Structural Analysis

Part I Basic tools and concepts; 1 Basic Equations of Linear Elasticity .1.1 The concept of stress; 1.1.1 The state of stress at a point; 1.1.2 Volume equilibrium equations; 1.1.3 Surface equilibrium equations; 1.2 Analysis of the state of stress at a point; 1.2.1 Stress components acting on an arbitrary face; 1.2.2 Principal stresses; 1.2.3 Rotation of stresses; 1.2.4 Problems; 1.3 The state of plane stress; 1.3.1 Equilibrium equations; 1.3.2 Stresses acting on an arbitrary face within the sheet; 1.3.3 Principal stresses;1.3.4 Rotation of stresses; 1.3.5 Special states of stress; 1.3.6 Mohr's circle for plane stress; 1.3.7 Lam??'s ellipse; 1.3.8 Problems; 1.4 The concept of strain; 1.4.1 The state of strain at a point; 1.4.2 The volumetric strain; 1.5 Analysis of the state of strain at a point; 1.5.1 Rotation of strains 1.5.2 Principal strains; 1.6 The state of plane strain; 1.6.1 Strain-displacement relations for plane strain; 1.6.2 Rotation of strains; 1.6.3 Principal strains; 1.6.4 Mohr's circle for plane strain; 1.7 Measurement of strains; 1.7.1 Problems; 1.8 Strain compatibility equations; 2 Constitutive Behavior of Materials; 2.1 Constitutive laws for isotropic materials; 2.1.1 Homogeneous, isotropic, linear elastic materials; 2.1.2 Thermal effects; 2.1.3 Problems; 2.1.4 Ductile materials; 2.1.5 Brittle materials; 2.2 Allowable stress; 2.3 Yielding under combined loading; 2.3.1 Tresca's criterion; 2.3.2 Von Mises' criterion; 2.3.3 Comparing Tresca's and von Mises' criteria;2.3.4 Problems; 2.4 Material selection for structural performance; 2.4.1 Strength design; 2.4.2 Stiffness design 2.4.3 Buckling design; 2.5 Composite materials; 2.5.1 Basic characteristics;2.5.2 Stress diffusion in a composite; 2.6 Constitutive laws for anisotropic materials; 2.6.1 Constitutive laws for a lamina in the fiber aligned triad; 2.6.2 Constitutive laws for a lamina in an arbitrary triad; 2.7 Strength of a transversely isotropic lamina; 2.7.1 Strength of a lamina under simple loading conditions; 2.7.2 The Tsai-Wu failure criterion; 2.7.3 The reserve factor; 3 Linear Elasticity Solutions; 3.1 Solution procedures; 3.1.1 Displacement formulation; 3.1.2 Stress formulation; 3.1.3 Solutions to elasticity problems; 3.2 Plane strain problems; 3.3 Plane stress problems; 3.4 Plane strain and plane stress in polar coordinates; 3.5 Problem featuring cylindrical symmetry; 3.5.1 Problems; 4 Engineering Structural Analysis; 4.1 Solution approaches; 4.2 Bar under constant axial force; 4.3 Hyperstatic systems; 4.3.1 Solution procedures; 4.3.2 The displacement or stiffness method; 4.3.3 The force or flexibility method; 4.3.4 Problems; 4.3.5 Thermal effects in hyperstatic system; 4.3.6 Manufacturing imperfection effects in hyperstatic system; 4.3.7 Problems; 4.4 Pressure vessels; 4.4.1 Rings under internal pressure; 4.4.2 Cylindrical pressure vessels; 4.4.3 Spherical pressure vessels; 4.4.4 Problems; 4.5 Saint-Venant's principle; Part II Beams and thin-wall structures5 Euler-Bernoulli Beam Theory; 5.1 The Euler-Bernoulli Assumptions; 5.2 Implications of the Euler-Bernoulli assumptions; 5.3 Stress resultants; 5.4 Beams subjected to axial loads; 5.4.1 Kinematic description; 5.4.2 Sectional constitutive law; 5.4.3 Equilibrium equations; 5.4.4 Governing equations; 5.4.5 The sectional axial stiffness; 5.4.6 The axial stress distribution; 5.4.7 Problems; 5.5 Beams subjected to transverse loads; 5.5.1 Kinematic description; 5.5.2 Sectional constitutive law; 5.5.3 Equilibrium equations; 5.5.4 Governing equations; 5.5.5 The sectional bending stiffness; 5.5.6 The axial stress distribution; 5.5.7 Rational design of beams under bending; 5.5.8 Problems; 5.6 Beams subjected to axial and transverse loads; 5.6.1 Kinematic description; 5.6.2 Sectional constitutive law; 5.6.3 Equilibrium equations; 5.6.4 Governing equations; 