Table of Contents

Cover

Preface

1 Introduction to Structural Analysis

1.1. Introduction
1.2. Concept of a structure
1.3. Structural analysis
1.4. History of structural analysis
1.5. Classification of structures
1.6. Static indeterminacy of structures
1.7. Sign convention
1.8. Conclusion
1.9. Problems

2 Actions on Structures

2.1. Loads
2.2. Permanent loads
2.3. Live loads
2.4. Snow loads
2.5. Wind loads
2.6. Seismic loads
2.7. Thermal loads
2.8. Hydrostatic loads and land thrust
2.9. Conclusion
2.10. Problems

3 External Analysis of Plane Structures

3.1. External equilibrium of structures
3.2. External and internal actions
3.3. Types of plane structure supports
3.4. Static determinacy, static indeterminacy and structural stability
3.5. Calculation of support reactions
3.6. Superposition principle
3.7. Matrix formulation
3.8. Concept of displacement
3.9. Concept of work
3.10. Principle of virtual work
3.11. Calculation of reactions by the virtual work method
3.12. Conclusion
3.13. Problems

4 Analysis of Trusses

4.1. Definition of a truss
4.2. Hypothesis of analysis
4.3. Sign convention and representation of internal forces
4.4. Degree of static indeterminacy and stability of trusses
4.5. Analysis methods of trusses
4.6. Composite trusses
4.7. Complex trusses
4.8. Space trusses
4.9. Conclusion
4.10. Problems

5 Internal Analysis of Beams and Frames

5.1. Normal force, shear force and bending moment
5.2. Sign convention
5.3. Beam analysis procedure
5.4. Diagrams of internal actions
5.5. Relationship between loading, shear force and bending moment
5.6. Static determinacy, static indeterminacy and instability of beams and frames
5.7. Plane frame analysis procedure
5.8. Conclusion
5.9. Problems

6 Deflections of Elastic Beams: Energy Methods

6.1. Elastic deflection of beams
6.2. Calculation of deflections
6.3. Superposition principle
6.4. Conclusion
6.5. Problems

7 Structural Deflections: Energy Methods

7.1. Work of external actions
7.2. Internal or strain energy
7.3. Principle of energy conservation
7.4. Principle of virtual work
7.5. Conservation of energy and strain energy
7.6. Castigliano’s theorem
7.7. Maxwell–Betti law
7.8. Conclusion
7.9. Problems

8 Cable Analysis

8.1. Introduction
8.2. Mechanical characteristics of cables
8.3. Hypothesis of cable analysis
8.4. Cable analysis
8.5. Cables with an inflection point outside the cable
8.6. Suspension bridges
8.7. Conclusion
8.8. Problems

9 Analysis of Arches

9.1. Introduction
9.2. Three-hinged arch
9.3. Bi-hinged arch
9.4. Conclusion
9.5. Problems

10 Influence Lines

10.1. Introduction
10.2. Influence line definition
10.3. Influence lines of a beam using the equilibrium method
10.4. Influence lines of a frame using the equilibrium method
10.5. Analysis steps
10.6. Influence lines of trusses
10.7. Influence lines using the Muller–Breslau principle
10.8. Influence lines of deflections
10.9. Conclusion
10.10. Problems

Appendices

A.1. Standard structural deflections
A.2. Evaluation of the integral

Bibliography

Index

End User License Agreement

List of Tables

2 Actions on Structures

Table 2.1. Unit weights of some materials
Table 2.2. Live load values
Table 2.3. Shape coefficients
Table 2.4. Characteristic values of snow on the ground
Table 2.5. Exceptional values of the snow load
Table 2.6. Reference velocities and pressures
Table 2.7. Values of K_r, z₀, z_min and K_l
Table 2.8. The maximum reference acceleration at ground level of class A
Table 2.9. Coefficient of importance
Table 2.10. Critical percentage of damping
Table 2.11. Vertical acceleration
Table 2.12. Vertical acceleration
Table 2.13. Inside temperature
Table 2.14. Outside temperature above ground level
Table 2.15. Outside temperature below ground level

