THERMAL STRESSES
2nd Edition
Naotake Noda
Department of Mechanical Engineering
Shizuoka University
Hamamatsu, Japan
Richard B. Hetnarski
Department of Mechanical Engineering
Rochester Institute of Technology
Rochester, New York, USA
Yoshinobu Tanigawa
Department of Mechanical Systems Engineering
Osaka Prefecture University
Sakai, Japan
Taylor & Francis
Taylor & Francis Group
NEW
YORK •
LONDON
CONTENTS
Preface to the Second Edition
Preface to the First Edition
Thermal Stresses in Bars
1.1
xi
xiii
1
Stress and Strain
1.1.1 Stress and Strain
1.1.2 Hooke'sLaw
1.1.3 Free Thermal Expansion and Free Thermal Strain
1.1.4 Action of External Force and Temperature Change
1.2 Thermal Stresses in Clamped Bars
1.2.1 Constant Temperature Change
1.2.2 Non-Uniform Temperature Change
1.3 Thermal Stresses in Partially Restrained Bars
1.4 Thermal Stresses in Bars Under Bending
Problems
1
1
3
4
5
5
5
13
15
22
24
Thermal Stresses in Beams
29
2.1
29
29
31
32
34
39
39
46
2.2
2.3
Thermal Stresses in Beams
2.1.1 Stresses in Beams Subjected to Mechanical Loads
2.1.2 Thermal Stresses in Clamped Beams
2.1.3 Thermal Stresses in Rectangular Beams
General Technique for Thermal Stresses in Beams
Thermal Stresses in Composite Beams
2.3.1 Thermal Stresses in a Two-Layered Beam
2.3.2 Thermal Stresses in Two Beams Clamped at Each End
vi
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Contents
2.3.3 Thermal Stresses in Multi-Layered Composite Beams
Thermal Stresses in Nonhomogeneous Beams
Thermal Stresses at Arbitrary Temperature
2.5.1 Homogeneous Beams
2.5.2 Nonhomogeneous Beams
2.6 Thermal Deflection in Beams
2.7 Curved Beams
2.8 Thermal Shearing Stresses in Thin-Walled Beams
2.9 Thermal Stresses in Beams on an Elastic Foundation
Problems
2.4
2.5
3 Heat Conduction
3.1
3.2
Heat Conduction Equation
Boundary and Initial Conditions
3.2.1 Boundary Conditions
3.2.2 Initial Condition
3.3 One-Dimensional Heat Conduction Problems in Cartesian Coordinates
3.3.1 One-Dimensional Heat Conduction Equation
3.3.2 One-Dimensional Temperature in the Steady State
3.3.3 One-Dimensional Transient Temperature; Separation of Variables
3.3.4 One-Dimensional Transient Temperature; Laplace Transform
3.4 One-Dimensional Heat Conduction Problems in Cylindrical Coordinates
3.4.1 Heat Conduction Equation in Cylindrical Coordinates
3.4.2 One-Dimensional Temperature in the Steady State
3.4.3 One-Dimensional Transient Temperature in a Solid Cylinder
3.4.4 One-Dimensional Transient Temperature in a Hollow Cylinder
3.5 One-Dimensional Heat Conduction Problems in Spherical Coordinates
3.5.1 Heat Conduction Equation in Spherical Coordinates
3.5.2 One-Dimensional Temperature in the Steady State
3.5.3 One-Dimensional Transient Temperature in a Solid Sphere
3.5.4 One-Dimensional Transient Temperature in a Hollow Sphere
Problems
4 Basic Equations of Thermoelasticity
4.1
4.2
4.3
4.4
Stress in a Cartesian Coordinate System
4.1.1 Equations of Equilibrium
4.1.2 Coordinate Transformation of Stress
Strain in a Cartesian Coordinate System
4.2.1 Components of Strain
4.2.2 Compatibility Conditions
4.2.3 Coordinate Transformation of Strain
Governing Equations of Thermoelasticity
4.3.1 Constitutive Equations
4.3.2 Displacement Equations
4.3.3 Compatibility Equations in Terms of Stress
4.3.4 Boundary Conditions
4.3.5 Body-Force Analogy
General Solution of Navier's Equations
4.4.1 Thermoelastic Displacement Potential
52
54
56
56
59
62
65
68
71
76
81
81
89
90
93
93
93
94
96
102
111
111
112
115
118
123
123
124
127
129
134
137
137
137
142
144
144
145
146
147
147
149
150
151
152
153
153
Contents
4.4.2 Displacement Potentials
4.4.3 General Solutions for the Potential Function
4.5 Cylindrical Coordinate System
4.5.1 Stress in a Cylindrical Coordinate System
4.5.2 Strain in a Cylindrical Coordinate System
4.5.3 Generalized Hooke's Law in a Cylindrical
Coordinate System
4.5.4 Navier's Equations of Thermoelasticity in a
Cylindrical Coordinate System
4.6 Spherical Coordinate System
4.6.1 Stress in a Spherical Coordinate System
4.6.2 Strain in a Spherical Coordinate System
4.6.3 Generalized Hooke's Law in a Spherical Coordinate System
4.6.4 Navier's Equations of Thermoelasticity in a
Spherical Coordinate System
4.7 Multiply Connected Bodies
4.7.1 Basic Equations for Multiply Connected Bodies
4.7.2 Temperature Distributions with Zero Thermal Stress
Problems
5 Plane Thermoelastic Problems
5.1
Plane Strain and Plane Stress
