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ANALYSIS of PLATE-BEAM STRUCTURES by JOHN TINSLEY ODEN Bachelor of Science Louisiana State University Baton Rouge

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ANALYSIS of PLATE-BEAM STRUCTURES by JOHN TINSLEY ODEN Bachelor of Science Louisiana State University Baton Rouge ANALYSIS OF PLATE-BEAM STRUCTURES By JOHN TINSLEY ODEN ,1 Bachelor of Science Louisiana State University Baton Rouge,· Louisiana 1959 Master of Science Oklahoma State University stillwater, Oklahoma 1960 Submitted to the faculty of the Graduate School of the Oklahoma State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY August, 1962 \~e_ s; s l '1 lo 21) (9-.2_ 3 5' ~ C.. O ~ i .;2., OKLAHOMA STATE UNIVERSITY LIBRARY NOV 8 1962 ANALYSIS OF PLATE-BEAM STRUCTURES Thesis Approved: a/11fz~~­ -2".9-~ ~~ .. -- Dean of the Graduate School 504614 ii PREFACE The flexibility approach to the structural analysis of systems of thin plates and elastic beams is presented in this dissertation. Defor­ mations of plate edges and supporting beams are developed in the form of Fourier series which have coefficients in terms of redundant forces and moments. Exact expressions are reduced to a form that allows term by term solutions; and compatibility between plate and beam elements is obtained through the use of "Edge- Deflection" and "Edge-Slope" equations. This research is the outgrowth of ideas expressed by Professor Jan J. Tuma in the summer of 1961. At that time, Professor Tuma suggested that the method of flexibilities used in the analysis of frames could be extended to the analysis of structural systems of plates and beams. In completing the final phase of his graduate study, the writer wishes to express his sincere appreciation to the fo�lowing individuals and organizations: To Professor Jan J. Tuma for his assistance and guidance throughout the preparation of this work. To Professors Roger L. Flanders, David M. MacAlpine, Olan H. Hamilton, Edwin J. Waller and Dr. Clark A. Dunn for their help­ ful advice and encouragement throughout the writer's graduate program. To Lisle Eddy and the staff of the Oklahoma State University Computing Center for their cooperation and their assistance with the iii electronic computation undertaken during the course of this research. To Mr. R. K. Munshi for his friendship and for the many hours that he spent checking parts of the calculations. and to Mr. S. B. Childs for proofreading part of the final manuscript. To the writer's parents. Mr. and Mrs. J. J. Oden. for their encouragement during these years of graduate work. To Mrs. Mary Jane Walters for her careful typing of the manu­ script and for her overtime hours taken to complete the task. July.12, 1962 Stillwater. Oklahoma John Tinsley Oden iv TABLE OF CONTENTS _Chapter Page I. INTRODUCTION 1 1-1. Statement" of the Problem . 1 1-2. Scope and Procedure of Investigation 3 1-3. Historical Study . 4 1-4 .. The Plate Equation . 6 1-5. Boundary Condftions . , . : . 10 1-6. General_ Solutiqn of the Plate Equation . 11 II. FUNCTIONS OF THE BASIC PLATE , .. 15 2-1. General . 15 2-2. Plate with Arbitrary Edge Forces , 16 2:;-:3-. Edge Forces and Moments on Free-Edged Plates� 32 2-4. Normally Loaded Plates Supported at Their Corners· . 50 2-:5, Displacement of Supports of Free:-Edged Plates 63 III. COMPATIBILITY RELATIONSHIPS 68 3-1. Plate Flexibilities . • , . 68 3-2. Deflection of Supporting Members . 87 3-3. Twist of Supporting Members . 91 3-4. The Edge-Slope_ and Edge-Deflection. Equations 97 IV. NUMERICAL APPLICATION OF THE THEORY 103 4-1. Calculation · of Flexibilities . 103 4-2. Numerical Example � . '. 110 A. Redundants . , . 111 B. Flexibilities and Load Functions 112 C�' Edge-Deflection Equations . 113 D. Edge-Slope Equation . 114 E. Deformation Equations for Supporting Beams . 114 F. Equation of Statics. • . , . · . 115 G. Final Forces and Moments . 115 v. SUMMARY AND CONCLUSIONS . 117 5-1. Summary. 117 5-2, Conclusions . 118 5-3. Extension . 119 A SELECTED BIBLIOGRAPHY 121 v LIST OF TABLES Table Page 2-L Deflection Functions . 23 2-2. Reaction Functions . .. ' .. 24 2-3A. Fourier Sine Series. Coefficients . 25 2-3B. Fourier Sine Series· Coefficients . 26 2-4. Coefficients P for One-Directional m Load Variations . 55 3-lA. Edge Deflections of the Basic Plate 71 3-lB. Edge Deflections of the Basic Plate 72 3-2A; Edge Slopes of the Basic Plate . 73 3-2B. Edge Slopes of the Basic Plate . • • 74 3-3. Components of Linear Flexibilities 78 '\·i� ..... 3-4. Components of Angular Flexibilities . 79 .4-1 . Elements of the Matrix �J 105 4-2. Matrices for a Uniformly Loaded Square Plate 107 4-:-3. Linear and Angular Flexibilities of a Square Plate . 