6 Three-Dimensional Beam Theory; 6.1 Kinematic description; 6.2 Sectional constitutive law; 6.3 Sectional equilibrium equations; 6.4 Governing equations; 6.5 Decoupling the three-dimensional problem; 6.5.1 Definition of the principal axes of bending; 6.5.2 Decoupled governing equations; 6.6 The principal centroidal axes of bending; 6.6.1 The bending stiffness ellipse; 6.7 Definition of the neutral axis; 6.8 Evaluation of sectional stiffnesses; 6.8.1 The parallel axis theorem; 6.8.2 Thin-walled sections; 6.8.3 Triangular area equivalence method; 6.8.4 Useful results; 6.8.5 Problems; 6.9 Summary of three-dimensional beam theory; 6.9.1 Examples; 6.9.2 Discussion of the results; 6.10 Problems; 7 Torsion; 7.1 Torsion of circular cylinders; 7.1.1 Kinematic description; 7.1.2 The stress field;7.1.3 Sectional constitutive law; 7.1.4 Equilibrium equations; 7.1.5 Governing equations; 7.1.6 The torsional stiffness; 7.1.7 Measuring the torsional stiffness; 7.1.8 The shear stress distribution; 7.1.9 Rational design of cylinders under torsion; 7.1.10 Problems; 7.2 Torsion combined with axial force or bending; 7.2.1 Problems; 7.3 Torsion of bars with arbitrary cross-sections; 7.3.1 Introduction; 7.3.2 Saint-Venant's solution; 7.3.3 Saint-Venant's solution for a rectangular cross-section; 7.3.4 Problems; 7.4 Torsion of a thin rectangular cross-section; 7.5 Torsion of thin-walled open sections; 7.5.1 Problems; 8 Thin-Walled Beams; 8.1 Basic equations for thin-walled beams; 8.1.1 The thin wall assumption; 8.1.2 Stress flows; 8.1.3 Stress resultants; 8.1.4 Local equilibrium equation; 8.2 Bending of thin-walled beams; 8.2.1 Problems; 8.3 Shearing of thin-walled beams; 8.3.1 Shearing of open sections; 8.3.2 Evaluation of stiffness static moments; 8.3.3 Shear flow distributions in open sections; 8.3.4 Problems; 8.3.5 Shear center for open sections; 8.3.6 Problems; 8.3.7 Shearing of closed sections; 8.3.8 Shearing of multi-cellular sections; 8.3.9 Problems; 8.4 The shear center; 8.4.1 Calculation of the shear center location; 8.4.2 Problems; 8.5 Torsion of thin-walled beams; 8.5.1 Torsion of open sections; 8.5.2 Torsion of closed section; 8.5.3 Comparison of open and closed sections; 8.5.4 Torsion of combined open and closed sections; 8.5.5 Torsion of multi-cellular sections; 8.5.6 Problems; 8.6 Coupled bending-torsion problems; 8.6.1 Problems; 8.7 Warping of thin-walled beams under torsion;8.7.1 Kinematic description; 8.7.2 Stress-strain relations; 8.7.3 Warping of open sections; 8.7.4 Problems; 8.7.5 Warping of closed sections; 8.7.6 Warping of multi-cellular sections; 8.8 Equivalence of the shear and twist centers; 8.9 Non-uniform torsion; 8.9.1 Non-uniform torsion: a classical approach; 8.9.2 Problems; 8.10 Structural idealization; 8.10.1 Lumping the thin-walled section into sheet and stringer components 8.10.2 Axial stress in the stringers; 8.10.3 Shear flow in the sheet components; 8.10.4 Torsion of sheet-stringer sections; 8.10.5 Problems; Part III Energy and variational methods9 Virtual Work Principles; 9.1 Introduction; 9.2 Equilibrium and work fundamentals; 9.2.1 Static equilibrium conditions; 9.2.2 Concept of mechanical work; 9.3 Principle of virtual work; 9.3.1 Principle of virtual work for a single particle; 9.3.2 Kinematically admissible virtual displacements; 9.3.3 Use of infinitesimals as virtual displacements; 9.3.4 Principle of virtual work for a system of particles; 9.3.5 Application of the principle of virtual work to mechanical systems; 9.3.6 Application of the principle of virtual work to trusses; 9.3.7 Generalized coordinates and forces; 9.3.8 Problems; 9.4 Principle of complementary virtual work; 9.4.1 Compatibility equations for a planar truss; 9.4.2 Principle of complementary virtual work for trusses; 9.4.3 Complementary virtual work; 9.4.4 Problems; 9.5 Unit load method for trusses; 9.5.1 