Examples in Structural Analysis

Description

This third edition of Examples in Structural Analysis uses a step-by-step approach and provides an extensive collection of fully worked and graded examples for a wide variety of structural analysis problems. It presents detailed information on the methods of solutions to problems and the results obtained. Also given within the text is a summary of each of the principal analysis techniques inherent in the design process and where appropriate, an explanation of the mathematical models used. The text emphasises that software should only be used if designers have appropriate knowledge and understanding of the mathematical assumptions, modelling and limitations inherent in the programs they use. It establishes the use of hand-methods for obtaining approximate solutions during preliminary design and an independent check on the answers obtained from computer analysis. What is New in the Third Edition: A new chapter covers the analysis and design of cables and arches subjected to concentrated loads and uniformly distributed loads. For cables without or with simply supported pinned trusses or steel girder beams through equally spaced hangers, tension forces, support reactions, sags and slopes in cables are determined. For two-pinned or three-pinned arches with parabolic, arched and semi-circular shapes, axial forces, radial shear forces and bending moments at various sections of arches are determined. An existing chapter has been expanded to the construction and use of influence lines for pin-pointed trusses and lattice girders. Also, the chapter Direct Stiffness Methods has been revisited and amended. Cover Half Title Title Page Copyright Page Contents Preface Acknowledgements About the Authors 1. Structural Analysis and Design 1.1. Introduction 1.2. Equilibrium 1.3. Mathematical Modelling 1.3.1. Line Diagrams 1.3.2. Load Path 1.3.3. Foundations 1.4. Structural Loading 1.5. Statical Indeterminacy 1.5.1. Indeterminacy of Two-Dimensional Pin-Jointed Frames 1.5.2. Indeterminacy of Two-Dimensional Rigid-Jointed Frames 1.6. Structural Degrees-of-Freedom 1.6.1. Problems: Indeterminacy and Degrees-of-Freedom 1.6.2. Solutions: Indeterminacy and Degrees-of-Freedom 2. Material and Section Properties 2.1. Introduction 2.1.1. Simple Stress and Strain 2.1.2. Young’s Modulus (Modulus of Elasticity) − E 2.1.3. Secant Modulus − Es 2.1.4. Tangent Modulus − Et 2.1.5. Shear Rigidity or Modulus (Modulus of Rigidity) − G 2.1.6. Yield Strength 2.1.7. Ultimate Tensile Strength 2.1.8. Modulus of Rupture in Bending 2.1.9. Modulus of Rupture in Torsion 2.1.10. Poisson’s Ratio − υ 2.1.11. Coefficient of Thermal Expansion − α 2.1.12. Elastic Assumptions 2.2. Elastic Cross-Section Properties 2.2.1. Cross-Sectional Area 2.2.2. Centre of Gravity and Centroid 2.2.3. Problems: Cross-Sectional Area and Position of Centroid 2.2.4. Solutions: Cross-Sectional Area and Position of Centroid 2.2.5. Elastic Neutral Axes 2.2.6. Second Moment of Area − I and Radius of Gyration − i 2.2.6.1. The Parallel Axis Theorem 2.2.7. Elastic Section Modulus − Wel 2.2.8. Problems: Second Moments of Area and Elastic Section Moduli 2.2.9. Solutions: Second Moments of Area and Elastic Section Moduli 2.3. Plastic Cross-Section Properties 2.3.1. Stress-Strain Relationships 2.3.2. Plastic Neutral Axis 2.3.3. Evaluation of Plastic Moment of Resistance and Plastic Section Modulus 2.3.4. Shape Factor 2.3.5. Section Classification 2.3.5.1. Aspect Ratio 2.3.5.2. Type of Section 2.4. Example 2.1: Plastic Cross-Section Properties − Section 1 2.5. Problems: Plastic Cross-Section Properties 2.6. Solutions: Plastic Cross-Section Properties 3. Pin-Jointed Frames 3.1. Introduction 3.2. Method of Sections 3.2.1. Example 