3 External Analysis of Plane Structures

Table 3.1. Kinematic conditions and support reactions

4 Analysis of Trusses

Table 4.1. Internal forces on the bars
Table 4.2. Forces on bars 1, 2 and 3
Table 4.3. Forces on the bars
Table 4.4. Normal force in the bar structure (Figure 4.24)
Table 4.5. Forces in the bars

5 Internal Analysis of Beams and Frames

Table 5.1. Degree of static indeterminacy beams and frames

6 Deflections of Elastic Beams: Energy Methods

Table 6.1. Beam deflections
Table 6.2. Supports of real and fictitious beams

7 Structural Deflections: Energy Methods

Table 7.1. Forces on the bars
Table 7.2. Unit forces
Table 7.3. Displacements on the bars
Table 7.4. Forces due to unit action
Table 7.5. Forces on the bars
Table 7.6. Forces on the bars

10 Influence Lines

Table 10.1. Forces in bars B1, B2, B3

List of Illustrations

1 Introduction to Structural Analysis

Figure 1.1. The conditions for defining a structure
Figure 1.2. Truss
Figure 1.3. Membranous structures
Figure 1.4. Spatial trusses
Figure 1.5. Beams
Figure 1.6. Internal actions
Figure 1.7. Frames
Figure 1.8. Three-dimensional frames
Figure 1.9. Crossbeam structures
Figure 1.10. A given arch
Figure 1.11. Types of support and associated reactions
Figure 1.12. Different structures of strusses
Figure 1.13. Given trusses
Figure 1.14. Reaction at fixed node
Figure 1.15. Single span frames
Figure 1.16. Single and multiple span frames
Figure 1.17. Given beams
Figure 1.18. Sign conventions of internal actions
Figure 1.19. Sign conventions of deflections

2 Actions on Structures

Figure 2.1. Given construction
Figure 2.2. Construction – live loads
Figure 2.3. Coefficient of internal pressure according to permeability
Figure 2.4. Given building and wind diagram
Figure 2.5. Structure and its modeling
Figure 2.6. Hydrostatic pressure and land thrust. For a color version of the figure, please see www.iste.co.uk/khalfallah/analysis1.zip

3 External Analysis of Plane Structures

Figure 3.1. Three-dimensional structure
Figure 3.2. Structure under concurrent forces
Figure 3.3. Structure subject to parallel forces
Figure 3.4. Internal actions
Figure 3.5. Statically determinate externally of structures
Figure 3.6. Statically indeterminate externally of structures
Figure 3.7. Unstable structures
Figure 3.8. Trusses
Figure 3.9. Given structures
Figure 3.10. Given beam
Figure 3.11. Free-body beam
Figure 3.12. Given frame
Figure 3.13. Free-body diagram
Figure 3.14. Statically indeterminate frame
Figure 3.15. Free-body frame
Figure 3.16. Truss
Figure 3.17. Free-body diagram
Figure 3.18. Beam under multiple loads
Figure 3.19. Beam deflections
Figure 3.20. Statically determinate beam
Figure 3.21. Free-body beam
Figure 3.22. Given beam
Figure 3.23. Beam decomposition
Figure 3.24. Deformation of a structure
Figure 3.25. “Spring” mechanical model
Figure 3.26. Force–displacement relationship
Figure 3.27. Force and displacement are independent
Figure 3.28. Linear behavior law
Figure 3.29. Displacements of the beam
Figure 3.30. Given beam
Figure 3.31. Free-body beam
Figure 3.32. Effect of horizontal forces
Figure 3.33. Effect of vertical forces
Figure 3.34. Effect of moments

