PLATE AND PANEL STRUCTURES OF ISOTROPIC, COMPOSITE AND PIEZOELECTRIC MATERIALS, INCLUDING SANDWICH CONSTRUCTION

Transcription

1 PLATE AND PANEL STRUCTURES OF ISOTROPIC, COMPOSITE AND PIEZOELECTRIC MATERIALS, INCLUDING SANDWICH CONSTRUCTION

2 SOLID MECHANICS AND ITS APPLICATIONS Volume 10 Series Editor: G.M.L. GLADWELL Department of Civil Engineering Universit of Waterloo Waterloo, Ontario, Canada NL 3GI Aims and Scope of the Series The fundamental questions arising in mechanics are: Wh?, How?, and How much? The aim of this series is to provide lucid accounts written bij authoritative researchers giving vision and insight in answering these questions on the subject of mechanics as it relates to solids. The scope of the series covers the entire spectrum of solid mechanics. Thus it includes the foundation of mechanics; variational formulations; computational mechanics; statics, kinematics and dnamics of rigid and elastic bodies: vibrations of solids and structures; dnamical sstems and chaos; the theories of elasticit, plasticit and viscoelasticit; composite materials; rods, beams, shells and membranes; structural control and stabilit; soils, rocks and geomechanics; fracture; tribolog; eperimental mechanics; biomechanics and machine design. The median level of presentation is the first ear graduate student. Some tets are monographs defining the current state of the field; others are accessible to final ear undergraduates; but essentiall the emphasis is on readabilit and clarit. For a list of related mechanics titles, see final pages.

3 Plate and Panel Structures of Isotropic, Composite and Piezoelectric Materials, Including Sandwich Construction b JACK R. VINSON Center for Composite Materials and College of Marine Studies, Department of Mechanical Engineering, Spencer Laborator,Universit of Delaware, Newark, Delaware, U.S.A.

4 A C.I.P. Catalogue record for this book is available from the Librar of Congress. ISBN (HB) ISBN (e-book) Published b Springer, P.O. Bo 17, 3300 AA Dordrecht, The Netherlands. Sold and distributed in North, Central and South America b Springer, 101 Philip Drive, Norwell, MA 0061, U.S.A. In all other countries, sold and distributed b Springer, P.O. Bo 3, 3300 AH Dordrecht, The Netherlands. Printed on acid-free paper All Rights Reserved 005 Springer No part of this work ma be reproduced, stored in a retrieval sstem, or transmitted in an form or b an means, electronic, mechanical, photocoping, microfilming, recording or otherwise, without written permission from the Publisher, with the eception of an material supplied specificall for the purpose of being entered and eecuted on a computer sstem, for eclusive use b the purchaser of the work. Printed in the Netherlands.

5 This tetbook is dedicated to m beautiful wife Midge, who through her encouragement and nurturing over these last two decades, has made the writing of this book possible.

6 Table of Contents Preface Part 1 Plates and Panels of Isotropic Materials v 1. Equations of Linear Elasticit in Cartesian Coordinates Stresses Displacements 1.3. Strains 1.4. Isotrop and Its Elastic Constants Equilibrium Equations Stress-Strain Relations Linear Strain-Displacement Relations Compatibilit Equations Summar References Problems 8. Derivation of the Governing Equations for Isotropic Rectangular Plates Assumptions of Plate Theor 11.. Derivation of the Equilibrium Equations for a Rectangular Plate 1.3. Derivation of Plate Moment-Curvature Relations and Integrated Stress Resultant-Displacement Relations Derivation of the Governing Differential Equations for a Plate 0.5. Boundar Conditions for a Rectangular Plate 3.6. Stress Distribution within a Plate 6.7. References 7.8. Problems 8 3. Solutions to Problems of Isotropic Rectangular Plates Some General Solutions of the Biharmonic Equations Double Series Solutions (Navier Solution) Single Series Solutions (Method of M. Lev) Eample of a Plate with Edges Supported b Beams Isotropic Plates Subjected to a Uniform Lateral Load Summar References Problems 46