3.1: Pin-Jointed Truss 3.3. Method of Joint Resolution 3.3.1. Problems: Method of Sections and Joint Resolution 3.3.2. Solutions: Method of Sections and Joint Resolution 3.4. Method of Tension Coefficients 3.4.1. Example 3.2: Two-Dimensional Plane Truss 3.4.2. Example 3.3: Three-Dimensional Space Truss 3.4.3. Problems: Method of Tension Coefficients 3.4.4. Solutions: Method of Tension Coefficients 3.5. Unit Load for Deflection 3.5.1. Strain Energy (Axial Load Effects) 3.5.2. Castigliano’s 2nd Theorem 3.5.3. Example 3.4: Deflection of a Pin-Jointed Truss 3.5.3.1. Fabrication Errors − Lack-of-Fit 3.5.3.2. Changes in Temperature 3.5.4. Example 3.5: Lack-of-Fit and Temperature Difference 3.5.5. Problems: Unit Load Method for Deflection of Pin-Jointed Frames 3.5.6. Solutions: Unit Load Method for Deflection of Pin-Jointed Frames 3.6. Unit Load Method for Singly Redundant Pin-Jointed Frames 3.6.1. Example 3.6: Singly Redundant Pin-Jointed Frame 1 3.6.2. Example 3.7: Singly Redundant Pin-Jointed Frame 2 3.6.3. Problems: Unit Load for Singly Redundant Pin-Jointed Frames 3.6.4. Solutions: Unit Load for Singly Redundant Pin-Jointed Frames 4. Beams 4.1. Statically Determinate Beams 4.1.1. Example 4.1: Beam with Point Loads 4.1.2. Shear Force Diagrams 4.1.3. Bending Moment Diagrams 4.1.4. Example 4.2: Beam with a Uniformly Distributed Load (UDL) 4.1.5. Example 4.3: Cantilever Beam 4.1.6. Problems: Statically Determinate Beams − Shear Force and Bending Moment 4.1.7. Solutions: Statically Determinate Beams − Shear Force and Bending Moment 4.2. McCaulay’s Method for the Deflection of Beams 4.2.1. Example 4.4: Beam with Point Loads 4.2.2. Example 4.5: Beam with Combined Point Loads and UDLs 4.3. Equivalent Uniformly Distributed Load Method for the Deflection of Beams 4.3.1. Problems: McCaulay’s and Equivalent UDL Methods for Deflection of Beams 4.3.2. Solutions: McCaulay’s and Equivalent UDL Methods for Deflection of Beams 4.4. The Principle of Superposition 4.4.1. Example 4.6: Superposition − Beam 1 4.4.2. Example 4.7: Superposition − Beam 2 4.4.3. Example 4.8: Superposition − Beam 3 4.4.4. Example 4.9: Superposition − Beam 4 4.4.5. Example 4.10: Superposition − Beam 5 4.5. Unit Load for Deflection of Beams 4.5.1. Strain Energy (Bending Load Effects) 4.5.2. Example 4.11: Deflection and Slope of a Uniform Cantilever 4.5.3. Example 4.12: Deflection and Slope of a Non-Uniform Cantilever 4.5.4. Example 4.13: Deflection and Slope of a Linearly Varying Cantilever 4.5.5. Example 4.14: Deflection of a Non-Uniform Simply-Supported Beam 4.5.6. Example 4.15: Deflection of a Frame and Beam Structure 4.5.7. Example 4.16: Deflection of a Uniform Cantilever Using Coefficients 4.5.8. Problems: Unit Load Method for Deflection of Beams and Frames 4.5.9. Solutions: Unit Load Method for Deflection of Beams and Frames 4.6. Statically Indeterminate Beams 4.6.1. Unit Load Method for Singly Redundant Beams 4.6.2. Example 4.17: Singly Redundant Beam 1 4.6.3. Example 4.18: Singly Redundant Beam 2 4.6.4. Problems: Unit Load Method for Singly Redundant Beams 4.6.5. Solutions: Unit Load Method for Singly Redundant Beams 4.7. Moment Distribution Method for Multi-Redundant Beams 4.7.1. Bending (Rotational) Stiffness 4.7.2. Carry-Over Moment 4.7.3. Pinned End 4.7.4. Free and Fixed Bending Moments 4.7.5. Example 4.19: Single-Span Encastré Beam 4.7.6. Propped Cantilevers 4.7.7. Example 4.20: Propped Cantilever 4.7.8. Distribution Factors 4.7.9. Application of the Method 4.7.10. Example 4.21: Three-Span Continuous Beam 4.7.11. Problems: Moment Distribution - Continuous Beams 4.7.12. Solutions: Moment Distribution - Continuous Beams 4.8. Redistribution of Moments 4.8.1. Example 4.22: Redistribution of Moments in a Two-Span Beam 