Poisson Theory of Elastic Plates 9789813342095, 9789813342101

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Springer Tracts in Mechanical Engineering

Kaza Vijayakumar Girish Kumar Ramaiah

Poisson Theory of Elastic Plates

Springer Tracts in Mechanical Engineering Series Editors Seung-Bok Choi, College of Engineering, Inha University, Incheon, Korea (Republic of) Haibin Duan, Beijing University of Aeronautics and Astronautics, Beijing, China Yili Fu, Harbin Institute of Technology, Harbin, China Carlos Guardiola, CMT-Motores Termicos, Polytechnic University of Valencia, Valencia, Spain Jian-Qiao Sun, University of California, Merced, CA, USA Young W. Kwon, Naval Postgraduate School, Monterey, CA, USA Francisco Cavas-Martínez, Departamento de Estructuras, Universidad Politécnica de Cartagena, Cartagena, Murcia, Spain Fakher Chaari, National School of Engineers of Sfax, Sfax, Tunisia

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Kaza Vijayakumar Girish Kumar Ramaiah •

Poisson Theory of Elastic Plates

123

Kaza Vijayakumar Department of Aerospace Engineering Indian Institute of Science Bangalore Bangalore, India

Girish Kumar Ramaiah M.S. Ramaiah Institute of Technology Bangalore, India Department of Mechanical Engineering GITAM University Visakhapatnam, Andhra Pradesh, India

ISSN 2195-9862 ISSN 2195-9870 (electronic) Springer Tracts in Mechanical Engineering ISBN 978-981-33-4209-5 ISBN 978-981-33-4210-1 (eBook) https://doi.org/10.1007/978-981-33-4210-1 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Foreword

In a recent article, ‘On the Kirchhoff-Love hypothesis’ (revised and vindicated), Ozenda and Virga rather hold, in the context of analysis of elastomeric materials, that it is just deceptively simple to presume that the theory of elasticity is a dead subject in a postmodern view. This monograph deals, exclusively, with the analysis of elastic plates and one finds, surprisingly, with new theories presented here for analysis of homogeneous and laminated plates with anisotropic plies conforms their view. Their statement that ‘It makes one believe that everything is understood and only routine computations need to be done, for which it suffices to devise the most appropriate algorithm (the distinguished job of computational mechanics)’ appears to be true even in the analysis of elastic plates within small deformation theory of elasticity. It is factual, perhaps, due to extensive use (even today) of Kirchhoff theory of plates in bending along with highly cited Reddy’s third-order shear deformation theory and systematic presentation of various orders of approximation of bending of elastic plates by Kienzler and his colleagues, apart from vast literature on extension and associated torsion problems. Science and Mechanics are full of examples where a scholar with repute and authority has made some prediction which did not live long up to its expectations. ‘Poisson-Kirchhoff Paradox’ is perhaps one among such predictions. This paradox (to use Reissner’s expression) will only be fully comprehended a little less than a century later (Reissner 1985, Reflections on the Theory Elastic Plates. Applied Mechanics Reviews 38, pp. 1453–1464). Scientific literature has many instances where swimming against the currents of contemporary thought, people have produced seminal works. Even the music of Thyagaraja and Mozart fall into the same paradigm and we would have missed out their music, had they listened to the critics too much. This monograph is a seminal work and unique addition to the existing literature on plate theories. Lacuna in the widely used Kirchhoff theory is due to in-plane displacements expressed as gradients of lateral displacement, resulting in fourth-order theory with two-edge conditions instead of a sixth-order theory with three-edge conditions as

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demanded by Poisson. This is what is known as ‘Poisson-Kirchhoff boundary condition paradox.’ It has not been properly resolved over the past 17 decades or so in spite of attempts by several mathematicians of high repute. This monograph contains a new way of resolving this paradox. Poisson himself never gave his name to the theory proposed by him for analysis of bending of elastic plates but his name got associated with it because of his stature, accomplishments, and the respect he commanded over his peers. Continuing this tradition and in the honor of Poisson, the authors have named the title of the monograph as ‘Poisson Theory of Elastic Plates.’ The main emphasis in the Chap. 1 of the monograph is on the analysis of bending of plates in the presence of prescribed in-plane shear stress along edges of the plate. It is based on a novel concept of w0(x, y) as face variable from prescribed zero transverse shear stresses along faces of the plate. In-plane displacements (u, v) from newly defined primary bending problem are determined by expressing them as combinations of gradients of two functions w and u with harmonic functions u and w governed by bi-harmonic operator. Edge condition w = 0 is replaced by ez = 0. Transverse shear stresses , independent of material constants and ‘z,’ are determined first from solution of auxiliary problem satisfying more practical prescribed constant along edges of the plate. They are used to determine w and u along with material constant dependent from satisfying both static and (usual) integrated in-plane equilibrium equations of 3-D infinitesimal element. Solution of a supplementary problem based on 15-decade-old Levy’s work is used to distinguish between neutral and face plane deflections. Dr. K. Vijayakumar attained superannuation in 1995 from the Aeronautical Engineering Department of the prestigious Indian Institute of Science, Bangalore and his academic career came to an abrupt end. However, his interest in resuming academic work post retirement was mainly due to prodding, support, and encouragement by his grandchildren and family. In fact, they introduced him and kindled his interest to modern computers and internet. Expertise in making use of computers and internet from the comfort of home ignited him to reactivate his research work after a gap of a decade or so. This second phase of research work was initiated with a technical note, ‘New Look at Kirchhoff’s Theory of Plates,’ based on the concept held in the year 1988. This phase of work resulted in a plethora of research publications on analysis of plates in reputed journals. This monograph describes an unusual problem, hither to unnoticed, in the analysis of extension problems with prescribed transverse shear stresses along edges of the plate. The need to apply Poisson theory is explained with the necessity of satisfying both static and usual integrated equilibrium equations of infinitesimal element. The Poisson theory appears to be the most suitable theory to overcome lacuna in the analysis of primary plate problems. The disadvantage in its application is in the development of software for the generation of necessary polynomial functions. This problem was avoided with the use of proper sinusoidal functions in Chap. 6.

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Dr. Vijayakumar was hesitant for working on the monograph due to his physical health issues. However, on assurance from his doctors and with the help and encouragement from his first doctoral student Dr. Girish Kumar Ramaiah, he initiated the work. Professor Vijayakumar spent his sabbatical year, 1980–1981, from the Indian Institute of Science (Bangalore, India), at the Georgia Institute of Technology (Atlanta, USA) collaborating with me on the problem of an elliptical crack, embedded in an infinite elastic solid, the faces of which are subjected to arbitrary crack-face tractions. His exceptional mathematical skills enabled him to produce a complete analytical solution to the problem. He presented this work at the International Conference on Theoretical and Applied Mechanics in 1980 in Toronto, Canada. This analytical solution played a seminal role in tackling important problems of that day in the topic of integrity of structures containing surface flaws, using finite element-alternating methods. I am happy to see that nearly 20 years after his retirement from the Indian Institute of Science, he persevered in his research to produce this seminal Monograph, in collaboration with his student Prof. G. K. Ramaiah. I am confident that applied mathematicians, researchers, and students from the aerospace, mechanical, and civil engineering departments will find this monograph very useful.

Articles of Authors Referred in the Foreword 1. Olivier Ozenda and Epifanio G. Virga: On the Kirchhoff-Love hypothesis (revised and vindicated), arXiv: 2005, 13412v1 [math-ph] 27 May 2020, p. 26 2. J. N. Reddy: A Simple Higher Order Theory for Laminated Composite Plates, Journal of Applied Mechanics (1984) 51:745–752 3. R. Kienzler: On consistent plate theories, Archive of Applied Mechanics (2002) 72: 229–247. doi: 10-1007/800419-002-0220-2 4. R. Kienzler, P. Schneider: Consistent Theories of Isotropic and Anisotropic Plates. J. Theor. Appl. Mech. 50 (2012) 755–768 5. P. Schneider, R. Kienzler, M. Böhm: Modeling of Consistent Second-order Plate Theories for Anisotropic Materials. Zeitschr. Angew. Math. Mech. 94 (2014) 21–42 6. P. Schneider, R. Kienzler: A Priori Estimation of the Systematic Error of Consistently Derived Theories for Thin Structures. Int. J. Sol. Structures (2020) 1–21. doi:10.1016/j.ijsolstr.2019.10.010 7. R. Kienzer, P. Schneider: Second order linear plate theories: Partial differential equations, stress resultants and displacements. Int. J. Sol. Structures 115–116 (2017) 14–26

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8. R. Kienzler, M. Kashtalyan: Assessment of the consistent second-order plate theory for isotropic plates from the perspective of the three-dimensional theory of elasticity. Int. J. Solids Structures 185–186 (2020) 257–271. September 2020

Prof. Dr. Satya N. Atluri Presidential Chair and University Distinguished Professor Department of Mechanical Engineering Texas Tech University Lubbock, Texas, USA

Preface

At age 76, Prof. Dr. Liviu Librescu was among the 32 people who were murdered in the Virginia Tech shooting on 16 April 2017. The revelation that Dr. Liviu Librescu blocked the door of his classroom in Norris Hall on the morning of April 16 so that his students could escape through the windows came as no surprise to his family, friends, and colleagues. The renowned aeronautical engineering educator and researcher had demonstrated profound courage throughout the 76 years of his life. A prolific researcher and wonderful teacher, he devoted himself to the profession, solely for the love of it. ‘It is a question of pleasures,’ Dr. Librescu said in 2005 when asked why he continued to work so hard. Dr. Liviu Librescu’s life is an inspiration to us and scores of other academicians around the world. The senior author of this monograph had had an opportunity to see Dr. Liviu Librescu while working with Prof. J. N. Reddy on a short visit to Virginia Tech University. It is a coincidence that Dr. Liviu Librescu was one of the Ph.D. thesis examiners of Dr. Ramaiah. The death of Dr. Liviu Librescu in a shooting incident at an academic institute was quite shocking and beyond comprehension. The senior author is an 85-year-old professor who resumed his research work after a gap of more than 12 years since his retirement in 1995 due to inducements from his grandchildren. It resulted in a large number of papers getting published on the formulation and development of Poisson and extended Poisson theories on the analysis of homogeneous and laminated composite plates with isotropic and anisotropic piles. His activity on the analysis of laminated composite plates during this phase of academic work was due to highly appreciative encouragement from Prof. Satya N. Atluri. The present monograph is the result of an unexpected and unsolicited invitation from several quarters immediately after the publication of Dr. Vijayakumar’s paper ‘Extended Poisson Theory of Plates with Fourier Sinusoidal Series.’ Dr. Vijayakumar was initially hesitant to take up writing of this monograph due to his indifferent health from cancer in the liver, but on assurances from Dr. T. Raja (Apollo Hospital, Madras, now Chennai) and on encouragement and help from ix

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Dr. Ramaiah, the work was initiated in late 2019. Lifetime achievement award by Marquis ‘Who’s Who’ to the senior author in 2017 was another motivating factor to remain academically active. Poisson’s name is attached to a wide variety of ideas, for example, Poisson integral, Poisson equation in potential theory (it forms the basis for our present work), Poisson brackets in differential equations, Poisson ratio in elasticity, and Poisson constant in electricity. Though he was not highly regarded by other French mathematicians, he was held in high esteem by foreign mathematicians who seemed to recognize the importance of his ideas. A Google search on ‘Poisson-Kirchhoff Boundary Conditions Paradox’ reveals the following statement: ‘It is zero along an edge of the plate if it is a specified condition. With zero transverse shear strains, specification of instead of zero is more appropriate along a supported edge since it implies zero tangential displacements along a straight edge which is the root cause of Poisson–Kirchhoff’s boundary conditions paradox.’ Above ‘boundary conditions paradox’ has not been resolved in a satisfactory manner over the past 17 decades or so in spite of attempts by several mathematicians of high repute. The present monograph is essentially an extension of the fundamental work on resolving the ‘Poisson–Kirchhoff boundary conditions’ paradox for analysis of the bending of plates. This has led to the proposal and development of ‘Poisson Theory of Plates’ including a significant formulation of a primary bending problem. This monograph contains a new way of resolving Poisson–Kirchhoff boundary conditions paradox and includes several advanced materials in the analysis of elastic plates within the small deformation theory of elasticity. Analysis of elastic plates is concerned with bending (flexure), extension (stretching), and (associated) torsion problems. In bending problems, Kirchhoff’s theory of plates is a single-variable theory of lateral deflection based on assuming zero transverse shear strains and neglecting transverse normal stress in constitutive relations. In extension problems with assumed zero transverse stresses, a single-variable theory is with Airy’s stress function. The mid-plane of the plate is treated as a neutral plane with zero lateral deflection in extension problems and zero transverse normal strain in bending problems. Vertical deflection in bending problems and Airy’s stress function in extension problems are governed by a fourth-order bi-harmonic differential operator in the classical theories. Several higher-order theories are reported over the past 17 decades or so to improve and rectify lacuna in Kirchhoff theory but not fully successful. Here, a proper rectification of lacuna in Kirchhoff’s theory is presented through a recently proposed and designated Poisson theory of elastic plates. An iterative procedure is adapted to generate a sequence of solutions of two-dimensional plate problems converging to the solution of three-dimensional problems. The monograph consists of six chapters. Chapter 1 considers the designated Poisson theory of isotropic plates in bending. Primary bending problems of homogeneous isotropic square plates are presented in this chapter. Kirchhoff’s theory, First-Order Shear Deformation Theory (FSDT) based on Hencky’s work,

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Reissner’s theory, and higher-order theories are briefly reviewed. Newly proposed Poisson theory is described. The concept of lateral displacement as a face variable is introduced so that it is uncoupled from the determination of in-plane displacements. Coupling of all three displacements through domain equilibrium equations is responsible for more than 17-decade-old Poisson–Kirchhoff boundary conditions paradox. This paradox is resolved, here, through Poisson theory. Supplementary problem based on 15-decade-old Levy’s work is used to distinguish between neutral and face plane deflections. New primary problems are defined with the edge condition of zero transverse normal strain. This edge condition is a more practical condition avoiding the need for vertical stress resultant in Kirchhoff theory. Poisson theory used for the analysis of these primary problems is designated, though not necessary, as Extended Poisson Theory (EPT) in earlier publications. Corresponding higher-order theories are presented through an iterative procedure. A solution of a simple textbook problem of a simply supported plate under doubly sinusoidal vertical load is presented to show the utility of Poisson theory even for thick plates with one-term representation of displacements. It is noted, however, that development of software for the generation of required polynomial functions is not easy for higher-order theories. Analysis of associated torsion problems requires only replacing in-plane gradient function with vertical deflection. The classical theory of extension problems is based on zero transverse stresses. The role of Airy’s stress function, parallel to lateral deflection in Kirchhoff theory, is discussed in Chap. 2. The displacement-based theory consists of coupled second-order differential equations governing two in-plane displacements. These displacements expressed as gradients of two functions lead to a bi-harmonic equation in either of these gradient functions replacing Airy’s stress function. It is to be noted that stresses from Airy’s function are independent of material constants, whereas solutions of the bi-harmonic equation in a gradient function are dependent on material constants. In this chapter, higher-order approximations through Poisson theory are presented. In Chap. 3, designated Poisson theory is applied for the analysis of homogeneous anisotropic plates with monoclinic symmetry for both bending and extension problems. In Chap. 4: Part I: Single-Layer Theories with stress resultants are presented for both symmetric and un-symmetric laminated plates. In Chap. 5, Part II: Layerwise Theories are presented. The designated Poisson theory with Fourier sinusoidal series without the need for higher-order polynomial functions is presented in the final Chap. 6. The procedure simplifies the iterative process for higher-order approximations in both bending and extension problems. The monograph is closed with an epilogue consisting of highlights of the present work along with the suggestions for future investigations with the application of Poisson theory to eigen-value problems like lateral buckling and free vibration problems. This monograph will be useful to researchers in the area of solid mechanics (elasticity, plate theory, anisotropic plates, layered plates), applied mathematicians, and to professionals in the field of aerospace, mechanical, and civil engineering. It

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will also be useful for graduate and doctoral students in aerospace, mechanical, and civil engineering. The authors are grateful to Prof. Satya Atluri, Presidential Chair and University Distinguished Professor in the Department of Mechanical Engineering at Texas Tech University and recipient of Padma Bhushan, from the Government of India for readily agreeing to write a Foreword to the monograph. The authors are thankful to Prof. K. P. Rao, Aerospace Engineering Department of Indian Institute of Science, Bangalore, for valuable discussions and also to Prof. K. Shivakumar of North Carolina Agricultural and Technical State University for his assistance and help in the first publication of Dr. K. Vijayakumar’s new phase of academic work as Technical Note entitled ‘New Look at Kirchhoff Theory of Bending of Plates’ in AIAA Journal which has drawn the attention of more than 600 academic people over the past two years. The help provided by Prof. Lewinski in sending copies of Levy’s work and copies of Jameilta’s work, ‘On Winding Paths of Plate Theories’ is gratefully acknowledged. The authors are thankful to the Editorial team of zbMATH in making available to them an electronic copy of the paper ‘Twenty-Second British Commonwealth Lecture’ Aeronautical Research in India by S. Dhawan (Journal of Royal Aeronautical Society 71(675); pp. 149–184, 1967). (Source for write up on Prof. Dr. Liviu Librescu is https://www.weremember.vt. edu/biographies/librescu.html). Bangalore, India Bangalore, India

Kaza Vijayakumar Girish Kumar Ramaiah

Introduction

Brief Historical Background In the context of present work, it is pertinent to mention the names of several prominent scientific personalities who contributed significantly in analyzing various types of problems associated with plates even within small deformation theory of elasticity. Notable among them are Euler (1707–1783), Joseph-Louis Lagrange (1736–1813), Pierre-Simon Laplace (1749–1827), Sophie Germain (1776–1831), Simon-Denis Poisson (1781–1840), Claude-Louis-Marie Navier (1785–1836), Augustin-Louis Cauchy (1789–1857), St. Venant (1797–1886), and Kirchhoff (1834–1887). All of them are of highest quality of applied mathematicians. Seventeenth and early eighteenth centuries were a productive period during which the mechanics of simple classical structural elements like beams and plates were developed. The investigations during this period was limited to two-dimensional theory and only later from 1820 onwards the analysis was extended for three-dimensional theory. Euler’s mathematics is distinguished by its quality and quantity. He is considered to be the greatest mathematician ever lived on earth. Poisson’s name is very familiar because of Poisson’s integral, Poisson’s equation in potential theory, Poisson’s brackets in differential equations, Poisson’s ratio in elasticity, and Poisson’s constant in electricity. The general theory of elasticity in a mathematical usable form was formulated by Navier in 1821. Navier is often considered to be the founder of structural mechanics. Cauchy’s main contribution was the foundation for the modern period of rigor in analysis. St. Venant in 1843 gave the correct derivation of Navier–Stokes equations. St. Venant is also known for providing solution for the torsion of non-circular shafts. For thin plates, Kirchhoff presented the correct fourth-order partial governing differential equation for governing transverse displacements along with two boundary conditions along the edge of the plate. Earlier, Poisson demanded a sixth-order partial governing differential equation along with three natural boundary conditions along the edge of the plate in contrast to Kirchhoff’s theory. This is what led to Kirchhoff–Poisson boundary condition paradox which has remained

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unresolved despite attempts by several mathematicians of repute to resolve it during the last 17 decades. This monograph contains a new way of resolving Poisson– Kirchhoff boundary conditions paradox in the analysis of bending of plates. For readers interested to know more about the ‘Historical Background’ on small deformation theory of elasticity, a list of references is appended as ‘Suggested Readings.’ Analysis of elastic plates is concerned with bending (flexure), extension (stretching), and (associated) torsion problems. In bending problems, Kirchhoff’s theory of plates is a single-variable theory of lateral deflection based on assuming zero transverse shear strains and neglecting transverse normal stress in constitutive relations. In extension problems with assumed zero transverse stresses, a single-variable theory is in terms of Airy’s stress function. The mid-plane of the plate is treated as a neutral plane with zero lateral deflection in extension problems and zero transverse normal strain in bending problems. Lateral deflection in bending problems and Airy’s stress function in extension problems are governed by a fourth-order bi-harmonic differential operator in the classical theories. Several higher-order theories are reported over the past 17 decades or so to improve and rectify lacuna in Kirchhoff theory but not fully successful. Here, a proper rectification of lacuna in Kirchhoff’s theory is presented through the recently proposed and designated Poisson theory of elastic plates within the small deformation theory of elasticity. An iterative procedure is adapted to generate a sequence of solutions of two-dimensional plate problems converging to the solution of three-dimensional problems. Practicing engineer’s terminology is mainly used without stress resultants and average displacements to bridge the gap with applied mathematicians. Analysis of plates with different geometries and material properties under different kinematic and loading conditions does not provide much scope for the development of new theories. Hence, the analysis of the primary problems of a square plate has been dealt with throughout this monograph. Primary bending problems of homogeneous isotropic square plates are presented in Chap. 1. Kirchhoff’s theory, First-Order Shear Deformation Theory (FSDT) based on Hencky’s work, Reissner’s theory, and higher-order theories are briefly reviewed. Newly proposed Poisson theory is described. The concept of lateral displacement as a face variable is introduced so that it is uncoupled from the determination of in-plane displacements. Coupling of all three displacements through domain equilibrium equations is responsible for more than 17-decade-old Poisson–Kirchhoff boundary conditions paradox. This paradox is resolved, here, through Poisson theory. Supplementary problem based on 15-decade-old Levy’s work is used to distinguish between neutral and face plane deflections. New primary problems are defined with zero transverse normal strain edge conditions. This edge condition is a more practical condition avoiding the need for vertical stress resultant in Kirchhoff theory. Poisson theory used for the analysis of these primary problems is designated, though not necessary, as Extended Poisson Theory (EPT) in earlier publications. Corresponding higher-order theories are presented through an iterative procedure. A solution of a simple textbook problem

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of a simply supported plate under doubly sinusoidal vertical load is presented to show the utility of Poisson theory even for thick plates with one-term representation of displacements. It is noted, however, that development of software for the generation of required polynomial functions is not easy for higher-order theories. Analysis of associated torsion problems requires only replacing the Poisson transverse shear function with lateral deflection. The classical theory of extension problems is based on zero transverse stresses. The role of Airy’s stress function, parallel to lateral deflection in Kirchhoff theory, is discussed in Chap. 2. The displacement-based theory consists of coupled second-order differential equations governing two in-plane displacements. These displacements expressed as gradients of two functions lead to a bi-harmonic equation in either of these gradient functions replacing Airy’s stress function. It is to be noted that stresses from Airy’s function are independent of material constants, whereas solutions of the bi-harmonic equation in a gradient function are dependent on material constants. In this chapter, higher-order approximations through Poisson theory are presented. It is shown that satisfaction of both static and thickness-wise integration of equilibrium equations is mandatory for the analysis of extension problems with polynomial functions used in the analysis. In Chap. 3, the designated Poisson theory is applied for the analysis of homogeneous anisotropic plates with monoclinic symmetry for both bending and extension problems. In Chap. 4: Part I: Single-Layer Theories with stress resultants are presented for both symmetric and un-symmetric laminated plates. Alternate form of Classical Laminate Plate Theory (CLPT) is also proposed to distinguish the effect of un-symmetry on transverse displacement due to in-plane displacements in bending problems and vice versa. In Chap. 5, Part II: Layerwise theories are presented with novel concept of ply analysis independent of lamination in place of reported zig-zig theories. The Poisson theory with Fourier sinusoidal series without the need for higher-order polynomial functions is presented in the final Chap. 6. The procedure simplifies the iterative process for higher-order approximations in both bending and extension problems. The need to include solutions of supplementary problems to distinguish lateral deflection of the face and neutral planes are eliminated. The monograph is closed with an epilogue consisting of highlights of the present work along with suggestions for future work, including the application of Poisson theory to eigenvalue problems like lateral buckling and free vibration problems.

Suggested Readings for Historical Background 1. Holm Altenbach and Victor A. Eremeyev, Thin-Walled Structural Elements: Classification, Classical and Advanced Theories, New Applications, Vol. 572, In book: Shell-like Structures (pp. 1–62), January 2017, CISM International Centre for Mechanical Sciences, Courses and Lectures (Taken From Research Gate).

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2. MacTutor History of Mathematics Archive, School of Mathematics and Statistics, University of St. Andrews, Scotland, July 2020 Edition. 3. Marquis de Condorcet, Eulogy to Mr. Euler, History of the Royal Academy of Sciences 1783, Paris 1786, pp. 37–68. 4. Nicolas Fuss, Eulogy of Leonhard (Translated by John S. D. Glaus), April 2005, Eulogy-The Euler Archive (The eulogy was written by Nicolas Fuss and delivered at the Imperial Academy of Sciences of Saint Petersburg on 23 October 1783. The translation by John S. D. Glaus appears to have been published only online. http://eulerarchive.maa.org/historica/fuss.pdf). 5. The History of the Planar Elastica: Insights into Mechanics and Scientific World, August 2008, Science and Education 18(8), 1057–1082 (Taken From Research Gate). 6. Martin J. Gander and Gerhard Wanner, From Euler, Ritz, and Galerkin to Modern Computing, SIAM REVIEW 2012 Society for Industrial and Applied Mathematics, Vol. 54, No. 4, pp. 1–40.

Contents

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1 2 3 3 4 5 8 8 9 10

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12 14 16 17 18

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2 Extension Problems: Higher-Order Approximations . . . . . . . . . . . . . 2.1 Analysis of Extension Problems . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Some Observations on the Classical Theories . . . . . . . . . . . . . . . .

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1 Poisson Theory of Isotropic Plates in Bending . . . . . . . . . . . . . . 1.1 Brief Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Kirchhoff Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Classical Sixth-Order Theories . . . . . . . . . . . . . . . . . . 1.2.2 Poisson Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 First-Order Shear Deformation Theory (FSDT) . . . . . . 1.2.4 Reissner’s Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 Associated Torsion Problem . . . . . . . . . . . . . . . . . . . . 1.3 Bending of Square Plates . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Bending Problem of Homogeneous Isotropic Plate . . . . 1.3.2 Resolution of Poisson–Kirchhoff Boundary Conditions Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Higher-Order Poisson Theories . . . . . . . . . . . . . . . . . . 1.3.4 Neutral Plane Deflection . . . . . . . . . . . . . . . . . . . . . . 1.3.5 Exact Solution with w(x, y, z) as Domain Variable . . . 1.3.6 Exact Solution with w0(x, y) as Face Variable . . . . . . . 1.3.7 Some Remarks on Solution of Kirchhoff’s Primary Bending Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 New Primary Bending Problems . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Neutral Plane Deflection . . . . . . . . . . . . . . . . . . . . . . 1.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 1.1—Derivation of Eq. (1.88) . . . . . . . . . . . . . . . . Appendix 1.2—Derivation of Eq. (1.90) from Eq. (1.88) . . . . Appendix 1.3—Derivation Eq. (1.91) from Eq. (1.90) . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2.2.1 Airy’s In-plane Stress Function U(x, y) . . . . 2.2.2 Analysis in Terms of Displacements . . . . . . 2.2.3 Related Bending Problem . . . . . . . . . . . . . . 2.3 Bi-harmonic Function in Extension Problems . . . . . 2.3.1 Sequence of Uncoupled 2-D Problems . . . . 2.4 Initial Solutions of Primary Extension Problems . . . 2.4.1 Unusual Problem: Need for Poisson Theory 2.5 Poisson Theory of Extension Problem . . . . . . . . . . 2.5.1 Thick Plate Analysis . . . . . . . . . . . . . . . . . 2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Homogeneous Anisotropic Plates . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Homogeneous Anisotropic Plates . . . . . . . . . . . . . . . . . . . . . 3.2.1 Stress–Strain and Strain–Displacement Relations . . . . 3.2.2 Equilibrium Equations in Terms of In-plane Strains . . 3.2.3 Displacements [u, v, w] . . . . . . . . . . . . . . . . . . . . . . 3.3 Analysis of Primary Bending Problems . . . . . . . . . . . . . . . . 3.3.1 Application of Poisson Theory . . . . . . . . . . . . . . . . . 3.3.2 Iterative Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Summary of Results from the First Stage of Iterative Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Associated Torsion Problems . . . . . . . . . . . . . . . . . . . . . . . 3.5 Extension Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Preliminary Analysis . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Higher-Order Corrections . . . . . . . . . . . . . . . . . . . . . 3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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4 Laminated Plates with Anisotropic Plies Part I: Single-Layer Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Geometry and Ply-Wise Constitutive Laws of Anisotropic Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Equilibrium Equations . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Equilibrium Equations in Terms of Strains . . . . . . . . . 4.3 Modified Homogeneous Plate Theories (MHPT) of Symmetric Laminates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Analysis of Primary Bending Problem . . . . . . . . . . . . 4.3.2 Analysis of Primary Extension Problem . . . . . . . . . . .

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4.4 Un-symmetrical Laminates: Alternate Form (ACLPT) of CLPT . 4.4.1 Bending Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Extension Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Contrast Between CLPT and ACLPT . . . . . . . . . . . . . . 4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Laminated Plates with Anisotropic Plies Part II: Layerwise Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Preliminaries in the Analysis of Symmetric Laminates . . . . . 5.2.1 fk(z) Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Displacements [u, v, w] . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Edge Conditions in Each Ply . . . . . . . . . . . . . . . . . . 5.3 Analysis of Primary Bending Problems . . . . . . . . . . . . . . . . 5.3.1 Corrective In-plane Displacements in the Face Ply . . . 5.3.2 Continuity of Displacements and Transverse Stresses Across Interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Vertical Deflection w(x, y, z) . . . . . . . . . . . . . . . . . . 5.4 Higher-Order Corrections in the Ply from the Iterative Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Correction to Vertical Displacement . . . . . . . . . . . . . 5.4.2 Corrective Displacements in the Face Ply . . . . . . . . . 5.5 Associated Torsion Problems . . . . . . . . . . . . . . . . . . . . . . . 5.6 Extension Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Preliminary Analysis of the Ply . . . . . . . . . . . . . . . . 5.6.2 Higher-Order Corrections in the Ply . . . . . . . . . . . . . 5.7 Un-symmetrical Laminates . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.1 Bending Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.2 Associated Extension Problem in Bending . . . . . . . . 5.7.3 Extension Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.4 Associated Bending Problem in Extension . . . . . . . . 5.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Poisson Theory of Plates with Fourier Sinusoidal Series . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Initial Sets of Solutions with fn(z) Functions . . . . . . . . . . . 6.3 Use of Fourier Sinusoidal Series . . . . . . . . . . . . . . . . . . . . 6.3.1 Analysis of Flexure (Bending) Problem . . . . . . . . . 6.3.2 Note on the Determination of Vertical Displacement in Bending Problems . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Associated Torsion Problem . . . . . . . . . . . . . . . . . . 6.3.4 Analysis of Extension Problem . . . . . . . . . . . . . . . .

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6.4 Symmetrical Laminated Plates . . . 6.5 Un-symmetrical Laminated Plates 6.5.1 Bending Problem . . . . . . . 6.5.2 Extension Problem . . . . . . 6.6 Conclusions . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .

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Data from Research Gate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

About the Editors

Dr. Kaza Vijayakumar is a retired Professor in Department of Aerospace Engineering at Indian Institute of Science (IISc), Bangalore, India. He has been selected for the 2017 Albert Nelson Marquis Lifetime Achievement Award. He was also honored by the Aerospace Engineering Department of IISc which presented him with the ‘Platinum Jubilee Distinguished Service Award’. His published work has been cited extensively by researchers all over the world. Dr. Vijayakumar is a reviewer for several journals of national and international repute. He is a research advisor at Nanyang Academy of Sciences, Singapore. Dr. Girish Kumar Ramaiah is a retired professor of Mechanical Engineering at GITAM University, Visakhapatnam and former Registrar (Academic), at M.S. Ramaiah Institute of Technology, Bangalore, India. He pursued his Ph.D. degree from the Indian Institute of Science, Bangalore, India. He has been awarded with the Alexander von Humboldt Fellowship at TH Darmstadt, and Leverhulme American/Commonwealth Fellowship at Queen’s University, Belfast. His areas of interest lie in structural mechanics (statics and dynamics), elastic stability, and mathematical methods for analysis. Dr. Ramaiah is an active reviewer for several journals of national and international repute.

xxi

Chapter 1

Poisson Theory of Isotropic Plates in Bending

Nomenclature a D E E e fi (z) G 2h q (x, y) Tx , Txy , Txz Ty , Txy , Tyz U, V, W (u, v, w) u, v w u1 , v1 , w0 w0f , w0n X, Y, Z (x, y, z)  α β2k β2n+1 εz , γxz , γyz μ ν ωz

Side length of a square plate Flexural stiffness of the plate/unit width Young’s modulus E/(1 – ν 2 ) (εx + εy ) in face parallel planes Polynomials in z, i = 1, 2, … Modulus of rigidity Plate thickness Applied load density Prescribed stresses at each of x = constant edges Prescribed stresses at each of y = constant edges Displacements in X, Y, and Z directions, respectively (U, V, W)/h (non-dimensional displacements) In-plane displacements in the x and y directions, respectively Vertical deflection Displacements in x, y, and z directions, respectively, in face parallel planes Face and neutral plane deflections, respectively Coordinates of a point in a Cartesian system (X/a, Y/a, Z/h) (non-dimensional coordinates) (∂ 2 /∂x2 + ∂ 2 /∂y2 ) Laplace operator in x–y plane (h/a) (half-plate thickness ratio) Transverse shear distribution correction factor Modification factor of σz in bending problems Transverse strains ν/(1 − ν) Poisson’s ratio α (v,x − u,y ), rigid body rotation about z-axis

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 K. Vijayakumar and G. K. Ramaiah, Poisson Theory of Elastic Plates, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-33-4210-1_1

1

2

1 Poisson Theory of Isotropic Plates in Bending

ϕ1 (x, y)  σx , σy , τxy τxz , τyz , σz τ*xz , τ*yz , σ*z

Harmonic function Poisson transverse shear stress function Bending stresses Transverse stresses Statically equivalent stresses

1.1 Brief Introduction Analysis of plates within the classical small deformation theory of elasticity is concerned with the problems of flexure, extension, and torsion. It is generally based on making suitable assumptions about the thickness-wise distribution of displacements and/or stresses (or strains) to derive two-dimensional plate equations from three-dimensional elasticity equations. There are many methods of deriving these equations using a series of functions in thickness coordinate and/or thickness parameter (asymptotic methods) of the plate. For a brief survey of most of these theories, one may refer to reviews by Reissner [1], Lo et al. [2, 3], Lewinski [4, 5], Blocki [6], Kienzler [7], and Batista [8]. Variational methods based on the principle of virtual work consist of virtual increments in displacement variables. Due to arbitrary virtual increments, one derives static equations governing real displacements independent of these virtual displacements which do not participate in these equations. If a 3-D variable, for example, u(x, y, z) is expressed in infinite series with the sum on (i, j, k = 0, 1, 2, …) u(x, y, z) =



ai xi



bj yj



ck zk

 (1.1)

then, determination of real constants (ai , bj , ck ) from relevant algebraic equations generated in the numerical methods such as finite difference methods and finite element methods (FEM) is presumably valid for solutions of such 3-D variables. However, the sequence of 2-D problems derived from variational principles do not converge to 3-D problems since δ



ai xi



bj yj



ck zk



=



zk δ



aik xi



(i, j, k = 0, 1, 2, . . .)

bjk yj

 , (1.2)

Hence, it is necessary to consider equilibrium equations of a 3-D infinitesimal element (instead of plate element) to generate a proper sequence of 2-D problems governing 2-D displacement variables [9]. The 3-D problem consists of determining three displacements, u, v, and w that satisfy three equilibrium equations in the interior of the plate and three specified surface conditions.

1.2 Kirchhoff Theory

3

1.2 Kirchhoff Theory Kirchhoff theory [10] is a simple classical theory that is continuously used even today to obtain design information. Kirchhoff’s assumption of zero transverse shear strains is, however, not a limitation of the theory as a first approximation to the exact 3-D solution. The main limitation in the theory is due to in-plane displacements expressed as gradients of lateral displacement, resulting in a fourth-order theory with two-edge conditions instead of a sixth-order theory with three-edge conditions as demanded by Poisson [1]. Poisson–Kirchhoff boundary condition paradox is well known and has not been properly resolved over the past 17 decades or so, with reported several sixth-order shear deformation theories in the literature. Primary variables are determined from integrated equilibrium equations, that is, plate element equilibrium equations with reduced stiffness coefficient [2E/3(1 − ν2 )]. The term [E/(1 − ν2 )] is from semi-inverted in-plane strain–stress relations neglecting σz in constitutive relations. Because of these assumptions, strain energy density is only due to in-plane stresses and strains. It can be easily shown that in-plane distributions of displacements thus obtained remain the same with displacements z2k w2k and [z2k+1 /(2k + 1)] [u, v]2k+1 (k ≥ 1) regarding the bending of a plate subjected to the same kinematic and stress resultant conditions in the Kirchhoff theory. Hence, z-distributions of displacements are not unique due to assumptions in Kirchhoff theory. One should note that vertical face deflection is the same for all k but [u, v] along faces are maximum from Kirchhoff theory [11]. The prescribed τxy along an edge is in the form of its tangential gradient contributing to an artificial additional vertical shear in Kelvin–Tait physical interpretation of Kirchhoff’s vertical stress resultant [12]. This additional shear is due to partial nullification of interior τxy in the bending problem [13]. Kirchhoff theory is, in fact, a zeroth-order shear deformation theory.

