Fundamentals ofStructural Mechanics and Analysis

M.L. GambhirFormerly

Professor and HeadDepartment of Civil Engineering

andDean, Planning and Resource Generation

Thapar University, Patiala

New Delhi-1100012011

FUNDAMENTALS OF STRUCTURAL MECHANICS AND ANALYSISM.L. Gambhir

© 2011 by PHI Learning Private Limited, New Delhi. All rights reserved. No part of this book maybe reproduced in any form, by mimeograph or any other means, without permission in writing fromthe publisher.

ISBN-978-81-203-4236-1

The export rights of this book are vested solely with the publisher.

Published by Asoke K. Ghosh, PHI Learning Private Limited, M-97, Connaught Circus,New Delhi-110001 and Printed by Baba Barkha Nath Printers, Bahadurgarh, Haryana-124507.

ToThe Society

to whom we owe a lot

Contents

Preface xv

1. Structures: An Introduction 1–211.1 Anatomy of Structures 11.2 Loads Acting on a Structure 21.3 Classification of Structures 3

1.3.1 Classification Based on Geometry 31.3.2 Classification Based on Stiffness 41.3.3 Classification Based on Materials of Construction 51.3.4 Classification Based on Load Transfer Mechanism 5

1.4 Basic Structural Elements and Systems 51.4.1 One-dimensional Element 5

1.5 Basic Requirements of Structures 101.5.1 Structural Stability 101.5.2 Internal Stress Resultants 11

1.6 Support Systems 111.7 Statically Determinate and Indeterminate Structures 141.8 Structural Idealization 151.9 Evolution of Structure 181.10 Methods of Structural Analysis 19

1.10.1 Linearity of Structural System 191.10.2 Modelling a Structure 201.10.3 Methods for Analysis of Skeleton Structures 21

Review Questions 21

2. Basic Concepts and Analysis Tools 22–1022.1 Introduction 222.2 Forces and Their Management 23

2.2.1 Specifying a Force 232.2.2 Parallelogram of Forces 24

v

vi ∑ Contents

2.2.3 Moments 272.2.4 The Resultant of a System of Parallel Forces 292.2.5 Equilibrant of a System of Concurrent Forces 302.2.6 Equivalent Force Systems 31

2.3 Equilibrium of Structures 312.3.1 Special Equilibrium Cases 332.3.2 Sign Convention 33

2.4 Free Body Diagrams 342.5 Reactive Forces 36

2.5.1 Support Conditions 372.5.2 Boundary Conditions 382.5.3 Structure with Cable or Elastically Supported End 442.5.4 Multi-Span Statically Determinate Beams or Cantilevered Beams 462.5.5 Plane Frame 49

2.6 Principle of Superposition 502.7 Stresses 51

2.7.1 Normal Stress 512.7.2 Shear Stress 51

2.8 Flexibility and Stiffness 522.9 Energy Methods 54

2.9.1 Conservation of Energy 542.9.2 The Work 542.9.3 Work of Externally Applied Forces 552.9.4 Complementary Work 562.9.5 Eigen Work and Displacement Work 572.9.6 Work of Internal Forces: Strain Energy 582.9.7 Strain Energy for Deformed Members 59

2.10 Real Work Equation 612.11 Virtual Work Methods 62

2.11.1 General 622.11.2 Virtual Work 642.11.3 Virtual Work and Complementary Virtual Work 642.11.4 Applications of Virtual Work Methods 692.11.5 Unit-Load and Unit-Displacement Methods 702.11.6 Graphical Integration 71

2.12 Important Theorems of Energy Methods 762.12.1 Reciprocal Relations 762.12.2 Cotterill–Castigliano’s Theorems 78

2.13 Principle of Stationary Total Potential Energy 822.14 Principles of Complimentary Virtual Work and Stationary Complimentary

Potential Energy 832.14.1 Complimentary Strain Energy and the Principle of Least Work 84

