1st Edition

# Introduction to Structural Analysis

512 Pages 282 B/W Illustrations
by CRC Press

512 Pages 282 B/W Illustrations
by CRC Press

Also available as eBook on:

Introduction to Structural Analysis covers the principles of structural analysis without any requirement of prior knowledge of structures or equations. Beginning with basic principles of equilibrium of forces and moments, all other subsequent theories of  structural analysis have been discussed logically. Divided into two major parts, this book discusses the basics of mechanics and principles of degrees of freedom upon which the entire paradigm rests, followed by analysis of determinate and indeterminate structures. The energy method of structural analysis is also included. Worked out examples are provided in each chapter to explain the concepts and solve real-life structural analysis problems along with a solutions manual. Aimed at undergraduate and  senior undergraduate students in civil, structural, and construction engineering, this book:

• Deals with the basic levels of structural analysis (i.e., types of structures and loads, materials and section properties up to the standard level, including analysis of determinate and indeterminate structures).

• Focuses on generalized coordinate systems and Lagrangian and Hamiltonian mechanics as an alternative method of studying the subject.

• Introduces structural indeterminacy and degrees of freedom with many worked out examples.

• Covers fundamentals of matrix theory of structural analysis.

• Reviews energy principles and their relationship for calculating structural deflections.

• Covers plastic analysis of structures.

Contents

Preface.................................................................................................................... xiii

Authors......................................................................................................................xv

PART I Introduction to Structural Analysis,

Chapter 1 Introduction to Structural Analysis.......................................................3

1.1 Introduction................................................................................3

1.2 Historical Background................................................................3

1.3 Importance of Structural Analysis.............................................8

PART II Analysis of Statically Determinate Structures

Chapter 2 Types of Structures and Loads............................................................ 13

2.1 Introduction.............................................................................. 13

2.2 Structural Classifications.......................................................... 13

2.2.1 Tension Structures....................................................... 13

2.2.2 Compression Structures.............................................. 14

2.2.3 Trusses......................................................................... 15

2.2.4 Shear Structures.......................................................... 16

2.2.5 Bending Structures...................................................... 16

2.9 Hydrostatic and Soil Pressure...................................................24

2.10 Thermal and Other Effects.......................................................25

2.12 Analytical Model......................................................................25

Chapter 3 Material and Section Properties..........................................................29

3.1 Introduction..............................................................................29

3.2 Simple Stress-Strain Relationship............................................29

3.3 Young’s Modulus or Modulus of Elasticity..............................30

3.4 Secant Modulus........................................................................ 31

3.5 Tangent Modulus...................................................................... 31

3.6 Shear Modulus or Modulus of Rigidity.................................... 32

3.7 Yield Strength........................................................................... 32

3.8 Ultimate Strength..................................................................... 33

3.9 Modulus of Rupture in Bending............................................... 33

3.10 Modulus of Rupture in Torsion................................................. 33

3.11 Poisson’s Ratio.......................................................................... 33

3.12 Coefficient of Thermal Expansion........................................... 33

3.13 Elastic Assumptions.................................................................34

3.14 Sturctural Nonlinearity............................................................34

3.15 Cross-Sectional Area................................................................ 35

3.16 Center of Gravity and Centroid................................................ 35

3.17 Elastic Neutral Axis.................................................................36

3.18 Second Moment of Area and Radius of Gyration....................36

3.19 Elastic Section Modulus...........................................................38

Chapter 4 Basic Concepts of Generalized Coordinates, Lagrangian,

and Hamiltonian Mechanics............................................................... 39

4.1 Introduction and Concept of Generalized Coordinates............ 39

4.2 Concept of Configuration Space and Phase Space...................40

4.3 Introduction to Lagrangian and Hamiltonian Formulation

of Mechanics............................................................................ 41

4.3.1 Hamilton’s Equation of Motion................................... 45

4.4 Concept of Symmetry and Conservation Laws........................46

Chapter 5 Equilibrium and Support Reactions.................................................... 47