1.2.1 Classical Sixth-Order Theories Minimum modification of Kirchhoff’s theory has to be a sixth-order theory rectifying lacuna in the theory along with the facility of providing maximum face distribution of [u1 , v1 ]. Reissner’s pioneering work [14] was an attempt in this direction but not fully successful. In Kirchhoff theory, [τxz0 , τyz0 ] are assumed to be zero due to zero face shear conditions. Normal stress σz , though identically zero from equilibrium Eq. (1.2), is assumed to satisfy load condition σz = ± q/2 along the top and bottom faces of the plate. It is based on f3 (z) = ½(z – z3 /3) distribution a priori in Reissner’s theory [14] and from statically equivalent (reactive) stress in Kirchhoff theory. In either case, transverse shear stresses correspond to parabolic f2 (z) = ½(1 – z2 ) distributions. These distributions are also used through shear correction factor k2 (= 5/6) in first-order shear deformation theory (FSDT) based on Hencky’s work [15]. FSDT

4

1 Poisson Theory of Isotropic Plates in Bending

is a displacement-based theory in which in-plane distributions [τxz0 , τyz0 ] are used through constitutive and strain–displacement relations, whereas they are independent variables in Reissner’s theory in applying Castigliano’s theorem of least work. There is no need to use the shear correction factor in Reissner’s theory, and its use is avoided in the displacement-based Reddy’s theory [16]. In stress-based Reissner’s theory, complementary strain energy due to σz involves cubic f3 (z) distribution of εz , whereas work done by the applied vertical load is associated with weighted vertical displacement. It must be noted here that Reddy’s shear deformation theory is equivalent through the change of variables to an earlier Reissner’s displacement-based theory [17] without the strain energy due to σz , and both of them are equivalent to Ambartsumyan’s theory [18] proposed much earlier. As such, σz = (z σz1 ) satisfying face load condition is ignored, whereas it plays an important role in rectifying lacuna in shear deformation theories as discussed later. These theories (which are based on plate element equilibrium equations with w0 (x, y) as domain variable) correspond to approximate associated torsion problems. In Reissner’s sixth-order theory [14] and FSDT, applied in-plane shear is combined with transverse shear resulting in ‘torsion-type’ problem in flexure instead of in-plane shear combining with reactive vertical shear implied in Kelvin and Tait’s [12] physical interpretation of contracted boundary condition in Kirchhoff’s theory. Torsion problem is associated, in fact, with flexure problem [13], whereas flexure problem (unlike directly or indirectly implied in the energy methods) is independent of the torsion problem. This torsion problem is, however, an approximation to the associated torsion problem in the bending of plates and results in an unrealistic vertical deflection of the plate. The solution of the associated torsion problem is to nullify the effect due to applied or reactive edge stress τxy in flexure problem (In this connection, pertinent observation is that Kirchhoff theory and FSDT are valid for the analysis of hard and soft simply supported plates, respectively). This can be inferred from numerical results reported by Lewinski [19], in which higher-order theories give decreasing values of vertical deflection. Recently, Kirchhoff theory is modified such that the resulting sixth-order theory of bending and St. Venant’s theory of torsion are mutually exclusive to each other [20].

1.2.2 Poisson Theory In the recently developed theory designated as ‘Poisson theory of plates in bending’ [9, 21], reactive transverse shear stresses are initially determined independent of material constants retaining their thickness-wise parabolic distributions. Dependence on material constants is through higher-order corrections to these stresses. These higher-order corrections are determined through an iterative procedure based on the equilibrium of 3-D infinitesimal element instead of the use of the stationary property of total potential in the energy methods. Dependence of analysis on vertical deflection w0 (x, y) is eliminated (One should note that the coupling of analysis with w0 in energy methods is due to the work done by the prescribed transverse

1.2 Kirchhoff Theory

5

stresses during deformation and also the root cause for Poisson–Kirchhoff boundary conditions paradox). Also, the coupling between the flexure problem and associated torsion problem is eliminated through an iterative method [22]. The problem at each stage of iteration is defined by a sixth-order system of equations using the solution at the preceding stage of iteration. In the earlier investigations [15, 20, 23], in-plane displacements are expressed in terms of gradients of two functions ψ and ϕ in which ψ is related to w, and ϕ is required to decouple bending and torsion problems. The function ϕ is an auxiliary plane harmonic function from zero rotation α(v,x − u,y ) denoted by ωz about the z-axis. This function ϕ was introduced earlier by Reissner [17] as ‘stress function’ governed by a ‘wave equation’ with an imaginary velocity of propagation. Kirchhoff’s theory, FSDT, Reissner’s theory, other shear deformation theories, and reported higher-order theories based on energy principles are not useful in the generation of a proper sequence of 2-D problems converging to the 3-D problem.

1.2.3 First-Order Shear Deformation Theory (FSDT) FSDT is a sixth-order theory with provision to satisfy three-edge conditions and maintains, unlike in Kirchhoff’s theory, independent linear thickness-wise distribution of tangential displacement even if the lateral deflection, w, is zero along a supported edge. However, each of the in-plane distributions of transverse shear stresses is used to replace parabolic reactive stress through a shear energy correction factor. A new concept of distribution correction factor [13] is introduced resulting in the same correction factor. Adapting the concept of shear correction factor in FSDT, higher-order transverse shear terms [23] with f2k+2 (z) may be expressed in terms of preceding shear terms with f2k (z) through distribution correction factors β2k so that     τxz , τyz = f2k τxz , τyz 2k (using strain-displacement relations)

(1.3)

    τxz , τyz = β2k f2k+2 τxz , τyz 2k (based on shear correction factors)

(1.4)

In such a case, solutions of plate element equations give shear strains [u1 + αw0,x , v1 + αw0,y ] tending to [0, 0] in the limit k → ∞ due to [τxz , τyz ] in the first set but not zero due to stresses in the second set. Obviously, the shear energy due to β2k does not belong to the physical problem. To generate a converging sequence of 2-D problems with w0 (x, y) as a domain variable, it is more convenient to express z in Fourier series. Due to zero face shear conditions, it is expressed in the form z = Ak sin kz(k = 1, 3, 5, 7 . . . .)

(1.5)

6

1 Poisson Theory of Isotropic Plates in Bending π

2 Ak = π

2 0

 π π π 2 1 2 sin k − k cos k z sin kzdz = π k 2 2 2

(1.6)

so that z-distribution g(z) of reactive transverse stresses is given by g(z) = − ( k1 ) Ak cos kz (Note that term by term differentiation is not valid in the series expansion of z). Displacements are assumed in the form with k = 1, 3, 5, 7 ….

1 w = w0 (x, y) −  Ak cos kz w2k (x, y) k

(1.7)

  

2 1 2 1 sin kz [u, v]2k+1 = {Ak [u, v]1 + [u, v]2k+1 } sin kz [u, v] = z[u, v]1 +  k k

(1.8) Transverse shear strains from strain–displacement relations and shear stresses are     γxz , γyz = [u, v]1 + α w0,x , w0,y   1  [u, v]2k+1 − Ak α w2k,x , w2k,y + cos kz k     τxz , τyz = G γxz , γyz

(1.9) (1.10)

Here, it is convenient to consider equations of equilibrium in the following form:

      τxz , τyz = − α σx,x + τxy,y dz, α σy,y + τxy,x dz σz = −

  α τxz,x + τyz,y dz

(1.11) (1.12)

After some algebra, the above equations of [τxz , τyz ] with obvious notation become 1  G{u1 + αw0,x +  u2k+1 − Ak αw2k,x cos kz} k  

2     α 1 cos kz σx,x + τxy,y 2k+1 = A(x, y) +  Ak σx,x + τxy,y 1 + k k 1  G{v1 + αw0,y +  v2k+1 − Ak αw2k,y cos kz} k  

2    α 1  cos kz = B(x, y) +  Ak σy,y + τxy,x 1 + σy,y + τxy,x 2k+1 k k

(1.13)

(1.14)

1.2 Kirchhoff Theory

7

Equations (1.13) and (1.14) give A(x, y) = G (u1 + αw0,x ) and B(x, y) = G(v1 + αw0,y ) (Note that A(x, y) and B(x, y) are zero in FSDT but not the corresponding strains. Shear energy correction factor k2 is through distribution correction factor expressed above in series form).  

2      1  σx,x + τxy,y 2k+1 (1.15) G u2k+1 − Ak αw2k,x = α Ak σx,x + τxy,y 1 + k  

2      1  σy,y + τxy,x 2k+1 (1.16) G v2k+1 − Ak αw2k,y = α Ak σy,y + τxy,x 1 + k Zero face shear conditions give [u1 , v1 ] = – α [w0,x , w0,y ] so that

   1 2 2 σz = −G e2k+1 − Ak α w2k sin kz k  

2    1  σx,xx + 2τxy,xy + σy,yy 2k+1 = −  Ak σx,xx + 2τxy,xy + σy,yy 1 + k  α 2 sin kz (1.17) k Due to Kirchhoff’s displacements, σz confining to the first expression of the above equation is given by 

2   4  1 σz =  Ak sin kz α E w0 k 

(1.18)

so that face load condition after some algebra gives Kirchhoff’s equation E Dα4 w0 = q. The above Kirchhoff’s equation is to be solved with two-edge conditions through the use of stress resultants from the stationary property of total potential. As such, the earlier statement that Kirchhoff’s theory is, in a way, zeroth-order shear deformation theory is justified. From Eqs. (1.15, 1.16), we have     α 2     1 2 2 q+ σx,xx + 2τxy,xy + σy,yy 2k+1 G e2k+1 − Ak α w2k = Ak k k (1.19)

8

1 Poisson Theory of Isotropic Plates in Bending

Three variables w2k , u2k+1 , and v2k+1 are governed by the above equation and two in-plane equilibrium equations. The above analysis clearly shows that the use of w0 (x, y) as domain variable is not suitable for the analysis of bending problems (Hence, detailed algebra involved in the analysis is omitted). Moreover, the use of stationary property of total potential leads to the solution of an associated torsion problem.

1.2.4 Reissner’s Theory Reissner [1] in his article felt that inclusion of transverse shear deformation effects was directly (or by implication) presumed to be the key in resolving Poisson–Kirchhoff boundary conditions paradox. In his pioneering work [14], he brought out the use of transverse shear stresses in deriving a sixth-order plate theory. Equations governing (average) displacements in his sixth-order theory [17] consist of Kirchhoff’s fourthorder equation and a second-order equation coupled through edge conditions. They correspond to plate element equations governing stress resultants consistent in the reduction of 3-D equations through the calculus of variations. After Reissner’s articles [17, 24], several sixth and higher-order shear deformation theories were reported. The governing equations derived either by using the calculus of variations or from statics correspond to errors in satisfying in-plane equilibrium equations and/or constitutive relations. In all these theories, transverse displacement w is a domain variable.

1.2.5 Associated Torsion Problem First-order and higher-order shear deformation theories do not provide proper corrections to the initial solution (from Kirchhoff’s theory) of primary flexure problems. In these theories, corrections are due to approximate solutions of associated torsion problems. One has to have a relook at the use of shear deformation theories based on a plate (instead of 3D) element equilibrium equations other than Kirchhoff’s theory in the analysis of flexure problems. In the earlier work [13], we have mentioned that this coupling of bending and torsion problems is nullified in the limit of satisfying all equations in the 3D problems. However, these higher-order theories in the case of primary flexure problems defined from Kirchhoff theory are about finding the exact solution of associated torsion problem only. One can obtain the exact solution of the torsion problem (instead of using higher-order polynomials in z by expanding z and f3 (z) in sine series and f2 (z) in cosine series [21]). In a pure torsion problem, normal stresses and strains are zero implying that [u, v, w] = [u(y, z), v(x, z), w(x, y)]

(1.20)

1.2 Kirchhoff Theory

9

Rotation ωz = 0 and warping displacements (u, v) are independent of vertical displacement w. In FSDT, ωz = 0 and corrections to Kirchhoff’s displacements are due to the solution of approximate torsion problem. In the isotropic rectangular plate, warping function u(y, z) is harmonic in the y–z-plane. It has to satisfy G α u,y = Tu (y, z) with prescribed Tu along an x = constant edge. By expressing u in a product form u = fu (y) gu (z), we have α2 fu,yy gu + fu gu,zz = 0

(1.21)

By taking gun = A un sin λn z where λn = (2n + 1) π/2 satisfying zero face shear conditions, we get fun = Aun cosh(λn αy) + Bun sinh(λn αy)

(1.22)

If prescribed Tu is fun with specified constants (Aun , Bun ), clearly there is no provision to satisfy zero in-plane shear condition along y = constant edges. It has to be nullified with the corresponding solution from the bending problem. That is why a torsion problem is associated with a bending problem but not vice versa. In a corresponding bending problem, all stress components are zero except τxy = −fun (y) gun (z). This solution is used only in satisfying edge conditions along y = constant edges in the presence of specified Tun along x = constant edges. It shows that τxy distribution in the flexure problem is nullified in the limit in shear deformation theories due to torsion. It is interesting to note that the previously mentioned deficiency, due to coupling with w0 in the plate element equilibrium equations in FSDT and higher-order shear deformation theories, does not exist if applied τxy is zero all along the closed boundary of the plate. It is complementary to the fact that the boundary condition paradox in Kirchhoff theory does not exist if tangential displacement and w0 are zero all along the boundary of the plate.

1.3 Bending of Square Plates Here, a proper resolution of the above-mentioned paradox is presented through designated Poisson theory without considering either shear energy due to transverse shear deformations or higher-order approximations to displacements. Analysis of plates with different geometries and material properties under different kinematic and loading conditions does not provide much scope for the development of new theories other than those with the analysis of primary problems of a square plate dealt with in this monograph (Fig. 1.1). A square plate is bounded within 0 ≤ X, Y ≤ a, Z = ± h planes with reference to Cartesian coordinate system (X, Y, Z). Coordinates X, Y, Z and displacements (U, V, W) in non-dimensional form x = X/a, y = Y/a, z = Z/h, (u, v, w) = (U, V, W)/h and half-thickness ratio α = (h/a) are used throughout this monograph.

10

1 Poisson Theory of Isotropic Plates in Bending

Fig. 1.1 3-D plate: (0 ≤ X, Y ≤ a), (−h ≤ Z ≤ h) with x = X/a, y = Y/a, and z = Z/h

With the above notation, equilibrium equations in stress components are (with 3-D stress components as functions of coordinates x, y, and z)   α σx,x + τxy,y + τxz,z = 0

(1.23)

  α σy,y + τxy,x + τyz,z = 0

(1.24)

  α τxz,x + τyz,y + σz,z = 0

(1.25)

in which suffix after ‘,’ denotes partial derivative operator.

1.3.1 Bending Problem of Homogeneous Isotropic Plate The material of the plate is homogeneous and isotropic with elastic constants E (Young’s modulus), ν (Poisson’s ratio), and G (shear modulus) that are related to each other by E = 2(1 + ν) G. The plate is subjected to asymmetric load σz = ±(q0 /2) and zero shear stresses along the z = ±1 faces. In displacement-based models, stress components are expressed in displacements, via six stress–strain constitutive relations and six strain–displacement relations. These relations within the classical small deformation theory of elasticity are: Constitutive Relations:   Eεx = σx − ν σy + σz

(1.26)

Eεy = σy − ν(σx + σz )

(1.27)

  Eεz = σz − ν σx + σy

(1.28)

1.3 Bending of Square Plates

11

    τxy , τxz , τyz = G γxy , γxz , γyz

(1.29)

Semi-Inverted Stress–strain Relations:   σx = E εx + νε y + μσz

(1.30)

  σy = E εy + νεx + μσz

(1.31)

εz = −μe + (1 − 2νμ)σz /E

(1.32)

in which e = (εx + εy ), E = E/(1 − ν2 ), and μ = ν/(1 − ν). Normal stress σz is neglected in Kirchhoff theory, FSDT, and even in most of the higher-order shear deformation theories reported in the literature. Strain–displacement Relations:   εx , εy , εz = [αu,x , αv,y , w,z ]

(1.33)

γxy = [αu,y + αv,x ]

(1.34)

    γxz , γyz = u,z + αw,x , v,z + αw,y

(1.35)

Equilibrium equations: Equilibrium equations in displacements in the 3-D problem are with ∇ = (∂ 2 /∂x2 + ∂ 2 /∂y2 + ∂ 2 /∂z2 ) α2 ∇u + u,zz +

  1 α αu,xx + αv,yx + w,zx = 0 1 − 2ν

(1.36)

α2 ∇v + v,zz +

1 α(αu,xy + αv,yy + w,zy) = 0 1 − 2ν

(1.37)

 1  αu,xz + αv,yz + w,zz = 0 1 − 2ν

(1.38)

α2 ∇w + w,zz +

In the case of zero ωz (= α v,x − α u,y ) decoupling bending and torsion problems, equilibrium equations in terms of (u, v, σz ) are with plane Laplace operator  = (∂ 2 /∂x2 + ∂ 2 /∂y2 ) E α2 u + μασz,x + τxz,z = 0

(1.39)

E α2 v + μασz,y + τyz,z = 0

(1.40)

12

1 Poisson Theory of Isotropic Plates in Bending

  α τxz,x + τyz,y + σz,z = 0

(1.41)

1.3.2 Resolution of Poisson–Kirchhoff Boundary Conditions Paradox In Kirchhoff theory, the basic variable is w0 (x, y), and [u, v] are from [γxz , γyz ] ≡ 0 in the plate. In FSDT, w0 is associated with z[u1 , v1 ] through [γxz , γyz ]. In these theories and other shear deformation theories, σz is neglected in constitutive relations. In the present analysis as in Kirchhoff’s theory and FSDT, σz is initially neglected in the semi-inverted constitutive relations for resolving Poisson–Kirchhoff boundary condition paradox in Kirchhoff’s primary bending problem. In-plane variables (u1 , v1 ) uncoupled from w0 are basic variables. Due to the condition ωz = 0 required to decouple bending and torsion problems, one obtains reactive transverse stresses from thickness-wise integration of equilibrium equations     τxz , τyz = E f2 (z)α 2e1,x , e1,y

(1.42)

σz3 = − E f3 (z)α2  e1

(1.43)

in which f2 = (1 − z2 )/2 and f3 = (z − z3 /3)/2 (Fig. 1.2). Constitutive relation gives εz = – μ f1 e1 with f1 = z and e1 = εz1 . From satisfying face load condition, one gets the equation governing e1 as 2  2 E α  e1 + q0 = 0 3

(1.44)

In Poisson’s theory [9], E e1 in the above equation is replaced by ψ(x, y) so that Fig. 1.2 Kirchhoff primary bending problem of isotropic plate: σz = ± (q0 /2), [τxz , τyz ] = [0, 0] along z = ±1 faces. [τxz , τyz ]* = f2 (z) [τxz2 , τyz2 ]* with f2 (z) = (1 − z2 )/2 and σz * = 3f3 (z) (q0 /2), with f3 (z) = (z − z3 /3)/2 are statically equivalent stresses

1.3 Bending of Square Plates

13

2 2 α ψ + q0 = 0 3

(1.45)

The above equation is solved with either ψ = 0 or ψ,n = Tn (s) along an edge (wall) of the plate in which Tn (s) is specified transverse shear stress. It is significant to note that Kirchhoff’s reactive transverse stresses are a priori determined here independent of [u1 , v1 ]. Equations governing u1 and v1 are given by, with e1 = α (u1 ,x + v1 ,y ), ωz = α (u1 ,y − v1 ,x ) and [E e1 , ωz ] = [ψ, 0],   E α2  u1 , v1 ] = α[ψ, x , ψ, y

(1.46)

Displacements u1 and v1 are coupled through conditions on u1 and v1 along x = constant edge (with analog conditions along y = constant edge) u1 = 0 or E u1,x = Tx (y)

(1.47a)

v1 = 0 or 2G v1,x = Txy (y)

(1.47b)

In the above equations, Tx and Txy are prescribed as stress distributions. From zero face shear conditions, one obtains w0 (x, y) in terms of known [u1 , v1 ] in the form of   α w0 (x, y) = − ∫ u1 dx + v1 dy

(1.48)

One should note that zero ωz is integrability condition and it is also Cauchy– Riemann relation for analytic w0 (x, y). Vertical deflection thus obtained corresponds to face deflection (note that zero face shear conditions do not participate in the 3-D domain equations unlike in Kirchhoff’s theory). It’s zero value along the edge that requires, simply, a support to prevent vertical deflection of the intersection of the face with the wall of the plate. Such support also ensures zero deflection of the neutral plane along its edge. It is because the face and neutral plane deflections (w0f , w0n ) are the same since w(x, y, z) with (w2 + εz1 ) = 0 can be expressed as 1 w(x, y, z) = w0f (x, y) + f2 (z)w2 = w0n (x, y)− z2 εz1 2

(1.49)

However, f2 (z) of second-order correction w2 to vertical deflection w is parabolic, thereby, applied or reactive transverse shear stress along an edge is parabolic. Equations (1.45) and (1.46) form a sixth-order system for determination of ψ, u1 , and v1 . Relevant three-edge conditions correspond to those required by Poisson. Hence, the theory is designated as ‘Poisson Theory of Plates’ in bending, parallel to Kirchhoff theory. Here, transverse shear stresses are in terms of gradients of ψ and the coefficient of f3 (z) is used to satisfy prescribed face load condition. It is to

14

1 Poisson Theory of Isotropic Plates in Bending

be noted that the condition w0 = 0 prescribed along segments of the edge does not affect the displacements [u1 , v1 , w0 ] obtained from the present Poisson theory. In the illustrative example of a simply supported square plate [9], q = q0 sin (πx) sin (πy), ν = 0.3, and α = 1/6 (i.e., plate thickness ratio 2h/a = 1/3), (2E/q0 ) w0 = 2.27 from Kirchhoff theory whereas Poisson theory gives a value of (2E/q0 ) w0f = 2,54 due to inclusion of εz in the analysis.

1.3.3 Higher-Order Poisson Theories The above-described Poisson theory is based on point-wise satisfaction of Eqs. (1.23– 1.25). Higher-order corrections to the solutions from this theory are obtained through an iterative procedure [9]. For this purpose, it is convenient to generate coordinate functions fn (z), n = 0, 1, 2, 3, … through recurrence relations with f0 = 1, f2n+1,z = f2n , f2n+2,z = − f2n+1 such that f2n+2 (±1) = 0 [21, 22]. They are up to n = 5 

 1  1 1 z − z3 [f1 , f2 , f3 ] = z, 1− z2 , 2 2 3 f4 =

1 [(5 − 6z2 + z4 )] 24

f5 = z(25 − 10z2 + z4 )/120

(1.50a) (1.50b) (1.50c)

In 2-D plate theories, one cannot avoid linear thickness-wise distributions of displacements but higher-order polynomial series may be replaced with trigonometric series like in Cheng’s theory [25] (or special functions like Bessel functions, orthogonal polynomials [26] like Legendre polynomials, etc.). However, the satisfaction of infinitesimal 3-D element equilibrium equations requires either expansion of z in sine series or use of each trigonometric term expressed in power series in z. Otherwise, one has to resort to plate element equilibrium equations which involve coupling with torsion problem. In Batista’s recent article [8], coordinate functions fn (z) are generated by the method of successive approximation satisfying zero transverse shear stress conditions at the faces of the plate. With zero vertical load condition, he used the plate element equilibrium equation so that σz is identically zero (physically invalid) in the domain. In obtaining a solution in the limit (with σz ≡ 0), he derived Cheng’s shear equation [25] which applies to St. Venant’s theory of torsion to rotation about x-axis or y-axis in the case of rectangular plates. This torsion problem is often used to illustrate sixthorder theories though the exact solution requires expansion of z in sine series. It is to be noted that Kienzler [7] used power series in ζ (= α z) and Gol’denveizer and Kolos [28] used power series in γ (= Z = h z) in the derivation of plate equations. By using the recurrence relations mentioned above, one can generate fn (α z) and fn (h z) replacing integration limit 1 by α and h, respectively, so that fn functions

1.3 Bending of Square Plates

15

in Eq. (1.25) become homogeneous polynomials of degree n in (α, ζ) and (h, Z). It implies that αn (or hn ) is the scale factor of fn (z) in Eq. (1.25). In-plane 2-D variables in [7] and [28] are connected to the present variables through linear transformations by equating coefficients of powers of z. Plate element equilibrium equations from using polynomials in z (e.g., power series, Taylor series, orthogonal polynomials [26], functions in the present analysis) in higher-order theories other than one-term representation of displacements in Kirchhoff’s theory, modified Kirchhoff theory (MKT) [13], and FSDT become equivalent to one other with appropriate change of 2-D variables. The present iterative procedure deals with point-wise equilibrium equations of a 3-D infinitesimal element. Iterative procedure: Displacements, strains, and stresses are expressed with summation on ‘n’ (n = 0, 1, 2, 3 …) in the form given below: [w, u, v] = [f2n w2n , f2n+1 u2n+1 , f2n+1 v2n+1 ]

(1.51a)

[εx , εy , γxy , εz ] = f 2n+1 [εx , εy , γxy , εz ]2n+1

(1.51b)

[σx , σy , τxy , σz ] = f 2n+1 [σx , σy , τxy , σz ]2n+1

(1.51c)

[γxz , γyz , τxz , τyz ] = f 2n [γxz , γyz , τxz , τyz ]2n

(1.51d)

In the iterative method [22], two functions ϕ and ψ are introduced in the following form [ψ, ϕ] = f2n+1 [ψ, ϕ]2n+1 (n = 0, 1, 2 . . . .)

(1.52)

in which ψ is related to w, and ϕ is required to decouple torsion and bending problems. Face-parallel distributions of in-plane displacements are assumed in the form of   u2n+1 = −α ψ2n+1,x −ϕ2n+1,y

(1.53a)

  v2n+1 = − α ψ2n+1,y + ϕ2n+1,x

(1.53b)

so that e2n+1 = −α2 ψ2n + 1 . The iterative scheme is to reduce errors in the analysis by increasing the order of reactive transverse stresses. The main limitation in Kirchhoff’s theory like in most of the theories based on plate element (instead of 3-D infinitesimal element) equilibrium equations is that reactive normal stress is zero through-thickness at locations of zeros of the applied load. To overcome this limitation, σz2n+3 (n ≥ 1) is kept as a free variable by modifying f2n+3 in the form of f∗2n+3 (z) = f2n+3 (z)−β2n+1 f2n+1 (z)

(1.54)

16

1 Poisson Theory of Isotropic Plates in Bending

in which β2n+1 = [f2n+3 (1)/f2n+1 (1)] so that f*2n+3 (±1) = 0. Coefficient of f2n+1 in σz denoted by σ*z2n+1 is given by σ*z2n+1 = σz2n+1 −β2n+1 σz2n+3

(1.55)

At nth stage of approximation, reactive transverse stresses and εz2n+1 are known in terms of solutions in the preceding stage. In-plane displacements u2n+1 , v2n+1 are modified in the form of Eqs. (1.56a, 1.56b) given below such that they are consistent with known stresses τxz2n , τyz2n and are free to obtain reactive stresses τxz2n+2 , τyz2n+2 , σz2n+3 and vertical deflection w2n+2 :   u∗2n+1 = −α ψ2n+1.x −ϕ2n+1,y + w2n,x + E α e2n−1,x /G

(1.56a)

  v∗2n+1 = −α ψ2n+1.y + ϕ2n+1,x + w2n,y + E αe2n−1,y /G

(1.56b)

in which w2n is required to obtain corrections to εz but there is no provision to determine w2n as a domain variable in the iterative method (finding w2n through energy methods involves coupling with torsion problem). However, secondary contribution w2n is the same as ψ2n+1 . As such, the above Eqs. (1.56a, 1.56b) are changed to   u∗2n+1 = −α 2ψ2n+1.x −ϕ2n+1,y + E αe2n−1,x /G

(1.56a)

  v∗2n+1 = −α 2ψ2n+1.y + ϕ2n+1,x + E αe2n−1,y /G

(1.56b)

After some lengthy algebra, we get for n = 1, with β2 = 2(απ)2 in the illustrative example, E )wface max = ( 2q0



   2 4−ν 2 6 1+ν 2 β β −1 5 1−ν 5 1−ν

(1.57)

In the illustrative example with ϕ ≡ 0, higher-order correction to (2E/q0 ) w0 uncoupled from torsion is 1.26. Total correction to (2E/q0 ) w0 = 2.27 from Kirchhoff’s theory is (0.27 + 1.26) giving (2E/q0 ) w0f (= (2E/q0 ) w0n ) = 3.80 so that (E/2q0 ) w face max = 3.80 [21].

1.3.4 Neutral Plane Deflection In Kirchhoff theory and shear deformation theories, including widely used FSDT, vertical deflection w0 (average displacement in Reissner’s sixth-order theory) is the same in the face parallel planes. But it is known physically that neutral plane deflection has to be higher than face deflection. This is because the elastic medium is on

1.3 Bending of Square Plates

17

either side of the neutral plane, whereas it is on one side of each of the top and bottom faces. Proper estimation of neutral plane deflection is dependent on normal strain εz ignored in Kirchhoff’s theory and shear deformation theories. In earlier investigation [9], it is shown that the expansion of displacements in polynomials of thickness coordinate z is not adequate for the proper estimation of the face and neutral plane deflections. This fact is overlooked in the analysis of even isotropic homogeneous plates through widely used FSDT and other shear deformation theories. The solution to a supplementary problem is required for obtaining neutral plane deflection. Correction to neutral plane deflection from a supplementary problem: Neutral plane deflection w0n is corrected from the solution of a supplementary problem based on 15-decade-old Levy’s work [27] with assumed displacements in the form of [u, v, w]s = [sin (π z/2) us3 , sin (π z/2) vs3 , (π/2) cos (π z/2) ws2 ]

(1.58)

They are added as corrections to [u*3 , v*3 ] obtained from the iterative method so that [u, v] in supplementary problem are given by     [u, v] = sin (πz/2) u∗3 + us3 , v∗3 + vs3

(1.59)

In the example, correction to (2E/q0 ) w0n is about 0.66 giving its value of 4.46 (= 3.80 + 0.66).

1.3.5 Exact Solution with w(x, y, z) as Domain Variable Exact values of w0n and w0f in the example problem are obtained earlier [9, 22] with displacements u = (A1 sinh βz + A2 z cosh βz)cosπx sin πy

(1.60)

v = (A1 sinh βz + A2 z cosh βz) cos πy sin πx

(1.61)

w = (C1 cosh βz + C2 z sinh βz) sin πx sin πy

(1.62)

One gets from equilibrium (1.36–1.38) in which w(x, y, z) is a domain variable and load conditions β=

√ √ 2απ and C2 = A2 = (2απ A1 − β C1 )/(3 − 4ν)

Zero shear stresses and vertical load condition along faces give

(1.63)

18

1 Poisson Theory of Isotropic Plates in Bending

C1 =

β tanh β + 2(1−ν) √ β sinh β + 2(1−ν) cosh β 2q0 2A1 = (1 + ν) β tanh β − (1−2ν) β(sinh β cosh β−β) E

From the above solutions with ν = 0.3, α = 1/6 and β =

(1.64)

√ 2 α π, one obtains

(E/2q0 )w(a/2, a/2, 0) = 3.49

(1.65)

(E/2q0 )w(a/2, a/2, 1) = 4.12

(1.66)

(It is found that the use of displacements (1.60–1.62) in Eqs. (1.39–1.41) gives the same estimates for vertical displacement). The above estimates correspond to those from the associate torsion problem since face deflection w0f is greater than neutral plane deflection w0n . Note that neutral plane deflection is lower than that of face deflection by about 15% which decreases with decreasing thickness ratio.

1.3.6 Exact Solution with w0 (x, y) as Face Variable Exact solution of the present illustrative example with w0 (x, y) as face variable was obtained earlier [9, 22] with assumed in-plane displacements (1.60, 1.61) and shear stresses, in place of w(x, y, z) in Eq. (1.62), in the following form τxz = (C1 cosh βz + C2 z sinh βz) cos πx sin πy

(1.67)

τyz = (C1 cosh βz + C2 z sinh βz) cos πy sin πx

(1.68)

Normal stress σz from Eq. (1.41) is used in Eqs. (1.39, 1.40). Due to zero face shear stresses, face deflection w0 (x, y) is evaluated by replacing [u,z , v,z ] with [u, v]z=1 in shear stress–strain relations so that   α w0 (x, y) = − (1.69) u dx + v dy z=1 Vertical displacement parameter (E/2q0 ) w of neutral and face planes at mid-point of the plate thus obtained is (E/2q0 )(w0f , w0n )max = (4.17, 4.49)

(1.70)

1.3 Bending of Square Plates

19

1.3.7 Some Remarks on Solution of Kirchhoff’s Primary Bending Problem Solution for w0 consists of satisfying (i) zero face shear conditions and (ii) edge support conditions on w0 . In Kirchhoff’s theory, both these requirements are met through the single variable w0 governed by a fourth-order equation. In Reissner’s theory [17], average vertical displacement satisfies condition (ii) and in FSDT, condition (i) is not satisfied but its effect is included through shear energy correction factor. In any case, both of them are approximations to associated torsion problems [13]. The present analysis provides some clarity about the preliminary solution of the primary flexure problem. It shows that the determination of in-plane displacements, bending stresses, and reactive transverse stresses from the integration of equilibrium equations is uncoupled from w0 . However, edge condition w0 = 0 or contracted stress resultant Vx = Vx0 in Kirchhoff’s theory is different from the condition εz = 0 or τxz2 = Txz

(1.71)

at each of x = constant edges (along with analogue conditions at each of y = constant edges). In the illustrative example, error in the estimated value (= 3.80) of w0f is relatively high compared to the accuracy achieved in neutral plane deflection. In the earlier work [9], estimation of w0f is further improved by modifying in-plane displacements (u*3 , v*3 ) such that (τxz2 , τyz2 ) are independent of εz1 . In the present example, correction to the face deflection changes to 1.43 so that face deflection value is 3.97 (= 2.54 + 1.43), which is under 4.7% from the exact value. The correction 1.43 to the face deflection is due to εz3 from the constitutive relation. Determination of εz3 in terms of (σz3 , u3 , v3 ) involves lengthy algebra and arithmetical work. The above analysis consists of a basic sixth-order system with the inclusion of the second-order effect due to σz3 and a supplementary fourth-order system. Since the primary second-order corrections have to be due to the inclusion of σz1 in the in-plane constitutive relations, it is much simpler to find its effect from Poisson theory of new primary bending problems in the next section without consideration of higher-order displacement components (u3 , v3 ).

1.4 New Primary Bending Problems New primary problems (designated as Poisson primary bending problems) are defined with zero transverse normal strain edge condition (εz = 0). This edge condition is more practical [9, 22] avoiding the need for vertical stress resultant in Kirchhoff theory. Here, primary variables are [u1 , v1 , ψ0 ] in which ψ0 (x, y) is a solution of an auxiliary problem governing transverse stresses. These transverse stresses are independent of material constants so that they are global solutions.

