Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

A Mindlin finite strip for the analysis of rectangular containers and continuous plates with elastically… Canisius, Tantirimudalige Don Gerard 1990

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1990_A1 C36.pdf [ 11.57MB ]
Metadata
JSON: 831-1.0050509.json
JSON-LD: 831-1.0050509-ld.json
RDF/XML (Pretty): 831-1.0050509-rdf.xml
RDF/JSON: 831-1.0050509-rdf.json
Turtle: 831-1.0050509-turtle.txt
N-Triples: 831-1.0050509-rdf-ntriples.txt
Original Record: 831-1.0050509-source.json
Full Text
831-1.0050509-fulltext.txt
Citation
831-1.0050509.ris

Full Text

A MINDLIN FINITE STRIP FOR T H E ANALYSIS OF R E C T A N G U L A R CONTAINERS A N D CONTINUOUS PLATES W I T H ELASTIC A L L Y R E S T R A I N E D SUPPORTS By TANTIRIMUDALIGE DON GERARD CANISIUS B.Sc.(Eng.) Hons., University of Moratuwa, Sri Lanka, 1981. M.A.Sc, The University of British Columbia, Canada, 1986. A THESIS SUBM ITTED IN PARTIAL F U L F I L L M E N T O F T H E REQU IREMENTS FOR T H E D E G R E E O F DO C T O R O F PH ILOSOPHY in T H E F A C U L T Y O F G R A D U A T E STUDIES D E P A R T M E N T O F CIVIL ENGINEER ING We accept this thesis as conforming to the required standard T H E UNIVERSITY O F BRITISH COLUMB IA September 1990 © TANTIRIMUDALIGE DON'GERARD CANISIUS, 1990 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. The University of British Columbia Vancouver, Canada Department of DE-6 (2/88) Abstract A first order shear deformable finite strip with support displacements is introduced. Both nonlinear geometric effects and initial deflections may be considered. Support displacements are introduced by the use of a set of basis functions for support de-grees of freedom. Each basis function is obtained by the solution of a Timoshenko beam under a unit displacement of the respective support degree of freedom. They are combined with the standard beam functions. The new finite strip can be ap-plied to the analysis of rectangular containers and continuous plates with elastically restrained supports. The elastic restraints are introduced with independent springs acting along supports and nodal lines. The finite strip is extended to the analysis of unsymmetrically laminated clamped composite plates by the definition of equivalent elasticity modulii to find the basis functions. The new finite strip is used in the analysis of rectangular containers. It is shown that compressive horizontal forces exist in the walls of flexible containers filled with a liquid. This can only be predicted by the simultaneous consideration of the movement of the wall corners and the geometric nonlinearities, as can be done with the present model. A 'mode transition finite strip' which has unequal numbers of modes in the nodal lines is introduced. It can be used to economize the finite strip analysis of plates with loads that need a large number of modes, but spread only across a few of the strips. Also a study of the determination of transverse shear stresses by the use of the equilibrium equations and the displacement solution is made, resulting in some important and interesting observations. ii Table of Contents Abstract ii List of Tables xii List of Figures xvii Acknowledgements xviii Dedication xx 1 Introduction 1 1.1 Contributions of this Thesis 2 1.2 Thesis Organisation 4 2 The Formulation of the Finite Strip 6 2.1 Introduction to the Basic Methods Used 7 2.1.1 Mindlin's Theory 8 2.1.2 The Use of the Finite Strip Method 11 2.1.2.1 The Finite Strip Method in Continuous Plate Analysis 12 2.2 Basic Displacement Interpolation 14 2.3 The Basis Functions for Displacements 18 2.3.1 The Basic Modes 19 2.3.1.1 Introduction 19 2.3.1.2 Equivalent Elasticity Modulii for Laminated Plates . 20 iii 2.3.1.2.1 Definition of Equivalent Modulii 21 2.3.1.3 Mode Shapes for a Pin-Pin Strip 22 2.3.1.4 Mode Shapes for a Fixed-Fixed Strip 23 2.3.1.5 Mode Shapes for a Fixed-Pinned Beam 24 2.3.2 Support Basis Functions and Additional Modes 25 2.3.2.1 Additional Modes 25 2.3.2.2 Pin-Pin Strip 26 2.3.2.3 Fixed-Fixed Strip 26 2.3.2.3.1 Bases due to Translation of Supports . . . . 26 2.3.2.3.2 Bases due to Rotation of Supports 28 2.3.2.4 The Fixed-Pinned Strip 28 2.3.2.4.1 Bases due to Displacement of Supports . . . 29 2.3.2.4.2 Bases due to the Rotation of the Fixed Support 29 2.3.2.5 Satisfaction of Boundary Conditions 30 2.4 Kinematics 30 2.4.1 The Displacement Approximation 31 2.4.2 The Strain Vector 33 2.5 Variational Formulation 44 2.5.1 Virtual Work Equations 44 2.5.2 Matrix Structure Equations 45 2.5.3 Continuity Requirements on Displacements 46 2.6 The Stiffness Matrices 47 2.6.1 The Secant Stiffness Matrix 47 2.6.2 The Tangent Stiffness Matrix 49 2.6.3 Symmetrical Secant Stiffness Matrix 54 2.7 Computer Implementation 56 iv 2.7.1 The Integration of Stiffness Matrices and Load Vector 56 2.7.1.1 Shear Locking and Reduced Integration 56 2.7.1.2 The Integration Orders Used 57 2.7.2 Calculation of Stresses 59 2.7.3 Solution Procedure 60 2.8 Penalty Function Formulation For A Rectangular Container 60 2.8.1 The Constraints 61 2.8.2 The Constraint Stiffness Matrix 65 2.9 Plates With Elastically Restrained Edges 66 2.9.1 Restraints at the End Supports of a Finite Strip 67 2.9.1.1 Modelling of Restraints 67 2.9.1.2 The Displacement and Elastic Force Vectors 68 2.9.1.3 Virtual Work and Stiffness Matrix 69 2.9.1.4 Form of the Stiffness Contribution 70 2.9.2 Restraints Along a Nodal Line of a Finite Strip 70 2.9.2.1 Modelling of Restraints 70 2.9.2.2 The Displacement Vector 71 2.9.2.3 The Elastic Forces in the Springs 72 2.9.2.4 Virtual Work and Contribution to the Stiffness Matrix 72 2.9.2.5 Form of the Stiffness Contribution 73 3 Numerical Verifications 74 3.1 Simply Supported Problems 74 3.1.1 The Linear Response of a Square Thick Orthotropic Plate . . 74 3.1.2 Geometrically Nonlinear Behaviour of a Moderately Thick Isotropic Plate 80 v 3.1.3 Linear Behaviour of Orthotropic, Symmetrically Laminated Thick and Thin Plates 83 3.1.4 Linear Analysis of Unsymmetrically Laminated Simply Sup-ported Plates 89 3.1.5 Nonlinear Analysis of a Thin Square Orthotropic Plate under Combined Loads 91 3.2 Verification of Finite Strips with Fixed-Ended Mode Shapes 95 3.2.1 Simply Supported Rectangular Thin Plate With Applied End-Moments 95 3.2.2 Linear Analysis of a Two-Span Continuous Plate 100 3.2.3 Nonlinear Analysis of Loosely Clamped Thin Unsymmetrically Laminated Plates 103 3.2.4 Nonlinear Analysis of Clamped Thick Unsymmetrically Lami-nated Plates 108 3.2.5 The Necessity of Equivalent Elasticity Modulii 112 3.2.5.1 Comparison of Results for the Clamped Thin Lami-nated Plate 115 3.2.5.2 Comparison of Results for the Clamped Thick Lami-nated Plate 119 3.2.5.2.1 Convergence Properties Under Different Mod-ulii 123 3.3 Analysis of Initially Deflected Plates 125 3.3.1 Initially Deformed Simply Supported Plate Loaded on Opposite Edges 127 3.3.2 Initially Deflected Plates Under Lateral Loads 130 3.3.2.1 Plate Loaded on the Concave Side 133 vi 3.3.2.2 Plate Loaded on the Convex Side 133 4 Analysis of Rectangular Containers 136 4.1 Some Previous Analyses of Rectangular Containers Under Hydrostatic Loads 137 4.2 Square Pinned Containers Under Internal Hydrostatic Pressure . . . . 139 4.2.1 Details of the Specimen 140 4.2.2 Experimental Procedure 142 4.2.3 Numerical Analysis 143 4.2.4 Results 144 4.2.5 Importance of Considering Corner Displacements and Inplane Forces 145 4.3 Eccentric Vertical Load Test on a Pinned Container 158 4.3.1 Details of the Specimen and Experimental Setup 158 4.3.2 Numerical Results 161 4.3.3 Discussion of Results 163 4.4 Analysis of Containers with Moment Resisting Corners 165 4.4.1 Analysis of a Cubic Container 165 4.4.1.1 Linear Analysis of the Container 167 4.4.1.2 Nonlinear Analysis of the Container 169 4.4.2 Analysis of a Rectangular Container 171 4.5 Summary 175 5 Mode Transition Finite Strips 177 5.1 Introduction 177 5.2 Formulation 178 5.3 Implementation 179 vii 5.4 Numerical Examples 180 5.4.1 Simply Supported Plate Under Central Point Load 180 5.4.2 Nonlinear Analysis of an Isotropic Plate Under End Load and Moment 184 6 Observations on the Calculation of Transverse Shear Stresses 186 6.1 Introduction 186 6.2 Equilibrium Equations and The Evaluation of Transverse Shear Stresses 188 6.2.1 Equilibrium Equations 188 6.2.2 The Constitutive Equations and the In-plane Stresses 188 6.2.3 Integration of Equilibrium Equations 189 6.2.4 Substitution of Displacements 190 6.3 Observations From The Equations 192 6.3.1 Inappropriateness of Linear Elements 192 6.3.1.1 Effect of Stress Gradients 194 6.3.2 Transverse Shear Stresses at the Plate Surfaces 195 6.3.2.1 The Case of a Linear Anlaysis 195 6.3.2.2 The Case of a Geometrically Nonlinear Analysis . . . 195 6.4 An Example of Transverse Shear Stress Calculation 196 6.4.1 Description of the Problem 196 6.4.2 Analysis with Fixed-Fixed Finite Strips 197 6.4.2.1 Shear Stress TXZ along the Centre of the Sixth Strip . 198 6.4.2.2 Shear Stress ryz Along The Centre Line Parallel To The j/-Axis 200 6.4.3 Analysis with Pin-Pin Finite Strips 201 6.4.3.1 Shear Stress TXZ Along the Centre Line of the Sixth Strip 208 viii 6.4.3.2 Shear Stress ryz Along the Integration Point Closest to the Centre Line Parallel to y-Axis 209 6.5 Conclusions 209 7 Conclusions and Suggestions 212 7.1 Conclusions 212 7.2 Suggestions for Future Research 213 Bibliography 215 Appendices 222 A Timoshenko Beam Mode Shapes 222 B Timoshenko Beams Under Support Displacement 226 B.l The Basic Equations 226 B. l . l Equilibrium Equations 226 B.2 The Solutions 229 B.2.1 Fixed-Fixed Beam 229 B.2.1.1 Fixed-Fixed Beam Under Support Translation . . . . 229 B.2.1.2 Fixed-Fixed Beam under Support Rotation 231 B.2.2 Fixed-Pinned Beam 232 B.2.2.1 Fixed-Pinned Beam: Translation of the Fixed Support 232 B.2.2.2 Fixed-Pinned Beam: Translation of the Pinned Support234 B.2.2.3 Fixed-Pinned Beam: Rotation of the Fixed Support . 235 B.2.3 Pinned-Pinned Beam 236 ix List of Tables 2.1 Integration Points for Integration in n 58 3.1 Linear Orthotropic Plate: Comparison of Response 76 3.2 Linear Orthotropic Plate: Convergence of Central Displacement . . . 77 3.3 Linear Orthotropic Plate: Convergence of Stresses 78 3.4 Nonlinear Response of Moderately Thick, Isotropic S-C-S-C Plate: 3 Strips and 4 Modes; 12 Points in rj 82 3.5 Linear Response of Thick Symmetrically Laminated Square Plate: Cen-tral Deflection (mm) 84 3.6 Linear Response of Thin Symmetrically Laminated Square Plate: Cen-tral Deflection (mm) 85 3.7 Linear Response of Thick Symmetrically Laminated Square Plate: Stresses (MPa) 87 3.8 Linear Response of Thin Symmetrically Laminated Square Plate: Stresses (MPa) 88 3.9 Linear Response of Thin Unsymmetrically Laminated Square Plate: Non-dimensional Central Deflection 90 3.10 Linear Response of Thick Unsymmetrically Laminated Square Plate: Non-dimensional Central Deflection 91 3.11 Central Deflection Under Combined Loads (A = l,/3 = 1) 95 3.12 Square Plate Response With and Without Additional Modes 98 3.13 Adequacy of 10 Linear Strips for Rectangular and Square Plates . . . 98 x 3.14 Rectangular Plate With End Moments: Convergence in Modes . . . . 99 3.15 Square Plate With End Moments: Convergence in Modes 99 3.16 Two-Span Continuous Plate: Convergence in Strips 102 3.17 Two-Span Continuous Plate: Convergence in Modes 104 3.18 Thin Two-Layer Rectangular Plate: Adequacy of Linear Strips . . . . 107 3.19 Loosely Clamped Thin Laminated Plates: Convergence in Modes . . 109 3.20 Clamped Thick Unsymmetrically Laminated Square Plates: Conver-gence in Displacement (w/h) with Modes I l l 3.21 Clamped Thick Unsymmetrically Laminated Square Plates: Nonlinear Displacement (w/h) Under Different Shear Correction Factors 114 3.22 Thin Two-Layer Square Plate: Effect of Different Equivalent Modulii 116 3.23 Thin Two-Layer Rectangular Plate: Effect of Different Equivalent Modulii 117 3.24 Thick Two-Layer Rectangular Plate Finite Strips: Frequency Param-eter 'fe' under Different Equivalent Modulii 120 3.25 Thick Two-Layer Square Plate: Effect of Different Equivalent Modulii Under Five Modes 121 3.26 Thick Two-Layer Square Plate: Effect of Different Equivalent Modulii Under Four Modes 122 3.27 Thick Two-Layer Square Plate: Percentage Error in Four Mode Results When Compared to Five Mode Results 124 3.28 Response of Edge Loaded Plate with Four Linear Strips 128 3.29 Response of Edge Loaded Plate with Four Quadratic Strips 129 3.30 Central Deflection, wo/h, of Plate with Load on Concave Face . . . . 133 3.31 Central Deflection, w/h, of Plate with Load on Convex Face 134 4.1 The Horizontal Force on Panel 1 Hinge under Eccentric Loading . . . 163 x i 4.2 The Load Point Deflections under Different Base Conditions 164 4.3 Square 'Fixed' Container: Effect of Penalty Parameter and Conver-gence of Displacements(mm) 168 4.4 Square 'Fixed' Container: Wall Lateral Displacement (mm) Under Dif-ferent Analyses 169 4.5 Square 'Fixed' Container: Internal Forces Along Vertical Centre Line 172 4.6 Square Thin-Walled(8mm) 'Fixed' Container: Wall Lateral Displace-ment (mm) Under Different Analyses 172 4.7 Square Thin-Walled(8mm) 'Fixed' Container: Horizontal Inplane Forces (Nx N/mm) Along Vertical Centre Line 174 4.8 Rectangular Container: Comparison of Bending Moments Mx (Nmm/mm)175 5.1 Plate with Point Load: Convergence of Central Responses With Ordi-nary Strips Having 13 Symmetrical Modes 181 5.2 Plate with Point Load: Convergence of Central Responses with Modes (Using 8 Linear Strips) 182 5.3 Plate with End Moment: Convergence of Response with Modes . . . 185 xii List of Figures 2.1 Finite Strip Reference Plane Displacements 8 2.2 Cross-sectional Displacements 9 2.3 Finite Strip Support Translation 17 2.4 Additional Rotational Mode due to Support Translation 19 2.5 Mode Shapes for Fixed-Pinned Beam 24 2.6 The Initial and Deflected Forms of the Plate 35 2.7 Compatibility at a Corner of a Container 61 3.1 Simply Supported Thick Orthotropic Plate 75 3.2 Simply Supported Plate Discretisation 75 3.3 Moderately Thick, Isotropic S-C-S-C Plate 80 3.4 Simply Supported Orthotropic Symmetrically Laminated Plate . . . . 83 3.5 Thin Simply Supported Orthotropic Plate under Combined Loading . 92 3.6 Simply Supported Plate with End-Moments 97 3.7 Two-Span Continuous Plate 100 3.8 Loosely Clamped Thin Laminated Plate 105 3.9 Loosely Clamped Thin Laminated Plates: Load vs. Central Deflection 108 3.10 Loosely Clamped Square Thin 2-Layer Plate: Load vs. Central Stress 110 3.11 Thick Unsymmetrically Laminated Square Plates: Comparison of Cen-tral Displacements 113 3.12 Loosely Clamped Thin Laminated Plate: Effect of Elasticity Modulus on Central Deflection 119 xiii 3.13 Initially Deflected Plate Under Edge Loads 126 3.14 Load Distribution in the Edge-Loaded Plate 131 3.15 Initially Deflected Plate Under Lateral Load 132 3.16 Load vs. Deflection Near Snap-Through Load 135 4.1 'Pinned' Container Model 1 140 4.2 Pinned Container: Points of Displacement Measurement Under Hy-drostatic Loading 142 4.3 Pinned Container: Displacement Boundary Conditions For Corners Under Symmetric Conditions . 144 4.4 400mm Pinned Container: Experimental and Analytical Results with 400mm of Water 146 4.5 400mm Pinned Container: Experimental and Analytical Results with 320mm of Water 147 4.6 Places of Response Sampling on the Pinned Container Wall 148 4.7 400mm Pinned Container: The Centre Line Lateral Displacements Un-der Various Analyses For Pinned Bottom 150 4.8 400mm Pinned Container: The Centre Line Lateral Displacements Un-der Various Analyses For Fixed Bottom 151 4.9 400mm Pinned Container: Centre Line Horizontal Bending Moment Under Various Analyses For Pinned Bottom 152 4.10 400mm Pinned Container: Centre Line Vertical Bending Moment Un-der Various Analyses For Pinned Bottom 152 4.11 400mm Pinned Container: Centre Line Horizontal Axial Force Under Various Analyses For Pinned Bottom 153 4.12 400mm Pinned Container: Centre Line Vertical Axial Force Under Various Analyses For Pinned Bottom 153 xiv 4.13 400mm Pinned Container: Centre Line Horizontal Bending Moment Under Various Analyses For Fixed Bottom 154 4.14 400mm Pinned Container: Centre Line Vertical Bending Moment Un-der Various Analyses For Fixed Bottom 154 4.15 400mm Pinned Container: Centre Line Horizontal Axial Force Under Various Analyses For Fixed Bottom 155 4.16 400mm Pinned Container: Centre Line Vertical Axial Force Under Various Analyses For Fixed Bottom 155 4.17 Pinned Bottom Pinned Containers: Lateral Displacement Profiles at Different Heights from Base, Under Movable Corner Nonlinear Analysis 156 4.18 Fixed Bottom Pinned Containers: Lateral Displacement Profiles at Different Heights from Base, Under Movable Corner Nonlinear Analysis 157 4.19 Pinned and Fixed Bottom Pinned Containers: Inward Movement of the Corners in the Direction of a Coordinate Axis 157 4.20 Pinned Container Under Vertical Eccentric Loading 158 4.21 Loading Arm Arrangement 160 4.22 Eccentric Loading of Pinned Container: Comparison of Response . . 166 4.23 A Panel of Cubic 'Fixed' Container 168 4.24 Horizontal Moment Mx in the 'Fixed' Containers with Different Wall Thicknesses 173 4.25 Horizontal Force Nx in the 'Fixed' Containers with Different Wall Thicknesses 173 4.26 Moment Resisting Rectangular Container 174 5.1 Plate Under Eccentric Loading 184 6.1 Positions of Stress Sampling 198 xv 6.2 Linear Fixed Strips: Convergence of rxz With Modes 202 6.3 Cubic Fixed Strips: Convergence of TXZ With Modes 202 6.4 Fixed Strips With 9 Modes: Change of TX2 With Strip Type 203 6.5 Linear Fixed Strips: Convergence of Residue Percentage of TXZ With Modes 203 6.6 Cubic Fixed Strips: Convergence of Residue Percentage of TXZ With Modes 204 6.7 Fixed Strips With 9 Modes: Change of Residue Percentage of TXZ With Strip Type 204 6.8 Linear Fixed Strips: Convergence of ryz With Modes 205 6.9 Cubic Fixed Strips: Convergence of ryz With Modes 205 6.10 Fixed Strips With 9 Modes: Change of ryz With Strip Type 206 6.11 Linear Fixed Strips: Convergence of Residue Percentage of ryz With Modes 206 6.12 Cubic Fixed Strips: Convergence of Residue Percentage of ryz With Modes 207 6.13 Fixed Strips With 9 Modes: Change of Residue Percentage of ryz With Strip Type 207 6.14 Pinned Strips With 9 Modes: Change of TXZ With Strip Type 210 6.15 Pinned Strips With 9 Modes: Change of Residue Percentage of TXZ With Strip Type 211 6.16 Pinned Strips With 9 Modes: Change of ryz With Strip Type 211 B.l Element of a Timoshenko Beam 227 B.2 Rotation and Shear Strain in a Timoshenko Beam 227 B.3 Fixed-Fixed Beam Under Support Translation 229 B.4 Fixed-Fixed Beam Under Support Rotation 231 xvi B.5 Fixed-Pinned Beam Under Fixed-Support Translation B.6 Fixed-Pinned Beam Under Pinned-Support Translation B.7 Fixed-Pinned Beam Under Fixed-Support Rotation xvii Acknowledgements I wish to thank my supervisor Dr. R.O. Foschi for his invaluable guidance, encour-agement and interest with regard to this work, for the many hours spent on valuable discussions and proof reading, and for his patience. I also thank Dr. M.D. Olson for his encouragement and valuable discussions which challenged me into solving some difficult issues. I wish to thank them and Dr. D.L. Anderson for going through my thesis and making constructive suggestions with regard to it. Also I wish to thank Drs. N.D. Nathan and B. Madsen for the advice and encouragement provided at various times. I also wish to thank all others, including my late parents, who taught me in varying capacities since my childhood. With this achievement being the cumulative effect of everything learnt during my whole life, this success is their's too. The University of British Columbia granted me a University Graduate Fellowship and the National Science and Engineering Research Council provided me with a Re-search Assistantship for the duration of this programme. It is this financial assistance which made it possible for me to undertake this study. With regard to the experimental work carried out, I wish to thank Connie Nicoletti for her participation during her 1989 summer employment and technicians Bernie Merkli and Paul Symons for their help and support. Pat Sheehan, Flora Lew and Anne Miele helped me in various capacities, and I thank them too. My friends here, especially Roy and Laura Wijenayake, Damika Wicremasinghe, Malik Ranasinghe, Upul Atukorala and Dan Dolan are thanked for their invaluable friendship, support and help provided at various times. Also Tim Battle, Dilip xviii Athaide, Angela Dempster, Reza Vaziri, Willie Yung and Fai Cheung are among those whose friendship and company I valued. It is only the need for brevity which prevents the mentioning of many others. I also thank the community of St. Mark's College at the University, especially Frs. Leo Klosterman and Paul Burns. The presence of this community and their services was a great solace to me. I could go to its Chapel knowing there is the One who will always comfort me, even during the most difficult of times. I cannot fail to thank my sisters and brothers who always provided me with their love and invaluable support in various forms. Only they could and would have provided it, and to them I will be ever grateful. Finally I wish to thank my fiancee, Dushanthi, who showed me her love and understanding during the short period I have known her. Her presence in my life gave added meaning to this whole endeavour. xix Dedication To the memory of my parents whose sacrifices, love and encouragement made it possible for me to study xx Chapter 1 Introduction This thesis describes the formulation and verification of a new finite strip for the ge-ometrically nonlinear analysis of orthotropic laminated plates with support displace-ment, using the Mindlin(1951) theory for thick plates and the von Karman(1910) theory of large deflections. Finite strips with free ends are not considered. This is an expansion of the work of Dawe and co-workers(Roufaeil and Dawe 1980, Azizian and Dawe 1985a, Dawe and Azizian 1986). The research was motivated by the need for a model for the static and dynamic analysis of large containers, with moderately thick solid or laminated sides, used in the transportation of liquids and granular material. This thesis focuses on the static analysis procedure. The walls of these containers may be classified as 'moderately thick' in the sense of plate terminology, thus requiring a shear deformation theory for layered material, while still being flexible as to need large deformation theory and the consideration of the displacements of the corners of the container. The analyses were implemented in computer codes written in FORTRAN, and these are completly self-contained. 1 Chapter 1. Introduction 2 1.1 Contributions of this Thesis The contributions made by this research to the literature fall into several categories and are described below. 1. Introduction of the ability for the finite strip supports to move in all the degrees of freedom. This makes it possible to extend the use of the finite strip to the analysis of the following types of problems. (a) Continuous and stiffened plates. (b) Plates with applied end forces (both bending moments and direct forces). (c) Plates on flexible foundations and plates with elastically or inelastically restrained supports. (d) Containers of polygonal cross-sections. Only rectangular containers are considered here, but extension into non-rectangular shapes is a simple matter. It is believed this is the first instance with nonlinear results from the analyses of flexible rectangular containers using any procedure, while considering the movement of the corners. As shown in Chapter 4, this analysis is able to predict a compressive horizontal force at certain locations in the walls under larger deformations. This cannot be predicted by other simple models. 2. Extension of the Mindlin finite strips to the analysis of laminated composite plates, including those laminated unsymmetrically, by the definition of equiva-lent elasticity modulii to find the basis functions. 3. Extension of Marguerre(1938) theory of initially deflected plates to Mindlin plates, and the subsequent implementation with finite strips. Although the Chapter 1. Introduction 3 former was carried out independently, it was realised that this had also been discussed recently by Minguet et a/.(1989). The use of finite strips for initially deflected plates had been made by Hancock(1981) using the thin plate theory, but the work in this thesis is believed to be the first application using Mindlin strips. 4. Introduction of the concept of Mode Transition Finite Strips for the analysis of problems with loads that extend only in a small area of the plate. These make it possible to use only a few finite strip modes in areas of low stress concentration, resulting in a saving of computer time which can be appreciable in the case of repeated analyses as in Monte Carlo Simulation. This finite strip corresponds to the transition elements in finite element analysis, and does not require extra computer work for the formulation. 5. A discussion on the determination of transverse shear stresses using the equi-librium equations and the displacement solution is made. Among other conclu-sions, it is shown that a finite strip with at least quadratic interpolation should be used for this purpose. A disadvantage, at present, of the new finite strip is the large bandwidth that occurs in its use with continuous plates and containers. This is due to the coupling of all the modes and the support degrees of freedom of each finite strip. Here it may be mentioned that with no support movement, it is possible to obtain uncoupled modes only in the case of the simply supported strips. Thus, there is no disadvantage arising simply because of the introduction of the support displacements. In the case of a continuous plate, the bandwidth could be reduced by having the strips parallel to the intermediate supports, instead of perpendicular to them as done in this thesis. But, here it had been considered so only with the idea of verifying the strips for Chapter 1. Introduction 4 the purpose of using them subsequently in stiffened plate analyses. In the case of containers, the specified use of the finite strips(horizontal) cannot be said to have any disadvantage with respect to the vertical placing of the strips as some researchers have done (Golley et al, 1987) as both present similar problems with coupling, except for the advantage of the latter when the tank brim contains no degrees of freedom. The main advantage of the horizontal finite strips will be in the analysis of a tank with a contained material that has to be discretised, for example a fluid or a granular material under dynamic conditions. Then, if the displacements of this material can be expressed as a Fourier series in the horizontal direction, the conditions of compatibility between the tank walls and the material can be easily satisfied. 1.2 Thesis Organisation The arrangement of the sequel of this thesis which comprises seven chapters will be as follows. • Chapter 2: This chapter will present the basic formulation of the finite strip stiffness matrices and load vectors. Penalty Function method for the extension of the finite strips to the analysis of rectangular tanks with moment resisting corners and the formulation necessary for their extension to the analysis of plates with elastically restrained supports are also presented. • Chapter 3: This chapter will deal with the verification of the basic finite strips by the use of already available results. • Chapter 4: The application of the finite strip for the analysis of rectangular containers will be presented in this chapter. Some of these applications have been verified experimentally. The details of the experiments are provided. Chapter 1. Introduction 5 • Chapter 5: This will introduce the concept of Mode Transition Finite Strips. It is shown how such finite strips can be formulated and applied without additional tedious computer work. Two examples are provided to show the advantage of these strips. • Chapter 6: This chapter will deal with the calculation of transverse shear stresses using the equilibrium equations and the displacement solution. A ba-sic difference between the transverse shear stress distributions under linear and nonlinear conditions is shown. Also it is shown that the linearly interpolated finite strip and the linear rectangular finite element used in Mindlin plate anal-yses cannot be used in such shear stress calculations. • Chapter 7: This chapter will present concluding remarks and suggestions for future research. Chapter 2 The Formulation of the Finite Strip This chapter deals with the theoretical displacement formulation of the finite strip and some aspects of its computer implementation. The finite strip being formulated belongs to the moderately thick plate category, but can be used also to analyse thin plates. The kinematics of deformation will be assumed according to Mindlin's The-ory(Mindlin, 1951). Thus, in contrast to the classical thin plate theory, the normals to the plate middle plane before deformation need no longer be so after deformation, while still remaining straight. The nonlinear effects are considered according to the von Karman theory(von Karman, 1910). Initial deflections of the plate are considered in accordance with the assumptions of Marguerre(1938) by extending them to Mindlin plates. These assumptions, where needed, will be explained at the appropriate places. This chapter is divided into several main sections as follows. The first section will provide a brief introduction to the shear deformation theories of laminated plates and to finite strips. The second and third sections will introduce the bases and shape functions used in expressing the displacements for the plate reference plane as func-tions of the two-dimensional plate coordinates and nodal line variables. The fourth section presents the kinematic relationships for the plate strip, while the standard variational formulation to obtain the stiffness matrices and the load vector is pre-sented in the fifth section. The sixth section presents the expression of the stiffness matrices in terms of the constitutive properties and the displacement interpolation functions. The numerical integration and solution procedure will be described in the 6 Chapter 2. The Formulation of tie Finite Strip 7 seventh section. Penalty function formulation used in the application of the finite strip to the analysis of rectangular containers with moment resisting corners is pre-sented in the eighth section. The ninth and the last section presents the additional theoretical development necessary for the application of the finite strip to the analysis of plates with elastically restrained edges. As far as possible the formulation is presented with the use of matrices, instead of the standard summation procedure used by others in the presentation of finite strip theory. It was felt that this would make the mathematics easier to follow, and also give a formulation that can be directly related to the notation used in standard finite element theory. This formulation is limited to the case of rectangular laminated plates made up of linear elastic, specially orthotropic laminae. The laminae are to be arranged in a cross-ply pattern, with the elasticity axes parallel to the geometric axes of the plate. Of course, angle-ply laminates can easily be considered by the use of a transformation matrix, but it is not considered to be within the bounds of this thesis. The following notation is followed with regard to the coordinate systems. (£, TJ, C) is used as the local coordinate system of a finite strip, (x, y, z) is used to express the local coordinates of a panel, with the directions of the coordinates corresponding to the local system of a finite strip. The global coordinates are denoted by (X, Y, Z). The use of a bar over a coordinate implies that it is a non-dimensional value. 2.1 Introduction to the Basic Methods Used This section presents a brief review of works of others with respect to the numerical analysis of shear deformable layered plates using the Mindlin theory. As a large number of good reviews of recent origin exists, this description is not desired to be thorough in its citings. Also given is a brief description of finite strip methods used Chapter 2. The Formulation of the Finite Strip 8 Figure 2.1: Finite Strip Reference Plane Displacements in the analysis of shear deformable plates. More citings of the works of others will be made elsewhere whenever appropriate. 2.1.1 Mindlin's Theory As mentioned earlier, the present finite strip formulation is based on the theory of Mindlin(1951) for moderately thick plates. This theory is the two-dimensional coun-terpart to the shear deformation beam theory presented by Bresse(1859)a and Tim-oshenko(1921 and 1922), and which is now known as the Timoshenko beam theory. According to this theory, the normals to the plate middle plane are considered to remain straight after deformation, but are no longer required to remain normal as in the classical Kirchoff assumptions. Mindlin's theory is the displacement based counterpart to the earlier presented stress formulation of Reissner(1945,1947) which Jquoted from Warburton(1983) and Kranys(1989) Chapter 2. The Formulation of the Finite Strip 9 Figure 2.2: Cross-sectional Displacements assumed a constant transvese shear stress at a section. A finite strip and its reference plane displacement components are shown in Fig-ure 2.1. Then, according to the assumptions of Mindlin's theory, the displacements at a cross-section of the plate can be described as shown in Figure 2.2. Because the cross-sectional rotations <f> and tp are considered as individual degrees of freedom, the transverse shear strains are given by 7 « = (2.1) dw . ,n . 7* = (2.2) where subscripts on 7 refer to the plane of shear (see Figure 2.2). Chapter 2. The Formulation of the Finite Strip 10 Then as can be seen from Figure 2.2, the displacements at any point in the plate can be written in terms of the reference plane displacements as U — UQ — Z(j) v = v0 — zip (2-3) w = Wo As mentioned previously, although it is a 'moderately thick plate' formulation, Mindlin's theory can be used also in the numerical analysis of thin plates where the transverse shear deformations are negligible. But, a problem that arises then is the phenomenon of 'shear locking' where the plate becomes stiffer due to the domination of the transverse shear strain energy, over that of flexure. This phenomenon, and ways to avoid it, has now been resolved. It is a consequence of the displacement field which provides transverse shear strains comparable (in energy) to flexural strains even where the former should be negligible. It can be eliminated by special precautions such as reduced integration or the use of higher order displacement fields. An advantage the Mindlin formulation offers over the classical approach is the need of only C° interelement continuity of the displacement field. It is this low order of compatibilty needed that made it possible to easily introduce the 'mode transition finite strip' presented in Chapter 5. Two notable limitations of this First Order Shear Deformation Plate Theory are the constant transverse shear strains that occur at a cross-section, and the non-satisfaction of shear-stress continuity at layer interfaces of a laminated plate. In order to overcome these shortcomings various researchers have presented higher order theories with different assumptions for the displacement field across the plate thick-ness. A generalisation of these displacement based theories has been provided by Reddy(1989). Chapter 2. The Formulation of the Finite Strip 11 Recent reviews of static analyses of laminated plates using shear flexible methods have been presented by Reddy(1985), Chia(1988), Noor and Burton(1989), and Ka-pania and Raciti(1989). Various methods, both analytical and numerical, have been described and the interested reader is asked to refer to them for details. 2.1.2 T h e U s e o f t h e F i n i t e S t r i p M e t h o d The finite strip method is a semi-analytical method introduced by Cheung(1968). The basic idea of the method is to use an analytical expression, usually a Fourier series, in the longitudinal direction of the strip, while using a finite element type piecewise displacement approximation in the transverse direction. An advantage of the method with respect to the linear analysis of plates is the uncoupling of the modes of simply supported strips. Although this does not occur with other boundary conditions, the ease of data preparation and the need for a lesser number of variables than in the case of a finite element analysis remain advantageous. The extension of the method to the analysis of classical plates with geometrical nonlinearities is due to Graves-Smith and Sridharan(1978). A review of the literature on the use of the finite strip method has been provided by Wiseman et o/.(1987). The application of the method to the analysis of shear deformable plates has been carried out by several researchers, among them Dawe and co-workers (Dawe 1978, Roufaeil and Dawe 1980, Azizian and Dawe 1985a and 1985b, Dawe and Azizian 1986) who introduced the non-simply supported Mindlin finite strips by the use of Timoshenko beam mode shapes. Of the above, Azizian and Dawe(1985b) applied the method to the geometrically nonlinear analysis of symmetrically laminated plates with pinned ends. These have also been applied to the vibration analysis of shear deformable symmetrically laminated plates by Craig and Dawe(1986 and 1987). It is their method that this thesis extends to the geometrically nonlinear analysis of Chapter 2. The Formulation of the Finite Strip 12 asymmetrically laminated composite continuous plates with support deflections and rectangular containers. For completeness it may be added that notable applications of the simply sup-ported Mindlin finite strips had been made by Mawenya and Davies(l974), Benson and Hinton(1976), Hinton(1977) and Ohate and Suarez(1983a and 1983b). 2.1.2.1 The Finite Strip Method in Continuous Plate Anal-ysis The application of finite strips to the analysis of continuous plates was made by several researchers using the classical theory. All applications were with respect to geometrically linear problems. Only Golley and Petrolito(1982) considered support displacements, but it was only with respect to the lateral translation. A description of these methods is given in the following paragraphs. M.S. Cheung et a/.