Shell Structures: Theory and Application

This book provides a complete derivation of the mathematical theory of shell structures, making it accessible to students of engineering mechanics and structural engineers. It is class-tested and ideal for those looking for a reference on shells.

Format: Hardback
Length: 353 pages
Publication date: 04 February 2022
Publisher: Springer Nature Switzerland AG

This text presents a comprehensive and intricate derivation of the mathematical framework governing shell structures. Numerous books on shells often provide only the essential equations or brief glimpses of theory, omitting the intricate mathematical processes necessary to derive these equations. This approach is understandable, considering the mathematical intricacy inherent in shell structures. Consequently, readers are left to accept the design equations blindly, without fully comprehending the underlying principles of shell theory. To address this gap, this book offers a comprehensive portrayal of shell theory. Extensively class-tested across three university postgraduate courses and one public course on shell structures, this book is mathematically rigorous yet accessible, making it an ideal resource for students pursuing engineering mechanics concentrations in civil and mechanical engineering, as well as practicing structural engineers seeking a reference on shells.

The mathematical theory of shell structures is a complex and intricate field that requires a deep understanding of mathematical principles and techniques. In this text, we will delve into the derivation of the mathematical theory of shell structures, covering key concepts and equations that are essential for understanding the behavior and design of these structures.

Shell structures are three-dimensional structures composed of thin shells or plates that are subjected to various external loads, such as pressure, tension, or shear. The mathematical theory of shell structures aims to describe the behavior of these structures and to predict their response to these loads.

The first step in deriving the mathematical theory of shell structures is to define the geometry of the shell. The geometry of a shell is defined by its thickness, radius, and length, as well as the number of edges and vertices it has. The thickness of a shell is the distance between its two adjacent faces, while the radius is the distance from the center of the shell to any edge. The length of a shell is the distance between its two ends.

The number of edges and vertices of a shell determines its topology, which is a fundamental concept in mathematics. The topology of a shell can be either planar, cylindrical, or spherical, depending on the shape of the shell. Planar shells have two faces that are parallel to each other, cylindrical shells have two faces that are parallel to each other and a cylindrical axis, and spherical shells have a spherical surface.

Once the geometry of the shell is defined, the next step is to derive the governing equations for shell structures. The governing equations for shell structures are derived from the principles of continuum mechanics and elasticity. Continuum mechanics is a branch of mathematics that deals with the study of continuous media, such as fluids, solids, and gases. Elasticity is a branch of mathematics that deals with the study of the behavior of materials under stress and strain.

The governing equations for shell structures are derived from the principles of continuum mechanics and elasticity. The equations that govern the behavior of a shell under pressure are the Euler-Bernoulli equations, which describe the relationship between the pressure and the strain in a shell. The equations that govern the behavior of a shell under tension or shear are the Timoshenko equations, which describe the relationship between the stress and the strain in a shell.

The Euler-Bernoulli equations are derived from the principles of continuum mechanics. They state that the stress in a shell is equal to the sum of the internal pressure and the external pressure. The internal pressure is the pressure that exists within the material of the shell, while the external pressure is the pressure that exists outside the shell.

The Timoshenko equations are derived from the principles of elasticity. They state that the stress in a shell is equal to the sum of the elastic strain and the shear strain. The elastic strain is the strain that occurs when a material is subjected to a force, while the shear strain is the strain that occurs when a material is subjected to a shear force.

Once the governing equations for shell structures are derived, they can be used to predict the behavior of shell structures under various external loads. The behavior of a shell under pressure is described by the Hooke's law, which states that the stress in a shell is proportional to the strain in the shell. The behavior of a shell under tension or shear is described by the von Mises stress, which states that the stress in a shell is proportional to the square of the strain in the shell.

In addition to the governing equations, the mathematical theory of shell structures also includes other important concepts and equations. These include the Poisson's ratio, which describes the relationship between the axial strain and the radial strain in a shell, and the Kirchhoff's theory, which describes the behavior of a shell under combined loads.

The Poisson's ratio is a fundamental concept in continuum mechanics. It describes the relationship between the axial strain and the radial strain in a shell. The Poisson's ratio is defined as the ratio of the axial strain to the radial strain in a shell. In a cylindrical shell, the Poisson's ratio is equal to 1/2, while in a spherical shell, the Poisson's ratio is equal to 0. In a planar shell, the Poisson's ratio can vary depending on the shape of the shell.

The Kirchhoff's theory is a fundamental concept in elasticity. It describes the behavior of a shell under combined loads, such as pressure and tension. The Kirchhoff's theory states that the stress in a shell is equal to the sum of the stresses in the individual members of the shell. The stresses in the individual members of the shell are determined by the geometry of the shell and the external loads that are applied to it.

In conclusion, the mathematical theory of shell structures is a complex and intricate field that requires a deep understanding of mathematical principles and techniques. This text has provided a comprehensive derivation of the mathematical theory of shell structures, covering key concepts and equations that are essential for understanding the behavior and design of these structures. By understanding the governing equations and other important concepts, engineers can design and analyze shell structures with greater accuracy and efficiency.

Weight: 711g
Dimension: 235 x 155 (mm)
ISBN-13: 9783030848064
Edition number: 1st ed. 2022