20

1 Poisson Theory of Isotropic Plates in Bending

Auxiliary problem is based on the equilibrium Eq. (1.25) in terms of transverse stresses with σz = z σz1 (= z q0 /2 satisfying prescribed face condition) neglected in most of the theories reported in the literature. It should be noted in the analysis of Kirchhoff’s primary bending problem that the term z σz1 in the Taylor series expansion of σz in odd powers of z is missing in the reported shear deformation theories. Here, transverse shear stresses are independent of z and assumed to be gradients of a function ψ0 so that [τxz0 , τyz0 ] = α [ψ0 ,x , ψ0 ,y ]. The function ψ0 (x, y) may be referred to as Poisson transverse shear stress function governed by the Poisson equation α2 ψ0 +

1 q =0 2 0

(1.72)

to be solved with edge condition either ψ0 = 0 or its normal gradient equal to prescribed transverse shear stress along the edge. They are more realistic edge conditions and more practical if they are independent of in-plane coordinates. Moreover, the function ψ0 is related to normal strain εz and ψ0 = 0 implies εz = 0. With the aid of above global solutions for transverse stresses, primary bending problem governing in-plane displacements is in terms of in-plane equilibrium Eqs. (1.23–1.24) along with statically equivalent transverse stresses (Fig. 1.3). With the aid of above global solutions for transverse stresses, primary bending problem governing in-plane displacements is in terms of in-plane equilibrium Eqs. (1.23–1.24) along with statically equivalent transverse stresses     τxz , τyz ] = [τxz0 , τyz0 + f2 (z) τxz2 , τyz2 σz =

1 z q0 + f3 (z)σz3 2

(1.73) (1.74)

in which (τxz2 , τyz2 , σz3 ) yet to be determined (satisfying edge conditions specified later, Eq. 1.89) are corrections due to (u1 , v1 ). Since the added terms are dependent Fig. 1.3 Poisson primary bending problem: auxiliary problem σz = z(q0 /2), [τxz0 , τyz0 ] = α [ψ0 ,x , ψ0 ,y ]. Bending problem: σz = [q/2 − β1 σz3 ] z + f3 σz3 , [τxz2 , τyz2 ] = [τxz0 , τyz0 ] + G [u1 , v1 ]

1.4 New Primary Bending Problems

21

on material constants, above transverse shear stresses are reactive stresses and cannot be prescribed stresses along the edges. In order to keep σz3 as a free variable, the function f3 (z) in Eq. (1.74) is replaced by f3 *(z) = (f3 – z/3) so that f3 *(±1) = 0. Then, the face load condition is satisfied with σz =

  1 zq 0 + f∗3 σz3 2

(1.75)

In further analysis, it is convenient to express σz with β1 = 1/3 in the form of

1 q − β1 σz3 z + f3 (z)σz3 σz = 2 0

(1.76)

The presence of σz3 in both f1 (z) (= z) and f3 (z) distributions gives the facility (believed to be a new one) of obtaining σz3 from static Eq. (1.25) and the usual thickness-wise integrated (z-integrated) equilibrium Eqs. (1.23–1.25). For this purpose, in-plane distributions [u1 , v1 ] are modified in the form with u∗1 = u1 + τxz0 /G−α w0,x

(1.77)

v∗1 = v1 + τyz0 /G−α w0,y

(1.78)

The introduction of w0 (x, y) term in the above equation is to avoid its participation in the static equilibrium equations through transverse shear stresses. Modified transverse shear stresses from stress–strain and strain–displacement relations are   τxz2 , τyz2 ] = G [u1 , v1 ] + [τxz0 , τyz0

(1.79)

One gets from Eq. (1.25), with the above shear stresses [τxz2 , τyz2 ] along with σz = z [q/2 − β1 σz3 ] and e1 = (εx1 + εy1 ) Ge1 = β1 σz3

(1.80)

For the use of [u1 *, v1 *] in the integration of equilibrium Eqs. (1.23–1.24), displacements [u1 , v1 ] due to the requirement of v1,x = u1,y to decouple bending and torsion problems are expressed in the form of   u1 , v1 ] = − α[ψ1,x , ψ1,y

(1.81)

in the case of hard and/or soft simply supported edge conditions. Contributions of ψ1 and w0 in [u1 , v1 ]* are the same (due to zero face shears) in giving corrections to w(x, y, z) and transverse stresses. In fact, the contribution of w0 is through strain–displacement relations in static equilibrium equations and

22

1 Poisson Theory of Isotropic Plates in Bending

constitutive relations in z-integration of equilibrium equations. Since w0 does not participate in the in-plane static equations, its contribution is through [u1 , v1 ] in the integrated in-plane equilibrium equations. Hence, w0 in [u1 , v1 ]* is replaced by ψ1 (to be independent of w0 used in strain–displacement relations) so that [u1 , v1 , εx1 , εy1 , γxy1 ]* are      u1 , v1 ∗ = (2 u1 + γxz0 ), 2 v1 + γyz0

(1.82)

      εx1 , εy1 ∗ = 2εx1 + αγxz0,x , 2εy1 + αγyz0,y

(1.83)

   ∗ γxy1 = 2γxy1 + α γxz0,y + γyz0,x

(1.84)

From integration of Eqs. (1.23 and 1.24) using the above strains in Eqs. (1.38 and 1.39) along with v1 ,x = u1 ,y , reactive transverse stresses [τxz2 , τyz2 ]* are   τ∗xz2 = α E e∗1 + μσz1 ,x

(1.85a)

  τ∗yz2 = α E e∗1 + μσz1 ,y

(1.85b)

α(τxz2,x + τyz2,y )∗ + σz3 = 0

(1.86)

One equation governing in-plane displacements (u1 , v1 ), noting that σz3 from Eq. (1.80) is negative of the one from Eq. (1.86) due to (f3 ,zz + f1 ) = 0, is given by  ∗ αβ1 τxz2,x + τyz2,y = Ge1

(1.87)

From Eqs. (1.76, 1.80, 1.87), one gets (see Appendix 1.1) E β1 α4

(2 ψ1 −ψ0 /G) + Ge1 − μ β1 α2 σz1 = 0

(1.88)

The above equation becomes a fourth-order equation in ψ1 to be solved with two in-plane conditions along constant x (and analog conditions y = constant edges) edges [σ∗1 , v∗1 ] = [0, 0]

(1.89)

√ After some algebra with G = E/2(1 + ν), μ = ν/(1 − ν), β = 2 απ, and ψ1 = c1 sin (πx) sin (πy), equation governing c1 becomes (see Appendix 1.2) [β2 +

3(1 − ν) (1 + ν) ]c1 − (2 − ν)(q0 /2E) = 0 4 8

(1.90)

From the above equation, one gets c1 = 0.26 (q0 /2E) with ν = 0.3 and α = 1/6.

1.4 New Primary Bending Problems

23

Face deflection w0 from the integration of transverse shear strain–displacement relations is    w0c = β2 (1 + ν)/2 q0 /2E + c1c (Appendix 1.3)

(1.91)

from which w0c = 1.39 (q0 /2E). Estimated face deflection w0fmax = 4.08 (2.27 + 0.27 + 1.39 + 0.15) (2q0 /E), which is fairly close to the exact value 4.17. It shows that the solution for w from the analysis provides proper correction to the estimation of face deflection. It is underestimated by 2.16% [9]. Here also, in-plane displacements [u1 , v1 ] are determined through the satisfaction of both static and integrated equilibrium equations with modified displacements u∗1 = u1 + γxz0 −α w0,x

(1.92)

v∗1 = v1 + γyz0 −α w0,y

(1.93)

along with conditions at x = constant edges (similar conditions at y = constant edges) u1 (y) = 0 or σx1 (y) = Tx1 (y)

(1.94a)

v1 (y) = 0 or τxy1 (y) = Txy1 (y)

(1.94b)

ψ2 (y) = 0 or τxz 2 (y) = τxz0 (y)

(1.95)

If q = 0 and in-plane edge conditions are homogeneous, edge conditions (1.95) and similar conditions at y = constant edges become the conditions used in obtaining a global solution.

1.4.1 Neutral Plane Deflection It is observed that error in the estimation of face deflection is much higher than that of neutral plane deflection in the primary bending problem in the Kirchhoff theory. But it is desirable to provide a uniform approximation to deformations through-thickness of the plate. It appears to the author’s knowledge that no suitable 2-D modeling which gives a more or less same percentage of approximation to the thickness-wise distribution of a displacement variable is reported till now except the one described below. Proper higher-order theories are to reduce only maximum error in the estimation of a physical variable to a desirable level.

24

1 Poisson Theory of Isotropic Plates in Bending

Corrective in-plane displacements in a supplementary problem are assumed in the following form:   [us , vs ] = u1s,V1s sin(πz/2)

(1.96)

In-plane distributions u1s and v1s are added as corrections to the known modified in-plane displacements (u1 *, v1 *), like in Poisson theory, so that (u, v) in the supplementary problem is given by [u, v] =

    ∗ u1 + u1s , v∗1 + v1s sin(πz/2)

(1.97)

in which u1 * = (u1c * + u1b ). By equating σz3s (from integration of Eqs. (1.1) with s variables) with β1 σz3 (from static equations with * variables), one equation governing [u1s , v1s ] is   (2/π)2 α2  E e1sj + μσz1 = β1 σz3

(1.98)

By expressing [u1s , v1s ] = − α [ψ1s ,x , ψ1s ,y ] due to the second equation v1s ,x = u1s ,y , Eq. (1.46) becomes a fourth-order equation in ψ1s to be solved with two inplane conditions along x = constant edges (and analog conditions along y = constant edges) in the present example σx1s = 0, v1s = 0

(1.99)

Here, ψ1s is related to e1c by the equation   E 4/π2 α4 ψ1s = Ge1c

(1.100)

After some algebra, c1s is related to c1c by the equation

   1−ν 2 2  2E/q0 c1s = π −β1 2E/q0 c1c 2

(1.101)

so that c1s = 0.55(q0 /2E) with ν = 0.3 and α = 1/6. In the present illustrative example, the estimated neutral plane deflection w0nmax = 4.36 (2q0 /E) (= 2.27 + 0.15 + 1.39 + 0.55), which is underestimated by 2.90% from the exact value 4.49. Evaluation of neutral plane deflection involves (u1 , v1 ) and (u1s , v1s ) associated with f2 (z) and cos (πz/2) distributions of transverse shear stresses, respectively. As such, the solution of the present auxiliary problem provides second-order corrections to the solutions of the primary problem. Moreover, the wide variation of percentage variations of errors (in Kirchhoff’s primary bending problem) from 0.67 in w0f to 4.80 in w0n at the first stage of iteration using (u3 , v3 ) in Poisson’s theory is reduced from 2.16 to 2.90 in the analysis of the new primary problem (see Table 1.1) without using

1.4 New Primary Bending Problems

25

Table 1.1 Homogeneous isotropic plate: face and neutral plane displacements wf = (E/2q0 ) w ( 21 , 1 2,

1), wn = (E/2q0 ) w ( 21 , 21 , 0); α = 1/6, ν = 0.3 1

2

3

4

5

6

7

8

wf

4.12

4.17

2.27

2.54

3.80

3.69

4.07

4.07

wn

3.49

4.49

2.27

2.54

3.80

3.41

4.36

4.46

1. Exact: w is domain variable, 2. Exact: w is face variable, 3. Kirchhoff’s theory, 4. Poisson theory with εz1 , 5. Poisson theory with σz1 and one iteration, 6. FSDT without k2 , 7. New primary problem: Poisson theory without εz1 , 8. New primary problem: Poisson theory with εz1

(u3, v3 ). One iteration using (u3 , v3 ) satisfying both static and integrated equilibrium equations is expected to reduce this gap even further along with decreased percentage errors. A significant implication of this observation is that the solution of the auxiliary problem is necessary to obtain a more or less uniform approximation to thicknesswise distributions of vertical displacement, like in Kirchhoff’s and shear deformation theories. Numerical results from Kirchhoff theory, FSDT, Poisson theory in the analysis of illustrative example are given in Table 1.1 for ready reference. One should note from the last two columns in Table 1.1 that the solution of Poisson’s primary bending problem with one-term in-plane displacements gives face deflection within 2.16% of exact value. Moreover, the use of solution of supplementary problem gives neutral plane deflection within 2.90% without using εz1 and within 0.7% using εz1 in the analysis. With reference to validity of small deformation theory of elasticity, Kirchhoff theory gives a lower bound of thickness ratio and the present Poisson theory gives upper bound of thickness ratio, depending on specified percentage error of maximum transverse displacement wmax in a given problem. It is necessary to work out several examples for inclusion in a textbook for graduates in aerospace, civil, mechanical engineering disciplines, and so on. The present work is essentially a monograph which is not the same as a classical textbook. The ‘Epilogue’ contains suggestions for future work. There are several approximate methods available for analysis of 2-D problems generated in the present monograph; for example, Rayleigh–Ritz method, Galerkin method, finite difference method, boundary element method, and so on. Finite element method is also an approximate method which is normally used when the geometry of the problem and loading are complicated. In the application of finite element method for solution of such problems, C(1) continuity elements has to be formulated and should be used here also, like in Kirchhoff theory.

1.5 Conclusions 3-D equations in displacements and sequence of 2-D problems with vertical displacement as domain variables correspond to associated torsion problems.

26

1 Poisson Theory of Isotropic Plates in Bending

• Kirchhoff’s theory is a zeroth-order shear deformation theory. • FSDT and higher-order shear deformation theories with shear correction factors deal with artificial torsion problems. • It is mandatory to satisfy thickness-wise integrated equilibrium equations for solutions of in-plane displacements. • Satisfying static equations improves solutions of these displacements. • The Poisson theory is based on the satisfaction of both static and integrated equilibrium equations using newly introduced Poisson transverse shear stress function  0 (x,y). • Thickness-wise distribution of displacements in terms of polynomials fn (z) is not adequate for finding interior solutions of these displacements. • Solutions of auxiliary and supplementary problems are necessary to rectify lacuna in Kirchhoff’s theory. • Analysis of associated torsion problems through Poisson theory is simply by replacing ψ with lateral displacement variable w0 (x, y). • The Poisson theory with proper modifications can also be used for the analysis of eigen-value problems, like lateral buckling and free vibration of elastic plates.

Appendices Appendix 1.1—Derivation of Eq. (1.88) ∗    From Eqs. (1.85a, 1.85b): α τxz2,x + τyz2,y = α2  E e∗1 + μσz1 ,  From Eq. (1.82): e∗1 = 2e1 + α γxz0,x + γyz0,y ,   e1 = α u1,x + v1,y = −α2 ψ1 , Gγxz0 = αψ0,x , Gγyz0 = αψ0,y .   α γxz0,x + γyz0,y = α2 ψ0 /G.     ∗  ∴ α τxz2,x + τyz2,y = α2  E −2α2 ψ1 + α2 ψ0 /G + μα2 σz1 From Eq. (1.87):  ∗ Ge1 = αβ1 τxz2z x + τyz2,y       = β1 α2  E −2α2 ψ1 + α2 ψ0 /G + μα2 σz1    = β1 α2  ψ0 /G − 2E α2 ψ1 + μα2 σz1

Appendices

27

From Eqs. (1.76, 1.80, 1.87), one gets E β1 α4 (2ψ1 − ψ0 /G) + Ge1 − μβ1 α2 σz1 = 0

(1.88)

Appendix 1.2—Derivation of Eq. (1.90) from Eq. (1.88)   E = E/ 1 − v2 , μ = v/(1 − v) · β1 = 1/3, ψ1 = c1 sin(πx) sin(πy), σz1 = q/2 √ α2 ψ0 + q/2 = 0, β = 2απ q = q0 sin(πx) sin(πy)   α2 ψ0 = −q/2 = − q0 /2 sin(πx) sin(πy) α2 ψ1 = −2(απ)2 c1 sin(πx) sin(πy), e1 = −α2 ψ1 = 2(απ)2 c1 sin(πx) sin(πy)   e1 = β2 c1 sin(πx) sin(πy), α2 σzl = −β2 q0 /2 sin(πx) sin(πy) Equation (1.88):     E β1 α2  −β2 2c1 + q0 /2G + Gβ2 c1 + μβ1 β2 q0 /2 = 0     i.e. E β1 2β2 c1 − q0 /2G + Gc1 + μβ1 q0 /2 = 0 With β1 = 1/3:     E 2β2 c1 − q0 /2G + 3Gc1 + μ q0 /2 = 0       E/ 1 − ν2 2β2 c1 − q0 (1 + ν)/E + 3Ec1 /[2(1 + ν)] + [ν/(1 − ν)] q0 /2 = 0      2/ 1 − ν2 β2 c1 + 3c1 /[2(1 + ν)] − q0 /[E(1 − ν)] + [ν/(1 − ν)] q0 /2E = 0  2  c1 : 2β /(1 − ν2 ) + 3/[2(1 + ν)]  of   2   4β + 3(1 − ν) 2[ 1 − ν2 ].   Coefficient of q0 /2E = (ν − 2)/(1 − ν) Coefficient

    ∴ β2 + (1 − ν)/4 c1 − [(1 + ν)/8](2 − ν) q0 /2E = 0

=

(1.90)

Appendix 1.3—Derivation Eq. (1.91) from Eq. (1.90) From the above equation, one gets, with ν = 0.3 and α = 1/6, β = [0.27625/(0.5483 + 0.75 × 0.7)=] 0.2574 (q0 /2E).

√ 2 α π, c1 =

28

1 Poisson Theory of Isotropic Plates in Bending

Face deflection w0c from the integration of transverse shear strain–displacement relations is α2 ψ0 = −q/2; q = q0 sin(πx) sin(πy)     τxz0, τyz0 = G γxz0 , γyz0 ; ψ0 = c0 sin(πx) sin(πy)     αw0c = (γxz0 − u1 )dx + γyz0 − v1 dy     = (τxz0 /G − u1 )dx + τyz0 /G − v1 dy      = αψ0,x /G − u1 dx + αψ0,y /G − v1 dy   ∴ w0c = ψ0 /G + c1 q0 /2E ψ0 /G = 2(1 + v)ψ0 /E

      1 1 ψ0 , , 1 /G = c0 β2 /4 2(1 + v)/E = β2 (1 + v)/2 2q0 /E 2 2       w0cmax = β2 (1 + v)/2 2q0 /E + c1c = 0.65β2 + 4c1 (2q0 /E)   = 1.39 2q0 /E (1.91)

References 1. Reissner E (1985) Reflections on the theory of elastic plates. Appl Mech Rev 38:1453–1464 2. Lo KH, Christensen RM, Wu EM (1977) A higher-order theory of plate deformation. J Appl Mech 44:663–676 3. Lo KH, Christensen RM, Wu EM (1978) Stress determination for higher-order plate theory. Int J Solids Struct 14:655–662 4. Lewinski T (1986) A note on recent developments in the theory of elastic plates with moderate thickness. Eng Trans 34(4):531–542 5. Lewinski T (1987) On refined plate models based on kinematical assumptions. Ing Arch 57(2):133–146 6. Blocki J (1992) A higher-order linear theory for isotropic plates-I, theoretical considerations. Int J Solids Struct 29(7):825–836 7. Kienzler R (2002) On consistent plate theories. Arch Appl Mech 72:229–247. https://doi.org/ 10.1007/s00419-002-0220-2 8. Batista M (2010) The derivation of the equations of moderately thick plates by the method of successive approximations. Acta Mech 210:159–168 9. Vijayakumar K (2013) On a sequence of approximate solutions: bending of a simply supported square plate. Int J Adv Struct Eng 5(1):18. https://doi.org/10.1186/2008-6695-5-18 10. Kirchhoff G (1850) Über das Gleichgewicht und die Bewegung einer elastischen Scheibe. Journal für Reins und Angewandte Mathematik 40(1850):51–88 11. Vijayakumar K (2014) Review of a few selected theories of plates in bending. J Appl Math 2014. Article ID 291478. https://doi.org/10.1155/2014/291478 12. Love AEH (1934) A treatise on mathematical theory of elasticity, 4th edn. Cambridge University Press, Cambridge, pp 458–463

References

29

13. Vijayakumar K (2011a) Modified Kirchhoff’s theory of plates including transverse shear deformations. Mech Res Commun 38(3):2011–2013 14. Reissner E (1944) On the theory of bending of elastic plates. J Math Phys 23:184–191 15. Hencky H (1947) Über die Berücksichtigung der Schubverzerrung in ebenen Platten. IngenieurArchiv 16:72–76. https://doi.org/10.1007/BF00534518 16. Reddy JN (1984) A simple higher order theory for laminated composite plates. J Appl Mech 51:745–752 17. Reissner E (1947) On bending of elastic plates. Q Appl Math 5(1):55–68 18. Ambartsumyan SA (1958) On a theory of bending of anisotropic plates (in Russian). Izv Akad. Nauk SSSR Otd. Tekh. Nauk 5:69–77 19. Lewinski T (1990) On the twelth-order theory of elastic plates. Mech. Res. Comm. 17(6):375– 382 20. Vijayakumar K (2009) New look at Kirchhoff’s theory of plates. AIIA J 47:1045–1046. https:// doi.org/10.2514/1.38471 21. Vijayakumar K (2013) Poisson’s theory for analysis of bending of isotropic and anisotropic plates. ISRN Civil Eng 8 pages. 562482. https://doi.org/10.1155/2013/562482 22. Vijayakumar K (2011b) A relook at Reissner’s theory of plates in bending. Arch App Mech 81:1717–1724. https://doi.org/10.1007/s00419-011-0513-4 23. Vijayakumar K (2015) Extended Poisson’s theory for analysis of bending of a simply supported square plate. SSRG Int J Civil Eng (SSRG-IJCE) 2(11). ISSN: 2348-8352. www.internationa ljournalssrg.org 24. Reissner E (1945) The effect of transverse shear deformations on the bending of elastic plates. J Appl Mech 12:A69–A77 25. Cheng S (1979) Elasticity theory of plates and a refined theory. J Appl Mech 46:644–650 26. Krenk S (1981) Theories for elastic plates via orthogonal polynomials. Transactions of the ASME 48:900–904 27. Levy M (1877): Memoire sur la Theorie des Plaques Elastiques Planes. Journal des MathematiquesPuresetAppliquees 30:219–306 28. Gol’denveizer AL, Kolos AV (1965) On the derivation of two-dimensional equations in the theory of thin elastic plates. PMM 29(1):141–155

Chapter 2

Extension Problems: Higher-Order Approximations

Nomenclature a E E’ e fn (z) G 2h q(x, y) Tx , Txy , Txz Ty , Txy , Tyz U, V, W (u, v, w) u, v w u0 , v0 , w1 X, Y, Z (x, y, z) α β2n+1 εx , εy , γxy εz , γxz , γyz  μ ν ωz (x, y)

Side length of a square plate. Young’s modulus. E/(1 – ν2 ). (εx + εy ) in face parallel planes. Thickness-wise (z-) distribution functions, n = 1, 2, … Shear modulus of rigidity. Plate thickness. Applied load density. Prescribed stresses at each of x = constant edges. Prescribed stresses at each of y = constant edges. Displacements in X, Y, and Z directions, respectively. (U, V, W)/h (non-dimensional displacements). In-plane displacements in the x and y directions, respectively. Vertical deflection. Displacements in x, y, and z directions, respectively, in face parallel planes. Coordinates of a point in a Cartesian system. (X/a, Y/a, Z/h) (non-dimensional coordinates). (h/a) (half-plate thickness ratio). Modification factor of transverse shear stresses in extension problems. In-plane strains. Transverse strains. (∂ 2 /∂x2 + ∂ 2 /∂y2 ) Laplace operator in x-y plane. ν/(1 − ν). Poisson’s ratio. α (v,x − u,y ), rigid body rotation about z-axis. Airy’s stress function.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 K. Vijayakumar and G. K. Ramaiah, Poisson Theory of Elastic Plates, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-33-4210-1_2

31

32

ϕ(x, y) σx , σy , τxy τxz , τyz , σz τ*xz , τ*yz , σ*z

2 Extension Problems: Higher-Order Approximations

Harmonic function. In-plane stresses. Transverse stresses. Statically equivalent transverse stresses.

2.1 Analysis of Extension Problems In extension (stretching) problems, the basic variables are u0 and v0 , and the classical theory in which transverse stresses are zero is without any major defect as a first approximation to the 3-D problem. This is mainly because static and thickness-wise (z-) integrated equations governing [u0 , v0 ] remain unaltered. An exception is in the error in transverse shear stress–strain relations due to εz0 from constitutive relation. A few of widely known books [1–5] on the theory of elasticity in the last century or before are from I. Todhunter (1820–1884), A. E. H. Love (1863–1940), I. S. Sokolnikoff (1901–1976), Nikoloz Muskhelishvili (1891–1976), S. P. Timoshenko (1878–1972), and so on. They contain wide coverage of analysis of primary extension problems. Because of the extensive use of C0 continuity elements in numerical methods like FEM, there are several books [vide, 6–11] now available in the market dealing with the analysis of a wide range of primary problems. However, the role of transverse stresses has not attracted much attention possibly due to the involvement of lateral displacement w(x, y, z) assumed to be insignificant in these problems. But they are important in the analysis of laminated composite plates. Higher-order theories involving transverse stresses are presented later in this chapter.

2.2 Some Observations on the Classical Theories In the classical theory of extension problems, the plate with its faces free of transverse stresses is in a state of plane stress. In-plane equilibrium equations neglecting interior distributions of transverse stresses are   α σx,x + τxy,y = 0

(2.1a)

  α σy,y + τxy,x = 0

(2.1b)

to be solved with two conditions at each of x = constant edges (with similar conditions at each of y = constant edges) in the primary problem (i) u0 = u˜ 0 (y) or σx0 (y) = Tx0 (y)

(2.2a)

2.2 Some Observations on the Classical Theories

(ii) v0 = v˜ 0 (y) or τxy0 (y) = Txy0 (y)

33

(2.2b)

2.2.1 Airy’s In-plane Stress Function Φ(x, y) (Complementary to Poisson Transverse Shear Stress Function in Chap. 1) If the edge conditions in Eqs. (2.2a, 2.2b) and similar conditions at each of y = constant edges correspond to stress components, it is convenient to express them in terms of second-order derivatives of well-known and widely used Airy’s stress function (x, y) such that Eqs. (2.1a, 2.1b) are identically satisfied. In such a case, stress components are independent of material constants, thereby, elastic deformations (Fig. 2.1). With stress components σx0 = α2 ,yy , τxy0 = −α2 ,xy , σy0 = α2 ,xx , Airy’s stress function (x, y) is governed by the bi-harmonic equation (widely known in several branches of science, see, for example, AMR Review article [12])  = 0

(2.3)

It has to be solved with two-edge conditions at each of x = constant edges (and similar conditions at each of y = constant edges) (i) ,y = Fx0 (y) or α2 ,yy = Tx0 (y)

(2.4a)

(ii) ,x = Fxy0 (y) or α2 ,xy = Txy0 (y)

(2.4b)

in which Fx0 (y), Fxy0 (y), Tx0 (y), and Txy0 (y) functions are prescribed distributions along edges of the plate (utility of these edge conditions on gradients of  needs future investigations). Each of the above set of problems may be considered as auxiliary extension problem complementary to auxiliary bending problem. Fig. 2.1 Classical theory of extension problem: [σz , τxz , τyz ] ≡ [0, 0, 0]. Airy’s stress function  (x, y). σx0 , τxy0 , σy0 =   α2 ,yy , −,xy , ,xx

34

2 Extension Problems: Higher-Order Approximations

Table 2.1 Analogy between (x, y) and (x, y) Poisson shear stress function: (x, y) along with Airy’s stress function: (x, y) with zero non-zero in-plane stresses transverse stresses Definition from the transverse equilibrium equation

Definition from in-plane equilibrium equations

τxz , τyz , σz are independent of material constants σx , σy , τxy are independent of material constants u, v are solutions of in-plane equilibrium equations

σx , σy , τxy are from the solution of compatibility condition

w is neutral plane variable

u, v are neutral plane variables

Face value of w from zero [γxz , γyz ]

Face values of [u, v] from [γxz , γyz ]

We have σz = q0 /2. (Note that a general bi-harmonic function was formulated in terms of two conjugate harmonic functions in the absence   Transverse   of σz [3].) shear stresses are τxz1 = α σx,x + τxy,y 0 ≡ 0, τyz1 = α σy,y + τxy,x 0 ≡ 0 from equilibrium equations. With known bi-harmonic  satisfying edge conditions (4), we make use of constitutive relations to determine three normal strains and in-plane shear strain:     Eεx0 = σx0 − ν σy0 + σz0 = α2 ,yy − ν,xx − ν q0 /2

(2.5a)

  Eεy0 = σy0 − ν(σx0 + σz0 ) = α2 ,xx − ν,yy − ν q0 /2

(2.5b)

  Eεz0 = σz0 − ν σx0 + σy0 = q0 /2 − ν α2n 

(2.5c)

G γxy = τxy0 = −α2 ,xy

(2.6)

 E w = E z w1 = E z εz0 = z

1 q − ν α2  2 0

 (2.7)

In-plane strains from constitutive relations are dependent on material constants, and displacements [u0 , v0 ] are determined from strain–displacement relations (Table 2.1).

2.2.2 Analysis in Terms of Displacements However, it is more convenient to analyze extension problems with displacements, particularly, in the application of numerical methods like finite element methods (FEM). Equilibrium equations corresponding to Eqs. (2.1a, 2.1b) in terms of displacements are

2.2 Some Observations on the Classical Theories

     2   1 E /3 α u0 − (1 + ν)α2 v0,x − u0,y ,y = 0 2      2   1 2 E /3 α v0 + (1 + ν)α v0,x − u0,y ,x = 0 2

35

(2.8a) (2.8b)

to be solved with edge conditions (2) and similar conditions at y = constant edges.

2.2.3 Related Bending Problem In the corresponding bending problem, in-plane displacements z[u1 , v1 ] are governed by the same set of above equations with [u1 , v1 ] replacing [u0 , v0 ]. In Kirchhoff’s theory, [u1 , v1 ] are expressed as gradients of a single function w0 (x, y) from zero transverse shear strains. As such, static Eqs. (2.1a, 2.1b) have to be ignored due to two equations governing a single variable w0 , and the required single equation through plate element equilibrium equations (PEEEs) is given by the bi-harmonic equation w0 = 0. However, the tangential edge condition is replaced by an artificial vertical stress condition. This edge condition ceases to exist if the prescribed in-plane shear stress is constant or zero all along the closed boundary of the plate. The required normal stress edge conditions to solve harmonic equations (u1 = 0, v1 = 0) are coupled in u1 and v1 . In such a case, strain energy stored in the plate is balanced through work done due to induced point loads normal to the plate at four corners of the plate. Hence, Kelvin–Tait physical interpretation of Kirchhoff vertical shear resultant is artificial through additional corner vertical load. (In fact, this corner reaction arises if τxy is zero on either side of a corner point.) As such, Poisson’s objection to Kirchhoff’s theory is justified in what is known as Poisson–Kirchhoff boundary conditions paradox, defying its proper resolution till now. This observation justifies the zero εz (instead of w) condition at these points. This problem does not arise in the extension problem due to the absence of w0 . However, it is possible to express [u0 , v0 ] in the form of gradients of a single function.

2.3 Bi-harmonic Function in Extension Problems Like in the Poisson theory proposed in Chap. 1, [u0 , v0 ] are expressed in terms of gradients of two functions ψ0 (x, y) and ϕ0 (x, y) in the form [u0 , v0 ] = α [ψ0,x + ϕ0,y , ψ0,y − ϕ0,x ]. By simple manipulations of Eqs. (2.1a, 2.1b) with derivatives of these equations, ψ0 (x, y) and ϕ0 (x, y) are governed by two uncoupled bi-harmonic equations. Since only two-edge conditions are prescribed, one obtains a solution for ψ0 with ϕ0 ≡ 0 or vice versa. Solutions thus obtained are one and the same for [u0 , v0 ] so that [u0 , v0 ] = α [ψ0,x , ψ0,y ] or α [ϕ0,y , − ϕ0,x ]. Note that the solution is a combination of that of the Laplace equation and the corresponding Poisson equation.

36

2 Extension Problems: Higher-Order Approximations

Due to the latter equation, the harmonic parts of ψ0 and ϕ0 need not be conjugate to each other. As such, the harmonic part of bi-harmonic operator function ψ1 and harmonic ϕ1 in Poisson theory in bending problems are not conjugate to each other, which is not explicitly stated in Reissner’s displacement-based theory. In fact, ϕ1 = 0 is not due to zero transverse strains assumed in Kirchhoff theory. Even if non-zero [τxz0 , τyz0 ] are prescribed at the faces of the plate, ϕ1 is still a harmonic function.

2.3.1 Sequence of Uncoupled 2-D Problems Here, the relevance of the sequence of uncoupled sets of 2-D static equations from expressing f(z) in the Fourier series of trigonometric functions is examined in both extension and bending problems. In the previous section, it is mentioned that the analysis is similar in extension and bending problems if the faces of the plate are free of transverse stresses. Such a similarity exists even in the case of prescribed stresses at the faces of the plate but slightly different in determining basic variables [u0 , v0 , w0 ]. It is known to be mandatory to satisfy thickness-wise (dz-) integrated equilibrium equations with assumed in-plane displacements [u, v] in bending problems (but advisable, and, in fact, essential in extension problems explained later) to satisfy static equations as well. This procedure can be reversed by the z-distribution of [u, v] obtained from integration of f(z) distribution of transverse shear stresses so that [u, v] are determined from static equilibrium equations. For this purpose, the possible expansions of σz in terms of cosine and sine functions in the extension and bending problems, respectively, with λn = 2/[(2n − 1)π], (sum n = 1, 2, …) are  σze = σz0e + cos(z/λn )σz2n (extension problem)

(2.9a)

 σzb = sin(z/λn )σz2n−1 (bending problem)

(2.9b)

In the extension problem, the face condition gives σz0e = q0 /2 and it is connected with other 2-D variables through constitutive relations only and σz2n need not be zero. In the bending problem, the face condition can be satisfied by any one 2-D variable σz2n−1 . The choice of σz2n−1 is dependent on the appropriate z-distribution of edge conditions. Like in the primary extension problem, relevant 2-D variables with w0 (x, y) as face variable satisfy both static and integrated equilibrium equations, thereby, in complete conformity with 3-D equations. The present Poisson theory is, however, based on σz = z q1 /2 associated with more practical reactive or prescribed transverse shear, in place of parabolic or cosine distribution, along the edges of the plate. In such a case, the above sinusoidal series becomes the expansion of z q1 /2. Note that σz0 and σz1 participate in the in-plane equilibrium equations through semi-inverted constitutive relations.

2.3 Bi-harmonic Function in Extension Problems

37

For convenience, transverse shear stresses by thickness-wise (dz-integration) of Eqs. (2.8a, 2.8b) are expressed as      τyz e =  λn sin(z/λn )  τyz 2n τxz , τxz ,        τxz , τyz b =  τxz , τyz 0b +  λn cos(z/λn )  τyz b2n−1 τxz ,

(2.10a) (2.10b)

problem used in the It is obvious that [τxz0b , τyz0b ] are from solutions   of auxiliary τxz , τyz , [u, v] apart from [u0 , v0 ] Poisson theory. By dz-integration of [u,z , v,z ] in  in the extension problems are assumed for convenience in the form [u, v]e =  λ2n [u, v]2n cos(z/λn )

(2.11a)

[u, v]b =  λ2n [u, v]2n−1 sin(z/λn )

(2.11b)

(Note that term by term differentiation of ub (or vb ) in Eq. (2.11b) is valid, whereas it is not valid in the expansion of z ub (or z vb ) in the Fourier expansion in terms of the corresponding sine   functions.) τyz ,z in the equilibrium equations are The components  τxz ,      τxz , τyz e,z =   τxz ,  τyz e2n cos(z/λn )

(2.12a)

     τyz b,z = −  τxz , τyz b2n−1 sin(z/λn ) τxz ,

(2.12b)

It is convenient to express 2-D variables in Eqs. (2.8a, 2.8b) in the following form   [u, v]n = (−1)n α ψn,x + ϕn,y , ψn,y − ϕn,x

(2.13)

    n,x , ψ n,y One gets equations governing 2-D variables with  τxz , τyz n = α ψ    1 n,x E α2 u − (1 + ν)α2 v,xy − u,yx n = αψ 2    1 n,y E α2 v + (1 + ν)α2 v,xy − u,yx n = αψ 2

(2.14a) (2.14b)

In Eqs. (2.10a, 2.10b), odd and even ‘n’ correspond to bending and extension problems, respectively. From Eqs. (2.1a, 2.1b), one gets E α2  en + σzn = 0, α2 ωzn = 0

(2.15)

Replacing en with α2 ψn , one gets bi-harmonic equation governing ψn to be solved along with harmonic function ϕn satisfying three-edge conditions.

38

2 Extension Problems: Higher-Order Approximations

Expansion of z-distribution of σz in terms of either power series or polynomials in z is artificial in the analysis based on PEEEs in the energy methods. This is due to ignoring σz = q/2 in extension problem and σz = z q/2 in the bending problem in the classical and higher-order theories. (Even the Poisson theory of bending of plates in Chap. 1 is based on neglecting σz = z q/2 to resolve Poisson–Kirchhoff boundary conditions paradox.) In bending problems, any f(z) function with f(1) = 1 can be used but restricted due to a priori prescribed practical transverse shear stresses along segments of the edge of the plate corresponding to the above-mentioned linear distribution of σz . Higher-order theories result only in the series expansion of neglected σz = z q/2 in the constitutive relations. It is clear now that the Poisson theory in conjunction with the adapted iterative procedure is the proper procedure for analysis of plates to generate a proper sequence of 2-D problems converging to a 3-D problem within the classical small deformation theory of elasticity.

2.4 Initial Solutions of Primary Extension Problems In a primary extension problem, the plate is subjected to symmetric normal stress σz0 = q0 (x, y)/2, asymmetric shear stresses [τxz1 , τyz1 ] = ± [Txz1 (x, y), Tyz1 (x, y)] along the top and bottom faces of the plate. Here, σz0 = q0 /2 satisfying face condition does not participate in equilibrium equation of transverse stresses and the corresponding  prescribed face  shears [Txz1 , Tyz1] are gradients of a given harmonic function ψ1 so    that Txz , Tyz = −α ψ1,x , ψ1,y (Fig. 2.2). shear   stresses and normal stress satisfying face conditions are  Transverse 1,x , ψ 1,y and σz0 = q0 (x, y)/2. τxz, , τyz = −α z ψ 1 remains as harmonic function even in the integrated equiIt is to be noted that ψ librium equation of transverse stresses so that the equilibrium equation of transverse 1 in normal stresses is ignored. With the inclusion of above gradients of the known ψ stresses, in-plane equilibrium Eqs. (2.1a, 2.1b) are Fig. 2.2 Primary extension problem: σz0 = q0 (x, y)/2, [τxz1 , τyz1 ] = ± [Txz1 (x, y), Tyz1 (x, y)] along z = ± 1 1 = 0 faces. 