(1970) used finite strips that spanned the whole continuous plate. The flexibilty method was combined with the stiffness method to achieve zero translations at the intermediate fine supports by the application of reactive forces. Wu and Cheung(1974) introduced the use of basis functions applicable throughout the span by the use of continuous functions obtained for a continuous beam. The application has been to the dynamic analysis. This method is said to be uneconom-ical for static analysis, especially under concentrated loads. Ghali and Tadros(1974) used single span strips in which interspan compatibility was satisfied only at discrete locations. They used the principle of superposition by a combination of flexibil-ity and stiffness methods, thus, preventing its use in nonlinear analyses. A similar method with superposition where the two methods were combined was presented by Szilagyi(1974). The method of Golley and Petrolito(1982) provided interelement com-patibility at the intermediate fine supports with respect to certain weight functions. Chapter 2. The Formulation of the Finite Strip 13 The bending moments along such a support were considered to be distributed accord-ing to an assumed piecewise linear function, and the transverse displacements along it were approximated by piecewise Hermite polynomials. The rotational compatibility and effective shear equilibrium at the strip junctions were satisified approximately by the use of the weighted residual method. Still, not enough degrees of freedom were provided to allow the consideration of, for example, a stiffener beam which may make the support translate both in the longitudinal and transverse directions of the strips, and also may provide a rotational constraint normal to the plate boundary. Another development was the 'finite strip-element'(Wang and Zhang, 1986; Golley and Petrolito, 1987) formed by the combination of a thin-plate rectangular finite element with the mode shapes of a simply supported beam. The method of latter authors provided partial uncoupling of the equations, but this was not so with the method of the former authors. Golley and Petrolito op. cit. still had to use La-grange's multiplier method to satisfy displacement conditions at fixed boundaries. M.S. Cheung and Li(1988a and 1988b) presented a finite strip method that used the beam functions of the corresponding continuous beam. Thus, a single finite strip ex-tended throughout all the spans. This method still lacked the ability to consider the translation of the supports or the rotation of a stiffener beam. The basis functions of the lateral w displacement and their derivatives were used as those for the transverse v and longitudinal u inplane displacements, respectively. The inplane displacements were considered for the purpose of analysing folded plates, and did not involve non-linear geometric considerations. Due to the presence of inplane displacements, all the modes were coupled in the stiffness matrix. Considering the above, it may again be stated that the present work is the first known instance of a finite strip displacement formulation that provides • The consideration of translation and rotation of fine supports which can be Chapter 2. The Formulation of the Finite Strip 14 elastically restrained explicitly or through structural members. • Continuity of displacements across intermediate line supports of a continuous plate, without the need of any approximation. • Ability to consider geometrical nonlinearities in a continuous plate problem. This section deals with the basic interpolation of the degrees of freedom used in the reference surface of the plate finite strip. The reference surface is usually considered as the middle plane of the plate due to the advantage of non-coupling between the inplane and bending deformations in the case of a linear analysis of a plate with homogeneous cross-section and also in the case of a symmetrically layered cross-ply plate. A finite strip is considered to have the five displacement degrees of freedom u 0 , v0, w, <f> , and xp, denned at its reference plane as shown in Figure 2.1. In the case of a linear bending analysis with no inplane-bending coupling, the inplane displacements u0 and v0 need not be considered. All the degrees of freedom are assumed to vary along the length of the strip according to basis functions of the coordinate £. These bases can be algebraic, trigonometric or hyperbolic functions, or their combinations. In 77, the transverse direction, they are assumed to vary according to a simple interpolation law. In the following, for simplicity, non-dimensional local strip coordinates £ and fj defined as, 2.2 Basic Displacement Interpolation 0 < t < 1 (2.4) Chapter 2. The Formulation of the Finite Strip 15 V = \ - 1 < 7 ? < 1 (2.5) will also be used. According to the above assumptions the displacements at any point ( £ , 7 / ) on the finite strip reference plane can be written as, M(,V) = {N(v)}T{UA}g^C) + {N(r,)}T{UB}g^C) M + Y,{N(r,)}T{um}g™(ti) (2.6) m = l MM = {N(v)}T{vA}g?(t) + {N{v)}T{vB}g*(t) M + E W r K i a ) (2-7) m = l *>{t,v) = {N(v)}T{WA}giU) + {N(V)f{wB}gB(0 M + EW)fK}ff:(™,o m = l + {N(V)}T {w*>A} gtA (0 + {N(V)}T {w*-B } gtB{() (2.8) <Kt,v) = Wi?)} T K}rf(0 + W i / ) } r K } r f ( 0 M + £ W i / ) } r W m } # ( 0 m = l 1>{t,v) = Wi? ) } T K }4 (0 + W»/)} r {^ B } r f (0 M m = l where A and 5 represent the £ = 0 and ( = I ends of the strip, respectively. These will be also referred to, respectively, as the Left and Right ends of the strip. {Ar(r/)} Chapter 2. The Formulation of the Finite Strip 16 is the interpolation vector between nodal lines, and is a function of rj only. This interpolation can be linear, quadratic or cubic in 77. The basis functions refered to above, which are functions of £ only, can be identified as follows. The subscripts and the superscripts of certain items are omitted for simplicity. The subscripts refer to the degrees of freedom, while the superscripts refer either to the mode (m) or to the end of the strip. Thus, 9w(€)i '• h^e displacement in w due to a unit movement at support A in the displacement to (only). (See Figure 2.3) 9w(() '• a s above, due to a support B displacement. 9w(0 '• the modal shape function for mode'm' in w, and 9wA(£) '• shape function introduced as an 'additional' mode in w when support A has a rotation <p. (These 'additional' modes are required to prevent the structure from being too stiff.) The other degree of freedom quantities can be identified in relation to the above. The degrees of freedom vectors, with which the interpolation vectors are to be post-multiplied, are defined as follows. As before, only the w terms will be explained, with the others becoming obvious through them. j WA j : the amplitudes of the degree of freedom at the support A, arranged in the order of the nodal fines. J V J / B J : a s above, at support B. {wm} : the amplitudes of the mth mode arranged in the nodal hne order. \w4>,A I . j.]je amplitude vector of the additional mode Chapter 2. The Formulation of the Finite Strip 17 Figure 2.3: Finite Strip Support Translation due to rotations at support A. {uAB} : as above, but due to rotation at support B. It is to be noted that in the case of a beam, the quantities <$>A of Equation 2.9 and xi^,A of Equation 2.8 should have the same value, but in this plate formulation where the basis functions have been obtained from the beam theory, they are kept independent and the latter is allowed to find its own value through the variational process. The interpolation vector in the r\ direction, {N}, which is based on equally spaced nodal lines, is as follows. • Linear Interpolation Ni = (l-v)/2 N2 = (1 + T/)/2 Chapter 2. The Formulation of the Finite Strip 18 • Quadratic Interpolation N, = rj(fj-l)/2 N2 = (l+fj)(l-rj) N3 = fj{fj+ l)/2 • Cubic Interpolation N l = ^ ( - ^ + ^ + ^ 9 - 1 / 9 ) 27 N3 = -(-f-f/3 + fj +1/3) N4 = 1(^ + ^ -^/9 - 1/9) As can be appreciated from the above formulae, the linear, quadratic and cubic strips have two, three and four nodal lines per strip, respectively. 2.3 The Basis Functions for Displacements This section presents the details of the basis functions introduced in the above section. Three sets of bases will be described, viz. the basic modes obtained by the use of the normal modes of vibration of a corresponding beam, the bases due to support displacements and the bases occuring as the 'additional' modes. The basic modes refer to the series used in the expression of the displacement along the strip with zero support displacements. The bases due to support displacements are used to express the displacements in a particular degree of freedom when the same degree of freedom is moved at a support. The additional modes are used to express the displacements in w that occur in the span when a support is rotated, or the rotations that occur Chapter 2. The Formulation of the Finite Strip 19 Figure 2.4: Additional Rotational Mode due to Support Translation in the span when a support is translated in w. Figure 2.4 shows the rotations that occur in the span when support A is translated laterally. It is to be noted that the basis functions should satisfy all the displacement bound-ary conditions of the problem, unless it is allowed for in the variational formulation. They are satified by those presented here. 2.3.1 The Basic Modes 2.3.1.1 Introduction As introduced by Roufaeil and Dawe(1980), for Mindlin finite strips of isotropic plates, the basic modes for the w and <f> degrees of freedom can be considered as the respective normal modes of a corresponding Timoshenko beam (Timoshenko 1921,1922), as they facilitate the satisfaction of the displacement boundary conditions of the Mindlin plate. These modes shapes had been obtained by Traill-Nash and Collar(1953) and Huang(1961), and were used by Dawe and co-workers (Dawe 1978, Roufaeil and Dawe Chapter 2. The Formulation of the Finite Strip 20 1980, Azizian and Dawe 1985a, Dawe and Azizian 1986, Craig and Dawe 1986) who expressed them in a slightly different form. Except in the case of a simply supported beam, the eigenvalues of these mode shapes are not in closed form and depend on the beam properties. Therefore, they have to be obtained for each new problem by the solution of the transcendental frequency equation. Only then can the mode shapes be calculated. These basic mode shapes and the related equations, including the frequency equation, for a fixed ended uniform isotropic Timoshenko beam are given in Appendix A, having been obtained from the work of Roufaeil and Dawe(1980). Here it may be noted that the Timoshenko beam would give rise only to the basis functions for w and cj), leaving the bases for the remaining three degrees of freedom to be obtained separately on the consideration of the displacement patterns. Based on the studies of Dawe(1978), Roufaeil and Dawe(1980) used the same basis functions for ijj as for w in their isotropic plate analyses. Following that, the same mode shapes as those of w were used for ip in this formulation, even in anisotropic problems. The agreement with the available results supported this choice. The basic modes for the inplane displacements were selected by the observation of the probable displacements and using the judgment of the author. 2.3.1.2 Equivalent Elasticity Modulii for Laminated Plates As mentioned above, the frequency equation for a non-simply supported Timoshenko beam depends on the beam properties. These mode shapes of the isotropic beam theory were used also in the present extension of the Mindlin finite strip to the nonlinear analysis of unsymmetrically laminated plates. This has been done also by Craig and Dawe(1986), but not much details are available as to what properties were used in the determination of mode shapes. Here, in order to be able to use the same equations as given in Appendices A and B, equivalent elasticity modulii were Chapter 2. The Formulation of the Finite Strip 21 defined for anisotropic plates as given below. The idea was to use a consistent set of values that depend on the elastic properties of the constituent layers. The plates are considered to be made up of specially orthotropic laminae placed in a cross-ply pattern with the plate axes and the principal elasticity axes of the laminae being in the same directions. 2.3.1.2.1 Definition of Equivalent Modulii Consider a beam formed by a portion of the plate spanning the direction of the finite strips. This beam will be compared with another beam of same dimensions, but with uniform elastic properties. The properties of the latter beam, as defined below, are the 'equivalent' modulii. For an orthotropic single layer plate the Young's modulus and the transverse shear modulus in the plane of the beam are used as the equivalent values. In the case of a layered plate, equivalent Young's modulus is calculated by con-sidering the deformation of a unit length of each beam under the action of a constant bending moment. The equivalent value is considered as that which gives the same flexural strain energy in the uniform beam as that in the beam with actual plate properties. In the case of the transverse shear modulus the equivalent value is defined similarly by considering the transverse shear strain energy under a given transverse shear force. In the above calculations, the cross-sections of both beams are assumed to remain straight after deformation. The equivalent modulii calculated as above can give rise to Poisson's ratios which are beyond the range for isotropic material. Still, it is better to calculate the mode shapes in this manner than with arbitrary values, as the modal pairs in w and <j) would determine the transverse shear deformation patterns that occur. In order to see the actual importance of these equivalent values several nonlinear analyses were carried out on two-layer cross-ply plates, the results of which are presented in Chapter 3. Chapter 2. The Formulation of the Finite Strip 22 The obtained results pointed to the need to use the equivalent values as a consistent and logical method, when compared to much more easily determined values such as the means of the individual layer modulii. The following presents the basic mode shapes used with a pin-pin strip. The modes used under symmetric conditions are denoted as 'symmetric'. When symmetry is ab-sent both symmetric and unsymmetric modes have to be used, and these are denoted as 'unsymmetric'. This notation will be followed also with the other types of strips to be described. For the pin-pin end condition, the Timoshenko beam mode shapes for the w and (j) displacements are as described below. 2.3.1.3 Mode Shapes for a Pin-Pin Strip sin m7r£ (unsymmetric) m = 1,..., M w cos mir( (unsymmetric) cos(2m — l)7r£ (symmetric) m — 1,... , M u sin m7r£ (unsymmetric) sin 2m7r£ (symmetric) > m = 1,... ,M v the same as the w mode the same as the w mode Chapter 2. The Formulation of the Finite Strip 23 2.3.1 .4 M o d e S h a p e s for a F i x e d - F i x e d S t r i p Details with regard to the determination of normal modes for the fixed-fixed Timo-shenko beam are given in Appendix A. Let these mode shapes be denoted by Wm(£) and 3>m(0 for the mode 'm' of the degrees of freedom w and respectively. Then, if bm is the mth root of the frequency equation (Equation A.4), the mode shapes are Wmtf) = Ssin bmB( + cos bm[3£ - - i£s inh bma£ - cosh bma( (2.11) « 2 $m(0 = Ski cos fcm/3£ — ki sin bmfl( — ki8 cosh fcmac; — k2 sinh bma£ (2.12) The other terms appearing in the above two equations are defined in Appendix A. Then the basic modes for the finite strip can be given as w Wm(l) (unsymmetric) W 2 m - i ( £ ) (symmetric) J m = 1,... ,M 4> <J>TO(£) (unsymmetric) $ 2 m - i ( £ ) (symmetric) m — 1, . . . ,M u : the same v : the same ip : the same as the (j) mode as the w mode as the w mode If desired, for study purposes, the u and v modes can be selected as the sine series given for the pin-pin case. This can be done by the use of the parameter IUCODE in the computer code. Chapter 2. The Formulation of the Finite Strip 24 Beam of Concern I I Figure 2.5: Mode Shapes for Fixed-Pinned Beam 2 .3 .1 .5 M o d e S h a p e s f o r a F i x e d - P i n n e d B e a m In the case of the fixed-pinned beam it is unnecessary to find the modes shapes sepa-rately as they correspond to the antisymmetric modes of a fixed-fixed beam of length 2/(Roufaeil and Dawe, 1980), where / is the length of the beam of concern (Fig. 2.5). Then, if the Timoshenko beam mode shapes of a beam of length 21 are denoted by W^(f) and $„(£ ) for the mth mode, then the basic modes for a finite strip with 'fixed' conditions at the end 'A' can be denoted as w: W2!(0 m = l , . . . , M <f> : *JK(«f) m = l , . . . , M it : sinm7r£ m = l , . . . , M v : the same as for w mode tp : the same as for w mode Chapter 2. The Formulation of the Finite Strip 25 2.3.2 Support Basis Functions and Additional Modes This section describes the basis functions due to support motion and, when they occur, the consequent 'additional' modes. As mentioned earlier, the support bases for w and <j) are obtained by finding the shape of the Timoshenko beam when the particular degree of freedom is displaced by a unit amount, while preventing the others from moving. The bases for the other three degrees of freedom, viz. u, v and ip, are considered in the simplest possible way by linear interpolation between the supports. 2.3.2.1 Additional Modes As discussed previously with regard to the basic modes of the finite strip, each w mode should have its corresponding <f> mode. Otherwise there will be very large transverse shear strains in the section, and the member will become stiffer. Similarly, when only one of w or (f> degrees of freedom is moved at a support, the other degree of freedom (</> or w, respectively) has to have non-zero values within the span in order to limit transverse shear strains (See Figure 2.4). Thus, for proper modelling, this means the introduction of a certain 'additional' mode whenever a support basis in w or <f> is introduced. The occurence of a very stiff member in the absence of additional modes will be shown in Chapter 3 with a numerical example of a plate with applied end moments. The solutions of the Timoshenko beam to obtain the support bases and the additional modes are given in Appendix B for the cases of interest. It is to be noted that in the analysis of a Timoshenko beam, the values of the lateral displacement and the rotation within the span due to the displacement of either one of the degrees of freedom w and </>, are related. But, as mentioned previously, in the finite strip analysis the shapes obtained from the beam theory will be considered Chapter 2. The Formulation of the Finite Strip 26 as independent. In other words, the displacement patterns were sought solely for the purpose of obtaining the shapes. In this manner, the basis for the support movement will have the support displacement as its amplitude, while the additional mode will have an independent amplitude given by the variational process. 2.3 .2 .2 P i n - P i n S t r i p The support basis functions for a pin-pin finite strip are as follows. at = 9t = st = 4 = l~t (2-13) 9u = 9v = 9* = 9? = £ (2.14) As described earlier, £ is the non-dimensional length coordinate of the finite strip or Timoshenko beam, and can have a value ranging from zero to unity. There is no basis for <b as it is not a support degree of freedom of this strip. But due to the w movement there is an additional mode given by 4 = i 2.3 .2 .3 F i x e d - F i x e d S t r i p The bases and the additional modes to be used with a fixed-fixed strip are given here. The support bases for u, v and rb are not presented as they are taken to be the same as that used for the pin-pin strip described above. 2.3.2.3.1 Bases due to Translation of Supports The bases due to the lateral displacement of supports A and B are given below. In each basis function the subscript (2.15) Chapter 2. The Formulation of the Finite Strip 27 refers to the degree of freedom. If a single superscript is present, it refers to a support which is displaced in the same direction as the subscript degree of freedom. If there are two superscripts, then the function is an additional mode. The first superscript refers to the degree of freedom, while the second refers to the support that is moved. This nomenclature is applicable also in the subsequent presentations. The functions denoted by W and P are those derived in Appendix B. In finding the modes, the beam is considered to be of the same height as the thickness of the plate. A is its cross-sectional area and I is its second moment of area about the centroidal axis. E and G are the equivalent Young's modulus and shear modulus, respectively. £ is the beam length coordinate, ranging from zero to I. & = Ktf(0 = 97A = 97B = - ^ ( 0 = Pt{() = ( | - l ) (2-18) where, I2 2 C = — . +—— 6EI K2AG and K 2 is the shear correction factor for an isotropic beam. In the above, it may be noted that the additional rotational bases due to the translation of each of the two supports are of the same shape. Thus, in order not to have a singular stiffness matrix only one of the two modes is to be used in an analysis. IK2AG (2.16) 2(Z-0 (/ IK2AG if *i 6EI + 1 (2.17) Chapter 2. The Formulation of the Finite Strip 28 2.3.2.3.2 Bases due to Rotation of Supports The basis functions due to the rotation of supports are as follows. 9* c - IL 2EI (2.19) 9$ Cl + I2 K2AG 2EI ZEI (2.20) 9tA 2EI K2AGP K2AGIC R R EIC1 J _ 2R 3P + ZEIIC (2.21) i + 2EI AR n2AGl2 K2AGIC\ e-o2 R .EICl where, in addition to the previous definitions, + (*-Oa J _ 2R 312 + 3EIIC\ (2.22) R I EI 3 K2AGI 2.3.2.4 T h e F i x e d - P i n n e d S t r ip In this case also the support bases for u, v and ip are not presented as they are taken to be the same as that used for the pin-pin strip. The basis functions used here have been derived in Appendix B. The pinned end is considered to be the end B as defined earlier. Chapter 2. The Formulation of the Finite Strip 2.3.2.4.1 Bases due to Displacement of Supports 29 A 9w 1 + '21 IK2AG 6EI \l 1 - 3 ^ (2.23) B W»(£) = 2 -C2i IK* AG 6EI \l , (2.24) where, 6 E 7 K M G 2.3.2.4.2 Bases due to the Rotation of the Fixed Support The following gives the bases due to a rotation of the fixed support A. As pinned ends are not considered to have independent rotational degrees of freedom of their own, such bases do not exist for the rotation of support B. 9<t> <^2l °21 ET + EI (2.26) W£{0 = 2A2l((-l) + B2l{(2l-C)2-e +D2l \e - ( 2 / - 0 3 (2.27) where , A2L = 1 + EI F 2K2AG12 K2AG12C2 Chapter 2. The Formulation of the Finite Strip 30 B21 F 2EIlC2l C21 D2l -1 F 12Z2 + 6EIl2C- 21 F 4/2 EI K 2 AG 2.3.2.5 Satisfaction of Boundary Conditions In a Rayleigh-Ritz type displacement formulation it is necessary to satisfy only the displacement boundary conditions. The homogeneous displacement boundary condi-tions to be satisfied are as follows(Mindlin 1951, Reddy and Khdeir 1989). • Pinned Boundary: u. = v = w = ip = 0; <j> ^ 0 • Fixed Boundary: u = v = w= x(} = <f> = § • Simple Support: w = ijj = 0; u = (f> ^ 0; D = 0 or v / 0 Due to the use of support bases, it is also possible to satisfy inhomogeneous boundary conditions. This section considers the expression of finite strip strains in terms of the nodal-line variables, bases and interpolation functions defined in the previous sections. 2.4 Kinematics Chapter 2. The Formulation of the Finite Strip 31 2.4.1 T h e D i sp lacement A p p r o x i m a t i o n The fundamental quantities u0,v0,w,<b, and \b at the reference surface of the finite strip can be expressed in terms of the strip nodal line variables as in Equations 2.6 to 2.10. Then, together with Equation 2.3, the displacements can be expressed in matrix form as, u V w N\ {d} (2.28) where, [N] is the matrix containing shape functions, and {d} is the vector of nodal-line displacement degrees of freedom. This can be re-written as j M {8} = [N.]{d.} + £[Ara]R} + £ [Nm]{dm} (2.29) a=l m=l where M is the number of modes. The value of j can be equal to.zero (in which case no terms exist), one or two, depending on the number of additional modes to be considered in the problem. [Na], [Na] and [iVm] are the shape function matrices for supports, additional modes and the basic modes, respectively, with {de},{da} and {dm} being their corresponding vectors of degrees of freedom. The equations to follow present the above shape function matrices and the dis-placement vectors. The expressions are given in the most general terms, and hence, depending on the support conditions not all the terms shown in the support and ad-ditional mode matrices may exist, or may have to be modified as indicated by their respective interpolation formulae. Chapter 2. The Formulation of the Finite Strip 32 gAN 0 0 0 gAN 0 0 0 gAN <9AN <9AN gBN 0 0 0 0 gBN 0 0 0 0 gBN 0 <9$N 0 o 0 (2.30) (shape functions corresponding to support terms) 0 0 0 -C9TB'N [Na] = 0 —* 0 and 0 0  9tBiN 0 (mode shapes corresponding to additional modes) [Nr, g?N 0 0 0 g™N 0 0 0 g™N <9%N 0 0 0 -C9™N —* 0 (2.32) (shape functions corresponding to mode m ) Note: g™'B will exist for any strip with even a single movable (in w) fixed end. That Chapter 2. The Formulation of the Finite Strip 33 is, it also stands for g^'A due to their hnear dependence. These terms exist depending on the respective support degree of freedom's existence and mobility. The displacement vectors are, {d,}T = (uA VA WA j>A $A UB VB Wf jB fB) (2.33) (support nodal-line variables) R =i} T = («?*«) (2.34) (for the additional mode from support A) {da=2}T = (w<t>b $wb) (2.35) (for the additional mode from support B) {dm}T - («m vm wm 4>m ifjj (2.36) (nodal-line variables corresponding to mode rn) 2.4.2 T h e S t r a i n V e c t o r The strain vector to be used is Green's strain tensor, obtained under the assumptions of the von Karman theory for large deflections and the Marguerre theory for initially deflected plates. As the former theory is well known its assumptions are not mentioned here. The application of the latter to obtain the strain vector from first principles will follow the procedure given for thin plates by Chia(1980). In the original theory of Chapter 2. The Formulation of the Finite Strip 34 Marguerre, the Kirchoff hypothesis of plate normals that remain normal to the middle plane has been used. But it is dropped here due to the Mindlin-type behaviour. Consider the initially deflected plate section in Figure 2.6. It shows the plate in its stress-free, natural, initially deflected shape. Also shown in the same figure is a deformed configuration of the plate element. The z axis points downward, and both the initial deflection Wi and the additional deflection w0 are measured in that direction. At the initial state the plate thickness t is perpendicular to the middle surface of the dished configuration. The reference cross-section on which the stresses are measured is considered to be normal to the x-y plane, and hence not generally so to the middle surface. A variation from the ordinary theory thus arises by the existence of transverse shear stresses at the surface of the deformed plate. Now consider a general point before deformation given by the position vector f0 ~ x i + y j + (wi + z) k (2.37) Then, if during deformation the cross-section is to rotate by <b (Figure 2.6) and ib in the x-z and y-z planes, respectively, then as presented earlier in Equation 2.3, the displacements are given by u = u0(x,y) - z(b(x,y) v = v0(x,y) - ztb(x,y) (2.38) w = w0(x,y) Therefore, the new position vector of the point can be expressed as f = [x + u0(x,y) - z<f>(x,y)} i + [y + v0{x,y) - zib(x,y)} j + [w0(x,y)+Wi+z] k (2.39) It should be noted that in the above Wi and z are functions of the coordinates x and y. Now, in order to form the strain tensor, consider a hnear differential element of Figure 2.6: The Initial and Deflected Forms of the Plate Chapter 2. The Formulation of the Finite Strip 36 the plate having an original length dsQ. Let it be expressed in the original state as dfQ, given by dr0 = dxi-r dy j + [(wi + z)iX dx + (u>i + z)>y dy + dz] k (2.40) Then, the length of the element before deformation, ds0, is given by dsl = df0.drQ = {l + [{wi + z\x)2} dx2 + {l + [(Wi + 2 ) , Y ] 2 } <fy2 + dz2 +2(wi + z)iX(wi + z)tV dx dy +2(wi + z)tV dydz +2(wi + z)tX dzdx (2.41) Now, considering the state after deformation, the infinitesimal element can be ex-pressed as dr = {[1 + n 0, x - (z<t>),x] dx + [{u0,y - (z<p),y] dy - <f)dz} i + {[VQ,X ~ (#),x] dx + [I + v0ty - (zil))ty\ dy - ijjdz) j + {(wi + z)tXdx + (wi -f-iuo-r- z)<ydy + dz} k (2.42) Then, as before, the deformed length of the element, ds, is given by ds2 = df.df = <fcc2 {[1 +u0,x - {z4>),x}2 + [v0,x - (ztp),x]2 + [(itf; + w0 + z),x]2} + i y 2 {[Tt 0 i y - (2<^ ),y]2 + [1 + V0jy - (2^),J/]2 + [fat + ™ 0 + 2),y]2} + dz 2{^ 2+V 2 + l} + 2dxdy {[1 + ti0,x - (^ ),x] No,y - (^ ),»] + [TJ 0, X - (r0),x] [1 + i ; 0 i J / - {zrb)M] +(wi + io 0 + z)iX(wi + w 0 + z)>y) Chapter 2. The Formulation of the Finite Strip 37 + 2dydz {(wi + w0 + z),y - (j> [u0jy - (z(j>)<y] -ib[l-r v0,y - {z^),y}} + 2dzdx {(wi + w0 + z),x - 4> [1 + u0iX - {z^>),x) -rb [v0,x - (2.43) Now, the difference between the squares of the original and subsequent lengths of the element is given by the difference of Equations 2.43 and 2.41. To obtain the expressions for strains, this difference can be compared with ds2 - dsl — 2(exdx2 + eydy2 + jxydxdy 4- i y z dy dz + -yzxdz dx) (2.44) where, the standard engineering shear strains have been used, and defines Green's Strain Tensor (Washizu, 1982). In Equation 2.44, in accordance with the standard plate theory, the direct z strain has been dropped as it is not to be derived from the assumed displacement field, but is obtained from the constitutive relations by the assumption of a zero direct stress in that direction(Fung, 1965). Then on comparison of terms it can be deduced that, e* = ^ {[1 + u0,x - {z <p)iX}2 + [v0>x - (z VO,*] 2 + [(Wi + w 0 + z),xf - 1 - \{wi + z),x]2} (2.45) X f 2 2 + [(Wi+Wo + z ) i y ] 2 - l - [ ( W i + z)iy]2} (2.46) 7*2/ = { [1 + «0,x - (Z </>),*] k>,y - (Z <f))J + [V0,x ~ {Z VO,*] [1 + V0,x - (zi>),y] +(w{ + w0 +z)iX(wi + w0 +z)<y - (wi + z)tX(wi +z)>y} (2.47) lyz = {W0,y - <t> [U0,y ~ {z<f>),y} ~ ^ [1 + « 0 , » ~ (*^ ),y]} (2.48) Izx = {wo,x - [^1 + UoiX - (z</>),x] - ij) [v0,x - {zrjj),x\} (2.49) Chapter 2. The Formulation of the Finite Strip 38 On further simplification, from Equation 2.45, the direct strain can be written as e» = uQ<x + ^  {ulx + v2Qx + wlx -2 [z<x <f> + z(biX]- 2u0,x [ztX (b + z cbiX] - 2v0jX [z<x ib + zxb<x] + [z,x<b +z<biX}2 + [ztXxb + zxbtX]2 +2 wi<xw0tX + 2 z>xw0tX} (2.50) Now, considering the assumption of Marguerre that the slopes of the initial shape are sufficiently small so as to make zyX negligible (i.e. having h constant), Equation 2.50 can be further simplified to give = u0,x + ^  {u20x + vlx + wlx} - z<biX- u0,x z <biX - v0iX zxb,x +\(z 2<t> 2x+z 2fx) + ™i,*™0,* (2-51) from which the second order terms in the derivatives of those other than w can be neglected as in the classical theory to give, ex = uo,x - zcbiX+ wiiX w0tX + ^  w20x (2.52) The third term in the above equation is the contribution of the initial deflection to this strain. Similarly, from Equation 2.46, the following expression can be derived for ey. I eV = V0,y ~ * V\y + ™i,y W0>y + - W2Qy (2.53) Again, starting from Equation 2.47 and following a similar procedure, and making similar assumptions, the in-plane shear strain ~fxy can be expressed as, 7xy = U0,y + VQ<X - Z (^ + xj) <x) + Wo,x™0,y + W0>xWi<y + W0>yWi<x (2.54) where the last two terms give the contribution of the initial deflection. Chapter 2. The Formulation of the Finite Strip 39 Considering Equation 2.49, the transverse shear strain jxz can be written as, Ixz = Wo,x - <t> ~ <fao,x - tpv0iX + z(<b(b<x + ibxbiX) (2.55) where the previously mentioned assumptions of negligible derivatives of z, and of the products of strain terms, have been made. Further, if it is assumed that the rotations and strains are small enough such that their products with each other are negligible, then, Equation 2.55 can be further simplified to give Ixz = wQiX - <f> (2.56) which is the same relationship as for an initially flat plate. Thus, no additional term exists for transverse shear strain in the new theory. Similar arguments for the other transverse shear strain jyz would give the following two equations as that what correspond to Equations 2.55 and 2.56, respectively. ~ w0,2/ — V-' (2.58) As can be appreciated from the above derivation, this theory is applicable only for plates that do not differ much from the shape of a plane, before and after deformation. It may again be noted here that the cross-sectional rotations are measured only from the vertical sections of the initially deformed plate. Now, if the strains, expressed in terms of reference plane displacements as given above, are put into matrix form, and those due to initial deflection are separated out, U} = {ef} + {ei} (2.59) where superscripts / and i refer to the strains in the case of a flat plate and to the additional contribution of the initial deflection, respectively. The two terms in the above equation are as follows. Chapter 2. The Formulation of the Finite Strip 40 {ef} = 4 4 Ixy » = < Ixz > = , ^0,x - Z <f>,x + \ W20x U0,y + VQ!X - Z ((f)>y + V>,x) + WQ,xW0,y WO,x - (f> Wi,x WO,* Wi,y W0>y (2.60) (2.61) Of the above two equations, the first gives the standard von Karman strains. From this point onwards the development of the strain vectors, and the subse-quent formulation of the stiffness matrix will, in general, be as follows. Always, at first the required development is carried out in the context of the flat plate theory. Then, any additional part due to the initial deflections will be considered and devel-oped, while using the pattern used for the fiat plate. Unless mentioned otherwise, all differentiations will be with respect to the non-dimensional inplane coordinates. The out-of-plane z (or £) coordinate will continue to be in the dimensional form. Of the above two components of the strain vector, {e-^ } consists of a linear part and a nonlinear part. If they are to be expressed in terms of the finite strip degrees of freedom {d}, • {e'} = [B0 + B1(d)] {d} (2.62) where the linear part is given by the first term, and the nonlinear part is given by the deflection dependent second term. The two terms inside the parentheses can be expressed as follows. Considering the fact that the displacements are given in terms of the degrees of Chapter 2. The Formulation of the Finite Strip 41 freedom by Equation 2.28, then if the matrix [Lo] is defined as a_ dx 0 0 I a a af 0 0 0 a By 0 0 i a ddfj 0 [Lo] = e a dx 0 i d ddfj I a a d£ 0 (2.63) a 8z 0 a_ dx a 0 1 d o d[ 0 a dz d 9y . 0 d dC 1 §_ d dfj then, [S„] {d} = [L0] [N]{d} (2.64) That is [Bo] = [Lo] [N] 1 9 adi 0 0 0 i a d dfj 0 i a d dfj 1 d 0 a ec 0 I a a 9£ 0 a 9C i a ddfj [N] (2.65) with the definition of [N] being implied by Equations 2.28 and 2.29. If the nonlinear part is denoted by {e^ ;}, then {i> = [Bi (d)] {d} (2.66) Chapter 2. The Formulation of the Finite Strip 42 where of which [J] is defined as [J] M^}T[^]T[J][NA] [NS]T[J)[X*] + {[#£]T[J][N* {°} T 0 0 0 0 0 0 0 0 1 (2.67) (2.68) The zero vectors in Equation 2.67 are equal in size to the vector of the degrees of freedom {d}. Using Equation 2.61, the additional strain terms due to the initial deflection (which are linear in the additional deflections) can be expressed as 4 > = < y Wi}y W0jy Chapter 2. The Formulation of the Finite Strip 43 {dw<}TN<xNiX{d™} {dWi}T N yN y{dw} {dw'}T(NTyN,x + NTxNj{d-} (2.69) where TV is the row vector of shape functions for the local w displacements. This row vector can be obtained from the third row of the matrix [N] of Equation 2.28 by the elimination of the terms corresponding to degrees of freedom other than w. {dw} and {dWi} are, respectively, the column vectors giving the response and initial displacement quantities in the w degrees of freedom. Here it has been assumed that the initial deflections are available in terms of the series expressions used in the displacement interpolation of the plate. This has been done purely for ease in programming. Re-expressing the above equation, N,XN,X { V } = {dw-y N..N. NIN* + NTJ. {dw} (2.70) Now, as in Equation 2.62 for the flat plate strains, let the additional strain from the initial deflections be expressed as {V} (2.71) with the degree of freedom vector now referring to all the degrees of freedom, and the strain vector referring to all the five engineering strains, including the transverse shear Chapter 2. The Formulation of the Finite Strip 44 strains. The matrix [AJ?o] can be obtained from Equation 2.69 by introduction of zeroes to positions corresponding to the non-u; degrees of freedom and the transverse shear stresses. Then, using Equations 2.59, 2.62 and 2.71, the total strains are given by {e} = [B0 + AB0 + B-i] {d} (2.72) = [B){d) (2.73) 2.5 Variational Formulation This section presents the variational formulation of the problem. The Principle of Virtual Work is used as the variational principle. 2.5.1 Virtual Work Equations The principle of virtual work in the finite displacement theory of elasticity can be written as(Washizu, 1982) / (<7A"Se v - Px8ux)dV - ( Fx8uxdS = 0 . (2.74) J V J S i In the above it is assumed that the prescribed boundary conditions are satisfied at the boundary surface 52 by the virtual displacement field. Si is the remainder of the boundary where the boundary conditions are prescribed force vectors (Fx). Px is the body force vector, and Se^ is the virtual strain vector corresponding to the compatible virtual displacements 8ux. ax>x is the real stress vector. In this work where a total Lagrangian formulation is used, the stress tensor is the Second Piola-Kirchoff (or simply Kirchoff) stress tensor and the strain tensor is the Green strain tensor. Chapter 2. The Formulation of the Finite Strip 45 2.5.2 Matrix Structure. Equations Expressing the displacements and strains in terms of the degrees of freedom vector {d}, the above equation can be re-written in matrix form as / 6({df[B}T) {<r}dV - j {8d}T[N]T{P(x,y,z)}dV J V v V - I {8d}T[N]T{F(x,y,z)}dS = 0 (2.75) JSi where [TV] is defined by Equation 2.28. {P(x,y, z)} and {F(x,y, z)} are the forces expressed as vectors in the coordinate directions. From this point onwards, unless mentioned otherwise, it will be assumed that the equations are written with respect to a single finite strip, and in its local coordinates which can be transformed into the global system of the structure. Considering a hnear elastic material with the constitutive relation W = \D]{e} (2.76) Equation 2.75 can be re-expressed as {Sd}T I [Bf [D][B]dV{d} - {8d}T j [N]T{P(x,y,z)}dV J V J V -{6d}T [ [N}T{F(x,y,z)}dS = 0 (2.77) In the above it has been assumed that the variation in strain can be expressed as follows. {Se} = 8([B]{d}) = [B]{6d} (2.78) The matrix [B] will be derived later. As the virtual displacement degrees of freedom are arbitrary (except for the need to satisfy the displacement boundary conditions), Equation 2.77 would give [Ks]{d}-{P}-{F} = 0 (2.79) Chapter 2. The Formulation of the Finite Strip 46 Here, [Ke] is the secant stiffness matrix, and is given by [K.] = l\B)T\D}\B)dV J V (2.80) {P} and {F} are the body force and surface traction vectors, respectively. They are defined as {P} Jv[N]T{P(x,y,z)}dV (2.81) I [N]T{F(x,y,z)}dS (2.82) Later in the appropriate sections the tangent stiffness matrix to be used in incre-mental analyses will be derived. 