LAMINATED BEAM THEORY BASED ON HOMOGENIZATION Yiu Mo Patton Chan A thesis submitted in conformity with the requirements for the degree of Master of Applied Science. Graduate Department of .-\erospace Science and Engineering, University of Toronto @Yiu Mo Patton Chan, 2000 National Library Bibliolhbque nationale 1*1 of Canada du Canada Acquisitions and Aquisitions et Bibliographie Services services bibliographiques 395 Wellington Stræt 395, rue Wellington OItawaON K1AW OttewaW K1AON9 Canade Canade The author has granted a non- L'auteur a accordé une Licence non exclusive licence allowing the exclusive permettant à la National Library of Canada to Bibliothèque nationale du Canada de reproduce, loan, distribute or seii reproduire, prêter, distribuer ou copies of this thesis in microfonn, vendre des copies de cette thèse sous paper or electronic formats. la forme de microfiche/fih, de reproduction sur papier ou sur format électronique. The author retains ownership of the L'auteur consewe la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantial extracts fiom it Ni la thèse ni des extraits substantiels may be printed or otherwise de celle-ci ne doivent être imprimés reproduced without the author' s ou autrement reproduits sans son permission. autorisation. Whatever your band finds to do, do it with ail your might, for in the grave, where you are going, there is neither working nor planning nor knowledge nor wisdom. Abstract A laminated beam theory similar to Timoshenko beam theory is proposed. It uses elasticity solutions of a beam to calibrate the beam's stiffness. By separating the kinematic response of the beam mode1 from the stresslstrain prediction of the actual beam, it can take into account the interlayer interaction of stresses using only three displacement variables. The proposed theory is applied for different test cases and compared with results giwn by Euler-Bernoulli and Timoshenko beam theor?. It is capable of predicting the shear stress and strain distributions exactly for the interior probleni of a caritilever beani made witli a symmetric cross-ply laniinate, subjected to constant shear and constant lateral pressure loads. Timoshenko beam t heory is unable to predict such distributioris correct lu. This theory is part icularly suit able for analyaing shear-deformed beüms witli clrast ic variat iim in the shear stiffness t hrough the thickness. iii Acknowledgement s My grateful thanks are due to my thesis supervisor, Prof. J.S. Hansen. Since our first meeting, discussion with him has been inspiring and refreshing. 1 apprrciate tiis patience. encouragement, advice and sharing of stories in his life throughout niy entire stay at UTIAS. 1 would also like to thank rny fellow students in the Structural hlechanics Group arid the Computational Gaç Dynamics Group for their Company. Special thanks to Guillaunie and Ravi. ivho have sacrificed so miich to setup the computers for our group. 1 am grateful for the prayer support arid fellowship of rny brotliers and sisters in niy church. During the entire period of my graduate studies, they have been a blessing to me. Finally, 1 am particularly grateful to my parents Wa Hing and Ngun Ngun. for their support, understanding and love. I am also thankful to my sisters SIoon and May. for making an effort to visit me in Toronto; and for having been rny friends for so long. Contents Abstract iii Acknowledgements iv Contents v List of Tables ix List of Figures xi 1 Introduction 1 1.1 Motivation for Beam Studies .......................... 1 1.2 Challenges .................................... -3 1.3.1 Emergence of hl ul t i-layered Beams .................. 9- 1.2 Sbear Deformation Effect ........................ 3 1.2.3 Layer Interactions ............................ 3 1.2.4 Number of Variables .......................... 4 1.3 Justification for Using Beam Theories ..................... 6 1.3.1 Load Configuration and Geonietcy .................. 6 1.3.2 Interior and Boundary Problems .................... 1 1.4 ThesisOutline .................................. 8 CONTENTS vi Review of Popular Beam Theories 9 2.1 Single-layered Theories ............................. 9 2.1.1 Euler-Bernoulli Beam Theory ..................... 9 2.1 $2 Tirnoshenko Beam Theory ....................... 11 2.1.3 Higlier Order Shear Deformation Theory ............... 13 2.2 Sliilti-layered Theories ............................. 13 2.2.1 Zig-zag Mode1 .............................. 14 2 4Iulti-layered Plate 'vlodel by Di Sciuva ................ 17 2.3 Discussion of Populsr Beam Theories ..................... 19 Polynomial Solution for Beams 21 3.1 Solution for the Interior Problem ....................... 21 3.2 Sign Convention ...................................3'1 3.3 Formulation ................................... 