Laminated Beam Theory Based on Homogenization

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 of accuracies (aluminum bearn. L/2h = 10) ...... 66 5.8 Cornparison of accuracies (laminated beam. L/2h = 5) ...... 67 LIST OF T4BLES x

5.9 Cornparison of accuracies (laminated beam. L/2h = 10) ...... 67

5.10 Ratio between solutions (aluminum beam. L/2h = 5) ...... 71 5.11 Ratio between solutions (aluminum beam. L/2h = 10) ...... 71 5-12 Ratio between solutions (laminated beam. L/2h = 5) ...... 72 5.13 Ratio between solutions (laminated beam. L/2h = 10) ...... 72 5.14 Ratio between solutions (larninated beam. L/2h = 50) ...... -42 5.15 Ratio between solutions (aluminum beam . L/2h = 5) ...... 74 5.16 Ratio between solutions (aluminuni beam . L/2h = 10) ...... 74 5 .l'ï Ratio between solutions (laminated beam . L/2h = 5) ...... --, a

I- 5.18 Ratio between solutions (laminated beam. L/2h = 10) ...... r û

-.- 5.19 Ratio between solutions (laminateci bearn. L/2h = 50) ...... 1 LI

C.1 Displacement coefficients for an aluminum beam ...... 106 C.2 Displacement coefficients for a laminated beam ...... 107 C.3 Strain coefficients for a laminated beam ...... 108 List of Figures

1.1 Layer-wise elastic properties ...... 3

1.2 Pure shear of a sandwich structure ...... a- 1.3 Beam coordinates system ...... 6

'1.1 Cornparison between Euler-Bernoulli and Timoshenko beani theories .... 12

'1 .2 Strain approximation by a higher order theory ...... 14 2.3 Zigzag Theory ...... 15 2.4 Through-thickness nodal point i ...... 16

3.1 Laminated Beam hlodel ...... -1 '1

3.2 Sign convention for the stress components ...... $1 *7 3.3 Sign convention for the stress moments ...... 23 3.4 Strain distributions for axial extension ...... 30 3.9 Stress distributions for axial extension ...... 30 3.6 Strain distributions for pure bending ...... 31 3.7 Stress distributions for pure bending ...... 31 3.8 Strain distributions for pure shear ...... 32 3.9 Stress distributions for pure shear ...... 32 3.10 Strain distributions for distributed pressure ...... 33 3.11 Stress distributions for distributed pressure ...... 33 3.12 Orientation-dependence of a composite plyTsproperties ...... 34 3.13 Strain distributions for axial extension ...... 37 LIST OF FIGURES xii

3.14 Stress distributions for axial extension ...... 37 3.15 Strain distributions for pure bending ...... 35 3.16 Stress distributions for pure bending ...... 38 3.17 Strain distributions for pure shear ...... 39 3.18 Stress distributions for pure shear ...... 39 3.19 Strain distributions for distributed pressure ...... 40

3.20 Stress distributions for distributed pressure ...... 40

4.1 External loads on a beam ...... 43

4.2 Yodal variable distribution for the i-th element ...... -19 .. 5.1 Axial Extension Load ...... ad

5.2 Cornparison of mial elongations (axial extension load) ...... 56 5.3 Cornparison of axial elongations (axial extension load) ...... 56 .- 4 Constant Bending Load ...... ;, 1 5.5 Comparison of lateral deflections (constant bending) ...... 58 5.6 Comparison of lateral deflections (constant bending) ...... 58

.c. a . r Constant Shear Load ...... 59 5.8 Cornparison of lateral deflectioris (constant shear ) ...... 61

5.9 Shear strain distribution in the laminated beam (constant shear) ...... 63 5.10 Shear stress distribution in the laminated beam (constant shear) ...... 63 5.11 Axial normal stress distribution at x = 1.35 (constant shear) ...... 64

5.12 Axial normal stress distribution at r = 3.75 (constant sliear) ...... 64 5.13 Constant lateral pressure on a cantilever ...... 65 5.14 Cornparison of transverse deflections (pressure on a cantilever) ...... 66 5.15 Shear stress distribution x = 1.25 (pressure on a cantilever) ...... 68 5.16 Shear stress distribution x = 3.75 (pressure on a cantilever) ...... 68 5.17 Axial normal stress distribution at x = 1.25 (pressure on a cantilever) ... 69 5.18 Axial normal stress distribution x = 1.25 (pressure on a cantilever) .... 69 LIST OF FIGURES xiii

5.19 Constant lateral pressure on a simply-supported beam ...... 70 5.20 Comparison of transverse deflections (simply-supported) ...... 71 5.21 Constant lateral pressure on a clamped beam ...... 73

5.22 Cornparison of transverse de flect ions (clarriped) ...... 76 5.23 Cornparison of transverse deflect ions (clamped) ...... 76

B.1 Geometry ...... 91 B.2 Stress Resultants ...... 92 Chapter 1

Introduction

1.1 Motivation for Beam Studies

Structural analysis is used to determine whether componrnts in aii engirieeririg desigii uill fail or not under the operating conditions of the design. In this context. ti coriipoiierit hils when it fractures or deforms excessively in operation. .-\proper structural analysis on the component reveals the component's deformation. strain and stress distributions. A design engineer can determine from such information whether a design is appropriate. When accurate prediction of this information is not available, the component rnay be under or over designed. While under design jeopardizes safety and reliability. over design hinders performance and increases cost . The theory of elasticity provides a complete set of governing equations an elastic body in static equilibrium has to satisfy. Theoretically speaking, structural analysis cm be performed by first seeking a displacement or stress distribution of the compotierit that satisfies the goveming equations. Then using the stress-strain and the strain-displacenient relationst al1 other required quantities can be obtained. However, the direct solution of the governing equations is not always easy. By taking advantage of the geometry and loading configuration of the problem. many t heories are developed to facilitate the solution process . Elementary structural members like beams, plates and shelis have their own specialized theories. They deserve intensive study because of their relatively simple geometries allow significant simplification of the governing equations. They can be assembled to emulate the elastic behavior of more complicated structures using the concept of structural idealization [9]. The widely used CHAPTER 1. INTRODUCTION -3 finite element method analyzes a complicated structure by modelling it with beam, plate or shell elements. Therefore the accuracy of such elements are important to the final result of the analysis. Among various elementary structural members, the beam deserves special attention. It is a versatile one-dimensional elementary member, capable of carryirig loads in various directions. It is also widely used in trusses and skeleton structures. Ribs iri airfoils and stringers in a fuselage can be seen as different manifestations of the bearn. Other than the bearn's frequent appearance in engineering applications. the fact that it resembles a one-dimensional plate also niake the study of it valuable. Sorrie pop~ilirrplate theories (for example. the Kirchhoff-Love plate theory and the Reissner-Miridlin plate the-

O-) are based on a bearn theor-y makirig corresponding assurnptions about the deformation. .l robust beam theory improves prediction accuracy by giving better niodelling to beani elements as dlas showing a possible way to improve plate elements.

1.2 Challenges

1.2.1 Emergence of Mult i-layered Beams

In the mid-1960's, fibre-reinforced composite materials begari to emerge. Such iiinterials. sometimes simply known as composite materials, are deliberately desigried and riianufac- tured to have orientation-dependent rnechanical/thermal properties. Csilally they have reinforcing fibres embedded in a matrix. Common examples are fibre-glass (glus tibres embedded in epoxy). gaphite/epo.xy, Iievlar/epoq boron/aluminum. etc. They possess high specific stiffness and are widely used in the aerospace indust. As the manufacturing technology for composite rnaterials matures? and more and more engineering applications cal1 for the use of such materials, the demand for accurate and efficient analysis techniques continues to mount. -4 component made of the composite material is designed to have the desired prop erties in different directions by stacking layers of anisotropic composite plys at different orientations. .A beam made of composite materials may exhibit a drastic variation in elastic properties through its thickness, and is known as a multi-layered beam or a laminated beam. Figure 1.1 shows the through-thickness cross-section of a multi-layered bearn made of orthotropic layers. Due to the variation in properties through the thickness, a multi- CH,LIPTER 1. INTRODUCTION layered beam has a different response to external loads as compared to a single-layered beam. For example, an unsymmetric beam twists when subjected to transverse load. Even when the beam is symmetric, the prediction of its response is more diffiçult than for that of a single-layered beam for the reasons discussed below.

Figure 1.1: Layer-wise elast ic propert ies

1.2.2 Shear Deformation Effect

The classical Euler-Bernoulli beam theory usually provides satisfüctory predictions for isotropic beams. but over-predicts the stiffness of bearns when shear deformation is significant (which is true when the beam has low span-to-thickness ratio)[l7]. The Tirnoshenko beam theory (also known as the first order shear deformation beam theory) remedies this deficiency by allowing angular rotation of the beam's transverse normal plane to be independent of the beam's curvature, thus permitting shear deformation to occur. It can better predict behavior of thicker beams. Since most fibre-reinforced composite materials have a high extensional modulus to shear modulus ratio, shear deformation in bearns made of such materials tend to be significant. The ability to take the shear deformation into account is important [la].

1.2.3 Layer Interactions

For a single-layered beam, the strain and stress distributions depend on external loads and the beam's cross-section only. When the elastic properties are not constant, distribut ions may be distorted by the layer interactions. The form of deformation in each layer can be different from one another. The resulting distributions of the displacement, stress and CHilPTER 1. INTRODUCTION strain depend on the through-thickness variation of the elastic properties, and is difficult to assume a priori based on the geometry or load configuration alone. It becornes impossible to specify the state through the thickness by the state on any reference plane. for example, the mid-plane. An example is shown in Figure 1.2 to demonstrate the effect of liiyer interactions. Suppose a three-layered beam is subjected to shear forces applied on its top aiid bottoni surfaces as shown. The top and bottom layers of the beani have a greater sliear stiffness than the middle layer. The resulting deforniacion is showri iu figure (aj, wliicli aliuws tiiii~ the shear deformation in the middle layer is larger than that in the top and bottorii layers. An analogy can be drawn between such a beam and a spring system. The shear stiffness of each layer is represented by the spring constant of a spririg. Figure (a) shows the actual deformation in which the layers interact as if tliey are springs connected in series. The forces in al1 spring connections are equal to the external force. In figure (a). the external sliear force equals the shear force at each layer interface. If the difference in shear stiffness between layers is not recognised. and the beam is niodeled as a single-layered beam, thcn the deformation will be as shown in figure (b). The shear deformations in al1 layers are the same. The layers interact as springs cotinectcd in parallel. Each spring carries a different force. .A beam deformed as in figure (b) does riot satisfy interply equilibrium requirements. Another consequence of the layer interactions is that higher stress resolution is needed. Stress transfer between layers sometinies causes sharp rises in stress or strairi discontinuities in very localized regions (like the layer interfaces). -4 competent bearn theory should have a high enough resolution to reveal the state of stress in such regions.

1.2.4 Number of Variables

The final challenge for a successful rnulti-layer beam theory is the number of variables involved to describe the state of a beam has to be small. It is not uncornmon to have beams consist of tens to hundreds of layers. In principle each individual layer presents a possible design variation. It is preferred if the t heory can handle efficiently a beam with many layers of difFerent stiffnesses. CHAPTER 1. INTRODUCTION

Figure 1.2: Pure shear of a sandwich structure CHAPTER 1. INTRODUCTION

1.3 Justification for Using Beam Theories

Even though it is commonly accepted that some sort of approximation is unavoidable in most engineering practice, it is worthwhile to state and examine explicitly the approximations used by the beam theories to reduce a physically three-dimensional body into a one-dimensional approximate model. In the following, a rectangular beam with its length. width and thickness aligned with the x-. y- and 2-axis as shown in Figure 1.3 will be considered.