2.5.3 Continuity Requirements on Displacements From an examination of the secant stiffness matrix together with the expressions for strains derived earlier, it can be seen that the maximum order of derivatives occuring in the integrand is unity. Therefore, for convergence it is required to have, as a minimum, a finite strip displacement field which is complete to the first order while providing C° interelement compatibilty in all the variables. This low order continuity is a distinct advantage of using Mindlin's theory also in the analysis of thin plates, instead of the classical theory. In Chapter 6 it will be shown that the order of interpolation within has to be complete upto the second order if the results are to be used in the calculation of transverse shear stresses by the use of the equilibrium equations and the displacement solution. But it is not a requirement with regard to the convergence and accuracy of the displacement solution and the calculation of inplane stresses. Chapter 2. The Formulation of the Finite Strip 47 2.6 The Stiffness Matrices 2.6.1 T h e S e c a n t S t i f fness M a t r i x As previously mentioned, this formulation is limited to the case where the material property axes of individual orthotropic layers coincide with the plate geometric axes. The general case of arbitrary axes may be easily introduced by the use of transfor-mation matrices. Now, for a laminated orthotropic plate with principal material axes (£,~n), the elasticity matrix of the layer Z, [Dl], under the condition of zero normal direct stress is (Jones, 1975) A1 GL GL BL 0 0 0 0 0 0 0 0 Cl 0 0 0 0 0 0 0 Dl 0 O O F ' (2.83) where, E: (2.84) BL = El (2.85) Cl Dl Fl = G K2G[( K2Gl (2.86) (2.87) (2.88) GL 1 - »\Ai (2.89) Chapter 2. The Formulation of the Finite Strip 48 K? = the correction factor for G, = the correction factor for G^ vl£v = the Poisson ratio giving the strain in rj direction due to a strain in £ In the above the shear correction factors are used to account for the fact that a con-stant shear strain is used at a cross-section of the plate. The factor for isotropic plates was derived by Mindlin(1951), and the factors for both single layer and laminated orthotropic plates have been derived by other researchers (Chow 1971, Whitney 1973, Noor 1975, Chatterjee and Kulkarni 1979) under different conditions. According to Noor and Burton(1989) it is better to calculate the correction factors after a prelim-inary analysis and then substitute them in for a second proper analysis. The reason for this is the dependence of the shear correction factors on the shear strain field in the plate. In the problems dealt with here, as have been done by other researchers (Reddy and Chao 1981a, Putcha and Reddy 1984, Reddy 1984c), the shear correction factors are considered to be equal and to have the same value as that of an isotropic plate, even in the case of an orthotropic layered plate, except when their effects on the results are studied. Now, from Equations 2.72 and 2.73, {e} = [Bo + A B 0 + B1 (d)} {d} = [B (d)} {d} (2.90) and, therefore {8e} = [B0 + AB0) {8d} + 8 ([£a (ci)] {d} ) (2.91) Now, from Equation 2.67, *([Bi(<*)] {<*}) = 2[B1(d)} {8d}= [BL(d)} {8d} (2.92) Chapter 2. The Formulation of the Finite Strip 49 with, [BL (d)] 2 [BX (d)} (2.93) Therefore, {6e} = [Bo + AB0 + 2 Bx (d)] {8d} (2.94) = [B0 + AB0 + B L {d)\ {5d} (2.95) [B (d)] {6d} (2.96) where, [B] = [BQ] + [ABo] + [BL] (2.97) Then, from Equation 2.80, the finite strip secant stiffness matrix, can be written as be symmetrised as shown later after the presentation of the tangent stiffness matrix. It is advantageous to have this in a symmetric form in the case of a direct iteration procedure, which was included in the computer code, although not utilised. Other-wise the only advantage is in the reduction of the array size of the element matrices that are needed in the determination of the resistive force in the case of a Newton-Raphson iteration. (2.98) / [Bo + AB0 + BL)T\D][Bo + AB0 + Bx] dV (2.99) The secant stiffness matrix, [Kt], as given above, is not symmetric. But this can 2.6.2 T h e T a n g e n t S t i f fness M a t r i x The following gives the derivation of the tangent stiffness matrix which will be needed in the case of a Newton-Raphson procedure. The procedure given by Zienkiewicz(1977) Chapter 2. The Formulation of the Finite Strip 50 is followed here. As the structure matrices can be obtained by the assembly of the strip matrices, again the following will be with respect to an individual strip, unless mentioned otherwise. Considering the virtual work equation of the static problem, as given by Equa-tions 2.75. 2.81 and 2.82 as / {8e}T{*} dV - {6d}T{P} - {Sd}T{F} = 0 (2.100) where, the external virtual work can be denoted by {5d}T{F} = {Sd}T{P} + {Sd}T{F} (2.101) with {F} being the total consistent load vector for the structure obtained on the summation of the strip contributions, the following can be written as the matrix equihbrium equation of the structure. * = J [B{d)]T {<r}dV -{F} = 0 (2.102) or, * = J[B(d)]T[D][B(d)]dV{d}-{F} = 0 (2.103) Now, the differential of \P can be written as <f¥ = J [B(d)]T d{{a}) dV + J d (jB (dj\T>) {tr}dV-d{F} (2.104) Here it may be noted that, d{F} = {0} if {F}^{F(d)} The first term of Equation 2.104 would give the linear stiffness matrix [K0 + A.K0] and the large displacement stiffness matrix [KL + AA't], while the second term would give the stress stiffness matrix [K^ + AK^}. The 'A' terms denote the additional terms Chapter 2. The Formulation of the Finite Strip 51 that occur due to the initial displacement, and have been so denoted because they will be dealt with separately. Then, J [B (d)]T d{<r} dV = [K] d{d} (2.105) where, [K] = [K0 + AK0] + [KL + AKL} = J[B{d)]T[D][B(d)]dV (2.106) gives the linear and large deflection stiffness matrices. On expansion using Equa-tion 2.97, the linear stiffness matrices can be writtens as [Ko] = / [B0]T[D][Bo]dV (2.107) and [AK0] = Jv({AB0}T[D}[AB0} + {ABo]T[D}[Bo} + [Bo]T{D}[AB0}) dV (2.108) with large deflection matrices given by [KL] = Jv([B0]T[D}[BL(d)) + [BL(d)]T \D)\BL{d)} + [BL(d)]T [D][B0]) dV (2.109) [AKL] = / ( [AB0f [D] [BL(d)] + [BL(d)f [D] [AB0]) dV (2.110) The stress stiffness matrix can be expressed in terms of the displacements as follows. Let, Jvd({B{d)f ){*} dV = Jvd[BL(d)\T{D][B{d)]{d}dV = [K„ + AK„]d{d} (2.111) Chapter 2. The Formulation of the Finite Strip 52 Considering Equations 2.67 and 2.93, d[BL]T = d }2mT[J)m{d}, ±[N*]T[J][N*]{d}, Id where [R}d{d}, [S]d{d], [T]d{d}, {0}, {0} (2.113) \S) = ^[N„]T[J][NA] (2.114) m = u[mT[m*]+[N*]T[w* It may be noted that each of [R], [S] and [T] is symmetric. Then, d[BL]T{*} = {a«[R]d{d} + <Tm[S}d{d} + Ttv[T]d{d}} (2.115) = [ <r« [R] + <rm [S] + riv [T] ) d{d} (2.116) Furthermore, defining a (5 X 1) column vector {Ni}, such that Nij = 1 (j = i) /Vi,- = 0 (j ^ t) Chapter 2. The Formulation of the Finite Strip 53 where j is the row number, the individual stresses can be expressed as follows. = {Nl}T[D][B]{d} = {d}T[B]T[D]{Nl} (2.117) °m = {^2}r{^} = {N2}T[D)[B]{d} = {d}T[B]T[D}{N2} (2.118) = {NZ}T[D][B]{d} = {d}T[Bf[D}{NZ} (2.119) Then, Equation 2.116 can be written as, d{\BL)T){cr} = \{d}T[B}T\D){Nl}[R]+{d}T\B]T[D}{N2}[S] +{d}T[B]T[D}{N3}[T}] d{d} (2.120) This can be further simplified to give, d {[BL]T) {cr} = {d}T [ [B]T[D]{N1} [R] + [B)T[D}{N2} [S] + [B}T[D){NZ} [T] } d{d} (2.121) Therefore, from Equations 2.111 and 2.121, [Kg- + A K „ ] d{d} = j d([BL{d})T){a}dV = {d}T Jy [ [Bf[D}{Nl} [R] + [Bf[D}{N2} [S] +[Bf[D]{N3} [T] ] dVd{d} (2.122) giving, as d{d} is arbitrary, [ K r + AKr] = {d}T [ [[B]T[D]{N1}[R] +[B]T[D}{N2}[S} + [B}T[D}{N3}[T}} dV (2.123) Chapter 2. The Formulation of the Finite Strip 54 Then, the tangent stiffness matrix of a finite strip, [K?], is given by [KT] = [K0 + AKo] + [KL + AKL] + [Kc + AKV] (2.124) 2.6 .3 S y m m e t r i c a l S e c a n t S t i f fness M a t r i x The unsymmetric secant stiffness matrix presented by Equation 2.98, and required by Equation 2.103, can be expressed in a symmetrical form as follows. According to Equations 2.98 and 2.99, [K.] = I [B0 + AB0 + BL}T[D} [B0 + AB0 + #i] dV (2.125) which on expansion gives [K.] = / [B0 + ABQ}t [D] [BO + ABo] dV +\ Jv ( [Bo + ABo]T [D] [BL] + [BLf [D] [BL] + [BL}T[D][Bo + AB0})dV + \ Jv[BL}T[D}[Bo + ABo]dV (2.126) = [K0 + AK0] + \ [KL + AKL] + \ Jv[BL]T[D)[Bo + ABo] dV (2.127) where \KQ + AKo\ and [KL + AKL] are the symmetric matrices defined earlier in Equations 2.107 and 2.109, respectively. It remains to make the third member of the equation symmetric. Considering the resistive force given rise to by this unsymmetrical part, and then Chapter 2. The Formulation of the Finite Strip 55 expanding using the definitions implied and given by Equations 2.113 and 2.114, ~Jv[BLf[D)[B0 + AB0}dV{d} = \jv[[R}{d} [S){d} [T]{d} {0} {0} [D] [B0 + A£?0] {d} dV (2.128) Now, noting that the multiplication of the last three matrices on the right hand side of the above equation gives rise to a (5 X 1) column vector, consider the term in the integrand obtained by the pre-multiplication of the first member of this column vector by [i?]{d}, which is the first column vector of the first matrix of the right hand side. Towards this let us denote the first row of the elasticity matrix by [Di]. Then, the resulting column vector is ±[R]{d}[D1][B0 + AB0]{d} = ^[D1][B0 + AB0]{d}[R]{d} (2.129) where, the first three terms, [DI][J50 + ABo]{d}, on multiplication give rise to the earlier referred to scalar. Expressing the other terms also in a similar manner by considering the appropriate rows of the elasticity matrix, the resistive force vector can be written as \ Jv[BL}T[D}[B0 + ABo] dV {d} = \jy( [Pi] [Bo + AB0]{d}[R] + [D2][B0 + AB0]{d}[S] + [£>3] [B0 + AB0}{d}[T]) dV{d}(2.m) giving the symmetrised part of the secant matrix as l- Jv[BLf[D)[B0 + ABo] dV = \JV( [Di] [B0 + AB0] {d}[R} +[D2][B0 + AB0]{d}[S} + [D3] [Bo + ABo] {d}[T]) dV (2.131) Chapter 2. The Formulation of the Finite Strip 56 2.7 Computer Implementation This section presents some details with regard to the implementation of the previous theory in a computer. 2.7.1 T h e I n t e g r a t i o n o f S t i f fness M a t r i c e s a n d L o a d V e c t o r The following gives details on the evaluation of the volume and surface integrals appearing in the stiffness matrices and load vectors presented earlier. The integration scheme used for the stiffness matrix was also used for the load vector even though the required order of integration is not as high. The volume integration of the stiffness matrix is performed in three stages, viz. the integration of the modes in £, the integration of the polynomial interpolation in 77, and the integration in thickness £. In the case of the simply supported strips where simple trigonometric terms are used, the £ integrations can be carried out manually without difficulty. But in order to allow for later inclusion of nonlinear material properties, numerical integration is being used even for that case. In finite element theory there are various criteria that can be used in the selection of an order of integration which would provide convergence and accuracy, and which are problem dependent. But in the case of Mindlin plates, during the integration, in addition to the accuracy of the solution, it is needed to consider the 'shear locking' phenomenon. 2.7.1 .1 S h e a r L o c k i n g a n d R e d u c e d I n t e g r a t i o n Reduced integration is a technique introduced by Zienkiewicz et a/.(1971) and by Pawsey and Clough(1971) to eliminate shear locking behaviour described below. Chapter 2. The Formulation of the Finite Strip 57 In the case of Mindlin plate finite elements, it had been observed that unless certain precautions are taken in the analysis of thin plates, the plate would become too stiff due to the dominance of transverse shear strain energy over that of flexural strain energy. This phenomenon is known as shear locking. As mentioned earlier, it is due to the fact that, as a result of the displacement field, transverse shear strains accompany flexural strains even in thin plates where the former should be negligible (in terms of strain energy). It has been discovered that this can be avoided either by reduced or selective integration or by the use of higher order displacement fields. The explanation for this has been provided by Zienkiewicz and Hinton(1976). The use of higher order interpolation would soften the structure, while the use of reduced or selective integration provides the needed singularity for the element matrices, thus softening it to avoid the locking behaviour. Here reduced integration means the use of a number of integration points one less than what is necessary for the exact integration of the whole stiffness matrix, while selective integration means the reduced integration of only that part of the stiffness matrix which arises from the transverse shear terms. Several studies on this shear locking behaviour and reduced integration are avail-able in the finite element literature, for example by Zienkiewicz and Hinton(1976), Hughes et a/.(1978), Pugh et a/.(1978), Kreja and Cywihski(1988), etc. In the case of finite strips, a similar study has been presented by Ohate and Suarez(1983b). There exact, selective and reduced integrations were considered with respect to 77, and the numbers of integration points given here in Table 2.1 were provided as necessary for exact and reduced integrations of the given orders of interpolation. 2.7.1 .2 T h e I n t e g r a t i o n O r d e r s U s e d While considering the above, the integrations in the three orthogonal coordinate directions, £, 77 and £, were carried out independently, as follows. Chapter 2. The Formulation of the Finite Strip 58 Element Type No. of Integration Points for: Exact Integration Reduced Integration Linear 2 1 Quadratic 3 2 Cubic 4 3 Table 2.1: Integration Points for Integration in v In the case of integration of mode, shapes in £, the sole criterion used in the selection of the number of integration points was the accuracy of the answer. At times a much larger number of points than necessary was used to avoid the need to interpolate or extrapolate the responses in this direction. In the case of the rj direction, reduced integration was used with all the three interpolation orders, viz. linear, quadratic and cubic, by the use of one, two and three integration points, respectively. This use of reduced integration even for the higher order quadratic and cubic interpolations was because it gives good results while being economical. Besides, exact integration has been found to be unreliable in both thin and thick plate analyses(Ohate and Suarez, 1983b). The integration across the plate thickness was carried out analytically. In the case of a layered plate this integration was carried out within each layer and then summed up in layers. The exact analytical integration can be used even in the case of a non-constant constitutive equation by dividing the material into an artificial set of layers, across each of which the properties can be assumed to be constant. But, in this case Gaussian integration should be superior, and preferably the method may be changed to it in case of the introduction of material properties that do not remain constant. Chapter 2. The Formulation of the Finite Strip 59 2.7.2 C a l c u l a t i o n o f S t r e s s e s After determining the displacement solution to a problem it is a simple matter to calculate the stresses present. But, in order to achieve convergence in stresses it is necessary to consider more modes and strips than what are adequate for the conver-gence in displacements. This is because the stresses are functions of the derivatives of displacements. According to finite element literature (Zienkiewicz, 1977), the best places to sam-ple the stresses are the Gauss integration points as they generally agree well with the analytical results at those points. At other places the stresses are to be obtained by simple bilinear interpolation of the integration point stresses(Zienkiewicz, op. cit.). This procedure was followed in the present analysis. Simple hnear extrapolation was used when the stresses were required outside of the points. As mentioned earlier, on certain occasions a very much larger number of integration points than that necessary for accurate integration was used in order not to have to interpolate or extrapolate the stresses. Where the problem was symmetric about the centre fines of the plate, and the response was needed at the centre, it was found that the average of interpolated and extrapolated values provides a better answer than either simple interpolation or extrapolation. More details of this will be presented in Chapter 3. The inplane direct stresses and the inplane shear stresses were calculated using the constitutive equations. But, in the case of the transverse shear stresses, such a calcu-lation would result in erroneous results due to the assumption of constant transverse shear strains across the thickness of the plate. Therefore, as has been done by others (Pryor and Barker 1971, Reddy and Chao 1981b, Reddy 1984a, Engblom and Ochoa 1985, Chaudhuri 1986, Putcha and Reddy 1986, Reddy 1989) the inplane equilibrium equations were used in the determination of these stresses. Some observations on such calculations are presented in Chapter 6. Chapter 2. The Formulation of the Finite Strip 60 2.7 .3 S o l u t i o n P r o c e d u r e The following description is with regard to the nonlinear problems. The same pro-cedure was followed for the linear problems by specifying a single iteration, and the non-consideration of the nonlinear matrices. Depending on the expected nonlinearity of the problem, or the need for intermedi-ate results, either a Newton-Raphson procedure or an Incremental Newton-Raphson procedure was used in the solution of the problems. Choleski decomposition was used in the solution of the resulting set of equations. A very strict convergence criterion was used in this research phase. Thus, a convergence of 0.01% to 0.001%, in each degree of freedom was generally used to terminate the iterations. The convergence was considered to be achieved if the criterion was satisfied in either the displacements or the residual force vector. 2.8 Penalty Function Formulation For A Rectan-gular Container In the case of a rectangular container certain compatibility conditions need to be sat-isfied at the vertical corners where the horizontally placed finite strips meet eachother. They are the satisfaction of the rotatory and translatory displacement compatibilities as shown in Figure 2.7. In the case of a pin-jointed container the rotational compati-bilities do not exist, as the <b rotations have to be independent, while the ip rotations are maintained at zero to have zero moment there. In the case of a rigidly connected container, the finite strips automatically provide the required compatibility of the <j> rotation due to the existence of the individual support rotation degrees of freedom that have the same sense. But the xb compatibility cannot be satisfied with the avail-able degrees of freedom. In the case of translatory displacements, the compatibility Chapter 2. The Formulation of the Finite Strip 61 Figure 2.7: Compatibility at a Corner of a Container is easily provided because of the existence of the support degrees of freedom. In order to satisfy the compatibility condition in xp when the fixed-ended strips are to be used, a constrained variational principle was utilised in the problem formu-lation. The Penalty Function method was selected over Lagrange's method as it does not involve additional unknowns. But, as will be shown by numerical examples pre-sented in Chapter 4 this constraint could also be satisfied sufficiently, in the problems considered, by specifying the tp rotation to be zero at the corners. 2.8 .1 T h e C o n s t r a i n t s The Penalty Function formulation introduced by Courant for a conservative problem is based on the minimisation of a new functional instead of the total potential energy. Although the previous derivation was with the use of the Virtual Work principle, here it will be based on the total potential energy approach, and will follow the method Chapter 2. The Formulation of the Finite Strip 62 presented by Zienkiewicz(1977). As mentioned above, the constraints to be satisfied with the Penalty Function method refer to the rotations ib at the corners. Let us consider the joint of the container shown in Figure 2.7. (The 'rotation' R shown in this figure is the £ derivative of the local v displacement.) If the panel on the left side of the joint, when viewed from outside, is to be denoted by L and that on the right side by R, the constraints to be satisfied along the joint can be written as MIL = ° <2-i32) *--{%) ^ = ° «2-i3) The £ coordinates in the above equations are local to the respective panels. The subscripts R and L refer to the panel, and £ = 0 is used to indicate that this constraint is being satisfied only at the reference surface. Now, as all the above quantities are to be evaluated at the corner, and also considering the fact that in the computer code it has been made required that at a given joint the local coordinate ( of the two meeting strips should not be the same, the following expressions can be written with respect to Equations 2.132 and 2.133. fa = Mt = 0) (2-134) fa = fa(t = l) (2-135) SL • SM. SL • SM. where the ib and v terms and derivatives are to be obtained from the shape function matrices of the respective finite strips of the respective panels as given below. Chapter 2. The Formulation of the Finite Strip 63 From this point onwards, unless mentioned otherwise, the treatment will be with respect to the two finite strips that correspond to each other in the two panels, and so meet at the common joint. This is on the assumption that all walls of the container are discretised in the same manner. The derived matrices can then be assembled in the finite element manner to get the total matrices. Towards this, let T/;^ refer to the •0 degrees of freedom of the strip R at its support A, which is the support of concern here. Similarly, let ?/>B be defined for the support B of strip L. Then, the values of ?/> terms of the above equations can be written as Mt = 0) = W r {V>*} (2-138) Mt = l) = W T K } (2-139) where {N} is the interpolation vector across the strip as defined by Equations 2.6 to 2.10. Also, given that, v = {Nv}T{d} (2.140) where the first vector is obtained from the second row of the shape function matrix [N] of Equation 2.28, the partial derivatives off are given by | = (2-141) In the above, / is the length of a finite strip. Now, as the two adjoining sides of the container need not be of the same length, let li and IR refer to the lengths of the left side and right side walls, respectively. Then, assuming that these containers will have only moment resisting corners so that the same shape function vectors and the same number of degrees of freedom are used in all strips, the constraint equations given above (Equations 2.132 and 2.133), can be re-written as {N}T {^i} + U N ^ O V {dL} = 0 (2.142) {N}T {^} + lR{N}(0,fj,0)}T{dR} = 0 (2.143) Chapter 2. The Formulation of the Finite Strip 64 where {dR} and {dL} are the local displacement degrees of freedom vectors for the right hand and left hand side strips, respectively. These can further be simplified as to be expressable purely in terms of the degrees of freedom vectors of the strips by defining a new shape function vector {JV^*} which gives the xb of the supports A and B on substitution of 0 and 1, respectively, for £. This can be obtained by taking the shape function vector Nv defined above for v, and substituting zero for all the terms that do not correspond to the support xb terms, and substituting unity for the — £ terms of the others. This will not be presented here, but will be clear on the examination of Equations 2.28 to 2.36. Then, the above constraint equations become {N*'(i=0)} T {dR} + lL{N}(l,fj,0)}T{dL} = 0 (2.144) {JV '^Ce = 1)}T {d^} + ± { ^ 0 , ^ 0)} r{*l , i} = 0 (2.145) which, on combination to give a single equation will be {N*-(£ = O)}T i{w«-(i,,-,o)}T " I {<"} ) > = < I W J 1 t o / (2.146) In the above the shape functions are in the local coordinates of the respective strips. For ease in the subsequent developments, Equation 2.146 will be written in a concise form as [C}{d} = {0} (2.147) where, (2.148) Chapter 2. The Formulation of the Finite Strip 65 and [C] = = 0 ) } r i{Jvj<i,^ o)}r £{^-(0,77, 0)}r {N*'({ = 1)}T (2.149) 2.8.2 T h e C o n s t r a i n t S t i f fness M a t r i x Following Zienkiewicz(1977), the functional 7r* to be minimised in the case of a penalty function formulation is given by TT* = TT + C* f {d}T[C}T[C){d}dn (2.150) where £2 is the domain that the constraint is applicable, a is the penalty parameter and 7T is the total potential energy of the conventional formulation. Now, the minimisation of the above functional would give the same terms as the virtual work formulation as the contribution of the potential energy term TT, plus additional terms due to the constraints, which will come totally from the second term on the right hand side. Only the latter will be dealt with here. Now consider the first variation of the Equation 2.150 to obtain the structure equihbrium equations. If the penalty term is denoted by 7r c, then on recognising the fact that the constraint integration is to occur only in the local 77 coordinate of a strip, and that even these matrices can be assembled in the finite element manner for the portion of the corner where the two strips meet, it can be written ftrc. = adj1 8({d}T[C]T[C}{d}) dfj (2.151) = {5d}T2adJ1 [C}T[C}dfj{d} (2.152) where d is the half-width of a finite strip, and appears here due to the normalizing of the limits of integration. Then, considering the arbitrariness of the variation of the Chapter 2. The Formulation of the Finite Strip 66 displacements, the additional stiffness matrix due to the constraints at the particular strip level at the given corner can be written as It is to be noted that the above stiffness matrix does not have any nonlinear parts. Also it has the contributions of two corresponding strips of adjoining panels, and is therefore of size (2n X 2n) where n is the size of a strip degrees of freedom vector. In the implementation of this procedure, the quantity 2a appearing in the above equation was considered as the penalty parameter to be provided by the user. 2.9 Plates With Elastically Restrained Edges This section presents the additional formulation needed for the analysis of plates with elastically restrained edges. The present finite strip can easily be extended for such analyses due to the presence of support degrees of freedom. Two sets of elastic restraints will be considered. The first set is applied at the end support and can, therefore, be directly related to the support degrees of freedom. The second type is applied along a nodal fine support of the plate, and hence, is related to the nodal fine variables. The phenomenon is modelled by considering each of the elastically restrained supports to have, in each of the restrained degrees of freedom, a spring with a given elastic stiffness. The springs are considered to be independent of each other. The formulation will be carried out as if all the restraints exist. Where the support is rigidly fixed with respect to a degree of freedom, its spring stiffness can be made infinite or the degree of freedom can be later eliminated from the problem. In the following development the two types of restraints will be considered sepa-rately. The modelling of restraints, the elastic forces generated, and the virtual work (2.153) Chapter 2. The Formulation of the Finite Strip 67 formulation to find the contribution to the stiffness matrix will be presented for each of the two cases. All the developments will be with respect to a single finite strip. 2.9.1 Restraints at the End Supports of a Finite Strip This set of restraints is that what comes at the end supports of a finite strip. They are connected directly to the support degrees of freedom. In this type of restraint, at present, the elastic constants for the support nodal points are to be specified. They will then be interpolated across the strip by the use of the same interpolation formulae as used for the displacements. It is possible to later include these in the form of a general function. As before, let the left side of the finite strip where £ is zero be denoted by A, and the right side be denoted by B. Let the stiffnesses of the restraint springs at the support 'S', with S being equal to either A or B, be given by the 5X1 vector {kTs} defined as In the above each of the terms refers to the restraint of a particular degree of freedom at the transverse coordinate value rj. The degree of freedom is identified by the second subscript. These values in 7/ are to be obtained by the use of the interpolation formulae and the support nodal point values. That is, if the u degree of freedom is 2.9.1.1 Modelling of Restraints {ks(v)} (2.154) Chapter 2. The Formulation of the Finite Strip 68 taken as an example, *s«fo) = XLiksUn Nn(r,) = {N(V)}T {kSu} (2.155) (2.156) where, ksUn is the nth member of the spring stiffness vector {fcsu} for u a t support S, and Nn(r]) is the nth member of the shape function vector {N} defined in Section 2.2. Now, for use in the work to follow, let the above defined vector of stiffnesses in Equation 2.154 be arranged as a diagonal 5 X 5 matrix, while using Equation 2.156, to give 0 0 0 0 0 0 {N}T{kSu} 0 0 0 0 0 0 krSw(v) 0 0 0 0 0 0 ksM 0 0 0 0 0 kSM (2.157) 0 0 {N}T{kSv} 0 0 0 {N}T{kSw} 0 0 0 {N}T{ks^} 0 0 0 0 0 0 0 {N}T{ks<i>} (2.158) 2.9 .1 .2 T h e D i s p l a c e m e n t a n d E l a s t i c F o r c e V e c t o r s Now, if the 5 X 1 vector of reference plane displacements at the support is denoted by {ds(r})}, the elastic forces in the restraining springs can be written as {Fs(v)} = \kTs(v)} {ds(v)} (2.159) The displacement vector used above can easily be obtained in terms of the strip degrees freedom of vector {d} and the shape function matrix N defined earlier in Chapter 2. The Formulation of the Finite Strip 69 Equation 2.28. It is to be carried out as follows. For the first three members u, v and w, substitute £ = 0 and £ = 0 for support A or £ = 1 for support B. That is, u(rj) v(n) = 0)] {d} (2.160) w(n) where, £s is the coordinate value for the support of concern. The two rotations (j> and N by if), respectively, can similarly be obtained from the first and second rows of eliminating their -^independent parts and then substituting -1 for £, in addition to the substitution for £. Let, the new matrix thus obtained be denoted by [./VRS], SO that {ds(v)} = [NRs(v)} {d} (2.161) Then Equation 2.159 can be re-expressed as {Fs(v)} = \ks(v)\ NRS(V)\ {d} (2.162) giving, as a function of 77, the elastic forces in the springs of support S. 2.9.1.3 Virtual Work and Stiffness Matrix Here a virtual work formulation is used to obtain the additional terms contributed by the elastic restraints to the finite strip stiffness matrix. Let there be a virtual displacement {8d} in the finite strip degrees of freedom. Then the virtual work of the restraining springs in support S can be written as rd -d~ giving the additional stiffness from this support as SW = {8d}T fd[NRS}T[krs}[NRS}dr,{d} J —dl (2.163). . [K*] = dJ\[NBs?[krs][NBs]drj (2.164) where the limits of integration have been non-dimensionalised. Chapter 2. The Formulation of the Finite Strip 70 2.9.1 .4 F o r m o f t h e S t i f fness C o n t r i b u t i o n An examination of Equation 2.161 will reveal that the matrix NRS\ has non-zero columns only with respect to the degrees of freedom of the particular support. There-fore, the additional contribution as given by Equation 2.164 will have zero terms with respect to all the degrees of freedom that do not correspond to the support of concern. 2.9.2 R e s t r a i n t s A l o n g a N o d a l L i n e o f a F i n i t e S t r i p This section considers the elastically restrained supports that can occur at the plate edges parallel to the finite strip nodal lines. As it is rare for a single finite strip to be used in the analysis of a full plate, it can be asserted that a finite strip will, in general, have at most only one restrained nodal line. 2.9.2.1 M o d e l l i n g o f R e s t r a i n t s Again, here, the restraints will be introduced as independent hnear elastic springs corresponding to each of the degrees of freedom. In this case the spring stiffnesses along the support are to be expressed as functions of the longitudinal coordinate £. Thus, let Kit) {*r(0> = { kut) \ (2-165) where each of the members of the vector are expressions in £ for the respective stiff-nesses along the edge. The subscript T has been introduced to indicate the fact that this is a case of restraints at a nodal hne. The second subscript refers to the degree Chapter 2. The Formulation of the Finite Strip 71 of freedom. As done before, these can be expressed in the form of a diagonal matrix kj denned by KH) o o o o kjw(o o o o 0 0 0 0 0 0 0 0 0 (2.166) 2.9.2.2 T h e D i s p l a c e m e n t V e c t o r Let the 5 X 1 vector of reference plane displacements be denoted by {d{\. The members of this vector can be obtained by the use of the shape function matrix and the displacement degrees of freedom vector defined in Equation 2.28 in a manner similar to that carried out for the supports. Then, the first three members of the vector will be given by v(0 = [N(t>Vi,0)\ {d} (2.167) where fji is the coordinate value of the strip nodal line of concern, and will be either -1 or 1. The two rotations (b and tb, respectively, can similarly be obtained from the first and second rows of |iV"j by eliminating their -^independent parts and then substituting -1 for £, in addition to the substitution of fji. Let the new matrix thus obtained be denoted by \NRI , so that {di(t)} = #„(0 {ci} (2.168) Here it may be noted that only the coefficients of {d} that correspond to the par-ticular nodal line will contribute terms towards the forming of {di}, and these will Chapter 2. The Formulation of the Finite Strip 72 form a Fourier series with the bases used in the interpolation. The displacements are expressed as shown above, instead of as an explicit Fourier series, so that the formulation could progress in the matrix form. 2.9.2.3 T h e E l a s t i c F o r c e s i n t h e S p r i n g s Using Equation 2.168, which expresses the displacements in each of the five degrees of freedom at any £ along the restrained nodal fine, and Equation 2.166 which gives the elastic stiffnesses, the elastic forces can be expressed as F: wl <t>l F, *r(o Nri(O {d} (2.169) (2.170) 2.9.2.4 V i r t u a l W o r k a n d C o n t r i b u t i o n t o t h e S t i f fness M a -t r i x Now, as before, let there be a compatible virtual displacement {6d}. Then the result-ing virtual work is given by SW = [{Sdi}T{FT}dC (2.171) J( = {Sd}T (\Nm]T{k\]\Nm)di{d} (2.172) Jo giving the additional stiffness from this support as [*,*]••= * [\Nm}T[kn[Nm}d^ (2.173) Jo In the above, the integration has been non-dimensionalised with respect to £. Chapter 2. The Formulation of the Finite Strip 73 2.9 .2 .5 F o r m o f t h e S t i f fness C o n t r i b u t i o n An examination of Equation 2.168 will reveal that the matrix iVjy(£)j has non-zero columns only with respect to the degrees of freedom of the particular nodal line. Therefore, the additional contribution as given by Equation 2.173 will have zero terms with respect to the degrees of freedom that do not correspond to the nodal line of concern. Chapter 3 Numerical Verifications This chapter presents the verification of the finite strip model and the computer code using results available in published works. The details of the numerical integration used in each analysis are presented with the results. The number of points for the £ direction integration is given first, while that for the 77 integration is given second, in pairs. As mentioned previously, in all the analyses, unless mentioned otherwise, the shear correction factor (/c2) was taken as 0.833 in both inplane coordinate directions. It is the value used in the analysis of isotropic plates. 3.1 Simply Supported Problems This section presents the verification of the simply supported/pinned finite strip model. 3.1.1 T h e L i n e a r R e s p o n s e o f a S q u a r e T h i c k O r t h o t r o p i c P l a t e This example was adopted from Reddy(1984a). Figure 3.1 shows the coordinate system used for the square simply supported plate having the following properties: a = b -- 1000 mm 74 er 3. Numerical Verifications -ir s.s. u=v=w=0 y=o y =o,v=o Finite Strips u=v=w=0, 0=0 S.S. Figure 3.2: Simply Supported Plate Discretisation Chapter 3. Numerical Verifications 76 CPT HSDPT FSDPT Exact Finite Strips* Displacement (mm) 0.1460 0.1677 0.1677 0.1672 0.1678 trm (MPa) 18.41 18.34 17.94 18.35 17.95 rxz (MPa) - 3.857 3.887 2.884 3.707 * 4 symmetric modes with 4 cubic strips per half-panel Table 3.1: Linear Orthotropic Plate: Comparison of Response h = 140 mm (h/a = 0.14) = 1.436 X10 5 MPa = 0.7543X105 MPa = 0.4206X105 MPa Gtc = 0.2558X105 MPa = 0.4268X105 MPa = 0.44 = 0.23 qo = 1.0 MPa (Uniformly Distributed Lateral Load) The boundary conditions were modelled as shown in Figure 3.2. The comparisons made were with the results from exact 3-D elasticity theory, exact classical plate theory (CPT), first order shear deformation plate theory (FSDPT) and higher order shear deformation plate theory (HSDPT) as provided by the reference. The 3-D elas-ticity results had been obtained by Reddy from the work of Srinivas and Rao(1970) who obtained a closed form series solution for the exact three-dimensional elasticity equations. The first order shear deformation theory corresponds to the Mindlin the-ory, while the higher order theory accounts for a parabolic transverse shear strain distribution across the thickness of the plate by considering a cubic expansion of the inplane displacements in the thickness variable. The latter theory requires no cor-rection factor for the transverse shear strains. The HSDPT and FSDPT results have Chapter 3. Numerical Verifications 77 Strip No. of Integration Central Deflection (mm) Type Symm. Points No. of strips/half-plate Modes 1 2 3 4 6** Linear 1 5 X 1 0.1753 0.1754 0.1726 0.1716 0.1710 2 7 X 1 0.1704 0.1717 0.1695 0.1686 0.1680 4 12 X 1 0.1709 0.1721 0.1698 0,1689 0.1683 Cubic 1 5 X 3 0.1703 0.1704 0.1704 0.1704 5X4* 0.1501+ 5X4* 0.1502* 3 9 X 3 0.1678 0.1680 0.1680 0.1680 4 40 X 3 0.1678 Exact integration in 77 40 X 1 integration for all modes + with the major axis in £ direction * with the major axis in 7/ direction Table 3.2: Linear Orthotropic Plate: Convergence of Central Displacement been obtained by the use of the Navier series solution procedure. The non-dimensional responses given in the above reference were converted to dimensional values for the purpose of comparison. A comparison of results is given below in Table 3.1 with respect to both deflections and stresses. The displacement is the central deflection. The direct stress, o~x, is at the surface of the strip at the centre of the plate, i.e. at (a/2, 6/2, h/2), and the transverse shear stress, TXZ, is at the middle surface at the centre of the x = 0 support, i.e. at coordinate (0,6/2,0) of the plate. Four symmetric modes with four cubic strips and an integration scheme of 40 X 3 were used in obtaining the results. As can be seen the agreement is very good in both the deflections and the stresses. Convergence studies of the deflections and the stresses are given in Tables 3.2 and 3.3, respectively. From the results presented in Table 3.2, it can be seen that the central lateral displacement is sufficiently accurate with just two modes and even a single hnear Chapter 3. Numerical Verifications 78 -a co OH co • ^ CO +-» I H - l IS t-l a as CO to CO CN co eo C N - 1 OH OH £ rt O i co H oo CN eo eo n i o oo oo eo co t - - CN T j * CN CO CO 00 l O C N N co n oo co i o CN CO CO © 00 t— H QO l O CN N q co co ^* co co co N CD N CO © CN i - H CN CN o CN 00 0O M r f 115 O oq TJJ co as as oo oo as 00 o as as CN io o o i o) o i ^ d oo oo oi oi CN t—I i—I i—I i—I o eo oo m i — i co IO *C o CN O O i-H i—I CN CN CN CN CN X x X X X X IO CN CN CN O .-H i—I i—I H CN rj< CO O l N i - H t - © i - H CO CO CO CO CN OO i-H CO CO CO l O o i-H eo co CO ^ H CO 00 © i o f-H © C75 01 oo s eo i-H CN *-* o i 06 eo CN as co oo co i - H © © OS CN f H CO CO CO lO OS o i-H CO "^ H rt O _d d o d ° - ! .rt « CO .2 .3 '•+3 , c- rt ~d rt d •H O » J A - + J .rt d <u J5 ""* - o d a, •2 « S x d o 2 S< "-^  rt rt rt —rt * J o CO rt =3 .8 . 