7 viii 4. Thermal Stress in Plates General Considerations Derivation of the Governing Equations for a Thermoelastic Plate Boundar Conditions General Treatment of Plate Nonhomogeneous Boundar Conditions Thermoelastic Effects on Beams Self-Equilibration of Thermal Stresses References Problems Circular Isotropic Plates Introduction Derivation of the Governing Equations Aiall Smmetric Circular Plates Solutions for Aiall Smmetric Circular Plates Circular Plate, Simpl Supported at the Outer Edge, Subjected to a Uniform Lateral Loading, p Circular Plate, Clamped at the Outer Edge, Subjected to a Uniform Lateral Loading, p Annular Plate, Simpl Supported at the Outer Edge, Subjected to a Stress Couple, M, at the Inner Boundar Annular Plate, Simpl Supported at the Outer Edge, Subjected to a Shear Resultant, Q 0, at the Inner Boundar Some General Remarks Laminated Circular Thermoelastic Plates References Problems Buckling of Isotropic Columns and Plates Derivation of the Plate Governing Equations for Buckling Buckling of Columns Simpl Supported at Each End Column Buckling With Other Boundar Conditions Buckling of Isotropic Rectangular Plates Simpl Supported on All Four Edges Buckling of Isotropic Rectangular Plates with Other Loads and Boundar Conditions The Buckling of an Isotropic Plate on an Elastic Foundation Subjected to Biaial In-Plane Compressive Loads References Problems 113

8 i 7. Vibrations of Isotropic Beams and Plates Introduction Natural Vibrations of Beams Natural Vibrations of Isotropic Plates Forced Vibration of Beams and Plates References Problems Theorem of Minimum Potential Energ, Hamilton s Principle and Their Applications Introduction Theorem of Minimum Potential Energ Analses of a Beam In Bending Using the Theorem of Minimum Potential Energ The Buckling of Columns Vibration of Beams Minimum Potential Energ for Rectangular Isotropic Plates The Buckling of an Isotropic Plate Under a Uniaial In-Plane Compressive Load, Simpl Supported on Three Sides, and Free on an Unloaded Edge Functions for Displacements in Using Minimum Potential Energ for Solving Beam, Column and Plate Problems References Problems Reissner s Variational Theorem and Its Applications Introduction Static Deformation of Moderatel Thick Beams Fleural Vibrations of Moderatel Thick Beams 150 Part - Plates and Panels of Composite Materials 10. Anisotropic Elasticit and Composite Laminate Theor Introduction Derivation of the Anisotropic Elastic Stiffness and Compliance Matrices The Phsical Meaning of the Components of the Orthotropic Elasticit Tensor Methods to Obtain Composite Elastic Properties from Fiber and Matri Properties Thermal and Hgrothermal Considerations 170

9 10.6. Time-Temperature Effects on Composite Materials High Strain Rate Effects on Material Properties Laminae of Composite Materials Laminate Analsis References Problems Plates and Panels of Composite Materials Introduction Plate Equilibrium Equations The Bending of Composite Material Laminated Plates: Classical Theor Classical Plate Theor Boundar Conditions Navier Solutions for Rectangular Composite Material Plates Navier Solution for a Uniforml Loaded Simpl Supported Plate - An Eample Problem Lev Solution for Plates of Composite Materials Perturbation Solutions for the Bending of a Composite Material Plate With Mid-Plane Smmetr and No Bending-Twisting Coupling Quasi-Isotropic Composite Panels Subjected to a Uniform Lateral Load A Static Analsis of Composite Material Panels Including Transverse Shear Deformation Effects Boundar Conditions for a Plate Including Transverse Shear Deformation Composite Plates on an Elastic Foundation or Contacting a Rigid Surface Solutions for Plates of Composite Materials Including Transverse Shear Deformation Effects, Simpl Supported on All Four Edges Some Remarks on Composite Structures Governing Equations for a Composite Material Plate with Mid-Plane Asmmetr Governing Equations for a Composite Material Plate with Bending-Twisting Coupling Concluding Remarks References Problems Elastic Instabilit (Buckling) of Composite Plates General Considerations The Buckling of an Orthotropic Composite Plate Subjected to In-Plane Loads Classical Theor Buckling of a Composite Plate on an Elastic Foundation References 5