Figure 1.3: Beam coordinates systeni

1.3.1 Load Configuration and Geometry

The first approximation is that the loads carried by the beam al1 lie on a sirigle plane through its longitudinal axis, and the beam's cross-section is symmetric about the same plane of the beam. For a symrnetric (in terms of cross-section geometry and laminate layup), multi-layered beam, t hese conditions ensure that no twisting results from the transverse load. In general, twisting will cause the state of stress to become three-dimensional. CHAPTER 1. INTROD UCTION

The next condition is related to the variation of stress or strain in the g direction. That is. does a state of plane stress or plane strain exist. In usual application of rnulti- layered beams. the width is large enough such that the state of plane strairi esists. Eveii though three-dimensional elasticity studies [8. 16, 151 have show boundary effects cause intensive stresses near the beam edge, such effects are a boundary phenomenoii. and are not considered in beam theories. For a beam with reasonable width, the boundary effect does not extend very far away from the beam's free edge. and should have minimal global cffccts.

1.3.2 Interior and Boundary Problems

An elasticity probleni can be separated into an "interior probleni" anci a .*boiiiiclary problem". While the solution to the interior problem satisfies only the gros edgc coritiitioiis. tlic solution to the boundary problem specifically satisfies the through-thickness prcscriptiori of the boundary conditions. The superposition of the interior and the bouriclary soliitions gives an accurate solution throughout the beam. When the esact throiigli thickriess boundary condition is unknown, the best one can ask for is tlir soliitiori of tlir iritcrior problem. In most practical situations. the exact stress or displacernent distribution througli the thickness of the beam is unknown and the boundary conditions at the edges are usually given in terms of a "gross edge boundary condition" (61. They are usually specified as a stress resultant across the thickness of the beam, or a displacement or an angular rotation at the middle surface of the beam. The inability to prescribe the exact boundary condition, especially on edges where external loads are applied, led to the development of Saint-Venant's principle [13]. Saint- Venant showed, using an energy argument, that the stress and strain fields induced in two identical bodies carrying statically equivalent loads differ only in the region close to the load-applied edges. Saint-Venant 's principle implies that the interior solution is an accurate solution everywhere in the beam except For regions close to the load-applied edges. While the principle is usually applicable to materials with isotropic elastic properties, pst investigations [2] indicate that for orthotropic materials, the principle should be applied with care. Nevertheless, for most beam problems, only the average boundary condition is known. Beam theories are therefore designed to solve the interior problem only. The solution of the boundary problem should be solved separately when necessary.

1.4 Thesis Outline

In the next chapter, two major categories of beam theories. the single-layered and the multi-layered theories, are reviewed. Their common characteristics are also discussed. In Chapter 3. a multi-lavered beam problem is solved using a coniplete two-dirriensioiial elasticity analysis. -4 unique, polynornial displacernent solution is obtaincd wi thiri the eii- tire beam. The solution for a single-layered beam made of aluminum and a [0"/9OU], laminated beam made of graphite epoxy (T300/N5208) are presentetl. The concept of a fundamental load is also introduced. In Chapter 4 the findings in Chapter 2 and 3 are used to develop a new enhariced beaiii t heory. The derivation of the governing equat ions using variat ional calculus arid a fini te elenient mode1 are presented. The various stiffnesses for the new theory are calculated. A procedure which gives the stress/strain distributions of the laminate based on calculated results from the theory is al1 described. Chapter 5 compares the enhanced beam theory with two other popular beam theories. Loads on beanis with different boundary conditions are consideretl for isotmpic and laminated beams with different aspect ratios. The mid-plane deformations and through- thickness stress/strain distributions calculated using different theories are comparecl. Dis- cussions for each test case. as well as for the new theory in general, are included. Chapter 6 concludes t his thesis and suggests possible future investigations. .\liscella- neous information, including the derivation of a three-dimensional polynomial solution for a multi-layered plate cm be found in the appendices. Chapter 2

Review of Popular Beam Theories

2.1 Single-layered Theories

Theories in this category are characterized by their representation of the bearn's displacement by a single set of generalized displacements applicable to al1 layers in the t~~arii.Tli~ resulting displacernents and strains are continuous through the tliickness. Tticsc tkicorics rvere originally developed for single-layered. isotropic beanis. but wre latcr exrericled to mult i-layered beam applications. The elastic propert ies of al1 layen are srrieareri irito orle elastic modulus that characterizes the beam as a whole.

2.1.1 Euler-Bernoulli Beam Theory

The Euler-Bernoulli beam theory makes two kinematic assumptions about the beam's de formation,

0 Plane sections initially normal to the bearn's longitudinal ais remain plane and normal to the ~uisafter deformation.

Any line initially parallel to the beam's longitudinal axis is deformed into an arc of a circle; that is, the curvature of the beam is constant.

These assumptions are valid for any beam under the action of pure bending moments applied at its ends. The axial displacernent u and transverse displacement w consistent with the above assumptions are,

provided the dope of the beam remains small.

Csing the strain-displacement relation. the following state of strain is obtained for the beam,

(x,2) = O

For a multi-layered beam. the elastic properties Vary betweeri layers. The asial norrnal stress is tvritten ris:

while aZand rZ2 are assurned to be zero everywhere.

The strain energy expression for an Euler-Bernoulli beam with thickness ecpal to 'Lh and length equal to L can be written as:

The transverse integration terms can be considered as the smeared or equivalent stiffness terms, while the derivatives of 1~0and coo cm be regarded as deformation terms. While the equivalent stiffness terms depend on the beam's elastic properties only. the deformation terms measure the beam's degree of deformation. Although the through-thickness variation of the stifhess is taken into account by the equivalent stiffness terms, the energy expression is exact only when the strain field is of the form in Eqs. 2.1 to 2.3. As shown above, this theory uses only two generalized displacement functions (uoand tuo) to describe the state of the beam, which includes the displacement, strain and stress. Since the transverse normal strain (6::) and the shear strain (y,,) are always zero, this theory neglects transverse normal and shear deformations. Such deformations are significant in a thick beam, or when the beam's extensional modulus is significantly larger than its shear modulus [IO, 171. In such cases the Euler-Bernoulli beam appears to be stiffer t han the actual beam is. As for thin, single-layered beams, this theory gives satisfactory predictions for the transverse displacement even wlien the tivo kinematic assumptions statecl above are not strictly satisfied.

2.1.2 Timoshenko Beam Theory

In order to include the effect of transverse shear strain, Timoshenko beam theory can be used. The two assumptions made by the Euler-Bernoulli beam theory are replacecl by.

a Plane sections initially normal to the beam's longitudinal avis rernitin plant. and normal to the asis after deformation.

In this case, the consistent axial and transverse displacements are found to be.

wliere &(x) is the rotation of the transverse normal about the L-asis. The state of strain is given by:

The physical interpretation for the assumed displacements in Eqs. 2.5 to 2.6 are compared with those in Euler-Bernoulli beam theory in Figure 2.1. A multi-layered beam is also treated as a single layer beam with a smeared layer. The number of generalized displacement is always three (uo, II> and wo),independent of how many layers the beam actually has. This theory gives a state of strain with a constant shear strain, as shown in Eqn. 2.9. Such shear strain distribution seldom reflects the actual distribution, even for a single-layered beam. For example, when an isotropic beam (a) Euler-BcrnoulIi Bcari

Figure 2.1: Coniparison between Euler-Bernoulli and Timoshenko beani t1ieoric.s

is subjected to a pure shearing load. the shear strain has a parabolic distribution in the transverse direction (Figure 3A). To âccount for the difference between the assumed and actual strain distributions. this theory uses a shear correction factor k in the shear stress-strain relation:

The energy expression can be shown to be equal to:

In the literature, the shear correction factor is usually regarded as a correction to the shear strain energies given by the assumed constant shear strain and the actual shear strain. For isotropic rectangular beams the value of the shear correction factor is found to be 516 [l]. For multi-layered bearns, the correlation of the strain energies should be calculated base on the particular layup and shear moduli of the layers of the beam. Similar to the Euler-Bernoulli beam, terms in the energy expression above may be classified as equivalent stiffness terms as well as deformation terms. The stiffness terms correspond to axial extension and bending are identical to those for the Euler-Bernoulli beam. The quadratic dependence on the deforrnation terms also enables the calculation of a local extremum of the energy expression. It should be noted that the additional sliear deformation term in Timoshenko beam theory enables the prediction of sliear deforrnation in the actuaI beam.

As indicated by Eqn. 2.8, the transverse normal straiii czL is neglected iri rhis ttieory. This theory gives better predictions for isotropic beams of small to medium thickness t han the Euler-Bernoulli beam theory.

2.1.3 Higher Order Shear Deformation Theory

Based on the findings from the previous theories, a possible improvement is to include everi higher order terms in the transverse direction to model the strain niore accuratel-. Christensen summarized the theory for a multi-layered plate in [3]. As for a beani. the assumed displacements become:

The equivalent stiffnesses For this model can be shown to be the rnonients of the layers' elastic moduli. Despite the improved capacity to model a more complicated strain distribution, a continuous polynoniial for the displacement is not a suit able representation of the displacement if the objective is to model the piecewise discontinuous strain distribution which occurs in mult i-layered beams (Figure 2.2). An example of the discontinuous strain field can be found in Figure 3.17, when a laminated beam is subjected to a pure shear load.

2.2 Multi-layered Theories

The multi-layered beam theories differ from their single-layered counterparts by individually prescribing the displacement state in each layer. As a result, elastic properties in each CHAPTER 2. REVlEW OF POPULAR BEAM THEORIES

Figure 2." Strain approximation by a higher order tlieory layer can be separately considered, instead of being used in an "averaged equivalent" façh- ion. The accuracy of the averaging process depends on t lie transverse strain distribution. and cannot be guaranteed. A brief overview about the development of layenvise theories can be found in [IO]. In the following the "zig-zag" theory and the theory developed by Di Sciuva are presented. Both theories are presented in their plate theory form. in which they were first developed.

This theory assumes a displacement field that is separable in the in-plane (x and y) ancl transverse (2) coordinates (Eqn.2.13). The t hrough-t hickness interpolation funct ions, CD' and a' in Eqn. 2.13, are piecewise Lagrangian interpolation functions non-zero over part of the thickness domain (Figure 2.3). LIj, and CVi are nodal values of u, v and w at the i-th node at (x, y). The use of Lagrangian interpolation functions guarantees the displacement field has only Co continuity through the thickness. It allow strain discontinuity (especialb CHAPTER 2. REVIEW OF POPULAR BE.4M THEORIES for F:,, -yy, and y,,) to occur when necessary.

where .V and .II are the number of nodes used through-thickness to interpolate iri-plane and transverse dis placements respect ively.

(a) riodd m.rilue~of u (b) tlirougli-ttiick~itxs iiiteq~olutionfurictiuii SV(:)

Figure 3.3: Zig-zag Theory

If Qiare chosen to be 1 for al1 i. the zig-zag theory simplifies to a partial Iayertvise theory. Compared with the single-layered theories, a partial layenvise theory rnodels in addition the transverse shear effects. A full layenvise theory considen transverse variation for

The governing equations can be derived by variational calculus and the result can he found in [IO].

Since the laminate is discretized in the transverse direct ion, t lie laminate const i t iit ive equations require the assembling of stiffnesses contributed by al1 lqers. The zig-mg theory avoids the evaluation of the shear correction factor. bccause al1 through-thickness strain distributions can be accurately represented (provided the nuniber of t hrough-t hickness interpolation points is large enough) . It also considers the transverse normal strain czz, which is neglected in al1 theories presented in previous sections. The major weakuess in this theory is the large number of variables involved. Consider a plate with L layers (Figure 2.4). If each layer-interface is assigned a variable for u, c and w respectively, so as to achieve linear variation of displacement in each layer. The number of variables at each nodal position in the q-plane is 3 (L + 1).

7 c:. ci. ri; X k

Figure 2.4: Through-thickness nodal point i CH-4PTER 2. REVIEW OF POPULAR BEAM THEORlES

2.2.2 Multi-layered Plate Mode1 by Di Sciuva

Introduction

Di Sciuva proposed a multi-layered anisotropic plate model [4]. It is kinematically defined by five generalized displacements uo. o", wo. g, and g, which are functions of x and y. This niodel satisfies both the geometric and stress continuity conditions at laver interfaces: yet unlike traditional "zig-zag" niodels. the number of generalized displacenients does not increase with the number of layers involved. This iayer-independence is açhicvetl by rtilaririg the displacement description of each layer with the shear stress ecluilibriuni at the iriterlayer interfaces. The transverse normal strain and stress are neglected in the riiodel. A plate element formulated base on this theory can be Found in [5].