5.1.1 Introduction
5.1.2 Plane Strain
5.1.3 Plane Stress
5.1.4 Governing Equations of Plane Problems
5.2 Thermal Stress Function
5.2.1 Thermal Stress Function
5.2.2 Displacements by Thermal Stress Function
5.2.3 Boundary Conditions in Terms of Thermal Stress Function
5.2.4 Multiply Connected Bodies
5.2.5 Temperature Distribution with Zero Thermal Stress
5.2.6 General Solution of Thermal Stress Function
5.3 Complex Variable Method
5.3.1 Complex Presentation of Thermal Stress Function
5.3.2 Complex Representation of Stress and Displacement
5.3.3 General Formulae for Multiply Connected Bodies
5.3.4 Conformal Mapping of the Formulae of Plane Theory
5.4 Potential Method
Problems
6 Thermal Stresses in Circular Cylinders
6.1
One-Dimensional Problems
6.1.1 Displacement Technique
6.1.2 Thermal Stress Function Technique
6.1.3 Thermoelastic Potential Technique
6.1.4 Complex Variable Technique
6.1.5 A Note on Reciprocal Theory Technique
6.1.6 Dislocation Technique
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vii
156
160
162
162
163
f
165
165
168
168
169
171
172
173
173
176
178
179
179
179
180
183
187
189
189
190
192
193
196
198
203
203
205
208
211
214
218
221
221
222
233
234
236
240
240
i
I
Contents
6.2
Plane Problems
6.2.1 Steady State Thermal Stresses in a Hollow Cylinder
6.2.2 Transient Thermal Stresses in a Solid Cylinder
6.3 Two-Dimensional Problems (Axisymmetric Problems)
6.3.1 Fundamental Equations for Two-Dimensional Problems
6.3.2 Steady State Thermal Stresses in a Semi-Infinite Body
6.3.3 Steady State Thermal Stresses in a Long Circular Cylinder
6.3.4 Transient Thermal Stresses in a Thick Plate
6.4 Three-Dimensional Problems
6.4.1 Solution Methods in Three-Dimensional Problems
6.4.2 Steady State Thermal Stresses in a Semi-Infinite Body
6.4.3 Steady State Thermal Stresses in a Long Circular Cylinder
Problems
244
244
250
259
259
261
266
269
273
273
276
284
293
Thermal Stresses in Spherical Bodies
295
7.1
7.2
Thermal Stresses in Spherical Bodies
One-Dimensional Problems
7.2.1 A Hollow Sphere
7.2.2 A Solid Sphere
7.2.3 Infinite Bodies
7.3 Two-Dimensional Problems (Axisymmetric Problems)
7.4 Illustrative Examples of Axisymmetric Problems in Spherical Bodies
7.4.1 Fundamental Solution of Temperature Field
7.4.2 Components of Displacement and Stress in Terms of Displacement
Functions <p and \jf
Problems
295
295
297
301
304
309
314
314
Thermal Stresses in Plates
345
316
341
8.1
Thermal Stress in a Plate Due to Uniform
Thermal Load
8.2 Basic Equations for a Rectangular Plate
8.2.1 Fundamental Relations of Thermal Bending
8.2.2 Boundary Conditions of the Plate
8.3 Fundamental Solutions for Rectangular Plates
8.3.1 Simply Supported Rectangular Plate
8.3.2 A Rectangular Plate with Two Opposite Edges Simply Supported
and the Other Two Edges Clamped
8.3.3 A Rectangular Plate with Built-in All Edges
8.4 Basic Equations for a Circular Plate
8.4.1 Fundamental Relations of Thermal Bending
8.4.2 Boundary Conditions for a Circular Plate
8.4.3 Axisymmetric Thermal Bending Problems
8.5 Fundamental Solutions for Circular Plates
8.5.1 Axisymmetric Problems
8.5.2 Non-axisymmetric Problems
Problems
367
371
376
376
379
380
381
381
387
392
Thermally Induced Instability
397
9.1
397
Instability of Beam-Column
345
355
355
360
365
365
Contents
9.1.1 Introduction
9.1.2 Beam-Column Problems by Axially Imposed External Load
9.2 Instability of Plate
9.2.1 Basic Equations for a Rectangular Plate
9.2.2 Illustrative Examples for a Rectangular Plate
9.2.3 Basic Equations for a Circular Plate
9.2.4 Illustrative Examples for a Circular Plate
Problems
10 Thermodynamics of Thermoelasticity
10.1 The Principle of Energy Conservation
10.2 The Second Law of Thermodynamics
10.2.1 Reversible and Irreversible Processes
10.2.2 Cycle of Carnot
10.2.3 Entropy
10.3 Thermodynamic Functions
10.4 Fundamental Differential Equations of Thermoelasticity
10.4.1 Heat Conduction Equation
10.4.2 Equation of Motion
10.4.3 Solution of Equation of Motion
10.5 The Variational Theorems of Thermoelasticity
10.6 Uniqueness Theorem
10.7 Reciprocal Theorem
10.7.1 Reciprocal Theorem
10.7.2 Practical Problem
Problems
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ix
397
399
409
409
413
419
424
429
431
431
434
434
435
437
439
446
446
447
449
451
454
457
457
463
466
A Inverse Laplace Transform
469
B Bessel Functions
473
C Legendre Functions
479
Books on Thermal Stresses
485
Index
489