10 i 1.5. Problems Linear and Nonlinear Vibration of Composite Plates Dnamic Effects on Panels of Composite Materials Natural Fleural Vibrations of Rectangular Plates: Classical Theor Natural Fleural Vibrations of Composite Material Plates Including Transverse Shear Deformation Effects Forced Vibration Response of a Composite Material Plate Subjected to a Dnamic Lateral Load Vibration Damping References Problems Energ Methods for Composite Material Structures Introduction A Rectangular Composite Material Plate Subjected to Lateral and Hgrothermal Loads In-Plane Shear Strength Determination of Composite Materials in Composite Panels Cantilevered Anisotropic Composite Plate Subjected to a Uniform Lateral Load Use of the Theorem of Minimum Potential Energ to Determine Buckling Loads in Composite Plates Trial Functions for Various Boundar Conditions for Composite Material Rectangular Plates Elastic Stabilit of a Composite Panel Including Transverse Shear Deformation and Hgrothermal Effects References Problems 93 Part 3 - Plates and Panels of Sandwich Construction 15. Governing Equations for Plates and Panels of Sandwich Construction Constitutive Equations for a Sandwich Plate Governing Equations for Sandwich Plates and Panels Minimum Potential Energ Theorem for Sandwich Plates Solutions to Problems Involving Sandwich Panels References Problems 300

11 ii 16. Elastic Instabilit (Buckling) of Sandwich Plates General Considerations The Overall Buckling of an Orthotropic Sandwich Plate Subjected to In-Plane Loads Classical Theor The Buckling of Honecomb Core Sandwich Panels Subjected to In-Plane Compressive Loads The Buckling of Solid-Core or Foam-Core Sandwich Panels Subjected to In-Plane Compressive Loads Buckling of a Truss-Core Sandwich Panel Subjected to Uniaial Compression Elastic Stabilit of a Web-Core Sandwich Panel Subjected to a Uniaial Compressive In-Plane Load Buckling of Honecomb Core Sandwich Panels Subjected to In-Plane Shear Loads Buckling of a Solid-Core or Foam-Core Sandwich Panel Subjected to In-Plane Shear Loads Buckling of a Truss-Core Sandwich Panel Subjected to In-Plane Shear Loads Buckling of a Web-Core Sandwich Panel Subjected to In-Plane Shear Loads Other Considerations References Problems Structural Optimization to Obtain Minimum Weight Sandwich Panels Introduction Minimum Weight Optimization of Honecomb Core Sandwich Panels Subjected to a Unidirectional Compressive Load Minimum Weight Optimization of Foam Core Sandwich Panels Subjected to a Unidirectional Compressive Load Minimum Weight Optimization of Truss Core Sandwich Panels Subjected to a Unidirectional Compressive Load Minimum Weight Optimization of Web Core Sandwich Panels Subjected to a Unidirectional Compressive Load Minimum Weight Optimization of Honecomb Core Sandwich Panels Subjected to In-Plane Shear Loads Minimum Weight Optimization of Solid and Foam Core Sandwich Panels Subjected to In-Plane Shear Loads Minimum Weight Optimization of Truss Core Sandwich Panels Subjected to In-Plane Shear Loads Minimum Weight Optimization of Web Core Sandwich Panels Subjected to In-Plane Shear Loads Optimal Stacking Sequence for Composite Material Laminate Faces for Various Sandwich Panels Subjected to Various Loads References 375