4 Analysis of Trusses

Figure 4.1. Connecting gusset
Figure 4.2. Plane truss
Figure 4.3. Space truss
Figure 4.4. Model of a truss
Figure 4.5. Sign convention
Figure 4.6. External analysis of trusses
Figure 4.7. Internal analysis of trusses
Figure 4.8. Truss
Figure 4.9. Free-body diagram
Figure 4.10. Normal force*
Figure 4.11. Given truss
Figure 4.12. Part of the structure proposed
Figure 4.13. Given structure
Figure 4.14. Part considered
Figure 4.15. Forces on bars 1, 2, and 3
Figure 4.16. Portion of a truss
Figure 4.17. Study of bar (AB)
Figure 4.18. Given structure
Figure 4.19. Given structure
Figure 4.20. Numbering of joints and bars
Figure 4.21. Cremona diagram
Figure 4.22. Forces around joint 1
Figure 4.23. Composite structure
Figure 4.24. Free-body diagram
Figure 4.25. Method of sections
Figure 4.26. Normal force diagram *
Figure 4.27. Numbering of joints and bars
Figure 4.28. Complex structures
Figure 4.29. Space truss
Figure 4.30. Spatial truss
Figure 4.31. Free-body diagram
Figure 4.32. Normal force on the bars *(F₀)

5 Internal Analysis of Beams and Frames

Figure 5.1. Studied beam
Figure 5.2. Free-body beam
Figure 5.3. Internal actions at any section
Figure 5.4. Sign convention
Figure 5.5. Given beam
Figure 5.6. Free-body beam
Figure 5.7. Reduction elements of section B
Figure 5.8. Given beam
Figure 5.9. Support reactions
Figure 5.10. Right portion
Figure 5.11. Diagram of internal actions
Figure 5.12. Study of an element Δx
Figure 5.13. Given beams and frames
Figure 5.14. Given frame
Figure 5.15. Support reactions
Figure 5.16. Bar (AB)
Figure 5.17. Bar (BC)
Figure 5.18. Bar (CD)
Figure 5.19. Diagram of internal actions
Figure 5.20. Given frame
Figure 5.21. Support reactions
Figure 5.22. Diagrams of internal actions

6 Deflections of Elastic Beams: Energy Methods

Figure 6.1. Studied beam
Figure 6.2. Deformed beam
Figure 6.3. Deflection of element Δx
Figure 6.4. Given beam
Figure 6.5. Given beam
Figure 6.6. Deformed beam
Figure 6.7. Relationships of sections A and B
Figure 6.8. Displacements of points A and B
Figure 6.9. Given beam
Figure 6.10. Bending moment diagram
Figure 6.11. Given cantilever
Figure 6.12. Bending moment diagram
Figure 6.13. Conjugate beam loading
Figure 6.14. Deformed cantilever
Figure 6.15. Given beam
Figure 6.16. Load force q
Figure 6.17. Load force P
Figure 6.18. Given beam

7 Structural Deflections: Energy Methods

Figure 7.1. Axial force
Figure 7.2. Work of force F
Figure 7.3. Bending moment
Figure 7.4. Mechanical work of moment M
Figure 7.5. Transversal or shear force
Figure 7.6. Work of transversal force
Figure 7.7. Angular deformation due to torque moment
Figure 7.8. Work of torque moment
Figure 7.9. Work of force F
Figure 7.10. Work of an action independent of deflection
Figure 7.11. Normal force
Figure 7.12. Bending moment
Figure 7.13. Normal force
Figure 7.14. Bar deflection
Figure 7.15. Bending
Figure 7.16. Beam deflection
Figure 7.17. Pure bending
Figure 7.18. Cantilever deflection
Figure 7.19. Deformable body
Figure 7.20. Deformed body
Figure 7.21. Internal force
Figure 7.22. Truss forces
Figure 7.23. Virtual deflection of the structure
Figure 7.24. Given structure
Figure 7.25. Given structure
Figure 7.26. Given structure
Figure 7.27. Deformation of the structure
Figure 7.28. Given beam
Figure 7.29. Unit action
Figure 7.30. Given beam
Figure 7.31. Virtual force
Figure 7.32. Unit structure
Figure 7.33. Deformed cantilever
Figure 7.34. Given beam
Figure 7.35. Free-body beam
Figure 7.36. Unit force
Figure 7.37. Unit moment
Figure 7.38. Deformed beam
Figure 7.39. Given frame
Figure 7.40. Unit moment
Figure 7.41. Effect of the virtual force
Figure 7.42. Deformed frame
Figure 7.43. Solid body
Figure 7.44. Given structure
Figure 7.45. Given cantilever
Figure 7.46. Given cantilever beam
Figure 7.47. Application of MC
Figure 7.48. Force at joint C
Figure 7.49. Deformable body
Figure 7.50. Given beam
Figure 7.51. Application of force in A
Figure 7.52. Application of force in B

