Chapter 1 Cartesian Tensors ..................................................... 1 1.1 Vectors .................................................................... 1 1.2 Dyads ..................................................................... 11 1.3 Definition and Rules of Operation of Tensors of the Second Rank ........... 12 1.4 Transformation of the Cartesian Components of a Tensor of the Second Rank upon Rotation of the System of Axes to Which They Are Referred ....... 20 1.5 Definition of a Tensor of the Second Rank on the Basis of the Law of Transformation of Its Components .......................................... 21 1.6 Symmetric Tensors of the Second Rank ...................................... 22 1.7 Invariants of the Cartesian Components of a Symmetric Tensor of the Second Rank ............................................................... 22 1.8 Stationary Values of a Function Subject to a Constraining Relation ........ 23 1.9 Stationary Values of the Diagonal Components of a Symmetric Tensor of the Second Rank ........................................................... 26 1.10 Quasi Plane Form of Symmetric Tensors of the Second Rank .................. 31 1.11 Stationary Values of the Diagonal and the Non-Diagonal Components of the Quasi Plane, Symmetric Tensors of the Second Rank ..................... 33 1.12 Mohr's Circle for Quasi Plane, Symmetric Tensors of the Second Rank ....... 37 1.13 Maximum Values of the Non-Diagonal Components of a Symmetric Tensor of the Second Rank ........................................................... 43 1.14 Problems .................................................................. 44 Chapter 2 Strain and Stress Tensors ............................................ 53 2.1 The Continuum Model ....................................................... 53 2.2 External Loads ............................................................ 53 2.3 The Displacement Vector of a Particle of a Body ........................... 55 2.4 Components of Strain of a Particle of a Body .............................. 56 2.5 Implications of the Assumption of Small Deformation ....................... 62 2.6 Proof of the Tensorial Property of the Components of Strain ............... 64 2.7 Traction and Components of Stress Acting on a Plane of a Particle of a Body .................................................................... 66 2.8 Proof of the Tensorial Property of the Components of Stress ............... 68 2.9 Properties of the Strain and Stress Tensors ............................... 71 2.10 Components of Displacement for a General Rigid Body Motion of a Particle ................................................................ 80 2.11 The Compatibility Equations ............................................... 82 2.12 Measurement of Strain ..................................................... 84 2.13 The Requirements for Equilibrium of the Particles of a Body ............... 88 2.14 Cylindrical Coordinates ................................................... 91 2.15 Strain-Displacement Relations in Cylindrical Coordinates .................. 93 2.16 The Equations of Compatibility in Cylindrical Coordinates ................. 94 2.17 The Equations of Equilibrium in Cylindrical Coordinates ................... 95 2.18 Problems .................................................................. 96 Chapter 3 Stress-Strain Relations ............................................. 107 3.1 Introduction ............................................................. 107 3.2 The Uniaxial Tension or Compression Test Performed in an Environment of Constant Temperature .................................................. 108 3.3 Strain Energy Density and Complementary Energy Density for Elastic Materials Subjected to Uniaxial Tension or Compression in an Environment of Constant Temperature ...................................... 115 3.4 The Torsion Test ......................................................... 119 3.5 Effect of Pressure, Rate of Loading and Temperature on the Response of Materials Subjected to Uniaxial States of Stress ......................... 121 3.6 Models of Idealized Time-Independent Stress-Strain Relations for Uniaxial States of Stress ................................................ 124 3.7 Stress-Strain Relations for Elastic Materials Subjected to Three-Dimensional States of Stress ....................................... 126 3.8 Stress-Strain Relations of Linearly Elastic Materials Subjected to Three-Dimensional States of Stress ....................................... 128 3.9 Stress-Strain Relations for Orthotropic, Linearly Elastic Materials ...... 130 3.10 Stress-Strain Relations for Isotropic, Linearly Elastic Materials Subjected to Three-Dimensional States of Stress .......................... 133 3.11 Strain Energy Density and Complementary Energy Density of a Particle of a Body Subjected to External Forces in an Environment of Constant Temperature .............................................................. 135 3.12 Thermodynamic Considerations of Deformation Processes Involving Bodies Made from Elastic Materials .............................................. 142 3.13 Linear Response of Bodies Made from Linearly Elastic Materials ........... 146 3.14 Time-Dependent Stress-Strain Relations .................................. 