12 iii Problems 376 Part 4 - Plates Using Smart (Piezoelectric) Materials 18. Piezoelectric Materials Introduction Piezoelectric Effect References Piezoelectric Effects Laminate of a Piezoelectric Material References Use of Minimum Potential Energ to Analze a Piezoelectric Beam Introduction References 393 Author Inde 395 Subject Inde 401

13 PREFACE Plates and panels are primar structural components in man structures from space vehicles, aircraft, automobiles, buildings and homes, bridges decks, ships, and submarines. The abilit to design, analze, optimize and select the proper materials and architecture for plates and panels is a necessit for all structural designers and analsts, whether the adjective in front of the engineer on their degree reads aerospace, civil, materials or mechanical. This tet is broken into four parts. The first part deals with the behavior of isotropic plates. Most metals and pure polmeric materials used in structures are isotropic, hence this part covers plates and panels using metallic and polmeric materials. The second part involves plates and panels of composite materials. Because these fiber reinforced matri materials can be designed for the particular geometr and loading, the are ver often anisotropic with the properties being functions of how the fibers are aligned, their volume fraction, and of course the fiber and matri materials used. In general, plate and panel structures involving composite materials will weigh less than a plate or panel of metallic material with the same loads and boundar conditions, as well as being more corrosion resistant. Hence, modern structural engineers must be knowledgeable in the more complicated anisotropic material usage for composite plates and panels. Sandwich plates and panels offer spectacular advantages over the monocoque constructions treated above. B having suitable face and core materials, isotropic or anisotropic, sandwich plates and panels subjected to bending loads can be 300 times as stiff in bending, with face stresses 1/30 of those using a monocoque construction of a thickness equal to the two faces of the sandwich. Thus, for onl the additional weight of the light core material, the spectacular advantages of sandwich construction can be attained. In Part 3, the analses, design and optimization of isotropic and anisotropic sandwich plates and panels are presented. In Part 4, the use of piezoelectric materials in beams, plates and panels are treated. Piezoelectric materials are those that when an electrical voltage is applied, the effects are tensile, compressive or shear strains in the material. Conversel, with piezoelectric materials, when loads cause tensile, compressive or shear strains, an electrical voltage is generated. Thus, piezoelectric materials can be used as damage sensors, used to achieve a planned structural response due to an electrical signal, or to increase damping. Piezoelectric materials are often referred to as smart or intelligent materials. The means to describe this behavior and incorporate this behavior into beam, plate and panel construction is the theme of Part 4. This book is intended for three purposes: as an undergraduate tetbook for those students who have taken a mechanics of material course, as a graduate tetbook, and as a reference for practicing engineers. It therefore provides the fundamentals of plate and panel behavior. It does not include all of the latest research information nor the complications associated with numerous comple structures but those structures can be studied and analzed better using the information provided herein.

14 vi Several hundred problems are given at the end of Chapters. Most if not all of these problems are homework and eam problems used b the author over several decades of teaching this material. Appreciation is epressed to Alejandro Rivera, who as the first student to take the course using this tet, worked most of the problems at the end of the chapters. These solutions will be the basis of a solutions manual which will be available to professors using this tet who contact me. Special thanks is given to James T. Arters, Research Assistant, who has tped this entire manuscript including all of its man changes and enhancements. Finall, man thanks are given to Dr. Moti Leibowitz who reviewed and offered significant suggestions toward improving Chapter 18, 19 and 0.