2.4 Initial Solutions of Primary Extension Problems

     2   1 E /3 α u + μ α σz0,x − (1 + ν)α2 v0,xy − u0,yy = 0 2      2   1 2 E /3 α v + μ α σz0,y + (1 + ν)α v0,xx − u0,xy = 0 2

39

(2.16a) (2.16b)

1 /G and v = v0 + ψ 1 /G. Note that f0 σz0 satisfying prescribed in which u = u0 + ψ σz0 along faces of the plate is neglected in the classical theory but its participation in the in-plane static equilibrium equations is necessary and mandatory to rectify errors in the constitutive relations. In a way, it justifies the need for auxiliary problems in the bending problems. The above static equilibrium Eqs. (2.16a, 2.16b) along with two conditions (Eq. 2.2) at each of x = constant edges (with similar conditions at each of y = constant edges) have to be solved for u0 and v0 . They remain the same in the zintegrated equations so that they are also independent of vertical displacement which, in a way, justifies Poisson theory proposed in Chap. 1.

2.4.1 Unusual Problem: Need for Poisson Theory Solutions of the above equations concerning 3-D problems are in error in transverse shear strain–displacement relations due to w = z εz0 (which is independent 1 ). To rectify this error, one considers higher-order in-plane of harmonic function ψ displacement terms f2 (z) [u2 , v2 ] which induce (or associate with prescribed) trans1 ). Assoverse shear stresses z[τxz1 , τyz1 ] (different from stresses due to harmonic ψ ciated with f2 (z) [u2 , v2 ] is induced f2 (z) σz2 from constitutive relations. Also, these in-plane displacements induce z w1 (x, y) (other than the known z εz0 ) due to strain– displacement relations in the domain of the plate. Note that w1 (x, y) is a domain variable here, whereas w0 (x, y) is a face variable in bending problems. Here, [τxz1 , τyz1 ] are expressed in terms of gradients of two functions [ψ1 , ϕ1 ] in the following form 

     τxzl , τyz1 = α ψ1,x + ϕ1,y , ψ1,x − ϕ1,x

(2.17)

Equilibrium equation in transverse stresses gives that ϕ1 is arbitrary (equal to zero without any loss of generality) and α2 ψ1 = σz2

(2.18)

The above equation is similar to the Poisson equation but with σz2 , not a priori is known (unlike σz1 = q1 /2 in bending problem). In-plane displacements [u2 , v2 ] are related from transverse shear–strain relations and constitutive relations to [τxz1 , τyz1 ], even in the absence of induced w1 , in the

40

2 Extension Problems: Higher-Order Approximations

form of 

     τxz1 , τyz1 = −G u2 − α εz0,x , v2 − α εz0,y

(2.19)

One should note that the three Eqs. (2.18 and 2.19) govern four variables u2 , v2 , ψ1 , and σz2 . Two in-plane equilibrium equations give only two higher-order transverse shear stresses [τxz3 , τyz3 ]. As such, the problem of three equations governing the above-mentioned four variables has not drawn attention in the literature till now. This is the peculiarity of the extension problem due to the absence of prescribed σz0 in equilibrium equations. This problem is also present in the bending problem if σz1 is prescribed along one face of the plate only (for example, plate bending under uniform load along top face the plate analyzed extensively in the literature). This problem is resolved here through the Poisson theory.

2.5 Poisson Theory of Extension Problem We consider the following displacements consistent with transverse stresses [τxz1 , τyz1, σz2 ] in Eqs. (2.18, 2.19), with εz0 = – μ e0 + (1 − 2 ν μ) σz0 /E from constitutive relation, w = z(εz0 + w1 ), u = u0 + f2 u2 , v = v0 + f2 v2 , σz = f2 σz2

(2.20)

In the vertical deflection, w1 (x, y) is added to facilitate determination of [u2 , v2 ] from satisfying both static and z-integrated equilibrium equations, like in bending problems (Fig. 2.3). In extending the Poisson theory to extension problems, transverse stresses have to be, initially, independent of vertical displacement. Hence, [u2 , v2 ] are modified as [u2 , v2 ]∗ = Fig. 2.3 Poisson theory: u∗2 = u2 − α(εz0 + w1 ),x , v∗2 = v2 − α (εz0 + w1 ),y

  u2 − α(εz0 + w1 ),x , v2 − α(εz0 + w1 ),y

(2.21)

2.5 Poisson Theory of Extension Problem

41

so that transverse shear stresses from strain–displacement relations and constitutive relations are ∗    τxz1 , τyz1 = −G u2, , v2

(2.22)

Normal stress σz2 from static equilibrium equation is   ∗ = −G α u2,x + v2,y σz2

(2.23)

To keep [τxz3 , τyz3 ] as free variables in the integrated equilibrium equations, f3 (z) is modified with β1 = 1/3 as f*3 (z) = f3 (z) − β1 z so that ∗∗    τxz , τyz = z τ∗xz1 , τ∗yz1 + f3 τxz3, , τyz3

(2.24)

   with τ∗xz1 , τ∗yz1 = (τxz1 − β1 τxz3 ), τxz1 − β1 τyz3 .

∗  From static equilibrium equation of transverse stresses, α τxz1,x + τyz1,y =  ∗ ∗∗ ∗ and α τxz3,x + τyz3,y = σz4 so that σz2 = σz2 − β1 σz4 from which one gets σz2   Gα u2,x + v2,y + β1 σz4 = 0 (coefficient of z)

(2.25)

Strain–displacement relations from Eq. (2.21) give ε∗x2 = εx2 − α2 (εz0 + w1 ),xx

(2.26a)

ε∗y2 = εy2 − α2 (εz0 + w1 ),yy

(2.26b)

∗ γxy2 = γxy2 − 2α2 (εz0 + w1 ),xy

(2.26c)

Here also, [u2 , v2 ] are expressed in terms of gradients of two functions [ψ2 , ϕ2 ], like in bending problems, in the following form     [u2 , v2 ] = −α ψ2,x + ϕ2,y , ψ2,y − ϕ2,x Note that the contribution of w1 is the same as ψ2 in [u2 , v2 ]* in the integration of equilibrium equations, since contributions of f1 and f2,z are of opposite sign in strain– displacement relations, whereas the corresponding contribution of f1 and z-integrated f2,z is of the same sign. In-plane strains become   ε∗x2 = −α2 2ψ2,xx + ϕ2,yx + α2 εz0,xx

(2.27a)

42

2 Extension Problems: Higher-Order Approximations

  ε∗y2 = −α2 2ψ2,yy − ϕ2,yx + α2 εz0,yy

(2.27b)

  ∗ γxy2 = −α2 4ψ2,xy + ϕ2,xx − ϕ2,yy + 2εz0,xy

(2.27c)

The corresponding in-plane stresses are       ∗ = −E α2 2 ψ2,xx + ν ψ2,yy + εz0,xx + ν εz0,yy + (1 − ν)ϕ2,xy + μ α σz2,x σx2 (2.28a)     ∗ σy2 = −E α2 2 ψ2,yy + ν ψ2,xx + εz0,yy + ν εz0,xx − (1 − ν)ϕ2,xy + μ α σz2,y (2.28b)   τ∗xy2 = −Gα2 4ψ2,xy + 2 εz0,xy + ϕ2,xx − ϕ2,yy (2.29) From the integration of equilibrium equations, reactive transverse stresses are ∗  τ∗xz3 = α σx2,x + τxy2,y

(2.30a)

∗  τ∗yz3 = α σy2,y + τxy2,x

(2.30b)

∗  σz4 = α τxz3,x + τyz3,y (coefficient of f3 )

(2.31)

Noting that σz4 from Eq. (2.25) is negative of the one from Eq. (2.31) due to (f3 + f1 ) = 0 along the faces of the plate, equation governing in-plane displacements (u2 , v2 ) is  ∗ α β1 τxz3,x + τyz3,y = G α2  ψ2

(2.32)

along with ϕ2 = 0. After some algebra (see Appendix), the equation governing ψ2 becomes   4(1 + ν)α4  ψ2 + 3 − 2ν2 α2 ψ2 − 2 ν(1 + ν)2 α2 (σz2 /E) = 0

(2.33)

The above equation is a fourth-order equation in ψ2 to be solved along with harmonic function ϕ2 with three conditions along x = constant edges (with analog conditions along y = constant edges). ∗ = 0, (ii) v2∗ = 0 or τ∗xy2 = 0, (iii) ψ2 = 0 or τ∗xz3 = 0 (i) u2∗ = 0 or σx2

(2.34)

With reference to the solution of a 3-D problem, the above analysis in the determination of [u2 , v2 , εz2 ] is in error in transverse strain–displacement relations due to [τxz , τyz ] = f3 (z) [τxz3 , τyz3 ], and in the constitutive relations due to f4 (z) σz4 .

2.5 Poisson Theory of Extension Problem

43

2.5.1 Thick Plate Analysis Here, displacements, strains, and stresses are expressed with summation on ‘n’ (n = 1, 2, 3, …):   [w, u, v] = f2n+1 w2n+1 , f2n+2 u2n+2 , f2n+2 v2n+2

(2.35)

    εx , εy , γxy , εz = f2n+2 εx , εy , γxy , εz 2n+2

(2.36a)

    σx , σy , τxy , σz = f2n+2 σx , σy , τxy , σz 2n+2

(2.36b)

   γxz , γyz , τxz , τyz = f2n+1 γxz , γyz , τxz , τyz 2n+1

(2.37)



Three-edge conditions to be satisfied are specified in the form: (i) u = 0 or σx = 0

(2.38a)

(ii) v = 0 or τxy = 0

(2.38b)

(iii) w = 0 or τxz = 0

(2.39)

at each of x = constant edges and similar conditions at each of y = constant edges. Iterative process (n ≥ 1) We note here that w = f2n+1 (εz2n + w2n+1 ) with virtual vertical displacement w2n+1 . In-plane displacements are modified to [u*, v*]2n+2 as    ∗ ∗ u , v 2n+2 = [u, v]2n+2 − α (εz2n + w2n+1 ),x , (εz2n + w2n+1 ),y

(2.40)

so that, from strain–displacement and constitutive relations,

τ∗xz, τ∗yz

2n+1

= −G[u, v]2n+2

(2.41)

Transverse shear stresses at the nth stage of the iteration are modified as τ∗xz2n+3 = (f2n+3 − β2n+1 f2n+1 )τxz2n+3

(2.42a)

τ∗yz2n+3 = (f2n+3 − β2n+1 f2n+1 )τyz2n+3

(2.42b)

44

2 Extension Problems: Higher-Order Approximations

in which β2n+1 = [f2n+3 /f2n+1 ]z = 1 so that [τxz , τyz ]2n+3 are free variables. Then, τ∗xz2n+1 = τxz2n+1 − β2n+1 τxz2n+3

(2.43a)

τ∗yz2n+1 = τyz2n+1 − β2n+1 τyz2n+3

(2.43b)

From static equilibrium equation of transverse stresses, α [τxz2n+1 ,x + τyz2n+1 ,y ]* = σ*z2n+2 and α [τxz2n+3 ,x + τyz2n+3 ,y ] = σz2n+4 so that σ*z2n+2 = (σz2n+2 − β2n+1 σz2n+4 ) from which one gets   G α u2n+2,x + v2n+2,y + β2n+1 σz2n+4 = 0 (coefficient of f2n+1 )

(2.44)

Strain–displacement relations from Eq. (2.40) give ε∗x2n+2 = εx2n+2 − α2 (εz2n + w2n+1 ),xx

(2.45a)

ε∗y2n+2 = εy2n+2 − α2 (εz2n + w2n+1 ),yy

(2.45b)

∗ γxy2n+2 = γxy2n+2 − 2α2 (εz2n + w2n+1 ),xy

(2.45c)

Here also, [u, v]2n+2 are expressed in terms of gradients of two functions [ψ2n+2 , ϕ2n+2 ] in the form of     [u, v]2n+2 = −α ψ2n+2,x + ϕ2n+2,y , ψ2n+2,y − ϕ2n+2,x

(2.46)

Noting that the contribution of w2n+1 is the same as ψ2n+2 in [u2n+2 , v2n+2 ]* in the integration of equilibrium equations, in-plane strains become ∗      εx2n+2 , εy2n+2 = −α2 2ψ2n + 2,xx + ϕ2n+2,yx , 2ψ2n+2,yy − ϕ2n+2,xy (2.47a)   ∗ γxy2n+2 = −α2 4ψ2n+2,xy + ϕ2n+2,xx − ϕ2n+2,yy

(2.47b)

The corresponding in-plane stresses are   ∗ σx2n+2 = −E α2 2ψ2n+2,xx + ν ψ2n+2,yy + εz2n,xx + ν εz2n,yy + (1 − ν)ϕ2n+2,xy + μ σz2n

(2.48a)

  ∗ σy2n+2 = −E α2 2ψ2n+2,yy + ν ψ2n+2,xx + εz2n,yy + ν εz2n,xx + (1 − ν) ϕ2n+2,xy + μ σz2n

(2.48b) τ∗xy2n+2

  = − Gα2 4ψ2n+2,xy + 2εz2n,xy + ϕ2n+2,xx −ϕ2n+2,yy

(2.49)

From the integration of equilibrium equations, reactive transverse stresses are

2.5 Poisson Theory of Extension Problem

45

∗  τ∗xz2n+3 = α σx2n+2,x + τxy2n+2,y

(2.50a)

 ∗ τ∗yz2n+3 = α σy2n+2,y + τxy2n+2,x

(2.50b)

∗  σz2n+4 = −α τxz2n+3,x + τyz2n+3,y (coefficient of f2n+3 )

(2.51)

Noting that σz2n+4 from Eq. (2.44) is negative of the one from Eq. (2.51) due to (f2n+3 + f2n+1 )z=1 = 0, another equation governing in-plane displacements (u2n+2 , v2n+2 ) is given by  ∗ α β1 τxz2n+3,x + τyz2n+3,y = G α2  ψ2n+2

(2.52)

The equation governing ψ2n+2 , after some algebra (like in the Appendix), becomes   4(1 + ν)α4  ψ2n+2 + 3 − 2ν2 α2  ψ2n+2 − 2ν(1 + ν)2 α2 (σz2n+2 /E) = 0 (2.53) The above equation is a fourth-order equation in ψ2n+2 to be solved along with harmonic function ϕ2n+2 with three conditions along x = constant edges (with analog conditions along y = constant edges). ∗ = 0, (ii) v∗2n+2 = 0 or τ∗xy2n+2 = 0, (i) u∗2n+2 = 0 or σx2n+2 ∗ (iii) ψ2n+2 = 0 or τxz2n+3 = 0

(2.54)

With reference to the solution of the 3-D problem, the above analysis in the determination of [u2n+2 , v2n+2 , εz2n ] is in error in the transverse strain–displacement relations due to [τxz , τyz ] = f2n+3 (z) [τxz2n+3 , τyz2n+3 ], and in the constitutive relations due to f2n+4 (z) σz2n+4 . In principle, one may continue the iterative procedure until specified accuracy is achieved. However, it is not easy to develop software for the generation of f(z) functions involved for the evaluation of necessary β2n−1 to keep face shears as free variables. If one ignores classical theory, the analysis presented here with prescribed z [w, τxz , τyz ]1 becomes with induced or reactive stresses [σx , σy , τxy , σz ]2 complementary to Kirchhoff theory of bending problem. With prescribed [w, τxz , τyz ] = [w, τxz , τyz ]1 along the z = ± 1 faces, induced or reactive σz2 is parabolic from equilibrium equation of transverse stresses, whereas in-plane displacements (u, v) or corresponding stresses are induced or prescribed parabolic distributions to be determined from z-integrated equilibrium equations.

46

2 Extension Problems: Higher-Order Approximations

2.6 Conclusions It may be useful to find a relation between Airy’s stress function (x, y) and ϕ0 (x, y) in the in-plane displacements expressed in the form [u0 , v0 ] = α [ψ0,x + ϕ0,y , ψ0,y − ϕ0,x ] so as to bring in the physical significance of edge conditions (4), in particular, on gradients of (x, y) which are believed to be new ones introduced now. (Stress conditions only are used in [13]in the extensive survey in [12].) Application of Poisson theory in the analysis of extension problems is necessary and essential to find the thickness-wise distribution of transverse stresses. (It is now clear that the supplementary problems presented in earlier publications are superfluous in extension problems.) Since the prescribed σz0 (x, y) does not participate in the transverse equilibrium equation, the problem of governing equations of in-plane displacements [u0 , v0 ] from classical theory may be treated as an auxiliary problem in the application of Poisson theory for determination of transverse stresses.

Appendix • Derivation of Eq. (2.33) ∗ σx2 = −E α2 [2(ψ2,xx + νψ2,yy ) + εz0xx + ν εz0,yy + (1−ν) ϕ2,xy ] + μα σz2,x (2.28a)     ∗ σy2 = −E α2 2 ψ2,yy + ν 2,xx + εz0,yy + ν εz0,xx − (1 − ν) ϕ2,xy + μ ασz2y (2.28b)   τ∗xy2 = −Gα2 4ψ2,xy + 2 εz0,xy + ϕ2,xx − ϕ2,yy (2.29)

τ∗xy2

   1 = −(1 − ν)E α 2ψ2,xy + εz0,xy + ϕ2,xx − ϕ2,yy 2  2

(A1)

From integration of equilibrium equations, reactive transverse stresses are ∗  τ∗xz3 = α σx2,x + τxy2,y

(2.30a)

∗  τ∗yz3 = α σy2,y + τxy 2,x

(2.30b)

Appendix

47

∗  ∗  α τxz3,x + τyz3,y = α2 σx2,xx + σy2,yy + 2 τxy2,xy = −E α4 2 ψ2,xxxx + ψ2,yyyy  + 4 ν2 ψ2,xxyy + εz0,xxxx + εz0,yyyy + 2 ν εz0,xxyy +(1 − ν) ϕ2,xyxx  − ϕ2,xyyy + μ α  σz2  −2(1−ν) E α4 2ψ2,xy + εz0,xy + 21 ϕ2,xx −ϕ2,yy xy = −E α4 [2 ψ2 −4(1 − ν)ψ2,xxyy +  εz0 − 2(1−ν)εz0,xxyy + +(1−ν) (ϕ2,xyxx − ϕ2,xyyy )] + μ α2 σz2  − (1−ν) E α4 [2(2ψ2,xy + εz0,xy )xy + ϕ2,xx −ϕ2,yy xy ] (A2) = −E α4 (2ψ2 + εz0 ) + μ α2 σz2

(A3)

From Eqs. (2.30a, 2.30b), we get   β1 2E α4 ψ2 − μα2  σz2 + Gα2  ψ2 = 0

(2.32)

With β1 = 1/3, 2G = E/(1 + ν) = (1– ν) E , we get [2E α4 ψ2 − μα2 σz2 ] + 3Gα2 ψ2 = 0 E [2α4  ψ2 + (1−ν)α2  ψ2 ] − μ α2  σz2 + G α2  ψ2 = 0 

   1 2α4 ψ2 + (1 − ν)α2 ψ2 − μ 1 − ν2 α2  σz2 /E + [ (1 + ν)]α2 ψ2 = 0 2 (A4) With μ = ν/(1 − ν), we get   4 1 2α ψ2 + (1 − ν)α2 ψ2 − ν(1 + ν)α2 σz2 /E + [ (1 + ν)]α2 ψ2 = 0 2 (A5)

2(1 + ν) [2α4 ψ2 + (1 − ν)α2 ψ2 ] − 2ν(1 + ν)2 α2 σz2 /E + α2 ψ2 = 0 (A6) 4(1 + ν)α4 ψ2 + 2(1 − ν2 )α2  ψ2 ] − 2 ν(1 + ν)2 α2 (σz2 /E) + α2  ψ2 = 0 (A7) 4(1 + ν)α4 ψ2 + (3 − 2ν2 )α2  ψ2 − 2ν(1 + ν)2 α2 (σz2 /E) = 0

(2.33)

48

2 Extension Problems: Higher-Order Approximations

References 1. Todhunter I (1886) A history of the theory of elasticity and of the strength of materials: Galilei to Saint-Venant, 1639–1850, vol 2, pt 1–2. Saint-Venant to Lord Kelvin, University Press 2. Love AEH (2018) A treatise on the mathematical theory of elasticity 1944, vol 1. Courier Corporation, Technology & Engineering. https://books.google.co.in, books 2015 3. Muskhelishvili NI (Nikola˘ı Ivanovich) 1891–1976, Some basic problems of the mathematical theory of elasticity (1949). English translation, 1953, P Noordhoff, N. V, of Groningen (from page 249 of [5]). 4. Timoshenko S, Goodier JN (1969) Theory of elasticity. McGraw-Hill 5. Sokolnikoff IS (1971) Mathematical theory of elasticity. McGraw-Hill, TMH Edition, New Delhi, 476 pp 6. Boresi AP, Chong KP (2000) Elasticity in engineering mechanics. Wiley, Technology & Engineering 7. Fung Y-c, Tong P (2001) Classical and computational solid mechanics. World Scientific, Technology & Engineering 8. Bisplighoff RL, Mar JW, Pan THH (2002) Statics of deformable solids, technology & engineering 9. Hetnarski RB, Ignaczak J (2010) Mathematical theory of elasticity, 2nd edn. Taylor & Francis (Contains a good account of creators of the theory of elasticity) 10. Reddy JN (2007) Theory and analysis of elastic plates and shells, 2nd edn. CRC Press/Taylor and Francis, Boca Raton 11. Sadd MH (2010) Elasticity: theory, applications, and numerics. Elsevier, Science, 480 pp 12. Meleshko VV (2003) Selected topics in the history of the two-dimensional biharmonic problem. Appl Mech Rev 56(1):33–85 (53 pp). https://doi.org/10.1115/1.152116 13. Gregory RD (1984) The semi-infinite strip x0, −1y1; completeness of the PapkovichFadle eigenfunctions when φxy (0, y), φyy (0, y) are prescribed. J Elast 14:27–64

Chapter 3

Homogeneous Anisotropic Plates

Nomenclature a fk (z) 2h q0 (x, y) Qij Qrs Sij Srs [Tx , Txy , Txz ] [Ty , Txy , Tyz ] U, V, W [u, v, w] O-X Y Z [x, y, z]  α (αn ) β2n+1 [γxz , γyz , εz ] [εx , εy , γxy ] [σx , σy , τxy ] [τxz , τyz , σz ] ωz

side length of a square plate thickness-wise (z-) distribution functions, k = 0, 1, 2, 3, … plate thickness prescribed face load intensity stiffness coefficients, (i, j = 1, 2, 3) stiffness coefficients, (r, s = 4, 5) elastic compliances, (i, j = 1, 2, 3, 6) elastic compliances, (r, s = 4, 5) prescribed stresses at each of x = constant edges prescribed stresses at each of y = constant edges displacements in X, Y, Z directions, respectively [U, V, W]/h Cartesian coordinate system [X/a, Y/a, Z/h]   plane Laplace operator ∂ 2 /∂ x 2 + ∂ 2 /∂ y 2 hn /a [f2n+1 (1)/f2n−1 (1)] ε3+i (i = 1, 2, 3), transverse stains εi (i = 1, 2, 3), in-plane strains σi (i = 1, 2, 3), in-plane stresses σ3+i (i = 1, 2, 3), transverse stresses α (v,x − u,y ), rigid body rotation about z-axis

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 K. Vijayakumar and G. K. Ramaiah, Poisson Theory of Elastic Plates, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-33-4210-1_3

49

50

3 Homogeneous Anisotropic Plates

3.1 Introduction Anisotropy of materials is sometimes called aeolotropy in old textbooks, for example, A. E. H. Love’s Classical book, A Treatise on the Mathematical Theory of Elasticity (Dover, New York, 1944). A general three-dimensional anisotropic elastic material consists of symmetric 6 × 6 stiffness matrix [C] with 21 elastic constants in [σ] = [C] [ε] in Voigt notation. In the present monograph, analysis is confined to anisotropic materials with monoclinic symmetry consisting of 13 elastic constants. There are several books and research articles on the analysis of anisotropic elastic plates. A few of them are listed [1–11], including the latest book, Paolo Vannucci’s Lecture Notes in Applied and Computational Mechanics: Anisotropic Elasticity. Analysis of plates reported in these books is more on laminated composite plates, and the numerical study is mainly with reported data of elastic constants from experiments. In theory, however, there are bounds on elastic constants based on the principle that the overall work done by the applied forces must be positive (see Sect. 2.4, pages 48– 53 [11]). All Young’s moduli, Ei (i = 1, 2, 3) and shear moduli Gij (i, j = 1, 2, 3) are strictly positive quantities and a rougher estimate [1] of sum of Poisson ratios (ν12 +ν 23 + ν31 ) < 3/2. It is well known that ν < ½ for isotropic materials, whereas the 3 , (E1 ≤ E2 ≤ E3 ). least Poisson ratio [12, page 175] νleast ≤ 1/E1/E 1 +1/E2 Here, the analysis of homogeneous anisotropic plates is presented through the application of designated Poisson theory in the previous two chapters as a prelude to the analysis of laminated plates in the next three chapters.

3.2 Homogeneous Anisotropic Plates A square plate is bounded within 0 ≤ X, Y ≤ a, Z = ±h planes with reference to Cartesian co-ordinate system (X, Y, Z). Coordinates x = X/a, y = Y/a, z = Z/h, displacements (u, v, w) = (U, V, W)/h, and half-thickness ratio α = (h/a) in non-dimensional form are used. With the above notation, equilibrium equations in stress components are (with 3-D stress components as functions of coordinates x, y, and z)   α σx,x + τxy,y + τxz,z = 0

(3.1a)

  α σy,y + τxy,x + τyz,z = 0

(3.1b)

  α τxz,x + τyz,y + σz,z = 0

(3.2)

in which suffix after ‘,’ denotes partial derivative operator.

3.2 Homogeneous Anisotropic Plates

51

3.2.1 Stress–Strain and Strain–Displacement Relations In displacement-based models, stress components are expressed in terms of displacements, via six stress–strain constitutive relations and six strain–displacement relations. In the present study, these relations are confined to the classical small deformation theory of elasticity. Here, it is convenient to denote displacements [u, v, w] as [ui ], (i = 1, 2, 3), inplane stresses [σx , σy , τxy ], and transverse stresses [τxz , τyz , σz ] as [σi ], [σ3+i ], (i = 1, 2, 3), respectively. With the corresponding notation for strains, strain–displacement relations are   [ε1 , ε2 , ε3 ] = α u,x , v,y , u,y + v,x

(3.3)

  [ε4 , ε5 , ε6 ] = u,z + α w,x , v,z + α w,y , w,z

(3.4)

The material of the plate is homogeneous and anisotropic with monoclinic symmetry. Strain–stress relations in terms of compliances [Sij ] with the usual summation convention of repeated suffix denoting summation over specified integer values are (Fig. 3.1): εi = Sij σi (i, j = 1, 2, 3, 6)

(3.5)

εr = Srs σs (r, s = 4, 5)

(3.6)

We have from semi-inverted strain–stress relations with [Qij ] as   σi = Qij εi − Sj6 σz (i, j = 1, 2.3)

(3.7)

y = Y/a z = Z/h

x = X/a 1 1 1 Fig. 3.1 Homogeneous anisotropic plate with monoclinic symmetry (13 elastic constants) E 1 ≤ E 2 ≤ E 3 , (α1 = 1/E 1 ≥ α2 = 1/E 2 ≥ α3 = 1/E 3 ), νleast ≤ [α3 /(α1 + α2 )]

52

3 Homogeneous Anisotropic Plates

σr = Qrs εs (r, s = 4, 5)

(3.8)

Normal strain εz is given by εz = S6j σj , j = 1, 2, 3, 6.

3.2.2 Equilibrium Equations in Terms of In-plane Strains With σi in Eq. (3.7), in-plane equilibrium Eq. (3.1a, 3.1b) in terms of strains become       α Q1j εj − Sj6 σz ,x + Q3j εj − Sj6 σz ,y + τxz,z = 0

(3.9a)

      α Q2j εj − Sj6 σz ,y + Q3j εj − Sj6 σz ,x + τyz,z = 0

(3.9b)

3.2.3 Displacements [u, v, w] We use thickness-wise (z-) distribution functions fn (z), n = 0, 1, 2, 3, … generated through recurrence relations with f0 = 1, f2n+1,z = f2n , f2n+2,z = − f2n+1 such that f2n+2 (±1) = 0. They are up to n = 5

   1 1 3 1 2 z− z [f1 , f2 , f3 ] = z, 1 − z , 2 2 3    1  5 − 6z2 + z4 f4 = 24 f5 =

 1  z 25 − 10z2 + z4 120

(3.10a) (3.10b) (3.10c)

Displacements [u, v, w] are expressed in the form [u, v, w] = fn (z)[un , vn , wn ], n = 0, 1, 2, . . .

(3.11)

To keep the associated 2-D variable as a free variable, it is necessary to replace f2i+1 by f∗2i+1 , and is given by f∗2i+1 = f2i+1 − β2i−1 f2i−1 , i = 1, 2, . . . in which β2i−1 = [f2i+1 (1)/f2i−1 (1)] so that f∗2i+1 (±1) = 0.

(3.12)

3.2 Homogeneous Anisotropic Plates

53

A priori prescribed conditions along faces and edges of the plate are adjusted such that vertical deflection w(x, y, z) is even in z in bending and odd in extension problems. Correspondingly, displacements [u, v] are odd and even functions in z in bending and extension problems, respectively.

3.3 Analysis of Primary Bending Problems 3.3.1 Application of Poisson Theory In primary bending problems, the plate is subjected to asymmetric load σz = ±(q0 /2) and zero shear stresses along z = ±1 faces. Displacements [u, v, w] are expressed in the form (n = 0, 1, 2, …)   [w, u, v] = f2n w2n , f2n+1 u2n+1 , f2n+1 v2n+1

(3.13a)

    εx , εy , γxy , εz = f2n+1 εx , εy , γxy , εz 2n+1

(3.13b)

    σx , σy , τxy , σz = f2n+1 σx , σy , τxy , σz 2n+1

(3.14a)

    γxz , γyz , τxz , τyz = f2n γxz , γyz , τxz , τyz 2n

(3.14b)

(Variables associated with f(z) are functions of (x, y) only)

Here also, transverse stresses are independent of material constants as in the auxiliary problem in Chap. 1. These equations necessary for the analysis of anisotropic plates are given below. Transverse shear stresses prescribed along relevant edges are     τxz0 , τyz0 = Txz (y), Tyz (x)

(3.15)

Because of Eq. (3.2), [τxz0 , τyz0 ] are expressed as     τxz0 , τyz0 = α ψ0,x , ψ0,y

(3.16)

so that ψ0 (x, y) with σz = z q0 /2 is governed by 1 α2 ψ0 + q0 = 0 2

(3.17)

to be solved with edge condition either ψ0 = 0 or its normal gradient equal to prescribed transverse shear stress along the edge. They are more realistic edge conditions and more practical if they are independent of in-plane coordinates. Assumed face stresses [Txz (y), Tyz (x)] correspond to [τxz0 , τyz0 ] in Eq. (3.16) consistent with

54

3 Homogeneous Anisotropic Plates

prescribed or reactive shears along the edges. Moreover, the function ψ0 is related to normal strain εz , and ψ0 = 0 implies εz = 0. With displacements [w0 , z u1 , z v1 ] and strain–displacement relations, w0f is given by α w0f (x, y) =

  γxz0 dx + γyz0 dy z=1 −



  u1 dx + v1 dy z=1

(3.18)

Since w0 (x, y) is from the satisfaction of zero face shear conditions, it is considered as face deflection, though it is the same for all face parallel planes. It is analytic in the domain of the plate if ωz = 0. In such a case, w0 is a domain variable in transverse shear–stress relations, and derivatives [ε1 ,y , ε2 ,x , ε3,y , ε3,x ] from strain–displacement relations become independent of cross derivatives given below:     ε1,y , ε2,x , ε3,y , ε3,x = α2 v,xx , u,yy , 2u,yy , 2v,xx Static equilibrium equations have to be independent of w0 (x, y) in bending problem to decouple from the torsion problem. For this purpose, [u1 , v1 ] are modified, with εr = Srs σs (r, s = 4, 5), as   u∗1 = u1 + ε4 − α w0,x

(3.19a)

  v∗1 = v1 + ε5 − α w0,y

(3.19b)

(Inclusion of transverse shear strains is necessary for finding face deflection w0f)

The introduction of w0 term in the above equations is to avoid its participation in the static in-plane equations. Note that [τxz0 , τyz0 ] through transverse shear strains εr (r = 4, 5) in Eq. (3.19a, 3.19b) are included due to the participation of σz1 in Eq. (3.9a, 3.9b). With displacements [u, v] = z [u1 , v1 ], determination of [u, v] satisfying equilibrium equations requires f2 [τxz2 , τyz2 ] along with σz = f3 σz3 . Since f3 (1) = 0, σz3 becomes a free variable by replacing f3 (z) with f*3 (z). Normal stress σz along with (z q1 /2) in the Poisson theory takes the form   1 σz = z q1 − β1 σz3 + f3 σz3 2

(3.20)

Transverse shear stresses along with those in the auxiliary problem are 

     τxz , τyz = τxz , τyz 0 + f2 (z) τxz , τyz 2

(3.21)

τxz2 = Q44 u1 + Q45 v1 + τxz0

(3.22a)

in which

3.3 Analysis of Primary Bending Problems

τyz2 = Q55 v1 + Q54 u1 + τyz0

55

(3.22b)

Note that [τxz0 , τyz0 ] in Eq. (3.21) are included due to the participation of σz1 in Eq. (3.1a, 3.1b). From Eqs. (3.1a, 3.1b, 3.2, 3.20, 3.22a, 3.22b), one obtains   α (Q44 u1 + Q45 v1 ),x + (Q54 u1 + Q55 v1 ),y = β1 σz3

(3.23)

(σ z3 is from the coefficient of f 1 . Note that one cannot prescribe zero τ xz2 (and τ yz2 ) along x (and y) constant edges since τ xz0 (and τ yz0 ) are independent of elastic deformations)

For the use of [u* , v* ]1 in the integration of equilibrium Eqs. (3.1a, 3.1b), it is convenient to express [u1 , v1 ] in terms of gradients of two functions [ψ, ϕ] in the form   [u1 , v1 ] = −α ψ1,x + ϕ1,y , ψ1,y − ϕ1,x

(3.24)

Contributions of ψ1 and w0 in [u* , v* ]1 are the same in giving corrections to w(x, y, z) and transverse stresses. Hence, w0 in [u* , v* ]1 is replaced by ψ1 so that [u* , v* ]1 are   u∗1 = −α 2ψ1,x + ϕ1,y + γxz0

(3.25a)

  v∗1 = −α 2ψ1,y − ϕ1,x + γyz0

(3.25b)

Correspondingly, in-plane strains [ε*x , ε*y , γ*xy ]1 with [εx1 , εy1 , γxy1 ] = −α2 [(2ψ1,xx + ϕ1,xy ), (2ψ1,yy − ϕ1,xy ), (4ψ1,xy + ϕ1,yy − ϕ1,xx )] are given by       ε∗x , ε∗y = εx1 + α γxz0,x , εy1 + α γxz0,y

(3.26a)

   ∗ γxy1 = γxyl + α γxz0,y + γyz0,x

(3.26b)

1

From integration of Eq. (3.2) and equilibrium Eq. (3.9a, 3.9b) using the strains in Eq. (3.26a, 3.26b), reactive transverse stresses are (with sum j = 1, 2, 3)       τ∗xz2 = α Q1j εj − Sj6 σz1 ,x + Q3j εj − Sj6 σz1 ,y

(3.27a)

      τ∗yz2 = α Q2j εj − Sj6 σz1 ,y + Q3j εj − Sj6 σz1 ,x

(3.27b)



σz3 = −α τxz2,x + τ∗yz2,y

(3.28)

Noting that σz3 (coefficient of f1 ) from Eq. (3.23) is negative of σz3 (coefficent of f3 ) from Eq. (3.28) due to (f3,zz + f1 ) = 0, the equation governing in-plane displacements is given by

56

3 Homogeneous Anisotropic Plates

α β1 (τ∗xz2,x + τ∗yz2,y ) = α[(Q44 u1 + Q45 v1 ),x + (Q54 u1 + Q55 v1 ),y ]

(3.29)

With the condition zero ωz (i.e., v,x = u,y ) required to decouple bending and torsion, Eq. (3.29) consists of Laplace equation ϕ1 = 0 and a fourth-order equation in ψ1 to be solved with the following three conditions at x = constant edges (with similar conditions at y = constant edges) 

  (i) u∗ or σ∗ 1 = 0, (ii) v∗ or τ∗xy = 0

(3.30a)

(iii) ψ1 or τ∗xz2 = 0

(3.30b)

1

• Corrective in-plane displacements: Transverse stresses in the plate are 

     τxz , τyz = τxz0 , τyz0 + f2 τxz2 , τyz2

(3.31)

σz = z σz1 + f3 σz3

(3.32)

Corrective in-plane displacements in the supplementary problem are assumed as [u, v]s = [u1 , v1 ]s sin(π z/2)

(3.33)

Correspondingly, in-plane stresses are σis = Qij ε1sj sin(π z/2) (i, j = 1, 2, 3)

(3.34)

From z-integration of equilibrium equations using in-plane stresses in Eq. (3.34) along with [τxz , τyz ] = [τxz2 , τyz2 ]s cos (π z/2) and σz3 = σz3s sin (π z/2), transverse stresses in the supplementary problem are given by   τxz2s = −(2/π)α Q1j ε1sj,x + Q3j ε1sj,y

(3.35a)

  τyz2s = −(2/π)α Q2j ε1sj,y + Q3j ε1sj,x

(3.35b)

  σz3s = (2/π)2 α2 Q1j ε1sj,xx + 2Q3j ε1sj,xy + Q2j ε1sj,yy

(3.36)

In-plane distributions u1s and v1s are added as corrections to the known in-plane displacements [u1 , v1 ] so that [u, v] in the supplementary problem are [u, v] = [(u1 + u1s ), (v1 + v1s )] sin(π z/2) From Eqs. (3.29, 3.35a, 3.35b, 3.36), one gets

(3.37)

3.3 Analysis of Primary Bending Problems

  β1 σz3 = (2/π)2 α2 Q1j ε1sj,xx + 2Q3j ε1sj,xy + Q2j ε1sj,yy

57

(3.38)

By expressing [u1s , v1s ] = − α [ψ1s,x , (ψ1s,y ], Eq. (3.38) becomes a fourth-order equation in ψ1s to be solved with two in-plane conditions at x = constant edges (with similar conditions at y = constant edges).   (i) (u1s or σxs1 ) = 0, (ii) v1s or τxys1 = 0.

