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Energy principles and variational methods in applied mechanics / J.N. Reddy.
Author
Reddy, J. N. (Junuthula Narasimha), 1945-
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Format
Book
Language
English
Εdition
Third edition.
Published/Created
Hoboken, NJ, USA : John Wiley & Sons, Inc., [2017]
©2017
Description
xxvi, 730 pages : illustrations ; 25 cm
Availability
Available Online
Ebook Central Perpetual, DDA and Subscription Titles
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Location
Call Number
Status
Location Service
Notes
Engineering Library - Stacks
TA350 .R393 2017
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Details
Subject(s)
Mechanics, Applied
—
Mathematics
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Calculus of variations
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Finite element method
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Force and energy
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Summary note
A comprehensive guide to using energy principles and variational methods for solving problems in solid mechanics. This book provides a systematic, highly practical introduction to the use of energy principles, traditional variational methods, and the finite element method for the solution of engineering problems involving bars, beams, torsion, plane elasticity, trusses, and plates. It begins with a review of the basic equations of mechanics, the concepts of work and energy, and key topics from variational calculus. It presents virtual work and energy principles, energy methods of solid and structural mechanics, Hamilton's principle for dynamical systems, and classical variational methods of approximation. And it takes a more unified approach than that found in most solid mechanics books, to introduce the finite element method. Featuring more than 200 illustrations and tables, this Third Edition has been extensively reorganized and contains much new material, including a new chapter devoted to the latest developments in functionally graded beams and plates. Offers clear and easy-to-follow descriptions of the concepts of work, energy, energy principles and variational methods. Covers energy principles of solid and structural mechanics, traditional variational methods, the least-squares variational method, and the finite element, along with applications for each. Provides an abundance of examples, in a problem-solving format, with descriptions of applications for equations derived in obtaining solutions to engineering structures. Features end-of-the-chapter problems for course assignments, a Companion Website with a Solutions Manual, Instructor's Manual, figures, and more. Energy Principles and Variational Methods in Applied Mechanics, Third Edition is both a superb text/reference for engineering students in aerospace, civil, mechanical, and applied mechanics, and a valuable working resource for engineers in design and analysis in the aircraft, automobile, civil engineering, and shipbuilding industries.
Bibliographic references
Includes bibliographical references and index.
Contents
1. Introduction and Mathematical Preliminaries
1.1 Introduction
1.1.1 Preliminary Comments
1.1.2 The Role of Energy Methods and Variational Principles
1.1.3 A Brief Review of Historical Developments
1.1.4 Preview
1.2 Vectors
1.2.1 Introduction
1.2.2 Definition of a Vector
1.2.3 Scalar and Vector Products
1.2.4 Components of a Vector
1.2.5 Summation Convention
1.2.6 Vector Calculus
1.2.7 Gradient, Divergence, and Curl Theorems
1.3 Tensors
1.3.1 Second-Order Tensors
1.3.2 General Properties of a Dyadic
1.3.3 Nonion Form and Matrix Representation of a Dyad
1.3.4 Eigenvectors Associated with Dyads
1.4 Summary
Problems
2. Review of Equations of Solid Mechanics
2.1 Introduction
2.1.1 Classification of Equations
2.1.2 Descriptions of Motion
2.2 Balance of Linear and Angular Momenta
2.2.1 Equations of Motion
2.2.2 Symmetry of Stress Tensors
2.3 Kinematics of Deformation
2.3.1 Green-Lagrange Strain Tensor
2.3.2 Strain Compatibility Equations
2.4 Constitutive Equations
2.4.1 Introduction
2.4.2 Generalized Hooke's Law
2.4.3 Plane Stress-Reduced Constitutive Relations
2.4.4 Thermoelastic Constitutive Relations
2.5 Theories of Straight Beams
2.5.1 Introduction
2.5.2 The Bernoulli-Euler Beam Theory
2.5.3 The Timoshenko Beam Theory
2.5.4 The von Ka'rma'n Theory of Beams
2.5.4.1 Preliminary Discussion
2.5.4.2 The Bernoulli-Euler Beam Theory
2.5.4.3 The Timoshenko Beam Theory
2.6 Summary
Problems.