147 3.15 The Creep and the Relaxation Tests ....................................... 148 3.16 Problems ................................................................. 150 Chapter 4 Yield and Failure Criteria .......................................... 155 4.1 Yield Criteria for Materials Subjected to Triaxial States of Stress in an Environment of Constant Temperature ................................... 155 4.2 The Von Mises Yield Criterion ............................................ 159 4.3 The Tresca Yield Criterion ............................................... 162 4.4 Comparison of the Von Mises and the Tresca Yield Criteria ................ 162 4.5 Failure of Structures — Factor of Safety for Design ...................... 167 4.6 The Maximum Normal Component of Stress Criterion for Fracture of Bodies Made from a Brittle, Isotropic, Linearly Elastic Material ................ 173 4.7 The Mohr's Fracture Criterion for Brittle Materials Subjected to States of Plane Stress .......................................................... 175 4.8 Problems ................................................................. 179 Chapter 5 Formulation and Solution of Boundary Value Problems Using the Linear Theory of Elasticity ................................ 185 5.1 Introduction ............................................................. 185 5.2 Boundary Value Problems for Computing the Displacement and Stress Fields of Solid Bodies on the Basis of the Assumption of Small Defor ............ 186 5.3 The Principle of Saint Venant ............................................ 193 5.4 Methods for Finding Exact Solutions for Boundary Value Problems in the Linear Theory of Elasticity .............................................. 196 5.5 Solution of Boundary Value Problems for Computing the Displacement and Stress Fields of Prismatic Bodies Made from Homogeneous, Isotropic, Linearly Elastic Materials ............................................... 196 5.6 Problems ................................................................. 216 Chapter 6 Prismatic Bodies Subjected to Torsional Moments at Their Ends ................................................................. 221 6.1 Description of the Boundary Value Problem for Computing the Displacement and Stress Fields in Prismatic Bodies Subjected to Torsional Moments at Their Ends ............................................................... 221 6.2 Relations among the Coordinates of a Point Located on a Curved Boundary of a Plane Surface ....................................................... 223 6.3 Formulation of the Torsion Problem for Prismatic of Arbitary Cross Section on the Basis of the Linear Theory of Elasticity .................. 224 6.4 Interpretation of the Results of the Torsion Problem ..................... 233 6.5 Computation of the Stress and Displacement Fields of Bodies of Solid Elliptical and Circular Cross Section Subjected to Equal and Opposite Torsional Moments at Their Ends .......................................... 236 6.6 Multiply Connected Prismatic Bodies Subjected to Equal and Opposite Torsional Moments at Their Ends .......................................... 241 6.7 Available Results ........................................................ 249 6.8 Direction and Magnitude of the Shearing Stress Acting on the Cross Sections of a Prismatic Body of Arbitrary Cross Section Subjected to Torsional Moments at Its Ends ............................................ 249 6.9 The Membrane Analogy to the Torsion Problem .............................. 251 6.10 Stress Distribution in Prismatic Bodies of Thin Rectangular Cross Section Subjected to Equal and Opposite Torsional Moments at Their Ends ..................................................................... 258 6.11 Torsion of Prismatic Bodies of Composite Simply Connected Cross Sections ................................................................. 261 6.12 Numerical Solutions of Torsion Problems Using Finite Differences ......... 263 6.13 Problems ................................................................. 268 Chapter 7 Plane Strain and Plane Stress Problems in Elasticity ................ 271 7.1 Plane Strain ............................................................. 271 7.2 Formulation of the Boundary Value Problem for Computing the Stress and the Displacement Fields in a Prismatic Body in a State of Plane Strain Using the Airy Stress Function .................................... 273 7.3 Prismatic Bodies of Multiply Connected Cross Sections in a State of Plane Strain ............................................................. 278 7.4 The Plane Strain Equations in Cylindrical Coordinates .................... 280 7.5 Plane Stress ............................................................. 287 7.6 Simply Connected Thin Prismatic Bodies (Plates) in a State of Plane Stress Subjected on Their Lateral Surface to Symmetric in x1 Components of Traction Tn2 and Tn3 ........................................ 290 7.7 Two-Dimensional or Generalized Plane Stress .............................. 295 7.8 Prismatic Members in a State of Axisymmetric Plane Strain or Plane Stress ................................................................... 306 7.9 Problems ................................................................. 