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E-raamat: Theories and Analyses of Beams and Axisymmetric Circular Plates [Taylor & Francis e-raamat]

(Texas A&M University, USA)
  • Formaat: 554 pages, 34 Tables, black and white; 147 Line drawings, black and white; 147 Illustrations, black and white
  • Sari: Applied and Computational Mechanics
  • Ilmumisaeg: 30-Jun-2022
  • Kirjastus: CRC Press
  • ISBN-13: 9781003240846
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  • Taylor & Francis e-raamat
  • Hind: 170,80 €*
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Theories and Analyses of Beams and Axisymmetric Circular PlatesSuurem pilt
  • Formaat: 554 pages, 34 Tables, black and white; 147 Line drawings, black and white; 147 Illustrations, black and white
  • Sari: Applied and Computational Mechanics
  • Ilmumisaeg: 30-Jun-2022
  • Kirjastus: CRC Press
  • ISBN-13: 9781003240846
Teised raamatud teemal:
Taylor & Francis e-raamatudRohkem infot Taylor & Francis e-raamatute kohta
Raamatu kodulehekülg: https://www.taylorfrancis.com/books/9781003240846
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.
Preface xvii
List of symbols used
xix
About the Author xxiii
Chapter 1 Mechanics Preliminaries
1(26)
1.1 General Comments
1(1)
1.2 Beams and Plates
2(2)
1.3 Vectors and Tensors
4(9)
1.3.1 Vectors and Coordinate Systems
4(1)
1.3.2 Summation Convention
5(1)
1.3.3 Stress Vector and Stress Tensor
6(3)
1.3.4 The Gradient Operator
9(4)
1.4 Review of the Equations of Solid Mechanics
13(6)
1.4.1 Green--Lagrange Strain Tensor
13(4)
1.4.2 The Second Piola--Kirchhoff Stress Tensor
17(1)
1.4.3 Equations of Motion
18(1)
1.4.4 Stress-Strain Relations
18(1)
1.5 Functionally Graded Structures
19(4)
1.5.1 Background
19(1)
1.5.2 Mori-Tanaka Scheme
20(1)
1.5.3 Voigt Scheme: Rule of Mixtures
21(1)
1.5.4 Exponential Model
21(1)
1.5.5 Power-Law Model
22(1)
1.6 Modified Couple Stress Effects
23(1)
1.6.1 Background
23(1)
1.6.2 The Strain Energy Functional
24(1)
1.7
Chapter Summary
24(3)
Chapter 2 Energy Principles and Variational Methods
27(34)
2.1 Concepts of Work and Energy
27(4)
2.1.1 Historical Background
27(1)
2.1.2 Objectives of the
Chapter
28(1)
2.1.3 Concept of Work Done
29(2)
2.2 Strain Energy and Complementary Strain Energy
31(4)
2.3 Total Potential Energy and Total Complementary Energy
35(1)
2.4 Virtual Work
36(3)
2.4.1 Virtual Displacements
36(3)
2.4.2 Virtual Forces
39(1)
2.5 Calculus of Variations and Duality Pairs
39(10)
2.5.1 The Variational Operator
39(2)
2.5.2 Functionals and Their Variations
41(1)
2.5.3 Fundamental Lemma of Variational Calculus
42(1)
2.5.4 Extremum of a Functional
43(1)
2.5.5 The Euler Equations and Duality Pairs
43(3)
2.5.6 Natural and Essential Boundary Conditions
46(3)
2.6 The Principle of Virtual Displacements
49(2)
2.7 Principle of Minimum Total Potential Energy
51(3)
2.8 Hamilton's Principle
54(4)
2.8.1 Preliminary Comments
54(1)
2.8.2 Statement of the Principle
55(1)
2.8.3 Euler--Lagrange Equations
56(2)
2.9
Chapter Summary
58(3)
Chapter 3 The Classical Beam Theory
61(90)
3.1 Introductory Comments
61(1)
3.2 Kinematics
62(1)
3.3 Equations of Motion
63(6)
3.3.1 Preliminary Comments
63(1)
3.3.2 Vector Approach
64(2)
3.3.3 Energy Approach
66(3)
3.4 Governing Equations in Terms of Displacements
69(5)
3.4.1 Material Constitutive Relations
69(1)