3. Work, Energy, and Variational Calculus
3.1 Concepts of Work and Energy
3.1.1 Preliminary Comments
3.1.2 External and Internal Work Done
3.2 Strain Energy and Complementary Strain Energy
3.2.1 General Development
3.2.2 Expressions for Strain Energy and Complementary Strain Energy Densities of Isotropic Linear Elastic Solids
3.2.2.1 Stain energy density
3.2.2.2 Complementary stain energy density
3.2.3 Strain Energy and Complementary Strain Energy for Trusses
3.2.4 Strain Energy and Complementary Strain Energy for Torsional Members
3.2.5 Strain Energy and Complementary Strain Energy for Beams
3.2.5.1 The Bernoulli-Euler Beam Theory
3.2.5.2 The Timoshenko Beam Theory
3.3 Total Potential Energy and Total Complementary Energy
3.3.1 Introduction
3.3.2 Total Potential Energy of Beams
3.3.3 Total Complementary Energy of Beams
3.4 Virtual Work
3.4.1 Virtual Displacements
3.4.2 Virtual Forces
3.5 Calculus of Variations
3.5.1 The Variational Operator
3.5.2 Functionals
3.5.3 The First Variation of a Functional
3.5.4 Fundamental Lemma of Variational Calculus
3.5.5 Extremum of a Functional
3.5.6 The Euler Equations
3.5.7 Natural and Essential Boundary Conditions
3.5.8 Minimization of Functionals with Equality Constraints
3.5.8.1 The Lagrange Multiplier Method
3.5.8.2 The Penalty Function Method
3.6 Summary
Problems
4. Virtual Work and Energy Principles of Mechanics
4.1 Introduction
4.2 The Principle of Virtual Displacements
4.2.1 Rigid Bodies
4.2.2 Deformable Solids
4.2.3 Unit Dummy-Displacement Method
4.3 The Principle of Minimum Total Potential Energy and Castigliano's Theorem I
4.3.1 The Principle of Minimum Total Potential Energy
4.3.2 Castigliano's Theorem I
4.4 The Principle of Virtual Forces
4.4.1 Deformable Solids
4.4.2 Unit Dummy-Load Method
4.5 Principle of Minimum Total Complementary Potential Energy and Castigliano's Theorem II
4.5.1 The Principle of the Minimum total Complementary Potential Energy
4.5.2 Castigliano's Theorem II
4.6 Clapeyron's, Betti's, and Maxwell's Theorems
4.6.1 Principle of Superposition for Linear Problems
4.6.2 Clapeyron's Theorem
4.6.3 Types of Elasticity Problems and Uniqueness of Solutions - 4.6.4 Betti's Reciprocity Theorem
4.6.5 Maxwell's Reciprocity Theorem
4.7 Summary
5. Dynamical Systems: Hamilton's Principle
5.1 Introduction
5.2 Hamilton's Principle for Discrete Systems
5.3 Hamilton's Principle for a Continuum
5.4 Hamilton's Principle for Constrained Systems
5.5 Rayleigh's Method
5.6 Summary
6. Direct Variational Methods
6.1 Introduction
6.2 Concepts from Functional Analysis
6.2.1 General Introduction
6.2.2 Linear Vector Spaces
6.2.3 Normed and Inner Product Spaces
6.2.3.1 Norm
6.2.3.2 Inner product
6.2.3.3 Orthogonality
6.2.4 Transformations, and Linear and Bilinear Forms
6.2.5 Minimum of a Quadratic Functional
6.3 The Ritz Method
6.3.1 Introduction
6.3.2 Description of the Method
6.3.3 Properties of Approximation Functions
6.3.3.1 Preliminary Comments
6.3.3.2 Boundary Conditions
6.3.3.3 Convergence
6.3.3.4 Completeness
6.3.3.5 Requirements on ɸ0 and ɸi
6.3.4 General Features of the Ritz Method
6.3.5 Examples
6.3.6 The Ritz Method for General Boundary-Value Problems
6.3.6.1 Preliminary Comments
6.3.6.2 Weak Forms
6.3.6.3 Model Equation 1
6.3.6.4 Model Equation 2
6.3.6.5 Model Equation 3
6.3.6.6 Ritz Approximations
6.4 Weighted-Residual Methods
6.4.1 Introduction
6.4.2 The General Method of Weighted Residuals
6.4.3 The Galerkin Method