322 Chapter 8 Theories of Mechanics of Materials .................................. 327 8.1 Introduction ............................................................. 327 8.2 Fundamental Assumptions of the Theories of Mechanics of Materials for Line Members ............................................................. 329 8.3 Internal Actions Acting on a Cross Section of Line Members ............... 337 8.4 Framed Structures ........................................................ 338 8.5 Types of Framed Structures ............................................... 340 8.6 Internal Action Release Mechanisms ....................................... 342 8.7 Statically Determinate and Indeterminate Framed Structures ............... 343 8.8 Computation of the Internal Actions of the Members of Statically Determinate Framed Structures ............................................ 346 8.9 Action Equations of Equilibrium for Line Members ......................... 355 8.10 Shear and Moment Diagrams for Beams by the Summation Method .............. 358 8.11 Stress-Strain Relations for a Particle of a Line Member Made from an Isotropic Linearly Elastic Material ...................................... 362 8.12 The Boundary Value Problems in the Theories of Mechanics of Materials for Line Members ......................................................... 365 8.13 The Boundary Value Problem for Computing the Axial Component of Translation and the Internal Force in a Member Made from an Isotropic, Linearly Elastic Material Subjected to Axial Centroidal Forces and to a Uniform Change in Temperature .......................................... 368 8.14 The Boundary Value Problem for Computing the Angle of Twist and the Internal Torsional Moment in Members of Circular Cross Section Made from an Isotropic, Linearly Elastic Material Subjected to Torsional Moments .................................................................. 378 8.15 Problems ................................................................. 384 Chapter 9 Theories of Mechanics of Materials for Straight Beams Made from Isotropic, Linearly Elastic Materials ......................................... 391 9.1 Formulation of the Boundary Value Problems for Computing the Components of Displacement and the Internal Actions in Prismatic Straight Beams Made from Isotropic, Linearly Elastic Materials .......................... 391 9.2 The Classical Theory of Beams ............................................ 405 9.3 Solution of the Boundary Value Problem for Computing the Transverse Components of Translation and the Internal Actions in Prismatic Beams Made from Isotropic, Linearly Elastic Materials Using Functions of Discontinuity ............................................................ 414 9.4 The Timoshenko Theory of Beams ........................................... 421 9.5 Computation of the Shearing Components of Stress in Prismatic Beams Subjected to Bending without Twisting .................................... 430 9.6 Build-Up Beams ........................................................... 444 9.7 Location of the Shear Center of Thin-Walled Open Sections ................ 448 9.8 Members Whose Cross Sections Are Subjected to a Combination of Internal Actions .................................................................. 454 9.9 Composite Beams .......................................................... 460 9.10 Prismatic Beams on Elastic Foundation .................................... 473 9.11 Effect of Restraining the Warping of One Cross Section of a Prismatic Member Subjected to Torsional Moments at Its Ends ........................ 477 9.12 Problems ................................................................. 486 Chapter 10 Non-Prismatic Members — Stress Concentrations ...................... 499 10.1 Computation of the Components of Displacement and Stress of Non-Prismatic Members ................................................... 499 10.2 Stresses in Symmetrically Tapered Beams ................................. 500 10.3 Stress Concentrations ................................................... 505 10.4 Problems ................................................................ 509 Chapter 11 Planar Curved Beams ................................................ 511 11.1 Introduction ............................................................ 511 11.2 Derivation of the Equations of Equilibrium for a Segment of Infinitesimal Length of a Planar Curved Beam ............................ 511 11.3 Computation of the Circumferential Component of Stress Acting on the Cross Sections of Planar Curved Beams Subjected to Bending without Twisting ................................................................ 514 11.4 Computation of the Radial and Shearing Components of Stress in Curved Beams ................................................................... 528 11.5 Problems ................................................................ 534 Chapter 12 Thin-Walled, Tubular Members ....................................... 537 12.1 Introduction ............................................................ 537 12.2 Computation of the Shearing Stress Acting on the Cross Sections of Thin-Walled, Single-Cell, Tubular Members Subjected to Equal and Opposite Torsional Moments at Their Ends ................................ 