Assumed Displacement

Di Sciuva assumes the following displacement field for a plate with :V layers:

where f (2) is a through-thickness modelling term:

hF and dT are "tracers" introduced to facilitate cornparison with other laminate tlieories. Briefly, they are equai to 1 in the current model and are zero in the classical plate theory. Yk are the Heaviside step function with the property:

O for z < y kj, = 1 forz2q zk is the z-coordinate of the interface between the k-th and k + 1-th layer. CHAPTER 2. REJVIEW OF POPULAR BEAM THEORIES

Interlayer Continuity Conditions

The geometric cont inui ty conditions

are aiways satisfied Ly t lit: dis ylaceiiieiit cuiiipuiieiits ajtiuiiiàd iii Eqs. 2.11 to 2.16. The stress continuity conditions are

Based on the fact that O,: is usually small when cornpared with other riorrri.CI L stress corri- ponents, it is neglected in t his model. The reniairiing two stress continiiity roriclitioris are satisfiecl if the Iayerwise correction terms chk and ;k are linearly relatcd to g, and y, ria:

where ak? bk? Q and dk depend on the shear elastic properties of layer k and k + 1 orily.

Cornparison with Classicai Plate Theory

The above assumed displacement components consist of two contributions, the terms corresponding to the classical plate theory ("on-superscripted terms) and the terms responsible for layerwise corrections (g,: g,, $k and t,bk-associated terms). For example' the first layer's displacemelit components are,

The in-plane components u and v for the k-th layer consist of ul and ul as above plus the first k - 1 layerwise correction terms (z - zk)t$k and (z - zk)$J~. Since dk and $J~are independent of z, the layerwise correction terms are linear in t. Notice that u and v for al1 layers are cubic in r, despite the fact that the cubic term is not individually but globally defined for al1 layers. The strains in the k-th layer are:

Notice that since w is independent of 2, the transverse normal strain ~,,kis zero for al1 k.

2.3 Discussion of Popular Beam Theories

The preceeding sections have given a brief picture about how various theories niodel a rnulti-layered beam. In this section, a general discussion will be given.

The single-layered theories have the advantage that no niatter how man' layrs the beam has, the number of generalized displacement variables remains unclianged. Their major drawback is inferior resolution for the local stress and strain in the transverse direction. In the shear deformation theories (i.e., Timoshenko beam theory and higher order shear deformation theory), the correct value of the shear correction factor is required to obtain a correct prediction of the global displacement (which is usually defined to be the displacement of the beam's mid-plane) . The zigzag mode1 allows high resolution of the displacement, strain and stress in each layer. However, the formulation is complicated and the number of generalized displacement variables increases with the number of layers. In the mode1 proposed by Di Sciuva, the number of variables is independent of the number of layers. This result is achieved by invoking shear stress equilibrium at al1 layer interfaces. Like other beam theories except the zigzag theory? the transverse normal strain and stress are iieglected. It is justified by the assumption that the transverse normal stress is small compared with other normal stresses. Chapter 3

Polynomial Solut ion for Beams

3.1 Solution for the Interior Problem

In this chapter. the solution of the two-dimensional elasticity probleni by means of' power series espansioris of the axial and transverse displacements is presentcd. .A sirriilar niethod is described by Timoshenko in [Id] for an isotropic beam. The extension to a multi-layered beam is briefly presented in the next section. Details may be found in Appendix A. The multi-layered beam to be considered is shown in Figure 3.1. Only the upper and lower surface. and the edge on the left hand side (r = 0) are prescribed with boundary conditions. The load on the upper and lower surfaces are represented by s polyiomials. The condition on the right side edge is determined by mechanical equilibriiim and can riot be prescribed arbitrarily The solution obtained using this method corresponds to the solution of the interior problem as described in Section 1.3.2, and will be called the self-siniilar solution for the applied ioad. Physically, the self-sirnilar solution is the state attaincd by a bearn with ari infinitely Long span, subjected to the same boundary conditions. Notice that the solution will be independent of the beam's length. This method is used to determine the deformation of two beams. The first is made from an isotropic material (aluminum) and the second from of a symmetric laminate. Both beams are subjected to four fundamental loads applied individually. The fundamental loads are defined in Section 3.4, along with the solutions obtained using the polynornial solution method. Figure 3.1: Laminated Beam .\Iode1

3.2 Sign Convention

The usual sign convention for stress is adopted: that is. a stress comporient is positive when the stress direction and the outward riornial of the plarie upoii which it acts are of the same sign. (see Figure 3.2.

Figure 3.2: Sign convention for the stress components CHAPTER 3. POLYNOMIAL SOL UTION FOR BEAM

The stress (Ro),shear (Qo)and bending moment (RI)resultants, defined as:

are positive in the directions show in Figure 3.3.

Figure 3.3: Sign convention for the stress nionirnts

3.3 Formulation

3.3.1 Notation

Consider a multi-layered beam made of L orthotropic layers. with an infinite Iength and a thickness of 2h. The self-similar displacement solution of the beam is composed of L pairs of polynomials in terms of x and z,

(3.1 where ui(x,t) and wi(x,z) are the x- and z-displacement in the 1-th layer (Figure 3.1). Let C = be the normalized transverse coordinate, the displacements in terms of x and C are:

where CHAPTER 3. POLYNOMIAL SOL UTION FOR BEAMS

The stress components in each layer, for notation purposes, are written as:

The stresses appiied oti the upper aiid luwrr surfaces are deriotecl by

and use the same sign convention as the stress. The stress. shear. and moment resultants applied on the left-sicle edgc are written as constants Ro. Qo and Rirespectively.

3.3.2 Governing Equations

In order to seek a closed form solution. the series exparisions of ui and wl in Eqn. 3.2 are first approsimated by complete polyiiomials of order .V in r and C. .A rricthod esists to determine whether a particular value of .V chosen is sufficieritly high. and is clescrit~cclin Section 3.3.3. When

it is shown in Appendix A that the stress coefficients (OF),ap;j), ?!il)) can be represented in terms of the displacement coefficients (u,('"),CV/'")) as

Eqn. 3.10 is obtained from the strain-displacement and stress-strain relations. The multi-layered beam problem also satisfies the following equations, equilibrium equations

for al1 x and C, within the domain of each layer.

boundai, conditions

(a) rigid body motion constraints

u,,, (O. O) = O:

w here the m- t h layer coiitains the niid-plane of the beani. (b) applied surface stresses

Moreover, the stress and displacement fields in each layer are related to those of the adjacent layer via

interface eauilibriurn

oIL J-1 (x,6) = GL,~(x~6); GZ,l- L(X,CL ) = G2,l(x?6 1 (3.17)

where Ci is the C-coordinate of the 1 - 1-st and 1-th layers' interface (Figure 3.1).

interface cont inuity

W(G(1) = udx76); wi-dx, 6)= wl (x,6) (3.18) CHrlPTER 3. POL\VOR/IIAL SOLUTION FOR BEAMS

3.3.3 Solution

Al1 equations from Eqn. 3.11 to 3.18 can be written in terms of displacement coefficients (Li,("'). I.Ç('")), by substituting O!;]), O?), from Eqn. 3.10. To seek the self-sirnilar solution, al1 equations are expanded by equating coefficients of x and C on both sides. From the analysis shown in Section A.11, it is found that for a beam with L layers. the number of coefficients be deterrnined is:

While the total number of equations is

ni = (N + 1)(N+ 2)L + 4

The result ing equat ions are expressecl in mat rix form yielding,

where the matris A is a (rn x n) matrix. 2 is a vector containing the unknown displacement coefficients and c'contains the independent terms. There are always four niore equatioris than unknowns because m is alwqs larger than n by 4. Since it is impossible to determine a priori which four equations are reclundant for an arbitrary load. the system of equations arc solved in the least square sense. that is.

ATAZ = A'+ B.? = (1 (3.2'2)

Zlevertheless, the solution F must satisfy al1 equations simultaneously. The validity of the solution is monitored by checking the magnitude of the residual vector. defined as

where

The magnitude of the residual represents how well the solution satisfies the governing equations. When Ilql is negligible compared with the magnitude of the unknown displacement coefficients, the solution descri bes the displacement accurately. Ot herwise, the displacements should be represented by higher-order polynomials. -4 larger value of N in Eqs. 3.8 to 3.9 is needed. CHAPTER 3. POLYNOR/fIAL SOLUTION FOR BEAM

3.4 Solutions for the Fundamental Loads

3.4.1 Fundamental Load

The terrn "fundamental load" is defined to designate a set of loads applied to a beam. In principle, when the number of fundamental loads defined is large enough. the linear combination of the fundamental loads can represent any conceivable load case. Throughout this thesis. four fi~nrlnmmtdloads are iised. They are defined in the table below.

Asi al Pure Pure Distributed Extension Bending Shcar Pressure

Tz- - (x) 0 0

Table 3.1: Fundamental Loads

The objective of defining such fundamental loads is that a structural stiffness of the beam correspond to each type of load can be determined. The term "fundamental mode" represents the states of displacement, strain and stress when the beam is subjected CO a corresponding fundamental load. In the following sections, solutions for the fundamental loads obtained by the rnethod of polynomial solution are presented. Two beams are studied. The first one, representing a single-layered beam, is made of aluminum. The second one, representing a rnulti-layered beam, is a [0°/900], laminated beam made of a graphite/epoxy composite T300/N5208. The displacement solutions will be presented as functions of x and C, where C is the CHAPTER 3. POLYNOhIL4L SOLUTION FOR BEAhIS normalized transverse coordinate, defineci as:

With respect to the state of strain and stress. the through-thickness distribution at x = O is presented. At this location the stress resultants Ra. RI and Qo are equal to their prescribed values specified by the fundamental load case in Table 3.1. The resultants' dependence on x is irrelevant to determine the beani's stiffness.

3.4.2 Fundamental Load Solutions for an Isotropie Beam

Elastic Properties

The elastic properties of an aluminum beam is surnmarized in Table 3.2. To ensure thc boundary effect at free edges does not affect the overall response of the bearn (as discussed in Section 1.3.1), a beam with a high width-to-thickness ratio is considered. .A state of plane strain is assumed to exist in the y-direction. The exact values for the width and length are irrelevant to the self-sirnilar solution obtained by this method.

I Young's ~iocîulusE l 73.0 GPa I 1 Poisson's Ratio v 1 O. 3 1 98.3 -42.1 O Elastic Modulus in the rz-plane E

Table 3.2: Elastic properties of an alurninum beam CHAPTER 3. POLYNOMIAL SOLUTlON FOR BEAM

Displacement s

The numerical values for the ci coefficients are tabulated in Appendix C.

4x7 c) 4.7 (2 Axial Extension C1.c ~2 C Pure Bending c3x< (;'+ c5.xP Pure Shear C~C~+ C~Z~ c~r + c9.ci2 + clor3

Table 3.3: Displacements for an aluminum beam

Strains and Stresses (at r = 0)

The strains at x = O are tabulated in Table 3.4. The strain and stress are plotted in Figures 3.4 to 3.11.

Q Axial Extension cl 2h O 9 - Pure Bending CJ i h O J 1 Pure Shear O O CS + (2+ cg) c' Distributed Pressure cllC+ c12c3 + hi3 0 Table 3.4: Transverse strain distributions for an aluminum beam (at r = 0) Figure 3.4: Strain distributions for axial extension

I Aiuminum

Figure 3.5: Stress distributions for axial extension CHAPTER 3. P0LYNOR:IIAL SOL UTlON FOR BEAMS

Figure 3.6: Strain distributions for pure bending

1 Aluminum Qummnding ( i; RI 1

Figure 3.7: Stress distribut ions for pure bending Aluminum Pun Shaw

Qo l

Figure 3.8: Strain distributions for piire shear

Aluminum Pun Show 00

Figure 3.9: Stress distributions for pure shear CH.4PTER 3. P0LYNOMIA.L SOLUTION FOR BE.4.kIS

Aluminum m4 ~istributwiPnruin

Figure 3.10: Strain distributions for clistributed pressure

Aluminurn Diltributed Prsuum

Figure 3.11: Stress distributions for distributed pressure CHAPTER 3. POLYNOMML SOL UTION FOR BE14,i1fS

3.4.3 Fundamental Load Solutions for a Laminated Beam

Elastic Properties

Normally for composite materials, only the in-plane elastic properties (properties iri the ry- plane) are given. In order to perform through-thickness analysis, the out-of-plane elastic propert ies are also needed.