4.9. Shear Force and Bending Moment Envelopes 5. Rigid-Jointed Frames 5.1. Rigid-Jointed Frames 5.1.1. Example 5.1: Statically Determinate Rigid-Jointed Frame 1 5.1.2. Example 5.2: Statically Determinate Rigid-Jointed Frame 2 5.1.3. Problems: Statically Determinate Rigid-Jointed Frames 5.1.4. Solutions: Statically Determinate Rigid-Jointed Frames 5.2. Unit Load Method for Singly Redundant Rigid-Jointed Frames 5.2.1. Example 5.3: Singly Redundant Rigid-Jointed Frame 5.2.2. Problems: Unit Load Method for Singly Redundant Rigid-Jointed Frames 5.2.3. Solutions: Unit Load Method for Singly Redundant Rigid-Jointed Frames 5.3. Moment Distribution for No-Sway Rigid-Jointed Frames 5.3.1. Example 5.4: No-Sway Rigid-Jointed Frame 1 5.3.2. Problems: Moment Distribution – No-Sway Rigid-Jointed Frames 5.3.3. Solutions: Moment Distribution – No-Sway Rigid-Jointed Frames 5.4. Moment Distribution for Rigid-Jointed Frames with Sway 5.4.1. Example 5.5: Rigid-Jointed Frame with Sway − Frame 1 5.4.2. Problems: Moment Distribution – Rigid-Jointed Frames with Sway 5.4.3. Solutions: Moment Distribution – Rigid-Jointed Frames with Sway 6. Buckling Instability 6.1. Introduction 6.1.1. Local Buckling 6.1.1.1. Class 1 Sections 6.1.1.2. Class 2 Sections 6.1.1.3. Class 3 Sections 6.1.1.4. Class 4 Sections 6.1.1.5. Section Classification 6.1.2. Flexural Buckling 6.1.2.1. Short Elements 6.1.2.2. Slender Elements 6.1.2.3. Intermediate Elements 6.2. Secondary Stresses 6.2.1. Effect on Short Elements 6.2.2. Effect on Slender Elements 6.2.3. Effect on Intermediate Elements 6.3. Critical Stress (σcr) 6.3.1. Critical Stress for Short Columns 6.3.2. Critical Stress for Slender Columns 6.3.3. Euler Equation 6.3.4. Effective Buckling Length (LE) 6.3.5. Critical Stress for Intermediate Columns 6.3.6. Tangent Modulus Theorem 6.4. Perry-Robertson Formula 6.5. European Column Curves 6.5.1. Non-dimensional Slenderness 6.6. Example 6.1: Slenderness 6.7. Example 6.2: Rolled Universal Column Section 6.8. Example 6.3: Compound Column Section 6.9. Built-Up Compression Members 6.9.1. Shear Stiffness for Laced Columns 6.10. Example 6.4: Laced Built-Up Column 6.11. Problems: Buckling Instability 6.12. Solutions: Buckling Instability 7. Direct Stiffness Method 7.1. Direct Stiffness Method of Analysis 7.2. Element Stiffness Matrix [k] 7.2.1. Beam Elements with Two Degrees-of-Freedom 7.2.2. Beam Elements with Four Degrees-of-Freedom 7.2.3. Local Co-Ordinate System 7.2.4. Beams Elements with Six Degrees-of-Freedom 7.3. Structural Stiffness Matrix [K] 7.4. Structural Load Vector [P] 7.5. Structural Displacement Vector [Δ] 7.6. Element Displacement Vector [δ] 7.7. Element Force Vector [F]Total 7.8. Example 7.1: Two-Span Beam 7.9. Example 7.2: Rigid-Jointed Frame 1 7.10. Transformation Matrices 7.11. Example 7.3: Rigid-Jointed Frame 2 7.12. Example 7.4: Pin-Jointed Frame 7.13. Problems: Direct Stiffness Method 7.14. Solutions: Direct Stiffness Method 8. Plastic Analysis 8.1. Introduction 8.1.1. Partial Collapse 8.1.2. Conditions for Full Collapse 8.2. Static Method for Continuous Beams 8.2.1. Example 8.1: Encastré Beam 8.2.2. Example 8.2: Propped Cantilever 1 8.2.3. Example 8.3: Propped Cantilever 2 8.3. Kinematic Method for Continuous Beams 8.3.1. Example 8.4: Continuous Beam 8.4. Problems: Plastic Analysis − Continuous Beams 8.5. Solutions: Plastic Analysis − Continuous Beams 8.6. Rigid-Jointed Frames 8.6.1. Example 8.5: Frame 1 8.7. Problems: Plastic Analysis − Rigid-Jointed Frames 1 8.8. Solutions: Plastic Analysis − Rigid-Jointed Frames 1 8.9. Example 8.6: Joint Mechanism 8.10. Problems: Plastic Analysis − Rigid-Jointed Frames 2 8.11. Solutions: Plastic Analysis − Rigid-Jointed Frames 2 8.12. Gable Mechanism 8.13. Instantaneous Centre of