108 4-4. Linear and Angular Load Functions For a Uniformly Loaded Square Plate .·. 109 4-5. Physical Properties of the Structural Elements 110 4-6. Linear and Twist Flexibilities of Edge Beams . 113 4-7. Final Forces and Moments . 116 vi LIST OF FIGURES Figure Page 1-1. Typical Plate Element . 7 2-1. Superposition Prqcedure for Obtaining Deflection Surface w 1 18 2-2. Development of Flr'ee-Edged Plate Under General Loading 19 2-3. Deflection Surface w 20 11 2-4. Deflection Surfac� w 28 12 Deflection Surface w 33 2-5. 2 2-6. Deflection Surface wR 34 2-7. Physical Interpretation of Terms of the Deflection Surface wR . 42 2-8. Deflection Surface w . 45 3 2-9. Deflection Surface w 46 M 2-10. Deflection Surface w0 Due to a One-Directional Load Variation • • • • • • • • a 54 2-11. Deflection Surface wL Due to a One- Directional Load Variation 58 2-12. Deflection Surface w O Due to a Concentrated Load . 61 2-13. Deflection Surface wL Due to a Concentrated Load . 63 2-14. Plate Bent by Concentrated Corner Loads 64 2-15. Plate with Corner Displacement 65 2-16. Corner Supported Plates . 67 3-1. Free Body of Typical Plate Element 69 vii Figure Page 3-2. Physical Interpretation of Linear Flexibilities . 80 3-3. Physical Interpretation of Angular Flexibilities . 82 3-4. Physical Interpretation of Linear and Angular Load Functions 84 3-5. Typical Edge Be1;1m AB 88 3-6. Torsional Moments Acting on Edge Beam AB 92 3-7. Portion of a Typical Plate-Beam Structure . 98 3-8. Solution by Superpos:i.tion of Harmonics . 100 3-9. Continuous Plate Supported by Elastic Beams 102 4-1. Square Plate Supported by Elastic Beams . 111 viii NOMENCLATURE a . Length of Plate. a a a 11··· if·· kk a . a Elements of Matrices u · . aif kk �!J, �� Respectively. b ...... Width of Plate . Elements of an Array. (3+µ )sh {3ma c A Constant, - ----- . m a f3m ( l-µ) 2 ) e · e . Fourier Series Coefficients. � �' mn . 1.) 2 ) f f Fourier Series Coefficients. mr' mr Fourier Series Coefficient, gmr h Thickness of Plate. j Summation Index. k Number of Terms Used in the Series, I ll \ s sin Q' x and \ s sin Q' x Ln n Ln n n n (3+µ )sh a b n k A Constant, n a b ( µ) n l- m, n • ISummation Indices. p Intensity of Normal Load Per Unit Area. Maximum Intensity of One-Directional Load Variation. q · · . Coefficient of Series for q(y). m · ix . Fourier Series Coefficient. q(y) . Applied Loading on Edge Beams. r . Summation Index, t(y) . Applied Torque Variation. Coefficient of Series for t(y). U, V. Coordinates of Concentrated Load. u - a. w. Deflection of Plate. Deflection Surfaces. x 1 . x - a. X, y, Z Coordinate Axes. Coefficient Matrices. A C D n, Bn, n, n, A B C D m' m, m, m, . Constants of Integration. A Diagonal Matrix. BR Coefficient of Series for Wide Beam Reaction yzm Due to Loads. Bz. Deflection Due to Loads of Beam Supporting 1 Plate's ith Edge. Twist Due to Loads of Beam Supporting BI/J.1 Plate's ith Edge. c . Eccentricity.of Edge Forces. D . Flexural Rigidity of Plate. E . Young's Modulus of Elasticity. • Fourier Series Coefficient. Matrices with Elements e!� and e;_·� , Respectively. x Reaction Functions. 2. Matrices with Elements f 1.) and f ), mr mr R espect. 1ve 1 y. Reaction Functions. Matrices with Elements g [a J. mr G . .. Modulus of Rigidity. G({:3 x . Reaction Function m ) . Coefficient of Series for Bl/Ji. Identity Matrix. I. 1 Moment of Inertia of Beam Supporting i th Edge of Plate. J ........ ... .1 Torsional Constant of Beam. Supporting i th Edge of Plate. M M ABx' BAx• M M End Moments of Edge Beam, AB. ABy'. BAy . M M Plate Bending Moments Per Unit Length. x' . y M' M" M' M" Edge Moments Per Unit Length of p th Plate. xp' xp' yp' yp (M" ) . xp m (M" ) Coefficients of Series for Edge Moments. p y m M g [ �m rn] rn -l g 1 . , [M' ( ) ,Reactive Force Matrices Due to M' . xm raj . x M M Plate Twisting Moments Per Unit Length. xy' yx p · Concentrated Load. p Coefficient of One-Directional Load Function m Series. p mJ Coefficient of Two-Directional Load Func­ tion Series. xi M�trix with Elements . .. qmr ( x . Reaction Function. Q /3m ) R, R Corner Reactions. L R R Plate Reactive Forces Per Unit Length. xz• yz R' · R" xzp• xzp R' R" . , . Edge Forces Per Unit Length of pth Plate. yzp' yzp (R' (R" ) xzp ) • xzp • m m ( , (R . Coefficients of Series for Edge Forces. R�zp) ;zp) m m . R' f 1 }l [ xzm m J • m . [R, ( - f 1 )1 Reactive Force Matrices Due to R' xzm 1 ) mJ xz I II . Column. Vector of Coefficients Sn and sn Coefficients of Series for Arbitrary Edge · Forces. Plate Shearing Forces Per Unit Length. v xz' v yz 4 x. y. Functions in Product Solution to v w = 0 .
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