Shear Deformable Beams and Plates: Relationships with Classical Solutions
By C.M. Wang and J.N. Reddy
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About this ebook
Equations governing shear deformation theories are typically more complicated than those of the classical theory. Hence it is desirable to have exact relationships between solutions of the classical theory and shear deformation theories so that whenever classical theory solutions are available, the corresponding solutions of shear deformation theories can be readily obtained. Such relationships not only furnish benchmark solutions of shear deformation theories but also provide insight into the significance of shear deformation on the response. The relationships for beams and plates have been developed by many authors over the last several years. The goal of this monograph is to bring together these relationships for beams and plates in a single volume.
The book is divided into two parts. Following the introduction, Part 1 consists of Chapters 2 to 5 dealing with beams, and Part 2 consists of Chapters 6 to 13 covering plates. Problems are included at the end of each chapter to use, extend, and develop new relationships.
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Shear Deformable Beams and Plates - C.M. Wang
Eastbourne
Preface
There exist many books on the theory and analysis of beams and plates. Most of the books deal with the classical (Euler-Bernoulli/Kirchhoff) theories but few include shear deformation theories in detail. The classical beam/plate theory is not adequate in providing accurate bending, buckling, and vibration results when the thickness-to-length ratio of the beam/plate is relatively large. This is because the effect of transverse shear strains, neglected in the classical theory, becomes significant in deep beams and thick plates. In such cases, shear deformation theories provide accurate solutions compared to the classical theory.
Equations governing shear deformation theories are typically more complicated than those of the classical theory. Hence it is desirable to have exact relationships between solutions of the classical theory and shear deformation theories so that whenever classical theory solutions are available, the corresponding solutions of shear deformation theories can be readily obtained. Such relationships not only furnish benchmark solutions of shear deformation theories but also provide insight into the significance of shear deformation on the response. The relationships for beams and plates have been developed by the authors and their colleagues over the last several years. However, this valuable information is dispersed in the literature. Therefore, the goal of this monograph is to bring together these relationships for beams and plates in a single volume.
The book is divided into two parts. Following the introduction, Part 1 consists of Chapters 2 to 5 dealing with beams, and Part 2 consists of Chapters 6 to 13 covering plates. Problems are included at the end of each chapter to use, extend, and develop new relationships. The book is suitable as a reference by engineers and scientists working in industry and academia. An introductory course on mechanics of materials and elasticity should prove to be helpful but not necessary because a review of the basics is included in the relevant chapters.
The authors gratefully acknowledge the support and encouragement of their respective universities in carrying out the collaborative research and the writing of this book. It is a pleasure to acknowledge the help of the following colleagues in proof reading of the preliminary manuscript: Kok-Keng Ang, Goy-Teck Lim, and Yang Xiang. Special thanks go to Poh Hong, Aruna and See Fong for their love and patience while their husbands were occupied with writing this book.
C.M. Wang, Singapore
J.N. Reddy, College Station, Texas
K.H. Lee, Singapore
Chapter 1
Introduction
1.1 Preliminary Comments
The primary objective of this book is to study the relationships between the solutions of classical theories of beams and plates with those of the shear deformation theories. Shear deformation theories are those in which the effect of transverse shear strains is included. Relationships are developed for bending, buckling, and free vibration solutions.
A plate is a structural element with plane form dimensions that are large compared to its thickness and is subjected to loads that cause bending deformation in addition to stretching. In most cases, the thickness is no greater than one-tenth of the smallest in-plane dimension. Because of the smallness of the thickness dimension, it is often not necessary to model the plate using 3D elasticity equations. Beams are one-dimensional counterparts of plates.
The governing equations of beams and plates can be derived using either vector mechanics or energy and variational principles. In vector mechanics, the forces and moments on a typical element of the plate are summed to obtain the equations of equilibrium or motion. In energy methods, the principles of virtual work or their derivatives, such as the principles of minimum potential energy or complementary energy, are used to obtain the equations. While both methods can give the same equations, the energy methods have the advantage of providing information on the form of the boundary conditions.
Beam and plate theories are developed by assuming the form of the displacement or stress field as a linear combination of unknown functions and the thickness coordinate. For example, in plate theories we assume
(1.1.1)
where φi is the ith component of displacement or stress, (x, y) are the in-plane coordinates, z is the thickness coordinate, t denotes the time, and φji are functions to be determined.
When φi are displacements, the equations governing φji are determined by the principle of virtual displacements
(1.1.2a)
or its dynamic version, i.e., Hamilton’s principle
(1.1.2b)
where (δU, δV, δW, δK) denote the virtual internal (strain) energy, virtual potential energy due to applied loads, the total virtual work done, and virtual kinetic energy, respectively. These quantities are determined in terms of the actual stresses and virtual strains, which depend on the assumed displacement functions φi and their variations. For plate structures, the integration over the domain of the plate is represented as the product of integration over the plane of the plate and integration over the thickness of the plate (volume integral=integral over the plane × integral over the thickness). This is possible due to the explicit nature of the assumed displacement field in the thickness coordinate. Thus, we can write
(1.1.3)
where h denotes the thickness of the plate and Ω0 denotes the undeformed mid-plane of the plate, which is assumed to coincide with the xy–plane. Since all undetermined variables are explicit functions of the thickness coordinate, the integration over plate thickness is carried out explicitly, reducing the problem to a two-dimensional one. Hence, the Euler–Lagrange equations associated with (x, y, tper unit length:
(1.1.4)
The stress resultants can be written in terms of φi with the help of the assumed constitutive equations and strain-displacement relations. More complete development of this procedure is presented in the forthcoming chapters.
The same approach is used when φi denote stress components, except that the basis of the derivation of the governing equations is the principle of virtual forces. In the present book, stress-based theories will receive very little attention. Readers interested in stress-based theories may consult the book by Panc (1975).
1.2 An Overview of Plate Theories
The simplest plate theory of bending is the classical plate theory (CPT). In the case of pure bending, the displacement of the CPT is given by (see Reddy 1984b, 1997a, 1999a)
(1.2.1)
where (u, v, w) are the displacement components along the (x, y, z) coordinate directions, respectively, and w0 is the transverse deflection of a point on the mid-plane (i.e., z = 0). The displacement field (1.2.1) implies that straight lines normal to the xy–plane before deformation remain straight and normal to the mid-surface after deformation. The Kirchhoff assumption amounts to neglecting both transverse shear and transverse normal strain effects, i.e., deformation is due entirely to