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Theories and Analyses of Beams and Axisymmetric Circular Plates

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This comprehensive textbook compiles cutting-edge research on beams and circular plates, covering theories, analytical solutions, and numerical solutions of interest to students, researchers, and engineers working in industry. Detailing both classical and shear deformation theories, the book provides a complete study of beam and plate theories, their analytical (exact) solutions, variational solutions, and numerical solutions using the finite element method.

Beams and plates are some of the most common structural elements used in many engineering structures. The book details both classical and advanced (i.e., shear deformation) theories, scaling in complexity to aid the reader in self-study, or to correspond with a taught course. It covers topics including equations of elasticity, equations of motion of the classical and first-order shear deformation theories, and analytical solutions for bending, buckling, and natural vibration. Additionally, it details static as well as transient response based on exact, the Navier, and variational solution approaches for beams and axisymmetric circular plates, and has dedicated chapters on linear and nonlinear finite element analysis of beams and circular plates.

Theories and Analyses of Beams and Axisymmetric Circular Plates will be of interest to aerospace, civil, materials, and mechanical engineers, alongside students and researchers in solid and structural mechanics.

Author(s): J. N. Reddy
Publisher: CRC Press
Year: 2022

Language: English
Pages: 577
City: Boca Raton

Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
Preface
List of Symbols Used
About the Author
Chapter 1: Mechanics Preliminaries
1.1. General Comments
1.2. Beams and Plates
1.3. Vectors and Tensors
1.3.1. Vectors and Coordinate Systems
1.3.2. Summation Convention
1.3.3. Stress Vector and Stress Tensor
1.3.4. The Gradient Operator
1.4. Review of the Equations of Solid Mechanics
1.4.1. Green–Lagrange Strain Tensor
1.4.2. The Second Piola–Kirchhoff Stress Tensor
1.4.3. Equations of Motion
1.4.4. Stress–Strain Relations
1.5. Functionally Graded Structures
1.5.1. Background
1.5.2. Mori–Tanaka Scheme
1.5.3. Voigt Scheme: Rule of Mixtures
1.5.4. Exponential Model
1.5.5. Power-Law Model
1.6. Modified Couple Stress Effects
1.6.1. Background
1.6.2. The Strain Energy Functional
1.7. Chapter Summary
Chapter 2: Energy Principles and Variational Methods
2.1. Concepts of Work and Energy
2.1.1. Historical Background
2.1.2. Objectives of the Chapter
2.1.3. Concept of Work Done
2.2. Strain Energy and Complementary Strain Energy
2.3. Total Potential Energy and Total Complementary Energy
2.4. Virtual Work
2.4.1. Virtual Displacements
2.4.2. Virtual Forces
2.5. Calculus of Variations and Duality Pairs
2.5.1. The Variational Operator
2.5.2. Functionals and Their Variations
2.5.3. Fundamental Lemma of Variational Calculus
2.5.4. Extremum of a Functional
2.5.5. The Euler Equations and Duality Pairs
2.5.6. Natural and Essential Boundary Conditions
2.6. The Principle of Virtual Displacements
2.7. Principle of Minimum Total Potential Energy
2.8. Hamilton’s Principle
2.8.1. Preliminary Comments
2.8.2. Statement of the Principle