Figure 3.12: Orientation-dependence of a composite plfs properties

With reference to Figure 3.12, it is round that Et, is identical to Ew, and G12 is identical to G,,. One more shear modulus, Ggz,is required for the analysis. Froni the figure, the composite ply is transvenely-isotropic in the yz-plane. The shear niodulus G,: can be written as:

However, vyZ cannot be calculated using the in-plane elastic properties. Based on the fact that the cross-section in the yz-plane is transversely-isotropic, u,: is assumed to be 0.3. The elastic properties for the [0/90],beam are tabulated below: CHAPTER 3. POLYNOMIAL SOL UTION FOR BEAMS 35

1 Total Beam Thickness (2h) 1 1 m 1 Longitudinal Young's Llodulus El 1 181.0 GPa 1 Transverse Young's Modulus E2*,E33 1 10.30 GPa 1 1 ~on~itud~alShear-iloduiusG~~~G~~~1 7.17 GPa 1 - - -. - Transverse Shear Modulus G23 4.02 GPa 1- - 1 1 Longitudinal Poisson's Ratio 43, 42 0.28 I 1 Transverse Poisson's Ratio 43, 1 0.3 1 I 153.3 4.17 O Elastic blodulus in the 12-plane E (O0 plys)

11.4 3.49 Elastic Modulus in the xz-plarie E (90" plys)

- - Table 3.5: Elastic properties of a T300/?j5208 beam with 1-up [0*/90°],

Dis placements

The numerical values for the bi and ci coefficients are tabulated in Appendix C.

Pure Bending bd b5 + b6<' + b7rï Pure Shear b8 + b9C + bloc3 + bllxzc blzr + b13rt;-t OI.,x3 I Distributed Pressure bI5x+ blbx< + bli~~3+ bl8z3C, bI9 + bZ0< + bzlc2 + 622~" +b23~2+ b24~2(2 + k5l4

Table 3.6: Displacements in the first layer of a laniinated beam CHAPTER 3. POLYNOR/I144LSOLUTION FOR BE-4kfS

1 Axial Extension 1 1 1

Table 3.7: Displacements in the second layer of a laminated beam

Strains and Stresses

The strains at J: = O are tabulated in Table 3.8. The numerical values for the di aticl e, coefficients are tabulated in Appendix C. The strain and stress are plottecl iri Figures 3.13 to 3.20.

- - --

40 cz1 ii) :dxz (4 Asid Extension 4 d2 0 Pure Bending d3C d-1C O 1 Pure Shear dsx< d6rC di + dH<' Distributed Pressure dg + dlo<+ d11c3 d13 + dl.& + 45~' dlix + &+<'

Table 3.8: Strain distributions in the first layer of a laminated beatn

1 Axial Extension ( e 1 1 e2 1 O 1 Pure Bending e3 c- 1 O

Table 3.9: Strain distributions in the second layer of a larninated beani yOm * h&Extension -j &

Figure 3.13: S train distribut ions for aial extension

Figure 3.14: Stress distributions for axial extension Figure 3.13: S train distributions for pure bending

Figure 3.16: Stress distributions for pure bending CH.4PTER 3. POLYNOh.ll4L SOLUTION FOR BE.4ICfS

ro0~07' Pum Shar

40 I

Figure 3.17: S train distributions for pure shear

Figure 3.18: Stress distributions for pure shear Figure 3.19: Strain distributions for distributed pressure

Figure 3.20: Stress distributions for dist ri bu ted pressure Chapter 4

An Enhanced Beam Theory

Insight from the Euler-Bernoulli and Timoshenko Beam Theories

In Chapter 2. the Euler-Bernoulli and Tirnoshenko beam theories are discussed. It is found that the strain energy expressions resulted from both theories (Eqn. 2.4 and Eqn. 2.10) are qudratic functions of some strain component terms. with constant coefficients knowri as the stiffness terms for the corresponding state of strain. These terms are summarized in the table below. Ex, and G,, are the elastic rnodulus and shear nioclulus of the material in the rz-plane.

1 Fundamental lod 1 term 1 Euler-Bernoulli 1 Tirnoslienko 1 duo duy Axial extension strain component & ff.K stiffness ph E&)d= E&)d: strain component -di2 Pure bending -3 Ilr st iffness $fh EZ,(~)ffdt5th Ez.(:) Pure shear strain component O ++&aclt stiffness n/a k $_hh Gr=(2) dr Table 4.1: Comparison of strain components and stiffness terms

Comparing the two theories, the relative success demonstrated by the Timoshenko beem theory when modelling a beam in shear is because: 1. shear deformation is allowed by the angular rotation term $; and

2. the correct value of the shear stiffness is known (through the knowleclge of the shear correction factor k)

-1s mentioned in Section 2.1.2, the value of k for a laniinated beam depends on t lie 1-up as well as the relative elastic properties of the constituting layers. To evaluate t tie esact value of the shear correction factor, the exact strain distribution through the t hirkness is requiretl jlj. It should be emphasized that in the Timoshenko beani theory. the slicar stiffncss turned out to be k~l~G12(4d2 because the kinematic assurnption riiade on tlie bearri's displacements (Xqs. 2.5 to 2.6) resulted in a constant t hrough-t hickness shear st rain distribution. For the reason described in Section 1.2.3, it is known that the shear stihess of each Iayer is in fact added in a "parallel" manner rather than in a "series" manner. Therefore defining the shear stiffness by an integration of the shear stiffness through the beam (shh G(:)dz) as in the Tirnoshenko beani theory is not sat.isfactory. The eriharicecl beam theory to be presented in this chapter defines the stiffness terms directly frorii the self-sirnilar solutions O bt ained for the fundaniental loads.

4.2 Governing Differential Equations

Imagine a hypo thetical, homogenized beam (H-beam) only capable of cleforrnat ion iri t lie form:

This beam can be calibrated sucli that the longitudinal and transverse displacements. as well as the angular rotation on its mid-plane are identical to those on the mid-plane of the actual beam when the sarne fundamental loads are applied. The strains of the H-beam are defined using the usual strain-displacement relations: The strain energy of the beam can be written in terms of stresses and strains as:

Substitute into Ü the terrns I,,' tzand from Eqs. 4.3 to 4.5, obtain:

where the homogeriized stiffness terms Hi*are defined as:

The deterniination of these terms are given in Section 4.3.

Figure 4.1: External loads on a beam

Consider the H-beam is subjected to external loads on its edges and surfaces (Figure 4.1). the work done becomes:

LL lhTZ + pt(rjY (x,h)dx - iLq-(x)

4,Rr and Qi are the externally applied stress resultants, defined as: Using the principle of virtual work, the H-beam is in equilibriuni when

Substitute Eqn. 4.6 and Eqn. 4.8 into Eqn. 4.9, the governing diffefercritiiil eqiiatioris are fourici to te:

and the appropriate boundary conditions are:

where Ro, Rl and Qo are the stress resultünts defined as:

bing Eqn. 4.7 aiid Eqn. 4.16, the governing differential equations (Eqs.4.10 to 4.12) can be written as:

Notice that these governing equations are the integrated form of the equilibrium equations given in Eqn. 3.11. 4.3 Homogenized Stiffness

The liomogenized stiffnesses defined in Eqn. 4.7 are ratios of ü stress moment and the corresponding strain component term of the H-beam. By equating the stress moments of the H-beam to those of the actual beam (Eqn. 4.20), the hornogenized stiffnesses can bc determined from the self-similar solutions of the fundamental loads (Section 3.4) as the followings.

The homogenized stiffness for ttie axial extension loütl Hl, is definecl as:

From Eqn. 4.3 it is found that 2 is the constant term of b,. In order to calibrate the stiffness of the W-beam with that of the actual beam. the constant terni of F,, on ttie mid- plane (C = O) in the actual beam subjected to the same fundamental load is needed. For this purpose. the "fundamental strain coniponents" Di are defined as the value of a strain component term wlien a unit of the corresponding fundamental load is applied. Therefore. froni Table 3.4 the fundamental strain components for the aluminurn bearri are:

DI = 0.1247 x IO-'' (axial extension) 4 = 0.1496 x IO-' (pure bending) D3 = 0.5342 x 1obL0 (pure shear)

Since ph5&)dt = phoXx(x)di = 1 for this load case, the axial extensional stiffness of this beani is:

Similarly, the stiffness for the pure bending and pure shear fundamental loads can be obtained for the aluminum beam. The stiffnesses for the laminated beam can be determined with the aid of Table 3.9. Notice that the value for the strain compouent term should be evaluated on the mid-plane (C = 0) and 2 = O. It is because at x = O only one fuidamental load is non-zero and equals to one. The homogenized stiffnesses for the two beams are tabulated in Table 4.2 and Ta- ble 4.3. The corresponding stiffnesses given by Euler-Bernoulli and Timoshenko beam theories are also listed. Their analytic expressions can be found in Table 4.1.

1 Fundamental load 1 Enhanced 1 Euler-Bernoulli 1 Tirnoshenko ( Axial extension 1 80.22 x log 1 80.22 x 10' ( 80.22 x 10" 1 Pure bending 1 6.685 x 10' 1 6.685 x 10' / 6.685 x log Pure shear 1 18.72 x 10' 1 n/a 1 '23.40 x 10" Table 4.2: .Uurninum beam's stiffnesses by different theories

The stiffnesses shown in the above table deserve some explanations. Since a state of plane strain is assumed, the modulus for extension becomes E*:

For aluminum. E* = 80.22 x 10" According to Euler-Bernoulli and Tinioshenko bearii theo- the extensional stiffness is 2hE'. This gives an identical stiffiiess as the homogenizecl stiffness HII. It is expected because in axial extension, c,, is constant. In the pure bending case, the stiffness given by the Euler-Bernoulli aiid Timoshenko beam theories is 12 . which is equal to 6.68 x 10"or aluminum. Since in this c;fie the axial strain É,, is linear in z, the Euler Bernoulli and Timoshenko beam theory capture al1 information about the strain field, and give an identical stiffness as the enhanced beani theory does. When comparing the stiffnesses correspond to the pure shear load, it is important to realize that in the Timoshenko beam theory, the parabolic shear strain distribution is represented by an average constant approximation. This average strain (7)is given by [l]:

In the enhanced beam t heory, the strain component used in the stiffness definition is CHAPTER 4. ,4N ENHANCED the shear strain on the mid-plane which is equal to:

The ratio bet\lreen the sliear stiffnesses piwn by t,he two hpnm theories is:

which is indeed the ratio between the values shown in Table 4.2. Table 4.3 shows the stiffnesses for the Iaminated bearn. The extensional stiffnesses For Euler-Bernoulli and Timoshenko beam theory are calculated by integrating the stress- strain relation through the thickness. The fact that c,, is constant and Poissoti's ratio for the state d plane strain are used. They are identical to the homogenized stiffness founci for the extensional load, for the same reason as stated for the aluminum beum. For referencc!. the fundamental strain components Di for latninated beam are:

Dl = 0.1041 x 10-l0 (auial extension) Dz = 0.7484 x 10-Io (pure bending) D3 = 0.3233 x IO-' (pure shear)

Fundamental load Enhanced Euler-Bernoulli Timoshenko hial extension 96.05 x log 96.05 x 109 96.05 x 10' Pure bending 13.36 x log 13.36 x 10' 13.36 x 10' Pure shear 3.09 x 10' da 4.66 x 10' Table 4.3: Laminated beam stiffnesses by different theories

Similarly, bending stiffnesses are calculated and compared with that from the enhanced beam theory. They are identical.

The shear stiffness given by Timoshenko beam theory is expected to be different from the one given by the enhanced beam theory. As noted in Section 1.2.3, defining the shear stiffness of a laminate as the arithmetic sum of the individual layer's shear stifiess is inappropriate. Also, a main advantage of the enhanced beam theory is that the shear correction factor k is not needed. The Timoshenko mode1 shear stiffness should be evaluated using the shear correction factor calculated for this particular laminate. If it is done, both theones will give a shear stiffness differing by the factor between the mid-plane shear strain and the average shear strain. The value shown in Table 4.3 used the value 516 for the shear correction factor.