5.1 Introduction.............................................................................. 47

5.2 Equilibrium of Structures......................................................... 47

5.2.1 Global and Local Equilibrium of Structures...............48

5.3 Free Body Diagrams.................................................................50

5.4 Sign Convention....................................................................... 51

5.5 External and Internal Forces.................................................... 53

5.6 Types of Supports for Structures.............................................. 58

5.7 Release of Internal Reactions or Member Forces..................... 58

5.7.1 Releasing Bending Moment........................................ 58

5.7.2 Releasing Shear Force................................................. 61

5.7.3 Releasing Axial Force................................................. 61

5.7.4 Releasing Axial Force and Bending Moment............. 61

5.7.5 Releasing Shear Force and Bending Moment............. 61

Chapter 6 Indeterminacy and Stability of Structure............................................63

6.1 Introduction..............................................................................63

6.2 Structural Indeterminacy.........................................................63

6.3 Static Indeterminacy and Stability...........................................63

6.3.1 Static Indeterminacy of Rigid Structures....................64

6.3.1.1 Shortcut Method for Determining

Internal Static Indeterminacy of Rigid

Structures.....................................................68

6.3.1.2 Degree of Statical Indeterminacy

When Load is Applied to the

Structure...................................................... 70

6.3.2 Static Indeterminacy of Pin-Jointed

Structures.................................................................... 72

6.3.2.1 Shortcut Method for Determining

Internal Static Indeterminacy of Pin-

Jointed Structures........................................ 73

6.3.3 External and Internal Stability of

Structures.................................................................... 74

6.3.3.1 External Stability of Structures................... 74

6.3.3.2 Internal Stability of Structures.................... 75

6.4 Kinematic Indeterminacy of Structures...................................77

6.4.1 Kinematic Indeterminacy of Pin-Jointed

Structures or Truss...................................................... 78

6.4.2 Kinematic Indeterminacy of Rigid-Jointed

Structures.................................................................... 79

6.4.3 Summary of All Formulations for Static and

Kinematic Indeterminacy............................................83

6.5 Principle of Superposition........................................................83

Chapter 7 Plane Trusses and Space Trusses........................................................85

7.1 Introduction..............................................................................85

7.2 Common Types of Trusses.......................................................85

7.3 Classification of Coplanar Trusses...........................................85

7.4 Assumptions on Analysis of Trusses........................................87

7.5 Arrangement of Members of Coplanar Trusses – Internal

Stability....................................................................................89

7.6 Static Determinacy, Indeterminacy, and Instability of

Coplanar Trusses and Their Solution Methods........................90

7.6.1 Method of Joints..........................................................92

7.6.2 Method of Sections......................................................95

7.6.3 Method of Tension Coefficients..................................97

7.6.4 Graphical Method of Truss Analysis........................ 101

7.6.4.1 Bow’s Notation.......................................... 101

7.6.5 Henneberg’s Method of Solution for Complex

Trusses....................................................................... 107

7.7 Compound Trusses................................................................. 111

7.8 Space Trusses......................................................................... 112

7.9 Zero-Force Members of Trusses............................................ 118

Chapter 8 Beams and Frames, Shear, and Bending Moments........................... 121

8.1 Introduction............................................................................ 121

8.2 Axial Force, Shear, and Bending Moments........................... 121

8.3 Shear and Bending Moment Diagrams for a Beam................ 122

8.4 Qualitative Discussion on the Deflected Shape of

Beams..................................................................................... 128

8.5 Relationships between Loads, Shear, and Bending

Moments................................................................................. 130

8.6 Shear and Bending Moment Diagram of Frames................... 136

Chapter 9 Deflections of Beams by Geometric Methods.................................. 143

9.1 Introduction............................................................................ 143

9.2 Deflected Shapes and Elastic Curve....................................... 143

9.3 Double Integration Method.................................................... 146

9.4 Moment-Area Method............................................................ 147

9.5 Conjugate Beam Method........................................................ 150

9.6 Macaulay’s Method................................................................ 154

Chapter 10 Energy Principles and Deflection of Beam....................................... 157

10.1 Introduction............................................................................ 157

10.2 Strain Energy and Pure Bending............................................ 157

10.3 Principle of Virtual Work....................................................... 162

10.3.1 Principle of Virtual Displacements for Rigid

Bodies........................................................................ 163