538 12.3 Computation of the Angle of Twist per Unit Length of Thin-Walled, Single-Cell, Tubular Members Subjected to Equal and Opposite Torsional Moment at Their Ends .................................................... 540 12.4 Prismatic Thin-Walled, Single-Cell, Tubular Members with Thin Fins Subjected to Torsional Moments .......................................... 546 12.5 Thin-Walled, Multi-Cell, Tubular Members Subjected to Torsional Moments ................................................................. 551 12.6 Thin-Walled, Single-Cell, Tubular Beams Subjected to Bending without Twisting ................................................................ 555 12.7 Thin-Walled, Multi-Cell, Tubular Beams Subjected to Bending without Twisting ................................................................ 565 12.8 Single-Cell, Tubular Beams with Longitudinal Stringers subjected to Bending without Twisting ................................................ 572 12.9 Problems ................................................................ 576 Chapter 13 Integral Theorems of Structural Mechanics .......................... 581 13.1 A Statically Admissible Stress Field and an Admissible Displacement Field of a Body ......................................................... 581 13.2 Derivation of the Principle of Virtual Work for Deformable Bodies ....... 582 13.3 Statically Admissible Reactions and Internal Actions of Framed Structures .............................................................. 587 13.4 The Principle of Virtual Work for Framed Structures ..................... 588 13.5 The Unit Load Method .................................................... 597 13.6 The Principle of Virtual Work for Framed Structures, Including the Effect of Shear Deformation ............................................. 606 13.7 The Strong Form of One-Dimensional, Linear Boundary Value Problems ...... 610 13.8 Approximation of the Solution of One-Dimensional, Linear Boundary Value Problems Using Trial Functions .................................... 613 13.9 The Classical Weighted Residual Form for Second Order, One-Dimensional, Linear Boundary Value Problems ......................... 615 13.10 The Classical Weighted Residual Form for Fourth Order, One-Dimensional, Linear Boundary Value Problems ......................... 617 13.11 Discretization of Boundary Value Problems Using the Classical Weighted Residual Methods ........................................................ 619 13.12 The Modified Weighted Residual (Weak) Form of One-Dimensional, Linear Boundary Value Problems ................................................. 620 13.13 Total Strain Energy of Framed Structures ................................ 629 13.14 Castigliano's Second Theorem ............................................ 630 13.15 Betti-Maxwell Reciprocal Theorem ........................................ 637 13.16 Proof That the Center of Twist of a Cross Section Coincides with Its Shear Center ............................................................ 639 13.17 The Variational Form of the Boundary Value Problem for Computing the Components of Displacement of a Deformable Body — Theorem of Stationary Total Potential Energy ....................................... 640 13.18 Comments on the Modified Gallerkin Form and the Theorem of Stationary Total Potential Energy .................................................. 651 13.19 Problems ................................................................ 651 Chapter 14 Analysis of Statically Indeterminate Framed Structures ............. 657 14.1 The Basic Force or Flexibility Method ................................... 657 14.2 Computation of Components of Displacement of Points of Statically Indeterminate Structures ................................................ 664 14.3 Problems ................................................................ 666 Chapter 15 The Finite Element Method .......................................... 671 15.1 Introduction ............................................................ 671 15.2 The Finite Element Method for One-Dimensional, Second Order, Linear Boundary Value Problems as a Modified Galerkin Method ................... 671 15.3 Element Shape Functions ................................................. 677 15.4 Assembly of the Stiffness Matrix for the Domain of One-Dimensional, Second Order, Linear Boundary Value Problems from the Stiffness Matrices of Their Elements .............................................. 680 15.5 Construction of the Force Vector for the Domain of One-Dimensional, Second Order, Linear Boundary Value Problems ............................ 683 15.6 Direct Computation of the Contribution of an Element to the Stiffness Matrix and the Load Vector of the Domain of One-Dimensional, Second Order, Linear Boundary Value Problems ................................... 685 15.7 Approximate Solution of Linear Boundary Value Problems Using the Finite Element Method ................................................... 689 15.8 Application of the Finite Element Method to the Analysis of Framed Structures .............................................................. 699 15.9 Approximate Solution of Scalar Two-Dimensional, Second Order, Linear Boundary Value Problems Using the Finite Element Method ................. 