(3.39)

3.3.2 Iterative Procedure The above solutions for displacements and transverse stresses are initial solutions in the iterative procedure in solving 3-D problems to generate a proper sequence of sets of 2-D equations. The only error concerning 3-D problems is in the transverse shear strain–displacement relations. Displacements f3 [u3 , v3 ] (thereby, εz3 ) consistent with f2 [τxz2 , τyz2 ] and reactive transverse stresses (τxz4 , τyz4 , σz5 ) have to be obtained from the first stage of the iterative procedure. Initial transverse shear strains from constitutive relations are γxz2 = S44 τxz2 + S45 τyz2

(3.40a)

γyz2 = S54 τxz2 + S55 τyz2

(3.40b)

Displacements [u3 , v3 ] are modified such that they are corrections to face parallel plane distributions of the preliminary solution so that w = f2 (z)(w2 − εz1 )

(3.41)

u∗3 = u3 + γxz2 − α(w2 − εz1 ),x

(3.42a)

v∗3 = v3 + γyz2 − α(w2 − εz1 ),y

(3.42b)

in which [u, v]3 = [u, v]1 + [u, v]3c with [u, v]3c denoting corrections due to transverse shear strain–displacement relations. Due to replacement of f5 by f*5 , one gets after some algebra     αβ3 τxz4,x + τyz4,y = α2 (Q44 u3 + Q45 v3 ),xx + (Q54 u3 + Q55 v3 ),yy + α2 σz1 (3.43)

58

3 Homogeneous Anisotropic Plates

With the condition zero ωz (v,x = u,y ) necessary for the analysis of bending problem, Eq. (3.43) consists of Laplace equation ϕ3 = 0 and a fourth-order equation in ψ3 to be solved with conditions at x = constant edges (with similar conditions at y = constant edges) (i) u∗3 or σ3∗ = 0, (ii) v3 or τ∗xy3 = 0, (iii) ψ3 or τ∗xz4 = 0

(3.44)

Corrective Displacements: Corrective displacements in the supplementary problems are assumed in the following form: w = w2s (π/2) cos(π z/2)

(3.45)

[us , vs ] = [u3s , v3s ] sin(πz/2)

(3.46)

σ3si = Qij ε3sj (i, j = 1, 2, 3)

(3.47)

Analysis here is a repetition of the corresponding analysis in the supplementary problem in the previous section. Necessary equations are given as follows:   τxz2s = −(2/π)α Q1j ε3sj,x + Q3j ε3sj,y

(3.48a)

  τyz2s = −(2/π)α Q2j ε3sj,y + Q3j ε3sj,x

(3.48b)

 ∗    u3 + u3s , v∗3 + v3s sin(πz/2)

[u, v] =

  β3 σz5 = (2/π)2 α2 Q1j ε∗sj,xx + 2Q3j ε∗sj,xy + Q2j ε∗sj,yy

(3.49)

(3)

(3.50)

Corrective displacements are w = w04 (x, y)−f4 εz3 + w04s (π/2) cos(πz/2)

(3.51)

  u = f3 u∗3 + u∗3 + u3s sin(πz/2)

(3.52a)

  v = f3 v∗3 + v∗3 + v3s sin(πz/2)

(3.52b)

α w04 = α w04s =

    (γxz2 −u3 )dx + γyz2 −v3 dy

(3.53)

    (γxz2s −u3s )dx + γyz2s −v3s dy

(3.54)

3.3 Analysis of Primary Bending Problems

59

One may add f4 term in shear components and f5 term in σz . These components are also dependent on material constants, but they need correction from the solution of a supplementary problem. The successive application of the previous iterative procedure leads to the solution of the 3-D problem in the limit.

3.3.3 Summary of Results from the First Stage of Iterative Procedure w = w0 −[f2 εz1 + f4 εz4 ]−εz3s (π/2) cos(πz/2)

(3.55a)

  u = f1 u1 + f3 u∗3 + u∗3 + u3s sin(πz/2)

(3.55b)

  v = f1 v1 + f3 v∗3 + v∗3 + v3s sin(πz/2)

(3.55c)

τxz = τxz0 + f2 τxz2 + τxzs2 (π/2) cos(πz/2)

(3.56a)

τyz = τyz0 + f2 τyz2 + τyzs2 (π/2) cos(πz/2)

(3.56b)

σz = z σz1 + f∗3 σz3

(3.56c)

3.4 Associated Torsion Problems Kirchhoff’s theory and FSDT due to their inherent deficiencies are not suitable for the analysis of primary bending problems, but w0 (x, y) as domain variable is required in the analysis of associated torsion problems. In the analysis, it is proper and convenient to include transverse shear stresses [τxz0 , τyz0 ] and σz = z q0 /2 used in the Poisson’s theory. Here, the analysis requires only modification of the analysis of bending problem replacing ψ1 with vertical displacement w0 (x, y) so that w = w0 (x, y)     [u, v] = −αz w0,x + ϕ1,y , w0,y − ϕ1,x

(3.57)

τxz = τxz0 + f2 (τxz0 + τxz2 )

(3.58a)

60

3 Homogeneous Anisotropic Plates

  τyz = τyz0 + f2 τyz0 + τyz2 σz3 =

1 z q + f∗3 σz3 2

(3.58b) (3.59)

Note that the transverse shear strains from strain–displacement relations are [−α ϕ1,y , α ϕ1,x ] which correspond to self-equating stresses. From zero face shear conditions, one gets an additional face deflection w0c (x, y) given by w0c =

  ϕ1,y dx − ϕ1,x dy

(3.60)

The w0c is a face variable that becomes zero with prescribed zero w0 at the edge of the plate by preventing vertical movement of the intersection of the face with the edge (sidewall) of the plate.

3.5 Extension Problems In a primary extension problem, the plate is subjected to symmetric normal stress σzo = q0 (x, y)/2, asymmetric shear stresses [τxz , τyz ] = ± [Txz (x, y), Tyz (x, y)] at the faces of the plate. Due to out-of-plane equilibrium equation, the prescribed face  shears [Txz , Tyz ]  1,x , ψ 1,y ]. 1 so that Txz , Tyz = −α[ψ have to be gradients of a harmonic function ψ Transverse shear stresses and normal stress satisfying face conditions are   1,x , ψ 1,y ], σz0 = 1 q0 (x, y) τxz , τyz = −α z [ψ 2

(3.61)

The above transverse stresses are independent of material constants and remain the same within the plate. One should note here that σz0 does not participate in the equilibrium equations but contributes to the in-plane constitutive relations. [τxz , τyz ] in the above equation are related to in-plane displacements [u0 , v0 ] through equilibrium Eq. (3.1a, 3.1b). From constitutive relation, 1 εz0 = S6j σj0 + S66 q0 (j = 1, 2, 3) 2

(3.62)

Correspondingly, the vertical deflection w is linear in z and cannot be prescribed to be zero along the edge of the plate due to S6j σj0 even if the faces are free of transverse stresses.

3.5 Extension Problems

61

3.5.1 Preliminary Analysis In-plane equilibrium equations in the preliminary analysis with [u, v] = [u0 (x, y), v0 (x, y)] are       α Q1j εj0 −Sj6 σz0 ,x +Q3j εj0 −Sj6 σz0 ,y = αψ1,x

(3.63a)

      α Q2j εj0 −Sj6 σz0 ,y +Q3j εj0 −Sj6 σz0 ,x = αψ1,y

(3.63b)

subjected to the edge conditions (3.15) along x (and y) constant edges. We recall ‘Unusual problem: Need for Poisson theory’ which was discussed and presented the Poisson theory for extension problems in Chap. 2. The same Poisson theory is extended here to rectify errors in transverse shear strain–displacement relations due to w = z εz0 . We consider higher-order in-plane displacement terms f2 (z) [u2 , v2 ] which induce transverse shear stresses z [τxz1 , τyz1 ], f2 (z) σz2 from constitutive relations, and z w1 (x, y) (other than the known z εz0 ) due to strain–displacement relations in the domain of the plate. In-plane displacements [u2 , v2 ] are related from transverse shear–strain relations and constitutive relations to [τxz1 , τyz1 ], even in the absence of induced w1 , as   τxz1 = −[Q44 (u2 −αεz0,x ) + Q45 v2 −αεz0,y ]

(3.64a)

  τyz1 = −[Q55 (v2 −αεz0,y ) + Q54 u2 −αεz0,x ]

(3.64b)

We consider the following displacements consistent with transverse stresses [τxz1 , τyz1, σz2 ], with εz = S6j σj , j = 1, 2, 3, 6 from constitutive relation, w = z(εz0 + w1 ), u = u0 + f2 u2 , v = v0 + f2 v2 , σz = f2 σz2

(3.65)

In the vertical deflection w, w1 (x, y) is added to facilitate determination of [u2 , v2 ] from satisfying both static and z-integrated equilibrium equations. In extending Poisson theory to extension problems, transverse stresses have to be independent of vertical displacement. Hence, [u2 , v2 ] are modified as [u2 , v2 ]∗ =

  u2 − α(εz0 + w1 ),x , v2 − α(εz0 + w1 ),y

(3.66)

so that transverse shear stresses from strain–displacement relations and constitutive relations are [τxz1 , τyz1 ]∗ = −[(Q44 u2 + Q45 v2 ), (Q55 v2 + Q54 u2 )] Normal stress σz2 from static equilibrium equation is

(3.67)

62

3 Homogeneous Anisotropic Plates ∗ σz2 = −α[(Q44 u2 + Q45 v2 ),x +(Q55 v2 + Q54 u2 ),y ]

(3.68)

To keep [τxz3 , τyz3 ] as free variables in the integrated equilibrium equations, f3 (z) is modified with β1 = 1/3 as f*3 (z) = f3 (z) − β1 z so that    ∗∗ τxz , τyz = z τ∗xz1 , τ∗yz1 + f3 [τxz3 , τyz3 ]

(3.69)

with [τ* xz1 , τ* yz1 ] = [(τxz1 − β1 τxz3 ), (τyz1 − β1 τyz3 )]. From static equilibrium equation of transverse stresses, α [τxz1 ,x + τyz1 ,y ]* = σ* z2 and α [τxz3 ,x + τyz3 ,y] = σz4 so that σ** z2 = σ* z2 − β1 σz4 from which one gets α[(Q44 u2 + Q45 v2 ),x +(Q55 v2 + Q54 u2 ),y ] + β1 σz4 = 0 (coefficient of z) (3.70) Strain–displacement relations from Eq. (3.66) give ε∗x2 = εx2 −α2 (εz0 + w1 ),xx

(3.71a)

ε∗y2 = εy2 −α2 (εz0 + w1 ),yy

(3.71b)

∗ γxy2 = γxy2 −2α2 (εz0 + w1 ),xy

(3.71c)

Here also, [u2 , v2 ] are expressed in terms of gradients of two functions [ψ2 , ϕ2 ], like in bending problems, in the form     [u2 , v2 ] = −α ψ2,x + ϕ2,y , ψ2,y − ϕ2,x Note that the contribution of w1 is the same as ψ2 in [u2 , v2 ]* in the integration of equilibrium equations since contributions of f1 and f2,z are of opposite sign in strain– displacement relations, whereas the corresponding contribution of f1 and z-integrated f2,z are of the same sign. In-plane strains become, with ε* i2 (i = 1, 2, 3) denoted by ε* x2 , ε* y2 , γ* xy2 , respectively,   ε∗x2 = −α2 2ψ2,xx + ϕ2,yx + α2 εz0,xx

(3.72a)

  ε∗y2 = −α2 2ψ2,yy − ϕ2,yx + α2 εz0,yy

(3.72b)

  ∗ γxy2 = −α2 4ψ2,xy + ϕ2,xx −ϕ2,yy + 2εz0,xy

(3.72c)

The corresponding in-plane stresses are

3.5 Extension Problems

63

 ∗ σi2 = Qij ε∗j2 − S6j σz0 (i, j = 1, 2, 3)

(3.73)

From the integration of equilibrium equations, the reactive transverse stresses are ∗  τ∗xz3 = α σ1,x + σ3,y 2

(3.74a)

∗  τ∗yz3 = α σ2,y + σ3,x 2

(3.74b)

∗  σz4 = α τxz3,x + τyz3,y (coefficient of f3 )

(3.75)

Noting that σz4 from Eq. (3.70) is negative of the one from Eq. (3.75) due to (f3 + f1 ) = 0 at the faces of the plate, the equation governing in-plane displacements (u2 , v2 ) is given by  ∗ αβ1 τxz3,x + τyz3,y = α[(Q44 u2 + Q45 v2 ),x +(Q55 v2 + Q45 u2 ),y ]

(3.76)

The above equation is a fourth-order equation in ψ2 to be solved along with harmonic function ϕ2 with three conditions at x = constant edges (with similar conditions at y = constant edges). ∗ =0 (i) u∗2 = 0 or σx2

(3.77a)

(ii) v∗2 = 0 or τ∗xy2 = 0

(3.77b)

(iii) ψ2 = 0 or τ∗xz3 = 0

(3.77c)

Concerning solution of a 3-D problem, the above analysis in the determination of [u2 , v2 , εz2 ] is in error in the transverse strain–displacement relations due to [τxz , τyz ] = f3 (z) [τxz3 , τyz3 ], and in the constitutive relations due to f4 (z) σz4 . With prescribed [w, τxz , τyz ] = ± [w, τxz , τyz ]1 along z = ± 1 faces, induced or reactive σz2 is parabolic from equilibrium equation of transverse stresses, whereas in-plane displacements (u, v) or corresponding stresses are induced or prescribed parabolic distributions to be determined from z-integrated equilibrium equations.

3.5.2 Higher-Order Corrections τ∗xz2n+1 = (τxz2n+1 −β2n−1 τxz2n−1 )

(3.78a)

  τ∗yz2n+1 = τyz2n+1 −β2n−1 τyz2n−1

(3.78b)

64

3 Homogeneous Anisotropic Plates ∗ σz2n+2 = σz2n+2 −β2n−1 σz2n

(3.79)

At the nth stage of iteration (n ≥ 1), transverse stresses [τxz , τyz ]2n−1 , and w2n−1 are known in the preceding stage. Concerning in-plane displacements, one should include additional terms such that they are consistent with known stresses [τxz , τyz ]2n−1 and are free to obtain stresses [τxz , τyz ]2n+1 , σz2n+2 , and w2n+1 . We have from constitutive relations, γxz2n−1 = S44 τxz2n−1 + S45 τyz2n−1

(3.80a)

γyz2n−1 = S45 τxz2n−1 + S55 τyz2n−1

(3.80b)

The modified displacements and the corresponding derived quantities denoted with * are with w2n−1 as correction to εz2n−2 due to [u, v]2n u∗2n = u2n −α(εz2n−2 + w2n−1 ),x + γxz2n−1

(3.81a)

v∗2n = v2n −α(εz2n−2 + w2n−1 ),y + γyz2n−1

(3.81b)

Strain–displacement relations give ε∗x2n = εx2n − α2 (εz2n−2 + w2n−1 ),xx + αγxz2n−1,x

(3.82a)

ε∗y2n = εy2n − α2 (εz2n−2 + w2n−1 ),yy + αγyz2n−1,y

(3.82b)

  ∗ γxy2n = γxy2n − 2α2 (εz2n−2 + w2n−1 ),xy + α γxz,y + γyz,x 2n−1

(3.82c)

∗ γxz2n−1 = γxz2n−1 −(u2n−2 + u2n )

(3.83a)

∗ = γyz2n−1 −(v2n−2 + v2n ) γyz2n−1

(3.83b)

In-plane stresses and transverse shear stresses from constitutive relations are    ∗ σi 2n = Qij ε∗j (i, j = 1, 2, 3)

(3.84)

τ∗xz2n−1 = τxz2n−1 −(Q44 u + Q45 v)2n

(3.85a)

τ∗yz2n−1 = τyz2n−1 −(Q54 u + Q55 v)2n

(3.85b)

2n

3.5 Extension Problems

65

One gets from Eqs. (3.1a, 3.1b, 3.2, 3.78a, 3.78b, 3.79) noting that σ* z2n = σz2n – β2n−1 σz2n+2   α (Q44 u2n + Q45 v2n ),x +(Q54 u2n + Q55 v2n ),y + β2n−1 σz2n+2 = 0

(3.86)

(Note that w2n-1 is not present in the above equation)

For the use of [u* , v* ]2n in the integration of equilibrium equations, displacements [u, v]2n are expressed in the form   [u, v]2n = −α ψ2n,x , ψ2n,y

(3.87)

The contributions of ψ2n and w2n−1 in [u* , v* ]2n are the same in giving corrections to w(x, y, z) and transverse stresses. (In fact, the contribution of w2n−1 is through strain–displacement relations in static equilibrium equations, and through constitutive relations in through-thickness integration of equilibrium equations.) Hence, w2n−1 in [u* , v* ]2n is replaced by ψ2n (to be independent of w2n−1 used in strain–displacement relations) so that [u* , v* , ε* x , ε* y , γ* xy ]2n are   u∗2n = 2u2n + γxz2n−1 −αεz2n−2,x

(3.88a)

  v∗2n = 2v2n + γyz2n−1 −αεz2n−2,y

(3.88b)

  ε∗x2n = 2εx2n + αγxz2n−1,x −α2 εz2n−2,xx

(3.89a)

  ε∗y2n = 2εy2n + αγyz2n−1,y −α2 εz2n−2,yy

(3.89b)

    ∗ γxy2n = 2γxy2n + α γxz2n−1,y + γyz2n−1,x − 2α2 εz2n−2,xy

(3.89c)

Note that the role of w2n−1 is in its contribution to the integrated equilibrium equations. From the integration of equilibrium equations using the strains in Eq. (3.88a, 3.88b), reactive transverse stresses are τ∗xz2n+1

 

  ∗ ∗ = α Q1j εj −Sj6 σz + Q3j εj −Sj6 σz , (j = 1, 2, 3)

(3.90a)

τ∗yz2n+1

 

  ∗ ∗ = α Q2j εj −Sj6 σz + Q3j εj −Sj6 σz , (j = 1, 2, 3)

(3.90b)

,x

,y

,y 2n

,x 2n



σz2n+2 = −α τ∗xz,x + τ∗yz,y

2n+1

(3.91)

66

3 Homogeneous Anisotropic Plates

One equation governing in-plane displacements (u, v)2n , noting that σz2n+2 from Eq. (3.86) is negative of the one from Eq. (3.91) due to (f2n+1,zz + f2n−1 ) = 0, is given by

 αβ2n−1 τ∗xz,x + τ∗yz,y

2n+1

  = α (Q44 u + Q45 v),x + (Q54 u + Q55 v),y 2n

(3.92)

With the second equation v2n,x = u2n,y , the above equation becomes a fourth-order equation in ψ2n to be solved along with harmonic function ϕ2n with three conditions along constant x = constant edges (with analogous conditions along y = constant edge) ∗  (i) (u2n or σ2n )∗ = 0, (ii) v2n or τxy2n = 0, (iii) τ∗xz2n+1 = 0

(3.93)

In principle, one may continue the iterative procedure until specified accuracy is achieved. However, it is not easy to develop software for the generation of f(z) functions involved in the evaluation of necessary β2n−1 to keep face shears as free variables.

3.6 Conclusions During the analysis of elastic plates within the small deformation theory of elasticity, the proposed Poisson theory is necessary and essential to generate a proper sequence of sets of 2-D problems even for monosymmetric anisotropic plates in bending, associated torsion, and extension problems which are mutually exclusive to one other in the analysis. Vertical displacement in the face and neutral planes are different in bending problems. There is no associated torsion problem in extension problems and no supplementary problems in both associated torsion and extension problems. It is, however, not easy to develop software for the generation of fn (z) and β2n+1 .

References 1. Lekhnitski˘ı SG (1963) Theory of elasticity of an anisotropic elastic body. Holden-Day, 404 pp 2. Ambartsumian SA (1970) Theory of anisotropic plates: strength, stability, vibration. Technomic Pub. Co., Anisotropy, 248 pp 3. Tarn J-Q (2002) A state space formalism for anisotropic elasticity. Part II: Cylindrical anisotropy. Int J Solids Struct 39:5157–5172. www.elsevier.com/ijsolstr 4. Rand O, Rovenski V (2007) Analytical methods in anisotropic elasticity: with symbolic computational tools. Springer Science & Business Media, 451 pp 5. Barber JR, Tin TCT (2007) Three-dimensional solutions for general anisotropy. J Mech Phys Solids 55:1993–2006

References

67

6. Ou Z-C, Chen Y-H (2007) General solution of the stress potential function in Lekhnitski˘ı’s elastic theory for anisotropic and piezoelectic materials. Adv Stud Theor Phys 1(8):357–366 7. Sadowski T, de Borst R (2008) Lecture notes on composite materials: current topics and achievements. Springer Science & Business Media, 237 pp 8. Hwu C (2010) Anisotropic elastic plates. Springer Science & Business Media, 673 pp 9. Vashakmadze TS (2013) The theory of anisotropic elastic plates. Springer Science & Business Media, 243 pp 10. Aghalovyan LA (2015) Asymptotic theory of anisotropic plates and shells. World Scientific, 376 pp 11. Vannucci P (2017) Lecture notes in applied and computational mechanics: anisotropic elasticity. Springer. ISBN-10981105438x ISBN-13978981105 4389 85, 426 pp 12. Dhawan S (1967) Twenty-second british commonwealth lecture: aeronautical research in India. J Roy Aeronaut Soc 71(675):149–184

Chapter 4

Laminated Plates with Anisotropic Plies Part I: Single-Layer Theories

Nomenclature a fn (z) fn (k) (z) 2h n O-X Y Z Qij Qrs q0 q1 Sij Srs [Tx , Txy , Txz ] [Ty , Txy , Tyz ] tk U, V, W [u, v, w] [x, y, z]  α (αn ) αk β2n-1 β2n-1 (k) [γxz , γyz , εz ] [εx , εy , γxy ] [σx , σy , τxy ] [τxz , τyz , σz ] ωz

side length of a square plate thickness-wise (z-) distribution functions, n = 0, 1, 2, 3, … ply-wise distribution of fn (z) plate thickness number of plies Cartesian coordinate system stiffness coefficients, (i, j = 1, 2, 3) stiffness coefficients, (r, s = 4, 5) prescribed face load intensity in extension problem prescribed face load intensity in bending problem elastic compliances, (i, j = 1, 2, 3, 6) elastic compliances, (r, s = 4, 5) prescribed stresses at each of x = constant edges prescribed stresses at each of y = constant edges (αk − αk-1 ), thickness of kth ply displacements in X, Y, Z directions, respectively [U, V, W]/h [X/a, Y/a, Z/h] plane Laplace operator (∂ 2 /∂x2 + ∂ 2 /∂y2 ) hn /a hk /a, upper-face of kth ply in the upper-half of the laminate [f2n+1 /[α 2 f2n−1 ]z=1 , n = 1, 2,… [f2n+1 /[αk2 f2n−1 ]z=αk ε3+i , (i = 1, 2, 3), transverse stains εi , (i = 1, 2, 3), in-plane strains σi , (i = 1, 2, 3), in-plane stresses σ3+i , (i = 1, 2, 3), transverse stresses α (v,x − u,y ), rigid body rotation about z-axis

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 K. Vijayakumar and G. K. Ramaiah, Poisson Theory of Elastic Plates, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-33-4210-1_4

69

70

4 Laminated Plates with Anisotropic Plies …

4.1 Introduction • Brief Introduction The use of fiber-reinforced composites in engineering structures has become quite common over the past decade. Investigations on analysis of laminated composite plates are initially based (see, e.g. [1]) on adapting classical plate theories modifying stiffness parameters due to ply-wise variation of Hooke’s constitutive materials. They are designated as modified homogeneous plate theories (MHPT) in the analysis of symmetric laminates. In the case of un-symmetrical laminates, they are referred to as classical laminated plate theories (CLPT) [2]. Higher-order homogeneous plate theories applied to the analysis of laminated plates are designated as single-layer theories (SLT) [3]. Models based on a priori derived lamination-dependent thicknesswise displacement distributions as coefficients of mid-plane variables are appropriately classified as smeared laminate models (SLM) [4–8]. Deformation profiles enriched with through-the-thickness warping effects by appropriate zig-zag functions in single-layer theories (SLT) are termed as equivalent single-layer models (ESLD) [9–11]. The Poisson theory is applied in the present chapter concerning basic single-layer theories.

4.2 Geometry and Ply-Wise Constitutive Laws of Anisotropic Material A square laminate bounded by 0 ≤ X, Y ≤ a, –hn ≤ Z ≤ hn with interfaces Z = hk in the Cartesian coordinate system (X, Y, Z) is considered. Coordinates X, Y, and Z and displacements U, V, and W in non-dimensional form [x, y] = [X, Y]/a, z = Z/hn , [u, v, w] = [U, V, W]/hn , and α = hn /a (half-thickness ratio) are utilized. The material of each ply is homogeneous and anisotropic with monoclinic symmetry. In the plate, z ≥ 0, upper-face of each ply is given by z = αk = hk /a. In face parallel planes, displacements [u, v, w] = [ui ], in-plane stresses [σx , σy , τxy ] = [σi ], transverse stresses [τxz , τyz , σz ] = [σ3+ i ], (i = 1, 2, 3) are used, for simplicity, wherever necessary. With corresponding notation for strains, strain–stress relations in each ply are:  k εi = Sij σj (i, j = 1, 2, 3, 6), (k = 1, 2, . . . n)

(4.1)

εr = [Srs ]k σs (r, s = 4, 5), (k = 1, 2, . . . n)

(4.2)

in which superscript ‘k’ denotes kth ply and usual summation convention of repeated suffix denoting summation over specified integer values (Fig. 4.1). We have from semi-inverted strain–stress relations in kth ply as

4.2 Geometry and Ply-Wise Constitutive Laws of Anisotropic Material

71

Fig. 4.1 Laminated plate with anisotropic plies: Material in each ply is homogeneous with mono3 clinic symmetry (13 elastic constants). Least Poisson ratio νleast ≤ 1/E1/E , (E1 ≤ E2 ≤ E3 ). 1 +1/E 2 Upper-half: inter faces αk (k = 0, 1, 2,…, n); tk = (αk − αk-1 ) thickness of kth ply

  k  σi = Qij εj − Sj6 k σz (i, j = 1, 2.3)

(4.3)

 k σr = Qrs εs (r, s = 4, 5)

(4.4)

Normal strain εz is given by  k εz = S6j σj , (j = 1, 2, 3, 6)

(4.5)

4.2.1 Equilibrium Equations Equilibrium equations in stress components are, with suffix denoting partial derivative operator,   α σx,x + τxy,y + τxz,z = 0

(4.6a)

  α σy,y + τxy,x + τyz,z = 0

(4.6b)

  α τxz,x + τyz,y + σz,z = 0

(4.7)

In the primary bending problem in Poisson theory, prescribed or reactive transverse stresses are independent of material constants in the auxiliary problem. In primary extension problems, these transverse stresses are induced to nullify errors in the solutions due to transverse normal strain from constitutive relation. Their determination is dependent on material constants. Hence one has to consider equilibrium equations in terms of displacements that are dependent on ply material constants.

72

4 Laminated Plates with Anisotropic Plies …

4.2.2 Equilibrium Equations in Terms of Strains With σi in Eqs. (4.3), in-plane equilibrium Eq. (4.6) in terms of strains with elastic constants in each ply (superscript ‘k’ omitted) become in the interval αk−1 ≤ z ≤ αk       αk Q1j εj − Sj6 σz ,x +Q3j εj − Sj6 σz ,y + τxz,z = 0

(4.8a)

      αk Q2j εj − Sj6 σz ,y +Q3j εj − Sj6 σz ,x + τyz,z = 0

(4.8b)

4.3 Modified Homogeneous Plate Theories (MHPT) of Symmetric Laminates 4.3.1 Analysis of Primary Bending Problem In primary bending problems, the plate is subjected to asymmetric load σz = ± (q1 /2) and zero shear stresses all along the z = ±1 faces. In Poisson’s theory, transverse stresses in the auxiliary problem presented in Chaps. 1 and 3 remain the same here. Transverse shear stresses prescribed along relevant edges are     τxz0 , τyz0 = Txz (y), Tyz (x)

(4.9)

Because of Eq. (4.7), [τxz0 , τyz0 ] are expressed as     τxz0 , τyz0 = α ψ0,x , ψ0,y

(4.10)

so that ψ0 (x, y) with σz = z q1 /2 is governed by 1 α2 ψ0 + q1 = 0 2

(4.11)

to be solved with edge condition either ψ0 = 0 or its normal gradient equal to prescribed transverse shear stress along the edge. Face stresses [Txz (y), Tyz (x)] correspond to [τxz0 , τyz0 ] in Eq. (4.9) consistent with prescribed or reactive transverse shears along the edges. Face deflection w0 (x, y) is from the satisfaction of zero face shear conditions. It is the same in the domain of the plate independent of lamination and analytic with ωz ≡ 0. As such, static equilibrium Eq. (4.8) with [u, v] = z [u1 , v1 ] become, with sum on k = 1, 2,…, n,       k tk Q1j εj − Sj6 σz ,x + Q3j εj − Sj6 σz1 ,y = 0

(4.12a)

4.3 Modified Homogeneous Plate Theories (MHPT) of Symmetric Laminates

73

      k tk Q2j εj − Sj6 σz ,y + Q3j εj − Sj6 σz1 ,x = 0

(4.12b)

The above equations become a fourth-order equation in ψ1 with [u1 , v1 ] = α [ψ1,x , ψ1,y ] to be solved with two in-plane edge conditions. The above analysis from the solution of static equilibrium equations consists of altered stiffness coefficients due to lamination from the usual stiffness coefficients of the homogeneous plate. Equations (4.12a, 4.12b) include σz1 from constitutive relations. However, reactive stresses due to in-plane stresses have to be obtained from thickness-wise (z-) integration of Eq. (4.8) satisfying continuity of transverse stresses across interfaces. Like in the homogeneous case, σz3 (k) has to be a free variable to reduce the analysis to that of a homogeneous plate in the absence of lamination. It is also necessary to determine [u, v]1 from satisfying both static and z-integrated equilibrium equations. Here, we consider the corresponding equation in the homogeneous plate problem and suitably modify it for the laminated plate. For this purpose, ply-wise distribution of f3 (k) (z) is modified to f∗3 (k) (z) such that ∗ (k) f3 (αk ) = 0 along the upper face (z = αk ) of each kth ply so that   f∗3 (z)(k) = f3 (k) (z) − β1 (k) f1 (k) (z)α2k

(4.13)

  with β1 (k) = f3 (k) (αk )/ f1 (k) (αk )αk2 . Noting that mid-plane variables in the homogeneous plate remain the same in the laminated plate here, we recall Eqs. (3.19a and 3.19b) in Chap. 3, and the modified equations for laminate are u∗ = zu∗1 = v∗ = zv∗1 =





zk u1 + ε(k) − α w k 0,x 4

(4.14a)





zk v1 + ε(k) 5 − αk w0,y

(4.14b)

Normal stress σz along with (z q1 /2) in the Poisson theory takes the form 1 (k) (k) + f3k σz3 q1 − β1k σz3 2      (k)  τxz , τyz = τxz , τyz 0 + f2k (z) τxz , τyz 2 σz =





zk

(4.15) (4.16)

in which τ(k) xz2 =

 (k) Q44 u1 + Q45 v1 + τxz0

(4.17a)

τ(k) yz2 =

 (k) Q55 v1 + Q45 u1 + τyz0

(4.17b)

74

4 Laminated Plates with Anisotropic Plies …

From Eqs. (4.6, 4.7, 4.15, 4.17), one obtains 

 (k)  αk (Q44 u1 + Q45 v1 ),x +(Q54 u1 + Q55 v1 ),y = β1k σz3

(4.18)

In the z-integrated equilibrium equations, u∗1 = − v∗1 = − τ∗xz2

=

τ∗yz2 =





  αk 2ψ1,x + ϕ1,y + γxz0

(4.19a)



  αk 2ψ1,y − ϕ1,x + γyz0

(4.19b)







(k) ∗ ∗ αk Q1j εj −Sj6 σz1 + Q3j εj −Sj6 σz1

(4.20a)





(k) αk Q2j ε∗j −Sj6 σz1 + Q3j ε∗j −Sj6 σz1

(4.20b)

,x



σz3 = −

,y

,y



,x



αk τ∗xz2,x + τ∗yz2,y (σz3 is coefficient of f 3 )

(4.21)

Due to (f3,zz + f1 ) = 0, elimination of σz3 from Eqs. (4.18) and (4.21) gives 



  (k) αk β1k τ∗xz2,x + τ∗yz2,y = αk (Q44 u1 + Q45 v1 ),x + (Q54 u1 + Q55 v1 ),y (4.22)

The above fourth-order equation in ψ1 to be solved along with harmonic function ϕ1 with the following three conditions at each of x = constant edges (with analogous conditions at each of y = constant edges)

  (i) u∗ or σ∗ 1 = 0, (ii) v∗ or τ∗xy = 0

(4.23a)

(iii) ψ1 or τ∗xz2 = 0

(4.23b)

1

In the associated torsion problem, ψ1 is replaced by w0 so that w0 becomes a domain variable. Details of the analysis are omitted.