4.4 Finite Element Formulation

In a finite element analysis, rnonotonic convergence to the exact solut ion can be guaranteecl when both the continuity and completeness requirements are satisfied. Since in the eriergy expression of the hypothetical beam (Eqn. 4.6)?the highest order of derivative is one. the requirements can be satisfied with an element having CUcontinuity at elerrierit interfaces and C1 continuity in the element interior. To ensure a state of zero shear strain cari be achieved natiirally the terms ul and 9 in -i,, need to be of the same orcler of r [Tl. For this reason. the r-order of the trial functions for the generalized displacements uo. ui and u;o are chosen to be:

a .u0 is lineür in z

0 u1 is linear in x

a wo is quadratic in x

With reference to Figure 4.2. consider the i-th element of length l,, the generalizecl displacements can be written in terms of the local nodal variables a,. The appropriate shape functions in the normalized z-coordinate (= t)are: CHAPTER 4. AN ENHANCED BEAM THEORY

Figure 4.2: Nodal variable distribution for the i-th element ivhere the superscript i designates the variables correspond to the i-t h elenlent. and:

The strain vector for the i-th element 6 is written as:

The stiffnesses for the three fundamental loads are written as: and the element stiffness matrix for the i-th element Kiis:

Ki = BTHB~

When the lateral pressure q(z) is equal to a constant qo, the consistent force vector for the i-th element becomes:

The stress resultants are functions of the local variable <,. Csing Eqs. 4.26 to 4.29. the global stiffness equation can be assembleci in the usuül manner to obtain:

where K is the global stiffness matrix, ü is the nodal displacernent vector and / is the consistent global force vector.

4.5 Strain and Stress Determination

In Chapter 1, it has been emphasized that the determination of the stress distribution is important to assess whether a design for an engineering component is suitable. In Chapter 3, the self-similar solution of the elasticity equations gave the displacement. strain and stress distributions accurately in regions not affected by boundav and load- application effects (Section 1.3). The traditional single-layered beam theories predict a continuous strain distribution through the beam's thickness, which is a weighted average of the actual strain in the beam. In this section a simple procedure is proposed to determine the exact strain and stress distributions using the st rain component terrns obtained from a fini te element analysis based on the enhanced beam theory. CH,-IPTER 4. AN ENHANCED BEAM THEORY 51

First calculate the "fundamental mode factors" (ai)by dividing the strain component ternis (2,%, ul + 2)from the finite element solution by the fundamental strain components (Di).The fundamental strain components are defined as the value of a deformation term when a unit of the corresponding fundamental load is applied (Section 4.3). Therefore, the fundamental mode factors are the multiplication factors by wliiçh the actual stress/strain distribution is related to the distribution of the fundamental loads.

The exact strain distribution can be expressed as a linear combination of the strain distributions of t hc fundamental modes, with the fundamental mode factors as the coefficients. That is:

where the subscripts 1.2 and 3 represent the axial extension. pure bending and pure shear fundamental mode respectively. For example. c 1 is the through-tliickness distribution of the strains (cIx(C), 6:- ((;) arid ni,, (C) ) of the beam su bjected to the axial estensiori Funclmienta1 load. Siniilarly, the exact stress distribution due to the transverse displacernerit loacl is:

This procedure is used to calculate the transverse stress and strain distributions presented in Chapter 5. Chapter 5

Results and Discussions

In the previous chapter. an enhanced beam t heory has been proposed. The iiiajor clifferr!rict~ betneen the present theory and the Timoshenko beam theory are:

1. a new interpretation of the kinematic assumption by the introductiori of the hypothetical H-beam :

2. definition of the shear stiffness; and

3. calculation of the through-t hickness stress and st rain distri butions.

The shear strain term ul+ 2 in the Timoshenko beam theory hlis been interpreted as the average shear strain through the thickness. By changing the terrn's interpretation to the value of the exact shear strain on the mid-plane. the present theory gives a different shear stiffness. Table 5.1 summarizes the shear stiffnesses for the aluniinuni and the larniriated beams. The extensional and the bending stiffnesses are identical for both theories.

Table 5.1: Shear stiffaesses by different t heories

The algorithm used to calculate the homogenized stiffnesses automatically considers the through-thickness distribution of the strain and stress, thus eliminated the need to calculate the shear correction factor. In Table 5.1, stiffnesses related to Timoshenko beam CHAPTER 5. RESULTS .4ND DISCUSSIONS 53 theory used has a shear correction factor equal to 5/6 For both materirls. Section 4.3 discussed the dif'ference between the shear stiffnesses from the two theories.

5.1 Test Cases

In order to assess the accuracy of the enhanced beam theory for beanis made of ditferent miitt+;ils and aspect ratios (the ratio between a beam's spaii and its tliickncss). aluniiiiuni and laminated beams of length five meters and ten meters are studied. Ali beanis have aii identical thickness equal to one meter, making the aspect ratios (LI%) 5 and 10 respectively. The beam elastic and geometric properties are given in Table 3.2 and Table 3.5. The finite element mode1 developed in the previous chapter is used to niodel the beams subjected to six different load cases. They are:

1. Axial Extension Load (Figure 5.1);

2. Constant Bending Load (Figure 5.4);

3. Constant Shear Load (Figure 5.7):

4. Constant Lateral Pressure on a Caiitilever (Figure 5.13):

3. Constant Lateral Pressure on a Simply-Supported Bearn (Figure 5.19): and

6. Constant Lateral Pressure on a Doubly-Clamped Beam (Figure 5.21)

5.2 Results Cornparison

For cornparison, the results based on the Euler-Bernoulli and the Timoshenko beam theory, as well as the exact elasticity solution are presented. r\nalytical results based on the Euler-Bernoulli beam theory shown in the following sections are found in [Il]. Results for Timoshenko and enhanced beam theory are obtained using the finite element method. Exact solut ions refer to the self-similar solut ion calculated using the polynomial solut ion technique described in Chapter 3. Such a solution disregards the boundary effects. and the local effects near the areas where extemal ioads are applied. It may argued that for a laminated beam, the effect caused by external loads at the ends extends over a considerable distance into the bearn [2]. Therefore the beams studied in the test cases, having aspect ratios equal to 5 and 10, may be too short for the end effects to be neglected. This argument is quite valid. But the objective of this thesis is to propose a new beam theory formulation. Xo beam theory attempts to include the effects due to boundaries or end conditions. The cornparison to be made is adequate to show how cvell each beam theory performs in the beam theory's context. There are extrenie probleni configurations in which no beam theory is competent to model. For such problems other solution techniques should be used. Notice that when the polynomial solution technique is used. al1 pressure loads are applied evenly on the upper and lower surfaces. obtainirig a synirnetric tleforitiatiori about the beam's mid-plane. For al1 beam theories considerecl, the proportion of the applied pressure on the surface does not affect the result. The resulting deformation is altwys symmetric about the mid-plane.

In order to quantify the rate of coiivergency for the finite eleniciit soltitions. th? rtiasimum deflections of the bearns are monitored. The accuracy of t lie solu tioii is cvaluatcti by determining the ratio of the finite element solution to the exact solution. Meri this ratio is less than 1. the rnodel is stiffer t han the actual beam is: otherwisc it is more flexible. When the beam is supported or clamped at both ends, the polynomial solution technique is not applicable. and the exact value for the maximum deflection is riot available unless other solution methods are used (for example, two diniensional finite element niodelling). For such load cases. the maximum deflection calculated using the Euler-Bernoulli beam theory is used to normalize the finite element solutions. The resulting ratio can be similarly used to monitor the convergence of the solution. When this ratio is less tkiari 1. the finite element model is stiffer than the Euler-Bernoulli beam, and vice versa. The transverse stress distribution resulted from a Timoshenko beam analysis is calculated according to the stress-strain relationship:

a,, = -

T', =

Eij are tabulated in Table 3.2 and Table 3.5; and Es, and Tzz are the strains obtained by the finite element model. CHAPTER 5. RESULTS AND DISCUSSIONS 55

When the enhanced beam theory is used, the stress and strain distributions are calculated by linear combination of the fundamental soliitions, as described in Section 4.5.

5.3 Axial Extension Load

Figure 5.1: Axial Extension Load

For both beams subjected to this load case. a constant axial strain (É,,) is intluccd. Thc shear strain remains zero. Euler-Bernoulli beam theory gives the exact lorigit udinal displacement on the mid-plane in such a situation to be

where E* is the axial extensional stiffness given in Table 4.2 and Table 4.3 For the aluminum and the laminated beam.

üsing Timoshenko beam theory and enhanced beam theory, a finite element mode1 using one element is sufficient to capture the exact solution. The solution is shown in Figure 5.2 and Figure 5.3. CK4PTER 5. IZESULTS AND DISCUSSIONS

- -

exacc Y present Timoshenko -Euler-Bernoullr ----

Figure 5.2: Comparison of axial elongations (axial extension load)

A Timoshenko Euler-Bernoulli

Figure 5.3: Comparison of axial elongations (axial extension load) CHAPTER 5. RESULTS AND DISCUSSIONS

5.4 Constant Bending Load

Figure 5.4: Constant Bending Load

When subjected to this Ioad. the beam's axial strain varies linearly t hrough the t hickriess. So shear strain is induced. Agairi. the Euler-Bernoulli. Timoshenko and enlianced bearri theories give the exact solution. They are shown in Figure 5.5 and Figure 5.6. CH,-IPTER 5. RESULTS -4ND DISCUSSIONS

1 exact I Tmoshenko ----- Euler -Bernoulli

Figure 5.5: Cornparison of lateral deflections (constant bencling)

-Timashenko \ Euler- Bernoul li

Figure 5.6: Cornparison of lateral deflections (constant bending) CHAPTER 5. RESULTS AND DISCUSSIONS

5.5 Constant Shear Load

Figure 5.7: Constant Shear Lod

When considerable shear strain arises in a beam. the Euler-Bernoulli beani theory becorncs inaccurate because it neglects the bearn's shear flexibility and under-estimates the lateral deflection. Thc rcsulting error on the alumiriurn bearn with aspect ratio equal to 5 is about Four percent. The error is much more significant for the larninated beani. as the results in Table 5.4 and Table 5.5 attest.

Comparing Table 5.2 and Table 5.3. it is found that the accuracy of che enharicecl beam theory is not affected as much by beam aspect ratio as Timoshenko beitrri theor?. Similar observation can be drawn from Table 5.4 and Table 5.5. except wlien only one element is used. It shows that the accuracy of the enhanced beam thwry in niodellingshear deformation is not affected by beam aspect ratio. Furtherniore. for the lamiriatecl beani with aspect ratio equal to 5, the Timoshenko beam mode1 converged to 0.585 tiines the actual solution, and is quite unsatisfactory. It is because the theory uses ari average shear strain. which does not correspond to the actual shear strain on the mid-plane. The resulting displacement on the rnid-plane becornes inaccurate. The enhanced beam theory used the shear strain on the mid-plaiie for calibration, and can reproduce the actual displacement on the mid-plane accurately. Figure 5.8 illustrztes how the finite elernent models of the two theories converge to different solutions. It is found that the enhanced beam theory gives a slightly poor accuracy for a longer beam than that for a shorter one when the same number of element are used. When only one element is used, the accuracy for the longer beam can be considerably worse than CH.4PTER 5. ktESULTS -4ND DISCUSSIONS that for the shorter beam. -4s for the Timoshenko beam model, the accuracy for the long beam is at least equal to (if not better than) that for the short beam. This is because in the Timoshenko beam model, the inaccuracy in modelling the shear deforrnation is very significant in a short beam. For the enhanced beam model, error due to shear deformation is minor. When the same nurnber of elements is used for both beams. each element in the long beam is longer than that in the short beam. The current formulation uses a quadratic interpolation furiction for wo, while the actual deflection is cubic. The longer eûch elernerit is. the larger the error is. 1 Theory 1 1 element 1 2 elements 1 4 elements 1 8 elenients 1 20 elements / 1 enhanced 1 0.760 1 0.940 1 0.983 1 0.996 1 0.999 1 1 Timoshenko r0.752 1 0.932 1 0.977 1 0.988 1 0.991 1 1 Euler-Bernoulli 1 0.959 1