6.4.4 The Least-Squares Method
6.4.5 The Collocation Method
6.4.6 The Subdomain Method
6.4.7 Eigenvalue and Time-Dependent Problems
6.4.7.1 Eigenvalue Problems
6.4.7.2 Time-Dependent Problems
6.5 Summary
7. Theory and Analysis of Plates
7.1 Introduction
7.1.1 General Comments
7.1.2 An Overview of Plate Theories
7.1.2.1 The Classical Plate Theory
7.1.2.2 The First-Order Plate Theory
7.1.2.3 The Third-Order Plate Theory
7.1.2.4 Stress-Based Theories
7.2 The Classical Plate Theory
7.2.1 Governing Equations of Circular Plates
7.2.2 Analysis of Circular Plates
7.2.2.1 Analytical Solutions For Bending
7.2.2.2 Analytical Solutions For Buckling
7.2.2.3 Variational Solutions
7.2.3 Governing Equations in Rectangular Coordinates
7.2.4 Navier Solutions of Rectangular Plates
7.2.4.1 Bending
7.2.4.2 Natural Vibration
7.2.4.3 Buckling Analysis
7.2.4.4 Transient Analysis
7.2.5 Lévy Solutions of Rectangular Plates
7.2.6 Variational Solutions: Bending
7.2.7 Variational Solutions: Natural Vibration
7.2.8 Variational Solutions: Buckling
7.2.8.1 Rectangular Plates Simply Supported along Two Opposite Sides and Compressed in the Direction Perpendicular to Those Sides
7.2.8.2 Formulation for Rectangular Plates with Arbitrary Boundary Conditions
7.3 The First-Order Shear Deformation Plate Theory
7.3.1 Equations of Circular Plates
7.3.2 Exact Solutions of Axisymmetric Circular Plates
7.3.3 Equations of Plates in Rectangular Coordinates
7.3.4 Exact Solutions of Rectangular Plates
7.3.4.1 Bending Analysis
7.3.4.2 Natural Vibration
7.3.4.3 Buckling Analysis
7.3.5 Variational Solutions of Circular and Rectangular Plates
7.3.5.1 Axisymmetric Circular Plates
7.3.5.2 Rectangular Plates
7.4 Relationships Between Bending Solutions of Classical and Shear Deformation Theories
7.4.1 Beams
7.4.1.1 Governing Equations
7.4.1.2 Relationships Between BET and TBT
7.4.2 Circular Plates
7.4.3 Rectangular Plates
7.5 Summary
8. The Finite Element Method
8.1 Introduction
8.2 Finite Element Analysis of Straight Bars
8.2.1 Governing Equation
8.2.2 Representation of the Domain by Finite Elements
8.2.3 Weak Form over an Element
8.2.4 Approximation over an Element
8.2.5 Finite Element Equations
8.2.5.1 Linear Element
8.2.5.2 Quadratic Element
8.2.6 Assembly (Connectivity) of Elements
8.2.7 Imposition of Boundary Conditions
8.2.8 Postprocessing
8.3 Finite Element Analysis of the Bernoulli-Euler Beam Theory
8.3.1 Governing Equation
8.3.2 Weak Form over an Element
8.3.3 Derivation of the Approximation Functions
8.3.4 Finite Element Model
8.3.5 Assembly of Element Equations
8.3.6 Imposition of Boundary Conditions
8.4 Finite Element Analysis of the Timoshenko Beam Theory
8.4.1 Governing Equations
8.4.2 Weak Forms
8.4.3 Finite Element Models
8.4.4 Reduced Integration Element (RIE)
8.4.5 Consistent Interpolation Element (CIE)
8.4.6 Superconvergent Element (SCE)
8.5 Finite Element Analysis of the Classical Plate Theory
8.5.1 Introduction
8.5.2 General Formulation
8.5.3 Conforming and Nonconforming Plate Elements
8.5.4 Fully Discretized Finite Element Models
8.5.4.1 Static Bending
8.5.4.2 Buckling
8.5.4.3 Natural Vibration
8.5.4.4 Transient Response
8.6 Finite Element Analysis of the First-Order Shear Deformation Plate Theory
8.6.1 Governing Equations and Weak Forms
8.6.2 Finite Element Approximations
8.6.3 Finite Element Model
8.6.4 Numerical Integration
8.6.5 Numerical Examples
8.6.5.1 Isotropic Plates
8.6.5.2 Laminated Plates
8.7 Summary
Problems.