3.4.2 Uniaxial Stress-Strain Relations
69(1)
3.4.3 Material Gradation through the Beam Height
70(1)
3.4.4 Beam Constitutive Equations
70(2)
3.4.5 Equations of Motion
72(1)
3.4.5.1 The general case (with FGM, VKN, and MCS)
72(1)
3.4.5.2 Homogeneous beams with VKN and MCS
73(1)
3.4.5.3 Linearized FGM beams with MCS
73(1)
3.4.5.4 Linearized homogeneous beams with MCS
74(1)
3.5 Equations in Terms of Displacements and Bending Moment
74(3)
3.5.1 Preliminary Comments
74(1)
3.5.2 General Case with FGM, MCS, and VKN
75(1)
3.5.3 Special Cases
76(1)
3.5.3.1 Homogeneous beams with VKN and MCS
76(1)
3.5.3.2 Linearized FGM beams with MCS
77(1)
3.5.3.3 Linearized homogeneous beams with MCS
77(1)
3.6 Cylindrical Bending of FGM Rectangular Plates
77(3)
3.6.1 Cylindrical Bending
77(1)
3.6.2 Governing Equations in Terms of Stress Resultants
78(1)
3.6.3 Governing Equations in Terms of Displacements
79(1)
3.7 Exact Solutions
80(22)
3.7.1 Bending Solutions
80(12)
3.7.2 Buckling and Natural Vibrations
92(1)
3.7.2.1 Buckling solutions
92(5)
3.7.2.2 Natural frequencies
97(5)
3.8 The Navier Solutions
102(11)
3.8.1 The General Procedure
102(1)
3.8.2 Navier's Solution of Equations of Motion
103(2)
3.8.3 Bending Solutions
105(3)
3.8.4 Natural Vibrations
108(1)
3.8.5 Transient Analysis
109(4)
3.9 Energy and Variational Methods
113(33)
3.9.1 Introduction
113(1)
3.9.2 The Ritz Method
114(1)
3.9.2.1 Background and model problem
114(1)
3.9.2.2 The Ritz approximation
115(3)
3.9.2.3 Requirements on the approximation functions
118(23)
3.9.3 The Weighted-Residual Methods
141(5)
3.10
Chapter Summary
146(5)
Chapter 4 The First-Order Shear Deformation Beam Theory
151(70)
4.1 Introductory Comments
151(1)
4.2 Displacements and Strains
152(1)
4.3 Equations of Motion
152(4)
4.3.1 Vector Approach
152(2)
4.3.2 Energy Approach
154(2)
4.4 Governing Equations in Terms of Displacements
156(4)
4.4.1 Beam Constitutive Equations
156(1)
4.4.2 Equations of Motion for the General Case
157(1)
4.4.3 Equations of Motion without the Couple Stress and Thermal Effects
158(1)
4.4.4 Equations of Motion for Homogeneous Beams
158(1)
4.4.5 Linearized Equations of Motion for FGM Beams
159(1)
4.4.6 Linearized Equations for Homogeneous Beams
159(1)
4.5 Mixed Formulation of the TBT
160(2)
4.6 Exact Solutions
162(15)
4.6.1 Bending Solutions
162(12)
4.6.2 Buckling Solutions
174(2)
4.6.3 Natural Vibration
176(1)
4.7 Relations between CBT and TBT
177(26)
4.7.1 Background
177(1)
4.7.2 Bending Relations between the CBT and TBT
178(1)
4.7.2.1 Summary of equations of the CBT
178(1)
4.7.2.2 Summary of equations of the TBT
178(1)
4.7.2.3 Relationships by similarity and load equivalence
179(8)
4.7.3 Bending Relationships for FGM Beams with the Couple Stress Effect
187(1)
4.7.3.1 Summary of equations of CBT and TBT
187(1)
4.7.3.2 General relationships
187(2)
4.7.3.3 Specialized relationships
189(11)
4.7.4 Buckling Relationships
200(1)
4.7.4.1 Summary of governing equations
200(2)
4.7.5 Frequency Relationships
202(1)
4.7.5.1 Governing equations of the CBT
202(1)
4.7.5.2 Governing equations of the TBT
202(1)
4.7.5.3 Relationship
203(1)
4.8 The Navier Solutions
203(10)
4.8.1 General Solution
203(2)
4.8.2 Bending Solution
205(2)
4.8.3 Natural Vibrations
207(6)
4.9 Solutions by Variational Methods
213(5)
4.10
Chapter Summary
218(3)
Chapter 5 Third-Order Beam Theories
221(82)
5.1 Introduction
221(2)