4.3.2 Analysis of Primary Extension Problem In primary extension problem, prescribed transverse stresses along faces of the laminate are [τxz , τyz ] = ± [Txz1 , Tyz1 ] and σz = q0 /2. Since σz0 does not participate in equilibrium Eq. (4.7), the prescribed   face shears

1 so that Txz , Tyz = [Txz , Tyz ] have to be gradients of a harmonic function ψ

1,x , ψ

1,y ]. Transverse shear stresses and normal stress satisfying face conditions −α[ψ are

4.3 Modified Homogeneous Plate Theories (MHPT) of Symmetric Laminates

75

   

1,x , ψ

1,y , σz0 = 1 q0 (x, y) τxz , τyz = − αk ψ 2

(4.24)



The above transverse stresses are independent of material constants and remain the same within the plate. From constitutive relation,  (k) 1 S = S σ + q (j = 1, 2, 3) ε(k) 6j j0 66 0 z0 2

(4.25)

Correspondingly, vertical deflection w is linear in z and cannot be prescribed to be zero along the edge of the plate due to S6j σj0 even if the faces are free of transverse stresses. In-plane equilibrium equations in the preliminary analysis with [u, v] = [u0 (x, y), v0 (x, y)] are 

     (k) 

1,x = αk ψ αk Q1j εj0 − Sj6 σz0 ,x + Q3j εj0 − Sj6 σz0 ,y

(4.26a)



     (k) 

1,y = αk ψ αk Q2j εj0 − Sj6 σz0 ,y + Q3j εj0 − Sj6 σz0 ,x

(4.26b)

‘Unusual problem: Need for Poisson theory’ is discussed and presented Poisson theory in Chaps. 2 and 3. It is adapted in the present analysis. We consider higher-order in-plane displacement terms f2 (k) (z) [u2 , v2 ] which induce transverse shear stresses f1 (k) [τxz1 , τyz1 ], f2 (k) (z) σz2 from constitutive relations, and f2 (k) w1 (x, y) (other than the known z εz0 ) due to strain–displacement relations in the domain of the plate. In-plane displacements [u2 , v2 ] in kth ply are related from transverse shear–strain relations and constitutive relations to [τxz1 , τyz1 ], even in the absence of induced w1 , in the form   (k)   τ(k) xz1 = − Q44 u2 − α εz0,x + Q45 v2 − α εz0,y

(4.27a)

  (k)   τ(k) yz1 = − Q55 v2 − α εz0,y + Q54 u2 − α εz0,x

(4.27b)

We consider the following displacements consistent with transverse stresses [τxz1 , τyz1, σz2 ](k) , with εz (k) = S6j (k) σj , j = 1, 2, 3, 6 from constitutive relation, (k) (k) (k) w = f(k) 1 (εz0 + w1 ), u = u0 + f2 u2 v = v0 + f2 v2 , σz = f2 σz2

(4.28)

In the vertical deflection, w1 (x, y) is added to facilitate determination of [u2 , v2 ] from satisfying both static and z-integrated equilibrium equations. In adapting Poisson theory to extension problems, transverse stresses have to be independent of domain variable w1 in transverse stresses. Hence, [u2 , v2 ] are modified as

76

4 Laminated Plates with Anisotropic Plies …

[u2 , v2 ]∗ =

  u2 − α(εz0 + w1 ),x , v2 − α(εz0 + w1 ),y

(4.29)

so that transverse shear stresses from strain–displacement relations and constitutive relations are 

τxz1 , τyz1

∗

 (k) = − (Q44 u2 + Q45 v2 ), (Q55 v2 + Q54 u2 )

(4.30)

Normal stress σz2 from static equilibrium equation in kth ply is  (k) ∗ σz2 = −αk (Q44 u2 + Q45 v2 ),x + (Q55 v2 + Q54 u2 ),y

(4.31)

In order to keep [τxz3 , τyz3 ] in kth ply as free variables in the integrated equilibrium equations, f3 (k) (z) is modified with β1k = 1/3 as f*3 (k) (z) = f3 (k) (z) − β1k f1 (k) so that   ∗∗   (k)  ∗ ∗ τ (4.32) = f(k) , τ τxz , τyz xz1 yz1 + f3 τxz3 , τyz3 1 with [τxz1 *, τyz1 *] = [(τxz1 − β1k τxz3 ), (τyz1 − β1k τyz3 )]. From static equilibrium equation of transverse stresses, αk [τxz1 ,x + τyz1 ,y ]*= σz2 * and αk [τxz3 ,x + τyz3 ,y ] = σz4 so that σz2 ** = σz2 * − β1k σz4 from which one gets (no sum on k)  (k) αk (Q44 u2 + Q45 v2 ),x + (Q55 v2 + Q54 u2 ),y + β1k σz4 = 0 (coefficient of z) (4.33) Strain–displacement relations from Eqs. (4.29) give ∗ = εx2 − α2k (εz0 + w1 ),xx εx2

(4.34a)

ε∗y2 = εy2 − α2k (εz0 + w1 ),yy

(4.34b)

∗ γxy2 = γxy2 − 2α2k (εz0 + w1 ),xy

(4.34c)

Here also, [u2 , v2 ] are expressed in terms of gradients of two functions [ψ2 , ϕ2 ], like in bending problems, in the form     [u2 , v2 ] = −αk ψ2,x + ϕ2,y , ψ2,y − ϕ2,x Note that the contribution of w1 is the same as ψ2 in [u2 , v2 ]* in the integration of equilibrium equations since contributions of f1 (k) and f2 (k) ,z are of opposite sign in strain–displacement relations, whereas the corresponding contribution of f1 (k) and z-integrated f2 (k) ,z are of the same sign.

4.3 Modified Homogeneous Plate Theories (MHPT) of Symmetric Laminates

77

In-plane strains become, with εi2 * (i = 1, 2, 3) denoted by εx2 *, εx2 *, εx2 *, respectively,   ε∗x2 = − α2k 2ψ2,xx + ϕ2,xy + α2 εz0,xx

(4.35a)

  ε∗y2 = − α2k 2ψ2,yy + ϕ2,yx + α2 εz0,yy

(4.35b)

  ∗ γxy2 = − α2k 4ψ2,xy + ϕ2,xx − ϕ2,yy + 2εz0,xy

(4.35c)

The corresponding in-plane stresses are

(k)  ∗ − S6j σz0 (i, j = 1, 2, 3) σi2∗ = Qij εj2

(4.36)

From the integration of equilibrium equations, reactive transverse stresses are ∗  τ∗xz3 = α σx2,x + σxy2,y

(4.37a)

∗  τ∗yz3 = α σy2,y + σxy2,x

(4.37b)

 ∗ σz4 = α τxz3,x + τyz3,y (coefficient of f3 )

(4.38)

Noting that σz4 from Eq. (4.33) is negative of the one from Eq. (4.38) due to (f3 + f1 ) = 0 along the faces of the plate, the equation governing in-plane displacements (u2 , v2 ) is 

 ∗(k)   (k) = αk (Q44 u2 + Q45 v2 ),x +(Q55 v2 + Q54 u2 ),y αk β1k τxz3,x + τyz3,y (4.39)

The above equation is a fourth-order equation in ψ2 to be solved along with harmonic function ϕ2 with three conditions along x = constant edges (with analog conditions along y = constant edges). ∗ =0 (i) u∗2 = 0 or σx2

(4.40a)

(ii) v∗2 = 0 or τ∗xy2 = 0

(4.40b)

(iii) ψ2 = 0 or τ∗xz3 = 0

(4.40c)

Note that expressions for transverse stresses [τxz3 , τyz3 , σz4 ] in each ply are used in the analysis so that they are determined from obtained solutions of in-plane displacements.

78

4 Laminated Plates with Anisotropic Plies …

Higher-order approximations: Corresponding higher-order single-layer theories are not presented in both extension and bending problems in spite of recent article [12] by Alexandre Loredo et al. since their utility in the estimation of in-plane stresses does not help much concerning exact solutions in each ply.

4.4 Un-symmetrical Laminates: Alternate Form (ACLPT) of CLPT Assumed bending displacements in CLPT are either from Kirchhoff’s theory or from first-order shear deformation theory (FSDT). In Kirchhoff’s theory, zero transverse shear conditions along the faces of the laminate are priori satisfied from strain– displacement relations. In the case of un-symmetrical laminates, however, transverse shears from integrated equilibrium equations do not satisfy either at one of the faces or maintain continuity across mid-plane of the laminate. Moreover, the edge conditions involve integrated stress resultants that have no unique point-wise distribution, thereby the need for post-processing for obtaining transverse stresses. An alternate form of CLPT is proposed to show that the bending stresses in the case of un-symmetrical laminate are uncoupled from primary stresses in the extension problem. Poisson’s theory is adapted to eliminate post-processing for finding transverse stresses and a supplementary problem is formulated to maintain continuity of these stresses across interfaces of the un-symmetrical layup of plies. Here, the laminate is bounded by –hm ≤ Z ≤ hn about the mid-plane Z = 0 in the thickness direction. It is convenient to consider the positive direction of the Z-axis in the upward and downward directions so that 0 ≤ Z ≤ hn and 0 ≤ Z ≤ hm in the upper and bottom halves of the laminate, respectively. Here, hm = hn but the number of plies ‘m’ need not be equal to ‘n’. Interfaces are given by z = αk = hk /hn (k = 1, 2, …, n−1) in the upper-half and αk = hk /hm (k = 1, 2, …, m−1) in the bottom-half of the laminate, respectively. Displacements in the classical theories are in the form [w0 , −z α w0 ,x , −z α w0 ,y ] and [u0 , v0 ] in pure bending and extension problems, respectively. Coupling between extension and bending problems is in the case of un-symmetrical laminates. This unsymmetrical lay-up of laminates may be classified into two groups: (1) Number of plies ‘m’ in the bottom-half is different from the number of plies ‘n’ in the upperhalf of the laminate. (2) Even if m = n, thickness and/or material properties in a kth ply are different from each other in these parts of the laminate. This coupling between extension and bending problems is through the ‘B’ matrix in PEEEs due to the thickness-wise integration of products of even and odd z-functions. This ‘B’ matrix imposes the same order of effect on extension and bending displacements. An alternate form of CLPT designated as ACLPT is proposed here to differentiate this effect in extension and bending problems. In-plane displacements are assumed, with δ = 1 but zero for symmetric laminates, in the form [u, v] = [u0 , v0 ] + (δ − z)[u1 , v1 ](bending)

(4.41)

4.4 Un-symmetrical Laminates: Alternate Form (ACLPT) of CLPT

79

   1 + z [u0 , v0 ] + z[u1 , v1 ](extension) [u, v] = 1 − δ 3

(4.42)

Equations governing [u0 , v0 ] and [u1 , v1 ] using Poisson’s theory are considered here.

4.4.1 Bending Problem Transverse stresses from the auxiliary problem with α2 ψ0 + q1 /2 = 0 are σz =

    1 z1 q, [ τ xz0 , τ yz0 ] = α ψ0,x , ψ0,y 2

(4.43)

In-plane displacements are represented in the form [u, v] = − (δ – z) [u1 , v1 ]* in which u∗1 = u1 + γxz0 − αw0,x , v∗1 = v1 + γyz0 − αw0,y   Transverse shear strains, with γxz0 , γyz0    S45 , τ yz0 ), (S55 τ yz0 + S54 , τ xz0 )], are

=

 ∗    ∗ = u1 + γxz0 , γyz = v1 + γyz0 γxz

(4.44)  [(S44 τ xz0 +

(4.45)

Correspondingly, transverse shear stresses are    τ∗xz = Q44 u1 + Q45 v1 + τ xz 0

(4.46a)

   τ∗yz = Q55 v1 + Q54 u1 + τ yz 0

(4.46b)

Transverse shear stress associated with [u1 , v1 ] is:     τxz0 = Q44 u1 + Q45 v1 , τyz0 = Q55 v1 + Q54 u1

(4.47)

Transverse stresses in each ply are:   ∗  k  τxz , τyz = τxz , τyz + f2k τxz2 , τyz2 σz =

1 z q + [f3k , σz3 ]k 2 1

The function f2k in Eq. (4.48) and f3k in Eq. (4.49) are

(4.48) (4.49)

80

4 Laminated Plates with Anisotropic Plies …

   f2k = (αk − z)δ − α2k − z2  f3k =

   1 2 1 3 2 αk z − z δ − α k z − z 2 3

(4.50a) (4.50b)

f3k (z) in each ply is replaced with f3k (z) given by 

f3k (z) = in which β1 = one gets

1 2

δ−

2 3



   1 2 1 3 2 − β1 α2k z αk z − z δ − αk z − z 2 3

(4.51)

so that f3k (αk ) = 0. From Eq. (4.7) in transverse stresses,

    (k) (k) , σz3 = αk τxz2,x + τyz2,y αk τxz0,x + τyz0,y = β1 α2k σz3

(4.52)

To satisfy integrated equilibrium equations, it is convenient to assume     [u1 , v1 ] = − ψ1,x + ϕ1,y , ψ1,y − ϕ1,x

(4.53)

Vertical deflection w0 in [u1 , v1 ]* is replaced by ψ1 in [2] so that   u1 ∗ = −α 2ψ1,x + ϕ1,y + γxz0

(4.54a)

  v1 ∗ = −α 2ψ1,y − ϕ1,x + γyz0

(4.54b)

Reactive transverse stresses are

τxz2 = τ˜yz2 =



     (k) αk Q1j ε˜ j − Sj6 σz1 ,x + Q3j ε˜ j − Sj6 σz1 ,y

(4.55a)



     (k) αk Q2j ε˜ j − Sj6 σz1 ,y + Q3j ε˜ j − Sj6 σz1 ,x

(4.55b)

τ∗xz2 = τ˜ xz2 + τxz0 , τ∗yz2 = τ˜ yz2 + τyz0

(4.56)

  ∗ σz3 = −α τ˜ xz2,x + τ˜ yz2,y , σz3 = σz3 + σz1

(4.57)

The equation governing in-plane displacements (u1 , v1 ) is given by     αβ1 α2k τ˜ xz2,x + τ˜ yz2,y = α τxz0,x + τyz0,y

(4.58)

4.4 Un-symmetrical Laminates: Alternate Form (ACLPT) of CLPT

81

Equation (4.58) is a fourth-order equation in ψ1 to be solved along with plane Laplace equation ϕ1 = 0. In the above analysis, ply-wise equilibrium equations are satisfied independent of lamination. In-plane displacements [u, v] thus obtained are dependent on material constants in each ply. In ACLPT, [u1 , v1 ] become dependent on laminate stiffness coefficients in place of ply material constants by assuming that they are the same in all plies. Here, it is not convenient to use the stationary property of total potential in the energy method. [u,v] are dependent on a different type of laminate stiffness coefficients. One needs stress resultants in plate element given by (with the sum on k)   1 Vx = − (αk − αk−1 ) δ − (αk + αk−1 ) (Q44 u1 + Q45 v1 )(k) 2   1 Vy = − (αk − αk−1 ) δ − (αk + αk−1 ) (Q55 v1 + Q54 u1 )(k) 2   1

vx2 = − α(αk − αk−1 ) δ − (αk + αk−1 ) 2      (k) Q1j εj − Sj6 σz1 ,x + Q3j εj − Sj6 σz1 ,y   1

vy2 = − α(αk − αk−1 ) δ − (αk + αk−1 ) 2      (k) Q2j εj − Sj6 σz1 ,y + Q3j εj − Sj6 σz1 ,x

(4.59a) (4.59b)

(4.60a)

(4.60b)

    vx2,x + vy2,y = α Vx,x + Vy,y which The equation governing ψ1 becomes: β1 α is again a fourth-order equation in ψ1 to be solved along with harmonic function ϕ1 subjected to the following three conditions (i, ii, iii) along x = constant edges (and analogous conditions along y = constant edges) 1 Tx1 (y), 3 1 = Txy1 (y) 3

x = (i) u1 (y) = 0 or M

xy (ii) v1 (y) = 0 or M

(iii) ψ1 (y) = 0 or Vx =

1 Txz0 (y) 2

(4.61a) (4.61b)

in which  (k)       1

i = − α  αk − αk−1 δ − αk + αk−1 Qij M εj − Sj6 σz1 (j = 1, 2, 3) (i = 1, 2, 3) 2

(4.62)

82

4 Laminated Plates with Anisotropic Plies …

Denote the in-plane displacements [u1 , v1 ] thus obtained by [ u1 , v1] which are

continuous across interfaces except across z = 0 plane. Moreover, τxz ,    τyz =  τxz2 , u1 , τyz2 are simply in terms of [ v1 ] and β1 σz3 = α τxz,x + τyz,y . (Postβ1 processing through equilibrium equations in CLPT for finding transverse stresses is eliminated. Note that this commonly used procedure does not ensure the satisfaction of either face conditions or continuity conditions across reference plane.) Bending displacements and transverse stresses in the above analysis from ACLPT are 1 u1 , v1 ] [u, v] = − (δ − z)[ 2          τxz , τyz = τ xz0 , τ yz0 + (αk − z)δ − α2k −z2 τxz2 , τyz2    1 2 1 3 2

σz3 αk z − z δ − αk z − z 2 3    αw0 (x, y) = −

u1 dx + v1 dy

1 σz = zq1 + 2

(4.63)



(4.64) (4.65)

Discontinuities across reference plane Bending displacements and transverse shear stresses along reference plane z = 0 are 1 [u, v](u,b) = − δ[ u1 , v1 ](u,b) 2   (u,b) α w(u,b) = −

u1 dx + v1 dy 0

(4.66) (4.67)

 1 τ(u,b) = τ xz0 + t1 (δ − t1 ) τ(u,b) xz xz2 2

(4.68a)

 1 = τ yz0 + t1 (δ − t1 ) τ(u,b) τ(u,b) yz xz2 2

(4.68b)

It can be seen from Eqs. (4.66, 4.67, 4.68), continuity of and trans  displacements v1 ] and τyz2 in the adjacent verse stresses requires only continuity of [ u1 , τxz2 , plies of the reference plane. For this purpose, one has to consider first the problem of reference plane subjected to  1 t1 (δ − t1 ) τ(u,b) xz2 − 2  1 t1 (δ − t1 ) = τ(u,b) yz2 − 2

= τ(u,b) xz

1 t1 (δ − t1 ) τ(u,b) xz2 2

τ(u,b) yz

1 t1 (δ − t1 ) τ(u,b) yz2 2

 (4.69a)  (4.69b)

4.4 Un-symmetrical Laminates: Alternate Form (ACLPT) of CLPT

83

It is convenient to introduce the coordinate z = (1− z) for (z ≥ 0) so that the reference plane z = 0 corresponds to z = 1. Consequently, hk  = 1− hk , αk  = (1– αk ) and [τxz , τyz ] are [τxz , τyz ] . Here, q = 0 along z = 1 and the faces z = 0 are free of transverse stresses. Replace [u1 , v1 ] with [u1  , v1  ] and δ = 0 and determine [u1  , v1  ], as before, in terms of laminate stiffness coefficients with three conditions along x = constant edges (with analog conditions along y = constant edges) MX = 0, MXY = 0, VX = TXZ (y). Displacements in the extension problem coupled with bending displacements now become.    1  v0 ] = [u0 , v0 ] − δ u1 + u1 , v1 + v1 u0 , [ 2

(4.70)

v0 ] are obtained from the earlier smeared Modified in-plane displacements [ u0 , u0 , v0 ] in the differential equations and laminate theory by replacing [u0 , v0 ] with [ edge conditions. The displacements thus obtained along with face variable w0 (x, y) from top and bottom halves of the laminate are continuous across the reference plane z = 0.

4.4.2 Extension Problem We consider [u, v] such that they do not affect displacements z[u1 , v1 ] (ignoring their relations with w0 ) in the presence of bending loads by assuming z-distribution orthogonal to z in the form    1 + z [u0 , v0 ] [u, v] = 1 − δ 3 Transverse strains are zero and        1 + z α u0,x , v0,y εx , εy = 1 − δ 3      1 + z α v0,x + u0,y γxy = 1 − δ 3

(4.71)

(4.72a) (4.72b)

Stress components in the kth ply are, with (i, j = 1, 2, 3), α(k) i

    (k) 1 +z Qij εj0 = 1−δ 3

(4.73)

84

4 Laminated Plates with Anisotropic Plies …

In smeared laminate theories, stress resultant in plate element is a sum of ply-wise stress resultant in k plies with k equal to ‘n’ and ‘m’ in the upper and bottom halves of the laminate, respectively. Hence, the summation sign on k in each half implies that k varies up to ‘n’ in the upper-half and up to ‘m’ in the bottom half of the laminate. Accordingly, stress resultants added together with z = −z in the bottom-half in the smeared laminate theory and PEEEs with (i, j = 1, 2, 3) are  Ni = 

 

  1 2 (k) 2 2 2 1 − δ tk − 2δ(1 − δ)(αk − αk−1 ) + δ αk − αk−1 Qij εj0 6 (4.74) Nx,x + Nxy,y = 0, Ny,y + Nxy,x = 0

(4.75)

The solution of the above equilibrium equations along with appropriately modified edge conditions gives [u0 , v0 ] denoted as [u0 , v0 ] same in all plies, thereby, continuous across interfaces. Transverse stresses dependent on material constants and sub-laminate stiffness coefficients are obtained from the usual post-processing from the integration of equilibrium equations in each ply maintaining continuity across interfaces of each ply. However, the procedure from one face to the other face is at the expense of not satisfying face conditions. If the process is used separately in each half, then these stresses are not continuous across the reference plane z = 0. The above transverse stresses from post-process in each ply adjacent to the reference plane z = 0 are, with    1 1  2 2 f1 (z) = 1 − δ (α1 − z) + δ α1 − z 3 2    1 1 2 2 1 − δ α1 (α1 − z) − α1 − z f2 (z) = 3 2     1 1 2 3 3 , + δ α1 (α1 − z) − α1 − z 2 3  (1) (u,b)

τ(u,b) = −αf (z) σ + σ 1 x,x xy,y xz

(4.76a)

 (1) (u,b)

τ(u,b) = −αf (z) σ + σ 1 y,y xy,x yz

(4.76b)

 (u,b)

σz(u,b) = α2 f2 (z) τxz,x + τyz,y

(4.77)

Note that the in-plane distributions of the above stress components are in terms of [u0 , v0 ]. Transverse stresses [τxz , τyz , σz ] along the reference plane z = 0 from upper-half and bottom-half of the laminate are, with

4.4 Un-symmetrical Laminates: Alternate Form (ACLPT) of CLPT

85

     1 1 1 1 1 − α1 , f2 (0) = 1 − δ + α1 α21 , 1 − δα1 3 2 3 2 2   (u,b)  (u,b) τxz , τyz = f1 (0) τxz , τyz , σz = f2 (0)σz (4.78) f1 (0) =

It can be seen from the above equations that continuity of transverse stresses  τyz0 in the adjacent plies of the reference plane. requires only continuity of τxz0 , For this purpose, one has to consider first the problem of reference plane subjected to

(b,u) (u,b)

(4.79a) = f (0) τ − τ τ(u,b) 1 xz xz2 xz2

τ(u,b) = f1 (0) τ(b,u) τ(u,b) yz yz2 − yz2

(4.79b)

It is convenient to introduce the coordinate z = (1− z) for (z ≥ 0) so that the reference plane z = 0 corresponds to z = 1. Consequently, hk  = 1− hk , αk  = (1– αk ) and [τxz , τyz , σz ] are [τxz , τyz , σz ] which are zero along z = 0 faces. Replace [u0 , v0 ] with [u0  , v0  ] and δ = 0 in Eqs. (4.61–4.65) and determine [u1  ,  v1 ], as before, in terms of laminate stiffness coefficients with three conditions Nx = 0, Nxy = 0, Vx = Txz (y)

(4.80)

at x = constant edges (with analog conditions along y = constant edges). The displacements uncoupled with bending displacements now become      1 +z

u0 + δu0 u0 = 1 − δ 3      1 +z

v0 + δv0 v0 = 1 − δ 3

(4.81a) (4.81b)

Displacements [u0 , v0 ], thereby, [τxz , τyz , σz ] thus obtained are continuous not only across z = 0 plane but also across all interfaces of plies. From the above analysis, [u, v]e in extension problem (denoted with suffix ‘e’) are      1 +z

u0e + δu0e + z (4.82a) u1e ue = 1 − δ 3      1 +z

v0e + δv0e + z ve = 1 − δ (4.82b) v1e 3

86

4 Laminated Plates with Anisotropic Plies …

On torsion associated with bending loads For determination of w0 (x, y) which is a primary variable in bending (associated torsion) problems, static Eqs. (4.1) and (4.2) are required and they are different from integrated equations. Kirchhoff’s theory is based on integrated equations normally used in bending problems. The author’s recent investigations, however, indicate that the sum of τxy in bending and τxy in torsion is zero in the exact solutions of 3D equations. This is due to the use of w0 as domain variable in torsion problems and as face variable in bending problems. Poisson’s theory recently proposed by the author brings out this distinction in the analysis of these problems.

4.4.3 Contrast Between CLPT and ACLPT Displacements [w0 , u0 , v0 ] in the classical theory of un-symmetrical laminates (CLPT) are coupled through B matrix in plate element equilibrium equations (PEEEs) arising due to thickness-wise integration of products of even and odd z-functions. This ‘B’ matrix imposes same order of effect on [u0 , v0 ] and α [w0 ,x , w0 ,y ]. Displacements in CLPT are assumed in the form w = w0 (x, y), u = u0 – z α w0 ,x , v = v0 – z α w0 ,y . Here, w0 is a domain variable and the coupling is between extension and torsion problems. An alternate form of CLPT denoted as ACLPT shows that the effect of un-symmetry in bending displacements is independent of [u0 , v0 ]. Poisson’s theory is used to satisfy both static and integrated equilibrium equations. A secondary problem is formulated governing induced second-order displacements of extension problem to maintain continuity of transverse stresses across interfaces of plies. The same analysis is applicable here also by replacing domain variable ψ1 in bending problem with domain variable w0 (x, y) in torsion problem. Hence, the corresponding analysis is not presented here. The associated torsion-type problem in extension problem requires the use of [u2 , v2 ] and the analysis of their influence on bending displacements needs to be carried out in the future investigation.

4.5 Conclusions The set of polynomials generated in z are necessary for satisfying both static and integrated equilibrium equations. Poisson’s theory is based on the satisfaction of both static and integrated equilibrium equations. This feature may be exploited in investigations on optimum ply lay-up, its utility in the analysis of associated eigenvalue problems of free vibration and buckling of plates, and even in the area of fracture mechanics. An alternate form of classical laminate theory is proposed for the analysis of unsymmetrical laminated plates using extended Poisson’s theory. There is no coupling

4.5 Conclusions

87

between extension and problems. The effect of un-symmetry is through linear variation of transverse stresses independent of each other in these problems. Analysis in extension and bending problems in ACLPT is based on assumed displacements (41) and (42). There is a need to quantify the amount of discontinuity in transverse stresses across the reference plane before using the solution of the auxiliary problem. It is dependent on the location of the reference plane which, in principle, can be any z = constant plane except either face of the laminate. The use of discontinuity in the corresponding strain energy density for this purpose gives wide scope for future investigations in finding the location of the reference plane with either minimum or minimum of maximum discontinuity. Coupling due to the ‘B’ matrix in CLPT imposes the same order of effect on extension and bending displacements but this coupling appears to be different in ACLPT. It is worthy of consideration in future investigations. The number of 2-D displacement variables is limited to the minimum number mainly because the in-plane distributions of these variables are not ply dependent and can never be equal to ply-dependent distributions in layer-wise theories. The normal requirement is a two-term representation of each in-plane displacement variable in the analysis of un-symmetrical laminates like in the present ACLPT. In CLPT, bending displacements are in terms of a single variable w0 (x, y) due to its use as a domain variable, and two-edge conditions (instead of three required in 3-D problem) are prescribed in each of extension and bending problems. In ACLPT, three-edge conditions in bending are prescribed due to the use of w0 (x, y) as a face variable in Poisson’s theory and extended Poisson’s theory. The corresponding in-plane displacements are determined by satisfying both static and integrated equilibrium equations. Such a facility is absent in extension problems since the even prescribed σz = q0 /2 along faces of the laminate does not disturb the equilibrium equations. If transverse shear stresses are prescribed all along the faces of the laminate, they have to be asymmetric in z and have to satisfy equilibrium equation in z-direction even in the case of the above-prescribed σz = q0 /2. They are governed by a static equilibrium equation. One gets from integrated equation σz = f2 (z)σz2 with unknown σz2 . Its determination is dependent on second-order [u2 , v2 ] displacements which have to be obtained from Poisson’s theory satisfying both static and integrated equilibrium equations. Higher-order displacement terms are to improve in-plane distributions whose utility may not be of much importance.

References 1. Lekhñitski˘ı SG (1963) Theory of elasticity of an anisotropic elastic body. Holden-Day, Elasticity, p 404 2. Reissner E, Stavsky Y (1961) Bending and stretching of certain types of heterogeneous aeolotropic elastic plates. J Appl Mech 28:402–408 3. Reddy JN (1990) A review of refined theories of laminated composite plates. Shock Vibr Dig 22:3–17

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4. Ambartsumyan SA (1958) On a theory of bending of anisotropic plates (in Russian) Izv. Akad Nauk SSSR Otd Tekh Nauk 5:69–77 5. Whitney JM (1969) The effect of transverse shear deformation on the bending of laminated plates. J Compos Mater 3:534–547 6. Valisetty RR, Rehfield LW (1983) A theory for stress analysis composite laminates. AIIA J 23(7):1111–1117 7. Ren JG (1986) Bending theory of laminated plate. Compos Sci Technol 27(3):225–248 8. Vijayakumar K, Krishna Murty AV (1988) Iterative modeling for stress analysis of composite laminates. Comp Sci Technol 32(3):165–181 9. Kienzler R, Schneider P (2012) Consistent theories of isotropic and anisotropic plates. J Theor Appl Mech 50(3):755–768 10. Abrate S, Di Sciuva M (2017) Equivalent single layer theories for composite and sandwich structures: A review. Compos Struct. https://doi.org/10.1016/j.compstruct.2017.07.090 11. Kreja I, Sabik A (2019) Equivalent single-layer models in deformation analysis of laminated multilayered plates. Acta Mech:1–25. https://doi.org/10.1007/s00707-019-02434-7 12. Loredo A et al (2019) A family of higher-order single layer plate models meeting C0 z - requirements for arbitrary laminates. Compos Struct: 1–25. https://doi.org/10.1016/j.compstruct.2019. 111146

Chapter 5

Laminated Plates with Anisotropic Plies Part II: Layerwise Theories

Nomenclature a fik (z) hk hm = hn m n O-X Y Z Qij Qrs q0 (x, y)/2 q1 (x, y)/2

side length of a square plate thickness-wise distribution functions in kth ply, i = 0, 1, 2, … interfaces, k = 1, 2,… plate half-thickness number of plies below the mid-plane z = 0 of the plate number of plies above the mid-plane z = 0 of the plate Cartesian coordinate system Stiffness coefficients, (i, j = 1, 2. 3) Stiffness coefficients, (r, s = 4, 5) prescribed σz (x, y) along a face of the plate in extension problems prescribed asymmetric σz (x, y) along the faces of the plate in bending elastic compliances, (i, j = 1, 2, 3, 6) Sij elastic compliances, (r, s = 4, 5) Srs Tx1 (y), Txy1 (y) prescribed stresses at each of x = constant edges (hk − hk-1 ), thickness of kth ply tk U, V, W displacements in X, Y, Z directions, respectively u, v, w U/a, V/a, W/hn , respectively [u, v, w]u displacements along upper face of kth ply displacements along lower face of kth ply [u, v, w]b x, y, z X/a, Y/a, Z/hn , respectively  −z, below mid-plane z = 0 of the plate z  plane Laplace operator (∂ 2 /∂x2 + ∂ 2 /∂y2 ) hn /a α(αn ) hk /hn αk   β2k-1 f2k+1 (αk )/ α2k f2k−1 (αk ) , εi , (i = 1, 2, 3) εx , εy , εz ε3+i , (i = 1, 2, 3) γxy , γxz , γyz © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 K. Vijayakumar and G. K. Ramaiah, Poisson Theory of Elastic Plates, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-33-4210-1_5

89

90

σx , σy , τxy τxz , τyz , σz τxzs , τyzs , σzs τxz , τyz ϕ, ψ ωz

5 Laminated Plates with Anisotropic Plies …

σi , (i = 1, 2, 3) σ3+i , (i = 1, 2, 3) transverse stresses in supplementary problems shear stresses in mid-plane of un-symmetrical laminates variables in face parallel x–y planes α(v,x – u,y)

5.1 Introduction In the case of symmetric laminates, several theories such as single-layer theories [vide references in Chap. 3], several review articles [1–11], layer-wise theories [12–18], zigzag theories [19–22], in the analysis of bending, extension [19, 23], and torsion problems are reported in the literature. The emphasis in these theories is for accurate estimation of stresses (more so for transverse stresses) all along the ply interfaces. Generally, distributions of displacements in face parallel planes are evaluated in most of these theories. Statically equivalent transverse stresses are obtained through postprocessing from the dz-integration of equilibrium equations ignoring transverse shear strain–displacement relations. It is mainly due to the use of vertical displacement as a domain variable. Second-order corrections in the estimation of these stresses are through proper estimation of displacements. Layer-wise theories are based on ply element equilibrium equations. They are coupled with adjacent plies through the equations-governing displacement and transverse stress variables along the common interface of the plies so that each ply analysis is dependent on lamination. They are solved by relating face conditions with symmetry conditions along the mid-plane in one half of the laminate through equations-governing common variables between adjacent plies. It is not a convenient procedure if the stacking sequence contains a large number of plies. (Moreover, there is a need to incorporate proper modifications in these theories in the analysis of un-symmetrical laminates.) The deficiencies in the above-mentioned layer-wise theories are avoided in the layer-wise theory using the present Poisson’s theory [14]. In this theory, ply analysis is independent of lamination and continuity of displacements and transverse stresses across interfaces are through the solution of a supplementary problem in the face ply along with recurrence relations. Moreover, solutions for displacements satisfy both static and integrated equilibrium equations. With a priori known transverse stresses (independent of elastic deformations) from the auxiliary problem, Poisson theory is a sixth-order theory in the analysis of bending problems. Necessity and mandatory use of such a theory in the preliminary analysis of extension problems as explained in Chap. 2 is further examined in the analysis of corresponding problems of laminated composite plates [23, 24]. Further, a novel procedure based on this theory is envisaged in the analysis of un-symmetrical laminates for which no proper procedure exists in the literature.

5.2 Preliminaries in the Analysis of Symmetric Laminates

91

5.2 Preliminaries in the Analysis of Symmetric Laminates For simplicity in presentation, a symmetric laminate bounded by 0 ≤ X, Y ≤ a, –hn ≤ Z ≤ hn with interfaces Z = hk in the Cartesian coordinate system (X, Y, Z) is considered. For convenience, the coordinates X, Y, and Z and displacements U, V, and W in non-dimensional form [x, y] = [X, Y]/a, z = Z/hn , [u, v, w] = [U, V, W]/hn and half-thickness ratio α = hn /a are utilized. The material of each ply is homogeneous and anisotropic with monoclinic symmetry. Interfaces are given by z = αk = hk /hn (k = 1, 2…, n − 1) in the upper-half of the laminate. Equilibrium equations in stress components are, with suffix denoting partial derivative operator,   α σx,x + τxy,y + τxz,z = 0

(5.1a)

  α σy,y + τxy,x + τyz,z = 0

(5.1b)

  α τxz,x + τyz,y + σz,z = 0

(5.2)

In displacement-based models, stress components are expressed in terms of displacements, via six stress–strain constitutive relations and six strain–displacement relations. In the present study, these relations are confined to the classical small deformation theory of elasticity. Upper face values of displacements [u, v, w]u and transverse stresses u b   τxz , τyz , σz in a ply are related to its lower face values [u, v, w]b and τxz , τyz , σz , respectively, through the solution of Eqs. (5.1a, 5.1b and 5.2) together with three conditions specified later at each of constant x (and y) edges and satisfaction of continuity conditions across interfaces. Here, it is convenient to denote displacements [u, v] as [ui ], (i = 1, 2), in-plane stresses [σx , σy , τxy ] and transverse stresses [τxz , τyz , σz ] as [σi ], [σ3+i ], (i = 1, 2, 3), respectively. With the corresponding notation for strains, strain–displacement relations are   [ε1 , ε2 , ε3 ] = α u,x , v,y , u,y + v,x

(5.3)

  [ε4 , ε5 , ε6 ] = u,z + α w,x , v,z + α w,y , w,z

(5.4)

Strain–stress and semi-inverted stress–strain relations with the usual summation convention of repeated suffix denoting summation over specified integer values: εi = Sij σj (i, j = 1, 2, 3, 6)

(5.5)

εr = Srs σs (r, s = 4, 5)

(5.6)

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5 Laminated Plates with Anisotropic Plies …

  σi = Qij εj − Sj6 σz (i, j = 1, 2.3)

(5.7)

σr = Qrs εs (r, s = 4, 5)

(5.8)

With σi in Eq. (5.7), in-plane equilibrium Eq. (5.1a, 5.1b) become       α Q1 j εj − Sj6 σz ,x + Q3j εj − Sj6 σz ,y + τxz,z = 0

(5.9a)

      α Q2j εj − Sj6 σz ,y + Q3j εj − Sj6 σz ,x + τyz,z = 0

(5.9b)

5.2.1 fk (z) Functions In expressing thickness-wise distribution of displacements, a complete set of fk (z) functions in each ply generated earlier [15] from recurrence relations with f0 = 1, f2k+1,z = f2k , f2k+2,z = − f2k+1 such that f2k+2 (±αk ) = 0 are used. They are (up to k = 5)   1 [f1 , f2 ] = z, α2k − z2 2 

1 2 1 αk z − z 3 f3 = 2 3

(5.10b)

 1 4 5 αk − 6α2k z2 + z4 24

(5.10c)

 1  z 25 α4k − 10 α2k z2 + z4 120

(5.10d)

f4 = f5 =

(5.10a)

5.2.2 Displacements [u, v, w] Displacements [u, v, w] are expressed in the form of [u, v, w] = fi (z)[ui , vi , wi ], i = 0, 1, 2, . . . . . .

(5.11)

To maintain continuity of a 3-D variable across interfaces and keep associated 2-D variable as a free variable, it is necessary to replace f2i+1 by f∗2i+1 given by

5.2 Preliminaries in the Analysis of Symmetric Laminates

f∗2i+1 = f2i+1 − β2i−1 f2i−1 , i = 1, 2, . . . . . .

93

(5.12)

  in which β2i−1 = f2i+1 (αk )/α2k f2i−1 (αk ) so that f∗2i+1 (αk ) = 0. With the above replacement of odd fi (z) functions, continuity of relevant variables across interfaces is ensured through the continuity of 2-D variables associated with f0 and f1 only. A priori prescribed conditions along faces and edges of the plate are adjusted such that vertical deflection w(x, y, z) is even in z in bending and odd in extension problems. Correspondingly, displacements [u, v] are odd and even functions in z in bending and extension problems, respectively.