3 d b0 rt i-i rt Table 3.3: Linear Orthotropic Plate: Convergence of Stresses Chapter 3. Numerical Verifications 79 strip. With one mode at least three strips are needed for good convergence, while even the single strip results are not too far off. Presented in Table 3.2 are also the results of a single mode analysis with a single strip, carried out for the purpose of checking the accuracy of the analysis if the orientation of the major elasticity axis is changed towards the y axis from the x direction. As can be seen there is excellent agreement between the two deflections. As exact integration was used in the ^-direction, the plate is 'stiffer' than when reduced integration is used. But it is not due to shear locking because this is a thick plate and also cubic interpolation is of a higher order than the minimum required. It just shows the flexibility provided by reduced integration. The convergence of stresses can be studied from Table 3.3. For upto nine inte-gration points, one of them falls on the centre line of the plate with respect to the x direction. Therefore no extrapolation or interpolation is necessary in that direction for the direct stresses. For all others, except the forty point integration with four modes, the results were obtained by extrapolation to the needed point in both x and y directions. For the forty point case the nineteenth and twentieth integration points are too close for extrapolation of the direct stress to the centre line. Therefore the stress was taken as that aj, the twentieth point extrapolated only in y to the other centre line, which in effect is an interpolation in x for the symmetric problem. Also presented in the table are the results for the direct stresses in the twelve point in-tegration for interpolation, extrapolation and their average in the x direction. Here interpolation means that extrapolation is omitted only in the x-direction. As can be seen the average values give better estimates than either the extrapolated or the interpolated values. Also it is to be noted that although the linear finite strip is not suitable for evaluation of the transverse shear stresses, the results presented here are not in much error. The reason for this is the stresses have been evaluated at the Chapter 3. Numerical Verifications 80 Figure 3.3: Moderately Thick, Isotropic S-C-S-C Plate centre line where the detrimental effect of the linear strip is minimal under uniformly distributed loading. This will be explained in detail in Chapter 6, 3.1 .2 G e o m e t r i c a l l y N o n l i n e a r B e h a v i o u r o f a M o d e r a t e l y T h i c k I s o t r o p i c P l a t e This example was taken from Azizian and Dawe( 1985a). This is being presented here as it will later be used in Chapter 6 for the transverse shear stress study. The problem is with regard to a square isotropic plate having pinned conditions along the supports parallel to the y-axis and clamped conditions along the other two supports, thus, S-C-S-C boundary conditions as defined in the reference. Clamped boundary conditions are introduced by specifying zero translational and rotational displacements. The plate properties are shown in Figure 3.3. The uniformly distributed lateral Chapter 3. Numerical Verifications 81 load (q) is related to the non-dimesionalised load Q of the above reference, as, _ ^ 12(1 - . n Q ~ h Eh* { 6 - 1 ] The results are presented with reference to Q. In this study three finite strips and four symmetric modes were considered. In order to study the convergence in strips, the interpolation order of the strips was varied from Hnear to cubic. No variation of the mode number was carried out as it was felt that four modes are sufficient for proper convergence of the (inplane)stresses as understandable from the reference and also verified by the work in Chapter 6. In the £ direction twelve integration points were used for the numerical integration. Reduced integration was carried out in the transverse direction. The comparison of results is provided in Table 3.4. The shown deflection and stress are at the centre of the plate. The latter was sampled at the tension edge. As can be seen from the tabulated results, the responses agree well with those of the reference. The stresses were obtained from the integration point values by various combinations of extrapolation and interpolation in the two inplane coordinates. Con-sider the results for cubic strips presented in the above table. As can be seen, the interpolated values are always lower than those obtained from the reference, while the fully extrapolated values are higher. The best results come from the values ob-tained by extrapolating in rj only(t), and from the average of fully extrapolated and interpolated values (see *$) with their values from the cubic strips being more closer to the values to compare with. As mentioned previously, the average of extrapolation and interpolation would give rise to logical values, and thus, may be selected as the criterion for stress determination at the plate centre of a symmetrical problem. Of course, no extrapolation is necessary in £ when the number of integration points is large enough so as to make the central points fall very close to the centre fine. Chapter 3. Numerical Verifications 82 Strip w (mm) o-y (MPa) Type Q = 10 Q = 50 Q = 100 Q = 10 Q =50 Q = 100 Linear 0.952 4.735 9.328 * 89.54 454.31 915.23 t 117.48 592.78 1185.56 t 124.69 628.86 1256.71 *t 103.51 523.54 1050.40 *t 107.12 541.58 1085.97 Quadratic 0.996 4.947 9.701 * 91.84 466.14 935.76 t 101.43 513.42 1026.77 % 107.53 543.99 1087.04 *t 96.64 489.78 981.26 *\ 99.68 505.06 1011.40 Cubic 0.995 4.944 9.694 * 92.50 469.38 941.91 t 94.11 477.29 957.07 % 99.64 505.16 1012.27 *f 93.30 473.34 949.49 *% 96.07 487.27 977.09 Az. &; Dawe 0.995 4.944 9.68 94.84 482.45 969.91 * Full interpolation, i.e. nearest integration point value f Extrapolation in 77 only X Extrapolation in both coordinates *t Average of * and f values, i.e. interp. in £, avg. of interp. & extrap. in 77 *X Average of* and f values, i.e. Avg. of full interp. and full extrap. Table 3.4: Nonlinear Response of Moderately Thick, Isotropic S-C-S-C Plate: 3 Strips and 4 Modes; 12 Points in 7/ Chapter 3. Numerical Verifications 83 Figure 3.4: Simply Supported Orthotropic Symmetrically Laminated Plate 3.1.3 Linear Behaviour of Orthotropic, Symmetrically Lam-inated Thick and Thin Plates These two examples were adopted from Reddy(1984b) and Putcha and Reddy(1984). The two plates differ only in their thicknesses, making one a thick plate and the other a thin one. The details of the plates are shown in Figure 3.4. The plates are made up of three orthotropic laminae of equal thicknesses, arranged in the 0°/90°/0° layout. A lamina is having the 0° orientation when its major elasticity axis (axis-1) coincides with the x-axis of the plate. The properties of the laminae and the plates are as given below. Ex = 1.746 X 10s MPa E2 = 7.0 X 103 MPa G12 = 3.5 X 103 MPa G13 = 3.5 X l O 3 MPa G 2 3 = 1.4 X l O 3 MPa Chapter 3. Numerical Verifications 84 Strip Type No. of. Modes Int'n Points w (mm) No. of strips per 1/2 jlate 1 2 3 4 Linear 1 2 3 4 5 X 1 7 X 1 9 X 1 12 X 1 0.1185 0.1110 0.1130 0.1122 0.0971 0.0930 0.0939 0.0936 0.0939 0.0897 0.0908 0.0903 0.0890 Quadratic 1 2 3 4 5 X 2 7 X 2 9 X 2 12 X 2 0.0909 0.0877 0.0883 0.0881 0.0915 0.0874 0.0884 0.0880 0.0915 0.0901 0.0884 0.0880 0.0880 Results for comparison Exact 3-D elasticity FSDPT HSDPT 0.0954 0.0881 0.0888 FSDPT: first order shear deformation plate theory HSDPT: higher order shear deformation plate theory Table 3.5: Linear Response of Thick Symmetrically Laminated Square Plate: Central Deflection (mm) ^12 = 0.25 "21 = 0.01 a = b — 1000 mm h/a 0.5 (thick plate) h a = 0.01 (thin plate) 9o 1.0 MPa (uniformly distributed lateral load) As mentioned previously a shear correction factor (K2) of 0.833 was used, although it is a value obtained for isotropic plates, and is the same as that used by Putcha and Reddy(1984) in their first order mixed finite element model. The higher order results of Reddy(1984b), where a model having a parabolic distribution of transverse shear strains through the thickness is utilised, have been Chapter 3. Numerical Verifications 85 Strip No. of. Int'n w (mm) Type Modes Points No. of Strips per 1/2 Plate 1 2 3 4 Linear 1 5 X 1 1359.5 1078.3 1010.7 2 7 X 1 1351.7 1074.8 1007.2 3 9 X 1 1352.2 1074.8 1007.5 4 12 X 1 1352.1 1074.8 1007.4 985.2 Quadratic 1 5 X 2 1004.6 963.5 962.5 2 7 X 2 1002.4 959.5 958.8 3 9 X 2 1002.4 959.8 958.9 4 12 X 2 1002.4 959.8 958.8 958.7 Results for comparison FSDPT 956.7 HSDPT 957.9 FSDPT: first order shear deformation plate theory HSDPT: higher order shear deformation plate theory Table 3.6: Linear Response of Thin Symmetrically Laminated Square Plate: Central Deflection (mm) derived using the Navier series solution procedure. The first order results are available from the mixed finite element mentioned above, but the series solution provided therein was selected for this comparison. When available, the exact results also have been presented. The direct stress crx was sampled at the centre of the plate at its surface, was sampled at the surface at a corner of the plate, i.e. at (0,0, h/2) , where h is the thickness of the plate. The results provided in a non-dimensional form in the references were dimension-alised by the inverse form of the following equations. (h\3 E2 „ , w = w[-\ — X 103 \aj q0a Chapter 3. Numerical Verifications 86 h2 xy T. xy q0a2 The results for the thick plate are presented in Tables 3.5 and 3.7, while those for the thin plate are given in Tables 3.6 and 3.8. Both hnear and quadratic strips were used, with the latter being preferred instead of having a larger number of hnear strips. The number of modes referred to is the number of symmetric modes. As the number of integration points used is too small to take the closest to the centre integration point stress value as that for the central stress, whenever the number of integration points was even, extrapolation of stresses was carried out. Also given are the results for the averages of interpolated and extrapolated values of the direct stress, with it providing better results than by pure extrapolation. Here interpolation is only in the £ direction with extrapolation being still carried out in TI . Use of purely interpolated values in the calculation of the averages would have lowered the results, giving better agreement, as happened with the example in the previous section. As can be observed from the results of the most refined analysis with four quadratic strips and four symmetric modes, the agreement of the deflection with those of the reference is excellent, when the FSDPT result is considered. The direct stress compar-ison also is very good, when the provided average value is considered. The agreement is better for the thicker plate, with the probable cause being the use of a thick plate theory. The agreement of the inplane shear stress also is good, although not as good as that for the direct stress. Chapter 3. Numerical Verifications 87 CM CO CM O 4i co H 0 t— © 0 1 00 i-H t— O i O CO CO T f o o o T f oo oo T f CM 00 CO N CJl CD T f r-H T f O i I O H CO CO T f o d d o CM © 00 CO oo CM l O CO O i-H i-H CM CM CM o d d CN CM d i-H CO eo co l O 1-H CN CO CM CM CM O 00 o O O i T f CM O i—( O i CN © I C 1H O CO CO T f CM CM CM CM CM O T f CO CM i-H i-H I d 00 K3 l O l O CO CM CM CM CN CN CM CO t— CM CO ID oo i f io o> H / i o oo oo o eo N S O ) H i f CO CM CM CO CO i— i CM eo T f eo »D T f 0O CD o O CM CO "HI i o T f o d d CO T f d eo CO o o CM i-H CO l O 00 00 T f ^4 T f o d d oo eo oo T f CD CN O ) O CM CN O i-H CO CD N CD o d d eo O i CM co CO CD CO O i r— co i-H CM CM CM CO CO CM T f CO i-H CD O i » C CD co T f co O i io CM O i »-H i-H CM CM i-H CM CM CM CM ID O t » - CD o T f t— eo co 1-H O i 1-H T f o co i-H CM CM eo CM CM CM CM CM ( O N O i CN O CO N M H I io T f o co co eo CO ID ID *D CO CM CN CM CM CM i-H CN eo T f T f no rt 3 cy CO CO t— eo d o 00 CN 1-H CM c o .m °C rt Cu e c o o I-l o i*5 co H OH "9 Q « •H •rt> c v O v .2 it "E. 'o rt & aJ .2 2 5 J J Is x a »—4 m . . •S *.s IS.2^ » o "5 rt CL w i o CC 4) o £ rt > X co < W &H H— +H-H OH Q Table 3.7: Linear Response of Thick Symmetrically Laminated Square Plate: Stresses (MPa) Chapter 3. Numerical Verifications 88 o t- o Oi Oi ID 0) T f id © «3 n T f Oi C M T f T f T f C M i - H C M Oi T f CO o CO T f CO o c o Oi © c q ID c o C O c o I-H oi C M C M i - H ci (-1 l O io T f i - H C M C M C M T f T f T f T f IO ID ID ID a OJ .OH T f C M I-H o i - H o ID T f b - Oi ID Oi 00 T f C O C O 'C C M i—1 id C O i - H i - H oi CO CO C O c o CO c o ID T f T f T f ID ID l O i O o No. C M o T f C M C O to e o Oi No. i - H °5 C M C M C O C O id ID T f i - H c d l O C O C O id t— C M i - H o o o oo oo C O 00 C O e o C O C O c o CO CO CO CO C O p t-- p C M id oi ? - H C M T f i - H 00 c o C M t — Oi C M ID o pla oo oo co 00 co pla C M l O l O C O i - H C O C M 1-H b -C O C O T f ci i - H oi T f o c o oo T—1 l O 00 c o c o T f l O c o C M e o oo <u C M Oi o i - H T f T f i - H C M C O l O Oi 00 Oi Oi Oi C O 00 00 00 oo ri b of strips ] O IO o C M IO IO CO T f of strips ] C M 10385 9992. 10056 10194 10497 8659. 8371. 8430. 8547. 8812. No. o oi o o o c o . - H oi i - H oi c o C M C O i - H Oi o Oi CO c o * c o i—I C M T f Oi C M C M o e o c o I-H Oi c o CO Oi Oi o c o r— C O o l O IO ID i C IO rt Oi Oi Oi 1-H G o oo CU "O o •H C M co H I-H CM eo T f i—I CM CO T f T f c6 IH -c Cf v IH Cl V O i i «u .2 "3. "3 J 2 CO 2 OH IH «3 M "3 2 « -3 ^ fc & o "2 5 ^ S c w "S £ o * H g £ CH 0) j3 Q > >< co < W Table 3.8: Linear Response of Thin Symmetrically Laminated Square Plate: Stresses (MPa) Chapter 3. Numerical Verifications 89 3.1.4 L i n e a r A n a l y s i s o f U n s y m m e t r i c a l l y L a m i n a t e d S i m -p l y S u p p o r t e d P l a t e s This problem has been adopted from Putcha and Reddy(1986). The model used in that analysis is a mixed finite element based on a higher order shear deformation theory having a cubic variation of the inplane displacements in the thickness coordi-nate. The square plate considered was made up of two equal thickness orthotropic laminae placed in the unsymmetrical cross-ply arrangement. In such unsymmetri-cal laminated plates inplane and bending deformations are coupled even under the hnear conditions, with the largest amount of coupling occurring in the two-ply lami-nates(Turvey, 1977). Two thicknesses are considered, giving a thin plate and a thick plate. The plate geometry, other than the number of layers, is as denoted for the previous example in Figure 3.4. The plate dimensions and properties considered are as follows. E1 1.746 X 105 MPa E2 6.984 X 103 MPa G12 = 3.492 X 103 MPa G 1 3 = 3.492 X 103 MPa ( ? 2 3 = 1.3968 X 103 MPa "12 = 0.25 "21 = 0.01 a = b = 1000 mm h/a = 0.5 (thick plate) h/a — 0.01 (thin plate) qo 1.0 MPa (uniformly distributed lateral load) Chapter 3. Numerical Verifications 90 No. of No. of Strips per 1/2 Plate Modes 1 2 3 4 1 1.5197 1.6999 1.7013 1.6999 2 1.5037 1.6957 1.6999 1.6992 3 1.5037 1.6957 1.6992 1.6992 4 1.5037 1.6964 1.7006 1.6999 Results for Comparison 1.6977 Table 3.9: Linear Response of Thin Unsymmetrically Laminated Square Plate: Non-dimensional Central Deflection A shear correction factor (K 2) of 0.833 was used. Due to symmetry only one-half of the plate was analysed. For the purpose of comparison, the central deflections were non-dimensionalised as wE2h3102 w = q0a* where it; and w are the real and non-dimensionalised values, respectively. In the analysis only linear finite strips were used. For the thin plate, from one to four strips per half-plate were considered, with the number of modes ranging from one to four symmetric modes. In the thick plate analysis, where the convergence was slower than that for the thin plate, upto five strips and five symmetric modes were considered. Reduced integration was carried out across the strips. Along the strips the numbers of integration points used were 5, 7, 9, 12 and 16 for 1, 2, 3, 4 and 5 symmetric modes, respectively. The results used in the comparisons had been provided as the 'exact' solution using the higher order theory, with the exact solution probably being the Navier solution, as suggested by other publications of Reddy and co-workers. The results of the present analyses are given in Tables 3.9 and 3.10. As can be seen from Table 3.9, the thin plate results are excellent and have a fast convergence too. But, as indicated by Table 3.10, for the thick plate the convergence is slower Chapter 3. Numerical Verifications 91 No. of No. of Strips per 1/2 Plate Modes 1 2 3 4 5 1 9.9784 8.6951 8.4943 8.4332 8.3983 2 9.2625 8.2848 8.0840 8.0141 7.9880 3 9.4546 8.3721 8.1887 8.1189 8.0927 4 9.3760 8.3459 8.1451 8.0840 8.0491 5 9.4022 8.3459 8.1538 8.0927 8.0665 Results for Comparison 6.9648 Table 3.10: Linear Response of Thick Unsymmetrically Laminated Square Plate: Non-dimensional Central Deflection than that for a thin plate, and also the results have converged to a value far away from the referenced value. As according to Noor and Burton(1989) its importance increases with the thickness of the plate, the reason for the difference may be the use of the isotropic shear correction factor. Its contribution may be important in the case of the thick unsymmetrically laminated plate, while not being so for the symmetric one presented previously. No study was carried out by varying the factor. 3.1 .5 N o n l i n e a r A n a l y s i s o f a T h i n S q u a r e O r t h o t r o p i c P l a t e u n d e r C o m b i n e d L o a d s The example presented here was adopted from Chia(1980), who used a generalised Fourier series solution procedure on the plate equations. The problem deals with the nonlinear response of a simply supported, square, thin orthotropic plate under the action of combined inplane and lateral loads. The uniformly distributed lateral load which acts throughout the plate was considered with different intensities. The inplane force was considered to act in the direction of the major elasticity axis and was taken as equal to the buckling load of the plate as given by Chia(op. cit.). Chapter 3. Numerical Verifications 92 y Lateral Load q 0 a„ E 2 i x Figure 3.5: Thin Simply Supported Orthotropic Plate under Combined Loading The material of the plate was assumed to have the following properties calculated from the ratios provided in the reference. They were obtained by first assuming a value for E\. Ex = 1.746 X 105 MPa E2 = 0.582 X 105 MPa G12 = 0.29106 X 10s MPa i/i2 = 0.25 i/2i = 0.0833 The plate was considered to be thin, with a thickness to breadth ratio, h/b, of 0.01. h is the thickness of the plate. In accordance with Chia, given that a 0 = 0.5a and 6Q = 0.56 Chapter 3. Numerical Verifications 93 the following non-dimensionalised quantities were defined. o nx , ny q0b ; k = -a ; Q n„ nx ExhA In the above, nx is the applied inplane load (per unit length) in the x-direction, and n„ is the corresponding buckling load, k is the ratio of the inplane loads. From Chia, if A is the plate aspect ratio, the critical load nXcr can be found as Tr 2 [D, + 2D 3A 2 + 7J2A4] fc2A2 [l + A ? ^ j where the applied in-plane stresses, px and py, are given, respectively, by n* , ny px = — and py h with Eih3 D2 = 12(1 - 1/12^21 ) Eoh? 12(1 - Vl2^2\) D2 = u12D2 + 2D4 For the present plate where py is zero, the critical load is 315.72 N/mm and it slightly differs from the already calculated value of 334.85 N/mm implied by Chia in his Table 6.1. As the results presented in the book may have been calculated using the latter, it was used in the present analyses also. Chapter 3. Numerical Verifications 94 All the results given below were obtained by the use of six linear strips per half-plate, except in the case of the six quadratic strips considered with seven symmetric modes as a further check on convergence. For the solution of the nonlinear problem, the Incremental Newton-Raphson procedure was used, but only the lateral load was incremented, while keeping the inplane load constant at the critical value.This is made possible by the use of the flag NUNFAC in the computer code, which incrementally applies only the specified loads. Reduced integration was used in the 77 direction. In the £ direction twelve points were used for three and four symmetric modes, while sixteen points were used for five symmetric modes. To compare with the finite strip analysis, results were obtained from the reference as follows. The deflections for non-dimensionahsed loads of 50, 150 and 250 were obtained from its Table 5.5. The deflections for Q of 100 and 200 were approximately measured out of the graph for Material 1 in its Fig. 5.10. The results from the finite strip analyses are given in Table 3.11. It shows the ac-curacy of the results and also their convergence with even three modes. The apparent error in the deflections for a load of 100 should be due to the approximate nature of the value obtained from the graph, while it is not so with its counterpart of Q = 200 due to its relative largeness and hence, the probable accuracy in measurement. Also it can be seen that the convergence is faster for the lower loads than it is for the higher loads. Chapter 3. Numerical Verifications 95 Type of No. of. Central Lateral Deflection (w/h) Strips Sym. Lateral Load (Q) Modes 50 100 150 200 250 Linear 3 3.30 4.27 4.97 5.54 6.02 4 3.30 4.28 5.00 5.58 6.08 5 3.30 4.28 5.00 5.59 6.09 Quadratic 7 6.18 Chia(1980) 3.29 4.38 5.01 5.61 6.11 Table 3.11: Central Deflection Under Combined Loads (A = = 1) 3.2 Verification of Finite Strips with Fixed-Ended Mode Shapes This section presents the verification of the finite strips having at least one moment resisting "end. They use the mode shapes obtained by the solution of the frequency equation for each new problem. These mode shapes, which are used as the basis functions, are evaluated at present only at the integration points. Therefore, sampling of displacements have to be confined to the positions of those points. 3.2.1 S i m p l y S u p p o r t e d R e c t a n g u l a r T h i n P l a t e W i t h A p -p l i e d E n d - M o m e n t s This example was obtained from Timoshenko and Woinowsky-Krieger(1970) where a Levy type solution procedure had been used. The problem deals with simply sup-ported rectangular and square thin isotropic plates having uniform moments applied along two opposite sides. Although the plates are simply-supported, application of the end-moments means that, if the finite-strips are in that direction, it is necessary to use the 'fixed-ended' modes and the appropriate support bases. This problem is Chapter 3. Numerical Verifications 96 a much more stringent test of the rotational support displacement of the finite strip than the continuous plate analysis presented next. This is because in the latter the rotation is only a secondary effect of the applied load. But in this case it is the end moment, or its support rotation, which causes the plate to deform, thus giving a good test for the assumed displacement field with respect to the rotational bases. The stringency of the test is shown by the fact that a large number of symmetric modes is required to achieve convergence. The problem is depicted in Figure 3.6, with some of the properties being given below. E = 1.436 X 105 MPa v = 0.3 G = 0.55231 X 105 MPa h = 4.5 mm b = 400 mm a — 400 mm (square plate) 800 mm (rectangular plate) M0 = 10 Nmm/mm The analysis carried out was a hnear one. No ability to input applied moments directly or as a stress that varies with the ( coordinate exists at present in the computer programme. Therefore the couples were applied to the plate as two equal and opposite line loads that act along the top and bottom edges of the boundary, while providing the same resultant moment as specified. The use of such resultants does not affect the potential energy or the virtual work of the loads due to the use of Kirchoff's assumption of straight normals. Besides, even if it does affect them, according to Saint-Venant's Principle(Timoshenko and Goodier, 1982), it would have only a very slight effect on the response at the centre Chapter 3. Numerical Verifications 97 Figure 3.6: Simply Supported Plate with End-Moments of the plate where the comparisons are made. The first set of results to be presented here in Table 3.12 is an exercise to show the necessity of having the 'additional' modes described in Chapter 2. The absence of the additional modes would cause the plate to suffer very large transverse shear strains if it has to deflect flexurally by even a very small amount. Consequently the plate becomes very stiff with most of the energy being absorbed in transverse shear. The fact that the stiffening effect is very dominant is clearly shown by the results. The number of modes referred to in the table is the number of eigenfunction modes, and excludes the additional modes. Only symmetric modes were used in the analyses. The remainder of the results will be presented with the number of modes referred to being made to include the additional modes. The adequacy of ten linear strips per half-plate was verified by comparison with the results from an analysis that used ten quadratic strips. These results, which were obtained with fourteen symmetric modes, are shown in Table 3.13. The deflections and the bending moments have converged Chapter 3. Numerical Verifications 98 'Additional' Response at Centre * Modes ? Deflection (mm) Ml Ml No 0.0011 0.0406 0.0795 Yes 0.0431 2.2297 3.4599 Exact 0.049 2.56 3.94 * with 8 symm. modes, 40 X 1 integration * Nmm/mm Table 3.12: Square Plate Response With and Without Additional Modes Plate Type w (mm) Mx (Nmm) My (Nmm) Lin. Quad. Lin. Quad. Lin. Quad. Rect. Sqr. 0.1269 0.0473 0.1265 0.0473 7.555 2.444 7.562 2.465 3.823 3.787 3.818 3.789 Table 3.13: Adequacy of 10 Linear Strips for Rectangular and Square Plates satisfactorily, when going from ten linear strips to ten quadratic strips, showing the adequacy of the former. In these analyses integration schemes of 64 X 1 and 64 X 2 were used for the linear and quadratic strips, respectively. Although such a large number of points may not be necessary in £, it was still used to avoid the need to check the accuracy of integration when a smaller number is used, and to eliminate the need to interpolate or extrapolate the moments. Presented in Tables 3.14 and 3.15 are the results obtained for the convergence of response with the number of modes used. All these analyses were carried out with ten linear strips for one-half of the plate. As can be seen, the new finite strip predicts the responses in a satisfactory manner when compared to the exact results. Fourteen symmetric modes (including the additional modes) are insufficient for the convergence of moments in the square plate as shown by the results in Table 3.15. Chapter 3. Numerical Verifications 99 No. of Modes Int'n Points Central Deflection (mm) Moment Mx (Nmm/mm) Moment My (Nmm/mm) 6 8 10 12 14 Exact 20 X 1 32 X 1 40 X 1 48 X 1 64 X 1 Results 0.086 0.112 0.121 0.125 0.127 0.128 5.15 6.67 7.23 7.45 7.56 7.70 2.74 3.46 3.70 3.79 3.82 3.87 Table 3.14: Rectangular Plate With End Moments: Convergence in Modes No. of Modes Int'n Points Central Deflection (mm) Moment Mx (Nmm/mm) Moment My (Nmm/mm) 6 8 10 12 14 Exact 32 X 1 32 X 1 40 X 1 48 X 1 64 X 1 lesults 0.024 0.037 0.043 0.046 0.047 0.049 1.22 1.91 2.23 2.38 2.45 2.56 1.91 2.97 3.46 3.68 3.79 3.94 Table 3.15: Square Plate With End Moments: Convergence in Modes Due to numerical problems in finding the higher roots of the Timoshenko beam, no more modes could be considered with this example. But, the trend of results shows that, as in the case of the rectangular plate, the convergence of these results will be to values less than the exact. It is felt that the transverse shear flexibility of the present model may be a cause for this, but it is inconclusive. A conclusion to be arrived at from these results is that when using these finite strips, it is always better to lay them in the direction of the longer span. This, anyhow, is normally done in all finite strip analyses to reduce the number of strips. Chapter 3. Numerical Verifications 100 u.d.l Y S.S. S.S. y y y y y y y y b/2 y y y 1 3 4 6 Lwwv 2 5 b/4 y y Finite Strips b/4 y y y t S.S. S.S. Figure 3.7: Two-Span Continuous Plate 3.2 .2 L i n e a r A n a l y s i s o f a T w o - S p a n C o n t i n u o u s P l a t e This section presents the results from the linear analysis of a thin isotropic plate which is continuous over two spans as depicted in Figure 3.7. This example was obtained from Golley and Petrolito(1984) who provided exact results said to have been obtained as suggested by Timoshenko and Woinowsky-Krieger(1970). The plate properties are as follows: E = 1.436 X 105 MPa h = 10 mm v = 0.2 b - 1000 mm The load is uniformly distributed laterally with an intensity of 0.01 MPa. The fixed-fixed and fixed-pinned types of strips were used for the left-hand and right-hand spans of the plate, respectively. As support rotations were to be allowed, Chapter 3. Numerical Verifications 101 the additional modes were considerd. The number of modes referred to later is the number of eigenfunction modes, and excludes the additional modes. Being an un-symmetric problem, all the.'io' modes were considered. The results to be compared are the bending moments in the x direction (Mx) at the Points 1 to 6 and the lateral deflection at Point 3. These points are defined in Figure 3.7. Being a fixed ended plate analysis, the deflections at positions in the £ coordinates were obtained only at the integration points and then simply interpolated between them. As a large number of points were used in the integration, extrapolation and averaging were unnecessary. As mentioned previously, the need to interpolate or extrapolate deflections is only a limitation of the computer code at its present state and is not due to any theoretical reasons. The moments also were sampled at the integration points and interpolated between them. The presentation of results would be in two stages. The first stage which is discussed here shows, using Table 3.16, the adequacy of ten linear strips per half-plate for the convergence of results with respect to the number of strips chosen. Shown in this table are the results obtained by the use of ten linear strips and ten quadratic strips. The same number of modes and £-integration points were used in both analyses. The moments at Points 1, 3, 4 and 6 were obtained by interpolating across the symmetric centre line, which means that they actually are the values along the line of integration closest to the centre line. Now, one and two integration points, respectively, are used in the n direction in the linear strip and quadratic strip analyses. Therefore, when compared to the linear analysis, the quadratic strip analysis has a line of integration points closer to the above mentioned points where the responses are needed. This means, that in addition to the higher accuracy from better discretisation, there is a further increase in the accuracy of the sampled moments in the former, when compared to the latter. Hence the shown slight difference in results Chapter 3. Numerical Verifications 102 Strip Typet No. of Modes Int'n Points Defl. at Pt. 3* Moment Mx Nmm/mm) at Point 1 2 3 4 5 6 Q* L** 6 6 40 X 2 40 X 1 1.517 1.521 -663.9 -663.1 -511.8 -509.6 318.7 318.1 -736.3 -735.0 -563.0 -560.5 372.3 371.2 Q L 8 8 48 X 2 48 X 1 1.506 1.510 -673.3 -672.6 -522.4 -520.3 311.2 310.6 -749.0 -747.7 -573.6 -573.5 367.5 366.4 Exact '. Results 1.474 -684.8 -537.6 311.7 -766.9 -597.8 373.3 * Q = Qudratic Strip + 10 strips for half-plate L = Linear Strip mm Table 3.16: Two-Span Continuous Plate: Convergence in Strips is not solely due to the increase in the number of nodal hnes. Thus, it may be concluded that the use of ten hnear strips is sufficiently accurate. The second stage is described here. It shows the convergence in terms of the number of modes and the accuracy in terms of the number of integration points, on having decided to discretise the problem with ten hnear strips per half-plate. The results are tabulated in Table 3.17. In addition to what was mentioned previously, the following points should be noted with regard to bending moment evaluation. No extrapolation of integration point responses was carried out, but interpolated where possible, even in the n coordinate. Thus, when the moment at an end-support(Points 1 and 2) was needed, it was considered as that at the nearest integration point in £. Thus, as the number of integration points is increased, there occurs an increase in the accuracy of the desired response, in addition to that what may occur through the increased accuracy of numerical integration. In the case of a point on the inner support (Points 4 and 5), the response was obtained by interpolating between the four integration points that straddle it across the two spans. Even here more integration points means that the sampling points are closer to the support. Hence, there is better interpolation and a consequent increase in accuracy. This can be appreciated Chapter 3. Numerical Verifications 103 from the poor results obtained for the support responses when four modes with nine integration points were used. But still, even in this case the responses at the mid-span points are as good as with forty points. This is so because the fourth integration point falls on Point 3 where the responses are required, thus, eliminating the need to interpolate. At present this observation regarding the sampling of moments is true with respect to the deflection also, as that too is evaluated only at the integration points instead of at the required place itself. In conclusion it may be said that the present method predicts the response of the continuous plate with very good accuracy. Engineering accuracy has been obtained with even six modes. 3.2.3 N o n l i n e a r A n a l y s i s o f L o o s e l y C l a m p e d T h i n U n s y m -m e t r i c a l l y L a m i n a t e d P l a t e s This section presents the responses of clamped, thin laminated plates under the action of a uniformly distributed lateral load. Two plate aspect ratios , viz. one and two, are considered. One set of plates is considered to be made up of two equal thickness antisymmetrically placed laminae of an orthotropic material. The other is a plate with six laminae of the same arrangement as the above two-layer plate and square in plan. The arrangement of the layers is to give a 0°/90° cross-ply arrangement, with the starting zero degree layer being at the negative £ side. As mentioned earlier, it is the antisymmetric two-layer laminated plate which gives the highest amount of inplane-bending coupling(nonlinearity) to the problem, and hence, the most critical test. As the number of layers is increased this coupling gets dramatically reduced, and the behaviour would get closer to that of an uncoupled plate(Chia and Prabhakara 1976, Turvey 1977, Putcha and Reddy 1986). The material is considered to be the Material II of Reddy and Chao( 1981a), the properties of which are given below. The longitudinal or filament direction of the Chapter 3. Numerical Verifications 104 No. of Int'n Deflection at Bending Moment (MX Nmm/mm) at: Modes Points Pt. 3 (mm) 1 2 3 4 5 6 4 9 X 1 1.530 -584.8 -442.7 301.6 -649.4 -488.4 354.6 10 X 1 1.470 -595.2 -450.8 293.5 -660.4 -496.9 349.5 12 X 1 1.488 -608.9 -461.4 296.0 -674.9 -508.1 351.2 40 X 1 1.526 -638.7 -484.6 301.0 -706.6 -532.6 354.3 6 10 X 1 1.461 -615.5 -471.4 298.5 -684.2 -520.1 356.3 12 X 1 1.480 -629.7 -482.8 304.2 -699.3 -532.0 360.6 16 X 1 1.499 -645.4 -495.4 310.6 -716.1 -545.4 365.5 20 X 1 1.508 -652.8 -501.4 313.7 -724.1 -551.8 367.9 32 X 1 1.519 -661.2 -508.0 317.3 -732.9 -558.8 370.6 40 X 1 1.521 -663.1 -509.6 318.1 -735.0 -560.5 371.2 8 12 X 1 1.465 -637.1 -491.6 301.8 -710.5 -543.5 360.8 16 X 1 1.487 -652.6 -503.9 306.4 -726.4 -556.2 363.5 20 X 1 1.496 -660.5 -510.4 308.1 -734.9 -563.0 364.7 32 X 1 1.506 -669.4 -517.6 310.0 -744.3 -570.76 365.9 40 X 1 1.508 -671.4 -519.3 310.4 -746.5 -569.3 366.2 48 X 1 1.510 -672.6 -520.3 310.6 -747.7 -573.5 366.4 10 16.X 1 1.477 -655.8 -508.1 306.3 -731.7 -561.8 365.1 20 X 1 1.486 -663.87 -514.6 309.0 -740.0 -568.6 367.0 32 X 1 1.496 -673.0 -522.3 312.1 -749.7 -576.5 369.3 40 X 1 1.499 -675.0 -524.0 312.9 -752.1 -578.5 369.9 48 X 1 1.500 -676.3 -525.0 313.3 -753.4 -579.5 370.2 Exact Results 1.474 -684.8 -537.6 311.7 -766.9 -597.8 373.3 Table 3.17: Two-Span Continuous Plate: Convergence in Modes Chapter 3. Numerical Verifications 105 2a b.c. as at bottom u * 0, v * 0 w =0=y= o b.c. as at left Plate Thickness = h u 4- 0, v + 0, w =0 =y = 0 2b Figure 3.8: Loosely Clamped Thin Laminated Plate materialis denoted as the Direction 1, while the transverse direction is denoted by 2. From the plate material axes, Direction 3 corresponds to the Z direction of the total plate, while if the lamina is in the 0° orientation, the Directions 1 and 2 would correspond to the X and Y plate axes directions, respectively. The material properties used are E2 = 0.582 X 105 MPa Ei/E2 = 40 G12/E2 = 0.6 G23/E2 = 0.5 G3l/E2 = 0.5 v12 = 0.25 v2X = 0.00625 To be consistent with Reddy and Chao (op. cit.) and their reference of Chia and Prabhakara(1976), the geometric properties were defined as follows (see Figure 3.8). 2a = 100 mm Chapter 3. Numerical Verifications 106 h = 1.25 mm (2a/h = 80) 2b = 100 mm (for square plate, a/b = 1) 26 = 50 mm (for rectangular plate, a/b = 2) If the uniformly distributed load is q, ?(2&) 4 v 1 A - 2 P = ^TTTVXIO E2h* defines the non-dimensionalised distributed load P. The non-dimensionalised deflec-tion w is obtained as the ratio of the central deflection to the plate thickness, i.e. w = w/h and the non-dimensionalised stress is defined as, a2 o~~ = tr, xE2h2 where o~x is the direct stress at the tension edge given by £ = h/2 for a positive lateral load. The boundary conditions applied in these problems are the 'loosely clamped' con-ditions where the plate is free to have inplane motions at the support. This a boundary condition that the previous finite strips, which lack support displacements, are unable to consider. The shear correction factor corresponding to isotropic plates was used, and the analyses were carried out with symmetric modes. For the rectangular plate the finite strips were placed in its longer direction. No convergence studies with regard to the number of strips were carried out, except to show the adequacy of the selected number (eight per half plate) with respect to the accuracy of deflections as shown in Table 3.18. As the deflections have sufficiently converged with even four linear strips, it was assumed that the stresses will be in a converged state with eight strips. Chapter 3. Numerical Verifications 107 Load (P) Central Deflection* (mm) 4 L. Strips 8 L. Strips 2 1.192 1.242 4 2.217 2.285 6 3.061 3.133 8 3.770 3.843 10 4.382 4.457 with eight symmetric modes Table 3.18: Thin Two-Layer Rectangular Plate: Adequacy of Linear Strips Eight hnear strips per half-plate with nine symmetric modes under an integration scheme of 40 X 1 for the square plate, and eight modes with 32 X 1 integration for the rectangular plate, were used to obtain the results presented in Figures 3.9 and 3.10, after having verified the adequacy of the number of modes from the results presented in Table 3.19. From this table it can be seen that the two-layer plate, where the couphng is high, needs more modes than the six-layer plate for the convergence of the displacements. As a large number of integration points was used in £, the central displacement was sampled at the £ integration point closest to the centre line and no extrapolation was performed, in this coordinate direction. The samphng point was on the other centre hne of the plate at 77 = 1 of the strip. Also the central stresses were sampled at the integration points closest to the centre and then taken as the average of the extrapolated and interpolated values in rj. In the provided results, the points that refer to the values read off the graphs of Reddy and Chao are only of low accuracy. It should be mentioned that values of the non-dimesionahsed displacements given by Reddy and Chao(1981a) are ten times higher than those of Chia and Prabhakara(1976). With the former results being too high for the validity of the plate theory while there is agreement of the present results Chapter 3. Numerical Verifications 108 3 -^ 2 7 ( Finite Strip Results Reddy&Chao(1981a) 2-layer, a/b = 2 ^ ' ... 2-layer, a/b = 1 Load P = [f2(2b/h)4/E]102 Figure 3.9: Loosely Clamped Thin Laminated Plates: Load vs. Central Deflection with the latter, there should have been an oversight in Reddy and Chao (op. cit.). From the results presented in Figure 3.9, it can be seen that the present finite strip displacement results provide very good agreement with those of the perturbation analyses of Chia and Prabhakara(1976), and the finite element analyses of Reddy and Chao(1981a). The comparison of central stresses in the 2-layer square plate is provided in Figure 3.10. It can be seen that the finite strip results are higher than the referenced values for smaller loads. 