Table 5.3: Cornparison of accuracies (aluniinurii beani. Ll2h = 5)

Theory 1 element 2 elements 4 elements 8 elemerits 20 elenlents en hanced 0.753 0.938 0.985 0.996 0.999 Timoshenko 0.751 0.936 0.982 O. 994 0.997

Table 5.3: Cornparison of accuracies (aluminuni beüm. LI-h = 10) CH.4PTER 5. RESULTS -4ND DISCUSSIONS

1 Theory 1 1 element 1 2 elements 1 4 elements 1 8 elenients 1 20 elements 1 / enhanced 1 0.835 1 0.959 1 0.990 1 0.997 1 1.000 1 1 Timoshenko 1 0.721 1 0.844 1 0.875 ( 0.883 1 0.883 ( 1 Euler-Bernoulli 1 0.659 1

Table 5.4: Comparison of accuracies (laminated beani. L/%h= 5)

Theory 1 1 element ( 2 elenlents 1 4 elements ( 8 elenients ( 20 eleilieiits 1 en hanced 0.779 0.945 0.986 0.997 0.999 : 1 1 Timoshenko 1 0.740 1 0.906 0.948 0.958 0.961

------Table 5.5: Comparison of accuracies (laminated beam. L/M = 10)

Laminated &am

- 4- - Timoshenko (3 element

Figure 5.8: Comparison of lateral deflections (constant shear) CH4PTER 5. RESULTS AND DISCUSSIONS 62

The strain distribution is examined for the laminated beam. The Timoshenko bearn theory gives a constant shear strain through the thickness, while the enhanced beani theory gives a piecewise quadratic strain distribution, corresponding to the actual situation (Figure 5.9: compare with Figure 1.2). The Euler-Bernoulli beam theory, howerer. does not allow any shear strain and thus fails to predict any shear stress. The sliear stress distributions predicted by Timoshenko and enhanced beam theory are compared mith the exact distribution in Figure 5.10. 3otice that Timoshenko beam theory gives a shear i tress distribution discontinuous at the laver intcrfaccs. It irnplics that mcchünical cqui- librium is not satisfied by the strain solution. The Timoshenko beam theory over predicts the maximum shear stress in the beam. and incorrectly suggested that the maximum shear stress will bc found in the outer layers of the beam.

As for the axial normal stress (O,,), Tinioshenko and erihanced beam theory give ail identical result. It is sfiown dong with the exact solution in Figure 5.11 and Figure 5.12 for the cross-sections at r = 1.25 and x = 3.75. Yotice that since 2 is not continuoiis across element interfaces, the value of the stress on both adjacent elernents (elenient numbcr 5 and 6 at x = 1.25 and element 15 and 16 at x = 3.75) are plotted. The values give the upper and lower bounds for the exact stress at the interface. CH4PTER 5. RES ULTS AND DISCUSSIONS

Larninated Beam 4.5E-10 x=u4 Timoshenko 4.OE-10 -Y present ; +exact 3.5E-10 - 3.OE-10 y - 2.5E-10

Figure 5.9: Shear strain distribution in the laminateci beam (constarit shear )

Laminated Beam x=u4 -T~moshenko I. presenc -exact

t .6W I '"t -400 t

Figure 5.10: Shear stress distribution in the laminated beam (constant shear) CH,-LPTER 5. RESULTS AND DISCUSSIONS

Laminated Beam x=U4

Timoshenko te=5) presenc ( e=5 1 Timoshenko l e= 6 presenc l e=6 1 exact

Figure 5.11: Asial normal stress distribution at r = 1.25 (constarit shear)

Laminaîed Beam xs3U4 -Timoshenko le=15 --e-- present le=15 1 - 4- - Trmoshenko (e=161 - -a- - presenc (ex161 exact

O y

-2 y

Figure 5.12: Axial normal stress distribution at x = 3.75 (constant shear) CHAPTER 5. WSULTS AND DISCUSSIONS

5.6 Constant Lateral Pressure on a Cantilever

Figure 5.13: Constant lateral pressure on a carit ilever

Compared with previous load cases. the major characteristic of this load is the introcluctioti of the transverse normal stress (a:,). When this stress is zero. the transwrse nornial straiti (e;.) is only induced by the aial normal strain by the Poisson's effect. arici is insigiiifirant. Therefore, the transverse nornial st rain has been neglected by al1 bearti t heories ciment ly considered. Figure 5.8 shows the performance of the various beam theories in predicting the lateral deflection of a laminated beam subjected to this load. Notice that the prediçtion given by the enhanced beam theory is quite good from r = O up to the middle of the beani. where the accuracy is 95%. From Tables 5.6 to 5.9. it is found that for an isotropic bearn with a small aspect ratio. the Euler-Bernoulli beam is not so inferior compared with the other two when it cornes to modelling the transverse deflect ion. However, the accuracy beconies iirisat- isfactory when a laminated beam is modelled. Timoshenko beam tlieory is superior to Euler-Bernoulli theory, for it is capable of handling shear deformation. It is still less accurate wlien compared with the enhanced beam theory. For the lrminated beam with an aspect ratio equal to 5, the enhanced bearn theory is about 8% stiffer than the actud bearn, while the Timoshenko and the Euler-Bernoulli beams are 21% and 45% stifFer. CHa4PTER 5. RESULTS AND DISCUSSIONS

Figure 5.14: Coniparison of transverse deflections (pressure on a cantilever)

Theory 1 element 4 elements 8 elenients 20 elements 40 elements enhanced 0.674 0.965 0.980 0.984 0.984 , 1 Tirnoshenko 1 0.663 1 0.955 ( 0.969 1 0.973 ( 0.974 ( ( Euler-Bernoulli 1 0.931 1

Table 5.6: Comparison of accuracies (aluminum beam, L/2h = 5)

Theory 1 elernent 4 elements 8 elements 10 elements 40 elernents en hmced 0.669 0.975 0.991 0.995 0.996

7- Timoshenko 1 0.666 0.973 0.988 0.992 0.993 : 1 Euler-Bernoulli 1 0.982 1

Table 5.7: Comparison of accuracies (aluminum beam, L/2h = 10) CHAPTER 5. RESULTS AND DISCUSSIONS 67

1 Theory 1 1 element 1 4 elements 1 8 elements 1 20 elements 1 40 elements 1 1 enhanced 1 0.735 1 0.904 1 0.912 1 0.915 1 0.915 1 1 Timoshenko 1 0.609 1 0.778 1 0.787 1 0.789 1 0.789 1 Euler-Bernoulli 1 0.541 Table 5.8: Cornparison of accuracies (laminated beam. L/2h = 5)

Theory 1 element 4 elements 8 elenients 20 elements 40 elements enhancecl 0.693 0.951 O. 963 0.967 0.967 Timos henko 0.645 0.903 0.915 0.919 0.920 1 Euler-Bernoulli 1 0.835 1 - --- Table 5.9: Cornparison of accuracies (laminated beam. Ll2h = 10)

Since the shear stress and the axial normal stress Vary linearly and c~uadratically along the beam's span. their distribution on the cross-sections at x = 1.25 arid r = 3.75 are plotted for the j-rneter laminated beani. From Figure 5.15 arid Figure 5.16. it is found that the enhanced beam theory gives a continuous distribution for the shear stress. Figure 5.17 and Figure 5.18 show the distribution of the axial normal stress in two different cross-sections. Comparing the two figures. it is found that the stress is seriously mispredicted at r = 3Ll-I = 3.75. It is because at this location. the stress is mainly induced y the pressure load on the beam. Since the load is not considered as a fundamental load. the enhanced beam theory cannot predict the stress caused by this load. In the cross- section at x = LI4 = 1.25, the dominating forces are the reactive bending moment and shear resultant. They are both fundamental loads and the stresses they induce ciln be predicted by the enhanced beam theory accurately. Therefore the theory gives a better stress prediction at this location. This explanation also applies to the higher accuracy for the deflection at locations away from the free end. CHAPTER 5. RESULTS AND DISCUSSIONS

Laminabd barn 0.0 x= U4

-1.0 - Timoshenko -presenc exact -2.0 -

CI -

-4.0

-5.0

Figure 5.15: Shear stress distribution r = 1.75 (pressure on a cantilever)

Laminated Eeam 0.0

-Timoshenko present -0.5 r -exact

n -

a h-1.3

-1.5

Figure 5-16: Shear stress distribution x = 3.75 (pressure on a cantilever) CHAPTER 5. RESULTS AND DISCUSSIONS

Laminated barn eu4 P

30.0E - 4- - ~imoshenkolez61 - *-- presenc (e=6J

Figure 5.17: Axial normal stress distribution at x = 1.25 (pressurta on a ca~it~ilever)

Laminated barn xdU4 -Timoshenko (e.15) --e-- presenc ie=lSl - 4- - Timashenko (e=lbl - -u- - present (en161 .+- .+- exact

Figure 5.18: Axial normal stress distribution x = 1.25 (pressure on a cantilever) CHAPTER 5. RES ULTS AND DlSCUSSlONS 70

5.7 Constant Lateral Pressure on a Simply-Supported Beam

il(L.0) = O t~*(L. O) = Il .\Ir (O) = o

Figure 5.19: Constant lateral pressure on s simply-supported beiini

The exact mmimurn deflection is not calculated for this load case. The transverse deflections by the three beam theories for the laminated beam with length equals to 5 nieters are plotted in Figure 5.20. In Tables 5.10 to 5.13, ratios of the finite elenient solution to the Euler-Bernoulli beam solution for the alurnirium and laminated beams with different aspect ratios are shown.

According to (121, the maximum deflection of an isotropie bearri according to the Timoshenko beam theory is C times that given by the Euler-Bernoulli beam theory, where C is given by: 8 211 l+v c=l+-(T)2(T)5 v is the Poisson's ratio and k is the shear correction factor for the bearn. C is found to be 1.110 and 1.027 when the aspect ratio (&) equals 5 and 10 respectively. Table 5.10 and Table 5.1 1 show t hat the Timoshenko finite element solution converged to t hese values. Again, the three theories give significantly different results for the laminated bearn with a srna11 aspect ratio. A laminated beam with an aspect ratio equal to 50 is considered. Close agreement between the results by the three theories is found (Table 5.14). It reconfirms that shear deformation only affects the transverse deflection significantly when the beam has a small aspect ratio. CHAPTER 5. RESULTS AND DISCUSSIONS

Laminated Beam b5

Euler-Bernoulli

Figure 5.20: Cornparison of transverse deflections (sirnply-supported)

[ Theory [ 1 element / 4 elements 1 8 elements 1 20 elements 1 40 elenlents 1 1 enhanced 1 0.937 1 1.087 1 1.124 1 1.135 1 1.137 / 1 Tirnoshenko 1 0.910 1 1.060 1 1.097 1 1.108 1 1.110 1

Tabie 5.10: Ratio between solutions (aluminum beam, Ll2h = 5)

Theory 1 element 4 elements 8 elements 20 elernents 40 elements enhanced 0.834 0.984 1.O22 1.O32 1 .O34

Table 5.11: Ratio between solutions (aluminum beam, L/2h = 10) CHAPTER 5. RESULTS AND DISCUSSIONS

Theory 1 element 4 elements 8 elements 20 elements 40 elements - enhanced '2.458 3.608 2.645 2.656 2,658 Timoshenko 1.901 2.051 2.088 2.099 2.100

Table 5.12: Ratio between solutions (Iaminated beani. Ll2h = 5)

Theory I element -4 elements 8 elernents 20 elements 40 elements 1 enhanced 1.214 1.364 1.402 1.41'2 1.414 Tinioshenko 1.075 1.225 1.263 1.273 1.275 1 Table 5.13: Ratio between solutions (laminated beam. Ll2h = 10)

Theory 1 element 4 elements 8 elements 20 elements 40 elements enhanced 0.816 0.966 1.004 1.O14 1.016 1 Timoshenko 1 0.811 1 0.961 0.998 1.O09 1.010 ------Table 5.14: Ratio between solutions (laminated beam, L/2h = 50) CHAPTER 5. RESULTS A"VD DISCUSSIONS