5.2.3 Edge Conditions in Each Ply Equilibrium Eqs. (5.1a, 5.1b and 5.2) in each ply have to be solved along with threeedge conditions and ensure continuity conditions of three displacements and three transverse stresses across interfaces. In the primary bending and associated torsion problems, the prescribed conditions at each of x = constant edge (with analogous conditions along y = constant edge) in the primary problems are (i) u1 (y) = 0 or σx1 (y) = Tx1 (y)

(5.13a)

(ii) v1 (y) = 0 or τxy 1 (y) = Txy1 (y)

(5.13b)

(iii) w0 (y) = 0 or τxz 0 (y) = Txz0 (y)

(5.14)

The contradiction between zero face shear conditions and prescribed transverse shears along the wall of the plate in the bending problem is resolved earlier [12]. In this earlier analysis, the derived 2-D problems are independent of lamination. Interface continuity of displacements and transverse stresses is through the solution of a supplementary problem defined from Levy’s work [13] in the face ply and recurrence relations across interfaces. It is to be noted that vertical displacement is a face (and interface) variable in bending problems and a domain variable in the associated torsion problems. The corresponding edge conditions in primary extension problems are (i) u = u˜ 0 (y) or σx0 (y) = Tx0 (y)

(5.15a)

(ii) v = v˜ 0 (y) or τxy0 (y) = Txy0 (y)

(5.15b)

(iii) w1 (y) = 0 or τxz1 (y) = Txz1 (y)

(5.16)

(It would be shown later that the condition w1 (y) = 0 cannot be imposed).

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5 Laminated Plates with Anisotropic Plies …

5.3 Analysis of Primary Bending Problems In a primary bending problem, the plate is subjected to zero transverse shear stresses and asymmetric normal stress σz = ±q0 (x, y)/2 along z = ± 1 faces. The primary transverse stresses in each ply in the initial analysis are     k  τxz , τyz = τxz0 , τyz0 + f2 τxz2 , f2 τyz2

(5.17a)

σz = zσz1 + [f3 σz3 ]k

(5.17b)

It is to be noted that [τxz0 , τyz0 , zσz1 ] are independent of lamination and material constants. The second expression consists of reactive stresses in the ply which are also independent of lamination due to the chosen fk (z) functions. In most of the reported layer-wise theories, [τxz0 , τyz0 , zσz1 ] are dependent on material constants due to lamination-dependent analysis. Because of Eq. (5.2), [τxz0 , τyz0 ] are expressed in the form of     τxz0 , τyz0 = α ψ0,x , ψ0,y

(5.18)

  so that ψ0 (x, y), with plane Laplace operator  = ∂ 2 /∂x2 + ∂ 2 /∂y2 , is governed by α2 ψ0 + σz1 = 0

  σz1 = q1 /2

(5.19)

with condition along x = constant edge (with analogous conditions along y = constant edge) ψ0 = 0 or α ψ0,x = Txz0 (y)

(5.20)

Solution for ψ0 from the above auxiliary problem is designated as a universal solution being independent of elastic deformations and material constants. With displacements [w0 , z u1 , z v1 ] and strain–displacement relations, w0f is given by αw0f (x, y) =

  γxz0 dx + γyz0 dy −



  u1 dx + v1 dy

(5.21)

Vertical deflection w0 (x, y) from the integration of εz is a known variable only with w0f (x, y). Otherwise, it remains as an unknown domain variable which is necessary for the associated torsion problem due to its participation in the static equilibrium equations. Static equilibrium equations have to be independent of w0 (x, y) in bending problem to decouple from the torsion problem. For this purpose, [u1 , v1 ] are modified as

5.3 Analysis of Primary Bending Problems

95

  u∗1 = u1 + γxz0 − α w0,x

(5.22a)

  v∗1 = v1 + γyz0 − α w0,y

(5.22b)

(Inclusion of transverse shear strains is necessary for finding face deflection w0f ). With displacements [u, v] = z [u1 , v1 ], determination of [u, v] satisfying equilibrium equations requires f2 [τxz2 , τyz2 ] along with σz = f3 σz3 . Since f3 (αk ) = 0, σz3 becomes a free variable by replacing f3 (z) with f∗3 (z). Normal stress σz along with (z q1 /2) in the extended Poisson’s theory takes the form 1 σz = z q1 − β1 σz3 + f3 σz3 2 

(5.23)

Transverse shear stresses along with those in the auxiliary problem are       τxz , τyz = τxz0 , τyz0 + f2 (z) τxz2 , τyz2

(5.24)

τxz2 = Q44 u1 + Q45 v1 + τxz0

(5.25a)

τyz2 = Q55 v1 + Q54 u1 + τyz0

(5.25b)

in which

Note that [τxz0 , τyz0 ] in Eq. (5.24) are included due to the participation of σz1 in Eqs. (5.1a, 5.1b). From Eqs. (5.1a, 5.1b, 5.2, 5.23 and 5.25a, 5.25b), one obtains  α (Q44 u1 + Q45 v1 ),x +(Q54 u1 + Q55 v1 ),y = β1 σz3

(5.26)

(σ z3 is from the coefficient of f 1 . Note that one cannot prescribe zero τ xz2 (and τ yz2 ) at x (and y) constant edgess since τ xz0 (and τ yz0 ) are independent of elastic deformations.) For the use of [u∗ , v∗ ]1 in the integration of equilibrium Eqs. (5.1a, 5.1b), it is convenient to express [u1 , v1 ] in terms of gradients of two functions [ψ, ϕ] in the form   [u1 , v1 ] = −α ψ1, x + ϕ1, y , ψ1, y − ϕ1, x

(5.27)

In the case of isotropic face ply, ψ1 is governed by the bi-harmonic operator, and ϕ1 by the plane Laplace equation (ϕ1 was originally introduced as ‘stress function’ in [2]). We note here that ϕ1 and harmonic part of bi-harmonic ψ1 are not conjugate to each other. The same set of harmonic functions in ψ1 has to be used in ϕ1 (with different multiplying algebraic coefficients) to replace tangential edge displacement from ψ1 with the normal gradient of ϕ1 . As such, in-plane displacements are in terms

96

5 Laminated Plates with Anisotropic Plies …

of gradients of [ψ1 , ϕ1 ] but without the need to use gradients of ϕ1 in transverse shear stresses. Contributions of ψ1 and w0 in [u∗ , v∗ ]1 are the same in giving corrections to w(x, y, z) and transverse stresses. Hence, w0 in [u∗ , v∗ ]1 is replaced by ψ1 so that [u∗ , v∗ ]1 are   u∗1 = −α 2 ψ1, x + ϕ1 , y + γxz 0

(5.28a)

  v∗1 = −α 2 ψ1, y − ϕ1, x + γyz 0

(5.28b)

    ∗ = Correspondingly, in-plane strains ε∗x , ε∗y , γxy with εx1 , εy1 , γxy1 1       2 −α 2 ψ1,xx + ϕ1,xy , 2 ψ1,yy − ϕ1,xy , 4 ψ1,xy + ϕ1,yy − ϕ1,xx are given by       ε∗x , ε∗y = εx1 + α γxz0,x , εy1 + α γxz0,y

(5.29a)

   ∗ γxy1 = γxy1 + α γxz0,y + γyz0,x

(5.29b)

1

From integration of Eq. (5.2) and equilibrium Eqs. (5.9a, 5.9b) using the strains in Eqs. (5.29a, 5.29b), reactive transverse stresses are (with sum j = 1, 2, 3)       τ∗xz2 = α Q1j εj − Sj6 σz1 ,x + Q3j εj − Sj6 σz1 ,y

(5.30a)

      τ∗yz2 = α Q2j εj − Sj6 σz1 ,y + Q3j εj − Sj6 σz1 ,x

(5.30b)



σz3 = −α τ∗xz2,x + τ∗yz2,y (σz3 is coefficient of f3 )

(5.31)

Noting that σz3 from Eq. (5.26) is negative of the one from Eq. (5.31) due to (f3,zz + f1 ) = 0, the equation-governing in-plane displacements is given by

   αβ1 τ∗xz2,x , +τ∗yz2,y = α (Q44 u1 + Q45 v1 ),x + (Q54 u1 + Q55 v1 ),y

(5.32)

With the condition zero ωz (i.e., v,x = u,y ) required to decouple bending and torsion, Eq. (5.32) consists of Laplace equation ϕ1 = 0 and a fourth-order equation in ψ1 to be solved with the following three conditions at x = constant edges (with similar conditions at y = constant edges) 

  (i) u∗ or σ∗ 1 = 0, (ii) v∗ or τ∗xy = 0

(5.33a)

(iii) ψ1 or τ∗xz2 = 0

(5.33b)

1

5.3 Analysis of Primary Bending Problems

97

5.3.1 Corrective In-plane Displacements in the Face Ply Transverse stresses in the face ply are       τxz , τyz = τxz0 , τyz0 + f2 τxz2 , τyz2

(5.34)

σz = z σz1 + f3 σz3

(5.35)

Corrective in-plane displacements in the supplementary problem are assumed as [u, v]s = [u1 , v1 ]s sin(π z/2)

(5.36)

In the case of homogeneous isotropic plate, single term sin(π z/2) implied in Levy’s work [13] is shown to be adequate for the proper estimation of [w(x, y, z)]z=0 (see Appendix). All shear deformation theories with a two-term representation of [u, v] give more or less same estimates around 3.48 to the maximum neutral plane deflection wn (see [15, 16]). In-plane stresses are σis = Qij ε1js sin(π z/2) (i, j = 1, 2, 3)

(5.37)

From integration of equilibrium equations using in-plane stresses in Eq. (5.37)    along with τxz , τyz = τxz2 , τyz2 s cos(π z/2) and σz3 = σz3s sin(π z/2), transverse stresses in the supplementary problem are given by   τxz2s = −(2/π)α Q1j ε1sj,x + Q3j ε1sj,y

(5.38a)

  τyz2s = −(2/π)α Q2j ε1sj,y + Q3j ε1sj,x

(5.38b)

  σz3s = (2/π)2 α2 Q1j ε1sj,xx + 2Q3j ε1sj,xy + Q2j ε1sj,yy

(5.39)

In-plane distributions u1s and v1s are added as corrections to the known in-plane displacements [u1 , v1 ] so that [u, v] in the supplementary problem are [u, v] = [(u1 + u1s ), (v1 + v1s )] sin(π z/2)

(5.40)

From Eqs. (5.32, 5.38a, 5.38b and 5.39), one obtains   β1 σz3 = (2/π)2 α2 Q1j ε1sj,xx + 2 Q3j ε1sj,xy + Q2j ε1sj,yy

(5.41)

  By expressing [u13 , v13 ] = −α ψ1s,x , ψ1s,y , Eq. (5.41) becomes a fourth-order equation in ψ1s to be solved with two in-plane conditions at x = constant edges (with similar conditions at y = constant edges)

98

5 Laminated Plates with Anisotropic Plies …

  (i) (u1s or σxs1 ) = 0, (ii) v1s or τxys1 = 0

(5.42)

5.3.2 Continuity of Displacements and Transverse Stresses Across Interfaces In-plane displacements and transverse stresses in the face ply, now, become u = zu1 + (u1 + u1s ) sin(π z/2)

(5.43a)

  v = zv1 + y1 + v1s sin(π z/2)

(5.43b)

τxz = τxz0 + f2 τxz2 + (τxz2 + τxz2s )(π/2) cos(π z/2)

(5.44a)

  τyz = τyz0 + f2 τyz2 + τyz2 + τyz2s (π/2) cos(π z/2)

(5.44b)

σz = z(q/2) + [f3 − sin(π z/2)]β1 σz3 − σz3s sin(π z/2)

(5.45)

In the interior plies, displacements [u1 , v1 ], thereby, in the neighboring plies are obtained from the solution of the sixth-order system of Eqs. (5.1a, 5.1b) governing [u1 , v1 ]. They are dependent on material constants but independent of lamination. Displacements [u1 , v1 ]s and transverse stresses [τxz2 , τyz2 , σz3 ]s are obtained from continuity conditions across interfaces. Continuity of u (with similar expressions for v) across interfaces is simply assured through the following recurrence relations       (k+1) (k+1) (k) (k+1) (k) u(k) sin α u + − u = α − u + sin α − u u [α ] k k k k 1s 1s 1 1 1 1

(5.46)

Since [τxz0 , τyz0 ] and σz1 = z q/2 are the same throughout the laminate, recurrence relations for τxz2 (with similar expressions for τyz2 ) and σz3 are f(k+1) = (αk )τ(k+1) 2 xz2

    (k+1) (k+1) cos αk τ(k) xz2s − τxz 2 s + τxz2 (k) − τxz2

(5.47)

      (k) (k+1) (k) (k+1) (k+1) (k) (k) σz3s − σz3s + β1 σz3 − σz3 (k + 1) sin αk = β1 f3 − f3 (αk )σz3 (αk )σz3

(5.48)

5.3 Analysis of Primary Bending Problems

99

5.3.3 Vertical Deflection w(x, y, z) With εz1 from constitutive relation, vertical deflection w(x, y, z) is given by w = w0 − f2 εz1 + (π/2)wos cos(π z/2)

(5.49)

Note that vertical deflections w0 and w0s are obtained from the integration of shear strain–displacement relations (whereas εz1 which does not participate in the determination of [u1 , v1 ]b is obtained from ε6 in the constitutive relations (5.5) in the interior of each ply). They are

  (ε40 − u1 )dx + (ε50 − v1 )dy

(5.50)

  (ε40 − u1 s )dx + (ε50 − v1s )dy

(5.51)

α w0 = α w0 s =

Due to zero face shear conditions, ε40 and ε50 are zero in the face ply and w0 corresponds to face deflection. To satisfy edge support condition, one needs only a support to prevent vertical movement of intersections of supported segments of the faces and wall of the plate. • Continuity across interfaces Continuity across interfaces gives the recurrence relation     (k+1) cos αk = α w(k+1) − [f2 (αk )εz1 ](k+1) − w(k) α w(k) 0s − w0s 0 0

(5.52)

5.4 Higher-Order Corrections in the Ply from the Iterative Procedure The above solutions for displacements and transverse stresses are initial solutions in the iterative procedure in solving 3-D problems to generate a proper sequence of sets of 2-D equations. The only error concerning 3-D problems is in the transverse shear strain–displacement relations. Displacements f3 [u3 , v3 ] (thereby, εz3 ) in consistent with f2 [τxz2 , τyz2 ] and reactive transverse stresses (τxz4 , τyz4 , σz5 ) have to be obtained from the first stage of the iterative procedure. Initial transverse shear strains from constitutive relations are γxz2 = S44 τxz2 + S45 τyz2 , γyz2 = S54 τxz2 + S55 τyz2

(5.53)

100

5 Laminated Plates with Anisotropic Plies …

Displacements [u3 , v3 ] are modified such that they are corrections to face parallel plane distributions of the preliminary solution so that w = f2 (z)(w2 − εz1 )

(5.54)

u∗3 = u3 + γxz2 − α(w2 − εz1 ),x

(5.55a)

v∗3 = v3 + γyz2 − α(w2 − εz1 ),y

(5.55b)

in which [u, v]3 = [u, v]1 + [u, v]3c with [u,v]3c denoting corrections due to transverse shear strain–displacement relations.

5.4.1 Correction to Vertical Displacement The corresponding correction to vertical displacement is w2 , which is treated as a virtual variable in strain–displacement relations due to its real contribution in the integrated equilibrium equations. Due to replacement of f5 by f∗5 , one gets after some algebra     α β3 τxz4,x + τyz4,y = α2 (Q44 u3 + Q45 v3 ),xx + (Q54 u3 + Q55 v3 ),yy + α2 σz1 (5.56) With the condition zero ωz (v,x = u,y ) necessary for the analysis of bending problem, Eq. (5.56) consists of Laplace equation ϕ3 = 0 and a fourth-order equation in ψ3 to be solved with conditions at x = constant edges (with analogous conditions along y = constant edges) (i) u∗3 or σ3∗ = 0 , (ii)v3 or τ∗xy3 = 0, (iii)ψ3 or τ∗xz4 = 0.

(5.57)

5.4.2 Corrective Displacements in the Face Ply Corrective displacements in the supplementary problems are assumed in the form of: w = w2s (π/2) cos(π z/2)

(5.58)

[us , vs ] = [u3s , v3s ] sin(π z/2)

(5.59)

5.4 Higher-Order Corrections in the Ply from the Iterative Procedure

101

σ3 si = Qij ε3sj (i, j = 1, 2, 3)

(5.60)

Analysis here is a repetition of the corresponding analysis in the supplementary problem in Section 5.3.1. It is to be noted that the in-plane strains corresponding to displacements in Eq. (5.62) are (ε* 3j + ε3sj ) and represented as ε* sj(3) in Eq. (5.63).   τxz2s = −(2/π)α Q1j ε3sj,x + Q3j ε3sj,y

(5.61a)

  τyz2s = −(2/π)α Q2j ε3sj,y + Q3j ε3sj,x

(5.61b)

 ∗    u3 + u3s , v∗3 + v3s sin(π z/2)

[u, v] =

  β3 σz5 = (2/π)2 α2 Q1j ε∗sj,xx + 2Q3j ε∗sj,xx + Q2j ε∗sj,yy

(5.62)

(3)

(5.63)

• Corrective displacements Corrective displacements are w = w04 (x, y)−f4 εz3 + w04s (π/2)cos(π z/2)

(5.64)

  u = f3 u∗3 + u∗3 + u3s sin(π z/2)

(5.65a)

  v = f3 v∗3 + v∗3 + v3s sin(π z/2)

(5.65b)

α w04 = α w04s =



    (γxz2 − u3 )dx + γyz2 − v3 dy

(5.66)

   (γxz2s − u3s )dx + γyz2s − v3s dy

(5.67)

Continuity of [τxz4 , τyz4 , σz5 ] across interfaces is not presented here since these stresses require further correction due to the next higher-order terms. Recurrence relations for continuity of displacement u (with similar expressions for v and analogs recurrence relations for w04 and w04s ) are given by     (k+1) u(k) sin(παk /2) = αk u1 (k+1) − u1 (k) 3s − u3s    + f3 (k+1) (αk ) + sin(παk /2) u∗3 (k+1) − u∗3 (k)

(5.68)

Successive use of Eq. (5.68) starting from top ply ensures continuity of displacements across the interfaces.

102

5 Laminated Plates with Anisotropic Plies …

In the face ply, the equations governing [ψ3 , ϕ3 ] and ψ3s consist of a sixth and fourth-order system of equations, respectively. Similar sets of equations govern [ψ, ϕ]2n+3 and ψ2n+3 s (n = 1, 2, 3 …), corresponding to higher-order displacement terms. In the interior plies, only a sixth-order system of equation govern [ψ, ϕ]2n+3 .

5.5 Associated Torsion Problems Kirchhoff’s theory and FSDT due to their inherent deficiencies are not suitable in the analysis of face ply but w0 (x, y) as domain variable is required in the analysis of associated torsion problems. In the analysis, it is convenient and proper to include transverse shear stresses [τxz0 , τyz0 ] and σz = z q/2 used in the Poisson’s theory. Here, the analysis requires only modification of the analysis of bending problem replacing ψ1 with vertical displacement w0 (x, y) so that w = w0 (x, y)

(5.69)

    [u, v] = −α z w0, x + ϕ1, y , w0, y − ϕ1, x

(5.70)

τxz = τxz0 + f2 (τxz0 + τxz2 )

(5.71a)

  τyz = τyz0 + f2 τyz0 + τyz2

(5.71b)

σz3 =

1 z q + f3 σz3 2

(5.72)

Note that the transverse shear strains from strain–displacement relations are [−α ϕ1,y , α ϕ1,x ] which correspond to self-equating stresses. From zero face shear conditions, one gets an additional face deflection w0c (x, y) given by αw0c =

  ϕ1, y dx − ϕ1, x dy

(5.73)

The w0c is a face variable that becomes zero with prescribed zero w0 at the edge of the plate by preventing vertical movement of the intersection of the face with the cylindrical surface of the sidewall of the plate. For obtaining second-order corrections, ψ3 is replaced by w2 which is treated as a virtual variable in strain–displacement relations with the real contribution in the integrated equilibrium equations.

5.6 Extension Problems

103

5.6 Extension Problems In a primary extension problem, the plate is subjected to symmetric normal stress σz0 = q0 (x, y)/2, asymmetric shear stresses [τxz , τyz ] = ± [Txz1 (x, y), Tyz1 (x, y)] along the top and bottom faces of the plate. Due to out-of-plane equilibrium equation, applied face shears [Txz , Tyz ] have to be gradients of a harmonic function ψ1 so that [Txz , Tyz ] = − α [ψ1,x , ψ1 ,y ]. Transverse shear stresses and normal stress satisfying face conditions are 

   1 τxz , τyz = −α z ψ1,x , ψ1,y , σz0 = q0 (x, y) 2

(5.74)

The above transverse stresses which are independent of material constants remain the same in the interior of the laminate. One should note here that σz0 does not participate in the equilibrium equations but contributes to the in-plane constitutive relations. [τxz , τyz ] in the above equation are related to in-plane displacements [u0 , v0 ] through equilibrium Eqs. (5.1a, 5.1b). From constitutive relation, 1 εz0 = S6j σj0 + S66 q0 (j = 1, 2, 3) 2

(5.75)

Correspondingly, vertical deflection w is linear in z and cannot be prescribed to be zero at the edge of the plate due to S6j σj0 even if the faces are free of transverse stresses. It is complementary to the fact that transverse shear strains in the bending problem from strain–displacement relations, thereby, transverse shear stresses cannot be prescribed to be zero along the wall of the plate even if the faces are free of transverse stresses. It, in a way, justifies the Poisson’s theory of bending of plates.

5.6.1 Preliminary Analysis of the Ply In-plane equilibrium equations in the preliminary analysis with [u, v] = [u0 (x, y), v0 (x, y)] are       Q1j εj0 − Sj6 σz0 ,x + Q3j εj0 − Sj6 σz0 ,y = αψ1,x

(5.76a)

      Q2j εj0 − Sj6 σz0 ,y + Q3j εj0 − Sj6 σz0 ,x = αψ1,y

(5.76b)

subjected to the edge conditions (15) at x (and y) constant edges. Solutions of [u0 , v0 ] from the above equations concerning the 3-D problem are in error in transverse shear strain–displacement relations due to w1 = εz0 (i.e., w = z εz0 ) from constitutive relation. In the analysis of laminated plates, transverse

104

5 Laminated Plates with Anisotropic Plies …

stresses along interfaces are important even in the case of applied face stresses, and edge transverse shear stresses are zero. Moreover, the analysis of the face and interior plies in the layer-wise theory adopted in the present work is independent of lamination. As such, one requires higher-order approximate solutions even in the homogeneous plate problem.

5.6.2 Higher-Order Corrections in the Ply One should note that transverse shear stresses are zero through-thickness at locations of zeros of gradients of ψ. To overcome this limitation, transverse stress components are assumed in the following form τ∗xz2n+1 = (τxz2n+1 − β2n−1 τxz2n−1 )

(5.77a)

  τ∗yz2n+1 = τyz2n+1 − β2n−1 τyz2n−1

(5.77b)

∗ σz2n+2 = σz2n+2 − β2n−1 σz2n

(5.78)

At the nth stage of iteration (n ≥ 1), transverse stresses [τxz , τyz ]2n-1 , and w2n-1 are known in the preceding stage. Concerning in-plane displacements, one should include additional terms such that they are consistent with known stresses [τxz , τyz ]2n-1 and are free to obtain stresses [τxz , τyz ]2n+1 , σz2n+2 , and w2n+1 . We have from constitutive relations, γxz2n−1 = S44 τxz2n−1 + S45 τyz2n−1

(5.79a)

γyz2n−1 = S54 τxz2n−1 + S55 τyz2n−1

(5.79b)

Modified displacements and the corresponding derived quantities denoted with * are with w2n-1 as correction to εz2n-2 due to [u, v]2n u∗2n = u2n − α(εz2n−2 + w2n−1 ),x + γxz2n−1

(5.80a)

v∗2n = v2n − α(εz2n−2 + w2n−1 ),y + γyz2n−1

(5.80b)

Strain–displacement relations give ε∗x2n = εx2n − α2 (εz2n−2 + w2n−1 ),,xx + αγxz2n−1,x

(5.81a)

ε∗y2n = εy2n − α2 (εz2n−2 + w2n−1 ),yy + α γyz2n−1,y

(5.81b)

5.6 Extension Problems

105

  ∗ γxy2n = γxy2n − 2α2 (εz2n−2 + w2n−1 ),xy + α γxz,y + γyz,x 2n−1

(5.81c)

∗ γxz2n−1 = γxz2n−1 − (u2n−2 + u2n )

(5.82a)

∗ = γyz2n−1 − (v2n−2 + v2n ) γyz2n−1

(5.82b)

In-plane stresses and transverse shear stresses from constitutive relations are    ∗ σi 2n = Qij ε∗j

2n

(i, j = 1, 2, 3)

(5.83)

τ∗xz2n−1 = τxz2n−1 − (Q44 u + Q45 v)2n

(5.84a)

τ∗yz2n−1 = τyz2n−1 − (Q54 u + Q55 v)2n

(5.84b)

∗ = One gets from Eqs. (5.1a, 5.1b, 5.2, 5.77a, 5.77b and 5.78) noting that σz2n σz2n − β2n−1 σz2n+2

  α (Q44 u2n + Q45 v2n ),x + (Q54 u2n + Q55 v2n ),y + β2n−1 σz2n+2 = 0 (Note that w2n−1 is not present in the above equation)

(5.85)

For the use of [u*, v*]2n in the integration of equilibrium equations, displacements [u, v]2n are expressed in the form   [u, v]2n = −α ψ2n,x , ψ2n,y

(5.86)

Contributions of ψ2n and w2n-1 in [u∗ , v∗ ]2n are the same in giving corrections to w(x, y, z) and transverse stresses. (In fact, the contribution of w2n-1 is through strain– displacement relations in static equilibrium equations, and through constitutive relations in through-thickness integration of equilibrium equations.) Hence, w2n-1 in of w2n-1 used in strain–displacement [u∗ , v∗ ]2n is replaced   by ψ2n (to be independent ∗ ∗ ∗ ∗ ∗ are relations) so that u , v , εx , εy , γxy 2n

  = 2u2n + γxz2n−1 − αεz2n−2,x

(5.87a)

  v∗2n = 2v2n + γyz2n−1 − α εz2n−2,y

(5.87b)

  ε∗x2n = 2εx2n + α γxz2n−1,x − α2 εz2n−2,xx

(5.88a)

  ∗ εy2n = 2εy2n + α γyz2n−1,y − α2 εz2n−2,yy

(5.88b)

u∗2n

106

5 Laminated Plates with Anisotropic Plies …

    ∗ γxy2n = 2 γxy2n + α γxz2n−1,y + γyz2n−1,x − 2α2 εz2n−2,xy

(5.88c)

Note that the role of w2n-1 is in its contribution to the integrated equilibrium equations where as it is a virtual quantity in transverse strain–displacement relations. From the integration of equilibrium equations using the strains in Eqs. (5.88a, 5.88b and 5.88c), reactive transverse stresses are  

 ∗ ∗ = α Q1j εj − Sj6 σz + Q3j εj − Sj6 σz , (j = 1, 2, 3)

(5.89a)

 

 τ∗yz2n+1 = α Q2j ε∗j − Sj6 σz + Q3j ε∗j − Sj6 σz , (j = 1, 2, 3)

(5.89b)

τ∗xz2n + 1

,x

,y



σz2n+2 = −α τ∗xz,x + τ∗yz,y

,y 2n,

,x 2n

2n+1

(5.90)

One equation-governing in-plane displacements (u, v)2n , noting that σz2n+2 from Eq. (5.85) is negative of the one from Eq. (5.90) due to (f2n+1,zz + f2n-1 ) = 0, is given by

 αβ2n−1 τ∗xz,x + τ∗yz,y

2n+1

  = α (Q44 u + Q45 v),x + (Q54 u + Q55 v),y 2n

(5.91)

With the second equation v2n,x = u2n,y , the above equation becomes a fourth-order equation in ψ2n to be solved along with harmonic function ϕ2n with three conditions along constant x = constant edges (with analogous conditions along y = constant edge)  ∗ (i) (u2n or σ2n )∗ = 0, (ii) v2n or τxy2n = 0 (iii)τ∗xz2n−1 = 0

(5.92)

• Continuity of displacements across interfaces Since [u, v, σz ] are in terms of even functions f2n (z), continuity of u, v, and σz across interface z = αk , (k = 1, 2, n-1), is ensured except u0 and v0 . Note that the analysis of face ply is independent of lamination. In the bending problem, the solution of a supplementary problem in the face ply is used to ensure continuity of displacements across interfaces. Here also, one should adopt the same procedure with the interchange of sine and cosine trigonometric terms used in the bending problem. Continuity of [u0 , v0 ] is achieved through the solution of the following supplementary problem.

5.7 Un-symmetrical Laminates

107

5.7 Un-symmetrical Laminates Here, suffix n in (−hn ) of the bottom face is replaced by m with the number of layers ‘m’ in the bottom-half z ≤ 0 need not be equal to the number of layers ‘n’ in the upper-half z ≥ 0. The initial set of solutions in the upper-half of the laminate in all the problems presented above are unaltered up to the reference plane z = 0. One has to consider the continuity of non-zero displacements and transverse stresses across the reference plane. They will be different due to asymmetry from a similar analysis in the bottom-half of the laminate. A novel procedure is proposed here to maintain the necessary continuity across z = 0 plane in bending problems and a similar procedure in extension problems. The corresponding procedures in torsion problems that involve simple modifications are not presented.

5.7.1 Bending Problem From the analysis of the upper-half the laminate, σz = 0 and transverse shear stresses including first-order corrections due to σz1 in the in-plane constitutive relations along the reference plane z = 0 are   (τxz )z=0 = τxz0 + f2 (0)τxz2 + (π/2) τ∗xz2 + τxz2s

(5.93a)

    τyz z=0 = τyz0 + f2 (0)τyz2 + (π/2) τ∗yz2 + τyz2s

(5.93b)

in which τ∗xz2 = τxz0 + (Q44 u1 + Q45 v1 )c τ∗yz2 = τyz0 + (Q54 u1 + Q55 v1 )c −

(5.94) −

Continuation of the same analysis with z (= − z) ≥ 0, one obtains along z = 0 − plane that normal stress σ z = 0 and   τxz = τxz0 + f2 (0)τxz2 + (π/2) τ∗xz2 + τxz2s

(5.95a)

  τyz = τyz0 + f2 (0)yz2 + (π/2) τ∗yz2 + τyz2s

(5.95b)

  τ∗xz2 = τ∗xz0 + Q44 u1 + Q45 v1 c

(5.96a)

  τ∗yz2 = τ∗yz0 + Q54 u1 + Q55 v1 c

(5.96b)

in which

108

5 Laminated Plates with Anisotropic Plies …

5.7.2 Associated Extension Problem in Bending In the initial set of solutions, transverse shear stresses obtained along z = 0 plane are sum of the stresses in Eqs. (5.110, 5.112). For continuity of these stresses across the z = 0 interface, one has to consider the adjacent plies above and below the interface subjected to shear stresses   τxz = ±[τxz − τxz ]z=0 ; τyz = ± τyz − τyz z=0

(5.97)

Continuity of these stresses is ensured by adding solutions of the laminate with free top and bottom faces along with the above stresses in the adjacent plies of the interface z = 0 to the solutions of problems in the initial set. Continuity of these stresses ensures also continuity of vertical displacement across z = 0 plane. It is convenient to introduce the coordinate z = (1 − z) for (z ≥ 0) so that the reference plane z = 0 corresponds to z = 1. hk = 1 − hk , interfaces αk = (1 − αk ). Here, q = 0 along z = 1 and the faces z = 0 are free of transverse stresses. It is inconvenient to use linear z τxz along edges satisfying the above face conditions since the corresponding solutions for in-plane displacements u0 , v0 from in-plane equilibrium equations are lamination-independent, thereby not satisfying continuity across interfaces. As such, the edge conditions at x = constant edges (and analogous conditions along y = constant edges) are assumed in the form   τxz = τxz (y) sin π z /2

(5.98)

    Since τxz , τyz are gradients α ψ1,x , ψ1,y of a harmonic function ψ1 and ∂/∂z =     −∂/∂z , in-plane static equilibrium equations governing u0 , v0 cos π z /2 are   α Q1j εj,x + Q3j εj,y = α (π/2)ψ1,x

(5.99a)

  α Q2j εj,y + Q3j εj,x = α (π/2)ψ1,y

(5.99b)

Solutions of theabove equations with zero bending and twisting stresses along edges give u0 , v0 in each ply independent of lamination. Continuity of these displacements across interfaces is through recurrence relations       (k+1) u(k) cos πα k/2 = u(k+1) − u(k) 0 − u0 0 0

(5.100)

  Normal strain εz = εz0 cos πz /2 from constitutive relation in which ε z0 is given by εz0 = Sij σj , (i, j = 1, 2, 3)

(5.101)

5.7 Un-symmetrical Laminates

109

  Vertical deflection w = −εz0 (2/π) sin π z /2 from the integration of εz in the interior of the ply. This deflection along the interface is obtained from shear stress– strain and strain–displacement relations in the form   α w = (π/2) cos π α k/2





    ε40 − u0 dx + ε50 − v0 dy

(5.102)

Its continuity across interfaces is through recurrence relations     α ε(k) − ε(k−1) z0 (2/π) sin π αk /2 = (π/2)



   ε40 − u0 dx+

 (k)    + ε50 − v0 dy cos π αk /2

(5.103)

A similar analysis is to be carried out with (= − z) ≥ 0 and z = (1 − z) so that z = 0 plane is z = 1 (this part of the analysis is omitted). By adding the above vertical displacements to the corresponding displacements obtained in the upperhalf and bottom-half of the relevant symmetric laminate ensures continuity across the reference plane. One can choose, in principle, anyone interface (excluding faces of the laminate) as reference plane but from consideration of limitations of small deformation theory, it is better to choose either mid-plane or its adjacent interface as a reference plane.

5.7.3 Extension Problem Along the reference plane z = 0, w and transverse shear stresses are zero and in-plane displacements and σz are (sum n ≥ 1)    u = u0 + u∗2 + u2s + f2n (0)u2n

(5.104a)

   v = v0 + v∗2 + v2s + f2n (0)v2n

(5.104b)

σz =

  ∗ 1 q (x, y) + σz2 + σz2s + f2n (0)σz2n 2 0

Continuation of the same analysis with z (= − z) ≥ 0, u, v and σ plane are

(5.105) z

along = 0

  u = u0 + u∗2 + u2s + f2n (0)u2n

(5.106a)

  v = v0 + v∗2 + v2s + f2n (0)v2n

(5.106b)

110

5 Laminated Plates with Anisotropic Plies …

σz =

  1 q0 (x, y) + σ∗z2 + σz2s + f2n (0)σz2n 2

(5.107)

Normal stress σz across the z = 0 plane has to be the same in the adjacent ply on each side of the z = 0 plane. Continuity of (u, v) is required only if the reference plane is not an interface of plies. For this purpose, one needs solutions to the associated bending problems.

5.7.4 Associated Bending Problem in Extension In the initial set of solutions, in-plane displacements obtained along z = 0 plane are the sum of the displacements in Eqs. (5.104a, 5.104b and 5.106a, 5.106b). For continuity of these displacements across z = 0 interface, one has to consider the adjacent plies above and below the interface z = 0 with u = ±[u − u]; v = ±[v − v]; σz = ±[σz − σz ]

(5.108)

Continuity of u, v, and σz is ensured by adding solutions of the laminate with free top and bottom faces along with above displacements and σz in the adjacent plies of the interface z = 0 to the solutions of problems in the initial set. It is convenient to introduce the coordinate z = (1 − z) for (z ≥ 0) so that z = 1 is reference plane z = 0, hk = 1−hk and αk = (1 − αk ) are interfaces. Here, σz = [σz − σz ] and u , v = [(u − u), (v − v)] along z = 1 plane and the faces z = 0 are free of transverse stresses. It is convenient to assume σz = [σz − σz ] sin πz /2 . Then, equation-governing ψ is α2 ψ 1 = σz with ∂/∂z = −∂/∂z and     τxz (π/2) cos πz /2 = α ψ1,x (π/2) cos πz /2

(5.109a)

    τyz (π/2) cos π z /2 = α ψ1,y (π/2) cos π z /2

(5.109b)

  The above equation is to be solved with zero normal gradient ψ1 n along the edge of the plate.     In-plane static equilibrium equations governing u0 , v0 sin πz /2 are   α Q1j εj,x + Q3j εj,y + α (π/2)2 ψ1,x = 0

(5.110a)

  α Q2j εj,y + Q3j εj,x + α (π/2)2 ψ1,y = 0

(5.110b)

Solutions of theabove equations with zero bending and twisting stresses along edges give u0 , v0 in each ply independent of lamination. Continuity of these

5.7 Un-symmetrical Laminates

111

displacements and σz  across interfaces is through recurrence relations       (k+1) (k+1) (k)  u(k) sin π α − u /2 = u − u k 0 0 0 0

(5.111a)

      (k+1) v(k) sin π αk /2 = v(k+1) − v(k) 0 − v0 0 0

(5.111b)

 (k)      σz − σz(k+1) sin παk /2 = σz(k+1) − σz (k)

(5.112)

A similar analysis is to be carried out with (z = − z) ≥ 0 and z = (1-z) so that z = 0 plane is z = 1 (this part of the analysis is omitted). By adding in-plane displacements and normal stress σz with those in the analysis of the upper-half and bottom-half of relevant symmetric laminates ensure continuity across reference plane.