3 .2 .4 N o n l i n e a r A n a l y s i s o f C l a m p e d T h i c k U n s y m m e t r i -c a l l y L a m i n a t e d P l a t e s This section presents the nonlinear analysis of clamped thick unsymmetrically lami-nated square plates whose properties other than the thickness are the same as that of the square thin plate presented in the previous section. Here the length to thickness •8 t o Load (P) Square Plate Rectangular Plate 2-Layer Plate 6-Layer Plate 2-Layer Plate Deflection (mm) Stress cTj. (MPa) Deflection (mm) Deflection (mm) 6 modes 8 modes 9 modes 6 modes 8 modes 9 modes 6 modes 8 modes 6 modes 8 modes 2 0.685 0.750 0.756 59.1 61.5 63.2 0.2619 0.2619 1.232 1.242 4 1.248 1.351 1.360 104.2 106.8 109.5 0.5200 0.5200 2.255 2.285 6 1.697 1.822 1.834 136.3 138.2 141.8 0.7713 0.7713 3.085 3.133 8 2.068 2.208 2.222 160.0 161.5 165.4 1.013 1.014 3.781 3.843 10 2.387 2.539 2.554 178.8 179.8 184.2 1.245 1.245 4.383 4.457 e 3 3. f s Co All Analyses with 8 Linear Strips for Half-Plate Numerical Integration: 32 points for 6 and 8 modes 40 points for 9 modes o to Chapter 3. Numerical Verifications 110 6 + Finite Strips 0 0 2 4 6 8 10 Load P = [PQ (2b/h)"/E]10 -2 Figure 3.10: Loosely Clamped Square Thin 2-Layer Plate: Load vs. Central Stress ratio is taken as ten, and the same shear correction factor as that for isotropic plates (0.833) is used except where mentioned otherwise. All the analyses were carried out with eight linear strips per half-plate. This number of strips was not checked for displacement convergence, but was accepted as sufficient. Five symmetric modes were used for the results presented in the compar-isons for accuracy. These were checked for convergence by comparison with four-mode results as shown in Table 3.20, and were found to be satisfactory. An integration scheme of 40 X 1 was used in all analyses. All the presented results have been non-dimensionalised. The load P is defined as and the deflection w is given by w/h. Here, except for a which now refers to the length of a side of the plate and so is equal to 100mm, the definitions of other symbols follow Chapter 3. Numerical Verifications 111 Load P 2-Layer Plate 8-Layer Plate 4 Modes 5 Modes 4 Modes 5 Modes 50 0.2758 0.2765 0.1929 0.1932 100 0.4425 0.4437 0.3450 0.3455 150 0.5557 0.5573 0.4605 0.4611 200 0.6423 0.6441 0.5519 0.5526 250 0.7130 0.7150 0.6274 0.6283 Table 3.20: Clamped Thick Unsymmetrically Laminated Square Plates: Convergence in Displacement (w/h) with Modes those of the thin laminated plate presented earlier. The results to compare were obtained from Putcha and Reddy(1986) who obtained their results using a mixed finite element based on a cubic distribution of the inplane displacements across the thickness of the plate. Figure 3.11 shows a comparison of the finite strip results with those of the refer-ence. The accuracy of the points of the referenced values are not high as they were read from the graphs. Both hnear and nonlinear results have been compared. As can be seen, the linear results agree very well, although the finite strip results are slightly flexible in the case of the two-layer plate. In numerical terms, the finite strip results are deflections of 16.266mm and 10.198mm under the full load, for two and eight layer plates, respectively, with the respective referenced results, on extrapola-tion from graphs, being 15.43mm and 10.10mm. It is felt that this slight difference in the two-layer plate result is due to the use of the isotropic shear correction factor, and that slightly stiffer results would have resulted if a slightly higher factor was used. This is suggested by deflections of 15.777mm and 9.712mm, respectively, obtained for the two and eight layer plates under hnear analyses with a common shear correction factor of 0.9025. Then their use will change the nonlinear results too. But, as shown by the results in Table 3.21, these changes in response are not so large as to change Chapter 3. Numerical Verifications 112 the graphs in Figure 3.11 appreciably. In the case of nonlinear analyses, there are large differences between the results from the present finite strip analysis and those of the reference. This difference cannot be due to the use of the isotropic shear correction factor, because as mentioned above, the results presented in Table 3.21 indicate that substantial changes in these factors cannot give rise to such large changes in the deflections. Therefore, of the two sets of results, viz. of finite strip analysis and those of the reference, the correct one can only be found from an independent analysis. Still, the fact that the graphs of Putcha and Reddy for the two plates intersect with each other casts doubts as to the accuracy of their results. 3.2 .5 T h e N e c e s s i t y o f E q u i v a l e n t E l a s t i c i t y M o d u l i i Although equivalent elasticity modulii were defined in Chapter 2 for the purpose of finding the mode shapes for laminated plate problems, there arise the questions as to whether they provide good results and whether it is not enough just to use an easily calculated value such as the mean value, if the effect of the difference in mode shapes is not to affect the results appreciably. For the first question, the answer has already been given in that it produced good results for both thin and thick laminated plates under nonlinear conditions. The answer to the second question is that it is necessary to use the equivalent modulii instead of the average modulii or any other arbitrary set of values such as those of a single layer, if consistently good results are to be obtained. In order to show this the following results are presented. The problems analysed are the same as those considered in the two previous sections, viz. the thin and thick unsymmetrically laminated plates. The two layer plates were selected from among them as they provide the highest amount of inhomogeneity in the £ direction. As the details of the plates Chapter 3. Numerical Verifications 113 LOAD P Figure 3.11: Thick Unsymmetrically Laminated Square Plates: Comparison of Cen-tral Displacements Chapter 3. Numerical Verifications 114 rt • oo CD rt I i CM rt CO CO •* C ) CM 00 CM CM CO l O T f O CM OO oo e o i o T f , — i f-j e o T f i o co © d o d o Tf o CM o CM CM CO O o i Tf e o co CM e o tO N CD e o T f i o o o o o o t — CM CM O CM CD IO ^ i — I CO T f N CN CO l O t - CO e o T f i o O O © O O © © II II eo CM CO O i to © CM e o ID ID r H T f CO I O e o T f i o o d d © N K5 OO (O CD e o t - T f i o CM I-H © i o oo CM OO O i 00 l O CM CO T f ID CD d © d © © eo T f © r H ID © eO © r H © i-H CO CM © r H N CO 115 CM T f © © _ T f r H 115 CD N d o © Tf O i CM 00 CM O i O i ID © t»- Tf i o O i O i CM 00 0O T f © T f r H T f 115 CO N o © © © © l O © e o c o CM ID T f 1-H © t— e o e o © O i CM 00 OO T f © T f 1-H T f m © t — © © © © © T f © CM t — © t — ID © CM © © CO N T f U5 T f U5 T f r H T f IT5 © © © © © © CM O i © © © O i © T f © CM U5 H CN t - (O O i © ID CN CM T f ID CD IS— d o d o © o © © © © S © ID © ID i-H i-H CM CM ID CM © O i CO e o T f c o © © d CQ Q " 5 CM © T f T f O i CO © o d © i , II II ID ID - CM CM Tf © © CO O i O i o d d « « « < o w Table 3.21: Clamped Thick Unsymmetrically Laminated Square Plates: Nonlinear Displacement (w/h) Under Different Shear Correction Factors Chapter 3. Numerical Verifications 115 were defined previously, they will not be presented here. Four types of analyses are considered, viz., with the equivalent modulii, with average modulii, with the modulii of the first layer, and with the modulii of the second layer. The results of the analyses are compared with each other, and it is shown that the equivalent moduhi is the best to use from among the four methods, especially when thick plates are considered. 3.2.5 .1 C o m p a r i s o n o f R e s u l t s f o r t h e C l a m p e d T h i n L a m -i n a t e d P l a t e Here, both rectangular and square plates are considered. It may be noted that the two layers differ only in their Young's modulii, with the shear moduhi being the same. Thus, the following study becomes one with respect to the equivalent E values only. The analyses used eight Hnear strips for the half-plate and eight symmetric modes. The integration scheme was 32 X 1. The results studied are those under the largest load of P = 25. The responses considered are the lateral deflection, direct stresses, bending moments and the normal inplane forces at the plate centre and the inplane shear stress, inplane shear force, and the twisting moment at a corner of the plate. The central responses are approximated with the results for the (16,1) integration point of the eighth strip, while the responses at the plate corner are approximated as those for the (1,1) integration point of the first strip. The stresses were sampled at three points of each cross section as given in the tables. Here, the question raised being as to which method gives consistent and best results for a given number of modes, a convergence study in modes is not carried out for this thin plate, but just a comparison of results with eight modes is provided. From the results shown in Tables 3.22 and 3.23 for the square and rectangular plates, respectively, it can be seen that the arbitrary use of the moduhi of one of the Response Position £ Equiv. Modulus Avg. Modulus Layer 1 Modulus Layer 2 Modulus (mm) E = 0.228 X 106* E = 1.1931 X 106* E = 0.0582 X 106* E = 2.328 X 106* G = 0.0291 X 106* G = 0.0291 X 106* G = 0.0291 X 106* G = 0.0291 X 106* Deflection** 2.5542 2.3947 (-6.2%) 2.3943 2.3974 K -0.6250 -1873.0 -1775.2 (-5.2%) -1835.4 -1909.6 -0.3125 237.7 301.5 (26.8%) 249.2 277.7 0.6250 181.3 179.1 (-1.2%) 177.8 186.8 -0.6250 -173.1 -159.9 (-7.6%) -157.2 -160.8 -0.3125 -102.1 -93.6 (-8.4%) -92.1 -93.7 0.6250 2837.8 2633.4 (7.2%) 2541.5 2641.0 Ml 117.7 102.8 (-12.6%) 113.34 115.8 N\ X 275.6 256.8 (-6.8%) 245.5 258.0 220.0 259.8 (18.0%) 226.4 248.0 1 w 254.4 242.7 (-4.6%) 219.9 245.9 T* xy -0.6250 -9.77 -13.39 (37.0%) -14.92 -7.42 -0.3125 -10.69 -14.47 (35.4%) -16.16 -7.89 0.6250 -13.46 -16.63 (23.6%) -19.88 -9.32 Aft xy -0.4805 -0.5636 (17.3%) -0.6451 -0.2476 11XV -14.52 -19.44 (33.9%) -21.75 -10.46 * MPa f N mm/mm : N/mm ** mm Chapter 3. Numerical Verifications 117 o X oo 3 o ^ C M CM eo Ui CM s 11 o CM T f CM T f T f O i i j S; *o H C ) «S CM CO 2 ^ S » r< S (O N 2 00 CR cq co eo ^ co i — i oo _JJ o TJ- O ^ < ? 2 ^ © l O T f 00 o T f CM i—i eo eo cq o 00 OS O 3 o os o t-i I CM i—l CO O i ID CM O O © © ID CM © CO © CM eo eo S ^  °° 2 o >o t"- <° " O O O l rt CM ID 00 O i eo T f t~ eo t - 2S ID © T f 2° i rt CM IO N N CN N eO ID O i "0 CO O i OS OS © © >rt eo O i < O) CN t—I © bb rt © < II II ^ S » J oo ^ ^ eo eo oo t ~ T f rt CM eo T f eo eo O i r— co CO O i S5 cq CM j^, T f eo or> rt ci-oT«> oo S © CM CO oo t — CM eo 2° 2 ^ CM © eo cq cq eo eo T f co & , CO O i © J o 2 3 ^ . CM g w W rjj P. t-eo T f ^ T f co LL o T f 3 rt eo " : eo 2 <» © ID i—I © CM N « to eo oo T f rt CO O i ID © CM i rt CM IN N H © oi CO CM CM ID O i eo I T 00 O OH ID _ rt 8 ° O O ID o JO CM § £J M CM © eo co © © o © ID ID CM CM rt © eo © © © ID CM CO © cu oo a o Cu oo & Table 3.23: Thin Two-Layer Rectangular Plate: Effect of Different Equivalent Modulii Chapter 3. Numerical Verifications 118 layers will give rise to results that differ from each other and also from the results for the average and equivalent modulii analyses, especially with respect to the shear stresses. Considering the results for the square plate, some of the conspicuously large dif-ferences occur between those from the equivalent and average modulii analyses with respect to the following. From Table 3.22 they are o~x at £ = -0.3125 mm (26.8%), Mx (-12.6%) and Nx (18.0%) at the centre, and all the quantities at the plate corner with differences in the range of 17.3% to 37.0%. All percentage differences were calculated with respect to the equivalent modulii results. Considering the central deflection, it can be seen that there is a reduction of 6.2% when changing from equivalent to aver-age modulii analysis, making the already slightly higher result(see Fig.3.12) slightly lower, thus making no contribution. In the case of the direct stress o~x at the top sur-face, it lowers the already low value (see Fig.3.10). Thus, with respect to the available results for the square plate, equivalent modulii results can be said to be better than those from the average modulii, although it is not yet conclusive from the presented results. In the case of the rectangular plate (Table 3.23) there is a lowering of the central displacement, making the average modulii result lesser than the well agreeing value from the equivalent modulii analysis. Thus, again these point to the appropriateness of using the equivalent modulii instead of the average values. In both the square and rectangular plates certain central responses, for example the deflection, can be seen to be having similar values under the average and individual layer modulii analyses. The reason for this is not known and is not pursued in this thesis. In the presented results, under different analyses, the responses that can be com-pared with the referenced values have only slight differences with eachother. But it Chapter 3. Numerical Verifications 119 Load P = [PQ (2b/hf/E]102 Figure 3.12: Loosely Clamped Thin Laminated Plate: Effect of Elasticity Modulus on Central Deflection is not so with the thick plate example to be presented next. 3 .2 .5 .2 C o m p a r i s o n o f R e s u l t s f o r t h e C l a m p e d T h i c k L a m -i n a t e d P l a t e Here, the square thick two-layer plate considered in Section 3.2.4 is used for the study of the use of the equivalent modulii approach. In this case the Young's modulii and the transverse shear modulii are different in the two layers. This and the fact that this is a thick plate problem makes this provide larger differences between the results of different approaches. (Still, as can be seen from Table 3.25, due to the equal thickness of the two layers the equivalent and the average shear modulii are having the same value.) As was done in the previous analyses of this problem, only up to five symmetric modes are considered. Although the stresses have not converged under Chapter 3. Numerical Verifications 120 Mode Equiv. Mod. Avg. Mod. Layer 1 Mod. Layer 2 Mod. 1 18.81 27.52 25.68 21.25 2 44.47 43.96 34.71 55.12 3 75.56 60.71 43.75 100.7 4 109.4 77.62 52.72 154.5 5 144.8 94.38 61.69 213.9 6 180.9 111.1 70.63 277.3 7 217.3 127.7 79.56 343.5 8 253.8 144.3 88.48 411.8 9 290.1 177.3 - 481.5 Table 3.24: Thick Two-Layer Rectangular Plate Finite Strips: Frequency Parameter 'tV under Different Equivalent Modulii this number of modes (see next section), it is unimportant as no referenced values are available for comparison. As before, also analyses with the modulii of individual layers are considerd. The details of discretisation are as given in Section 3.2.4, and the results presented here are with respect to the maximum load considered in that nonlinear analysis. The results from this study are presented in Table 3.25. In the case of the Layer-1 modulii analysis, the ninth mode frequency parameter 'tb'(see Equation A.4 of Ap-pendix A) did not converge during the frequency analysis, preventing the consider-ation of five symmetric modes. Therefore no responses have been presented with respect to it. In the meantime, it may be mentioned here that the equivalent modulii analyses in Section 3.2.4 were limited to a maximum of five modes not only because of the convergence of the central displacement (which is what was wanted there), but also because of the inability of the mode finding procedure to find the frequency of the next needed mode due to non-covergence. Thus, it is necessary to further study the solution procedure for the frequency equation as it stands at present. Chapter 3. Numerical Verifications 121 -2 o S -d o CN l-i I X CN CO o X o i-H OS CN O T f c o c o ° - CO J3 o o> £3 l O CN t - H « T f T f HH, w 0 0 TT . ro Ol O „ IO CN o -r «? ° Ol ° co £• o i CN <£> CN CN 2 « CO to O ^ O t-H O X X KH 3 1(5 (O lO i-H lO T f I- l I r J .4 <=> • co <o o 3 r H ^ * ° © o < " II K) N ® Ol "5 S CN « ^ rH IO 2 c o c o O i-H CN d "? £ s I £ ^  £ S ! CO c o ^ § fe 1^ O '"I eo ^ - H i-H r H t -CO OS w » O .3 2 -^ X * . _ £ CN > "I o •g ° d ^ 11 r H ~T »C e o CN O r H CN c o eo i-H eo lO CO oo ^ « & £3 w m H 0 O "5 s ! J £ £ -7 s ° g S CN Ol oo g> o i O l t— CO CO CO OO CN T f r - i N (O T f T f CO CN H H oq CN IO r H c o fl -o co O OH O lO IO 0 O lO o >?t?115 O lO o "? C? l O cu co fl o OH SfS.B rt OH Table 3.25: Thick Two-Layer Square Plate: Effect of Different Equivalent Modulii Under Five Modes Chapter 3. Numerical Verifications 122 • CD ! ° 2 o CM IH I CM oo »D © x o 1—< Oi CM © © II O © d N . o o n Oi Oi ^ Oi i—I CO lO ° ° . T f 1 S § 8 £2 T f CM — « TJ- rt 25 eo rt °> » ^ -* 22 co <=> CM oo CO T f rt T f rt CM co o X ID kD T f IH rt CD o X T f co 5 K5 (O ^ lO <—I rt o d II O co cq t— CN I O CD O rt lO 00 T f Oi 01 rt I ffl _ o • Oi • to • eo £5 "2 ID rt ^ CM to m s J j cq co o CO © ^ CO rt ID 00 rt P rt eo m fc rt CM o to o 3 X X w O © •rt CO ^ *D bb d < II eo CM O I O eo to P o o o o 00 CM lO O O CO o o o © © © o d d oo eo CM eo t— Oi o ID rt CM O O co © © © o d d © CO rt § o o o o o P d d P O o P d * 03 <» » © 3 2 --5 X rt CO CO >• £ • 3 o cr || W os ^: eo t s ! © a 00 CO Oi CM co co • rt eo 0 - ~ T f 52 eo eo 2 ^ H ^ ID O t^; T f CO 5£5 00 ID — T f O ^ eo rt ID CM o Oi Oi © ^ <=> T f CM CO fM CM rt T Oi d Oi CO CM a • o CQ o Cu, O ID o ID CM © »D © kD CM © ID o ID CM CO o CM as V «3 s Table 3.26: Thick Two-Layer Square Plate: Effect of Different Equivalent Modulii Under Four Modes Chapter 3. Numerical Verifications 123 As can be seen from the results in Table 3.25, the response of the average mod-ulii analysis is very poor, with it being very stiff and providing a negative central displacement, thus, certainly pointing out to numerical difficulties with the modes. In the case of the Layer-2 modulii analysis, the central deflection is better, while there are wide differences in the stresses, with a maximum of 40.9% in the case of ux at the bottom surface of the plate. As mentioned previously, it is inappropriate to consider an arbitrary layer because of the unpredictability of the results as shown by the non-convergence of the Layer-1 analysis here, and the very poor results given by it under analysis with four symmetric modes as shown in the next section. Further, given in Table 3.24 are the frequencies obtained under the analyses. They show a wide variation for the four methods considered. In the case of the average modulus, it seems that the mode found as the ninth is in actuality the tenth because of its larger difference from the eighth mode frequency, when compared to the differences between the others. This should probably be the reason for the presence of a negative central displacement in this case. From the above presented results it can be concluded that the equivalent modulii as defined in Chapter 2 are the best to be used in the analysis of fixed-ended laminated plates. This will be further proven by the results presented next where the convergence properties under the equivalent and Layer-2 modulii analyses are compared. 3.2.5.2.1 Convergence Properties Under Different Modulii This section presents a study of the convergence properties of the solutions obtained under differ-ent modulii by the comparison of responses obtained with four and five symmetric modes. These results further vindicate the use of the equivalent modulii instead of the Layer-2 modulii (which still can be chosen only arbitrarily) as the former has bet-ter convergence than the latter. The results under the average and Layer-1 modulii, although presented, are not used in the convergence comparison because of their large Chapter 3. Numerical Verifications 124 Response Position £ Equiv. Modulus Layer 2 Modulus (mm) E = 0.1666 X 106* E = 0.0582 X 106* G = 0.02037 X 106* G = 0.02910 X 106* Deflection -0.3 -0.7 o-x -5.0 -90.4 -112.8 -2.5 15.8 32.1 5.0 2.4 5.9 °y -5.0 -2.2 -4.0 -2.5 -4.8 -9.0 5.0 -0.4 -0.9 Mx 35.7 77.3 My -0.5 -1.1 Nx 14.2 28.9 Ny -0.2 -0.6 TXy -5.0 -8.7 -20.4 -2.5 -9.0 -21.1 5.0 -7.5 -17.8 Mxy -8.3 -19.4 -10.3 -24.5 * MPa Table 3.27: Thick Two-Layer Square Plate: Percentage Error in Four Mode Results When Compared to Five Mode Results errors. Presented in Table 3.26 are the results obtained under four symmetric modes using the four types of modulii described earlier. Otherwise all the details of the analyses and the plate are the same as provided earlier. From the results presented in the table, it can again be seen that the use of the Layer-1 modulii and average modulii is not advisable. As mentioned earlier, the purpose of this section is to show that the equivalent modulii results are better than the Layer-2 modulii results. For this purpose, the percentage differences in the presented results under four and five symmetric modes Chapter 3. Numerical Verifications 125 when using the equivalent and Layer-2 modulii are shown in Table 3.27. The per-centages have been calculated using the ratio of results given by (Four Mode - Five Mode)/Five Mode, under the respective analyses. As can be noted the percentage errors for the equivalent modulii results are always about one-half of those for the Layer-1 modulii, except in the case of crx at the bottom surface where both are in error by large amounts. The reason for the latter is not known. Still, from the above results it can be concluded that, from among the methods considered, the equivalent modulii are the best to be used in the calculation of the mode shapes. 3.3 Analysis of Initially Deflected Plates This section presents the verification of the formulated theory and its implementation with respect to plates having initial lateral deformations. Two types of problems are considered. The first deals with the behaviour of a plate loaded in the inplane direction, and was obtained from Yamaki(1959). Although the results available there dealt also with the post-buckled range, the present verification was kept within the pre-buckled range. The second type deals with an initially deformed simply supported plate loaded laterally. The load can be either on the concave side or on the convex side. The latter provides snap-through buckling, while the former does not have this phenomenon. The examples were obtained from Rushton(1970 and 1972). All the plates are considered to be square, isotropic and of a length to thickness ratio of 20. The plate thickness is denoted by h. Chapter 3. Numerical Verifications Figure 3.13: Initially Deflected Plate Under Edge Loads Chapter 3. Numerical Verifications 127 3.3.1 I n i t i a l l y D e f o r m e d S i m p l y S u p p o r t e d P l a t e L o a d e d o n O p p o s i t e E d g e s This comparison is with results of Yamaki(1959), where a square isotropic pin-ended plate is being acted upon two opposite edges by compressive loads, while constraining these sides to be straight. The plate and its boundary conditions are shown in Figure 3.13. The plate properties are as follows. E = 2.05 X 10s MPa v = 0.33333 b = 1000 mm h = 50 mm b is the length and h is the thickness of the plate. The edge load was considered to be of an average value of 1729.895 MPa, which is 91.2 percent of the critical load. The initial deflection has been provided as the first sine modes in the two coordinate directions. The solutions of Yamaki had been obtained as a double Fourier series by the solution of the plate differential equations by the Galerkin procedure. In the presented results the numerical intergration was carried out with 32 X 1 and 32 X 2 points for the hnear and quadratic strips, respectively, for all the numbers of modes considered. In Yamaki's analyses, the unloaded edges had been kept stress free by allowing free movement of those edges in the inplane degrees of freedom. In the loaded edges, the sides have been kept straight, with the given load being only the average inplane compressive stress on a side. This boundary condition and the loading were achieved as follows. Noting that the load is applied at the ends of the finite strips, the loaded edge was kept straight by specifying the equality of the inplane u support degrees of freedom there. In the case of the load, it was first assumed to be distributed uniformly and the problem solved to obtain the inplane stresses at the integration points closest to the loaded edge. Assuming sufficient integration points have been used to make Chapter 3. Numerical Verifications 128 No. of Symm. Modes w* (mm) Nx (N/mm) at Strip No. 1 2 3 4 1 10.268 - 0.8645 - 0.8732 - 0.8733 - 0.8698 2 10.386 - 0.8702 - 0.8751 - 0.8749 - 0.8684 3 24.484 - 0.9141 - 0.9588 - 0.9437 - 0.9213 4 24.488 - 0.9148 - 0.9598 - 0.9439 - 0.9209 5 24.489 - 0.9151 - 0.9603 - 0.9440 - 0.9207 6 24.489 - 0.9152 - 0.9606 - 0.9440 - 0.9206 * Yamaki (1959) = 24.798 mm Table 3.28: Response of Edge Loaded Plate with Four Linear Strips them close enough to the edge, it was then assumed that the load is distributed in the same manner as these inplane stresses. Then a new analysis was carried out by distributing the inplane load accordingly on to the strips. This distribution was achieved by multiplying the inplane stresses by a factor. This factor was found as that, on numerical integration of the stress along the loaded end, would give the same total load as applied. An iterative process was carried out until this factor converged to an accuracy of one percent. In all the analyses only two iterations were required to achieve this convergence. In the present analyses the initial deflection was provided through the specification of the amplitude of the first mode at each nodal line. The results of the analyses are shown in Tables 3.28 and 3.29, the first of which shows the results from linear strip analyses while the second has results of the quadratic strip analyses. It can be seen from Table 3.28 that the solution has converged sufficiently with three modes as it was with Yamaki. But comparison with displacements from quadratics strips given in Table 3.29 shows that four linear strips are not sufficient for strip-wise convergence, and that the result is more flexible than that of Yamaki. This flexibility at a first glance may be attributed to the presence of some amount of transverse shear flexibility Chapter 3. Numerical Verifications Nx X 10s  (N/mm) at Int. Points of Strip No. Pt. 2 - 0.9270 - 0.9205 - 0.9190 - 0 9186 - 0.9183 - 0.9182 Nx X 10s  (N/mm) at Int. Points of Strip No. r—\ OH - 0.9303 - 0.9273 - 0.9264 - n Q961 - 0.9260 - 0.9259 Nx X 10s  (N/mm) at Int. Points of Strip No. eo Pt. 2 - 0.9326 - 0.9344 - 0.9342 - 0 9342 - 0.9342 - 0.9341 Nx X 10s  (N/mm) at Int. Points of Strip No. Pt. 1 - 0.9463 - 0.9545 - 0.9554 - 0 9558 - 0.9559 - 0.9560 Nx X 10s  (N/mm) at Int. Points of Strip No. CN CN CM - 0.9430 - 0.9627 - 0.9655 - 0 Q6fi5 - 0.9669 - 0.9671 Nx X 10s  (N/mm) at Int. Points of Strip No. OH - 0.9370 - 0.9613 - 0.9649 - 0 Q661 - 0.9667 - 0.9670 Nx X 10s  (N/mm) at Int. Points of Strip No. Pt. 2 - 0.9233 - 0.9509 - 0.9548 - 0 Q'ifiO - 0.9566 - 0.9570 Nx X 10s  (N/mm) at Int. Points of Strip No. i-H OH - 0.8744 - 0.9002 - 0.9035 - 0 Q049 - 0.9044 - 0.9044 w* (mm) 25.375 25.982 26.008 26.012 26.014 26.014 No. of Symm. Modes H cs n TC m <o O i O i Table 3.29: Response of Edge Loaded Plate with Four Quadratic Strips Chapter 3. Numerical Verifications 130 in the plate whose thickness to span ratio is 0.05. But it cannot be the reason as the analysis of a plate with this ratio equal to 0.01 also provided similar results. Not presented is a set of analyses carried out with four quadratic strips under exact integration in n using an integration scheme of 32 X 3 points for all modes. It gave almost identical results to that of the reduced integration presented here in Table 3.29, indicating that the integration scheme is not a cause for the above observation. Whatever the cause is, still, the results are close enough for practical purposes. Also provided in the tables are the axial forces at the end of the plate at the first integration point in £. This point is at a distance of only 0.00137 times the plate length from the end of the plate. The forces have achieved the standard distribution observed in such plates when the unloaded edges are free to move and stress free while the loaded edges are kept straight(Coan 1951, Allen and Bulson 1980). That is, the load intensity at the middle and outer portions of the plate is lesser than at the other areas. Figure 3.14 shows a diagram of this stress distribution plotted with the results from the quadratic strip analysis with five modes. 3.3.2 I n i t i a l l y D e f l e c t e d P l a t e s U n d e r L a t e r a l L o a d s This section considers the response of an initially deflected plate under a lateral uniformly distributed load. Two sets of loads are considered in the analysis of these problems obtained from the works of Rushton(1970 and 1972) who solved them using a finite difference, dynamic relaxation technique on the thin plate equations. The first problem deals with a load applied in the same direction as that of the initial deflection, resulting in only an increase in the deflections in that direction. This problem and its solution is available in Rushton(1970), and is called 'a problem with the concave side of the plate loaded'. The second problem has a load acting in the direction opposite to that of the initial deflection, giving rise to 'snap-through' behaviour at a Chapter 3. Numerical Verifications 131 0.0 -0-2-co Q. -1.0 H 1 1 1 1 1 1 1 1 r 0 200 400 600 800 1000 Distance from Plate Side (mm) Figure 3.14: Load Distribution in the Edge-Loaded Plate critical load. This problem and its solution is available in Rushton(1972), and it has been called a problem with the convex side of the plate loaded. Although Rushton had considered behaviour closer to the snap-through buckling load, and had obtained results, the present method failed to reproduce those results near that load. For both of the problems presented, the initial deflection was considered to be in the form of a first sine mode in both the x and y directions with a central value of 0.1 h. The boundary conditions of the plate were of the pinned type as shown in Figure 3.15. All the plates were square in plan and had a thickness to span ratio (h/b) of 0.05, resulting in a slight amount of transverse shear deformation. The geometric and material properties used are given below, supplementing those in Figure 3.15. E = 2.05 X 10s MPa v = 0.3 b = 1000 mm h = 50 mm Chapter 3. Numerical Verifications 132 Figure 3.15: Initially Deflected Plate Under Lateral Load The lateral load q is nondimensionalised to Q, defined as V Eh4 If the initial deflection is denoted by u>,- and the response of the plate under the load is given by w0, then the total deflection w defined as W = Wi + w0 is the distance of the plate middle surface after deformation, from that of an unloaded flat plate. All the problems were analysed by the use of four quadratic strips for a half-plate, with one to three symmetric modes. An integration scheme of 12 X 2 was used. As evident from the results presented in Tables 3.30 and 3.31, the results have converged well with three modes. No convergence study in strips was carried out as Chapter 3. Numerical Verifications 133 Load (Q) Rushton (1970) No. of Modes 1 2 3 9.16 0.316 0.318 0.314 0.314 36.6 0.747 0.757 0.742 0.744 146.5 1.38 1.413 1.367 1.374 586 2.03 2.387 2.278 2.295 2344 3.73 3.890 3.681 3.716 9377 5.98 6.257 5.897 5.960 Table 3.30: Central Deflection, Wo/h, of Plate with Load on Concave Face past experience, including the previous analyses of the edge-loaded plate, suggested that four quadratic strips are sufficient. 3.3 .2 .1 P l a t e L o a d e d o n t h e C o n c a v e S i d e The results are presented as the non-dimensionalised additional deflection w0/h at the centre of the plate. As can be seen from Table 3.30, the agreement is excellent with three symmetric modes, except in the case of a load of 586. The agreement is very good even with just two modes. 3.3 .2 .2 P l a t e L o a d e d o n t h e C o n v e x S i d e The results are presented as the non-dimensionalised total deflection w/h at the centre of the plate. Only symmetric modes were considered in these analyses as according to Rushton(1972), for a square plate the symmetrical modes give the least buckling load. In Table 3.31, the change of sign of the total deflection indicates the snapping-through stage. According to Rushton's results the snap-through buckling load lies between values of 2.02 and 2.29 of the load Q. When three symmetric modes are considered, the agreement of the present results is very good at loads away from the Chapter 3. Numerical Verifications 134 Initial Defl. Load (Q) Rushton (1972) No. of Modes 1 2 3 O.lh 0.92 0.052 0.059 0.059 0.059 1.37 0.039 0.038 0.039 0.039 1.83 0.018 0.017 0.018 0.018 2.02 0.012 NS NS NS 2.29 - 0.018 -0.004 -0.003 -0.003 4.58 - 0.133 -0.108 -0.105 -0.106 6.87 - 0.193 - 0.203 - 0.200 -0.200 9.16 - 0.277 - 0.287 - 0.282 - 0.283 36.6 - 0.80 - 0.807 - 0.782 - 0.794 146.5 - 1.47 - 1.495 - 1.449 - 1.456 586 - 2.42 - 2.480 - 2.371 - 2.388 2344 - 3.86 - 3.985 - 3.778 -3.813 9377 - 6.15 - 6.356 -5.995 -6.059 NS : Not sampled Table 3.31: Central Deflection, w/h, of Plate with Load on Convex Face buckling load. Figure 3.16 shows a plot of Q against the total deflection w/h. Results from both the analysis of Rushton and the present finite strip method are shown. The finite strip results presented in this figure were obtained by the consideration of forty load steps upto a load Q of 9.16. As could be seen Rushton had obtained a snap-through behaviour, whereas the finite strips provide a gradual transition from negative to positive deflection. With respect to the above non-production of the snap-through stage by the finite strips, it may first be stated that there is a difference in the way the initial deflections are specified in the n direction in the two methods of analyses. Rushton used smooth sine shapes in both the transverse and longitudinal directions. In the finite strip analysis, a smooth sine shape is specified only in the longitudinal direction. In the transverse direction a piecewise curve of the order of the displacement interpolation Chapter 3. Numerical Verifications 135 0.1 0.0 i % •0.2 --0.3 \ \ — 'V, \ " Legend: • Rushton(1972) \ Finite Strips 1 1 -[-T-I 1 1 1 1 1 20 40 60 80 100 Load QX10 916 Figure 3.16: Load vs. Deflection Near Snap-Through Load is used. This is achieved by specifying the ordinates of the transverse sine shape at the nodal line positions as the amplitudes of the longitudinal sine curves. These are then interpolated between the nodal lines. As the plate thickness to length ratio is 0.05, it was suspected that the presence of the transverse shear flexibility in the present model may have caused the phenomenon. But it was not so as a plate with a thickness to length ratio of 0.01 also provided similar results, on considering even a very large number of load steps, both in the loading and unloading regimes. This phenomenon is not investigated any further in this thesis. Chapter 4 Analysis of Rectangular Containers This chapter deals with the use of the finite strips in the analysis of rectangular containers. Presented first is a short account of some of the analytical and numerical methods used so far by others in the analysis of such containers. The second section considers the analysis of a square container having pinned wall-wall junctions with the wall bottom either pinned or fixed to a rigid base. The analysis is verified with an experiment. Also presented are the differences between the results from analyses under various assumptions. It will be shown that to obtain good results for this flexible 'pinned'(at wall-wall joints) square container, it is necessary to consider a nonlinear analysis that has the ability to incorporate the movement of the vertical corners. The third section presents results from an analysis of the same container with eccentric vertical loads applied at the centres of the top edges of two opposite sides. This also is verified with an experiment. It is felt that this example will demonstrate the ability of the model to predict the behaviour of the container under vertical loads such as those likely to come from stacked containers. The fourth section presents the analysis of containers with moment resisting vertical joints, also to be referred to as 'fixed' containers. Two examples will be described. The first is a container square in plan, with equal properties for all sides, so that the vertical corners will not rotate under vertically symmetrical loadings. The second is a rectangular container where the corners will rotate about a vertical axis. The importance of nonlinear analysis with movable corners will, again, be shown. 136 Chapter 4. Analysis of Rectangular Containers 137 4.1 Some Previous Analyses of Rectangular Con-tainers Under Hydrostatic Loads The analysis of water containers has a long history and a good review of work prior to the mid-seventies was provided by Wilby(1977). Two analyses of flexible containers since then are those of Golley and Petrolito(1984) and Golley et a/.(1987). According to Wilby(1977) the methods of analysis used with respect to rectan-gular water containers could be divided into three, namely analytical, analogy and numerical methods. Among the analytical, the works of Davies(1962a, 1962b, 1963, 1964) and Grigorian(1966) must be cited. Davies(1962a) considered the support con-ditions on the behaviour of long rectangular containers by assuming two-dimensional behaviour. The effect of different support conditions of the flexible base was analysed by considering a container strip that behaved two-dimensionally. Davies( 1962b) pre-sented an approximate method of analysis of a container square in plane and having square panels. The interaction between the flexible wall and the flexible base was considered, but that between the adjoining walls was not, assuming instead clamped conditions. Although this is adequate with respect to the rotation about the verti-cal axis, it is not so with respect to other degrees of freedom if geometric nonlinear effects are to be considered. The approximate linear solution was obtained by the Kantorovich method. The wall and base problems were solved separately and the in-terconnection between them was treated in an approximate manner by superposition, while considering the compatibilty of rotations and the equilbrium of moments only at the centre of the connection, instead of along the entire length. Various support conditions were studied by Davies, and experimental verifications were provided, e.g. containers supported on edges( 1962b), nonlinear effect of lifting bottom of a con-tainer supported on a rigid plate(1963), rigid base, flexible base supported at corners, Chapter 4. Analysis of Rectangular Containers 138 flexible base supported on granular soil etc.