9. Mixed Variational and Finite Element Formulations
9.1 Introduction
9.1.1 General Comments
9.1.2 Mixed Variational Principles
9.1.3 Extremum and Stationary Behavior of Functionals
9.2 Stationary Variational Principles
9.2.1 Minimum Total Potential Energy
9.2.2 The Hellinger-Reissner Variational Principle
9.2.3 The Reissner Variational Principle
9.3 Variational Solutions Based on Mixed Formulations
9.4 Mixed Finite Element Models of Beams
9.4.1 The Bernoulli-Euler Beam Theory
9.4.1.1 Governing Equations And Weak Forms
9.4.1.2 Weak-Form Mixed Finite Element Model
9.4.1.3 Weighted-Residual Finite Element Models
9.4.2 The Timoshenko Beam Theory
9.4.2.1 Governing Equations
9.4.2.2 General Finite Element Model
9.4.2.3 ASD-LLCC Element
9.4.2.4 ASD-QLCC Element
9.4.2.5 ASD-HQLC Element
9.5 Mixed Finite Element Analysis of the Classical Plate Theory
9.5.1 Preliminary Comments
9.5.2 Mixed Model I
9.5.2.1 Governing Equations
9.5.2.2 Weak Forms
9.5.2.3 Finite Element Model
9.5.3 Mixed Model II
9.5.3.1 Governing Equations
9.5.3.2 Weak Forms
9.5.3.3 Finite Element Model
9.6 Summary
10. Analysis of Functionally Graded Beams and Plates
10.1 Introduction
10.2 Functionally Graded Beams
10.2.1 The Bernoulli-Euler Beam Theory
10.2.1.1 Displacement and strain fields
10.2.1.2 Equations of motion and boundary conditions
10.2.2 The Timoshenko Beam Theory
10.2.2.1 Displacement and strain fields
10.2.2.2 Equations of motion and boundary conditions
10.2.3 Equations of Motion in terms of Generalized Displacements
10.2.3.1 Constitutive Equations
10.2.3.2 Stress Resultants of BET
10.2.3.3 Stress Resultants of TBT
10.2.3.4 Equations of Motion of the BET
10.2.3.5 Equations of Motion of the TBT
10.2.4 Stiffiness Coefficients
10.3 Functionally Graded Circular Plates
10.3.1 Introduction
10.3.2 Classical Plate Theory
10.3.2.1 Displacement and Strain Fields
10.3.2.2 Equations of Motion
10.3.3 First-Order Shear Deformation Theory
10.3.3.1 Displacement and Strain Fields
10.3.3.2 Equations of Motion
10.3.4 Plate Constitutive Relations
10.3.4.1 Classical Plate Theory
10.3.4.2 First-Order Plate Theory
10.4 A General Third-Order Plate Theory
10.4.1 Introduction
10.4.2 Displacements and Strains
10.4.3 Equations of Motion
10.4.4 Constitutive Relations
10.4.5 Specialization to Other Theories
10.4.5.1 A General Third-Order Plate Theory with Traction-Free Top and Bottom Surfaces
10.4.5.2 The Reddy Third-Order Plate Theory
10.4.5.3 The First-Order Plate Theory
10.4.5.4 The Classical Plate Theory
10.5 Navier's Solutions
10.5.1 Preliminary Comments
10.5.2 Analysis of Beams
10.5.2.1 Bernoulli-Euler Beams
10.5.2.2 Timoshenko Beams
10.5.2.3 Numerical Results
10.5.3 Analysis of Plates
10.5.3.1 Boundary Conditions
10.5.3.2 Expansions of Generalized Displacements
10.5.3.3 Bending Analysis
10.5.3.4 Free Vibration Analysis
10.5.3.5 Buckling Analysis
10.5.3.6 Numerical Results
10.6 Finite Element Models
10.6.1 Bending of Beams
10.6.1.1 Bernoulli-Euler Beam Theory
10.6.1.2 Timoshenko Beam Theory
10.6.2 Axisymmetric Bending of Circular Plates
10.6.2.1 Classical Plate Theory
10.6.2.2 First-Order Shear Deformation Plate Theory
10.6.3 Solution of Nonlinear Equations
10.6.3.1 Times approximation
10.6.3.2 Newton's Iteration Approach
10.6.3.3 Tangent Stiffiness Coefficients for the BET
10.6.3.4 Tangent Stiffiness Coefficients for the TBT
10.6.3.5 Tangent Stiffiness Coefficients for the CPT
10.6.3.6 Tangent Stiffiness Coefficients for the FSDT
10.6.4 Numerical Results for Beams and Circular Plates
10.6.4.1 Beams
10.6.4.2 Circular Plates --10.7 Summary
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ISBN
1119087376 ((paperback))
9781119087373 ((paperback))
OCLC
967365306
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Energy principles and variational methods in applied mechanics / J.N. Reddy, (Texas A&M University, USA).
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99125352937206421