15 CHAPTER 1 EQUATIONS OF LINEAR ELASTICITY IN CARTESIAN COORDINATES References [ ] * derive in detail the formulation of the governing differential equations of elasticit. Those derivations will not be repeated here, but rather the equations are presented and then utilized to sstematicall make certain assumptions in the process of deriving the governing equations for rectangular plates and beams. 1.1 Stresses Consider an elastic bod of an general shape. Consider the material to be a continuum, ignoring its crstalline structure and its grain boundaries. Also consider the continuum to be homogeneous, i.e., no variation of material properties with respect to the spatial coordinates. Then, consider a material point anwhere in the interior of the elastic bod. If one assigns a Cartesian reference frame with aes, and z, shown in Figure 1.1, it is then convenient to assign a rectangular parallelepiped shape to the material point, and label it a control element of dimensions d, d and dz. The control element is defined to be infinitesimall small compared to the size of the elastic bod, et infinitel large compared to elements of the molecular structure, in order that the material can be considered a continuum. On the surfaces of the control element there can eist both normal stresses (those perpendicular to the plane of the face) and shear stresses (those parallel to the plane of the face). On an one face these three stress components comprise a vector, called a surface traction. It is important to note the sign convention and the meaning of the subscripts of these surfaces stresses. For a stress component on a positive face, that is, a face whose outer normal is in the direction of a positive ais, that stress component is positive when it is directed in the direction of that positive ais. Conversel, when a stress is on a negative face of the control element, it is positive when it is directed in the negative ais direction. This procedure is followed in Figure 1.1. Also, the first subscript of an stress component on an face signifies the ais to which the outer normal of that face is parallel. The second subscript refers to the ais to which that stress component is parallel. In the case of normal stresses the subscripts are seen to be repeated and often the two subscripts are shortened to one, i.e. where i =, or z. i ii * Numbers in brackets refer to references given at the end of chapters.

16 Figure 1.1. Control element in an elastic bod showing positive direction of stresses. 1. Displacements The displacements u, v and w are parallel to the, and z aes respectivel and are positive when in the positive ais direction. 1.3 Strains Strains in an elastic bod are also of two tpes, etensional and shear. Etensional strains, where i =, or z, are directed parallel to each of the aes respectivel and are a measure of the change in dimension of the control volume in the subscripted direction due to the normal stresses acting on all surfaces of the control volume. Looking at Figure 1., one can define shear strains. The shear strain (where i and j =, or z, and i j) is a change of angle. As i j an eample shown in Figure 1., in the -- plane, defining to be (in radians), (1.1) then, 1. (1.)

17 3 Figure 1.. Shearing of a control element. It is important to define the shear strain to be one half the angle in order to use tensor notation. However, in man tets and papers the shear strain is defined as. Care must be taken to insure awareness of which definition is used when reading or utilizing a tet or research paper, to obtain correct results in subsequent analsis. Sometimes ij is termed tensor strain, and is referred to as engineering shear strain (not a tensor quantit). The rules regarding subscripts of strains are identical to those of stresses presented earlier. 1.4 Isotrop and Its Elastic Constants An isotropic material is one in which the mechanical and phsical properties do not var with orientation. In mathematicall modeling an isotropic material, the constant of proportionalit between a normal stress and the resulting etensional strain, in the sense of tensile tests is called the modulus of elasticit, E. Similarl, from mechanics of materials, the proportionalit between shear stress and the resulting angle ij described earlier, in a state of pure shear, is called the shear modulus, G. One final quantit must be defined the Poisson s ratio, denoted b. It is defined as the ratio of the negative of the strain in the j direction to the strain in the i direction caused b a stress in the i direction, ii. With this definition it is a positive quantit of magnitude 0 0.5, for all isotropic materials.

18 4 The well known relationship between the modulus of elasticit, the shear modulus and Poisson s ratio for an isotropic material should be remembered: E G. (1.3) (1 ) It must also be remembered that (1.3) can onl be used for isotropic materials. The basic equations of elasticit for a control element of an elastic bod in a Cartesian reference frame can now be written. The are written in detail in the following sections and the compact Einsteinian notation of tensor calculus is also provided. 1.5 Equilibrium Equations A material point within an elastic bod can be acted on b two tpes of forces: bod forces ( F i ) and surface tractions. The former are forces which are proportional to the mass, such as magnetic forces. Because the material is homogeneous, the bod forces can be considered to be proportional to the volume. The latter involve stresses caused b neighboring control elements. Figure 1.3. Control element showing variation of stresses.