5.8 Constant Lateral Pressure on a Clamped Beam

y= -1

Figure 5.21: Constant lateral pressure on a clamped beam

Conipared with results obtained for the other loacl cases. Euler-Bernoulli beani theor' appears to be incapable of modelling a pressure load on a clampecl beam. For esaniplt.. for a laminated beam with aspect ratio equal to 5 clamped on both ends. the Tir~ioshcriko and enlianced beam theories give maximum deflections 6.5 and 9.3 tirries larger ttian tliat given by Euler-Bernoulli beam t heory (Table 5.17). The t raiisverse dcflcct ion plots in Figure 5.22 and Figure 5.23 niay offer an explanation. From the figures, it rnay be noticed that the solution slopes at the beam ends are zero for using Euler-Bernoulli theory. However. if the other beam theorics are adoptecl. the slopes are non-zero. The relevant boundary conditions for the clamped ends are 2 on the mid-plane at both ends (Figure 5.21). That is. the claniped ends have identical axial displacement across the entire cross-section (which is set to be zero by the bounclary conditions u(0,O = O and u(L,O) = O). Since the shear strain (y) is:

the above boundary conditions lead to 2 = y at x = O and x = L. While Euler-Bernoulli theory does not allow shear deformation, and has zero sliear strain everywhere, the above boundary conditions are translated to 2 = O at the ends. As for other beam theories, shear strain caused by the shear force acting at the ends are allowed. The deflection curves thus have non-zero siope at the ends. When a beam has a high shear stiffness or aspect ratio, the effect of the bounday condition on the maximum deflection decreases. For such beams Euler-Bernoulli theory CHAPTER 5. RESULTS AND DISCUSSIONS may be used to approximate the maximum deflection. Comparing the results from Tini- oshenko and enhanced beam theory, the maximum deflection given by the former is 70% of the latter. Unless an exact value for the maximum deflection is compared. it is difficult to tell which theo~is better. But one can at least conclude that the evaluation of an accurate shear stiffness is important.

Theory 1 element 4 elemeiits 8 elements 20 elements 40 elernents enhanced 0.686 1A36 1.623 1.676 1.653 Timoshenko 1 0.549 1 1.299 1.486 1.539 1.a46 1 Table 5.15: Ratio between solutions (aluminurii beam. L/% = Z)

1 Theory 1 1 elernent 1 4 elernents 1 8 elernents 1 20 elenients 1 40 elerrierits 1 enhanced O. 171 0.92 1 1.109 1.161 1.169 Timoshenko 0.131 0.881 1.O75 1.121 1.135

Table 5.16: Ratio between solutions (alurninuni beim. L/2h = 10) CHAPTER 5. RESllLTS AND DISCUSSIONS

Theory 1 element 4 elements 8 elements 20 elements 40 elerrients enhanced 8.292 9.042 9.229 9.282 9.289 Timoshenko 5.505 6.254 6.442 6.494 6.502 ,

Table 5.17: Ratio between solutions (laminated beam. L/2h = S)

Theory 1 element 4 elements 8 elements 20 elements 40 elenients 3 enhanced 2.073 2.823 3.010 3.063 3.070 Timoshenko 1.376 2.126 '2.313 2.366 2.313 1 Table 5.18: Ratio between solutions (laminated beani. L/Zh = 10)

Theory 1 element 4 elements 8 elements 20 elements 40 elements enhanced 0.0829 0.833 1 .O20 1 .O73 1.080 Timoshenko 0.0550 0.805 O ,992 1.O45 1.052

Table 5.19: Ratio between solutions (laminated beam, L/2h = 50) CH.4PTER 5. RESULTS -41YD DISCUSSIONS

Aluminum Beam !

Figure 5.22: Comparisori of transverse deflectioris (clarripcd)

Figure 5.23: Comparison of transverse deflections (ciamped) CHi1PTER 5. RESULTS AND DISCUSSIONS 74

5.9 General Discussion about the Enhanced Beam The-

The success of the enhanced beam theory illustrated in previous sections relies on the introduction of a hypothetical, hornogenized beam. Such a beam has the property tliat its stiffnesses wit h respect to different fundamental loads can be calibrated against ari act ual beam using exact elasticity solutions. The following explains why such ari üpproacli is adopted. The implications of this approach is elaborated.

5.9.1 Different Interpretation of the Prescribed Displacement hlotivated by Tirnoslienko beam tlieory. the kinematics of the hornogenized beani (H-boani) is prescribed as (Eqn. 4.1 and Eqn. 4.2):

Contrasted to the traditional iriterpretation that the assumed displacement field approsi- mates that of the actual beam, the enhanccd beüm theory's perspective is that the H-beam deforms exactly according to the prescribed forni above. Csing this interpretation. the H- beam's strein energy expression naturally gives rise to the extensional. beriding and shex stiffnesses as in Eqn. 4.6. Note that the prescribed displacements are ais0 the Taylor approximations of the axial and transverse displacements in the thickness direction about the mid-plane (to the first and zero-th order respectively). In the finite element limit the H-beam displacement approaches that of the actual beam at the niid-plane. The notion of a H-beam being a separate entity from the actual beürn also clarifies the difference between the stresslstrain distribution in the H-beam and the predicted distribution in the actual bearn. To determine the strains (or stresses) in the actual beam, the displacement of the H-beam is converted to the strain component terms using the usual st rain-displacement relations. The stress in the H-beam is obtained using the stress-strain relation and the homogenized stiffnesses. The resulting stress and strain are only used to predict the mid-plane displacernent of the H-beam using the principle of virtual work, and are not related to the stress and strain in the actual beam. The actual stress and strain distributions are obtained using the strain cornponent terms. Shese terms define the fun- CHAPTER 5. RESULTS AND DISCUSSIONS 18 daniental mode factors, which are the coefficients to be used when linearly combining the fundamental mode solutions as in Eqn. 4.31 and Eqn. 4.32. Notice that the strain energy in the H-beam is not necessarily equal to that in the actual beam. In this respect, the present theory is very different from the Timoshenko beam t heory.

5.9.2 Interpretation of the Homogenized Stiffness

In traditional bearn theories the strain distribution through tlie cross-section of the bearii must be characterised (or at lest be approximated). In a sense. this approach is vcry logical becnuse the stiffness of the beam is expressed in terrns of the tnaterial stiffriess. Sincc the material stiffness is al~vaysexpressed iri ail infinitesinial rrianner (i.r.. a stress is applied to a cube of material. where the cube is so small that tlie applietl stress can be regarded as uniform), the strain distribution niust be closely approsiniatccl over tlic. eritirv cross-section. This approach is convenient when the strain distribution has thforni of a single polynomial in any cross-section of the beani. When the distributio~iis coriiplicated and cannot be approsimated by a sniall number of variables. the bcarri motlcl bcconics increasingly cornples. This is the case for most laminated beums made of il large nuriiber of composite plys at different ply angles. Csing the H-beam hypothesis, the strains of a beüm at any cross-section cari be characterised using a unique set of strain component terms. These ternis rernain constant over any cross-section. Stiffnesses can be defined in a "cross-sectional" manner. wit h respect to each fundamental load. hother way to understand the relationship between the strain component terins and the fundamental loads is from the standardized experimental measurement of Young's modulus of a material. In order to perform such a measurement, a known estensiorial force must be applied on a uniform test sample and the resulting deformation rnust be rneasured. Instead of using a test sample of a single material, suppose a multi-layered beani is used. The application of an extensional force will affect a certain state of strain at the mid-plane (the reference plane), however complicated the strain is distributed in the cross- section. When the strain at the mid-plane can be used to characterise the strain over the cross-section, the stiffness of the beam can be defined in a cross-sectional manner. When several loads are appiied simultaneously, multiple strain component terms must be chosen to allow unambiguous characterisation of the strain for the cross-section. CH*APTER 5. RESULTS AND DISCUSSIONS 79

It is also possible to define the stiffness of a beam, using moments of the strain as the strain component terms. The transverse (2) moments of the strain can be regarded as averages of the strain in a cross-section, and can be used to characterise the transverse strain distribution of the bearn. In view of the fact that the external force specified in beam problems are usually expressed in terrns of the transverse moments of the stresses. it would be interesting to investigate whether a beam theory based on the moments of strain and stress cari be more general than those based on the strain and stress themselves.

5.9.3 About the Reference Plane

A remark is needed to explain why the mid-plane is chosen to be the reference plane. Quite frequently the displacement boundary conditions are specified on the beani's rnicl-plane. When the H-beam is calibrated against this reference plane. such boundary conditions cari be applied directly to the rnodel. A disadvantage resulted from this choice of the reference plane is that work clone by external forces applied on a beam's top and bottom surfaces are not exactly niodelled. The reason being that the H-beam is not calibrated to give the exact displacenierit oii its surfaces. Chapter 6

Conclusions and Suggested Future Investigations

6.1 Advantages of the Enhanced Beam Theory

From the results presented. the proposed enharicecl bearri tlieary lias the foilomirig advari- tages:

1. it models the extensional, bending and shear deformation accurately for single-layerd and symrnetric, mu1 ti-layered beams subjected to various boundary condit ions:

2. the number of generalized displacement variables needed is three. independent of the number of layers in the beam;

3. the exact solutions for the fundamental loads give:

i. the homogenized stiffness; and

ii. the exact stresslstrain distribution

with respect to each fundamental load.

4. when the load is a linear combination of the fundamental loads, the through-thickness stress and strain distributions (in particular the shear stress and strain) can be accurately predicted. CHAPTER 6. CONCL USlONS -4ND SUGGESTED FUTURE INVESTIGATIONS 81

6.2 Conclusions

An enhanced beam theory is formulated and presented. It has a kinematic assumption similar to that of the Timoshenko beam theory, but the interpretations for the displacement. strain and stress distributions are different due to the introduction of a hypothetical. hornogenized beam. The present theory accurately predicts the mid-plane displacements. and the stresslstrain distribution for both single-layered and syrnmetric. niulti-layered bcanis subjected to avial ~stmsional,h~nding iind shwr loads. The ;ict:tirary in predictirig slieiir stress and strain is impressire. This theory is suitable for modelling stiear deforrriatiori of a sandwich beam and any beani with drastic through-thickness variation in the sliear stiffness. A distributed pressure load is also analyzed and the effect of a rio~i-FiincLriierital load on the accuracy of the present theory is examined. The accuracies for deflectiori aiid stress distribution are less satisfactory than those obtainecl for load statcs consisting of combinations the furidamental loads only.

Since the present theory is compared with the interior solution of tlie beams. careful esamination should be made before applying the results to cases where tlie boundary and end effects are significarit.

6.3 Future Investigations

1. The results shown in Chapter 5 are for a syrnmetric [0°/90"],laminated beam. The performance of the theory on non-symrnetric laminated beams should be esaminecl.

2. The current formulation lacks the ability to predict the stress and strain when a pressure load is applied. It is desirable to develop a hierarchical, systematic procedure to derive the corresponding homogenized stiffnesses for increasingly complicated fundamental loads (e.g., linear varying pressure load on the surfaces). Such a theory may require more terms in the kinematic prescription for the H-bearn, or using the beam surface as the reference plane for cali brat ion.

3. .4s mentioned in Section 5.9.2, a stress/strain moment beam theory may also possess the same advantages of the present theory. More investigations in this direction might reveal a profound relation between the physical meaning of the moments of stress and strain. Appendix A

Derivation of the Two Dimensional Solut ion

A.1 Geometry

Consider a laminated beam lis in Figure -4.1 corisisting of L orttiotropic lqers. aricl wiiich is subjected to surface stress and shear on the upper and lower surfaces (known as buunduries) and stresslshear resultants at its ends (known as edges).

Figure Al: Geometry APPENDI. A. DERIVATlON OF THE TWO DIR/IENSION,-IL SOL UTION 83

The origin of the coordinate system is located on the beam's midplane. The laminate is restrained from translation and rotation at the origin. The normalized coordinate C is defined as:

where 2h is the beam's thickness.

A.2 Displacement Forms

The displacements of the 1-th laver can be ivritten as powr serics oF r am1 <.