(1964). In all the above, the plates were considered to be thin and isotropic. Grigorian(1966) extended this method to con-sider the wall-wall interaction, but without considering the wall-base interaction. It seems according to the wording that the consideration of this interaction was not approximate. Only the Hnear behaviour had been considered. Among the numerical analyses methods, the finite differences work of GhaH(1957) found difficulties with boundary conditions and the need for various approximations with regard to them, concluding that the method was unsuitable for use on containers. Cheung and Zienkiewicz(1965) presented a finite element analysis of a container in the shape of a cube, resting on an elastic foundation. In this Hnear elastic analysis, from the limited information available, it seems that the waU-waU corners had been taken as clamped on considering the symmetry of the problem. Davies and Che-ung(1967) applied the finite element method to long rectangular containers, while considering the corners and the bottoms of the waUs to be clamped. They were able to show that the bending moments at and near the waU-wall corners are not as neg-ligible as had been assumed up to then. Cheung and Davies(1967) appHed the finite element method to the analysis of rectangular containers on elastic foundations, while considering the wali-wall-base interaction. However, these were also Hnear analyses, with the only wall-wall interaction effect considered being that what comes from the rotations and moments. Davies et af.(1970) had appHed this method to the analysis of a long rectangular container with a roof. More recent works on apphcation of numerical methods to the analysis of rect-angular containers are those of GoUey and PetroHto(1984) and GoUey et a/.(1987). Even these do not consider the effect of the inplane displacements, although the ro-tations that occur at corners of rectangular containers were considered. The method of Golley and PetroHto(1984) is a semi-analytical approach, where each of the panels Chapter 4. Analysis of Rectangular Containers 139 was taken as a single element. The panel boundaries that do not meet at corners were considered to be rigidly connected to beams. The ratio of the flexural rigidity of each beam to that of its connected plate was to have the same value for all the plates that met at a corner. The distribution of moments along the wall corners was assumed apriori. Then a solution was sought to satisfy the linear thin plate bending equation within the element. Golley et o/.(1987) used a vertically placed finite strip element for the analysis of rectangular containers. The element was formulated by the combination of the shape functions of a thin rectangular finite element and the mode shapes used in the simply supported thin-plate finite strip. The boundary conditions for fixed ends were satisfied by the use of Lagrange's multipliers. The method of analysis proposed in this thesis will differ from those described above mainly in that it considers the inplane deformations and the geometrically nonlinear effects, and also the movement of the wall-wall corners. While it may be justifiable to neglect inplane deformations in a concrete container whose lateral deformations are quite small, it will be studied in this chapter as to how they affect the behaviour of a more flexible container, especially in the light of comments of Wilby(1977) on how analogy methods had indicated the importance of inplane forces. 4.2 Square Pinned Containers Under Internal Hy-drostatic Pressure The examples deal with the analysis of a container made up of square panels, con-nected to each other by flexible hinges and encastred at their bottoms to a rigid base. The container is internally loaded with the pressure of the contained water. The solu-tion to this was experimentally verified with the model container depicted in Fig. 4.1. The model dimensions were selected purely for experimental ease and had not been Chapter 4. Analysis of Rectangular Containers 140 Figure 4.1: 'Pinned' Container Model made to represent a real container by dimensional similitude. 4.2.1 Details of the Specimen The container was made up of polymethyl methacrylate (Plexiglass) panels of nominal thickness 4.5mm. The container was square in plan with centre line dimensions of 400mm X 400mm. The internal height was 400mm. The bottom was made up of a 5-ply plywood base of 18mm thickness. The walls were encastred into the base using a groove laden with epoxy glue to obtain 'fixed' conditions. But, this did not provide a perfect fixed support as was desired, probably due to the inadequacy of the thickness of the base and the length encastred, and also the wetting of the base during the hydrostatic test. The last despite the fact that a polythene bag was pasted to the inside of the container. The container walls were joined to each other by the use of long brass hinges. The connections between the hinges and plates were through Chapter 4. Analysis of Rectangular Containers 141 both mechanical and adhesive jointing. The use of brass hinges, which are more rigid than the. imaginary perfect Plexiglass hinges the model assumes, turned out to be somewhat detrimental to the eccentric load test (without water) to be described later. A shortcoming of polymethyl methacrylate is the wide variability of the elasticity modulus and its temperature and duration of load dependency. The Young's modulus of the material was given by McClintock and Argon(1966) to be in the range of 0.35 to 0.45 X 106 psi, while Warnock and Benham(1965) gave an approximate value of 0.6 X 106 psi. Therefore simple bending tests were performed on rectangular specimens obtaining a value of 0.47 X 106 psi from loadings that lasted for about ten to fifteen minutes before the deflections were recorded. Here it may be noted that Davies(1963) had provided a value of 0.4 X 106 psi which had been obtained from loadings that lasted for about two hours. A Poisson's ratio of 0.375, as was reported by Davies op. cit., was used. Due to apparent poor quality control of the material, the plate thicknesses varied considerably even within the same panel of 400mm X 400mm dimensions. This varia-tion was, unfortunately, realised only after the construction of the container, making it impossible to have simple direct measurements of the thicknesses at a range of places throughout the plates for the purpose of using average values. Therefore they were measured only at a range of places along the brims of the panels. Even along the brims the thicknesses varied. For example Panel 2 of the container gave mea-surements of 4.54, 4.40, 4.42, 4.43, 4.36 and 4.53 mm. (Panel numbers are given in Figure 4.2.) The average values for thicknesses of Panels 1 to 4 were 4.40, 4.38, 4.77 and 4.43 mm, respectively. It is to be noted that these panel numberings were so made as to reduce the bandwidth of the stiffness matrix when a complete container is analysed. Chapter 4. Analysis of Rectangular Containers 142 Figure 4.2: Pinned Container: Points of Displacement Measurement Under Hydro-static Loading Although it was intended to build a symmetrical container with equal sides, the above variation of thickness prevented its achievement. The use of plexiglass, instead of a quality controlled material like steel or aluminium, was with the intention of mak-ing the plate deflections under the small hydrostatic load measurable with LVDTs, while being small enough to not prevent the applicability of standard plate theory in the analysis by having deformations very large compared to the plate thickness. 4.2.2 Experimental Procedure The loading of the container consisted of simple pumping of water into it at a rate slow enough to avoid waves at the surface. As the pump was reversible, the reverse procedure was carried out for unloading. Chapter 4. Analysis of Rectangular Containers 143 The measurements made were the water height and the lateral deflections at the top centres of all four sides and at the centre of Panel 2. The water height was measured by the use of a simple gauge made of a graduated rod on a stand. The deflections were measured by the use of both dial-guages and LVDTs. The LVDT readings were fed through an AD converter into an IBM AT computer where the data were stored. The dial guages were used for the purpose of checking the LVDT results, as the latter were later to be used in measurements on the eccentrically loaded container described in the next section. The measurements were made at different water heights, during both loading and unloading, by suspending the pumping and allowing the dial gauge readings to settle down somewhat. It should be mentioned that the dial gauge readings never completely settled down for the durations of pausing due to the probable continuous creep of the walls and of the wetted plywood base. 4 .2.3 N u m e r i c a l A n a l y s i s The numerical analysis of the problem was carried out by the use of finite strips placed horizontally along the sides of the container. Although the average thicknesses of the four sides of the container differed somewhat from each other, preliminary analyses showed that they are too small as to affect much the results of a particular side, if the other three sides also were assumed to be of the same thickness. Thus, just a single panel was analysed assuming symmetric conditions. Under these assumptions the corners of the container are considered to translate in a diagonal direction, making the magnitudes of the U and V displacements there to be the same. Figure 4.3 shows the boundary conditions used under such symmetric conditions. Ten quadratic strips with six symmetric modes were used to obtain the presented results. The geometric nonhnearities were considered. Chapter 4. Analysis of Rectangular Containers 144 Figure 4.3: Pinned Container: Displacement Boundary Conditions For Corners Under Symmetric Conditions 4.2 .4 R e s u l t s Figure 4.4 shows the experimental results for the lateral deflection of Panel 2 measured at mid height and the top of the plate at its vertical centre line. The container was full of water for those measurements. Also shown are the results from the nonlinear analyses with the bottom of the container considered as fixed and pinned to the base, in turn. As expected, the experimental results fall between the two sets of analytical results, given that the experimental panel to base connection could only be said to be between fixed and pinned. Figure 4.5 shows the same type of plot for the instants when the container was having water to a height of 320mm. Both the loading and unloading results have been plotted, with the latter showing a permanent set probably due to the creep of the material and the bottom joint. These results show that the computer model is capable of predicting the experimental results in an acceptable Chapter 4. Analysis of Rectangular Containers 145 manner. 4.2.5 I m p o r t a n c e o f C o n s i d e r i n g C o r n e r D i s p l a c e m e n t s a n d I n p l a n e F o r c e s This section presents some graphical results from analyses carried out with different assumptions for the behaviour of the container. The purpose of the analyses was to see whether the consideration of the movement of the corners of the container gives rise to different results than what can be obtained from a simple hnear analysis as had been done so far or under nonlinear analysis where the movement of the supports is neglected. Figures 4.7 and 4.8 are for pinned and fixed bottomed containers, respectively, and show the lateral deflections along the vertical centre hne A-A of Figure 4.6 from various analyses. The results correspond to the nonlinear analysis with movable corners which was presented earher, the nonhnear analysis with immovable corners and a simple hnear analysis aUowing for the movement of the corners. The container was considered to be full of water upto its brim. As can be seen from the results, the deflections under the nonhnear analyses with movable and immovable corners differ from each other considerably, making the latter highly unreahstic. Further results from the analyses, giving the bending moments and axial forces very near the vertical centre hne of the panel (hne A-A of Figure 4.6) have been presented in Figures 4.9 to 4.12 for the case of the pinned bottom container. The results were sampled in every strip at its (16,1) integration point under the integration scheme of 32 X 2. Thus, they were not sampled exactly at the centre hne and also do not include the very top and the very bottom of the panel. As can be seen from these graphs, the moments and the forces differ widely in the three analyses, showing the importance of the consideration of the nonlinear movable Chapter 4. Analysis of Rectangular Containers 146 0 4 a 12 Lateral Deflection (mm) Figure 4.4: 400mm Pinned Container: Experimental and Analytical Results with 400mm of Water Chapter 4. Analysis of Rectangular Containers 147 0 2 4 6 Lateral Deflection (mm) Figure 4.5: 400mm Pinned Container: Experimental and Analytical Results with 320mm of Water Chapter 4. Analysis of Rectangular Containers 148 corner behaviour, compared to the linear movable corner and nonlinear immovable corner analyses. It can be seen that the linear simply supported plate analysis pro-vides conservative results for the lateral deflections and bending moments, but as to be expected, the prediction of the axial forces is very poor. Another important observation to be made is the compressive horizontal inplane force at the top of the container shown by the movable supports, something not seen with the other cases. The reason for the existence of compressive forces may be understood by the examination of the simpler case of a simply supported plate under a uniformly dis-tributed lateral load. Consider one half of the plate obtained by sectioning it along a centre line parallel to a support. If the plate boundaries are free of inplane tractions, then under the regime of large deformations, both compressive and tensile internal inplane forces occur along this section. This is because the occurrence of only tensile forces will destroy inplane equilibrium of the plate as there are no external horizontal forces to nullify them. Thus, it is necessary for these internal compressive forces to be present and balance the internal tensile forces. Naturally, the tensile forces will be Chapter 4. Analysis of Rectangular Containers 149 generated in the central section of the plate where the deflections are larger, leaving the compressive forces to occur near the boundaries. This change from tensile to com-pressive forces will occur due to the presence of inplane shear forces. Now, considering a panel of the container, it can be seen to be similar to the above case with a few differences, such as the boundary conditions at the top and the bottom of the plate and the elastic inplane forces at the vertical boundaries generated by the adjoining panels. Thus, this may be seen as a problem similar to the simply supported plate, with the reactions from the adjoining walls being not large enough to neutralise the compression. From the above it may be appreciated that this phenomenon can be observed only under the simultaneous consideration of geometric nonlinearities and the translation of the corners of the container. A similar set of results for the case of the fixed-bottomed container has been presented in Figures 4.13 to 4.16. With regard to the bending moments, the same comments as were made before are applicable. As can be seen from Figures 4.15 and 4.16, the axial forces obtained from the linear analysis are unrealistic. From a comparison of Figures 4.12 and 4.16, the difference between the nonlinear analyses with or without movable corners on the vertical axial forces can be seen to be very much dependent on the assumed fixity for the bottom edge. The immovable corner nonlinear analysis gives large values for the vertical force at the bottom. It also shows larger absolute values for Nx, but does not show compressive to tensile changes shown by the movable corner analysis. The vertical axial force tends to zero at the fixed bottom under the movable corner analysis. Immovable corner case again provides moments smaller than the realistic movable corner case. For completeness, shown in Figures 4.17 and 4.18 are the lateral deflections of the containers having pinned and fixed bottoms, respectively, under 400mm of wa-ter. The deflection profiles shown are at heights of 180 and 380mm from the base, Chapter 4. Analysis of Rectangular Containers 150 0 5 10 15 Lateral Deflection (mm) Figure 4.7: 400mm Pinned Container: The Centre Line Lateral Displacements Under Various Analyses For Pinned Bottom Chapter 4. Analysis of Rectangular Containers 151 Figure 4.8: 400mm Pinned Container: The Centre Line Lateral Displacements Under Various Analyses For Fixed Bottom Chapter 4. Analysis of Rectangular Containers 152 Figure 4.9: 400mm Pinned Container: Centre Line Horizontal Bending Moment Un-der Various Analyses For Pinned Bottom Figure 4.10: 400mm Pinned Container: Centre Line Vertical Bending Moment Under Various Analyses For Pinned Bottom Chapter 4. Analysis of Rectangular Containers 153 CD Oo L. c o o I Water Height: 400mm Container Ht.: 400mm Bottom: Pinned.-N Immovable Corner Movable Corner Linear 8.46 88.46 168.46 248.46 Distance from Base (mm) 328.46 Figure 4.11: 400mm Pinned Container: Centre Line Horizontal Axial Force Under Various Analyses For Pinned Bottom 4 - N y 3 - — ^ Water Height: 400mm Movable Corner 2 - Container Ht.: 400mm Bottom: Pinned 1 - Immovable Corner ' \ . \ y Linear 0 T 1 1 1 1 1 1 r 8.46 88.46 168.46 248.46 328.46 Distance from Base (mm) Figure 4.12: 400mm Pinned Container: Centre Line Vertical Axial Force Under Var-ious Analyses For Pinned Bottom Chapter 4. Analysis of Rectangular Containers 154 Figure 4.13: 400mm Pinned Container: Centre Line Horizontal Bending Moment Under Various Analyses For Fixed Bottom -30 " M y ^ Movable Corner / ; # Immovable Corner " / / Linear Water Height: 400mm Container Ht.: 400mm -— r • i i Bottom: Fixed i i i i i 8.46 88.46 168.46 248.46 328.46 Distance from Base (mm) Figure 4.14: 400mm Pinned Container: Centre Line Vertical Bending Moment Under Various Analyses For Fixed Bottom Chapter 4. Analysis of Rectangular Containers 155 5 -1 8.46 88.46 168.46 248.46 328.46 Distance from Base (mm) Figure 4.15: 400mm Pinned Container: Centre Line Horizontal Axial Force Under Various Analyses For Fixed Bottom Figure 4.16: 400mm Pinned Container: Centre Line Vertical Axial Force Under Var-ious Analyses For Fixed Bottom Chapter 4. Analysis of Rectangular Containers 156 Distance from Panel Corner (mm) Figure 4.17: Pinned Bottom Pinned Containers: Lateral Displacement Profiles at Different Heights from Base, Under Movable Corner Nonlinear Analysis i.e. along lines B-B and C-C of Figure 4.6, respectively. It is to be noted that, by diagonal symmetry, the corner lateral displacements shown are also the plate in-plane displacements. That is these points move inward diagonally. The inward movement of the corner in the panel inplane direction, throughout the height of each container, is shown in Figure 4.19. As a conclusion, it may be stated that for the design of a container of the type used in the above analyses, it is important to use the movable corner nonlinear method, as the linear and immovable nonlinear cases do not give consistently conservative results for all the responses considered. Chapter 4. Analysis of Rectangular Containers 157 10 E E S 6 tS CD CB Q 4 1 CD 3 2 H 0 -1 At 380mm At 180mm Bottom: Fixed Water Height = 400mm 40 80 120 160 Distance from Panel Corner (mm) 200 Figure 4.18: Fixed Bottom Pinned Containers: Lateral Displacement Profiles at Dif-ferent Heights from Base, Under Movable Corner Nonlinear Analysis 0.6 E ° - 5 E 0.4-0 0.3 t3 CD 3= Q 0.2 TJ CO 1 0.1 Height of Water = 400mm Pinned Bottom Height from Container Base (mm) Figure 4.19: Pinned and Fixed Bottom Pinned Containers: Inward Movement of the Corners in the Direction of a Coordinate Axis Figure 4.20: Pinned Container Under Vertical Eccentric Loading 4.3 Eccentric Vertical Load Test on a Pinned Con-tainer This section deals with the problem of a pinned panel container, two facing sides of which are loaded eccentrically by equal vertical loads as shown in Figure 4.20. 4.3 .1 D e t a i l s o f t h e S p e c i m e n a n d E x p e r i m e n t a l S e t u p In this experiment the same container as described earlier was used. The loads were applied on two opposite sides of the container at an eccentricity of 10mm from the middle plane of the wall as shown in Figure 4.20. The loaded panels were Panels 1 and 4 defined in Figure 4.2. The Plexiglass loading arm, which is depicted schematically in Figure 4.20 and in detail in Figure 4.21, had the panel top fitted into it. A ball bearing kept on a well lubricated spherical dent in the arm provided a frictionless pivotal base for the Chapter 4. Analysis of Rectangular Containers 159 loading beam. Between the loading beam and the ball bearing was a well lubricated roller to allow free horizontal movement of the top of the plate. The loading was performed in a displacement controlled manner through the MTS 810 testing machine available in the Timber Engineering Laboratory of The University of British Columbia. The reading of the total load and the actuator movement (which is equal to the displacement of the load point of the specimen) were measured through the data aquisition system of the MTS machine and recorded through an HP X-Y Plotter. The load signal was also fed through an AD converter together with the displacement measurements made with the LVDTs, and then stored in an AT computer by the use of a data aquisition programme. The LVDT readings were not utilised in this study. The loading rates used were 0.3mm/min and 3mm/min. The faster test was performed to see the change in response with the rate of loading. Unloading was performed at the same rate as that for the loading, and always showed a path different to that of loading. As the loading arrangement itself experienced some deflection, a separate test was performed to measure it. This provided a correction curve to be used to obtain the actual load point displacements of the container, from that measured by the data aquisition system. All load-deflection curves are shown in Figure 4.22. The jump in the unloading curve near the zero load end indicates the actual termination of the experiment, the residual being the weight of the loading arrangement. No deflections were measured after the loading arrangement was taken off, thus resulting in this jump. Chapter 4. Analysis of Rectangular Containers Note: Figures Not To Scale Loading Beam Loading Arm Isometric View Loading Arm \ Panel Load I Loading Beam w r> r> n 10 mm Roller Ball Bearing 2 Symmetric View A Figure 4.21: Loading Arm Arrangement Chapter 4. Analysis of Rectangular Containers 161 4.3.2 N u m e r i c a l R e s u l t s The numerical results were obtained with finite strips placed horizontally in each panel. Each panel was discretised with ten linear strips and eleven symmetric modes, which were shown to provide adequate convergence of displacements in comparison with ten quadratic strips and thirteen symmetric modes. Before presenting the experimental-numerical comparison, the method of analysis, including the boundary conditions, will be discussed. As this problem had only two facing sides of the container loaded, and the con-tainer properties themselves were not equal in the four panels, an analysis of the complete container was made when considering the corners movable. The displace-ment 6mm at the machine loading point was calculated as the average of that obtained for the loading points at the ends of the two arms. The latter were obtained from the vertical displacement v and rotation xb at the top centre of each panel using the formula 8 = v + Ixb where Z, the length of the loading arm, was 10mm. The average of the load points values were calculated as they differed due to the difference in the thicknesses of the two panels. These average values have been plotted in Figure 4.22. The plotted finite strip results have been augmented by the loading arrangement deflections obtained from the graph, and so are higher than programme results. (This correction could have been made on the experimental curve, but the above was found to be easier.) The results for the corners movable analysis are shown in the above figure, and are seen to overpredict the experimental response. A further set of analyses was carried out by assuming the container corners to be immovable, as it was felt that the brass hinges restrained their movement. Towards this, the two panels were analysed separately assuming pinned behaviour and the responses predicted by them averaged Chapter 4. Analysis of Rectangular Containers 162 to obtain the machine load point displacements. These also have been plotted in Figure 4.22, and can be seen to be within the loading and unloading curves of the slower test. Also the figure indicates that the trend of the experimental curve is somewhat between the immovable and movable corner analysis curves implying that the real behaviour was in between them. As can be seen, for this example, there is not much difference between the above two types of analyses under the lower loads. The aforementioned assertion of immobilty of the hinges may seem to contradict the previous experiment with water where the movable corners provided proper re-sults. But in that case the hydrostatic load acted as a primary cause of mobility, while in this case there is no such force. Still, an investigation was carried out on the horizontal axial forces that have to occur to prevent movement. The results under loads of 100 N and 200 N per panel are shown in Table 4.1. The forces were obtained at the (1,1) integration point of each strip (hne D-D of Figure 4.6) in the analysis that used ten quadratic strips with thirteen symmetric modes under an integration scheme of 64 X 2. From them what can be noted is that the forces change from tensile at the top to compressive at the bottom of the hinge. So, what could be seen is that the panel is actually pushing the hinge out at the bottom, while trying to pull it at the top. Thus, in the absence of the immovable hinge, the plate would try to deform the hinge in its own plane, instead of trying to pull the whole hinge in as a rigid body. Thus, it can be successfully argued that the container behaved somewhat in an immovable hinge s ense with respect to this problem. All the results presented for discussion have been based on the assumption of clamped conditions for the bottom of the panel. But this can give rise to the question about the non-fixity at the bottom described in the previous experiment. The answer to this lies in the fact that the container bottom is so far away from the point of loading and response measurement, that its effect is negligible to this study. This Chapter 4. Analysis of Rectangular Containers 163 Height (mm) from Base Load on Panel (N/mm) 100 200 8.46 -0.03 -0.06 48.46 -0.07 -0.14 88.46 -0.08 -0.17 128.46 -0.10 -0.20 168.46 -0.11 -0.23 208.46 -0.12 -0.26 248.46 -0.11 -0.25 288.46 -0.05 -0.08 328.46 0.15 0.49 368.46 0.55 1.77 Table 4.1: The Horizontal Force on Panel 1 Hinge under Eccentric Loading can be seen from the results presented in Table 4.2 where the displacements under the two conditions are presented. The analyses presented were carried out assuming immovable corners and using ten hnear strips with eleven symmetric modes. All anal-yses were performed on Panel 1. Another set of analyses made under the assumption of movable corners and the loading of all four panels provided similar results. Also it may be noted from the presented results that the discrepancy between the pinned and fixed bottom cases increases with the increase of the load due to the deformations that spread to the bottom of the container. But within the range of importance to this exercise, the differences in the results are negligible. 4.3.3 D i s c u s s i o n o f R e s u l t s As mentioned previously, the experimental and numerical results for comparison have been provided in Figure 4.22. Three experimental curves have been shown in it. Two of them are the deflections measured directly by the MTS machine during the container tests, and the third is an experimental curve giving the displacements of Chapter 4. Analysis of Rectangular Containers 164 Load on Panel (N) Load Point Deflection (mm) Pinned Bottom Fixed Bottom 200 1.0937 1.0891 400 2.6692 2.6286 600 4.9070 4.8389 Table 4.2: The Load Point Deflections under Different Base Conditions the loading arrangement. It can be seen from this figure that the numerical results are more flexible than the experimental results. But they indicate the nonlinear path in an acceptable manner. The rigidity of the experimental model may be attributed to several reasons. The first is the rate of loading effect as shown by the faster rate experiment. Thus, a much slower quasi static loading that corresponded to the experiment to measure the Young's modulus would have given higher deflections. (In the present case, the slower rate experiment took approximately only 12 minutes for each of the loading and unloading segments.) The second reason is the friction in the rollers used to provide free lateral motion to the panel at its top, and in the vertical brass hinges. This should have lowered the experimental deflections when loading, and increased them when unloading. The constant deflection segment in the experimental curve, which was obtained on the reversal of loading, indicates the presence of friction. The reversal was performed immediately after reaching the maximum response, precluding as the cause creep that would have occured if the deflection was left constant for sometime at its maximum in this deflection controlled experiment. (The effect of creep is the difference between the loading and unloading curves at zero load.) As the increasing load will increase the normal forces at the metal-metal contacts, frictional forces also should become larger, and hence have larger effects. This can be seen from the larger widths of the hysteresis loops at higher loads. Now, it can be noted that the present Chapter 4. Analysis of Rectangular Containers 165 immovable corner analysis curve falls closer to the average of the experimental loading and unloading curves. Thus, it is further clear from the above that an appreciable amount of friction was present during the experiment, and that the finite strips have predicted the response well if the corners did not move at all. Among other reasons for discrepancies should be the differences in the panel thicknesses as mentioned in the previous section. As a conclusion it may be said that the finite strip model has predicted the results in satisfactory manner, considering the margin of error in the experiment. 4.4 Analysis of Containers with Moment Resist-ing Corners This section considers containers with moment resisting corners. Although the analy-ses and the presentation of results will not be as detailed as for the 'pinned' containers, they are sufficient to show the importance of the movable corner nonhnear analysis. Two types of containers will be considered. The first has the shape of a cube, and is assumed to be filled with water. The second is rectangular in plan, also containing water up to its brim. The two problems differ from each other due to the fact that the corners of the square container would not rotate about the vertical axis due to symmetry, while they would in the rectangular case. Only a few results will be presented with respect to the latter, the purpose of which is only to show the abihty of the finite strip model to consider that case. 4.4.1 A n a l y s i s o f a C u b i c C o n t a i n e r This section presents the analysis of a cubic container with moment resisting corners and internal dimensions of 610mm X 610mm X 610mm. This example was obtained Chapter 4. Analysis of Rectangular Containers 166 1.2 1.0 -0.8 -z o 0.6 0.4 0.2 0.0 Loading Arrangement Deflection Expt. (Slow) Expt. (Fast) • Immovable Corner (F. Strips) + Movable Corner (F. Strips) 1 1 r 2 3 4 Actuator Deflection (mm) Figure 4.22: Eccentric Loading of Pinned Container: Comparison of Response Chapter 4. Analysis of Rectangular Containers 167 from Davies(1964) who used the approximate linear analytical solution procedure described in Section 4.1. Here the comparisons will be made with his experimental results. Also presented are some results with different penalty parameters to study the importance of satisfying the constraint on ^-rotation at the corners. Due to symmetry, only one panel of the container was analysed while considering appropriate displacement boundary conditions at the corners as shown in Figure 4.23. As usual, the finite strips were placed horizontally. In all of the following, the number of modes excludes the additional modes which were taken into account due to the movement of both supports. All panels were assumed fixed at the bottom to a rigid base. The plates were 9.525mm thick. The modulus of elasticity was 2715 MPa, and Poisson's ratio 0.375. 4.4.1.1 L i n e a r A n a l y s i s o f t h e C o n t a i n e r For the purpose of studying the importance of the penalty parameter, it was varied from zero to IO10. Another analysis was carried out with no penalty parameter while specifying the xb rotations to be zero at the vertical corners. Forty-eight integration points were used along the finite strip so that interpolation/extrapolation of the displacements and forces be not needed in that direction. Table 4.3 provides results obtained from analyses performed for the purpose of studying convergence in displacement under linear conditions and satisfaction of the constraint. The displacements are those sampled at points A and B of Figure 4.23. As can be seen, the results from the three symmetric modes analysis have converged sufficiently to give more than the engineering accuracy. Also it can be noted that the present results are slightly greater than those of Davies. A penalty parameter of IO10 can be seen as a proper value for the satisfaction of the constraint. It can also be concluded from the results obtained by specifying zero Chapter 4. Analysis of Rectangular Containers 168 Figure 4.23: A Panel of Cubic 'Fixed' Container Discretisation Displacement V (mm) at Penalty Parameter 0.0 IO1 0 0.0* 3 Modes Lin. Strips A 3.0518 3.0449 3.0497 B 2.3608 2.3551 2.3550 5 Modes Quad. Strips A 3.0303 3.0299 3.0299 B 2.3174 2.3153 2.3153 Davies(1964) A 2.911 B 2.172 * %p = 0 at supports Table 4.3: Square 'Fixed' Container: Effect of Penalty Parameter and Convergence of Displacements(mm) Chapter 4. Analysis of Rectangular Containers 169 Position Linear Nonlinear Davies A 3.0425 2.9602 2.911 B 2.3365 2.2801 2.172 Table 4.4: Square 'Fixed' Container: Wall Lateral Displacement (mm) Under Differ-ent Analyses V> at the support(with no penalty parameter) that it is also adequate to obtain proper displacement results for the container of concern under the present analysis because the rotation of the adjoining panel in its own plane remained negligible. 4.4 .1 .2 N o n l i n e a r A n a l y s i s o f t h e C o n t a i n e r This section presents results from a nonhnear analysis of the above container which is under the regime of small deformations. A convergence study showed that the use of six symmetric modes, i.e. a total of eight modes if the additional modes are considered, will give results of sufficient accuracy also with respect to bending moments. The axial forces needed at least seven modes for their convergence. The following results were obtained under the latter number of modes. Ten quadratic strips were seen to be adequate for the discretisation in the direction of the height of the container. An integration scheme of 48 X 2 was used due to the previously mentioned reason. AU analyses were carried out with a penalty parameter of IO10. Shown in Table 4.4 are the hnear and nonhnear lateral displacements obtained at the centre and the top-centre of the plate (Points A and B, respectively, of Fig-ure 4.23). The hnear results are shghtly different from those presented in Table 4.3 as the present analysis used seven modes. It may be noted that the nonhnear results are closer to Davies' experimental results where such effects should have been present. A cause for the shghtly higher results may be the shear flexibility of the present model. Chapter 4. Analysis of Rectangular Containers 170 As noted, the displacements agree well with the work of Davies in both linear and nonlinear analyses due to the small deformations that occur (less than 1/3 of the plate thickness). But it is in some of the forces that the present nonlinear analysis differs from results of a linear analysis. Also linear analysis referred to here considers the movement of the supports. (If not, the plate cannot be made to experience the forces that arise from the adjoining sides. Thus, it is the present model with movable corners that makes this possible as the standard clamped-clamped finite strip cannot have an inplane displacement.) The forces along the vertical centre line of a panel (L-L of Figure 4.23) are shown in Table 4.5. As can be expected, the differences between the moments under linear and nonlinear analyses are small as this is a small deflection problem. But the dif-ferences in the inplane forces are conspicuous. Among them the most interesting is the horizontal compressive force, (Nx), at the top of the container, even in this 'small deflection' problem, under movable corner nonlinear analysis. As it was with the pinned container problems presented earlier, this force cannot be observed under any other assumptions. The cause of this can be explained with the arguments presented earlier with respect to the pinned container. In order to further investigate this horizontal compressive inplane force with re-spect to the 'fixed' container, more analyses were carried out considering thicknesses of 8mm and 6mm for the walls, so that the deflections are larger and the interactions much more. Both containers had the same internal dimensions as earlier, and were completly full of water. The discretisations were the same as those for the previous analysis. All the analyses were performed with the use of the penalty parameter. The displacements obtained for the 8mm container are shown in Table 4.6. As can be seen, they are still not 'large' compared to the plate thickness, but the effect of the nonlinearities is more. The horizontal force along the vertical centre line L-L Chapter 4. Analysis of Rectangular Containers 171 of this 8mm container has been shown in Table 4.7. The length along which the compressive forces act at the top of the panel has now increased. Also, as was seen in Figure 4.13 for the fixed bottom pinned panel container, the horizontal force towards the bottom has become compressive. As earlier, the predictions of linear analysis and the immovable corner nonlinear analysis are highly unrealistic when compared with the movable corner nonlinear analysis. Shown in Figures 4.24 and 4.25, respectively, are the distributions of the horizontal moment Mx and the horizontal inplane force Nx along the line L-L in the three containers considered. From them it can be seen that the moments generally become smaller and the inplane forces larger while the wall thicknesses decrease and hence the deflections increase. (For completness it may be stated that the deflections at A and B of the 6mm container were 9.539mm and 9.591mm, respectively). Also it may be noted from the figures that the changes in the moments are very much smaller than the increases in the inplane forces. Therefore, again, the movable corner nonlinear analysis can be seen to be important if a container is to be designed against failure under large deflections. 4.4 .2 A n a l y s i s o f a R e c t a n g u l a r C o n t a i n e r This section presents the linear analysis of the rectangular container shown in Fig-ure 4.26. The problem was obtained from Golleyei a/.(1987) who used a finite strip-element in their linear analysis (see Section 4.1). As in the reference, Poisson's ratio was chosen as 0.15. Other properties, including the container dimensions, were assumed. The thickness of container walls and the modulus of elasticity were taken as 9.525mm and 2715 MPa, respectively. The results provided in a non-dimensionalised form were converted to a dimensional form for comparison. Chapter 4. Analysis of Rectangular Containers 172 Ht. from Base(mm) H.Moment(Mx)J V.Moment(Mv)} H.Force(/Y,)" V.Force( A y -Lin. Nonlin. Lin. Nonlin. Lin. Nonlin. Lin. Nonlin. 