كتاب Mechanics and Strength of Materials

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Mechanics and Strength of Materials
Vitor Dias da Silva
Department of Civil Engineering
Faculty of Science & Technology
University of Coimbra
Polo II da Universidade - Pinhal de Marrocos

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Introduction 3
I.1 General Considerations 3
I.2 Fundamental Definitions 4
I.3 Subdivisions of the Mechanics of Materials 6
II The Stress Tensor 9
II.1 Introduction 9
II.2 General Considerations 9
II.3 Equilibrium Conditions 12
II.3.a Equilibrium in the Interior of the Body 12
II.3.b Equilibrium at the Boundary 15
II.4 Stresses in an Inclined Facet 16
II.5 Transposition of the Reference Axes 17
II.6 Principal Stresses and Principal Directions 19
II.6.a The Roots of the Characteristic Equation 21
II.6.b Orthogonality of the Principal Directions 22
II.6.c Lam´e’s Ellipsoid 22
II.7 Isotropic and Deviatoric Components
of the Stress Tensor 24
II.8 Octahedral Stresses 25
II.9 Two-Dimensional Analysis of the Stress Tensor 27
II.9.a Introduction 27
II.9.b Stresses on an Inclined Facet 28
II.9.c Principal Stresses and Directions 29
II.9.d Mohr’s Circle 31
II.10 Three-Dimensional Mohr’s Circles 33
II.11 Conclusions 36
II.12 Examples and Exercises
III The Strain Tensor 41
III.1 Introduction 41
III.2 General Considerations 41
III.3 Components of the Strain Tensor 44
III.4 Pure Deformation and Rigid Body Motion 49
III.5 Equations of Compatibility 51
III.6 Deformation in an Arbitrary Direction 54
III.7 Volumetric Strain 58
III.8 Two-Dimensional Analysis of the Strain Tensor 59
III.8.a Introduction 59
III.8.b Components of the Strain Tensor 60
III.8.c Strain in an Arbitrary Direction 60
III.9 Conclusions 63
III.10 Examples and Exercises 64
IV Constitutive Law 67
IV.1 Introduction 67
IV.2 General Considerations 67
IV.3 Ideal Rheological Behaviour – Physical Models 69
IV.4 Generalized Hooke’s Law 75
IV.4.a Introduction 75
IV.4.b Isotropic Materials 75
IV.4.c Monotropic Materials 80
IV.4.d Orthotropic Materials 82
IV.4.e Isotropic Material with Linear Visco-Elastic
Behaviour 83
IV.5 Newtonian Liquid 84
IV.6 Deformation Energy 86
IV.6.a General Considerations 86
IV.6.b Superposition of Deformation Energy
in the Linear Elastic Case 89
IV.6.c Deformation Energy in Materials
with Linear Elastic Behaviour 90
IV.7 Yielding and Rupture Laws 92
IV.7.a General Considerations 92
IV.7.b Yielding Criteria 93
IV.7.b.i Theory of Maximum Normal Stress 93
IV.7.b.ii Theory of Maximum Longitudinal Deformation 94
IV.7.b.iii Theory of Maximum Deformation Energy 94
IV.7.b.iv Theory of Maximum Shearing Stress 95
IV.7.b.v Theory of Maximum Distortion Energy 95
IV.7.b.vi Comparison of Yielding Criteria 96
IV.7.b.vii Conclusions About the Yielding Theories 100
IV.7.c Mohr’s Rupture Theory for Brittle Materials 101
IV.8 Concluding Remarks
IV.9 Examples and Exercises 106
Part II Strength of Materials
V Fundamental Concepts of Strength of Materials 119
V.1 Introduction 119
V.2 Ductile and Brittle Material Behaviour 121
V.3 Stress and Strain 123
V.4 Work of Deformation. Resilience and Tenacity. 125
V.5 High-Strength Steel 127
V.6 Fatigue Failure 128
V.7 Saint-Venant’s Principle 130
V.8 Principle of Superposition 131
V.9 Structural Reliability and Safety 133
V.9.a Introduction 133
V.9.b Uncertainties Affecting the Verification
of Structural Reliability 133
V.9.c Probabilistic Approach. 134
V.9.d Semi-Probabilistic Approach 135
V.9.e Safety Stresses 136
V.10 Slender Members 137
V.10.a Introduction 137
V.10.b Definition of Slender Member 138
V.10.c Conservation of Plane Sections 138
VI Axially Loaded Members 141
VI.1 Introduction 141
VI.2 Dimensioning of Members Under Axial Loading 142
VI.3 Axial Deformations 142
VI.4 Statically Indeterminate Structures 143
VI.4.a Introduction 143
VI.4.b Computation of Internal Forces 144
VI.4.c Elasto-Plastic Analysis 145
VI.5 An Introduction to the Prestressing Technique 150
VI.6 Composite Members 153
VI.6.a Introduction 153
VI.6.b Position of the Stress Resultant 153
VI.6.c Stresses and Strains Caused by the Axial Force 154
VI.6.d Effects of Temperature Variations 155
VI.7 Non-Prismatic Members 157
VI.7.a Introduction 157
VI.7.b Slender Members with Curved Axis 157
VI.7.c Slender Members with Variable Cross-Section 159
VI.8 Non-Constant Axial Force – Self-Weight
VI.9 Stress Concentrations 161
VI.10 Examples and Exercises 163
VII Bending Moment 189
VII.1 Introduction 189
VII.2 General Considerations 190
VII.3 Pure Plane Bending 193