8 Cable Analysis

Figure 8.1. Cables
Figure 8.2. Given cable
Figure 8.3. Free-body diagram
Figure 8.4. Cable structure under concentrated forces
Figure 8.5. Support reactions
Figure 8.6. Normal force in cable sections
Figure 8.7. Cable under concentrated force
Figure 8.8. Cable under distributed force
Figure 8.9. Cable portion
Figure 8.10. Given cable
Figure 8.11. Spotting the structure
Figure 8.12. Given cable
Figure 8.13. Given structure
Figure 8.14. Given cable
Figure 8.15. Support forces
Figure 8.16. Given cable
Figure 8.17. Suspension bridge
Figure 8.18. Forces on the pulley
Figure 8.19. Arrangement at the end of the pylon
Figure 8.20. Given suspension bridge

9 Analysis of Arches

Figure 9.1. Types of arch structure
Figure 9.2. Three-hinged arch
Figure 9.3. Dividing the arch
Figure 9.4. Given arch
Figure 9.5. Free-body arch
Figure 9.6. Variation of V_A and V_B
Figure 9.7. Variation of H_A and H_B
Figure 9.8. Section θ₁ ≤ θ
Figure 9.9. Section
Figure 9.10. Section
Figure 9.11. Diagrams of internal actions
Figure 9.12. Given arch
Figure 9.13. Free-body arch
Figure 9.14. Section
Figure 9.15. Diagrams of internal actions
Figure 9.16. Parabolic arch
Figure 9.17. Support reactions
Figure 9.18. Section 0 ≤ x ≤ a
Figure 9.19. Section a ≤ x ≤ L
Figure 9.20. Section L ≤ x ≤ 2L
Figure 9.21. Diagram of internal actions
Figure 9.22. Parabolic arch
Figure 9.23. Free-body arch
Figure 9.24. Section 0 ≤ x ≤ L
Figure 9.25. Section L ≤ x ≤ 2L
Figure 9.26. Diagram of internal actions
Figure 9.27. Arch with level supports
Figure 9.28. Identifying the arch
Figure 9.29. Section 11.53°≤ θ ≤ 90°
Figure 9.30. Section 90°≤ θ ≤ 135.56°
Figure 9.31. Section 135.56°≤ θ ≤ 156.42°
Figure 9.32. Diagram of internal actions
Figure 9.33. Bi-hinged arch
Figure 9.34. Free-body arch
Figure 9.35. Section θ₁ ≤ θ
Figure 9.36. Section θ ≤ θ₁ ≤ π
Figure 9.37. Diagrams of internal actions

