Examples in structural analysis

Preface

Acknowledgements

About the Author

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)

2.1.3. Secant Modulus

2.1.4. Tangent Modulus

2.1.5. Shear Rigidity (Modulus of Rigidity)

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

2.1.11. Coefficient of Thermal Expansion

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 and Radius of Gyration

2.2.6.1. The Parallel Axis Theorem

2.2.7. Elastic Section Modulus

2.2.8. Problems: Second Moment of Area and Elastic Section Modulii

2.2.9. Solutions: Second Moment of Area and Elastic Section Modulii

2.3. Plastic Cross-Section Properties

2.3.1. Stress/Strain Relationship

2.3.2. Plastic Neutral Axis

2.3.3. Evaluation of Plastic Moment 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 1 st 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

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 UDL's

4.3. Equivalent Uniformly Distributed Load Method for the Deflection of Beams

4.3.1. Problems: McCaulay's and Equivalent VOL 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 Uniform Cantilever using Coefficients

4.5.8. Problems: Unit Load Method for Deflection of Beams/Frames

4.5.9. Solutions: Unit Load Method for Deflection of Beams/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 Encastre 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 5.3: 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 5.4: 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

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

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

7.2.1. Beams Elements with Two Degrees-of-Freedom

7.2.2. Beams 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

7.4. Structural Load Vector

7.5. Structural Displacement Vector

7.6. Element Displacement Vector

7.7. Element Force Vector

7.8. Example 7.1: Two-span Beam

7.9. Example 7.2: Rigid-Jointed Frame

7.10. Problems: Direct Stiffness Method

7.11. 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

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

9.2.3. Influence Line for the Bending Moment

9.3. Muller-Breslau Principle for the Influence Lines for Beams

9.4. Influence Lines for a Statically Determinate Beam

9.5. Example 9.3: Influence Line for a Statically Indeterminate Beam

9.6. The use o f 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. Problems: Influence Lines for Beams

9.10. Solutions: Influence Lines for Beams

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-joinied Frames Using Sub-frames

10.6.3. Simple Portal Frames with Pinned Bases Subjected to Horizontal Loads

10.6.3.1. Example 1 0.6: 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: Multistorey 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

Appendices

Appendix 1. Elastic section properties of geometric figures

Appendix 2. Beam reactions, bending moments and deflections

Appendix 3. Matrix algebra

Index