19 5 Figure 1.1 is repeated above, but in Figure 1.3, the provision for stresses varing with respect to space is provided. Thus on the back face the stress is shown, while on the front face that stress value differs because is a function of ; hence, its value is ( / ) d. Also shown are the appropriate epressions for the shear stresses. The bod forces per unit volume, F i (i =,, z) are proportional to mass and, as stated before, because the bod is homogeneous, are proportional to volume. The summation of forces in the direction can be written as d d dz d d dz z z dz d d d dz d dz z d d F d d dz 0. z F (1.4) After cancellations, ever term is multiplied b the volume, which upon division b the volume, results in z z F 0. (1.5) Likewise, in the and z direction, the equilibrium equations are: z z F 0 (1.6) z z z z F z 0. (1.7) In the compact Einsteinian notation, the above three equilibrium equations are written as ki, F i 0 (i, k =,, z) (1.8) k where this is the ith equation, and the repeated subscripts k refer to each term being repeated in, and z, and where the comma means partial differentiation with respect to the subsequent subscript.

20 6 1.6 Stress-Strain Relations The relationship between the stresses and strains at a material point in a three dimensional bod mathematicall describe the wa the elastic material behaves. The are often referred to as the constitutive equations and are given below without derivation, because eas reference to man tets on elasticit can be made, such as [ ]. 1 1, E (1.9), (1.10) E 1 z, E 1 G (1.11), (1.1) 1 G, 1 z z G (1.13), (1.14) z z From (1.9) the proportionalit between the strain and the stress is clearl seen. It is also seen that stresses and z affect the strain, due to the Poisson s ratio effect. Similarl, in (1.1) the proportionalit between the shear strain and the shear stress is clearl seen, the number two being present due to the definition of given in (1.). In the compact Einsteinian notation, the above si equations can be written as (1.15) ij aijkl kl where a ijkl is the generalized compliance tensor. 1.7 Linear Strain-Displacement Relations The strain-displacement relations are the kinematic equations relating the displacements that result from an elastic bod being strained due to applied loads, or the strains that occur in the material when an elastic bod is phsicall displaced. u, v (1.16), (1.17) w 1 u v z, zz (1.18), (1.19)

21 7 1 u w 1 v w z, z z (1.0), (1.1) z In compact Einsteinian notation, these si equations are written as: 1 ij ( ui, u, ) j j (i, j =,, z) (1.) i 1.8 Compatibilit Equations The purpose of the compatibilit equations is to insure that the displacements of an elastic bod are single-valued and continuous. The can be written as: z z z z (1.3) z z z z (1.4) zz z z z z (1.5) z zz, zz zz (1.6), (1.7) z zz zz zz (1.8) In compact Einsteinian notation, the compatibilit equations are written as follows: 0 (i, j, k, l =,, z). (1.9) ij,,,, kl kl ij ik jl jl ik However, in all of what follows herein, namel treating plates and beams, invariabl the governing differential equations are placed in terms of displacements, and if the solutions are functions which are single-valued and continuous, it is not necessar to utilize the compatibilit equations.

