Analysis and Design of Elastic Beams – l Methods

WALTER D. PILKEY is the Morse Professor of Engineering at the University of Virginia. He is the author or coauthor of numerous articles and books, including Formulas for Stress, Strain, and Structural Matrices and Peterson's Stress Concentration Factors, Second Edition, both published by Wiley.

PREFACE.1 BEAMS IN BENDING.1.1 Review of Linear Elasticity.1.1.1 Kinematical Strain-Displacement Equations.1.1.2 Material Law.1.1.3 Equations of Equilibrium.1.1.4 Surface Forces and Boundary Conditions.1.1.5 Other Forms of the Governing Differential Equations.1.2 Bending Stresses in a Beam in Pure Bending.1.3 Principal Bending Axes.1.4 Axial Loads.1.5 Elasticity Solution for Pure Bending.References.2 BEAMELEMENTS.2.1 Fundamental Engineering Theory Equations for a Straight Beam.2.1.1 Geometry of Deformation.2.1.2 Force-Deformation Relations.2.1.3 Equations of Equilibrium.2.1.4 Boundary Conditions.2.1.5 Displacement Form of the Governing Differential Equations.2.1.6 Mixed Form of the Governing Differential Equations.2.1.7 Principle of Virtual Work: Integral Form of the Governing Equations.2.2 Response of Beam Elements.2.2.1 First-Order Form of the Governing Equations.2.2.2 Sign Conventions for Beams.2.2.3 Definition of Stiffness Matrices.2.2.4 Determination of Stiffness Matrices.2.2.5 Development of an Element by Mapping from a Reference Element.2.3 Mass Matrices for Dynamic Problems.2.3.1 Consistent Mass Matrices.2.3.2 Lumped Mass Matrices.2.3.3 Exact Mass and Dynamic Stiffness Matrices.2.4 Geometric Stiffness Matrices for Beams with Axial Loading.2.5 Thermoelastic Analysis.References.3 BEAM SYSTEMS.3.1 Structural Systems.3.1.1 Coordinate System and Degrees of Freedom.3.1.2 Transformation of Forces and Displacements.3.2 Displacement Method of Analysis.3.2.1 Direct Stiffness Method.3.2.2 Characteristics of the Displacement Method.3.3 Transfer Matrix Method of Analysis.3.4 Dynamic Responses.3.4.1 Free Vibration Analysis.3.4.2 Forced Response.3.5 Stability Analysis.3.6 Analyses Using Exact Stiffness Matrices.References.4 FINITE ELEMENTS FOR CROSS-SECTIONAL ANALYSIS.4.1 Shape Functions.4.2 Transformation of Derivatives and Integrals.4.3 Integrals.4.4 Cross-Sectional Properties.4.5 Modulus-Weighted Properties.References.5 SAINT-VENANT TORSION.5.1 Fundamentals of Saint-Venant Torsion.5.1.1 Force Formulation.5.1.2 Membrane Analogy.5.2 Classical Formulas for Thin-Walled Cross Sections.5.2.1 Open Sections.5.2.2 Closed Sections, Hollow Shafts.5.3 Composite Cross Sections.5.4 Stiffness Matrices.5.4.1 Principle of Virtual Work.5.4.2 Weighted Residual Methods.5.4.3 Isoparametric Elements.5.5 Assembly of System Matrices.5.6 Calculation of the Torsional Constant and Stresses.5.7 Alternative Computational Methods.5.7.1 Boundary Integral Equations.5.7.2 Boundary Element Method.5.7.3 Direct Integration of the Integral Equations.References.6 BEAMS UNDER TRANSVERSE SHEAR LOADS.6.1 Transverse Shear Stresses in a Prismatic Beam.6.1.1 Approximate Shear Stress Formulas Based on Engineering Beam Theory.6.1.2 Theory of Elasticity Solution.6.1.3 Composite Cross Section.6.1.4 Finite Element Solution Formulation.6.2 Shear Center.6.2.1 y Coordinate of the Shear Center.6.2.2 Axis of Symmetry.6.2.3 Location of Shear Centers for Common Cross Sections.6.2.4 z Coordinate of the Shear Center.6.2.5 Finite Element Solution Formulation.6.2.6 Trefftz's Definition of the Shear Center.6.3 Shear Deformation Coefficients.6.3.1 Derivation.6.3.2 Principal Shear Axes.6.3.3 Finite Element Solution Formulation.6.3.4 Traditional Analytical Formulas.6.4 Deflection Response of Beams with Shear Deformation.6.4.1 Governing Equations.6.4.2 Transfer Matrix.6.4.3 Stiffness Matrix.6.4.4 Exact Geometric Stiffness Matrix for Beams with Axial Loading.6.4.5 Shape Function-Based Geometric Stiffness and Mass Matrices.6.4.6 Loading Vectors.6.4.7 Elasticity-Based Beam Theory.6.5 Curved Bars.References.7 RESTRAINED WARPING OF BEAMS.7.1 Restrained Warping.7.2 Thin-Walled Beams.7.2.1 Saint-Venant Torsion.7.2.2 Restrained Warping.7.3 Calculation of the Angle of Twist.7.3.1 Governing Equations.7.3.2 Boundary Conditions.7.3.3 Response Expressions.7.3.4 First-Order Governing Equations and General Solution.7.4 Warping Constant.7.5 Normal Stress due to Restrained Warping.7.6 Shear Stress in Open-Section Beams due to Restrained Warping.7.7 Beams Formed of Multiple Parallel Members Attached at the Boundaries.7.7.1 Calculation of Open-Section Properties.7.7.2 Warping and Torsional Constants of an Open Section.7.7.3 Calculation of the Effective Torsional Constant.7.8 More Precise Theories.References.8 ANALYSIS OF STRESS.8.1 Principal Stresses and Extreme Shear Stresses.8.1.1 State of Stress.8.1.2 Principal Stresses.8.1.3 Invariants of the Stress Matrix.8.1.4 Extreme Values of Shear Stress.8.1.5 Beam Stresses.8.2 Yielding and Failure Criteria.8.2.1 Maximum Stress Theory.8.2.2 Maximum Shear Theory.8.2.3 Von Mises Criterion.References.9 RATIONAL B-SPLINE CURVES.9.1 Concept of a NURBS Curve.9.2 Definition of B-Spline Basis Functions.9.3 B-Spline and Rational B-Spline Curves.9.4 Use of Rational B-Spline Curves in Thin-Walled Beam Analysis.References.10 SHAPE OPTIMIZATION OF THIN-WALLED SECTIONS.10.1 Design Velocity Field.10.2 Design Sensitivity Analysis.10.2.1 Derivatives of Geometric Quantities.10.2.2 Derivative of the Normal Stress.10.2.3 Derivatives of the Torsional Constant and the Shear Stresses.10.3 Design Sensitivity of the Shear Deformation Coefficients.10.4 Design Sensitivity Analysis for Warping Properties.10.5 Design Sensitivity Analysis for Effective Torsional Constant.10.6 Optimization.Reference.APPENDIX A: USING THE COMPUTER PROGRAMS.A.1 Overview of the Programs.A.2 Input Data File for Cross-Section Analysis.A.3 Output Files.APPENDIX B: NUMERICAL EXAMPLES.B.1 Closed Elliptical Tube.B.2 Symmetric Channel Section.B.3 L Section without Symmetry.B.4 Open Circular Cross Section.B.5 Welded Hat Section.B.6 Open Curved Section.B.7 Circular Arc.B.8 Composite Rectangular Strip.References.INDEX.