12.88 -24.05 -23.66 -64.69 -63.64 0.08 0.09 0.141 0.382 73.78 -14.09 -13.14 -13.94 -13.41 0.26 0.10 0.062 0.462 134.68 14.58 14.30 13.05 13.07 0.43 0.48 -0.003 0.552 195.58 24.55 23.92 24.66 24.20 0.57 0.98 -0.052 0.629 256.48 29.48 28.59 26.96 26.18 0.65 1.32 -0.080 0.655 317.38 30.55 29.57 24.06 23.20 0.67 1.33 -0.085 0.606 378.28 29.01 28.09 18.67 17.95 0.64 1.07 -0.072 0.484 439.18 26.05 25.29 12.47 12.01 0.58 0.67 -0.049 0.319 500.08 22.84 22.27 6.59 6.39 0.51 0.25 -0.024 0.152 560.98 20.41 20.18 0.40 1.77 0.39 -0.12 0.004 0.034 * N/mm; t Nmm/mm Table 4.5: Square 'Fixed' Container: Internal Forces Along Vertical Centre Line Position Linear (Movable) Nonlinear (Movable) Nonlinear (Immovable) A 5.1392 4.8190 4.4668 B 3.9599 3.8721 3.3421 Table 4.6: Square Thin-Walled(8mm) 'Fixed' Container: Wall Lateral Displacement (mm) Under Different Analyses Chapter 4. Analysis of Rectangular Containers 173 30 200 400 Distance from Base (mm) 600 Figure 4.24: Horizontal Moment Mx in the 'Fixed' Containers with Different Wall Thicknesses Figure 4.25: Horizontal Force Nx in the 'Fixed' Containers with Different Wall Thick-nesses Chapter 4. Analysis of Rectangular Containers 174 Height from Linear Nonlinear Nonlinear Base (mm) (Movable Corner) (Movable Corner) (Immovable Corner) 12.88 0.085 0.098 0.605 73.78 0.264 -0.078 0.641 134.68 0.434 0.550 1.491 195.58 0.570 1.495 2.681 256.48 0.652 2.102 3.553 317.38 0.676 2.088 3.796 378.28 0.649 1.554 3.471 439.18 0.588 0.764 2.797 500.08 0.511 -0.047 1.975 560.98 0.436 -0.729 1.161 596.12 0.402 -1.000 0.624 Table 4.7: Square Thin-Walled(8mm) 'Fixed' Container: Horizontal Inplane Forces (Nx N/mm) Along Vertical Centre Line Figure 4.26: Moment Resisting Rectangular Container Chapter 4. Analysis of Rectangular Containers 175 Position Finite Strips Golley et al. C -28.61 -31.51 D 61.03 62.76 Table 4.8: Rectangular Container: Comparison of Bending Moments Mx (Nmm/mm) Ten quadratic strips with seven symmetric modes were used to discretise each panel. Due to symmetric conditions only two adjoining panels were considered by providing proper boundary conditions. As assumed by Golley et al., the corners were prevented from translating while allowing for rotation about the vertical axis. As the penalty function analysis was not implemented for the general rectangular container case, the constraint was approximated by specifying zero ip rotations at the corners. The moments were extrapolated across the finite strips to obtain the values at the required points. No extrapolation and averaging were performed along the length of the finite strips. Shown in Table 4.8 are the horizontal bending moment Mx sampled at the points C and D of the longer side of the container (see Figure 4.26). As can be seen the agreement is satisfactory, while the difference in the moment at C is more than at D. The slight difference may again be due to the shear deformability of the finite strips. This agreement verifies the ability of the model to predict the response of a 'fixed' container when the corners may rotate. No nonlinear analysis was performed with respect to this problem. 4.5 Summary In this chapter a brief survey of analytical and numerical methods available in the lit-erature was presented. By the use of experiments, the validity of the pinned container Chapter 4. Analysis of Rectangular Containers 176 model was verified under both internal hydrostatic pressure and vertical external loads. The case of a cubic container with moment resisting corners, filled completely with water, was verified with experimental results available in the literature. A similar rectangular container was verified with available numerical results. In both pinned and fixed containers a compressive horizontal force was observed at the top of the panel. In the case of fixed bottom containers such forces were observed also near their bases. These could be observed only under a movable corner nonlinear analysis, thus, indicating its indispensability to the understanding of the behaviour of a flexible container. Chapter 5 Mode Transition Finite Strips This chapter introduces what was decided to be called a 'Mode Transition Finite Strip'. This can be used in the economical analysis of structures with loads that spread only across a few of the finite strips used. 5.1 Introduction It is a well known fact that the application of concentrated loads and other compli-cated loads calls for a higher number of finite strip modes than what is necessary for a regular load, such as one uniformly distributed on the plate surface. The need for more modes means additional costs in both the forming and the assembly of the strips, and also in the solution of equations due to the higher number of unknowns and the resulting larger sizes of the matrices. But, here it is shown that if one is to consider such a load that extends only across a few finite strips (compared to the total number used in the analysis), the need for the higher modes is less important for the strips far away from the load. Therefore it becomes possible to use only a fewer modes in those regions. This has been implemented by the use of 'Mode Transition Finite Strips'. They make this possible by having dissimilar numbers of modes in different nodal lines, while still preserving the required displacement continuity. The strip is presented below in general terms, and its usefulness has been demonstrated by two numerical examples. 177 Chapter 5. Mode Transition Finite Strips 178 These transition strips can be considered as the counterpart to the use of larger elements at non-critical areas in finite element and finite strip analyses. The biggest advantage of these strips will be in the case of a problem with many repeated analyses, such as Monte Carlo Simulation. Then, a few preliminary analyses to discover the best combination of modes will be more than compensated by the economical analyses to follow. 5.2 Formulation Consider a Mindhn plate finite strip of interpolation order (n — 1), which would have n number of nodal fines. Let nodal hne number / have M/ number of modes. Then, the values of the fundamental reference surface variables u0, v0, w, <f> and ip at any position in the strip can be written as Mt>v) In the above equation the following notation was used, as exemplified by the w quan-tities given below. Er = 1 [ZmUurF?(O\ Ni{V) zr=i [iZmU^mo] Ni(v) (5.1) Chapter 5. Mode Transition Finite Strips 179 I — the nodal line number m = the mode number F™(£) = the w basis function for mode m Ni(r)) = the shape function corresponding to nodal line I w™ = the coefficient for mode m at nodal line I The above relationship can also be put in to matrix form as was done in Chapter 2 for the case of standard strips. Then, the same procedure as that presented there can be followed to formulate load vectors and stiffness matrices. It is not repeated here. It is to be noted that as it is C° continuity which is needed in Mindlin plate strips, this discretisation fulfils it provided that the adjacent strips have the same number of modes for the connecting nodal lines. Also, the necessary homogeneous displacement boundary conditions remain satisfied by the basic modes as each of them does so. 5.3 Implementation In the case of a linear problem with similar strip properties, where usually only one stiffness matrix is formulated, the use of transition strips does not mean the need for extra formulations. This is because, during the assembly procedure, the unnecessary parts can be dropped off from the strip matrix with the highest number of modes. This gives rise to several transition strips of different orders from just a single formulation. Thus, it leads to no disadvantage, while providing a lesser number of degrees of freedom, and in the case of uncoupling of modes in linear simply supported problems, giving the additional benefit of smaller problem sizes and, at times, smaller bandwidths for the higher modes that spread across only a few of the strips. In a non-linear problem where the stiffness matrix of each strip has to be made anew at each stiffness re-forming step, the above procedure is not possible because Chapter 5. Mode Transition Finite Strips 180 of the different displacements each strip will generally have. In this case each strip matrix can be re-formed with the use of only the necessary numbers of modes. But, in the present implementation, which was carried out only for the purpose of verifying the usefulness of transition strips, such a procedure was not followed. Instead, for ease of programming, all the matrices were formed assuming the largest number of modes and zero displacements for the non-existing modes. Then the terms corresponding to the unnecessary modes were dropped off at the assembly stage. Thus, it was not the most economical analysis because the only economy over the full analysis came from the reduced sizes of the structure matrices. Thus, no comparisons of CPU times are given for the nonlinear problem presented below as it would not do justice to the new finite strip. As this study was performed through the computer code written for the earlier presented generally coupled (in modes) analysis procedure, the uncoupling of modes that may occur was not utilised. Therefore, the CPU time comparison provided for the linear problem below does not show the economy that arises from the reduction in the problem sizes of uncoupled higher modes. The transition strips are to be provided in a given structure so that the numbers of modes in the adjoining sides of adjacent strips are the same. This will preserve the interelement continuity of displacements. 5.4 Numerical Examples 5.4.1 S i m p l y S u p p o r t e d P l a t e U n d e r C e n t r a l P o i n t L o a d The problem concerns a simply supported, thin, isotropic plate with a central point load. A linear analysis is carried out with the intention of finding the central bending moments. Deflection, although presented, is not the best response to consider as, Chapter 5. Mode Transition Finite Strips 181 4 Strips 8 Strips Exact Deflection (mm) 0.2188 0.2182 0.2177 M x (Nmm) 267.65 310.42 M y (Nmm) 240.60 296.79 Table 5.1: Plate with Point Load: Convergence of Central Responses With Ordinary Strips Having 13 Symmetrical Modes generally, only a small number of modes is necessary for its convergence, thus, not making the advantage of the new finite strip evident. The exact result for the displacement was obtained from Timoshenko and Woinowsky-Krieger(1970). Plate properties etc. are not given here as their knowledge is of no importance to this exercise. Presented in Table 5.1 are the results from analyses with thirteen symmetric modes, using four and eight Hnear finite strips for one half of the plate. From them it is evident that four strips are not sufficient to obtain convergence in moments. It is needed to point this out as the importance of transition strips would decrease with the decrease in the number of strips used. As far as possible, the same number of integration points, 40 X 1, was used for all the different numbers of modes, so that the comparison of moments wiU not be biased by the closeness of the points of stress sampfing to the centre Hne under higher order integration. Table 5.2 presents the results obtained with eight ordinary strips per panel and different numbers of modes. The numbers of degrees of freedom given are those needed for a linear analysis (w, cf> and ip), although the actual analyses were carried out with the presence of the inplane degrees of freedom. From this table it can be seen that four modes having 96 degrees of freedom give a deflection with an error of 0.5% from the converged answer, while three modes having 72 degrees of freedom Chapter 5. Mode Transition Finite Strips 182 Modes Degs. of Central Response (symm.) Freedom0 Deflection (mm) M x (Nmm) My (Nmm) 1 24 0.2024 152.16 189.75 2 48 0.2136 210.92 235.76 3 72 0.2161 242.09 257.55 4 4+ 96 63 0.2170 0.2164 261.54 258.80 270.00 270.68 4* 63 0.2167 259.72 270.68 5 120 0.2174 274.59 277.79 6 144 0.2177 283.66 282.89 7 168 0.2178 290.05 286.32 8 192 0.2179 294.56 288.64 9 216 0.2180 297.67 290.18 10 240 0.2181 299.73 ' 291.17 10** 141 0.2178 298.78 291.23 11 264 0.2181 300.99 291.76 11" 264 0.2181 307.97 295.68 °w, d>, anc ib only 1 1,1,1,2,3,3,4,4,4, modes * 1,1,2,2,2,3,4,4,4, modes *" 1,1,2,4,6,8,9,10,10, modes * 48 X 1 integration Table 5.2: Plate with Point Load: Convergence of Central Responses with Modes (Using 8 Linear Strips) Chapter 5. Mode Transition Finite Strips 183 give an answer in error by 0.9%. Now, consider the use of a maximum of four modes with transition strips having 1,1,2,2,2,3,4,4,4 modes in the nodal lines. (The order of modes starts from the first nodal line which is at the support.) This gives a total of only 63 degrees of freedom, which is less than the number for the three mode analysis with the full number of modes. It provides the same accuracy as the full four mode analysis having 96 degrees of freedom, showing the advantage of the mode transition strip. The analysis with 1,1,1,2,3,3,4,4,4 modes in the nodal lines, which also has 63 degrees of freedom, gives an answer only as good as the full three mode analysis, showing the importance of selecting the number of modes for each nodal line judiciously. In the case of the central bending moments, Table 5.2 indicates that the use of ten modes has given an accuracy of approximately 0.3% in both bending moments when compared to the values from eleven modes with 40 X 1 integration. (In the case of 48 X 1 integration the results become different due to the closeness of the integration point to the centre and the increased accuracy of integration.) Also, it may be noted that the full mode analysis with six modes, which has 144 degrees of freedom, has errors of approximately 5.3% and 2.8% in Mx and My, respectively, compared to the ten-mode values. But, the use of transition strips with a mode arrangement of 1,1,2,4,6,8,9,10,10 in the nodal lines, and which has only 141 degrees of freedom, gives the same accuracy as that given by the full ten-mode analysis with a total of 240 degrees of freedom. In CPU time this reduction in degrees of freedom meant a saving of approximately 20.5% from that needed for a full mode analysis, or the need of 25.7% more CPU time for the full mode analysis from that for the transition strip analysis, thus, showing the advantage of the mode transition strips. If the analyses were carried out by considering the uncoupled modes separately, then the saving in CPU time would have been higher due to the previously mentioned reasons. Chapter 5. Mode Transition Finite Strips 184 Nodal Line 3 Nodal Line 2 Nodal Line 1 Figure 5.1: Plate Under Eccentric Loading 5.4.2 N o n l i n e a r A n a l y s i s o f a n I s o t r o p i c P l a t e U n d e r E n d L o a d a n d M o m e n t The definition of this problem is shown in Figure 5.1. This is the same loading condition as was used in the experimental verification of a container provided in Chapter 4. The side boundaries are immovably pinned, thus, allowing only the normal to support rotation (<f>). The properties of the plate are as defined earlier, except for the use of a Poisson's ratio of 0.25 here. In this section it will be assumed that the responses of concern are the displace-ments at the top of the plate at its line of symmetry. Transition strips will be used to reduce the size of the problem by having a progressively fewer number of modes as the fixed support, where the responses are minimal, is approached. Ten linear strips were used to obtain the results in this nonlinear analysis. In Table 5.3 the modes for the respective nodal lines under transition strip analyses Chapter 5. Mode Transition Finite Strips 185 Symm. Modes Intg'n Points Degs. of Freedom Response At Centre Line of Top of Plate v (mm) w (mm) v> 4 16 X 1 200 -1.2279 - 12.7499 -0.2862 5 20 X 1 250 -1.3124 -13.0575 -0.3050 6 32 X 1 300 -1.3625 -13.2238 -0.3169 7 32 X 1 350 -1.3890 -13.3116 -0.3243 8 32 X 1 400 -1.4031 -13.3540 -0.3286 8+ 32 X 1 310 -1.4011 -13.3460 -0.3275 8* 32 X 1 305 -1.4008 -13.3449 -0.3274 T 4,4,4,4,6,6,6,8,8,8,8 modes in lines 1-11 1 3,3,4,4,5,6,7,8,8,8,8 modes in lines 1-11 Table 5.3: Plate with End Moment: Convergence of Response with Modes are given starting from nodal line one, which is at the fixed support of the plate, to the eleventh nodal line which is at the free end where the load is applied. As can be seen from the table, good accuracy of the displacements can be obtained with eight symmetric modes with standard strips. The analysis with six modes using standard strips, which results in a total of 300 degrees of freedom, has errors of 2.9%, 1.0% and 3.6% in v, w and xb values, respectively, when compared with those from the analysis with eight modes which has 400 degrees of freedom. But, the use of transition strips with only 305 degrees of freedom has given errors of only 0.2%, 0.1%, and 0.4%, and these results are even better than those obtained from analysis with standard strips and seven modes that have 350 degrees of freedom. Thus, again, the advantage of the transition strips is demonstrated. (No CPU time comparison was carried out here due to the previously mentioned reason.) Chapter 6 Observations on the Calculation of Transverse Shear Stresses This chapter deals with some observations on the calculation of transverse shear stresses in Mindlin plates by the use of the equilibrium equations and the displace-ment solution. Several observations are made with regard to the Mindlin finite strips in general, to the present displacement interpolation, and to a difference between the linear and geometrically nonlinear analyses with respect to the distribution of transverse shear stresses across the plate thickness. The presentation will be as follows. The first section will give a brief introduction. Starting from the equilibrium equations, the second will derive the equations needed for the determination of the transverse shear stresses in terms of the plate deflections. The third will present the observations to be made based upon the derived equations. The fourth will present an example. 6.1 Introduction The Mindlin plate theory(Mindlin, 1951) is an approximation to the real behaviour of a plate in that it constrains the plate to have constant transverse shear strains at a cross-section. Therefore the equilibrium equations have been used in the calculation of the transverse shear stresses (Pryor and Barker 1971, Reddy and Chao 1981b, Reddy 1984a, Engblom and Ochoa 1985, Chaudhuri 1986, Putcha and Reddy 1986, 186 Chapter 6. Observations on the Calculation of Transverse Shear Stresses 187 Reddy 1989). This provides a more realistic stress distribution than that provided by the constitutive equation. This use of the equihbrium equations has been made even with the classical plate theory (Reddy 1984a. Lajczok 1986). Of the many methods available for solution of Mindlin plate problems, only the variational methods are considered here. In many linear analyses(Pryor and Barker 1971, Putcha and Reddy 1986, Reddy 1989) transverse shear stress distributions that achieved zero values at the plate sur-faces have been presented. While some have nonlinear analyses (Azizian and Dawe 1985a and 1985b, Dawe and Azizian 1986, Pica et al. 1980, Putcha and Reddy 1986, Reddy and Chao 1981b), no transverse shear stress distributions across the thickness have been presented although some of them provided individual point values. In this chapter one of the problems used by Dawe and co-workers for geometrically nonlinear analysis is utilised to present some observations with regard to transverse shear stress calculations. This problem was presented in Section 3.1.2 of this thesis, and hence will not be described again in much detail. It has two opposite boundaries clamped and the other two pinned. It is acted upon by a uniformly distributed load. In the analyses presented here, the inplane mode shapes for the fixed strip have been considered as the sine functions used in the pinned strip analyses. This assumption has no effect on the conclusions to be arrived at. Although the analyses are with respect to finite strips, some of the observations will be applicable also to finite element analyses. The discussion is made with respect to homogeneous isotropic plates, but the conclusions should be generally valid for any plate symmetric about its middle plane. The observations made are with respect to the calculations made using the displacement solution. The finite difference technique on the stresses presented by Lajczok op.cit. is not dealt with here. Chapter 6. Observations on the Calculation of Transverse Shear Stresses 188 6.2 Equilibrium Equations and The Evaluation of Transverse Shear Stresses This section presents the equilibrium equations and their manipulation to obtain the transverse shear stresses in terms of the inplane stresses, which in turn can be ex-pressed in terms of the displacement solution to a problem. 6.2.1 E q u i l i b r i u m E q u a t i o n s The equilibrium equations in the in-plane x and y directions, for the Lagrangian de-scription, are given by Fung(1965, pg.466), and have the same forms as the respective Cauchy equations for Eulerian description. They, respectively, are UO~x OTvx OT~, , . , Ox dy dz and do-y ,dTxy.dry2 -dy- + ^x- + -dr+pif»-a«] = 0 (6-2) where, / = the body force per unit volume, a — the inertial force per unit volume, and p = the mass density of the material of the plate. a and r refer to the standard engineering stress nomenclature, with their sub-scripts denoting the directions. 6.2.2 T h e C o n s t i t u t i v e E q u a t i o n s a n d t h e I n - p l a n e S t r e s s e s During a particular analysis, on having solved the problem for the displacements, the three in-plane stress quantities in the above equations can be expressed in terms of Chapter 6. Observations on the Calculation of Transverse Shear Stresses 189 the former and the constitutive properties as M = [D] {e} where [D] is the elasticity matrix. ey Hxy (6.3) and are the inplane stresses and strains respectively. It is this equation which is used to find the in-plane stresses that are to be substituted into the equihbrium equations. 6.2.3 I n t e g r a t i o n o f E q u i l i b r i u m E q u a t i o n s Now consider substitution of the in-plane stresses, which are in terms of the displace-ment degrees of freedom, into the equihbrium equations and the latter's integration across the plate thickness in order to find the required transverse shear stresses. The use of the x equilibrium equation to find the transverse shear stress rxz is presented here. Due to their similarity to the y equihbrium equation and r„ 2, respectively, the latter will not be dealt with here. Assuming that the transverse shear stress at the bottom fibre (i.e. at z — -h/2) is Txz J ^ n e transverse shear stress TXZ at any position of the cross-section can be obtained on the integration of the equihbrium equation (Equation 6.1), as (6.4) for a static problem with no body forces. In the above — | and z refer, respectively, to the bottom-most coordinate of the plate and to the coordinate of the point at Chapter 6. Observations on the Calculation of Transverse Shear Stresses 190 which the stress is calculated, assuming the definition of the coordinates to be from the middle plane of the plate of thickness h. Now, due to the interchangeability of differentiation and integration, the above equation can be re-expressed as r" {x>y>z) = ~k ( / - i d z ) ~ h ( / - i T x y d z ) + T-0 x z (6.5) In what follows the bottom-most fibre transverse shear stress will be considered to be zero as required for the problem considered. Also if one is to find the transverse shear stress at the top surface of the plate, the upper integration limit will be h/2 and the stress is given by T« (x'y' = ~Tx ( /_| dz) ~ ly (/Ir- dz) (6-6) 6.2.4 S u b s t i t u t i o n o f D i s p l a c e m e n t s For the purpose of this study, unhke in the computer model where items are expressed in terms of the nodal degrees of freedom, here the expressions for the inplane stresses will be kept at the level of middle plane displacement degrees of freedom, viz. u0, v0, w, <f> and rp, of Mindlin plate theory, prior to discretisation. Now, for the purpose of substitution into the equilibrium equations, assuming the initial deflections to be non-existent in Equations 2.52 to 2.54, the inplane strains can be expressed in terms of the above displacements as e* = u0iX - z (j)tX + 2X (6.7) £y = vo,y ~ ztp,y + ^w2y (6.8) Txy = U0<y + l>o,x - z (cf>ty + V>,x) + wtXwty (6.9) Then, substitution into the constitutive equation would give the two required inplane stresses and their x derivatives as, <rx = Eex + (j^—^}Eey Chapter 6. Observations on the Calculation of Transverse Shear Stresses 191 E (u 0 i I + uv0,y) - zE (>,x + i?V,y) + \ E ( V X + vw2y) (6.10) ' x,x E (w0,xx + ^vo ! X y) - zE [<btXX + l /^ .xy ) + ( W , X W X X + VW^W^y) (6.11) r x y — G Ixy = G(u 0 , y + v0iX) - zG (<£iy -f V>,x) + Gw^Wj (6.12) with T x y , y = G (U0 , y y + V0>Xy) - ZG ((btyy + lb,Xy) + G(w x y ™ y + W,xWtyy) (6.13) 1 - z/2 Then, substituting into Equation 6.5 and integrating with respect to z, Similarly, -E(z + O.bh) w0,xx + Mow + 2 (w*w** + p u , * w . "w) +0 .5£ (z 2 - 0.25/i2) (</>,xx + i>V,xy) -G(Z + 0.5fc)(uOlW + ^O.xy + ™,xy™,y + W , x ™ , y y ) +0.5G (z2 - 0.25/i2) + ib,xy) -E(z + 0.5/i) u 0 , y ! / + Pu 0, yx + 2 (^.y^-yy + v™*™**) +0.5E (z 2 - 0.25/i2) (^, y y + i^, x y) -G{Z + 0.5/l)(vo,xx + « 0 , x y + W ' ,xy^ ,x + ™ , y ™ , x * ) +0.5G (z 2 - 0.25ft2) {ibiXX + 4>,xy) (6.14) (6.15) Chapter 6. Observations on the Calculation of Transverse Shear Stresses 192 If the integration is up to the top surface of the plate, then the above equations will give the transverse shear stresses at this surface as r„(h/2) = -hE 1 / U0<XX + W0tXy + - {WiXWiXX + VWtyW,Xy) -kG [U0lyy + V0<xy + W^yWj, + w >xw m] (6.16) Tyz(h/2) = ~hE V0iyy + VU0<yx + ^ (w <YW > Y Y + PWTXW,Xy) -hG [v0tXX + u0iXy + wiXywiX + w>ywtXX} (6-17) 6.3 Observations From The Equations From the observation of the above derived Equations 6.4 to 6.6 and 6.14 to 6.17, sev-eral conclusions can be made with regard to the transverse shear stress distributions and their determination. Some practical observations of these may be made through the analyses presented later. In the following, it is assumed that the finite strips are parallel to the x axis of the plate so that the piecewise discretisation is in the y direction. 6.3.1 Inappropriateness of Linear Elements Consider Equations 6.14 and 6.15 for the two transverse shear stresses rxz and ryz at any z. 1. The transverse shear stress rxz. (a) Consider a finite strip analysis. Considering the T X Z , it can be seen that the expression contains the second derivatives in y of UQ, W and <f>. These will vanish in the case of hnear finite strips. It may be noted that these terms come from those derived from the inplane shear terms (and hence Chapter 6. Observations on the Calculation of Transverse Shear Stresses 193 connected to the shear modulus G). In a problem such as will be consid-ered, bending and stretching actions dominate over the twisting moments along the centre lines where the stresses are evaluated. Therefore the in-plane direct stresses should dominate over the shear stresses, and hence the effect of the vanishing derivatives on the transverse shear stress T X Z should be minimal. It will be seen that it is so from the examples to be provided. (b) Consider a finite element analysis using a hnear rectangle. In this case one half of the terms form each source (viz. inplane direct and inplane shear stresses) vanish giving results poorer than those from a finite strip analysis. 2. The transverse shear stress ryz. (a) Consider a finite strip analysis. Considering the ryz, it can be seen that the expression contains the second derivatives in y of D o , w and ip. These will vanish in the case of hnear finite strips. Further, it may be noted that these terms come from those derived from the inplane direct stress terms (and hence connected to the Young's modulus E). Due to the previously mentioned reasons on the domination by the inplane direct stresses that are due to both bending moments and the inplane forces, they should dominate over this shear stress also. Thus, the effect of the vanishing derivatives on the transverse shear stress ryz should be much more than that on rxz. This should be evident as a larger difference between the proper results and those predicted by linear strips. This will be seen to be so in the results to be presented. Chapter 6. Observations on the Calculation .of Transverse Shear Stresses 194 (b) Consider a finite element analysis using a linear rectangle. As before, in this case also, one half of the terms from each source (viz. inplane direct and inplane shear stresses) vanish giving results that will be poorer than that from a finite strip analysis. But here the relative error should be the same as for the other transverse stress rxz, as similar terms vanish on both occassions. 6.3.1.1 Effect of Stress Gradients As was seen from the above presentation, during finite strip analysis with linear elements, the terms in the second y derivative vanish, generally resulting in erroneous values. Therefore it was pointed out that the effect of this on ryz will be greater than that on T X Z , if the inplane direct stresses are to dominate. A further such observation may be made with regard to the stress gradients present under finite strip analyses. Consider a finite strip analysis of a rectangular plate with the conditions on the -^constant boundaries being taken in turn as pinned and fixed. In the case of the fixed boundary, the stress gradients in the transverse y direction will be much higher than those in the pinned case. Thus, when a linear strip analysis is carried out, the resulting errors in the transverse shear stresses calculated for the plate with fixed boundaries in the transverse direction should be somewhat more than those for the pinned boundary case. This also will be seen in the presented results which have been found by switching the finite strip directions from the fixed-fixed to pinned-pinned boundaries in a square plate with two opposite edges clamped and the others pinned. Although different types of modes are to be used in the two analyses, viz. fixed-fixed and pinned-pinned, the results should not be affected as the mode direction does not affect this study. Chapter 6. Observations on the Calculation of Transverse Shear Stresses 195 6.3.2 T r a n s v e r s e S h e a r S t r e s s e s a t t h e P l a t e S u r f a c e s 6 .3 .2 .1 T h e C a s e o f a L i n e a r A n l a y s i s In the case of a hnear analysis with no apphed inplane forces, the middle plane of the plate acts as a neutral plane, due to the assumption of straight normals used in the Mindhn theory. Therefore, each of the two inplane stresses o~x and rxy in Equation 6.6 will be antisymmetric about that plane, and will have a hnear distribution across the thickness. Thus, the integration of the stresses across the total thickness would cause each of the two terms to vanish on their own, giving rise to a zero transverse shear stress at the surface, as required. This result is purely a consequence of the assumed displacement field across the thickness, and has nothing to do with the correctness or the convergence of the solution obtained for the problem of concern. The same can be observed from Equations 6.16, for which all terms vanish in a Hnear problem, giving a zero value for the surface stress. Thus, independent of aU other causes, the transverse shear stress distribution wiU be symmetric about the middle plane with zero values at the extreme fibres. In the case of a Hnear analysis with appHed constant direct inplane forces, the above antisymmetric distribution of stresses would not occur, but stiU the additional contributions from the appHed inplane forces wiU vanish due to their constancy in the inplane coordinates, which makes the partial derivatives vanish. Thus, zero transverse shear stresses are obtained at the extreme fibres also in this case. 6.3 .2 .2 T h e C a s e o f a G e o m e t r i c a l l y N o n l i n e a r A n a l y s i s In the case of a geometricaUy nonhnear analysis which considers the presence of inplane forces, the independent vanishing of both of the inplane stress terms, on integration across the thickness, would not generally occur. This is because either Chapter 6. Observations on the Calculation of Transverse Shear Stresses 196 the two integrals or the partial derivatives cannot, in general, become null on their own. The result will only depend on how well the equihbrium is satisfied, which alone makes the sum of the two terms of Equation 6.6 become zero. Thus, zero transverse shear stress at the top fibre will occur only if equihbrium has been exactly satisfied by the displacement field used in the solution to the problem. It is known that in variational analyses the equihbrium equations are satisified in the domain only in an integral sense and not in a point-wise sense(Washizu, 1982, pg.22). The satisfaction of the equihbrium at each point of the domain occurs only in the limit of a very fine discretisation in the case of a Finite Element analysis or a very large number of terms in the case of a Rayleigh-Ritz analysis with a Fourier Series type of displacement approximation. Thus there is, in general, a non-zero transverse shear stress at the top-most fibre in the case of a geometrically nonlinear analysis. This residual will tend towards zero as the fineness of the discretisation increases, but exact vanishing may not be obtained due to numerical errors from round-off. 6.4 A n Example of Transverse Shear Stress Cal-culation This section presents the example problem used to show some of the items introduced above. 6.4.1 D e s c r i p t i o n o f t h e P r o b l e m The example used is the analysis of a square, moderately thick, isotropic plate with two opposite edges clamped and the other two pinned, under a uniformly distributed lateral load. As mentioned earlier, this example was obtained from the work of Dawe and co-workers, and is the same as that presented in Section 3.1.2. In the present Chapter 6. Observations on the Calculation of Transverse Shear Stresses 197 context two sets of analyses were carried out. The first was by the use of fixed ended finite strips that spanned the clamped boundaries and the second was with a set of pin ended finite strips that spanned the pinned boundaries. Due to symmetry, only one-half of the plate was considered in the analyses which almost always used either six hnear strips or six cubic strips. A single pinned strip analysis was carried out with quadratic strips having nine modes to verify the conver-gence in strips as the convergence of transverse shear stresses with six inappropriate hnear strips may not guarantee that of six cubic strips. This check was performed on the pinned strips because they would need a finer discretisation in strips than the fixed strips as the former has higher stress gradients in the transverse direction (to the strips) in the present problem. Although the results have not been presented here, the convergence of six hnear strips was verified with analyses using higher numbers of strips. Here it may be pointed out that the convergence of transverse shear stresses may require a finer discretisation than that by the inplane stresses as the former is derived by the use of the derivatives of the latter. Hence, a convergence check with the inplane stresses is not sufficent to be confident of the convergence of the transverse shear stresses. Only symmetric modes were used for the displacement approximation, and the numbers of modes mentioned will refer to those. The presented results are with regard to the transverse shear stresses at the points shown in Figure 6.1. At each point the stresses were evaluated at the middle plane and the top surface of the plate. 6.4.2 A n a l y s i s w i t h F i x e d - F i x e d F i n i t e S t r i p s This set of analyses was performed with fixed-fixed finite strips placed across the clamped boundaries of the plate. Chapter 6. Observations on the Calculation of Transverse Shear Stresses 198 a/2 6 Points of Stress Sampling a/2 Strip No. a/2 a/2 Figure 6.1: Positions of Stress Sampling 6.4 .2 .1 S h e a r S t r e s s TXZ a l o n g t h e C e n t r e o f t h e S i x t h S t r i p Figures 6.2 to 6.4 show the values of the transverse shear stress TXZ along the centre line of sixth finite strip which is the closest to the centre line of the plate. The distances along this mid-line of the strip are given from the centre line of the plate by non-dimensionalising it with respect to one-half of the length of a plate side. As shown by Figures 6.2 and 6.3, the eight-mode results have converged in the sense that they are not significantly different from those for six modes. Both linear and cubic strips provide very similar results as shown by Figure 6.4 which has been plotted for nine modes. The reason for this is the above mentioned lesser effect of the vanishing derivatives in y on this transverse shear stress. Also presented here is the residue (i.e. the value that should be zero but is not) of the shear stresses at the top-most fibre of the plate as a percentage of the maximum Chapter 6. Observations on the Calculation of Transverse Shear Stresses 199 (sufficiently converged) transverse shear stress obtained at the middle plane of the plate from among the points involved in the particular study. (The sufficiently con-verged values were taken as those obtained with six cubic strips and nine symmetric modes.) This ratio, as a percentage, will henceforth be called the residue percentage, and its distribution for the residues of rX 2 is shown in Figures 6.5 to 6.7. As Figures 6.2 and 6.3 show that the transverse shear stress has somewhat con-verged by the time eight symmetric modes is used, nine-mode values were considered as sufficiently converged. Four modes is far from being converged. Also it can be noticed that the convergence is oscillatory. From Figures 6.5 and 6.6 it is clear that even the residue has converged a great amount with nine modes, while for four modes it oscillates with a large amphtude in both hnear and cubic strips. For the former, the converged value does not tend towards the required zero value, and the oscilla-tions do not occur about the zero axis due to the inappropriateness of the strips. The plot of residue percentage for nine modes in Figure 6.7 shows the difference in the percentage obtained with the two strip types, with values much lesser for cubic strips than for the hnear strips. Here it may be mentioned that the residues obtained for this symmetric problem will have opposite signs about the centre hne, as for the transverse stress itself, and will get cancelled on addition. As can be seen from those plots, the residue in the cubic strip is smallest closer to the centre of the plate, where the stress itself is smaller and reaches zero. Also it can be seen from Figures 6.5 to 6.7 that for any number of modes with any type of strip, the residue at the support has the same non-zero value. The reason for this, even with the hnear strip, is not known. But it is beheved to be due to the basis functions used in the analyses, and may be related to the Gibb's phenomenon. It is not investigated in this thesis due to the comphcations that may arise from the large number of terms present. Chapter 6. Observations on the Calculation of Transverse Shear Stresses 200 6.4.2 .2 S h e a r S t r e s s ryz A l o n g T h e C e n t r e L i n e P a r a l l e l T o T h e y-Axis A set of plots similar to the previously described ones for T X Z , but now drawn with respect to the ryz vaules, are shown in Figures 6.8 to 6.13. The stress sampling has been done at the line of integration points at the centre of each strip at the position closest to the centre line of the plate. As forty integration points were used for the x direction integration of nine modes, and only thirty-two points were used for the lesser numbers of modes, the point of stress samphng for the former will he closer to the centre line than those for the latter. As can be seen from Figures 6.8 and 6.9, the stresses at the middle plane have sufficiently converged for both linear and cubic strips with the use of nine modes. It can also be seen that the stresses for nine modes have a larger increase from the value for eight modes, than that what occurrs when going from six to eight modes or from four to six modes. The reason for this is not the non-convergence of the solution, but, as mentioned above, the closeness to the centre line of the point of stress samphng with nine modes. Figure 6.10 shows a very large change in the stress when changing from linear to cubic strips, something not observable in the previously considered Figure 6.4 for T X Z . The reason for this is, as was explained earlier, the greater effect of the vanishing derivatives on this stress when the inappropriate linear strip is used. The residue percentages for linear and cubic strips with nine modes are shown in Figure 6.13. It shows that they have become lesser very fast for the cubic strips due to the inappropriateness and the coarseness of discretisation of the linear strips analyses. In the case of this cubic strip the residue percentage has become lesser towards the centre line(see Figure 6.12) while for the linear strip the residue percentage holds a different distribution as shown in Figure 6.11. From Figure 6.12 it can be seen that the residue percentage for cubic strips has switched signs when going from eight to Chapter 6. Observations on the Calculation of Transverse Shear Stresses 201 nine modes. But those values are very small at this stage, making it possibile for numerical errors to become prominent. Also it may be seen that these are much smaller when compared to the values obtained for rxz(Figures 6.6 and 6.7). 