22 8 1.9 Summar It can be shown that both the stress and strain tensor quantities are smmetric, i.e., ij and (i, j =,, z). (1.30) ji ij ji Therefore, for the elastic solid there are fifteen independent variables; si stress components, si strain components and three displacements. In the case where compatibilit is satisfied, there are fifteen equations: three equilibrium equations, si constitutive relations and si strain-displacement equations. For a rather complete discussion [1.7] of the equations of elasticit for anisotropic materials, see Chapter 10 of this tet References 1.1. Sokolnikoff, I.S. (1956) Mathematical Theor of Elasticit, McGraw-Hill Book Compan, nd Edition, New York. 1.. Timoshenko, S. and Goodier, J.N. (1970) Theor of Elasticit, McGraw-Hill Book Compan, New York Green, A.E. and Zerna, W. (1954) Theoretical Elasticit, Oford Universit Press Love, A.E.H. (1934) Mathematical Theor of Elasticit, Cambridge Universit Press Muskhelishvili, N.I. (1953) Some Basic Problems in the Mathematical Theor of Elasticit, Noordhoff Publishing Compan Love, A.E.H. (1944) A Treatise on the Mathematical Theor of Elasticit, Dover Publications, Fourth Edition, New York Vinson, J.R. and Sierakowski, R.L. (00) The Behavior of Structures Composed of Composite Materials, Second Edition, Kluwer Academic Publishers, Dordrecht, The Netherlands Problems 1.1. Prove that the stresses are smmetric, i.e., ij ji. (Suggestion: take moments about the, and z aes.) 1.. When v = 0.5 a material is called incompressible. Prove that for v = 0.5, under an set of stresses, the control volume of Figure 1.1 will not change volume when subjected to applied stresses An elastic bod has the following strain field: z 3z zz z z 3z z

23 Does this strain field satisf compatibilit? Note: compatibilit is not satisfied if an one or more of the compatibilit equations is violated. 9

24 CHAPTER DERIVATION OF THE GOVERNING EQUATIONS FOR ISOTROPIC RECTANGULAR PLATES This approach in this chapter is to sstematicall derive the governing equations for an isotropic classical, thin elastic isotropic rectangular plate. Analogous derivations are given in [.1 -.8]..1 Assumptions of Plate Theor In classical, linear thin plate theor, there are a number of assumptions that are necessar in order to reduce the three dimensional equations of elasticit to a two dimensional set that can be solved. Consider an elastic bod shown in Figure.1, comprising the region 0 a, 0 b and h z h, such that h << a and h << b. This is called a plate. The following assumptions are made. Figure.1. Rectangular plate. 1. A lineal element of the plate etending through the plate thickness, normal to the mid surface, -- plane, in the unstressed state, upon the application of load: a. undergoes at most a translation and a rotation with respect to the original coordinate sstem; b. remains normal to the deformed middle surface.. A plate resists lateral and in-plane loads b bending, transverse shear stresses, and in-plane action, not through block like compression or tension in the plate in the

25 1 thickness direction. This assumption results from the fact that h/a << 1 and h/b << 1. From 1a the following is implied: 3. A lineal element through the thickness does not elongate or contract. 4. The lineal element remains straight upon load application. In addition, 5. St. Venant s Principle applies. It is seen from 1a that the most general form for the two in-plane displacements is: u(,, z) u0 (, ) z (, ) (.1) v(,, z) v0 (, ) z (, ) (.) where u0 and v 0 are the in-plane middle surface displacements (z = 0), and and are rotations as et undefined. Assumption 3 requires that z 0, which in turn means that the lateral deflection w is at most (from Equation 1.18) w = w((, ). (.3) Also, Equations (1.11) is ignored. Assumption 4 requires that for an z, both z constant and z constant at an specific location (, ) on the plate middle surface for all z. Assumption 1b requires that the constant is zero, hence z z 0. Assumption means that 0 z in the stress strain relations. Incidentall, the assumptions above are identical to those of thin classical beam, ring and shell theor.. Derivation of the Equilibrium Equations for a Rectangular Plate Figure. shows the positive directions of stress quantities to be defined when the plate is subjected to lateral and in-plane loads. The stress couples are defined as follows:

26 13 Figure.. Positive directions of stress resultants and couples. h M z d (.4) h h M z dz (.5) h h M z dz (.6) h M h z dzz M. (.7) h Phsicall, it is seen that the stress couple is the summation of the moment about the middle surface of all the stresses shown acting on all of the infinitesimal control elements through the plate thickness at a location ((, ). In the limit the summation is replaced b the integration. Similarl, the shear resultants are defined as, h Q dz (.8) z h