VII.4 Pure Inclined Bending 196
VII.5 Composed Circular Bending 200
VII.5.a The Core of a Cross-Section 202
VII.6 Deformation in the Cross-Section Plane 204
VII.7 Influence of a Non-Constant Shear Force 209
VII.8 Non-Prismatic Members 210
VII.8.a Introduction 210
VII.8.b Slender Members with Variable Cross-Section 210
VII.8.c Slender Members with Curved Axis 212
VII.9 Bending of Composite Members 213
VII.9.a Linear Analysis of Symmetrical Reinforced
Concrete Cross-Sections 216
VII.10 Nonlinear bending 219
VII.10.a Introduction 219
VII.10.b Nonlinear Elastic Bending 220
VII.10.c Bending in Elasto-Plastic Regime 221
VII.10.d Ultimate Bending Strength
of Reinforced Concrete Members 226
VII.11 Examples and Exercises 228
VIII Shear Force 251
VIII.1 General Considerations 251
VIII.2 The Longitudinal Shear Force 252
VIII.3 Shearing Stresses Caused by the Shear Force 258
VIII.3.a Rectangular Cross-Sections 258
VIII.3.b Symmetrical Cross-Sections 259
VIII.3.c Open Thin-Walled Cross-Sections 261
VIII.3.d Closed Thin-Walled Cross-Sections 265
VIII.3.e Composite Members 268
VIII.3.f Non-Principal Reference Axes. 269
VIII.4 The Shear Centre 270
VIII.5 Non-Prismatic Members 273
VIII.5.a Introduction 273
VIII.5.b Slender Members with Curved Axis 273
VIII.5.c Slender Members with Variable Cross-Section 274
VIII.6 Influence of a Non-Constant Shear Force 275
VIII.7 Stress State in Slender Members 276
VIII.8 Examples and Exercises
IX Bending Deflections 297
IX.1 Deflections Caused by the Bending Moment 297
IX.1.a Introduction 297
IX.1.b Method of Integration of the Curvature Equation 298
IX.1.c The Conjugate Beam Method. 302
IX.1.d Moment-Area Method 304
IX.2 Deflections Caused by the Shear Force 308
IX.2.a Introduction 308
IX.2.b Rectangular Cross-Sections 311
IX.2.c Symmetrical Cross-Sections 312
IX.2.d Thin-Walled Cross-Sections. 312
IX.3 Statically Indeterminate Frames Under Bending 315
IX.3.a Introduction 315
IX.3.b Equation of Two Moments 317
IX.3.c Equation of Three Moments 317
IX.4 Elasto-Plastic Analysis Under Bending 320
IX.5 Examples and Exercises 323
X Torsion 347
X.1 Introduction 347
X.2 Circular Cross-Sections 347
X.2.a Torsion in the Elasto-Plastic Regime. 353
X.3 Closed Thin-Walled Cross-Sections 356
X.3.a Applicability of the Bredt Formulas 361
X.4 General Case 362
X.4.a Introduction 362
X.4.b Hydrodynamical Analogy 364
X.4.c Membrane Analogy. 365
X.4.d Rectangular Cross-Sections 367
X.4.e Open Thin-Walled Cross-Sections 368
X.5 Optimal Shape of Cross-Sections Under Torsion 369
X.6 Examples and Exercises 371
XI Structural Stability 389
XI.1 Introduction 389
XI.2 Fundamental Concepts 391
XI.2.a Computation of Critical Loads 391
XI.2.b Post-Critical Behaviour 393
XI.2.c Effect of Imperfections 396
XI.2.d Effect of Plastification of Deformable Elements 399
XI.3 Instability in the Axial Compression
of a Prismatic Bar 401
XI.3.a Introduction 401
XI.3.b Euler’s Problem. 402
XI.3.c Prismatic Bars with Other Support Conditions
XI.3.d Safety Evaluation of Axially Compressed Members405
XI.3.e Optimal Shape of Axially Compressed
Cross-Sections 409
XI.4 Instability Under Composed Bending 409
XI.4.a Introduction and General Considerations 409
XI.4.b Safety Evaluation 414
XI.4.c Composed Bending with a Tensile Axial Force 416
XI.5 Examples and Exercises 416
XI.6 Stability Analysis by the Displacement Method 439
XI.6.a Introduction 439
XI.6.b Simple Examples 440
XI.6.c Framed Structures Under Bending 445
XI.6.c.i Stiffness Matrix of a Compressed Bar 445
XI.6.c.ii Stiffness Matrix of a Tensioned Bar 451
XI.6.c.iiiLinearization of the Stiffness Coefficients 452
XI.6.c.ivExamples of Application 455
XII Energy Theorems 465
XII.1 General Considerations 465
XII.2 Elastic Potential Energy in Slender Members 466
XII.3 Theorems for Structures with Linear Elastic Behaviour 468
XII.3.a Clapeyron’s Theorem 468
XII.3.b Castigliano’s Theorem 469
XII.3.c Menabrea’s Theorem or Minimum Energy
Theorem 473
XII.3.d Betti’s Theorem 473
XII.3.e Maxwell’s Theorem 477
XII.4 Theorems of Virtual Displacements and Virtual Forces 479
XII.4.a Theorem of Virtual Displacements 479
XII.4.b Theorem of Virtual Forces 482
XII.5 Considerations About the Total Potential Energy 485
XII.5.a Definition of Total Potential Energy 485
XII.5.b Principle of Stationarity of the Potential Energy 486
XII.5.c Stability of the Equilibrium 486
XII.6 Elementary Analysis of Impact Loads 489
XII.7 Examples and Exercises 491
XII.8 Chapter VII 517
XII.9 Chapter IX 518
References 523
Index

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