Review Questions 92Problems 93

3. Cables and Suspension Bridges 103–1543.1 Introduction 1033.2 Suspended Cable 104

3.2.1 Cable Subjected to Concentrated Loads 104

Contents ∑ vii

3.2.2 Cable Subjected to Uniformly Distributed Load 1083.2.3 Length of Cable 1123.2.4 Temperature Stresses in the Cable 113

3.3 Cable Supports 1163.3.1 Cable Passing Over Guide Pulleys 1173.3.2 Cable Clamped to Saddle on Smooth Rollers Mounted on Top of Pier 117

3.4 Suspension Bridge 1193.5 Cable with Three-hinged Stiffening Girder 119

3.5.1 Analysis of Three-hinged Stiffening Girder 1193.5.2 Influence Line Diagrams for Three-hinged Stiffening Girder 125

3.6 Cable with Two-hinged Stiffening Girder 1403.6.1 Analysis of Two-hinged Stiffening Girder 1413.6.2 Temperature Stresses in Two-hinged Girder 1413.6.3 Influence Lines Diagrams 142

Problems 151

4. The Plane Trusses 155–2114.1 Introduction 1554.2 Components of a Truss 1564.3 Member Forces 1564.4 Classification of Trusses 1564.5 Analysis of Simple Plane Trusses 158

4.5.1 Assumptions 1584.5.2 Truss Notation 1584.5.3 Geometric Stability 1594.5.4 Static Determinacy 1594.5.5 The Principle of Analysis 1614.5.6 Methods for Analysis and Sign Conventions 1624.5.7 Member Force Notation 163

4.6 Method of Joints Equilibrium 1644.6.1 Simplifying Conditions 167

4.7 Method of Sections or Moments 1704.7.1 Effect of Height of Truss on the Member Forces 1754.7.2 Trusses with Sub-divided Panels 176

4.8 Method of Tension Coefficients 1774.9 Influence Lines for Simple Trusses 180

4.9.1 Parallel Chord Trusses 1814.9.2 Non-parallel Chord Trusses 1814.9.3 Trusses with Sub-divided Panels 181

4.10 Determination of Maximum Forces 1814.11 Influence Lines for a Truss with Subdivided Panels 190

Problems 193

5. Three-dimensional or Space Trusses 212–2305.1 Introduction 2125.2 Basic Principles 213

5.2.1 Equations of Static Equilibrium 2145.2.2 Stability of Space-trusses 2155.2.3 Types of Support 215

viii ∑ Contents

5.3 Analysis of Space-trusses 2165.3.1 Method of Joints 2175.3.2 Method of Tension Coefficients 2225.3.3 Method of Sections 224

Problems 226

6. Analysis of Arches 231–2726.1 Introduction 2316.2 Three-hinged Arches 232

6.2.1 Geometry of Three-hinged Arch 2336.2.2 Reactions and Forces at the Connections 2366.2.3 Reactions and Bending Moments 2396.2.4 Reactions and Forces in Framed Arches 2486.2.5 Normal Thrust and Radial Shear 249

6.3 Influence Lines for Three-hinged Arches 2526.3.1 Horizontal Thrust H 2536.3.2 Bending Moment at a Section 2546.3.3 Normal Thrust and Radial Shear at a Section 2556.3.4 Maximum Bending Moments Due to the Moving Loads 255

6.4 Analysis of Three-hinged Tied Arches 2646.5 Two-hinged and Fixed Arches 268

Problems 268

7. Influence Lines and Rolling Loads 273–3367.1 Introduction 2737.2 Influence Lines for Beams 274

7.2.1 Simple Beams 2747.2.2 Statically Determinate Beams with Hinges 2797.2.3 Statically Determinate Frames Reduced to Simple Beams 2847.2.4 Statically Determinate Beams Subjected to Indirect Loads 289