5.1.1 Why a Third-Order Theory?
221(1)
5.1.2 Present Study
222(1)
5.2 A General Third-Order Theory
223(7)
5.2.1 Kinematics
223(2)
5.2.2 Equations of Motion
225(3)
5.2.3 Equations of Motion without Couple Stress Effects
228(1)
5.2.4 Constitutive Relations
228(2)
5.3 A Third-Order Theory with Vanishing Shear Stress on the Top and Bottom Faces
230(14)
5.3.1 The General Case
230(2)
5.3.2 The Reddy Third-Order Beam Theory (RBT)
232(1)
5.3.2.1 Kinematics
232(1)
5.3.2.2 Equations of motion using Hamilton's principle
233(2)
5.3.2.3 Constitutive relations
235(3)
5.3.2.4 Equations of motion in terms of the generalized displacements: the general case
238(1)
5.3.2.5 Equations of motion in terms of the generalized displacements: the linear case
239(2)
5.3.3 Levinson's Third-Order Beam Theory (LBT)
241(1)
5.3.3.1 Equations of motion
241(1)
5.3.3.2 Equations of motion in terms of the displacements
242(1)
5.3.3.3 Equations of motion for the linear case
243(1)
5.3.3.4 Equations of equilibrium for the linear case
243(1)
5.3.3.5 Linearized equations without the couple stress effect
244(1)
5.4 Exact Solutions for Bending
244(15)
5.4.1 The Reddy Beam Theory
244(4)
5.4.2 The Simplified RBT
248(1)
5.4.2.1 FGM beams
248(2)
5.4.2.2 Homogeneous beams
250(1)
5.4.3 The Levinson Beam Theory
251(1)
5.4.3.1 FGM beams
251(2)
5.4.3.2 Homogeneous beams
253(6)
5.5 Bending Relationships for the RBT
259(21)
5.5.1 Preliminary Comments
259(1)
5.5.2 Summary of Equations
259(1)
5.5.3 General Relationships
260(3)
5.5.4 Bending Relationships for the Simplified RBT
263(2)
5.5.5 Relationships between the LBT and the CBT
265(2)
5.5.6 Numerical Examples
267(6)
5.5.7 Buckling Relationships
273(1)
5.5.7.1 Summary of equations of the CBT
273(1)
5.5.7.2 Summary of equations of the RBT
274(6)
5.6 Navier Solutions
280(12)
5.6.1 The Reddy Beam Theory (RBT)
280(4)
5.6.1.1 Bending analysis
284(1)
5.6.1.2 Natural vibration
285(1)
5.6.2 The Levinson Beam Theory (LBT)
285(1)
5.6.3 Numerical Results
286(6)
5.7 Solutions by Variational Methods
292(6)
5.8
Chapter Summary
298(5)
Chapter 6 Classical Theory of Circular Plates
303(52)
6.1 General Relations
303(3)
6.1.1 Preliminary Comments
303(1)
6.1.2 Kinematic Relations
304(1)
6.1.2.1 Modified Green-Lagrange strains
304(1)
6.1.2.2 Curvature tensor
304(1)
6.1.3 Stress--Strain Relations
305(1)
6.1.4 Strain Energy Functional
305(1)
6.2 Governing Equations of the CPT
306(8)
6.2.1 Displacements and Strains
306(1)
6.2.2 Equations of Motion
306(4)
6.2.3 Isotropic Constitutive Relations
310(1)
6.2.4 Displacement Formulation of the CPT
311(1)
6.2.5 Mixed Formulation of the CPT
312(2)
6.3 Solutions for Homogeneous Plates in Bending
314(9)
6.3.1 Governing Equations
314(1)
6.3.2 Exact Solutions
315(1)
6.3.3 Numerical Examples
316(7)
6.4 Bending Solutions for FGM Plates
323(9)
6.4.1 Governing Equations
323(1)
6.4.2 Exact Solutions
324(8)
6.5 Buckling and Natural Vibration
332(8)
6.5.1 Buckling Solutions
332(3)
6.5.2 Natural Frequencies
335(5)
6.6 Variational Solutions
