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2024 | Book

Introduction to Plate and Shell Structures Read chapter Read first chapter

Thermal Stresses in Plates and Shells

Author: Mohammad Reza Eslami

Publisher: Springer Nature Switzerland

Book Series : Solid Mechanics and Its Applications

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik"

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About this book

This book covers thermal stresses in plates and shells, offering a cutting-edge exploration of this critical field. Tailored for a diverse audience, including graduate and postgraduate students, dedicated researchers, and scientists in both industrial and government sectors, as well as engineers specializing in mechanical, aerospace, and civil engineering. The book unfolds over eight meticulously crafted chapters, providing a detailed examination of thermal stresses in rectangular and circular plates, along with an array of shell geometries. Circular cylindrical, spherical, conical, and shells of revolution undergo rigorous analysis under various load conditions. A focal point of the text lies in the exhaustive treatment of tensor analysis within a curvilinear coordinate system. This framework lays the foundation for the derivation of precise strain-displacement relations for an array of shell configurations. The book further elucidates the transformation of Codazzi and Gauss conditions from surface continuity to compatibility conditions within elasticity theory. Chapter 5 introduces analytical solutions for diverse thermal loads affecting cylindrical, spherical, and conical shells. Chapters 6 and 7 delve into the intricate domain of coupled thermoelasticity, particularly in plates and shells subjected to shock loads. The book culminates in Chapter 8, where the intriguing phenomenon of thermal-induced vibrations in plates and shells takes center stage. With a commitment to accessibility, this self-contained volume presents mathematical concepts and numerical methods in an approachable manner, ensuring ease of comprehension for the reader. However, a foundational understanding of classical mathematics, mechanics, and elasticity theory is recommended for optimal engagement.

Table of Contents

Frontmatter
Chapter 1. Introduction to Plate and Shell Structures
Abstract
Plates and shell structures, as a division of solid mechanics, are frequently used in the structural design problems. The variety of different plate and shell theories are developed and this chapter is intended to give the milestones of these historic development.
Mohammad Reza Eslami
Chapter 2. Tensor Analysis
Abstract
Tensor analysis is an essential tool for derivation of the basic laws of physics, whenever representation of the law in a different coordinates system is required. The shell structures, due to the different geometries and different shell theories, fall into this category. To write the kinematical relations of a shell geometry, either the reader should accept a given relation or he/she has to be able to derive the required relations. This is why this chapter is devoted to the tensor analysis. The chapter begins with simple definitions of tensor laws in the rectangular Cartesian coordinates and then follows to discuss the tensor laws in the general curvilinear coordinates. The covariant and contravariant tensors and their coordinate transformations are given. The covariant derivatives are replaced with the partial derivatives, giving the requirement for considering the Christoffel symbols, and base vectors take the role of unit vectors (but not unit in magnitude). The chapter comes to an end with the explanation of surface theory and the basic definitions of the Codazzi and Gauss conditions, which are replaced with the compatibility conditions of the theory of elasticity.
Mohammad Reza Eslami
Chapter 3. Thermal Stresses in Plates
Abstract
This chapter presents the derivations of the equilibrium equations of rectangular and circular plates in terms of the lateral deflection, using the classical first-order theory under the mechanical and thermal loads. The classical solutions of the plates under the simply supported boundary conditions are given by Navier and Levy. These two methods of solutions for rectangular plates are discussed in this chapter. The chapter continues to derive the equilibrium equations of composite rectangular plates using the classical plate theory. Assuming the simply supported boundary conditions at two opposite edges, the Levy solution is applied to obtain the lateral deflection. The first-order shear deformation theory is then considered with five partial differential equations for five dependent functions and, with the assumption of two opposite simply supported boundary conditions, the analytical Levy solution is employed to obtain the lateral deflection. It is therefore concluded that once two opposite edges are simply supported, the Levy solution to derive analytical solutions is applicable for the rectangular plates. Plate’s material properties are indifferent for application of the Navier or Levy solutions, as long as the types of boundary conditions are those assumed by the Navier or Levy. When none of the boundaries are assumed according to the Navier or Levy assumptions, the numerical solution methods may be essential for the analysis, such as the problem given in Sect. 3.12 of this chapter.
Mohammad Reza Eslami
Chapter 4. Theory of Shells
Abstract
This chapter begins with the shell geometry using the surface theory. Defining the Lamè parameters, derivations continue to discuss the continuity of surface upon satisfying the Codazzi and Gauss conditions. These continuity conditions are derived in detail. The geometry of the shells of revolution is discussed and the Euclidean metric tensor, associated metric tensor, and the Christoffel symbols are derived in the cylindrical and spherical coordinates. Derivation of general strain–displacement relations, using the general expression for strain tensor in curvilinear coordinates, is presented in detail. Different classical second-order shell theories are described and their associated expressions for the strains and curvatures are derived. Assuming the first-order shear deformation theory including the effect of normal stress and strain, the general strain–displacement relations are derived. These equations are reduced to those for the spherical, conical, and cylindrical shells. Considering Hooke’s law and the strain–displacement relations, the stresses are derived in terms of the displacement components. Considering the second-order shell theory, Hamilton’s principle is employed to derive the most general form of the equations of motion for shells of revolution. These equations are reduced for different shell geometries under the axisymmetric loading conditions.
Mohammad Reza Eslami
Chapter 5. Thermal Stresses in Shells
Abstract
This chapter presents a number of analytical solutions for thermal stresses in different shell geometries. Thermal stresses in cylindrical, spherical, and conical shells under simple thermal loads are derived by analytical methods. The analytical solution methods for thermal or mechanical stresses for different shells are given in literature. Their discussion, however, in a classical book on the shell theory may not be quite justified.
Mohammad Reza Eslami
Chapter 6. Coupled Thermoelasticity of Plates
Abstract
Recently developed functionally graded materials (FGMs) show promise for their adaptability to high-temperature environments and have thus attracted attention. Therefore, it is desirable to analyze FGM structures subjected to thermal loadings, such as thermal shock, which have a wide range of applications in engineering problems.
Mohammad Reza Eslami
Chapter 7. Couple Thermoelasticity of Shells
Abstract
Shell structures under thermal shock loads are frequently encountered in the structural design problems. This chapter presents the coupled and generalized thermoelasticity of the cylindrical shells, spherical shells, conical shells, and shells of revolution. The shell material is assumed to be made of functionally graded, where by proper substitution for the power law index, the response of shells of homogeneous material is obtained.
Mohammad Reza Eslami
Chapter 8. Thermal Induced Vibrations
Abstract
Thermal induced vibrations occur in flexural elements such as beams, plates, and shells. Whenever structures are exposed to transient thermal conditions, thermal induced vibrations may occur. This phenomenon was first observed in the satellites behavior in orbit. The occurrence of thermal induced vibrations depends upon the geometrical properties of the element in consideration. This chapter presents thermal induced vibrations of rectangular plates and the spherical and conical shells. The analysis is based on the solution of transient heat conduction equation along with the equations of motion of plates and shells. Limitations for the occurrence of induced vibrations are discussed in detail.
Mohammad Reza Eslami
Backmatter
Metadata
Title
Thermal Stresses in Plates and Shells
Author
Mohammad Reza Eslami
Copyright Year
2024
Electronic ISBN
978-3-031-49915-9
Print ISBN
978-3-031-49914-2
DOI
https://doi.org/10.1007/978-3-031-49915-9

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