7.3 Quantitative Influence Lines 2967.3.1 Müller–Breslau Principle 296

7.4 Applications of Influence Lines 3007.5 Moving Loads 311

7.5.1 Maximum Bending Moments and Shearing force Due to the Moving Loads 3117.6 Maximum Shearing Force Diagrams for Dead and Live Loads 3267.7 Equivalent Uniformly Distributed Load 330

Problems 331

8. Elastic Deflections 337–432

8.1 Introduction 3378.2 Deformed Shapes of the Structures 338

8.2.1 Members 3388.2.2 Joints 338

Contents ∑ ix

8.3 Beam Deflections by Direct Integration 3418.3.1 Governing Differential Equation 3428.3.2 Boundary Conditions 3428.3.3 Macaulay Procedure 349

8.4 Semi-geometrical Methods 3538.5 The Moment–Area Method for Symmetrical Bending 353

8.5.1 Application of Moment–Area Theorems 3558.5.2 Application with the Principle of Superposition 3648.5.3 Application to Statically Indeterminate Beams 367

8.6 Elastic Load Method 3708.7 Application of Moment–Area and Elastic-load Methods 3738.8 Conjugate-beam Method 376

8.8.1 Basis of Development 3768.8.2 Boundary Conditions of the Conjugate Beam 377

8.9 Work–Energy Methods 3848.9.1 Real Work Method 3848.9.2 The Virtual Work Method 3908.9.3 The Principle of Superposition of Mechanical Work 399

8.10 Energy Theorems of Elastic Systems 4008.11 Deflections Due to Unsymmetrical Bending 407

Problems 413

9. An Introduction to Statically Indeterminate Structures 433–4539.1 General 4339.2 Determinacy of Beams and Frames 434

9.2.1 Degree of Statical Indeterminacy 4349.2.2 General Relationship 4399.2.3 Degree of Kinematical Indeterminacy 439

9.3 Determinacy of Pin-jointed Frames 4409.3.1 Plane Truss 4409.3.2 Pin-jointed Space Frame 441

9.4 Structural Analysis 4429.4.1 Force–Displacement Relationship 4429.4.2 Methods of Analysis 443

9.5 Principle of Superposition 4449.6 Fixed End Moments 4459.7 Concept of Symmetry 448

Problems 451

10. Flexibility Method (Force or Consistent Deformation Method) 454–54510.1 Introduction 45410.2 Force or Flexibility Method 455

10.2.1 Formulation of Elastic Equations 45510.2.2 Solution Procedure for Elastic Equations of the Structure 459

10.3 Analysis of Indeterminate Beams 45910.3.1 Propped Cantilever Beam 45910.3.2 Continuous Beams 463

x ∑ Contents

10.4 Settlement of Supports 46810.5 Elastic Supports 47110.6 The Three-moment Equation (Clapeyron Theorem) 479

10.6.1 Derivation of Three Moment Equation 47910.6.2 Application of Three-moment Equations 483

10.7 Continuous Beams with Supports at Different Levels 48510.8 Indeterminate Trusses 492

10.8.1 Analysis of Indeterminate Truss 49310.8.2 Temperature Changes 49510.8.3 Lack of Fit or Fabrication Errors 49510.8.4 Composite Trusses 504

10.9 Two-hinged Arches 50710.9.1 Secant Variation of Moment of Inertia of the Cross-Section 50910.9.2 Applications to Different Types of Load Systems 51110.9.3 Tied Arch or Bowstring Girder 519

10.10 Fixed-ended or Hingeless Arches 52310.11 The Suspension Bridge 53110.12 Choice of Force or Displacement Method 533

Problems 533

11. Force–Displacement Methods (Slope–Deflection, Three-Moment Equations,Moment Distribution and Stiffness Methods) 546–59911.1 Introduction 54611.2 Fundamental Force–Displacement Relationships 546

11.2.1 Derivation by Integration of Flexural Differential Equation 54611.2.2 Derivation Based on the Moment-Area Theorems 551