340(12)
6.6.1 Introductory Comments
340(1)
6.6.2 Variational Statement
340(1)
6.6.3 The Ritz Method
341(3)
6.6.4 The Galerkin Method
344(4)
6.6.5 Natural Frequencies and Buckling Loads
348(1)
6.6.5.1 Variational statement
348(4)
6.7
Chapter Summary
352(3)
Chapter 7 First-Order Theory of Circular Plates
355(40)
7.1 Governing Equations
355(5)
7.1.1 Displacements and Strains
355(1)
7.1.2 Equations of Motion
356(1)
7.1.3 Plate Constitutive Relations
357(1)
7.1.4 Equations of Motion in Terms of the Displacements
358(1)
7.1.4.1 The general case
358(1)
7.1.4.2 Nonlinear equations of equilibrium
359(1)
7.1.4.3 Linear equations of equilibrium without couple stress
360(1)
7.1.4.4 Linear equations of equilibrium without couple stress and FGM
360(1)
7.2 Exact Solutions of Isotropic Circular Plates
360(5)
7.3 Exact Solutions for FGM Circular Plates
365(7)
7.3.1 Governing Equations
365(1)
7.3.2 Exact Solutions
366(2)
7.3.3 Examples
368(4)
7.4 Bending Relationships between CPT and FST
372(7)
7.4.1 Summary of the Governing Equations
372(1)
7.4.2 Relationships
372(2)
7.4.3 Examples
374(5)
7.5 Bending Relationships for Functionally Graded Circular Plates
379(13)
7.5.1 Introduction
379(1)
7.5.2 Summary of Equations
379(2)
7.5.3 Relationships between the CPT and FST
381(11)
7.6
Chapter Summary
392(3)
Chapter 8 Third-Order Theory of Circular Plates
395(20)
8.1 Governing Equations
395(5)
8.1.1 Preliminary Comments
395(1)
8.1.2 Displacements and Strains
395(1)
8.1.3 Equations of Motion
396(2)
8.1.4 Plate Constitutive Equations
398(2)
8.2 Exact Solutions of the TST
400(3)
8.3 Relationships between CPT and TST
403(11)
8.3.1 Bending Relationships
403(1)
8.3.1.1 Classical plate theory (CPT)
403(1)
8.3.1.2 Third-order shear deformation plate theory (TST)
404(1)
8.3.2 Relationships
405(4)
8.3.3 Buckling Relationships
409(1)
8.3.3.1 Governing equations
409(1)
8.3.3.2 Relationship between CPT and FST
410(1)
8.3.3.3 Relationship between CPT and TST
411(3)
8.4
Chapter Summary
414(1)
Chapter 9 Finite Element Analysis of Beams
415(72)
9.1 Introduction
415(8)
9.1.1 The Finite Element Method
415(2)
9.1.2 Interpolation Functions
417(5)
9.1.3 Present Study
422(1)
9.2 Displacement Model of the CBT
423(4)
9.2.1 Governing Equations and Variational Statements
423(2)
9.2.2 Finite Element Model
425(2)
9.3 Mixed Finite Element Model of the CBT
427(3)
9.3.1 Variational Statements
427(2)
9.3.2 Finite Element Model
429(1)
9.4 Displacement Finite Element Model of the TBT
430(3)
9.4.1 Governing Equations and Variational Statements
430(1)
9.4.2 The Finite Element Model
431(2)
9.5 Mixed Finite Element Model of the TBT
433(2)
9.5.1 Governing Equations and Variational Statements
433(2)
9.5.2 Finite Element Model
435(1)
9.6 Displacement Finite Element Model of the RBT
435(4)
9.6.1 Governing Equations
435(1)
9.6.2 Weak Forms
436(1)
9.6.3 Finite Element Model
437(2)
9.7 Time Approximation (Full Discretization)
439(3)
9.7.1 Introduction
439(1)
9.7.2 Newmark's Method
439(1)