Statement of the unit load method for trusses; 9.5.2 Application to trusses; 9.5.3 Problems .; 9.6 Unit load method for beams; 9.6.1 Beam deflection due to bending; 9.6.2 Beam deflection due to torsion; 9.6.3 Application to beam problems; 9.6.4 Deflections of beams with unsymmetric cross sections; 9.6.5 Problems; 9.7 Application of the unit load method to hyperstatic problems; 9.7.1 Force method for trusses; 9.7.2 Force method for beams; 9.7.3 Combined truss and beam problems; 9.7.4 Multiple redundancies; 9.7.5 Problems; 10 Energy Methods; 10.1 Conservative forces;10.1.1 Potential for internal and external forces; 10.1.2 Calculation of the potential functions; 10.2 Principle of minimum total potential energy;10.2.1 Nonconservative external forces; 10.3 Strain energy in springs; 10.3.1 Rectilinear springs; 10.3.2 Torsional springs; 10.4 Problems; 10.5 Strain energy in beams; 10.5.1 Beam under axial loads; 10.5.2 Beam under transverse loads;10.5.3 Beam under torsional loads; 10.6 Strain energy in solids; 10.6.1 General three-dimensional stress state; 10.6.2 Beams under multi-axis bending and axial load; 10.7 Applications to trusses; 10.7.1 Problems;10.7.2 Development of a finite element formulation for trusses;10.7.3 Problems;10.7.4 Applications to beams;10.8 Principle of minimum complementary energy;10.8.1 The potential of the prescribed displacements;10.8.2 Constitutive laws for elastic materials; 10.8.3 The principle of minimum complementary energy; 10.8.4 The principle of least work; 10.8.5 Examples using the PMCP/LWP; 10.8.6 Problems; 10.9 Energy theorems;10.9.1 Clapeyron's theorem; 10.9.2 Castigliano's first theorem; 10.9.3 Crotti-Engesser theorem; 10.9.4 Castigliano's second theorem; 10.9.5 Applications of energy theorems; 10.9.6 The dummy load method; 10.9.7 Unit load method revisited; 10.9.8 Conclusions; 10.9.9 Problems; 10.10Reciprocity theorems; 10.10.1Betti's theorem; 10.10.2Maxwell's theorem; 10.10.3Problems; 11 Variational and Approximate Solutions; 11.1 Approach; 11.2 Approximations based on the principle of minimum total potential energy; 11.2.1 Application to a bending of a beam; 11.2.2 Examples; 11.2.3 Problems; 11.3 The strong and weak statements of equilibrium; 11.3.1 The weak form for beams under axial loads; 11.3.2 Approximate solutions for beams under axial loads; 11.3.3 Problems; 11.3.4 The weak form for beams under transverse loads; 11.3.5 Approximate solutions for beams under transverse loads; 11.3.6 Problems 11.3.7 Equivalence with energy principles;11.3.8 The principle of minimum total potential energy; 11.3.9 Treatment of the boundary conditions; 11.4 Formal procedures for the derivation of approximate solutions; 11.4.1 Basic approximations; 11.4.2 Principle of virtual work; 11.4.3 The principle of minimum total potential energy; 11.4.4 Examples; 11.4.5 Problems; 11.5 A Finite Element formulation for beams; 11.5.1 Formulation of an Euler-Bernoulli beam element; 11.5.2 Examples; 11.5.3 Summary; 11.5.4 Problems; 12 Variational and Energy Principles; 12.1 Variational formulation of beam problems; 12.2 Mathematical Preliminaries; 12.2.1 Stationary point of a function; 12.2.2 Lagrange multiplier method; 12.2.3 Stationary point of a definite integral; 12.3 Variational and Energy Principles; 12.3.1 Review of the equations of linear elasticity; 12.3.2 The principle of virtual work;12.3.3 The principle of complementary virtual work; 12.3.4 The strain energy density function; 12.3.5 The stress energy density function; 12.3.6 The principle of minimum total potential energy; 12.3.7 The principle of minimum complementary energy; 12.3.8 Examples; 12.3.9 The principle of least work;12.3.10Hu-Washizu's principle; 12.3.11Hellinger-Reissner's principle; 12.3.12Problems; 12.4 Applications of Variational and Energy Principles; 12.4.1 The shear lag problem; 12.4.2 The non-uniform torsion