11.3 Slope–Deflection Method 55511.3.1 Sign Convention 55511.3.2 Member End Moments for Single Span Beams 55611.3.3 Joint Equilibrium Equations 55711.3.4 Fixed-end Moments 557

11.4 Analysis of Statically Indeterminate Beams 55911.4.1 General Procedure for Continuous Beams 574

11.5 Analysis of Frames without Side Sway 57911.6 Frames with Side or Lateral Sway 585

Problems 595

12. System Approach 600–65312.1 General 60012.2 Analysis of Beams and Frames 602

12.2.1 Analysis of Beams 60212.2.2 Analysis of Rigid Frames 608

12.3 Analysis of Pin-jointed Plane Frames (Trusses) 61812.4 Analysis of Structures with Elastic Connections 62512.5 Analysis of Grid 628

Problems 637

Contents ∑ xi

13. Moment Distribution 654–69613.1 Introduction 65413.2 Concept of Moment Distribution 65413.3 Concept of Symmetric and Skew-symmetric Deformations 655

13.3.1 Symmetric Case 65513.3.2 Skew-symmetric Case 655

13.4 Member Stiffness and Joint Distribution Factors 65513.4.1 Absolute and Relative Stiffness of Members 65613.4.2 Joint Distribution Factors 65713.4.3 Sign Convention 660

13.5 Analysis Procedure 66013.6 Analysis for Settlement at Supports 66813.7 Analysis of Frames 674

13.7.1 Nonsway Frames 67413.7.2 Frames with Sidesway 67613.7.3 Analysis of Non-sway Frames 67613.7.4 Analysis of Sway Frames 67813.7.5 Analysis for Temperature Variation 686

13.8 Special Frames 68713.8.1 Rectangular Liquid Tank 68713.8.2 Box Culvert 689

13.9 Comparison between MDM and Matrix Methods 690

Problems 691

14. Direct Stiffness Method 697–77114.1 Introduction 69714.2 Basic Concepts 69714.3 Notation 699

14.3.1 Global and Local Coordinate Systems 69914.3.2 Joint and Member Notation 70014.3.3 Global and Local Displacements 70014.3.4 Global and Local Forces 701

14.4 Direct Stiffness Method 70214.4.1 Kinematic Degrees of Freedom 70414.4.2 Subdivision or Breaking the Structure 704

14.5 Formulation of Local Stiffness Matrix 70414.6 Formulation of the Global Stiffness Matrix 708

14.6.1 Transformation of Forces and Displacements 70914.6.2 Global Force–Displacement Relations 71114.6.3 Member Global Stiffness Matrix 71114.6.4 Interpretation of Stiffness Coefficients 71314.6.5 Compatibility Conditions 71314.6.6 Joint Equilibrium Equations 714

14.7 Construction of Stiffness Matrix, Force Vector and Displacement Vector 71614.7.1 Problem Definition 71614.7.2 Assembly of Structure Stiffness Matrix, and Displacement

and Force Vectors 71614.7.3 Assembly Rules 717

xii ∑ Contents

14.8 Formulation of System of Equations 72014.8.1 Introduction of Boundary (Support) Conditions 72014.8.2 Computation of Displacements and Reactions 72214.8.3 Computation of Member Forces 722

14.9 Stiffness Matrix for the Frame Members 73614.9.1 Member Oriented Along Reference Axis Subjected to Pure Bending 73714.9.2 Arbitrarily Oriented Member Subjected to Pure Bending 75014.9.3 General Plane Frame Member with Combined Bending and Axial loads 75314.9.4 Three-dimensional Beam Element 755

14.10 Assembly of Structure Stiffness Matrix 75614.11 Computer Implementation of the Stiffness Method 763

14.11.1 Structural Analysis Software 764

Problems 764

15. An Introduction to Finite Element Method 772–83815.1 Introduction 77215.2 Basis of the Finite Element Method 77315.3 Analysis Procedure 776