5.8 Conclusions An attempt is made here to present a proper sequence of sets of 2-D problems necessary for the analysis of laminated plates within the small deformation theory of elasticity. Emphasis is on the usage of vertical displacement variable. If it is used as a domain variable, analysis corresponds to the solution of associated torsion problem in which normal strains are not zero unlike in the St. Venant’s torsion problem. In bending and extension problems, it cannot be used as a domain variable. In the interior of the domain, it is from the thickness-wise integration of normal strain z . Zero vertical displacement along the edge of the plate is to be replaced by zero z . Displacement w(x, y) arising out of the integration of z is to be obtained as a face variable from the integration of transverse strain–displacement relations. Zero w(x, y) along the prescribed edge condition requires only the prevention of vertical movement of the line segment of the intersection of face and wall of the plate. 2-D variables associated with assumed polynomials in z of vertical displacement are virtual quantities in transverse shear strain–displacement relations but contribute to higher-order corrections in stress components from the integration of equilibrium equations. The set of polynomials generated in z is necessary for satisfying both static and integrated equilibrium equations. (It is, however, not simple to develop software for generation of fk (z) functions and β2k+1 , necessary for the application of the theory with thickness ratio varying up to unit value.) This problem is avoided in the next chapter. One significant feature of the present work is that the ply analysis is independent of lamination. Moreover, corrections to the solutions at each stage of the adapted iterative procedure are determined without disturbing solutions in the preceding stages of iterations. This facility is not available with solutions from PEEEs.

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5 Laminated Plates with Anisotropic Plies …

The present theory needs exploitation in investigations on optimum ply lay-up, its utility in the analysis of associated eigenvalue problems of free vibration and buckling of plates, and even in the area of fracture mechanics. However, polynomials in z are not adequate for proper solutions to 3-D problems. The solution of a supplementary problem based on appropriate trigonometric function in z representing each of displacement and stress components is required. A solution to the additional similar problem is required in the analysis of un-symmetrical laminates. Highlights of the present work • 3-D equations in displacements and sequence of 2-D problems with vertical displacement as domain variable correspond to associated torsion problems. • Kirchhoff’s theory is a zeroth-order shear deformation theory. • Poisson’s theory is necessary to rectify lacuna in Kirchhoff’s theory. • Poisson’s theory is based on the satisfaction of both static and integrated equilibrium equations. • Solutions of auxiliary and supplementary problems are necessary. • fk (z) functions are chosen such that ply analysis is independent of lamination. • A novel procedure is proposed for the analysis of un-symmetrical laminates. • More or less uniform accuracy of face parallel plane deflections along each normal to the plane is achieved through the solution of this secondary problem. Such approximation to deformations of parallel planes is useful in the analysis of laminates embedded with piezoelectric actuators.

References 1. Liew KM, Pan ZZ, Zhang LW (2019) A overview of layerwise theories for composite laminates and structures: development, numerical implementation, and application. Compos Struct 216:240–259 2. Birman V, Kardomateas GA (2018) Review of current trends in research and applications of sandwich structures. Compos B 142:221–240 3. Herakovich CT (2012) Mechanics of composites: a historical review. Mech Res Commun 41:1–20 4. Rasoul K et al (2012) The development of laminated composite plate theories: a review. J Mater Sci 47:5901–5910 5. Ahn J, Basu P, Woo K (2011) Hierarchic layer models for anisotropic laminated plates. KSCE J Civil Eng 15(6):1067–1080. https://doi.org/10.1007/s12205-011-1142-8 6. Carrera E, Brischetto S (2009) A survey with numerical assessment of classical and refined theories for the analysis of sandwich plates. Appl Mech Rev 62(1):010803. https://doi.org/10. 1115/1.3013824 7. Chen Wanji and Wu Zhen (2008) A selected review on recent development of displacementbased laminated plate theories. Recent Pat Mech Eng 1:29–44 8. Ghugal YM, Shimpi RP (2002) (2002): A review of refined shear deformation theories of isotropic and anisotropic laminated plates. J Reinf Plast Compos 21:775. https://doi.org/10. 1177/073168402128988481 9. Kant T, Swaminathan K (2000) Estimation of transverse/inter-laminar stresses in laminated composites—a selective review and survey of current developments. Compos Struct 49:65–75

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10. Altenbach H (1998) Theories for laminated and sandwich plates—a review. Mech Compos Mater 34:243–252 11. Reddy JN, Robbins DH Jr (1994) Theories and computational models for composite laminates. App Mech Rev 47(6):147–169 12. Reddy JN (1990a) A review of refined theories of laminated composite plates. Shock Vib Dig 22(7):3–17 13. Madenci E, Ozotok A (2020) Variational approximate for higher-order bending analysis of laminated composite plates. Struct Eng Mech 73(1):97–108. https://doi.org/10.12989/sem.2020.73. 1.097 14. Vijayakumar K (2013) On uniform approximate solutions in bending of symmetric laminated plates. CMC: Comput Mater Contin 34(1):1–25 15. Vijayakumar K (2011) Layer-wise theory of bending of symmetric laminates with Isotropic plies. AIAA J 49(9):2073–2076 16. Reddy JN (1990b) On refined theories of composite laminates. Meccanica 25:230–238 17. Noor AK, Scott W, Burton. (1989) Assessment of shear deformation theories for multilayered composite plates. App Mech Rev 42(1):1–13 18. Rakesh k Kapania, Stefano Raciti, (1989) Recent advances in analysis of laminated beams and plates, part I: shear effects and buckling. AIAA J 27(7):923–934 19. Hashin Z (1983) Analysis of composite materials. Transactions of ASME. J Appl Mech 50:481– 505 20. Tessler A, Di Sciuva M, Gherlone M (2010) Refined Zigzag theory for homogeneous, laminated composite, and sandwich plates: a homogeneous limit methodology for Zigzag function selection. NASA/TP 292010216214:1 21. Demasi L (2008) ∞6 mixed plate theories based on the generalized unified formulation, part IV: zig-zag theories. Compos Struct. https://doi.org/10.1016/j.compstruct.2008.07.010 22. Carrera E (2003) Historical review of zig-zag theories for multilayered plates and shells. Appl Mech Rev 56(3):287–308. https://doi.org/10.1115/1.1557614 23. Mittelstedt C, Becker W (2007) Free-edge effects in composite laminates. Appl Mech Rev 60(5):217. DOI: https://doi.org/10.1115/1.2777169 24. Vijayakumar K (2016) Exact analysis of laminated plates with anisotropic plies. JMEST 3(12), ISSN: 2458–9403 pdf 42351959

Chapter 6

Poisson Theory of Plates with Fourier Sinusoidal Series

Nomenclature a fn (z) fn (k) (z) 2h n O-X Y Z Qij Qrs q0 q1 Sij Srs tk [Tx , Txy , Txz ] [Ty , Txy , Tyz ] U, V, W [u, v, w] [x, y, z]  α(αn ) αk β2n − 1 β2n − 1 (k) [εx , εy , γxy ] [γxz , γyz , εz ] [σx , σy , τxy ] [τxz , τyz , σz ] ωz

Side length of a square plate Thickness-wise (z-) distribution functions, n = 0, 1, 2, 3, … Ply-wise distribution of fn (z) Plate thickness Number of plies Cartesian coordinate system Stiffness coefficients, (i, j = 1, 2, 3) Stiffness coefficients, (r, s = 4, 5) Prescribed face load intensity in extension problem Prescribed face load intensity in bending problem Elastic compliances, (i, j = 1, 2, 3, 6) Elastic compliances, (r, s = 4, 5) (αk − αk−1 ), thickness of kth ply Prescribed stresses at each of x = constant edges Prescribed stresses at each of y = constant edges Displacements in X, Y, Z directions, respectively [U, V, W]/h [X/a, Y/a, Z/h] Plane Laplace operator (∂ 2 /∂x2 + ∂ 2 /∂y2 ) hn /a hk /a, upper-face of kth ply in the upper-half of the laminate [f2n+1 /[α 2 f2n−1 ]z=1 , n = 1, 2, … [f2n+1 /[αk2 f2n−1 ]z=αk , (k = 1, 2,…, n), n = 1, 2, … εi , (i = 1, 2, 3), in-plane strains ε3+i , (i = 1, 2, 3), transverse stains σi , (i = 1, 2, 3), in-plane stresses σ3+i , (i = 1, 2, 3), transverse stresses α(v,x − u,y ), rigid body rotation about z-axis

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 K. Vijayakumar and G. K. Ramaiah, Poisson Theory of Elastic Plates, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-33-4210-1_6

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6 Poisson Theory of Plates with Fourier Sinusoidal Series

6.1 Introduction Despite several review articles on plate theories reported in the literature, the present work is essentially due to Jemielita’s 217-page survey [1] in which an attempt was made to answer the general question, ‘To study or to create.’ In this context, the review article of a few selected theories of plates in bending [2] and the development of present Poisson theory in the analysis of laminated composite plates with anisotropic plies within the small deformation theory of elasticity are presumably significant contributions. In the commonly used energy methods, equations governing 2-D variables correspond to plate element equilibrium equations. The need to use these equations is eliminated through an adapted iterative procedure in the Poisson theory of elastic plates. It appears that one cannot avoid initial solutions of displacements with one- and two-term representation of in-plane displacements (u, v) with transverse (vertical) displacement w from dz-integration constant w0 (x, y) of εz (x, y, z) and linear z w1 (x, y) [=z εz (x, y) from constitutive relation] in bending and extension problems, respectfully. Displacement w is used as a face variable in bending problems and a domain variable in extension problems. It is shown through a simple example in Chap. 1 that one-term solution of a new primary problem in the bending of the plate is quite adequate even for moderately thick plates. A two-term solution of the primary extension problem is mandatory for the determination of transverse stresses due to non-participation of σz0 (x, y) satisfying prescribed normal stress along faces of the plate in equilibrium equation of transverse stresses. Analysis of plates with different geometries and material properties under different kinematic and loading conditions does not provide much scope for the development of new theories other than those with the analysis of primary problems of a square plate. As mentioned by Ghugal and Shimpi in their review article [3], the development of refined structural theories for laminated plates (made up from advanced fiberreinforced composite materials) has their origins in the refined theories of isotropic plates. The Poisson theory appears to be the most suitable theory to overcome lacuna in the classical theories of primary plate problems. The disadvantage in its application is in the development of software for the generation of polynomial fk (z) functions necessary for the analysis of plates with thickness ratio varying up to unit value. New theories of plates are proposed here to overcome this problem of software development replacing fk (z) functions with the Fourier series in terms of proper sinusoidal functions. They are presented, initially, in the analysis of primary bending and extension problems of homogeneous isotropic plates. They are extended, later, to the analysis of anisotropic plates. Part A: Homogeneous Isotropic Plates Preliminaries from Chaps. 1 and 2 A square plate bounded within 0 ≤ X, Y ≤ a, − h ≤ Z ≤ h with reference to the Cartesian coordinate system (X, Y, Z) is considered. The material of the plate is

6.1 Introduction

117

homogeneous and isotropic with elastic constants E (Young’s modulus), ν (Poisson’s ratio), and G (shear modulus) that are related to one other by E = 2(1 + ν) G. For convenience, coordinates X, Y, Z and displacements (U, V, W) in the non-dimensional form x = X/a, y = Y/a, z = Z/h, and half-thickness ratio α = (h/a) are used. With the above notation, equilibrium equations in terms of stress components are:   α σx,x + τxy,y + τxz,z = 0

(6.1a)

  α σy,y + τxy,x + τyz,z = 0

(6.1b)

  α τxz,x + τyz,y + σz,z = 0

(6.2)

in which suffix after ‘,’ denotes partial derivative operator. In-plane equilibrium equations in terms of displacements are   1 2 2 E α u − (1 + ν)α (v,xy −u,yy ) + μ α σz,x + τxz,z = 0 2   1 E α2 v + (1 + ν)α2 (v,xx −u,xy ) + μ α σz,y + τyz,z = 0 2 

(6.3a) (6.3b)

The prescribed upper and bottom face conditions along with edge conditions can be modified such that even functions f2n (z) and odd functions f2n+1 (z) in the zdistribution of (u, v) are for analysis of extension and bending problems, respectively. Correspondingly, vertical displacement w(x, y, z) is odd in extension and even in bending problems due to transverse shear strain–displacement relations.

6.2 Initial Sets of Solutions with fn (z) Functions In the Poisson theory of primary plate problems, in-plane displacements [u, v] require two-term representation in extension problems and one-term representation in bending (or associated torsion) problems. Prescribed conditions along each of x = constant edge (with analogous conditions along y = constant edge) in the primary problems are u = un (y)

σxn (y) = Txn (y)

(6.4a)

v = vn (y) or τxyn (y) = Txyn (y)

(6.4b)

or

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6 Poisson Theory of Plates with Fourier Sinusoidal Series

in which ‘n ’ is ‘0 ’ in extension and ‘1 ’ in bending problems. Similarly, the prescribed transverse stresses along z = ± 1 faces of the plate are [Txz (x, y), Tyz (x, y), Tz (x, y)]n . Due to odd and even z-distribution, however, [Txz1 , Tyz1 , Tz0 ] correspond to extension problems and vice versa in bending problems. In auxiliary problems, transverse shear stresses in bending and extension problems are expressed in the form of     τxz , τyz 0b = −α ψ0,x , ψ0,y

(6.5a)

    τxz , τyz 1e = −α ψ1,x , ψ1,y

(6.5b)

Note that α2 ψ0 = q1 (x, y)/2 in bending problems and ψ1 = 0 in extension problems. In the classical plate theories, w is linear in ‘z’ in extension problems, whereas εz is linear in bending problems. In extension problems, it has become mandatory to use Poisson theory for determination of [u, v] satisfying both static and dz-integrated equilibrium equations. We noted that the analysis of extension problems with Airy’s stress function is likely to be complementary to that of bending problems with Poisson stress function but could not succeed due to lack of relations between gradients of Airy’s stress function and in-plane strains. We would like to show, here, that analysis of bending and extension problems are indeed complementary to each other with the use of thickness-wise Fourier sinusoidal functions.

6.3 Use of Fourier Sinusoidal Series [4] Here, we consider expansion of z in Fourier sine series in the form, with λ2n+1 = 2/[(2n + 1)π], z=



A2n+1 sin(z/λ2n+1 ) (sum on n = 0, 1, 2, . . .)

(6.6)

in which A2n+1 =

1

sin(z/λ2n+1 ) z dz = λ2n+1 2

(6.7)

0

Successive integrations of f1 (z) = z gives, with (sum on n = 0, 1, 2, …), for k = 1, 2, …,  λ2k (6.8) f2k−1 (z) = 2n+1 sin(z/λ2n+1 ) f2k (z) = −



λ2k+1 2n+1 cos(z/λ2n+1 )

(6.9)

6.3 Use of Fourier Sinusoidal Series [4]

119

6.3.1 Analysis of Flexure (Bending) Problem In bending problems, displacements [u, v] are assumed in the form (with sum n = 0, 1, 2, …) [u, v]b =



[u, v]2n+1 λ22n+1 sin(z/λ2n+1 )

(6.10)

The corresponding components of in-plane strains from strain–displacement relations, normal stresses, and transverse normal strain from constitutive relations are coefficients of [λ2 2n+1 sin (z/λ2n+1 )]. Due to [u, v],z in transverse shear stress–strain relations, transverse stresses are assumed as  τxz2n λ2n+1 cos(z/λ2n+1 ) (6.11a) τxzb = τyzb =



τyz2n λ2n+1 cos(z/λ2n+1 )

(6.11b)

so that equilibrium of transverse stresses is satisfied with σzb =



σz2n+1 λ22n+1 sin(z/λ2n+1 )

(6.12)

from constitutive relation. With [τxz , τyz ]0 = − α [ψ0 ,x , ψ0 ,y ], the auxiliary problem with first terms in Eqs. (6.11a, 6.11b) and (6.12) is the same one presented in Chap. 1 so that α2  ψ0 = q1 (x, y)/2 but thickness variation of ψ0 and its gradients is cos (z/λ1 ). Due to [τxz , τyz ],z in the in-plane equilibrium equations, the corresponding [τxz , τyz ] are τxzb = τyzb =

 

τxz2n+2 λ32n+1 cos(z/λ2n+1 )

(6.13a)

τyz2n+2 λ32n+1 cos(z/λ2n+1 )

(6.13b)

so that one gets from transverse equilibrium equation σzb =



σz2n+3 λ42n+1 sin(z/λ2n+1 )

(6.14)

In-plane equilibrium equations with (v, x – u, y ) = 0 and using Eqs. (6.10, 6.13a, 6.13b) are E α2 u2n+1 + μ α σz2n+1,x + τxz2n+2,z = 0

(6.15a)

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6 Poisson Theory of Plates with Fourier Sinusoidal Series

E α2 v2n+1 + μ α σz2n+1,y + τyz2n+2,z = 0

(6.15b)

Here, (v,x – u,y ) = 0 is not due to [τxz , τyz ]0 ≡ 0 in Kirchhoff theory explained in Chap. 2 in the analysis with [u, v] = but due to ϕ1  = 0 as  −α ψ,x − ϕ,y , ψ,y + ϕ,x . Equilibrium equation of transverse stresses from Eqs. (6.13a, 6.13b, 6.14) is   α τxz2n+2,x + τyz2n+2,y + σz2n+3 = 0

(6.16)

    With [u, v]2n+1 = −α ψ2n+1,x − ϕ2n+1,y , ψ2n+1,y + ϕ2n+1,x , equation governing ψ2n+1 from Eqs. (6.15a, 6.15b, 6.16) becomes E α4 ψ2n+1 + μ α2 σz2n+1 = λ22n+1 σz2n+3

(6.17)

to be solved along with ϕ2n+1 = 0 subjected to the edge conditions along x = constant edges (and analog conditions along y = constant edges) u2n+1 (y) = 0 or σx2n+1 (y) = Tx2n+1 (y)

(6.18)

v2n+1 (y) = 0 or τxy2n+1 (y) = Txy2n+1 (y)

(6.19)

ψ2n (y) = 0 or τxz2n (y) = Txz2n (y)

(6.20)

6.3.2 Note on the Determination of Vertical Displacement in Bending Problems In the bending problem, one should note that the first term σz1 in σzb is due to in-plane stresses from constitutive relations and the second term is due to the equilibrium equation in the z-direction. On substitution of first terms in the above transverse stresses in the equilibrium equation, one gets the governing equation in the auxiliary problem as   α τxz0,x + τyz0,y + σzl = 0

(6.21)

due to face load condition with Tz1 = q1 (x, y) satisfied with σz1 = q1 (x, y) from   σz1 λ21 sin(z/λ1 ) z=1 = λ21 q1 (x, y)

(6.22)

6.3 Use of Fourier Sinusoidal Series [4]

121

The above Eq. (6.21) of the auxiliary problem was imbedded first time (to our knowledge) in Touratier’s work [5]. However, he used transverse shear stresses in the energy method with w0 (x, y) as a domain variable instead of face variable used in the present monograph. Here, 3-D variables [u, v, σx, σy, σz , εx , εy , εz , γxy , γxz , γyz ] are determined independent of vertical displacement. Face deflection has to be determined from the stationary property of total potential involving strain energy (I) stored in the plate and work done (W) due to prescribed or reactive surface stresses and displacements, whereas neutral plane deflection is from shear strain– displacement relations. This problem is the replacement of supplementary problems of earlier analysis with polynomial functions. For this purpose, one has to consider form δ(I – W) corresponding ˜ to variation due to (δ w) only,    that 

˝variational δ σz εz + τxz γxz + τyz γ yz (dV) − δ σz w + α τxz w,x + α τyz w,y (dS) , is, 21 which generates equations governing w(x, y, z) in face parallel planes. In the case of bending of square plates considered in this monograph, these supplementary problems form a sequence of one set of 2-D Poisson equations corresponding to Eq. (6.21) of the auxiliary problem and another set of equations corresponding to Eq. (6.17) of static equilibrium equation related to in-plane equilibrium equations. It can be seen that auxiliary problems are about an error in the constitutive relations in the semiinverted relations. Normal stress σz in higher-order auxiliary problems, unlike in the first auxiliary problem, is dependent on material constants through a priori determined in-plane displacements (e.g., σz3 from the determination of [u1 , v1 ]). Here, [τxz , τyz ] are zero along faces of the plate, and [u, v] which are not zero along the faces are determined from static equilibrium equations. Face deflection is obtained from zero face [τxz , τyz ] stresses, which in a way justifies Kirchhoff’s assumptions. Neutral plane deflection is from strain–displacement relations of transverse strains with zero in-plane displacements. One has to note that only static equilibrium equations are used in the analysis.

6.3.3 Associated Torsion Problem Analysis of associated torsion problem requires only replacement of ψ with w(x, y) in the in-plane displacements [6]. Details of the analysis of a sequence of 2-D problems are not presented. The corresponding analysis of the bending of homogeneous anisotropic plates and symmetric laminated plates with anisotropic plies is omitted. These torsion problems do not arise in problems associated with extension problems.

6.3.4 Analysis of Extension Problem Here, we recall the classical theory presented in Chap. 2 in which the prescribed σz along faces of the plate does not present in the in-plane equilibrium equations. We note that vertical displacement w(x, y, z) is anti-symmetric in z. As such, work done

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6 Poisson Theory of Plates with Fourier Sinusoidal Series

by the load corresponding to σz during deformation in variational methods is due to w(x, y, 1). In the analysis with Fourier sinusoidal series, w(x, y, z) is assumed initially in the form of w(x, y, z) = w1 (x, y) sin (πz/2). With εz0 (x, y) for constitutive relation from classical theory, w(x, y, z) = εz0 (x, y) sin (πz/2). Vertical displacement we based on the above observation is assumed in the form (sum on n ≥ 0) we =



w2n+1 λ22n+1 sin(z/λ2n+1 )

(6.23)

One obtains from w,z = εz and [u, v] from the corresponding normal strains εz = [u, v] =

 εz2n λ2n+1 cos(z/λ2n+1 ) 

[u, v]2n λ2n+1 cos(z/λ2n+1 )

(6.24) (6.25)

The auxiliary problem is defined here as the initial problem of [u, v]0 with prescribed σz0 in the constitutive relations (neglected in the classical theories) in the in-plane equilibrium equations with zero transverse shear stresses. Procedure for analysis is complementary to that of the auxiliary problem in the bending of plates (see Table: Auxiliary problems). With (v,x – u,y ) = 0 as explained in Chap. 2, the problem involves two Laplace equations [u, v]0 = [0, 0] coupled through two-edge conditions. With known [u, v]0 from the auxiliary problem, the constitutive relation gives non-harmonic εz0 with prescribed σz0 along the faces of the plate. Transverse shear stresses are determined from thickness-wise (dz-) integration of in-plane equilibrium equations. Procedures for the analysis of auxiliary problems in bending and extension problems are, in fact, complementary to each other. Table: Auxiliary problems Bending problems

Extension problems

[τxz , τyz ]0 are independent of z

[u, v]0 with prescribed σz0 are independent of z

[u, v, σz ] in sine series

[τxz , τyz , w] in sine series

τxz , τyz , σz are independent of material constants σx , σy , τxy are dependent on material constants [u, v] from static equilibrium equations

[τxz , τyz , σz ] from the integration of equilibrium equations

w0 is face variable

u, v are neutral plane variables

w0 from transverse strain–displacement relations Neutral plane value of εz from constitutive relation Procedures for analysis of bending and extension problems are complementary to each other

6.3 Use of Fourier Sinusoidal Series [4]

123

Here, the basic auxiliary problem in which the prescribed surface condition σz0 = q0 (x, y) participates as domain variable through constitutive relations is independent of the above cosine series. As such, the basic problem in Chap. 2 is altered from classical theory with the inclusion of σz0 . Analysis based on the above cosine series is to nullify errors due to εz0 in the constitute relations by replacing w1 by εz0 in Eq. (6.23). Note that the higher-order εz2n at the neutral plane gives face deflection w(x, y). Similarly, the corresponding [τxz1 , τyz1 ] along faces remain the same. Part B: Layer-Wise Theories of Plates with Anisotropic Plies

6.4 Symmetrical Laminated Plates A symmetric laminate bounded by 0 ≤ X, Y ≤ a, –hn ≤ Z ≤ hn with interfaces Z = hk in the Cartesian coordinate system (X, Y, Z) is considered. For convenience, coordinates X, Y, and Z and displacements U, V, and W in non-dimensional form [x, y] = [X, Y]/a, z = Z/hn , [u, v, w] = [U, V, W]/hn and half-thickness ratio α = hn /a are utilized. The material of each ply is homogeneous and anisotropic with monoclinic symmetry. Interfaces are given by z = αk = hk /hn (k = 1, 2…, n – 1) in the upper-half of the laminate. Upper face values of displacements [u, v, w]u and transverse stresses [τxz , τyz , σz ]u in a ply are related to its lower face values [u, v, w]b and [τxz , τyz , σz ]b , respectively, through the solution of Eqs. (6.1a, 6.1b, 6.2) together with three conditions specified later at each of constant x (and y) edges and satisfaction of continuity conditions across interfaces. Here, we consider expansion of z in Fourier sine series in the form throughthickness of kth ply, zk−1 ≤ z ≤ zk , with λ(k) 2n−1 = 2/[(2n – 1)αk π], z=



(k) (sum on n) A(k) 2n−1 sin z/λ2n−1

(6.26)

in which A(k) 2n−1 = [λ(2) 2n−1 ](k) . It is convenient to introduce ply variable z(k) = (z – zk–1 )/ (zk – zk–1 ) so that 0 ≤ z(k) ≤ 1. With the above notation, analysis of both bending and extension problems of symmetric laminates is exactly the same with that of a homogeneous plate by replacing α to αk [z, λ2n−1 , A2n−1 ], with superscript [z, λ2n−1 , A2n−1 ](k) in trigonometric functions.

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6 Poisson Theory of Plates with Fourier Sinusoidal Series

6.5 Un-symmetrical Laminated Plates The number of layers in the bottom-half (z ≤ 0) of the laminate is replaced by m which need not be equal to the number of layers ‘n’ in the upper-half (z ≥ 0) but hn + hm = 0. The initial set of solutions in the upper half of the laminate in all the problems presented above are unaltered up to the reference plane z = 0. Analysis with coordinate z using z + z = 0 in the bottom half of the unsymmetrical laminate is unaltered with first terms in bending and extension problems in Chap. 5 in which higher-order corrections are not presented due to complexity of algebra involved. Here, higher-order corrective solutions with Fourier sinusoidal series are presented.

6.5.1 Bending Problem From the analysis of the upper-half of the laminate, σz = 0 and transverse shear stresses including first-order corrections due to σz1 in the in-plane constitutive relations along the reference plane z = 0 are (τxz )z=0 =

 (1) Q44 (u2n+3 + μ u2n+1 ) + Q45 (v2n+3 + μ v2n+1 )

(6.27a)

(τyz )z=0 =

 (1) Q54 (u2n+3 + μ u2n+1 ) + Q55 (v2n+3 + μ v2n+1 )

(6.27b)

Continuation of the same analysis with z(= − z) ≥ 0, one obtains along z = 0 plane that normal stress σ z = 0 and the corresponding (τxz )z=0 and (τyz )z=0 are −

 − − − − − − [Q44 (u2n+3 + μu2n+1 ) + Q45 (v2n+3 + μv2n+1 )]

(6.28a)

−

 − − − − − − [Q54 (u2n+3 + μu2n+1 ) + Q55 (v2n+3 + μv2n+1 )]

(6.28b)

( τ xz )z=0 = ( τ yz )z=0 =

Associated extension problem in bending In the initial set of solutions, transverse shear stresses obtained along z = 0 plane are sum of the stresses in Eqs. (6.27a, 6.27b, 6.28a, 6.28b). For continuity of these stresses across the z = 0 interface, one has to consider the adjacent plies above and below the interface subjected to shear stresses 

−



−

τxz = ±[ τ xz − τxz ]z=0 ; τyz = ±[ τ yz − τyz ]z=0

(6.29)

The analysis is the same presented earlier with the corresponding analysis in Chap. 5.

6.5 Un-symmetrical Laminated Plates

125

6.5.2 Extension Problem Along the reference plane z = 0, w, the transverse shear stresses are zero and in-plane displacements and σz are (sum n ≥ 1) [u, v] = [u, v]0 − σz =



[u, v]2n+2 λ2n+1

(6.30)

 1 σz2n+2 λ32n+1 q0 (x, y) + 2 − −

(6.31) −

Continuation of the same analysis with z(= −z) ≥ 0, u, v and σz along z = 0 plane are given by the following equations [u, v] = [u, v]0 − − σz

=



[u, v]2n+2 λ2n+1

− 1 σz2n+2 λ32n+1 q0 (x, y) + 2

(6.32) (6.33)

Here also, normal stress σz across the z = 0 plane has to be the same in the adjacent ply on each side of z = 0 plane. Continuity of (u, v) is required only if the reference plane is not an interface of plies. Analysis along with associated bending problem is the same as the corresponding analysis presented in Chap. 5.

6.6 Conclusions In Chaps. 1 and 2, it was demonstrated that analysis with one-term representation in bending and two-term representation in extension problems is quite adequate for the practicing engineers even for thick plates. However, the difficulty for higher-order approximations in the analysis of thick plates is eliminated in this chapter. It has become simpler with the use of Fourier sinusoidal series to generate a sequence of two-dimensional problems converging to three-dimensional problems. It is shown that the procedures for analysis of primary bending and extension problems are complementary to each other. Analysis through Airy’s stress function is complementary to that of Poisson stress function in bending. Due to a lack of relations of gradients of Airy’s stress function to in-plane strains, we have used the corresponding equations in terms of in-plane displacements.

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6 Poisson Theory of Plates with Fourier Sinusoidal Series

References 1. Jemielita G (1991) On the winding paths of the theory of plates (in Polish). PraceNaukowe Poltechniki Warszawskiej, Budownictwo, p 117 2. Vijayakumar K. Review of a few selected theories of plates in bending. Hindawi Publishing Corporation. International Scholarly Research Notices, vol 2014, Article ID 291478, 9 pp. https://doi.org/10.1155/2014/291478 3. Ghugal YM, Shimpi RP (2002) A review of refined shear deformation theories of isotropic and anisotropic laminated plates. J Reinf Plast Compos 21:775. https://doi.org/10.1177/073168402 128988481 4. Vijayakumar K (2019) Extended Poisson theory with Fourier sinusoidal series. J Multidiscipl Eng Sci Technol (JMEST) 6(8). ISSN: 2458-9403 5. Touratier M (1991) An efficient standard plate theory. Int J Eng Sci 29(8):901–916 6. Vijayakumar K (2016) Extended Poisson theory for analysis of laminated plates. J Multidiscip Eng Sci Technol (JMEST) 3(2):17. ISSN: 3159-0040

Data from Research Gate

Main text of the monograph is based on the publications listed below. Kaza Vijayakumar (years 2009–2019) Articles from resumed academic work after a gap of more than 12 years since retirement in the year 1995 due to inducement and help from Grand-Children (Nikhil, Tanya & Jabili). (Reads and citations updated from Research Gate as on Fourth January 2021). 1.

2.

3.

4. 5. 6. 7.

8.

New Look at Kirchhoff Theory of Plates, AIAA Journal 2009/4, Vol. 47, Issue 4, pp. 1045–1046 (Based on the idea held in the year 1988) (649 Reads 13 citations, mostly over the past two years). Modified Kirchhoff’s theory of plates including transverse shear deformations, Mechanics Research Communications 38 (3), Apr 2011, pp. 211–213 (88 Reads 14 citations). A relook at Reissner’s theory of plates in bending, Archive of Applied Mechanics 81, Nov 2011, pp. 1717–1724 (288 Reads 16 citations, mostly over the past two years). New Primary Bending problem with proper corrections in Kirchhoff’s theory (47 Reads from Archives of AAM). Layer wise Theory of Bending of Symmetric Laminates with Isotropic Plies, AIAA Journal 49 (9), Sep 2011, pp. 2073–2076 (36 Reads 6 citations). Poisson’s Theory for Analysis of Bending of Isotropic and Anisotropic Plates, ISRN Civil Engineering, Hindawi, 8 Jul 2013 (66 Reads 9 citations). On Uniform Approximate Solutions in Bending of Symmetric Laminated Plates, CMC-Computers, Materials & Continua 34 (1), Aug 2013, pp. 1–25 (54 Reads 10 citations). On a sequence of approximate solutions: bending of a simply supported square plate, International Journal of Advanced Structural Engineering 5 (1), 18, Aug 2013 (50 Reads 2 citation).

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 K. Vijayakumar and G. K. Ramaiah, Poisson Theory of Elastic Plates, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-33-4210-1

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Data from Research Gate

Review of a Few Selected Theories of Plates in Bending, International Scholarly Research Notices, 28, 2014:291478. https://doi.org/10.1155/2014/291478 (37 Reads 8 citation). Smeared Laminate Theory of Unsymmetrical Laminated Plates: Use of Extended Poisson Theory, Journal of Multidisciplinary Engineering Science and Technology 2 (11), 9 Nov 2015 (JMEST Open access) (113 Reads 3 citations). Extended Poisson’s Theory for Analysis of Bending of a Simply Supported Square Plate, SSRG International Journal of Civil Engineering 2 (11), 5 Nov 2015 (SSRG-IJCE) (20 Reads 2 citations). Modifications of Extended Poisson Theory of Plates, Journal of Multidisciplinary Engineering Science and Technology 2 (12), 5 Dec 2015 (JMEST Open access) (38 Reads 3 citations). Extended Poisson Theory for Analysis of Laminated Plates, Journal of Multidisciplinary Engineering Science and Technology, Feb 2016 (JMEST Open access), pp. 4085–4101 (47 Reads 2 citations). Extended Poisson Theory (EPT): Sequential Development, Journal of Multidisciplinary Engineering Science and Technology, Vol. 3 Issue 3, March2016 pp. 4217–4222 (JMEST Open access) (21 Reads). Extension Problem: A Note On Proper Modification Of Extended Poisson Theory, Journal of Multidisciplinary Engineering Science and Technology 3 Nov 2016 (JMEST Open access) (24 Reads). Exact Analysis of Laminated Plates with Anisotropic Plies, Journal of Multidisciplinary Engineering Science and Technology 12 (3), 9 Dec 2016 (JMEST Open access) (50 Reads). (Vijayakumar Received the Abstracts of the Above Articles Prepared by the Editorial Staff, MARCUS WHO’S WHO for the Distinguished Service Award 2017.) Extended Poisson Theory with Fourier Sinusoidal Series, Journal of Multidisciplinary Engineering Science and Technology (JMEST) ISSN: 2458-9403 Vol. 6 Issue 8, August-2019 (12 Reads).

Epilogue

In this monograph, the analysis of elastic plates within the small deformation theory of elasticity is presented through a newly developed and designated as ‘Poisson Theory of Elastic Plates.’ Proper primary problems in bending, associated torsion, and extension problems are formulated. Auxiliary problems are defined in both bending and extension problems. Solutions of these problems are used for accurate analysis of plates. It is shown that the procedures for analysis of bending and extension problems are complementary to each other. Basically, transverse displacement is a face variable in bending problems whereas it is a neutral plane variable in extension problems. It is a domain variable in the associated torsion problem. Complementary to this observation, in-plane displacements may be considered as face variables in the corresponding associated bending problem due to thickness wise odd distribution of transverse shear stresses which needs proper explanation in future work. In a way, classical extension problem may be considered as 0th associated bending problem. In future investigations, it is necessary to express gradients of Airy’s stress function (governing in-plane stresses) and gradients of Poisson function (governing transverse shear stresses) with those of gradients of ψ and ϕ of displacements in bending problems. In contrast to classical theories, analysis based on polynomial representation with one term in bending and two terms in extension problems of displacements is more than enough for the practicing engineers. It is, in principle, possible to obtain corrections to the initial solutions of primary problems with higher-order polynomial functions through a sequence of solutions of more or less uncoupled sets of 2-D problems by the adapted iterative procedure. A significant contribution to the analysis of laminated plates is that the ply analysis is independent of lamination. One interesting observation is that Kirchhoff theory in bending of plates gives only a lower bound for the thickness ratio of the plate for the validity of small deformation theory of elasticity whereas the present Poisson theory gives upper bound to the thickness ratio dependent on a priory specified percentage error in the estimation of maximum transverse displacement. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2021 K. Vijayakumar and G. K. Ramaiah, Poisson Theory of Elastic Plates, Springer Tracts in Mechanical Engineering, https://doi.org/10.1007/978-981-33-4210-1

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Epilogue

However, it is not easy to develop the necessary software to generate polynomial functions used in the analysis. This problem is overcome with the use of Fourier sinusoidal series in Chap. 6 to obtain solutions without any difficulty for problems with the thickness ratio of the plate varying up to unit value. The procedure simplifies the iterative process for higher-order approximations in both bending and extension problems. Finite element method out of several approximate methods available for the analysis of 2-D problems is normally used when the geometry of the plate and loading are complicated. In its application for solution of such problems, C(0) continuity elements have to be formulated and used for solving Poisson equations and C(1) continuity elements for solving fourth-order partial differential equations. The monograph is closed with suggestions for future work with the application of Poisson theory to eigen-value problems like lateral buckling and free vibration problems.