problem; 12.4.3 The non-uniform torsion problem (closed sections); 12.4.4 The non-uniform torsion problem (open sections); 12.4.5 Problems; Part IV Advanced topics; 13 Introduction to Plasticity and Thermal Stresses; 13.1 Yielding under combined loading; 13.1.1 Introduction to yield criteria; 13.1.2 Tresca's criterion; 13.1.3 Von Mises' criterion; 13.1.4 Problems; 13.2 Applications of yield criteria to structural problems; 13.2.1 Problems; 13.2.2 Plastic bending; 13.2.3 Problems; 13.2.4 Plastic torsion; 13.2.5 Examples; 13.3 Thermal stresses in structures; 13.3.1 The direct method; 13.3.2 Examples; 13.3.3 Problems; 13.3.4 The constraint method; 13.4 Application to bars and trusses; 13.4.1 Examples; 13.4.2 Problems; 13.4.3 Application to beams; 13.4.4 Examples; 13.4.5 Problems; 14 Buckling of Beams; 14.1 Rigid bar with root torsional spring; 14.1.1 Analysis of a perfect system; 14.1.2 Analysis of an imperfect system; 14.2 Buckling of beams; 14.2.1 Equilibrium equations; 14.2.2 Buckling of a pinned-pinned beam (Equilibrium approach); 14.2.3 Buckling of a pinned-pinned beam (Imperfection approach); 14.2.4 Work done by the axial force; 14.2.5 Buckling of a pinned-pinned beam (Energy approach); 14.2.6 Examples; 14.2.7 Buckling of beams with various end conditions; 14.2.8 Problems; 14.3 Buckling of sandwich beams; 15 Shearing Deformations in Beams; 15.1 Introduction; 15.1.1 A simplified approach; 15.1.2 An equilibrium approach; 15.1.3 Examples; 15.1.4 Problems; 15.2 Shear deformable beams: an energy approach; 15.2.1 Shearing effects on static deflections; 15.2.2 Examples; 15.2.3 Shearing effects on buckling; 15.2.4 Shearing and rotary inertia effects on vibration; 15.2.5 Problems; 16 Kirchhoff Plate Theory; 16.1 Governing equations of Kirchhoff plate theory; 16.1.1 Kirchhoff assumptions; 16.1.2 Stress resultant; 16.1.3 Equilibrium equations; 16.1.4 Constitutive laws; 16.1.5 Summary of Kirchhoff plate theory; 16.2 The bending problem; 16.2.1 Typical boundary conditions; 16.2.2 Simple plate bending solutions; 16.2.3 Problems; 16.3 Anisotropic plates; 16.3.1 Laminated composite plates; 16.3.2 Constitutive laws for laminated composite plates; 16.3.3 The in-plane stiffness matrix; 16.3.4 Problems; 16.3.5 The bending stiffness matrix; 16.3.6 The coupling stiffness matrix; 16.3.7 Problems; 16.3.8 Directionally stiffened plates; 16.3.9 Governing equations for anisotropic plates; 16.4 Solution techniques for rectangular plates; 16.4.1 Navier's solution for simply supported plates; 16.4.2 Examples; 16.4.3 Problems; 16.4.4 Levy's solution; 16.4.5 Problems; 16.5 Circular plates; 16.5.1 Governing equations for the bending of circular plates; 16.5.2 Circular plates subjected to loading presenting circular symmetry; 16.5.3 Examples; 16.5.4 Problems; 16.5.5 Circular plates subjected to arbitrary loading; 16.5.6 Examples;;16.5.7 Problems 16.6 Energy formulation of Kirchhoff plate theory; 16.6.1 The virtual work done by internal forces and moments; 16.6.2 The virtual work done by the applied loads; 16.6.3 The principle of virtual work for Kirchhoff plates; 16.6.4 The principle of minimum total potential energy for Kirchhoff plates; 16.6.5 Energy based approximate solutions for Kirchhoff plates; 16.6.6 Examples;16.6.7 Problems; 16.7 Shear deformable plates; 16.7.1 Problems;16.8 Buckling of plates; 16.8.1 Problems; 17 Appendix: Mathematical Tools;17.1 Coordinate systems and transformations; 17.1.1 The rotation matrix; 17.1.2 Rotation of vector components; 17.1.3 The rotation matrix in two dimensions; 17.1.4 Rotation of vector components in two dimensions; 17.2 Least-square solution of linear systems with redundant equations; 17.3 Arrays and matrices; 17.3.1 Partial derivatives of a linear form; 17.3.2 Partial derivatives of a quadratic form; 17.4 Orthogonality properties of trigonometric functions; References; Index