2.8.3. Euler–Lagrange Equations
2.9. Chapter Summary
Chapter 3: The Classical Beam Theory
3.1. Introductory Comments
3.2. Kinematics
3.3. Equations of Motion
3.3.1. Preliminary Comments
3.3.2. Vector Approach
3.3.3. Energy Approach
3.4. Governing Equations in Terms of Displacements
3.4.1. Material Constitutive Relations
3.4.2. Uniaxial Stress–Strain Relations
3.4.3. Material Gradation through the Beam Height
3.4.4. Beam Constitutive Equations
3.4.5. Equations of Motion
3.4.5.1. The general case (with FGM, VKN, and MCS)
3.4.5.2. Homogeneous beams with VKN and MCS
3.4.5.3. Linearized FGM beams with MCS
3.4.5.4. Linearized homogeneous beams with MCS
3.5. Equations in Terms of Displacements and Bending Moment
3.5.1. Preliminary Comments
3.5.2. General Case with FGM, MCS, and VKN
3.5.3. Special Cases
3.5.3.1. Homogeneous beams with VKN and MCS
3.5.3.2. Linearized FGM beams with MCS
3.5.3.3. Linearized homogeneous beams with MCS
3.6. Cylindrical Bending of FGM Rectangular Plates
3.6.1. Cylindrical Bending
3.6.2. Governing Equations in Terms of Stress Resultants
3.6.3. Governing Equations in Terms of Displacements
3.7. Exact Solutions
3.7.1. Bending Solutions
3.7.2. Buckling and Natural Vibrations
3.7.2.1. Buckling solutions
3.7.2.2. Natural frequencies
3.8. The Navier Solutions
3.8.1. The General Procedure
3.8.2. Navier’s Solution of Equations of Motion
3.8.3. Bending Solutions
3.8.4. Natural Vibrations
3.8.5. Transient Analysis
3.9. Energy and Variational Methods
3.9.1. Introduction
3.9.2. The Ritz Method
3.9.2.1. Background and model problem
3.9.2.2. The Ritz approximation
3.9.2.3. Requirements on the approximation functions
3.9.3. The Weighted-Residual Methods
3.10. Chapter Summary
Chapter 4: The First-Order Shear Deformation Beam Theory
4.1. Introductory Comments
4.2. Displacements and Strains
4.3. Equations of Motion
4.3.1. Vector Approach
4.3.2. Energy Approach
4.4. Governing Equations in Terms of Displacements
4.4.1. Beam Constitutive Equations
4.4.2. Equations of Motion for the General Case
4.4.3. Equations of Motion without the Couple Stress and Thermal Effects
4.4.4. Equations of Motion for Homogeneous Beams
4.4.5. Linearized Equations of Motion for FGM Beams
4.4.6. Linearized Equations for Homogeneous Beams
4.5. Mixed Formulation of the TBT
4.6. Exact Solutions
4.6.1. Bending Solutions
4.6.2. Buckling Solutions
4.6.3. Natural Vibration
4.7. Relations between CBT and TBT
4.7.1. Background
4.7.2. Bending Relations between the CBT and TBT
4.7.2.1. Summary of equations of the CBT
4.7.2.2. Summary of equations of the TBT
4.7.2.3. Relationships by similarity and load equivalence
4.7.3. Bending Relationships for FGM Beams with the Couple Stress Effect
4.7.3.1. Summary of equations of CBT and TBT
4.7.3.2. General relationships
4.7.3.3. Specialized relationships
4.7.4. Buckling Relationships
4.7.4.1. Summary of governing equations
4.7.5. Frequency Relationships
4.7.5.1. Governing equations of the CBT
4.7.5.2. Governing equations of the TBT
4.7.5.3. Relationship
4.8. The Navier Solutions
4.8.1. General Solution
4.8.2. Bending Solution
4.8.3. Natural Vibrations
4.9. Solutions by Variational Methods
4.10. Chapter Summary
Chapter 5: Third-Order Beam Theories
5.1. Introduction
5.1.1. Why a Third-Order Theory?
5.1.2. Present Study
5.2. A General Third-Order Theory