15.3.1 Properties of Stiffness Matrices 77915.4 Formulation of Finite Element 780

15.4.1 Modelling of Various Parameters 78015.4.2 Approaches for Formulation of Finite Elements 781

15.5 Displacement Interpolation or Shape Functions 78115.5.1 Requirements for a Shape Function 78315.5.2 Derivation of Shape Function 783

15.6 Axial Force Rod or Truss Elements 78415.6.1 Element Shape Function 78415.6.2 Element Stiffness Matrix in Local Coordinates 78715.6.3 Element Stress Matrix in Local Coordinates 78915.6.4 Transformation of Element Stiffness and Stress Matrices to

Global Coordinates 79015.6.5 Formation of Structural Governing Equation and Assembled

Stiffness Matrix 79015.7 Evaluation of Displacements and Reactions 79215.8 Simple Beam Element 798

15.8.1 Element Stiffness Matrix in Local Coordinates 80115.8.2 Transformation of Element Stiffness Matrix to Global Coordinates 80315.8.3 Beam Element with Combined Bending and Axial Loads 80415.8.4 Element Stress Matrix in Local Coordinates 80415.8.5 Element Stress Matrix in Global Coordinates 80515.8.6 Formation of Global or Structural Governing Equation 806

15.9 Structural Loads 81015.9.1 Work Equivalence Method 81015.9.2 Distributed Loads 81115.9.3 Concentrated Load Acting between the Nodes 813

15.10 Static Condensation 81515.10.1 Beam Element with Nodal Hinge 816

15.11 Bar of Varying Cross-section 819

Contents ∑ xiii

15.12 Membrane Elements 82015.12.1 Plane Stress Condition 82015.12.2 Plane Strain Condition 82015.12.3 Constant Strain Triangle 82115.12.4 Formulation of Element Stiffness Matrix 82515.12.5 Four Node Quadrilateral Plane Stress Element 82915.12.6 3-D Solid Elements 832

15.13 Convergence 83215.14 Types of Error 83315.15 Structural Analysis Resources 833

Questions 833Problems 835

16. Approximate Analysis of Indeterminate Structures 839–87116.1 Introduction 83916.2 Analysis of Indeterminate Trusses 84016.3 Analysis of Portals and Industrial Frames 843

16.3.1 Analysis 84316.4 Analysis of Frame Subjected to Vertical Loads 84716.5 Building Frame Subjected to Lateral Loads 851

16.5.1 Cantilever Method 85216.5.2 Portal Method 85616.5.3 Wilbur’s Factor Method 862

Problems 866

Appendix A Review of Matrix Algebra 873–882A.1 Introduction 873A.2 Definitions 873

A.2.1 Terminology and Notations 874A.3 Basic Operations 874

A.3.1 Matrix Addition 874A.3.2 Scalar Multiplication 874A.3.3 Matrix Multiplication 874A.3.4 Transpose 875

A.4 Square Matrices and Related Definitions 876A.4.1 Properties of Invertible Matrices 876

A.5 Determinant 877A.5.1 Values of Determinants of 2-by-2 Matrices 877A.5.2 Values of Determinants of 3-by-3 Matrices 877A.5.3 Other Properties of Determinants 879

A.6 The Minors and Cofactors of a Matrix 879A.7 Methods of Matrix Inversion 880A.8 Application of Matrices 882

A.8.1 Solution of System of Linear Equations 882A.8.2 Eigenvalues 882

xiv ∑ Contents

Appendix B Shear Force and Bending Moment Diagrams,and Deflection Formulae 883–897

B.1 Simply Supported Beam 883B.2 Cantilever Beam 888B.3 Beam Fixed at One End and Simply Supported at the Other 890B.4 Beam Overhanging at One Support 892B.5 Beam Overhanging at Both the Supports 895B.6 Beam Fixed at Both the Supports 895

Bibliography 899–901Suggested Further Reading 900

Index 903–906

Fundamentals Of Structural MechanicsAnd Analysis

Publisher : PHI Learning ISBN : 9788120342361 Author : Gambhir

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