736 15.10 Problems ................................................................ 758 Chapter 16 Plastic Analysis and Design of Structures .......................... 763 16.1 Strain-Curvature Relation of Prismatic Beams Subjected to Bending without Twisting ........................................................ 763 16.2 Initiation of Yielding Moment and Fully Plastic Moment of Beams Made from Isotropic, Linearly Elastic, Ideally Plastic Materials ............. 765 16.3 Distribution of the Shearing Component of Stress Acting on the Cross Sections of Beams Where M2Y<M2<M2P ........................................ 769 16.4 Location of the Elastoplastic Boundaries — Moment-Curvature Relation .... 772 16.5 Computation of the Deflection of Beams Made from Isotropic, Linearly Elastic, Ideally Plastic Materials ...................................... 778 16.6 Effect of Stress Concentrations on the Design of Line Members ........... 780 16.7 Elastic and Plastic Design for Strength of Statically Determinate Structures .............................................................. 782 16.8 Plastic Analysis and Design of Planar Statically, Indeterminate Beams and Frames .............................................................. 785 16.9 Direct Computations of the Collapse Load of Beams and Frames ............ 790 16.10 Derivation of the Equations of Equilibrium for a Structure Using the Principle of Virtual Work ............................................... 793 16.11 Theorems for Limit Analysis ............................................. 795 16.12 Systematic Procedure for Plastic Analysis of Structures ................. 797 16.13 Problems ................................................................ 802 Chapter 17 Mechanics of Materials Theory for Thin Plates ...................... 807 17.1 Introduction ............................................................ 807 17.2 Fundamental Assumptions of the Theories of Mechanics of Materials for Thin Plates ............................................................. 809 17.3 Internal Action Intensities Acting on an Element of a Plate ............. 812 17.4 Internal Action Intensities Acting on Planes Which Are Inclined to the x1 and x2 Axes ....................................................... 815 17.5 Equations of Equilibrium for a Plate .................................... 816 17.6 Boundary Conditions for Plates .......................................... 819 17.7 Analysis of Simply Supported Rectangular Plates Subjected to a General Distribution of Transverse Forces ....................................... 825 17.8 The Method of Levy for Computing the Deflection of Rectangular Plates Having a Simply Supported Pair of Parallel Edges ........................ 832 17.9 Bending of Circular Plates .............................................. 839 17.10 Use of the Weighted Residual Methods to Construct Approximate Expressions for the Deflection of Plates ................................ 845 17.11 The Theorem of Total Stationary Potential Energy for Plates ............. 856 17.12 Problems ................................................................ 858 Chapter 18 Instability of Elastic Structures .................................. 861 18.1 States of Unstable Equilibrium of Structures ............................ 861 18.2 The Non-Linear Theory of Elasticity and the Theory of Moderate Rotations ............................................................... 872 18.3 Criterion for the Stability or Instability of an Equilibrium Configuration of Structures ............................................. 875 18.4 Investigation of the Beginning of Buckling .............................. 875 18.5 Buckling of Structures Having One Degree of Freedom ..................... 876 18.6 Buckling of Structures Having Infinite Degrees of Freedom — The Direct Equilibrium Approach .................................................... 888 18.7 Buckling of Structures Having Infinite Degrees of Freedom — The Stationary Total Potential Energy Approach .............................. 895 18.8 Determination of the Critical Load at Buckling of Infinite Degree of Freedom Structures by Investigating the Beginning of Buckling ........... 897 18.9 Columns Subjected to Eccentric Axial Compressive Forces at Their Ends ... 900 18.10 Local Buckling of Columns ............................................... 904 18.11 Problems ................................................................ 904 Appendices .................................................................... 905 A Mechanical Properties of Materials .......................................... 907 B Stress-Strain Relations for Orthotropic and Isotropic Materials ............. 909 C Centroid, Moments and Products of Inertia of Plane Surfaces ................. 919 D Method of Finite Differences ................................................ 929 E Elements of Calculus of Variations .......................................... 943 F Derivation of the Expression for the Plane Stress Functions X(x1, x2, x3) .... 951 G Functions of Discontinuity .................................................. 957 H Properties of Rolled Shapes ................................................. 961 Index ......................................................................... 967