5.2.1. Kinematics
5.2.2. Equations of Motion
5.2.3. Equations of Motion without Couple Stress Effects
5.2.4. Constitutive Relations
5.3. A Third-Order Theory with Vanishing Shear Stress on the Top and Bottom Faces
5.3.1. The General Case
5.3.2. The Reddy Third-Order Beam Theory (RBT)
5.3.2.1. Kinematics
5.3.2.2. Equations of motion using Hamilton's principle
5.3.2.3. Constitutive relations
5.3.2.4. Equations of motion in terms of the generalized displacements: the general case
5.3.2.5. Equations of motion in terms of the generalized displacements: the linear case
5.3.3. Levinson's Third-Order Beam Theory (LBT)
5.3.3.1. Equations of motion
5.3.3.2. Equations of motion in terms of the displacements
5.3.3.3. Equations of motion for the linear case
5.3.3.4. Equations of equilibrium for the linear case
5.3.3.5. Linearized equations without the couple stress effect
5.4. Exact Solutions for Bending
5.4.1. The Reddy Beam Theory
5.4.2. The Simplified RBT
5.4.2.1. FGM beams
5.4.2.2. Homogeneous beams
5.4.3. The Levinson Beam Theory
5.4.3.1. FGM beams
5.4.3.2. Homogeneous beams
5.5. Bending Relationships for the RBT
5.5.1. Preliminary Comments
5.5.2. Summary of Equations
5.5.3. General Relationships
5.5.4. Bending Relationships for the Simplified RBT
5.5.5. Relationships between the LBT and the CBT
5.5.6. Numerical Examples
5.5.7. Buckling Relationships
5.5.7.1. Summary of equations of the CBT
5.5.7.2. Summary of equations of the RBT
5.6. Navier Solutions
5.6.1. The Reddy Beam Theory (RBT)
5.6.1.1. Bending analysis
5.6.1.2. Natural vibration
5.6.2. The Levinson Beam Theory (LBT)
5.6.3. Numerical Results
5.7. Solutions by Variational Methods
5.8. Chapter Summary
Chapter 6: Classical Theory of Circular Plates
6.1. General Relations
6.1.1. Preliminary Comments
6.1.2. Kinematic Relations
6.1.2.1. Modified Green–Lagrange strains
6.1.2.2. Curvature tensor
6.1.3. Stress–Strain Relations
6.1.4. Strain Energy Functional
6.2. Governing Equations of the CPT
6.2.1. Displacements and Strains
6.2.2. Equations of Motion
6.2.3. Isotropic Constitutive Relations
6.2.4. Displacement Formulation of the CPT
6.2.5. Mixed Formulation of the CPT
6.3. Solutions for Homogeneous Plates in Bending
6.3.1. Governing Equations
6.3.2. Exact Solutions
6.3.3. Numerical Examples
6.4. Bending Solutions for FGM Plates
6.4.1. Governing Equations
6.4.2. Exact Solutions
6.5. Buckling and Natural Vibration
6.5.1. Buckling Solutions
6.5.2. Natural Frequencies
6.6. Variational Solutions
6.6.1. Introductory Comments
6.6.2. Variational Statement
6.6.3. The Ritz Method
6.6.4. The Galerkin Method
6.6.5. Natural Frequencies and Buckling Loads
6.6.5.1. Variational statement
6.7. Chapter Summary
Chapter 7: First-Order Theory of Circular Plates
7.1. Governing Equations
7.1.1. Displacements and Strains
7.1.2. Equations of Motion
7.1.3. Plate Constitutive Relations
7.1.4. Equations of Motion in Terms of the Displacements
7.1.4.1. The general case
7.1.4.2. Nonlinear equations of equilibrium
7.1.4.3. Linear equations of equilibrium without couple stress
7.1.4.4. Linear equations of equilibrium without couple stress and FGM
7.2. Exact Solutions of Isotropic Circular Plates
7.3. Exact Solutions for FGM Circular Plates
7.3.1. Governing Equations
7.3.2. Exact Solutions
7.3.3. Examples
7.4. Bending Relationships between CPT and FST