23 3.3.1 Notation ................................. 23 3.3.2 Governing Equations .......................... 24 3.3.3 Solution ................................. 26 3.4 Solutions for the Fundamental Loads ..................... I21 . 3.4.1 Fundamental Load ........................... 27 3.4.2 Fundamental Load Solutions for an Isotropie Beam ........ 23 3.4.3 Fundamental Load Solutions for a Laminated Bcam ......... 3-1 4 An Enhanced Beam Theory 4.1 Insight frorn the Euler-Bernoulli and Timosbenko Beam Theories .......................... 41 4.2 GoverningDifferentialEquations ....................... 42 4.3 Homogenized Stiffness ............................. 45 4.4 Finite Element Formulation .......................... 48 4.5 Strain and Stress Determination ........................ 50 CONTENTS 5 Results and Discussions 52 5.1 Test Cases .................................... 53 5.2 Results Cornparison ............................... 53 - w 5.3 -4xial Extension Load .............................. XI .I 5.4 Constant Bending Load ............................ a ( 5.5 Constant Shear Load .............................. 59 5.6 Constant Lateral Pressure on a Cantilever .................. 65 5.7 Constant Lateral Pressure on a Simply-Supported Beam .......... 70 5.8 Constant Lateral Pressure on a Clamped Beam ............... 73 -- 5.9 General Discussion about the Enhanced Beam Theu::; ............ i i -- 5.9.1 Different Interpretation of the Prescribed Displacement ....... i i 3.9.2 Interpretation of the Homogenized Stiffness ............. 78 5.9.3 About the Reference Plane ....................... 79 6 Conclusions and Suggested Future Investigations 80 6.1 Advantages of the Enhanced Beam Theory .................. 80 6.2 Conclusions ................................... 81 6.3 Future Investigations .............................. 81 A Derivation of the Two Dimensional Solution 82 A.1 Geometry .................................... 82 A.2 Displacernent Forms .............................. 83 h.3 Governing Equations .............................. 53 4 Stress and Strain Components ......................... 84 A.5 Rigid Body Motion ............................... 85 .4.6 StressjShear on the Boundary ......................... 86 A.7 Resultants on the Edge ............................. 87 .4.8 IntralayerEquilibrium ............................. 88 A.9 InterfaceEquilibrium .............................. 89 ... CONTENTS vlu h.10 Interface Continuity ............................... 89 Al1 System of Equations .............................. 90 B Derivation of the Three Dimensional Solution 91 B.1 Geometry .................................... 91 B.2 Displacement Forxns .............................. 92 B . 3 Governing Eqiiations .............................. 93 B.4 Stress and Strain Components ......................... 94 B.5 Rigid Body Motion ............................... 96 B.6 StressJShear on the Boundary ......................... 97 B .i Resultants on the Edges ............................ 99 8.8 Intralaminar Equilibrium ............................ 102 B.9 Interlaminar Equilibrium ............................ 103 B .10 Interlaminar Continuity ............................ 104 B.11 System of Equations .............................. IO5 C Numerical Values for the Elasticity Solutions 106 C.1 .A luminum Beam ................................ 106 C.2 Laminated Beam ................................ 107 References List of Tables Fundamental Loads ............................... .2, . Elastic properties of an aluminum bearn ................... 28 Displacements for an aluminuni bearn ..................... 29 Transverse strain distributions for an aluminuni bearn (at r = 0) ...... 29 Elastic properties of a T300/N5208 beam with Iayup [0'/90°], ....... 33 Displacements in the first layer of a laniiriated beani ............. 35 Displacements in the second 1-r of a laminated beani ........... 3G Strain distributions in the first layer of a laniinated beam .......... 36 Strain distributions in the second 1-r of a laminated beani ........ 36 4.1 Comparison of strain components and stiffness ternis ............ 41 4.2 Aluminum beam's stiffnesses by different t heories .............. 46 4.3 Laminated bearn stiffnesses by different theories ............... -47 5.1 Shear stiffnesses by different theories ..................... 52 5.2 Comparison of accuracies (aluminum beam. L/2h = 5) ........... GO 5.3 Comparison of accuracies (alumiriuni beum. L/2h = 10) ........... 60 5.4 Cornparison of accuracies (laminated beam. L/2h = 5) ........... 61 5.5 Cornparison of accuracies (laminated beam. L/2h = 10) ........... 61 5.6 Cornparison of accuracies (aluminum beam. L/2h = 5) ........... 66 5.7 Cornparison