If terms of order up to .V are included. then the displacements are represented as:

A.3 Governing Equations

For an orthotropic material in the rC-plane, the stress-strain relatioii is:

and the strain-displacement relations, in normalized coordinates, are: For two dimensional problems, the governing equilibriurn equations are:

-da, +- 1 arzz =O ax h a~

Csing equations -4.3 to -1.6.the equilibrium equatioris can be written as

A.4 Stress and Strain Components

Frorn the strain-displacement relations. Substituting czZlr, e,,,~arid yZL,iinto the stress-strain relation Eqn. A.3,

A.5 Rigid Body Motion

Since the beam is constrained at the origin. the following boundiiry coriditioris apply:

where the rn-th layer coiitains the origin of the coordinate system.

Substitute equation -1.1 and A.2 into A.10 and A. Il. it is foiiiicl th

(A.13)

(A.l-l)

Substitute equation A. 1 into equation A. 12.

At = O and C = 0, .4PPENDI,Y A. DERIVATION OF THE TWO DIRfENSION,.IL SOL UTION 86

Therefore, from the rotation constraint ,

ql)= 0

A.6 StressIShear on the Boundary

The normal and shear stresses on the top and bottom surfaces are given by Eqn. A.9 witli

From equations -4.16 to 1.19, coefficients for the stress on boiindaries can represented N-1-i (i+l.j) + 1Lv/i,j+l)) [(Il ( + 1) 1 + c33,l (J= ---T"'- - (-4.21) J=O

for i from O to iV- 1.

A.7 Resultants on the Edge

The k-th moment of a,, is. from Eqn. A.9: APPENDIX A. D ERIVATION OF THE T WO DIR/IENSIONAL SOL UTION

Therefore, Ro(0)and RL(0)can be espressed as:

The O-th moment of z,, is. from Eqn. A.9:

- C55 ,I (i,j+l) -[(j + 1) L + (i + 1) ~cc;'~'")

SO Qa(O) becomes:

A. 8 Int ralayer Equilibrium

Substitute equations A.1 and A.2 into the equilibrium equations (equations A.7 and A.8): -4PPENDIX A. DERIVATION OF THE TWO DIRIENSIONAL SOL UTION 89

For equation -4.27 to hold at arbitrary (E, C), al1 coefficients of <'CI rnust equal zero. That is, for i = O, ..N - 2 and j = O,..N- 2 - i.

Similarl- for equation -1.28.

A.9 Interface Equilibrium

Consider the equilibriurn at layer-interfaces at C = Cr+ 1, 1 = 1. ..L - 1.

That is, for i = O. ..:V - 1,

Similarl-

T,;J (1,Cl+,) = 'r:.l+l(~. Cltd That is, for i = O, ..:V - 1.

A.10 Interface Continuity

Consider the displacement continuity at layer-interfaces at C = Ci+l, 1 = 1, ..L - 1. APPENDIX A. DERIV4TION OF THE T WU DIibIENSION.4 L SOL UTlON

'ihus for 1 = l,..L- 1, i =O. ..iV,

Similarly. for 1 = 1, .. L - 1, i = O, ..N.

A.11 System of Equations

-411 previously derived equations are summarized in the follorvi~igtable:

Equation Number of Equations Equation Nurnber * Rigid Body Motion 3 .4.13-.\. 1s Stress on the Boundary 4:V .\.20--1.23 ( Resultantç at J = O 1 3 ( -1.24-A26 1 ------1 Intralarninar ~~uilibriurnl (V- 1) L ( .\.29-.\.30 1 Iriterlaminar Equilibrium 2!Y(L - 1) -4.31--4.32 Interlaminar Continuity U(.V + 1)(L - 1) -1.33-A.3-4

Let .VN be the total number of equations and JI AI be the total nuniber of iinknown coefficients, then it can be shown that

That is, the system of equation is always over-determined. The systeni contains four redundant equat ions. Appendix B

Derivation of the Three Dimensional Solut ion

B.1 Geometry

Consider a laminate consists of L orthotropic layers, and is subjected to siirfacc. stress and shear on the upper and lower surfaces (known as boundan'es). as well as stress/slicar resultants on its edges (Figure B.1). The origin of the coorclinates systern is located on

Figure B.1: Geometry the laminate's midplane. It is restrained from translation and rotation. .4PPENDI.X B. DEMWTION OF THE THREE Dlh/fENSIONAL SOL UTION 92

QG-

Figure B.?: S trcss Resultaiits

B.2 Displacement Forms

It is desirable to write the displacements of the 1-th Iaj-er as power series of x. y aiid :.

Introduce a normalized transverse coordinate i.such that

where 2h is the laminate's thickness. Therefore,

w here

Similarly, APPENDIX B. DEEUVATION OF THE THREE DIMENSIOiV=IL SOL CTmN 93 w here

B.3 Governing Equations

For an orthotropic material in the rc y<-space, the stress-strain relation is:

and the strain-displacernent relations are:

For a three dimensional problem, the governing equilibrium equations are:

Using Eqs. 8.4 to B.10,the equilibrium equations can be written in terms of dis- ,4PPEiVDN B. D ERIVATION OF THE THREE DIiZ/fENSION.4 L SOL UTION 94 placements as

B.4 Stress and Strain Components

From the strain-displacement relations.

B.5 Rigid Body Motion

If the laminate is constrained as if cliimped on edge number 4 in Figure B.1. die followirig conditions apply:

cm (O. O, O) = 0 w, (O. o. O) = 0 durn -(O. O. O) = O at au, -(O. O. O) = O 32 au, -(0, O, O) = O

where the m-th layer contains the origin of the coorclinate systeni.

Substitute Eqs. B.l to B.3 into Eqs. B.26 to 8.25. obtain: B.6 Stress/Shear on the Boundary

The normal and shear stresses at the laminatek upper and lower surfaces are specified as input by:

Notice that the functions T;, T:;. Ts, TG, T1; and TA are defined to have the same sign convention as the stress. They are positive when their directions and the plane on which they act are both positive or both negative.

By equating coefficients of equal power of 2 and y in the above equations with those obtained by substituting C = Il into Eqs. B.14 to B.19. the following equations are .UTENDI. B. DERIVUTON OF THE THREE DIkIENSIONAL SOLUTION 98 obtained: APPENDIX B. DERIVATION OF THE THREE DlhlENSIONAL SOL UTION 99

B.7 Resultants on the Edges

On edge 1,

cr; APPENDZX B. DERIVATION OF THE THREE DIlbIENSI0N.U SOL CITION

Similarly, on edge 4,

APPENDIX B. DERiV4TION OF THE THREE DIfiIENSION.4 L SOL LlTIOY

B.8 Int ralarninar Equilibrium

Substitute Eqs. B.l to B.3 into the equilibrium equations (Eqs. B.11 to B.13):

For Eqn. B.46 to hold at arbitra. (x,y. C), al1 coefficients of J'IJJ<~miist eqiial zero. That is. for i = O,..N - 2, j = O...N- 2 - i and k = O,..iV - 2 - i - j. -4PPENDIX B. DERltl4TlON OF THE THREE DIICIENSIONAL SOL UTION 103

Similarly, Eqn. B.47 becomes,

As above. Eqn. B.48 can be written as.

B.9 Int erlaminar Equilibrium

Consider the equilibrium at layer-interfaces at C = ci+ 1 = 1. ..L - 1.

Tfiat is. for 1 = 1...L - 1. i =O...- V- 1?j = 0 ...-V- 1 - i.

Similarly, That is, for 1 = 1, ..L- 1, i = O, ..N - 1, j = O, ..iV - 1 - i,

B.10 Interlaminar Continuity

Consider the displacenierit continuity at laver-interfaces at < = cl+,.1 = 1. ..L - 1.

cor (.~?y,G+i) = ~+i(x-~+Ci+i)

Thus for 1 = 1. ..L - 1. i =O. ..X. j = OI..!V - i. B.11 System of Equations

Equat ion Number of Equations Equation Nuniber Rigid Body Motion 6 B.16-13.31 1 Stresses on the Boundary 1 3X(X + 1) 1 B.32-B.37 1 Resultants on Edges 8 ;V B.38-B.45 IV -2 Intralaminar Equilibrium L x 1 (i + 1) (i + 2) B.49-B.51 i=O = L (p3- - ?Y)- Interlaminar Equilibrium a- (L- 1) LV (M+ 1) B.5'2-B.54 Interlaminar Continuity [L- 1) (Y + 1) (:V + 2) B.55-B.57

Let NZi be the total nuniber of equations and .id.CI be the total nuniber of unkiiown coefficients.

1).v (.Y + 1)

Therefore, there are always 5iV + 3 niore equations then unknown variables. Appendix C

Numerical Values for the Elasticity Solut ions

Aluminum Beam

1 coefficient 1 numerical value 1

Table C.l: Displacement coefficients for an aluminum beani -4PPENDIX C. NUkIERYCAL VALUES FOR THE ELASTICITY SOL UTIONS 107

C.2 Laminated Beam

Table C.2: Displacement coefficients for a laminated beam I coefficient 1 numerical value 1 coefficient 1 numerical value ]

------dis .9487E-9

Table C.3: Strain coefficients for a laminated beam References

[l] K.J. Bathe. Finite Element hacedures tri Engineering ;Inalysis. Prent ice Hall. En- glewood Cliffs, N. J., 1996. pp.399.

[2] 1. Choi and C.O. Horgan. Saint-venant's principle and end effects in anisotropic elasticity. AShIE Journal of ilpplied Meehunics. 4-kpp.424--430, 1977.

[3] Richard 51. Christensen. Mechanics of Composite ibhteriuls. CViley. Sew York. 1979. pp. 174-179.

[4] M. Di Sciuva. Multilayered anisotropic plate models with continuous interlaminar stresses. Composite Structures. 22:pp. 149-167. 1992.

[5] M. Di Sciuva. A general quacirilateral niultilayered plate element mith roritiniioiis interlaniinar stresses. Cornputers €4 Structures. 47( 1):pp.g1 - 105. 1993.

[6] Lloyd Hamilton Donnell. Beams, Plutes. and Shells. SIcGraw-Hill Book Conipariy. New York, 1976. pp.284-285.

['il D.C.D. Oguamanam, Hansen J.S., and G.R. Heppler. The simplest plate beridirig finite elernents. Institute of IlIechu~~icalEngineering, rlalborg University, 1997. Report No. 86.

[8] R. Byron Pipes and N.J. Pagano. Interlaminar stresses in composite larninates under uniform axial extension. Journal of Composite Materiah, kpp.538-548, 1970.

(91 Peter C. Powell. Engineering wzth Fibre-Po$mer Laminates. Chapman & Hall, Lon- don, 1994. p.52.

[IO] J.N. Reddy. Mechanics of Laminated Composite Plates: Theory and Analgsis. CRC Press, New York, 1997. pp.653-666. [Il] Raymond J. Roark. Fornulas for Stress and Stmin. McGraw-Hill Book Company, New hrk, 1965.

[12] Irving H. Sharnes and Clive L. Dym. Energy and Fznite Element Methoh in Structurctl hIechanics. Hemisphere Publishing Corporation, Washington, 1985. pp.204.

[13] Ivan Stephen Sokolnikoff. Mathematical Theory of Elasticity, 2nd. Edition. 'IlcGraw- Hill Book Company, New York, 1956. pp.89-90.

(1-11 S. Timoshenko and J.N. Goodier. Theonj of elasticity. 'rlcGraw-Hill. Yew Ebrk. 1% 1. pp.29-44.

[IS] A.S.D. CVaiig and Frank W.Crossnian. Edge effects on thernially induced stresses iti composite laminates. Journal of Composite Materiuls. 1 1:pp.3OU-319. 1977.

[16] -4.S.D. Wang and Frank W. Crossman. Sorrie new resuits on edge effcct in syiriirtric coniposite laminates. Journal of Composite içluteriuls. 1l:pp.92-106. 1077.

[li] J.M. Whitney. Structural rlnulgsis of Laminuted Anisutrupic Plutes. Techiioiiiic Pub- lishing Company, Lancaster, Pennsyliania. 1957. pp.43.

(181 .J.M. Whitney and N. J. Pagano. Shear deformation in heterogeneous anisot ropic plates. ASME Journal of Applied Mechoriics. 92pp.1031-1036, L970.