9.7.3 Fully Discretized Equations
440(2)
9.8 Solution of Nonlinear Algebraic Equations
442(9)
9.8.1 Preliminary Comments
442(1)
9.8.2 Direct Iteration Procedure
443(1)
9.8.3 Newton's Iteration Procedure
444(6)
9.8.4 Load Increments
450(1)
9.9 Tangent Stiffness Coefficients
451(4)
9.9.1 Definition of Tangent Stiffness Coefficients
451(1)
9.9.2 The Displacement Model of the CBT
451(2)
9.9.3 The Mixed Model of the CBT
453(1)
9.9.4 The Displacement Model of the TBT
453(1)
9.9.5 The Mixed Model of the TBT
454(1)
9.9.6 The Displacement Model of the RBT
454(1)
9.10 Post-Computations
455(3)
9.10.1 General Comments
455(1)
9.10.2 CBT Finite Element Models
455(1)
9.10.3 TBT Finite Element Models
456(1)
9.10.4 RBT Displacement Model
457(1)
9.11 Numerical Results
458(24)
9.11.1 Geometry and Boundary Conditions
458(1)
9.11.2 Material Constitution
458(1)
9.11.3 Examples
459(23)
9.12
Chapter Summary
482(5)
Chapter 10 Finite Element Analysis of Circular Plates
487(38)
10.1 Introductory Remarks
487(1)
10.2 Displacement Model of the CPT
488(3)
10.2.1 Weak Forms
488(2)
10.2.2 Finite Element Model
490(1)
10.3 Mixed Model of the CPT
491(3)
10.3.1 Weak Forms
491(1)
10.3.2 Finite Element Model
492(2)
10.4 Displacement Model of the FST
494(4)
10.4.1 Weak Forms
494(2)
10.4.2 Finite Element Model
496(2)
10.5 Displacement Model of the TST
498(6)
10.5.1 Variational Statements
498(3)
10.5.2 Finite Element Model
501(3)
10.6 Tangent Stiffness Coefficients
504(2)
10.6.1 Preliminary Comments
504(1)
10.6.2 The Displacement Model of the CPT
504(1)
10.6.3 The Mixed Model of the CPT
504(1)
10.6.4 The Displacement Model of the FST
505(1)
10.6.5 The Displacement Model of the TST
505(1)
10.7 Numerical Results
506(16)
10.7.1 Preliminary Comments
506(1)
10.7.2 Linear Analysis
507(3)
10.7.3 Nonlinear Analysis without Couple Stress Effect
510(7)
10.7.4 Nonlinear Analysis with Couple Stress Effect
517(5)
10.8
Chapter Summary
522(3)
References 525(10)
Papers with Collaborators 535(6)
Answers 541(8)
Index 549
J. N. Reddy, the O'Donnell Foundation Chair IV Professor in J. MikeWalker '66 Department of Mechanical Engineering at Texas A&M University, is a highly-cited researcher, author of 24 textbooks and over 750 journal papers, and a leader in the applied mechanics field for more nearly 50 years. He is known worldwide for his significant contributions to the field of applied mechanics through the authorship of widely used textbooks on mechanics of materials, continuum mechanics, linear and nonlinear finite element analyses, energy principles and variational methods, and composite materials and structures. His pioneering works on the development of shear deformation theories of beams, plates, and shells (that bear his name in the literature as the Reddy third-order plate theory and the Reddy layerwise theory), nonlocal and non-classical continuum mechanics have had a major impact, and have led to new research developments and applications.
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