7.4.1. Summary of the Governing Equations
7.4.2. Relationships
7.4.3. Examples
7.5. Bending Relationships for Functionally Graded Circular Plates
7.5.1. Introduction
7.5.2. Summary of Equations
7.5.3. Relationships between the CPT and FST
7.6. Chapter Summary
Chapter 8: Third-Order Theory of Circular Plates
8.1. Governing Equations
8.1.1. Preliminary Comments
8.1.2. Displacements and Strains
8.1.3. Equations of Motion
8.1.4. Plate Constitutive Equations
8.2. Exact Solutions of the TST
8.3. Relationships between CPT and TST
8.3.1. Bending Relationships
8.3.1.1. Classical plate theory (CPT)
8.3.1.2. Third-order shear deformation plate theory (TST)
8.3.2. Relationships
8.3.3. Buckling Relationships
8.3.3.1. Governing equations
8.3.3.2. Relationship between CPT and FST
8.3.3.3. Relationship between CPT and TST
8.4. Chapter Summary
Chapter 9: Finite Element Analysis of Beams
9.1. Introduction
9.1.1. The Finite Element Method
9.1.2. Interpolation Functions
9.1.3. Present Study
9.2. Displacement Model of the CBT
9.2.1. Governing Equations and Variational Statements
9.2.2. Finite Element Model
9.3. Mixed Finite Element Model of the CBT
9.3.1. Variational Statements
9.3.2. Finite Element Model
9.4. Displacement Finite Element Model of the TBT
9.4.1. Governing Equations and Variational Statements
9.4.2. The Finite Element Model
9.5. Mixed Finite Element Model of the TBT
9.5.1. Governing Equations and Variational Statements
9.5.2. Finite Element Model
9.6. Displacement Finite Element Model of the RBT
9.6.1. Governing Equations
9.6.2. Weak Forms
9.6.3. Finite Element Model
9.7. Time Approximation (Full Discretization)
9.7.1. Introduction
9.7.2. Newmark’s Method
9.7.3. Fully Discretized Equations
9.8. Solution of Nonlinear Algebraic Equations
9.8.1. Preliminary Comments
9.8.2. Direct Iteration Procedure
9.8.3. Newton’s Iteration Procedure
9.8.4. Load Increments
9.9. Tangent Stiffness Coefficients
9.9.1. Definition of Tangent Stiffness Coefficients
9.9.2. The Displacement Model of the CBT
9.9.3. The Mixed Model of the CBT
9.9.4. The Displacement Model of the TBT
9.9.5. The Mixed Model of the TBT
9.9.6. The Displacement Model of the RBT
9.10. Post-Computations
9.10.1. General Comments
9.10.2. CBT Finite Element Models
9.10.3. TBT Finite Element Models
9.10.4. RBT Displacement Model
9.11. Numerical Results
9.11.1. Geometry and Boundary Conditions
9.11.2. Material Constitution
9.11.3. Examples
9.12. Chapter Summary
Chapter 10: Finite Element Analysis of Circular Plates
10.1. Introductory Remarks
10.2. Displacement Model of the CPT
10.2.1. Weak Forms
10.2.2. Finite Element Model
10.3. Mixed Model of the CPT
10.3.1. Weak Forms
10.3.2. Finite Element Model
10.4. Displacement Model of the FST
10.4.1. Weak Forms
10.4.2. Finite Element Model
10.5. Displacement Model of the TST
10.5.1. Variational Statements
10.5.2. Finite Element Model
10.6. Tangent Stiffness Coefficients
10.6.1. Preliminary Comments
10.6.2. The Displacement Model of the CPT
10.6.3. The Mixed Model of the CPT
10.6.4. The Displacement Model of the FST
10.6.5. The Displacement Model of the TST
10.7. Numerical Results
10.7.1. Preliminary Comments
10.7.2. Linear Analysis
10.7.3. Nonlinear Analysis without Couple Stress Effect
10.7.4. Nonlinear Analysis with Couple Stress Effect
10.8. Chapter Summary
References
Papers with Collaborators
Answers
Index