27 14 h Q dz. (.9) z h Again the shear resultant is phsicall the summation of all the shear stresses in the thickness direction acting on all of the infinitesimal control elements across the thickness of the plate at the location ((, ). Finall, the stress resultants are defined to be: h N d (.10) h h N dz (.11) h h N dz (.1) h N h dzz N (.13) h These then are the sum of all the in-plane stresses acting on all of the infinitesimal control elements across the thickness of the plate at,. Thus, in plate theor, the details of each control element under consideration are disregarded when one integrates the stress quantities across the thickness h. Instead of considering stresses at each material point one reall deals with the integrated stress quantities defined above. The procedure to obtain the governing equations for plates from the equations of elasticit is to perform certain integrations on them. Proceeding, multipl Equation (1.5) b z dzd and integrate between h/ and +h/, as follows: h h z z z z z dz 0 h h z dz h z dz h h h z z z dz 0 M M z z h h h dz 0. z h

28 15 In the above, the order of differentiation and integration can be reversed because and z are orthogonal one to the other. Looking at the third term, z z 0 when there are no shear loads on the upper or lower plate surface. If there are surface shear stresses then defining 1 z ) and z ), the results are shown below in Equation (.14). It should also be noted that for plates supported on an edge, z ma not go to zero at h, and so the theor is not accurate at that edge, but due to St. Venant s Principle, the solutions are satisfactor awa from the edge supports. M M h ( 1 ) Q 0. (.14) Likewise Equation (1.6) becomes M M h ( 1 ) Q 0 (.15) where 1 z ( h ) and z ( h ). These two equations describe the moment equilibrium of a plate element. Looking now at Equations (1.7), multipling it b dz, and integrating between h/ and +h/, results in h h z z z dz 0 z Q Q h z h 0 Q Q p1(, ) p (, ) 0 (.16) where p1(, ) z ( h ), z ). One could also derive (.16) b considering vertical equilibrium of a plate element shown in Figure.3.

29 16 Figure.3. Vertical forces on a plate element. z One ma ask wh use is made of in this equation and not in the stress-strain relation? The foregoing is not reall inconsistent, since z does not appear eplicitl in Equation (.16) and once awa from the surface the normal surface traction is absorbed b shear and in-plane stresses rather than b z in the plate interior, as stated previousl in Assumption. Similarl, multipling Equations (1.5) and (1.6) b dz and integrating across the plate thickness results in the plate equilibrium equations in the and directions respectivel, in terms of the in-plane stress resultants and the surface shear stresses. N N N N ( ) 0 (.17) ( 1 ( ) 0. (.18) ( 1.3 Derivation of Plate Moment-Curvature Relations and Integrated Stress Resultant-Displacement Relations Now, the plate equations must be derived corresponding to the elastic stress strain relations. The strains, and will not be used eplicitl since the stresses have been averaged b integrating through the thickness. Hence, displacements are utilized. Thus, combining (1.9) through (1.1) gives the following, remembering that z has been assumed zero in the interior of the plate and ecluding Equation (1.11) for reasons given previousl. u 1 [ ] E (.19)

30 17 v 1 [ ] E (.0) 1 u v 1 G (.1) 1 v w 1 z z G z 1 w u 1 z. zz G (.) (.3) Net, recall the form of the admissible displacements resulting from the plate theor assumptions, given in (.1) through (.3): u u (, ) z (, ) (.4) 0 v v (, ) z (, ) (.5) 0 w w(, ) onl. (.6) In plate theor it is remembered that a lineal element through the plate will eperience translations, rotations, but no etensions or contractions. For these assumptions to be valid, the lateral deflections are restricted to being small compared to the plate thickness. It is noted that if a plate is ver thin, lateral loads can cause lateral deflections man times the thickness and the plate then behaves largel as a membrane because it has little or no bending resistance, i.e., D 0. The assumptions of classical plate theor require that transverse shear deformation be zero. If 0 then from Equations (1.0) and (1.1) z z 1 u w 0 zz or u w, zz likewise v w. zz Hence, from Equations (.4) through (.6) and the above, it is seen that the rotations are w (.7)