ll 6.7 Effect of Aspect Ratio and Thickness-to-Width Ratio .... 117 VII. DISCUSSION........................................................................................... 120 7.1 Summary and Conclusions .................................................. 120 7.2 Recommendations for Further Research ........................... 121 APPENDICES A. MODE SHAPES FOR POISSON'S RATIO 0 3 122 B. MODE SHAPES FOR OTHER POISSON'S RATIOS....................... 198 REFERENCES ...................................................................................................... 222 - vii - it LIST OF FIGURES Rectangular plate (cartesian coordinates). 6 General coordinates used in the analysis 77 SS-2 a/b=l, h/b=0.5 .......................... 109 SS-5 a/b=l, h/b=0.^ .......................... 110 Mode shape coordinates ........................... 122 Mode shape SS-1 , a/b =1 .................. 123 Mode shape SS-2 , a/b =1 ................. 124 Mode shape SS-3 , a/b = 1 .................... 125 Mode shape SS-4 , a/b =1 .................... 126 Mode shape SS-5 , a/b =1 .................... 127 Mode sha])e SS-6 , a/b =1 .................... 128 Mode shape SS-7 , a/b =1 .................... 129 Mode shape SS-8 , a/b =1 .................... 130 Mode shape SS-9 , a/b =1 .................... 131 Mode shape SS-10, a/b = 1 .................... 132 Mode shape SS-11 , a/b = 1 .................... 133 Mode shape SA-1 , a/b =1 .................... 134 Mode shape SA-2 , a/b =1 .................... 135 Mode shape SA-3 , a/b =1 .................... 136 Mode shape SA-4 , a/b =1 ........................ 137 A.17 Mode shape SA-5 , a/b -= ] ............................................................... 138 A.18 Mode shape SA-6 , a/b = 1 ............................................................... 139 A.l 9 Mode shape SA-7 , a/b = 1 ............................................................... 140 A.20 Mode shape SA-8 , a/b = 1 ............................................................... 141 A.21 Mode shape SA-9 , a/b =1 .............................................................. 142 A.22 Mode shape SA-10 , a/b =1 143 A.23 Mode shape SA-11 , a/b =1 144 A.24 Mode shape AA-1 , a/b = 1 ................................................................