6.4.3 Analysis with Pin-Pin Finite Strips Some graphs similar to those presented for the analysis with fixed-fixed strips are presented in Figures 6.14 to 6.16 for the case of the analysis of the same problem using pin-pin finite strips. As the structure is now rotated with reference to the x-y coordinate system, so that the finite strips are still along the x direction, here rxz will correspond to ryz of the previous case, and similarly, ryz here will correspond to T X Z of the fixed-fixed analysis. Still the different nomenclature is used for the two types to indicate the directions of the finite strips when compared to the stresses. In this analysis the stress sampling and numerical integration were carried out in the same manner as for the fixed strip analysis. Also shown on certain occasions are the results obtained using six quadratic strips. As mentioned before, these are provided to show that the stresses have converged by the time the discretisation is refined to the use of cubic strips, as the results from the hnear strips are not appropriate. Now, the two-point integration in 77 of the quadratic strip does not provide a hne of points along its centre line. Therefore, in order to obtain the required stresses at the centre lines of the strips, they were calculated at the two integration points and averaged. The residue values that can be so obtained, cannot be used in this study due to their nullifying effect when added. Chapter 6. Observations on the Calculation of Transverse Shear Stresses 202 0.0 0.2 0.4 0.6 0.8 1.0 Distance From Centre Line Figure 6.2: Linear Fixed Strips: Convergence of T X Z With Modes OH i i i i i i i i i 0.0 0.2 0.4 0.6 0.8 1.0 Distance From Centre Line Figure 6.3: Cubic Fixed Strips: Convergence of rxz With Modes Chapter 6. Observations on the Calculation of Transverse Shear Stresses 203 H I I I 1 I 1 I I I 0.0 0.2 0.4 0.6 0.8 1.0 Distance From Centre Line Figure 6.4: Fixed Strips With 9 Modes: Change of rxz With Strip Type o -6 T (z=h/2) at Centre of Strip 6 / \ xz / \ / •*' \ / • v' / ' \ • \' "'••s'J \ /' "/••/ \ \ >\ X Legend V — 4 Modes 6 Modes 8 Modes 9 Modes i i i i i Strip Details \ / Type: Fixed \ / Order: Linear i i i i 1 1 1 1 1 I 1 1 1 1 1 0.0 0.2 0.4 0.6 0.8 1.0 Distance From Centre Line Figure 6.5: Linear Fixed Strips: Convergence of Residue Percentage of r X 2 With Modes Chapter 6. Observations on the Calculation of Transverse Shear Stresses 204 Figure 6.6: Cubic Fixed Strips: Convergence of Residue Percentage of rxz With Modes TJ (z=h/2) at Centre of Strip 6 S 0-ID O) ra c co o <5 a. <t> •g ' « <o Legend Strip Details Linear Strips Cubic Strips 1 r 0.2 Type: Fixed Modes: 9 — i — 0.8 0.0 0.4 0.6 1.0 Distance From Centre Line Figure 6.7: Fixed Strips With 9 Modes: Change of Residue Percentage of rxz With Strip Type Chapter 6. Observations on the Calculation of Transverse Shear Stresses 205 Figure 6.9: Cubic Fixed Strips: Convergence of ryz With Modes Chapter 6. Observations on the Calculation of Transverse Shear Stresses 206 80 0 Strip Details Type: Fixed Modes: 9 Legend — Linear Strips Cubic Strips T (z=0) Along Strip Centres yz 1 1 1 3 i 5 Strip Number Figure 6.10: Fixed Strips With 9 Modes: Change of ryz With Strip Type Figure 6.11: Modes Linear Fixed Strips: Convergence of Residue Percentage of ryz With Chapter 6. Observations on the Calculation of Transverse Shear Stresses 207 Strip Details Type: Fixed Order: Cubic A Modes 6 Modes 8 Modes 9 Modes Strip Number Figure 6.12: Cubic Fixed Strips: Convergence of Residue Percentage of ryz With Modes 2 -6 - \ T (z=h/2) Along Strip Centres \ y z Strip Details Type: Fixed Modes: 9 Legend — Linear Strips Cubic Strips i i i i -] 1 1 1 r 1 3 5 Strip Number Figure 6.13: Fixed Strips With 9 Modes: Change of Residue Percentage of ryz With Strip Type Chapter 6. Observations on the Calculation of Transverse Shear Stresses 208 As the same discretisation and the numbers of modes as in the fixed-fixed strip case were used here, it is pertinent to note the following. In the problem being studied, the stress gradients will be higher when going from one fixed support to the other, than when moving from one pinned support to the other. Therefore the discretisation, in terms of either modes or strips, should be so as to provide the ability to model these high gradients. Now, from the results presented with respect to the analysis with fixed-fixed strips it was seen that nine symmetric modes were sufficient for the approximation in the fixed-fixed direction. Now, considering the pin-pin case, even before carrying out any analysis, it can be expected from previous results that nine modes will be more than sufficient for convergence in the strip direction, due to the use of mode shapes that somewhat correspond to the deformations to be expected under the particular boundary conditions. This was found to be so from results that have not been presented here. Further, as the higher stress gradients are now in the transverse direction (i.e. y direction), the convergence in strips as shown by this example acts also as a test of the convergence in strips of the previous problem where the stress gradients were smaller. Also due to such higher transverse stress gradients and the corresponding effect of the missing terms in the hnear strip, the change from hnear to cubic strips means a bigger change in the evaluated transverse shear stresses in this problem than in the case of the previous analysis. This is shown by Figures 6.4 and 6.14 of fixed and pinned strips, respectively. 6.4.3.1 S h e a r S t r e s s TXZ A l o n g t h e C e n t r e L i n e o f t h e S i x t h S t r i p The following refers to the transverse shear stress r x z for the orientation of the plate considered in this analysis. This corresponds to ryz of the previous analyses. Figure 6.14 presents the transverse shear stresses calculated at the middle plane of Chapter 6. Observations on the Calculation of Transverse Shear Stresses 209 the centre of the sixth strip. The stresses for the quadratic and cubic strips are almost the same, indicating their convergence with six quadratic strips. As mentioned pre-viously, the convergence of the inappropriate linear strips was verified independently. Figure 6.15 shows the residues under nine modes plotted for both linear and cubic strips. Again, these residues converge to the same value near the support. 6.4.3 .2 S h e a r S t r e s s ryz A l o n g t h e I n t e g r a t i o n P o i n t C l o s e s t t o t h e C e n t r e L i n e P a r a l l e l t o y - A x i s As before the transverse shear stress values are presented in Figure 6.16 for the two types of strips. As can be seen, the stresses obtained with the linear strips are very much lesser than those from the cubic strips. This was not so in the case of the previous fixed-fixed strip due to the smaller transverse stress gradients and the resulting effects on the inappropriate linear strip. (Also it may be noted that in the case of rxz obtained in this analysis, this difference between the linear and cubic strip results is not as large as that for the ryz. This is because the major part of TXZ comes from the deflection pattern in x direction in which the mode shapes occur, whereas most of the y-derivative effect on it comes from the shear terms which are not dominant as indicated by Equations 6.14 and 6.15.) 6.5 Conclusions From the exposition in this chapter, it can be concluded that 1. The linear Mindlin finite strip and the linear rectangular Mindhn finite element should not be used in analyses that require the calculation of transverse shear stresses from the displacement solution by the use of the equihbrium equations. Chapter 6. Observations on the Calculation of Transverse Shear Stresses 210 Figure 6.14: Pinned Strips With 9 Modes: Change of rxz With Strip Type For this purpose finite strips or finite elements with at least quadratic interpo-lation of displacements should be used. 2. Although in a linear case it is possible to obtain zero transverse shear stresses at the outer plate surface on the use of the equilibrium equations, it may be possible in the case of a nonlinear analysis only in the limit of infinite discretisation. But the residuals are not large enough as to preclude the use of the calculated shear stresses if they have converged appreciably at the mid-part of the section. 3. With finite strip bases used in the present analyses, the residue at the support converges to a non-zero value. The reason for this may be related to the used, or the use of, basis functions as it does not occur in the transverse direction in which a piece-wise discretisation is performed. Chapter 6. Observations on the Calculation of Transverse Shear Stresses 211 Figure 6.15: Pinned Strips With 9 Modes: Change of Residue Percentage of TXZ With Strip Type 150" 0. — iooi Pinned Strips Legend Linear Strips Cubic Strips Strip Details Type: Pinned Modes: 9 IS 10 50-~U (z=0) Along Strip Centres _ J 2 Strip Number Figure 6.16: Pinned Strips With 9 Modes: Change of ryz With Strip Type Chapter 7 Conclusions and Suggestions This chapter presents concluding remarks and some suggestions for future research. 7.1 Conclusions The proposed finite strip provides a satisfactory method for the analysis of rectangular containers and continuous plates. With the equivalent elasticity modulii as defined for the purpose of finding the mode shapes, the theory has been extended to composite plates, including those laminated unsymmetrically, with clamped boundaries. In the case of a rectangular container with free upper boundaries, and filled with a liquid, the model predicts a horizontal compressive forces in its walls. This cannot be seen with either linear analyses or nonlinear analyses with immovable corners. Thus, a proper nonlinear analysis requires the consideration of the movement of the corners. For the moment resisting containers considered, the %p rotation compatibility at the vertical corners could approximately be satisfied by specifying zero rotations there. The main usefulness of the proposed model will lie in analyses that require the discretisation of a contained material. While a single or two plate analysis of a symmetric structure is advantageous due to the smaller number of degrees of freedom that will be needed, the analysis of a complete container is not, due to the large bandwidth that will occur. The proposed Mode Transition Finite Strips can be used to economically analyse 212 Chapter 7. Conclusions and Suggestions 213 problems with loads that extend only across a few finite strips. The economy occurs in both linear and nonlinear problems due to the smaller number of degrees of freedom necessary, while in the former there is the additional advantage of smaller uncoupled problem sizes and bandwidths for the higher modes if the strips are simply supported. The linear finite strip and the linear rectangular finite element are not to be used in determining the transverse shear stresses by the use of the equihbrium equations and the displacement solution. For this purpose the displacement interpolation should be at least quadratic. 7.2 Suggestions for Future Research The following may be suggested as some of the work to be carried out in the future. Although the presented explanation is adequate to understand the reason for the horizontal compressive forces that occur in the container walls under hydrostatic loading, experimental verification of the same would give further confirmation of the phenomenon. According to the literature, the flexibilty of the base of a container can affect its behaviour. Therefore, the introduction of the capability to consider such a flexible base may shed further hght into the behaviour of containers under geometrically nonlinear conditions. It is not known how important it is to apply the constraint at the corners using the penalty parameter in the case of rectangular containers. Therefore, a thorough study of this, and also of square containers under non-symmetric conditions, may be performed. This model may now be extended to consider the dynamic interaction between a container and its contained material. The model is especially economical for the anal-ysis of the vertical excitation problem because only a single panel need be considered Chapter 7. Conclusions and Suggestions 214 if the container is symmetric. In this thesis a formulation for the extension of the computer programme to the analysis of plates with elastically restrained edges was presented. This may be imple-mented in the future. Incorporation of stiffeners into the plates is another possible extension of the presented theory. Bibliography Allen, H.G., and Bulson, P.S. (1980); Background to Buckling, McGraw-Hill, London,1980. Azizian, Z.G., and Dawe, D.J. (1985a); Geometrically Nonlinear Analysis of Rectangular Mindlin Plates Using the Finite Strip Method, Comput. Struct., 21(l),423-426, 1985. Azizian, Z.G., and Dawe, D.J. (1985b); Analysis of the Large Deflection Be-haviour of Laminated Composite Plates Using the Finite Strip Method pp.677-691, in Composite Structures 3, Marshall, I. H. Editor, Elsevier, England, 1985. Benson, P.R., and Hinton, E. (1976); A Thick Finite Strip Solution for Static, Free Vibration and Stability Problems, Int. J. Num. Meth. Eng., 10, pp.665-678, 1976. Bishop, R.E.D., and Johnson, D.C. (1960); The Mechanics of Vibration, Cam-bridge University Press, England, 1960. Bresse, M (1859); Cours de Mecanique Appliguee, p.126, Mallet-Bachelier, Paris, France, 1859. Chatterjee, S.N., and Kulkarni, S.V. (1979); Shear Correction Factors for Lam-inated Plates, AIAA J., 17(5), pp.498-499, May 1979. Chaudhuri, R.A. (1986); An Equilibrium Method for Prediction of Transverse Shear Stresses in a Thick Laminated Plate, Comput. Struct., 23(2), pp.139-146, 1986. Cheung, M.S., Cheung, Y.K., and Ghali, A. (1970); Analysis of Slab and Girder Bridges by the Finite Strip Method, Bldg. Sci., 5, pp.95-104, 1970. Cheung, M.S., and Li, W. (1988a); Finite Strip Analysis of Continuous Struc-tures, Can. J. Civ. Eng., 15, pp.424-429, 1988. Cheung, M.S., and Li, W. (1988b); Analysis of Haunched, Continuous Bridges by the Finite Strip Method, Comput. Struct., 28(5), pp.621-626, 1988. Cheung, Y.K. (1968); The Finite Strip Method in the Analysis of Elastic Plates with Two Opposite Simply Supported Ends, Proc. Inst. Civil Eng., 40, pp-1-7, 1968. Cheung, Y.K., and Davies, J.D. (1967); Analysis of Rectangular Tanks: Use of Finite Element Technique, Concrete, 1(5), pp.169-174, May 1967. Cheung, Y.K., and Zienkiewicz, O.C. (1965); Plates and Tanks on Elastic Foun-dations - An Application of Finite Element Method, Int. J. Solids Struct., 1, pp.451-461, 1965. 215 Bibliography 216 [15] Chia, C Y . (1980); Nonlinear Analysis of Plates, McGraw-Hill, New York, 1980. [16] Chia, C Y . (1988); Geometrically Nonlinear Behaviour of Composite Plates: A Review, Appl.Mech.Rev., 41(12), pp.439-451, Dec. 1988. [17] Chia, C Y . and Prabhakara, M.K. (1976); Large Deflection of Unsymmetric Cross-Ply and Angle-Ply Plates, J.Mech.Eng.Sc, 18(4), pp.179-183, 1976. [18] Chow, T.S. (1971); On the Propagation of Flexural Waves in an Orthotropic Laminated Plate and Its Response to an Impulsive Load, J. Comp. Mat., 5, pp.306-319, July 1971. [19] Clough, R.W. and Penzien, J. (1982); Dynamics of Structures, Intl. St. Ed., McGraw-Hill, Singapore, 1982. [20] Coan, J.M. (1951) ; Large-Deflection Theory for Plates With Small Initial Curva-ture Loaded in Edge Compression, J.Appl.Mech., 18, Trans.ASME, 73, pp.143-151, June 1951. [21] Craig, T.J., and Dawe, D.J. (1986); Flexural Vibrations of Symmetrically Lam-inated Composite Rectangular Plates Including Transverse Shear Effects, Int. J. Solids Struct., 22(2), pp.155-169, 1986. [22] Craig, T.J., and Dawe, D.J. (1987); Vibration of Shear-Deformable Laminated Plate Structures by the the Finite Strip Method, Comput. Struct., 27(1), pp.61-72, 1987. [23] Davies, J.D. (1962a); Influence of Support Conditions on the Behaviour of Long Rectangular Tanks, J.ACI, 59(4), pp.601-608, April 1962. [24] Davies, J.D. (1962b); Bending Moments in Edge Supported Square Concrete Tanks, The Struct.Engr., 40(5), pp.161-166, May 1962. [25] Davies, J.D. (1963); Bending Moments in Square Concrete Tanks Resting on Flat Rigid Supports, The Struct.Engr., 41(12), pp.407-410, Dec. 1963. [26] Davies, J.D. (1964); The Influence of Support Conditions on the Behaviour of Square Concrete Tanks, Mag. Cone. Res., 16(48), pp.153-165, Sept. 1964. .[27] Davies, J.D., and Cheung, Y.K. (1967); Bending Moments in Long Walled Tanks, ACI J., 64(10), pp.685-690, Oct. 1967. [28] Davies, J.D., Cheung, Y.K., and Gorecki, A. (1970); Analysis of Long Rectan-gular Tanks, Water and Water Eng., 74, pp.510-516, Dec. 1970. [29] Dawe, D.J. (1978); Finite Strip Models for Vibration of Mindlin Plates, J.Sound Vib., 59, pp.441-452, 1978. [30] Dawe, D.J., and Azizian, Z.G. (1986); The Performance of Mindlin Plate Finite Strips in Geometrically Nonlinear Analysis, Comput. Struct., 23(1), pp.1-14, 1986. [31] Engblom, J.J. and Ochoa, O.O. (1985); Through the Thickness Stress Predictions for Laminated Plates of Advanced Composite Materials, Int. J. Num. Meth. Eng., 21, pp.1759-1776, 1985. BibHography 217 Fung, Y.C. (1965); Foundations of Solid Mechanics, Prentice-Hall, USA, 1965. Ghali, A. (1957); The Structural Analysis of Circular and Rectangular Concrete Tanks, Ph.D. Thesis, The Universit}' of Leeds, Leeds, England, 1957. Ghali, A. and Tadros, G.S. (1974); On Finite Strip Analysis of Continuous Plates, Pubhcations, Int. Assoc. for Bridge and Struct. Eng., 34(1), pp.25-34, 1974. Golley, B.W. and Petrohto, J. (1982); An Alternative Finite Strip Approach for the Static Analysis of Single-Span, Multi-Span and Cantilever Plates, Proc. Int. Conf. on Finite Element Methods, Shanghai, China, 2, pp.125-130, 1982. Golley, B.W. and Petrohto, J. (1984); Method for Analysing Tanks and Continous Plates, Comput. Struct., 18(6), pp.1141-1151, 1984. Golley, B.W., Grice, W.A. and Petrolito, J. (1987); Plate-Bending Analysis Using Finite Strip-Elements, J.Struct.Eng.,ASCE, 113(6), pp.1282-1296, June 1987. Graves-Smith, T.R., and Sridharan, S. (1978); A Finite Strip Method for the Post-Locally-Buckled Analysis of Plate Structures, Int.J.Mech.Sci., 20, pp.833-842, 1978. Grigorian, M. (1966); The Analysis and Design of Rectangular and Circular Containers and Related Structures, Ph.D. Thesis, Oxford University, Oxford, England, 1966. Hancock, G.J. (1981); Nonlinear Analysis of Thin Sections in Compression, J.Struct.Eng.,ASCE, 107(3), pp.455-471, Mar. 1981. Hinton, E. (1977); Flexure of Composite Laminates using the Thick Finite Strip Method, Comput. Struct., 7, pp.217-220, 1977. Huang, T.C. (1961); The Effect of Rotatory Inertia and of Shear Deformation on the Frequency and Normal Mode Equations of Uniform Beams with Simple End Conditions, J.Appl.Mech., Trans. ASME, 28, pp.579-584, 1961. Hughes, T.J.R., Cohen, M. and Haroun, M. (1978); Reduced and Selective In-tegration Techniques in the Finite Element Analysis of Plates, Nucl.Eng.Desg., 46, pp.203-222, 1978. Jones, R.M. (1975); Mechanics of Composite Materials, Intl. Stud. Ed., McGraw-Hill Kogakusha, Tokyo, Japan, 1975. Kapania, R.K., and Raciti, S. (1989); Recent Advances in Analysis of Laminated Beams and Plates, Part I: Shear Effects and Buckling, AIAA J., 27(7), pp.923-934, July 1989. Kranys, M. (1989); Causal Theories of Evolution and Wave Propagation in Math-ematical Physics, Appl.Mech.Rev., 42(11), pp.305-322, Nov. 1989. Kreja, I. and Cywihski, Z. (1988); Is Reduced Integration Just A Numerical Trick, Comput. Struct., 29(3), pp.491-496, 1988. Bibliography 218 Lajczok, M.R. (1986); New Approach in the Determination of Interlaminar Shear Stresses from the Results of MSC/NASTRAN, Comput. Struct., 24(4), pp.651-656, 1986. Marguerre, K. (1938); Zur Theorie Der Gekrummten Platte Grosser Forman-derung, Proc. 5th Intl. Cong. Appl. Mech., John Wiley & Sons, Inc., New York, pp.93-101, 1938. Mawenya, A.S., and Davies, J.D. (1974); Finite Strip Analysis of Plate Bending Including Transverse Shear Effects, Build. Sci., 9, pp.175-180, 1974. McClintock, F.A. and Argon, A.S. (1966); Mechanical Behaviour of Materials, Addison-Wesley, USA, 1966. Mindlin, R.D. (1951); Influence of Rotatory Inertia and Shear on Flexural Mo-tions of Isotropic, Elastic Plates, J. Appl. Mech., Trans. ASME,18, pp.31-38, 1951. Minguet, P.J., Dugundji, J., and Lagace, P. (1989); Postbuckling Behaviour of Laminated Plates Using a Direct Energy-Minimization Technique, AIAA J., 27, pp.1785-1792, Dec. 1989. Noor, A.K. (1975); Stability of Multilayered Composite Plates, Fibre Sc. and Tech., 8, pp.81-89, 1975. Noor, A.K., and Burton, W.S. (1989); Assessment of Shear Deformation The-ories for Multilayered Composite Plates, Appl.Mech.Rev., 42(1), pp.1-13, Jan. 1989. Ohate, E. and Suarez, B. (1983a); A Unified Approach for the Analysis of Bridges, Plates and Axisymmetric Shells Using The Linear Mindlin Strip El-ement, Comput. Struct., 17(3), pp.407-426, 1983. Ohate, E. and Suarez, B. (1983b); A Comparison of the Linear, Quadratic and Cubic Mindlin Strip Elements for the Analysis of Thick and Thin Plates, Com-put. Struct., 17(3), pp.427-439, 1983. Pawsey, S.F. and Clough, R.W. (1971); Improved Numerical Integration of Thick Shell Finite Elements, Int.J.Num.Meth.Eng., 3, pp.575-586, 1971. Pica, A., Wood, R.D. and Hinton, E. (1980); Finite Element Analysis Of Ge-ometrically Nonlinear Plate Behaviour Using A Mindlin Formulation, Comput. Struct., 11, pp.203-215, 1980. Pryor Jr., C.W., and Barker, R.M. (1971); A Finite-Element Analysis Including Transverse Shear Effects for Applications to Laminated Plates, AIAA J., 9(5). 912-917, 1971. Pugh, E.D.L., Hinton, E. and Zienkiewicz, O.C. (1978); A Study of Quadrilateral Plate Bending Elements with 'Reduced' Integration, Int.J.Num.Meth.Eng., 12, pp.1059-1079, 1978. Putcha, N.S. and Reddy, J.N. (1984) ; A Mixed Shear Flexible Finite Element for the Analysis of Laminated Plates, Comp. Meth. in App. Mech. and Eng., 44, pp.213-227, 1984. Bibliography 219 [63] Putcha, N.S. and Reddy, J.N. (1986) ; A Refined Mixed Shear Flexible Finite Element for the Nonlinear Analysis of Laminated Plates, Comput. Struct., 22(4), pp.529-538, 1986. [64] Reddy, J.N. (1984a); A Refined Nonlinear Theory of Plates with Transverse Shear Deformation, Int. J. Solids Struct., 20(9/10), 881-896, 1984. [65] Reddy, J.N. (1984b); A Simple Higher-Order Theory for Laminated Composite Plates, J.App.Mech., 51, pp.745-752, Dec. 1984. [66] Reddy, J.N. (1984c); Energy and Variational Methods in Applied Mechanics, John Wiley & Sons, 1984. [67] Reddy, J.N. (1985); A Review of the Literature on Finite-Element Modelling of Laminated Composite Plates, Shock and Vib. Digest, 17(4), pp.3-8, April 1985. [68] Reddy, J.N. (1989); On the Generalization of Displacement-Based Laminate The-ories, Appl.Mech.Rev., 42(11), Pt.2, pp.S213-S222, Nov. 1989. [69] Reddy, J.N., and Chao, W.C. (1981a); Non-Linear Bending of Thick Recatngular, Laminated Composite Plates, Int.J.Non-Linear Mech.,16(3/4), pp.291-301, 1981. [70] Reddy, J.N., and Chao, W.C. (1981b); Large-Deflection and Large-Amplitude Free Vibrations of Laminated Composite-Material Plates, Comput. Struct., 13, pp.341-347, 1981. [71] Reddy, J.N., and Khdeir, A.A. (1989); Buckling and Vibration of Laminated Composite Plates Using Various Plate Theories, AIAA J.. 27(12), pp.1808-1817, Dec. 1989. [72] Reissner, E. (1945); The Effect of Transverse Shear Deformation on the Bending of Elastic Plates, J.Appl.Mech., Trans.ASME, 67,pp.A69 - A77, 1945. [73] Reissner, E. (1947); On Bending of Elastic Plates, Quart.J.Appl.Math., 5, pp.55-68, 1947. [74] Roufaeil, O.L. and Dawe, D.J. (1980); Vibration Analysis of Rectangular Mindlin Plates by the Finite Strip Method, Comput. Struct., 12, pp.833-842, 1980. [75] Rushton, K.R. (1970) ; Large Deflexion of Plates with Initial Curvature, Int.J.Mech.Sci., 12, pp.1037-1051, 1970. [76] Rushton, K.R. (1972) ; Buckling of Laterally Loaded Plates having Initial Cur-vature, Int.J.Mech.Sci.,' 14, pp.667-680, 1972. [77] Srinivas, S. and Rao, A.K. (1970); Bending, Vibration and Buckling of Simply Supported Thick Orthotropic Rectangular Plates and Laminates, Int. J. Sohds Struct., 6, 1463-1481, 1970. [78] Szilagyi, G.Y. (1974); Quelques applications de la methode des bandes finies, Pubhcations, Int. Assoc. for Bridge and Struct. Eng., 34(11), pp.149-168, 1974. [79] Timoshenko, S.P. (1921); On the Correction for Shear of the Differential Equa-tion for Transverse Vibrations of Prismatic Bars, Phil. Mag. (Series 6) 41, pp.744-746, 1921. Bibliography 220 [80] Timoshenko, S.P. (1922); On the Transverse Vibrations of Bars of Uniform Cross-Section, Phil. Mag. (Series 6) 43, pp.125-131, 1922. [81] Timoshenko, S.P. , Young, D.H. and Weaver, W. (1974); Vibration Problems in Engineering, John Wiley & Sons, U.S.A., 1974. [82] Timoshenko, S.P. and Goodier, J.N. (1982); Theory of Elasticity, 3rd Ed., McGraw-Hill,Japan, 1982. [83] Timoshenko, S.P. and Woinowsky-Krieger, S. (1970); Theory of Plates and Shells, 2nd Ed., McGraw-Hill Kogakusha, Japan, 1970. [84] Traill-Nash, R.W. and Collar, A.R. (1953); The Effects of Shear Flexibilty and Rotatory Inertia on the Bending Vibrations of Beams, Q.J.Mech.Appl.Math., 6, Pt.2, pp.186-222, 1953. [85] Turvey, G.J. (1977); Bending of Laterally Loaded, Simply Supported, Moderately Thick, Antisymmetrically Laminated Rectangular Plates, Fibre Sc. and Tech., 10, PP.211-232,"l977. [86] von Karman, T. (1910); Festigkeitsprobleme im Maschinenbau, Encyklopadie der Mathematischen Wissenchaften, rV, B.G. Teubner, Leipzig, Germany, 1910. [87] Wang, H., and Zhang, J. (1986); The Finite Strip-Element Method, Proc. Int. Conf. Comput. Mech., Tokyo, Japan, I, pp. 151-156, Springer Verlag, 1986. [88] Warnock, F.V., and Benham, P.P. (1965); Mechanics of Solids and Strength of Materials, Sir Issac Pitman, London, England, 1965. [89] Warburton, G.B. (1983); Rayleigh's Contribution to Modern Vibration Analysis, J. Sound and Vib., 88(2), pp.162-173, 1983. [90] Washizu, K. (1982); Variational Methods in Elasticity and Plasticity, 3rd Edition, Pergamon, England, 1982. [91] Whitney, J.M. (1973); Shear Correction Factors for Orthotropic Laminates Un-der Static Load, J. Appl. Mech., Trans. ASME, 40, pp.302-304, 1973. [92] Wilby, C A . (1977); Structural Analysis of Reinforced Concrete Tanks, J. ASCE, Struct. Div., 103(ST5), pp.989-1004, May 1977. [93] Wiseman, D.L., Puckett, J.A. and Chong, K.P. (1987); Recent Developments of the Finite Strip Method, in "Dynamics of Structures", Roesset, J.M., Edit., Proc. of Struct. Dyn. Section, Struct. Cong., ASCE, pp.292-309, 1987. [94] Wu, C L , and Cheung, Y.K. (1974); Frequency Analysis of Rectangular Plates Continuous in One or Two Directions, Earth. Eng. and Struct. Dyn., 3, pp.3-14, 1974. [95] Yamaki, N. (1959) ; Postbuckling Behaviour of Rectangular Plates With Small Initial Curvature Loaded in Edge Compression, J. App. Mech., Trans. ASME, 26, pp.407-414, Sept., 1959. [96] Zienkiewicz, O.C (1977); The Finite Element Method, 3rd Edition, McGraw-Hill, London, England, 1977. Bibliography 221 [97] Zienkiewicz, O.C. and Hinton, E. (1976); Reduced Integration, Function Smooth-ing and Non-conformity in Finite Element Analysis ( with Special Reference to Thick Plates), J. Franklyn Inst., 302, pp.443-461, 1976. [98] Zienkiewicz, O.C, Taylor, R.L. and Too, J.M. (1971); Reduced Integration Tech-nique in General Analysis of Plates and Shells, Int. J. Num. Meth. Eng., 3, pp.275-290, 1971. Appendix A Timoshenko Beam Mode Shapes As mentioned in Chapter 2, the bases used in the analysis of the fixed-ended strips are the mode shapes of a beam made up of a cross section of the plate of concern. In thick plate theory, these modes are from Timoshenko beam theory which considers both rotatory inertia and transverse shear deformation. The mode shapes for the Timoshenko beams under various boundary conditions had been derived by Traill-Nash and Collar(1953) and Huang(1961). The normal modes consists of two sets of spectra(Traill-Nash and Collar op. cit., Huang op. cit., Roufaeil and Dawe 1980). One set deals with the flexural vibration of the beam and has the slope of the deflection curve and the rotation in phase, thus giving rise to little shear deflection, compared to the other set where the two quantities are out of phase with high shear strains and low deflections. The present theory is concerned with the behaviour of moderately thick plates where the flexural behaviour still dominates over that of shear deformation. Therefore, as suggested by Roufaeil and Dawe(op. cit.), only the the flexural modes are considered. The modes shapes as found by Huang, and presented by Dawe in a slightly rear-ranged form, were used in the present analysis and are presented below. The rotation of the cross-section has been defined here so that it is consistent with the general plate theory presented in the body of the thesis. 222 Appendix A. Timoshenko Beam Mode Shapes 223 Let the non-dimensionahsed coordinate along the beam, £, be defined by £=| (o < e < i) where £ and L are the coordinate along the beam and the length of the beam, re-spectively. Also let the following denote the respective quantities with regard to the beam properties. u) = circular frequency of vibration, A = area of cross-section, / = second moment of area of cross-section, n2 — transverse shear correction factor, p = density of material, E = Young's modulus G = shear modulus (Here it may be noted that the density of the material does not affect the normal mode shapes.) Then, defining the following quantities, I r2 AL' 2 EI sl -K2AGL2 K/32 - s2) PL b(a2 + s2) k2 = ctL Appendix A. Timoshenko Beam Mode Shapes 224 where V2 ( r 2 - s 2 ) 2 + b2 i I 1 2 (A.l) then the normal mode pairs can be written, while omitting the mode number suffix, as - Jfe, _ _ W = 6 sin b(3£ A cos 6/3£ - — .5 sinh 6a£ - cosh 6a£ (A.2) « 2 $ = 8ki cos 6/3£ — ki sin 6/3£ — kiS cosh ba( — k2 sinh 6a£ (A.3) where r5 = cos b/3 — cosh ba sinh 6a — sin 6/3 with the values of b being obtained as the roots of the frequency equation b\b2s2(r2 - s2)2 4- (3s2 - r2)} 2 - 2 cosh ba cos 6/3 + J . ; K ^ sinh 6a sin 6/3 = 0 (A.4) V I — b2r2s2 It is to be noted that the above spectrum of flexural modes would exist only under the condition of ( r 2 - s 2 ) 2 + ^f >(r2 + s2) The orthogonality condition of the above normal modes as given by Huang(1961) is £(wmwn + j$m $ „ K ~ = o and it was used at the initial stages to check the accuracy of the solutions. The solution of the frequency equation has been carried out by the use of a scheme that makes use of the root finding subroutine DRZFUN provided by the UBC Com-puting Centre. This routine needs the approximate position of the root to be provided. Appendix A. Timoshenko Beam Mode Shapes 225 Therefore as the first mode root, the value of b for the classical beam, given by Bishop and Johnson(1960) as (2m + l ) 2 7 r 2 bm = with m = 1, was used. Then on finding the first mode frequency, the approximate value was gradually incremented until the required number of roots were found. There are two shortcomings present in the numerical procedure adopted so far. First is the inability to find the higher roots on certain occassions. The second is the numerical errors in the mode shape for high modes when £ is close to unity. It should be possible to avoid these difficulties by re-expressing the equations in a different form, but this was not attempted as it was of no immediate concern. In problems symmetrical about the plate centre hne the second error is avoided by the use of numerical integration only on one half of the plate. Appendix B Timoshenko Beams Under Support Displacement This appendix presents the solution of the static Timoshenko Beam equations under support displacements. The purpose of this exercise was to find the support bases and the additional modes used in Chapter 2. B. l The Basic Equations Following the procedure of Timoshenko et <z/.(1974) with a sign convention and nomenclature that is consistent with that used in this thesis, the basic equations can be derived as follows. B . l . l E q u i l i b r i u m E q u a t i o n s Rotational equilibrium of the element shown in Figure B.l gives v = m Vertical equilibrium gives dV The total slope of the middle axis can be written as (see Figure B.2) 226 (B.l) (B.2) Appendix B. Timoshenko Beams Under Support Displacement Figure B.2: Rotation and Shear Strain in a Timoshenko Beam Appendix B. Timoshenko Beams Under Support Displacement 228 + (B.S) where, w = the total displacement of the neutral axis due to both bending and shear <j> = the rotation of cross-section due to bending, and /3 = the shear strain at the section. Now from elementary beam theory, (Equation (d) of pg. 433 of Timoshenko et al., 1974) M = -EI% (B.4) di V = k'pAG (B.5) where , k'(— K2) is a correction to the shear modulus to account for the non-constant shear strain acting at a cross-section. The generally accepted value for this constant in the case of an isotropic beam is 0.833 (Clough and Penzien 1982, Timoshenko et al. 1974). Using Equations B.l to B.5, the following expressions can be obtained for <f> and w, <Kt) = ~ JMdiA-C (B.6) W{i) = I ^ A G d i + Imdt + D (B-7) where C and D are constants of integration. Appendix B. Timoshenko Beams Under Support Displacement 229 B.2 The Solutions This section presents the solutions obtained for the above equations under different boundary conditions. The bases, which are obtained from the solutions by considering support displace-ments of unity are presented here with the following code. The letter W or P refers, respectively, to w or <}>, which will be the degree of freedom of the base. The sub-script refers to the degree of freedom that is moved, and the superscript refers to the support that is moved, with A for cf = 0 and B for cf = /. B . 2 . 1 F i x e d - F i x e d B e a m B . 2 . 1 . 1 F i x e d - F i x e d B e a m U n d e r S u p p o r t T r a n s l a t i o n For the beam shown in Figure B.3, the boundary conditions are w(0) = AA Appendix B. Timoshenko Beams Under Support Displacement 230 and (b(0) = 4(1) = w(l) = 0 The solution for the equations, then, are w(0 = AA-WA(() <Kt) = AA.PA(£) (B.8) (B.9) where, wA(() = i + - -2* + £2 IK2AG 6EI \ I 2£ ' -r - 3 (B.10) ^ ( 0 = T - 1 (B.ll) (B.12) with I2 2 6EI K2AG The bases for translation of support B can be obtained from the above by the sub-stitution of (/ — £) for £, giving w*(0 = i c (B.13) p£it) = (B.14) Appendix B. Timoshenko Beams Under Support Displacement Figure B.4: Fixed-Fixed Beam Under Support Rotation B . 2 . 1 . 2 F i x e d - F i x e d B e a m u n d e r S u p p o r t R o t a t i o n For the beam shown in Figure B.4, the displacement boundary conditions are and w(0) = u;(/) = <j>(l) = 0 The solutions are, w(0 = eA-w*(o (B (B where, c it 2EI (B Appendix B. Timoshenko Beams Under Support Displacement 232 Witt) = I 1 + 2EI K2AGP K2AGIC R r R i [ E I C \ 1 2R 3Z2 3E/ZCJ (B.18) with, in addition to the previous definitions, „ 1 EI 3 K2AG I Similarly, in the case of the rotation of support B, the following bases can be obtained. ci i( i2 K2AG 2EI 3EI 2EI ( / - 0 ( i + - AR 2AGl2 K2AGIC (B.19) - ( J - 0 a R EIC1 3P + 3EIIC (B.20) B . 2 . 2 F i x e d - P i n n e d B e a m B . 2 . 2 . 1 F i x e d - P i n n e d B e a m : T r a n s l a t i o n o f t h e F i x e d S u p -p o r t The solution for this problem can be obtained from the solution for a fixed-fixed beam. Considering Figure B.5, it can be seen that the solution is the same as that of the left-half of a fixed-fixed beam with the same properties but twice the length. Appendix B. Timoshenko Beams Under Support Displacement 233 Figure B.5: Fixed-Pinned Beam Under Fixed-Support Translation This gives, «>(0 = AA-WA(() m = AA-Pa(O (B.21) (B.22) where, C21 + IK2 AG 6EI \ I (B.23) w K Z ) EIC2t\2l J (B.24) with, Appendix B. Timoshenko Beams Under Support Displacement 234 Figure B.6: Fixed-Pinned Beam Under Pinned-Support Translation B .2 .2 .2 Fixed-Pinned Beam: Translation of the Pinned Sup-port As shown in Figure B.6, this also can be solved by the use of the solution for a fixed-fixed beam, giving the bases as, Of course, the previously derived bases for fixed support translation also could have been used to obtain the same by transforming them. 2 i - j (2/-o 2 n ; IK2 AG 6EI \l \ (B.25) Appendix B. Timoshenko Beams Under Support Displacement 235 Figure B.7: Fixed-Pinned Beam Under Fixed-Support Rotation B.2.2.3 Fixed-Pinned Beam: Rotation of the Fixed Sup-port As shown in Figure B.7, the solution for this can be obtained from a fixed ended beam of twice the length by the application of antisymmetric rotations to its supports. Then, by the use of the subscript 21 to refer to the items defined with respect to a fixed-fixed beam of length 21, where / is the length of the beam of concern, * ( 0 = *A[{PtMt) + {P?Ut)] (0 < e < 0 (B.27) «K0 = *A [(w}ut) + (wfuo] (o < e < 0 (B.28). giving, Pt(0 = ir 1_ C21 C " - E I + EI (B.29) Appendix B. Timoshenko Beams Under Support Displacement 236 Wj(t) = 2A2l(C-l) + B2l[(2l-i)2-e} +Z>2- [C3 - (21 - 03] (B.30) where E I F A2L = 1 + B->i = 2K2AGP K2AGPC2 F 2EIlC2l D, = F 12P 6EIPC 21 C2L - ~ R Al2 EI F = — — h 3 K2AG with R being as defined earlier. B.2.3 Pinned-Pinned Beam This is the easiest to solve giving the following bases as the result of respective unit displacements at the supports. All bases are due to translations as there can be no applied rotations at such supports. Wtf(0 = l - f (B.31) P*(() = -1 (B.32) W*(0 = j (B-33) P*(0 = 1 (B.34) 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0050509/manifest

Comment

Related Items