10 Influence Lines

Figure 10.1. Moving mass on a bridge
Figure 10.2. Moving unit force on a beam
Figure 10.3. Influence lines of support reactions
Figure 10.4. Beam studied
Figure 10.5. Influence line of a shear force
Figure 10.6. Influence line of a bending moment
Figure 10.7. Given beam
Figure 10.8. Influence line of support reaction V_B
Figure 10.9. Influence line of moment M_C
Figure 10.10. Influence line of T_C
Figure 10.11. Given frame
Figure 10.12. Influence line of V_A
Figure 10.13. Influence line of M_Q
Figure 10.14. Influence line of T_Q
Figure 10.15. Given structure
Figure 10.16. Influence lines at points A and D
Figure 10.17. Given beam
Figure 10.18. Beam with a hinge
Figure 10.19. Influence line of V_E
Figure 10.20. Influence line of moment M_B
Figure 10.21. Influence line of T_C
Figure 10.22. Given frame
Figure 10.23. Bar (AB)
Figure 10.24. I.L.H_A
Figure 10.25. I.L.T_D
Figure 10.26. I.L.M_D
Figure 10.27. Truss
Figure 10.28. Truss
Figure 10.29. Numbering of bars
Figure 10.30. I.L.V_A
Figure 10.31. I.L.B₁
Figure 10.32. I.L.B₂
Figure 10.33. I.L.B₃
Figure 10.34. Given beam
Figure 10.35. Given beam
Figure 10.36. Virtual unit displacement
Figure 10.37. Given beam
Figure 10.38. Effect of reaction V_A
Figure 10.39. Unit force 0 ≤ α ≤ L / 2
Figure 10.40. Unit force
Figure 10.41. I.L.V_A
Figure 10.42. Shear force at point C
Figure 10.43. Tc for α ≤ x
Figure 10.44. Tc for x ≤ α ≤ L
Figure 10.45. I.L.T_C
Figure 10.46. Shear force
Figure 10.47. Shear force
Figure 10.48. Shear force
Figure 10.49. Influence line of a shear force relating to point D.
Figure 10.50. Given beam
Figure 10.51. Beam deflection
Figure 10.52. I.L.M_C
Figure 10.53. Force
Figure 10.54. Force
Figure 10.55. Force
Figure 10.56. I.L.M_D
Figure 10.57. Given beam
Figure 10.58. Maxwell–Betti law
Figure 10.59. Given cantilever beam
Figure 10.60. Bending moment diagram
Figure 10.61. Conjugate beam
Figure 10.62. Influence lines of δ_B and ω_B.

First published 2018 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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The rights of Salah Khalfallah to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

Library of Congress Control Number: 2018943087

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

ISBN 978-1-78630-338-7

Preface

The main objective of this volume is to provide students of civil, mechanical, aeronautical and marine engineering as well as those interested in structural analysis with the essentials for analyzing statically determinate structures. This book allows them to acquire sufficient knowledge to study and analyze statically determinate structures. The reader will find a series of exercises at the end of each chapter that can be used to deepen their knowledge and improve their ability to master statically determinate structure analysis methods.

This volume covers aspects of structural analysis such as trusses, beams, frames, cable structures and arch structures. The last chapter looks at influence lines for live loads, which play a role in the design phase of statically determinate structures.

This book is the first of two volumes. It consists of two parts. Each chapter of the book is constructed in a specific way: an illustration of the objectives and the parts covered, a general introduction, a theory of the proposed approach, a numerical study of some examples and a summary at the end. Each chapter ends with a series of problems and exercises.

The first part includes a general introduction to structural analysis (Chapter 1) and illustrates in detail the different types of loads that can be applied to a structure (Chapter 2). The load magnitude is evaluated according to European standards, including those adopted by France. The second part describes statically determinate structure analysis methods.

This part is divided into eight separate chapters. Chapter 3 offers an overall analysis of determinate structures and looks into their external equilibrium. Chapter 4 looks at how to analyze trusses and several methods are given, such as the method of joint equilibrium, the method of sections, the graphical method and the matrix method. In this context, complex and spatial trusses are analyzed. In the same way, the analysis of beams and frames is developed in Chapter 5. Chapter 6 discusses the calculation of beam deflections using geometric methods. Here, elastic line deflection, direct integration and fictitious beam methods are reviewed. For the same purpose, Chapter 7 also discusses the calculation of deflections using energy methods, emphasizing the virtual work method. Chapter 8 describes the methodology for analyzing cables acted upon by concentrated and distributed loads. Also, arc analysis is described in Chapter 9, in which two types of arches are selected: semicircular and parabolic. The loads proposed here are concentrated and uniformly distributed. The last chapter illustrates the influence lines of the internal or external actions that are built under the effect of a moving unitary load along a beam, a frame or a truss.

Finally, we hope that our approach in this book’s publication will meet the needs of students interested in this scientific and technical subject. Nevertheless, we are very aware that the work presented is not exempt from mistakes. For this reason, we warmly welcome any corrections and comments, which will improve future editions of this book. Comments or suggestions can be sent to the email addresses found on the website www.freewebs.com/khalfallah/index.htm.

Salah KHALFALLAH
May 2018

Statically Determinate Structures