Rotation 8.14. Example 8.7: Pitched Roof Frame 8.15. Problems: Plastic Analysis − Rigid-Jointed Frames 3 8.16. Solutions: Plastic Analysis − Rigid-Jointed Frames 3 9. Influence Lines for Beams, Pin-Jointed Trusses and Lattice Girders 9.1. Introduction 9.2. Example 9.1: Influence Lines for a Simply Supported Beam 9.2.1. Influence Lines for the Support Reactions 9.2.2. Influence Line for the Shear Force at Point B 9.2.3. Influence Line for the Bending Moment at Point B 9.3. Müller-Breslau Principle for the Influence Lines for Beams 9.4. Example 9.2: Influence Lines for a Statically Determinate Beam 9.5. Example 9.3: Influence Line for a Statically Indeterminate Beam 9.6. The Use of Influence Lines 9.6.1. Concentrated Loads 9.6.2. Distributed Loads 9.6.3. Example 9.4: Evaluation of Functions for Statically Determinate Beam 1 9.6.4. Example 9.5: Evaluation of Functions for Statically Determinate Beam 2 9.7. Example 9.6: Evaluation of Functions for a Statically Indeterminate Beam 9.8. Train of Loads 9.8.1. Example 9.7: Evaluation of Functions for a Train of Loads 9.9. Influence Lines for Pin-Jointed Trusses and Lattice Girders 9.9.1. Example 9.8: Lattice Girder 9.10. Problems: Influence Lines for Beams, Pin-Jointed Trusses and Lattice Girders 9.11. Solutions: Influence Lines for Beams, Pin-Jointed Trusses and Lattice Girders 10. Approximate Methods of Analysis 10.1. Introduction 10.2. Example 10.1: Statically Indeterminate Pin-Jointed Plane Frame 1 10.3. Example 10.2: Statically Indeterminate Pin-Jointed Plane Frame 2 10.4. Example 10.3: Statically Indeterminate Single-Span Beam 10.5. Example 10.4: Multi-Span Beam 10.6. Rigid-Jointed Frames Subjected to Vertical Loads 10.6.1. Example 10.5: Multi-Storey Rigid-Jointed Frame 1 10.6.2. Approximate Analysis of Multi-Storey Rigid-Jointed Frames Using Sub-Frames 10.6.2.1. Simplification into Sub-Frames 10.6.2.2. Alternative Simplification for Individual Beams and Associated Columns 10.6.2.3. ‘Continuous Beam’ Simplification 10.6.2.4. Asymmetrically Loaded Columns 10.6.3. Simple Portal Frames with Pinned Bases Subjected to Horizontal Loads 10.6.3.1. Example 10.6: Simple Rectangular Portal Frame – Pinned Bases 10.6.4. Simple Portal Frames with Fixed Bases Subjected to Horizontal Loads 10.6.4.1. Example 10.7: Simple Rectangular Portal Frame – Fixed Bases 10.7. Multi-Storey Rigid-Jointed Frames Subjected to Horizontal Loads 10.7.1. Portal Method 10.7.1.1. Example 10.8: Multi-Storey Rigid-Jointed Frame 2 10.7.1.2. Approximate Analysis of Vierendeel Trusses Using the Portal Method 10.7.1.3. Example 10.9: Vierendeel Truss 10.7.2. Cantilever Method 10.7.2.1. Example 10.10: Multi-Storey Rigid-Jointed Frame 3 11. Cables and Arches 11.1. Introduction to Cables 11.2. Types of Cable 11.3. Cables Subjected to Concentrated Loads 11.4. Example 11.1: Cable Subjected to Concentrated Loads 11.5. Example 11.2: Cable Subjected to Concentrated Loads with Uneven Supports 11.6. Problems: Cables Subjected to Concentrated Loads 11.7. Solutions: Cables Subjected to Concentrated Loads 11.8. Cables Subjected to Uniformly Distributed Loads 11.9. Example 11.3: Cable Subjected to Uniformly Distributed Load 11.10. Example 11.4: Cable Subjected to UDL from the Simply Supported Beam 11.11. Problems: Cables Subjected to Uniformly Distributed Loads 11.12. Solutions: Cables Subjected to Uniformly Distributed Loads 11.13. Introduction to Arches 11.14. Example 11.5: Three-Pinned Segmental Arch 11.15. Example 11.6: Two-Pinned Parabolic Arch 11.16. Example 11.7: Two-Pinned Semi-Circular Arch 11.17. Problems: Arches 11.18. Solutions: Arches Appendix 1: Elastic Section Properties of Geometric Figures Appendix 2: Beam Reactions, Bending Moments and Deflections Appendix 3: Matrix Algebra Index