10.3.2 Principle of Virtual Forces for Deformable

Bodies........................................................................ 166

10.4 Deflection of Trusses by Virtual Work Method..................... 167

10.5 Deflection of Beams by Virtual Work Method...................... 170

10.6 Deflection of Frames by Virtual Work Method..................... 171

10.7 Castigliano’s Theorem............................................................ 173

10.8 Maxwell-Betti Law of Reciprocal Deflections....................... 177

Chapter 11 Rolling Loads and Influence Lines and Their Applications............. 181

11.1 Introduction............................................................................ 181

11.2 Influence Lines for Beams and Frames by Equilibrium

Method.................................................................................... 181

11.3 Qualitative Influence Lines and Müller-Breslau’s

Principle.................................................................................. 188

11.3.1 Müller-Breslau Principle........................................... 188

11.4 Influence Lines for Floor Girders........................................... 190

11.5 Influence Lines for Trusses.................................................... 194

11.6 Maximum Influence at a Point Due to a Series of

11.7 Maximum Influence at a Point Due to a Uniformly

11.8 Absolute Maximum Shear and Moment................................205

11.8.1 Absolute Maximum Bending Moment for

11.9 Influence Lines for Deflections.............................................. 214

Chapter 12 Cables, Arches, and Suspension Bridges.......................................... 217

12.1 Introduction............................................................................ 217

12.2 Cables..................................................................................... 217

12.3 Cables Subjected to Concentrated Loads............................... 217

12.4 Cable Subjected to a Uniformly Distributed Load................. 221

12.5 Arches.....................................................................................226

12.6 Three-Hinged Arches.............................................................227

12.7 Three-Hinged Stiffening Girders........................................... 235

Chapter 13 Analysis of Symmetric Structures.................................................... 243

13.1 Introduction............................................................................ 243

13.2 Symmetric and Antisymmetric Components of

13.3 Symmetric and Antisymmetric Components of

13.4 Behavior of Symmetric Structures under Symmetric

PART III Analysis of Statically

Indeterminate Structures

Chapter 14 Introduction to Statically Indeterminate Structures.......................... 251

14.1 Introduction............................................................................ 251

14.2 Advantages of Indeterminate Structure................................. 252

14.3 Disadvantages of Indeterminate Structure............................. 253

Chapter 15 Approximate Analysis of Statically Indeterminate

Structures.......................................................................................... 255

15.1 Introduction............................................................................ 255

15.2 Assumptions for Approximate Analysis................................ 255

15.2.1 Assumptions about the Location of Points of

Inflection...................................................................256

15.2.2 Assumptions about the Distribution of Forces

and Reactions............................................................ 258

15.3 Vertical Loads on Building Frames....................................... 258

15.4 Lateral Loads on Building Frames: Portal Method................ 259

15.5 Lateral Loads on Building Frames: Cantilever Method.........265

Chapter 16 Method of Consistent Deformations................................................. 271

16.1 Introduction Force Method of Analysis: General

Procedure................................................................................ 271

16.2 Structures with a Single Degree of Indeterminacy................ 271

16.3 Method of Least Work............................................................ 273

16.4 Structures with Multiple Degree of Indeterminacy............... 276

16.4.1 Shear and Bending Moment Diagrams of

Three-Span Continuous Beams................................ 279

16.5 Support Settlements, Temperature Changes, and

Fabrication Errors................................................................... 279

16.5.1 Temperature Changes and Fabrication Errors...........282

Chapter 17 Influence Lines for Statically Indeterminate Structures...................285

17.1 Introduction of Influence Lines for Statically

Indeterminate Structures........................................................285

17.2 Influence Lines for Beams......................................................285

17.3 Influence Lines for Trusses....................................................292

17.4 Qualitative Influence Lines by the Müller-Breslau’s

Principle and Influence Line for Frames................................294

17.5 Alternate Approach for Finding Influence Line Diagrams

for Indeterminate Beams........................................................298

Chapter 18 Slope Deflection Method...................................................................305

18.1 Introduction............................................................................305

18.2 Slope Deflection Equations and Analysis of Continuous

Beams.....................................................................................305

18.3 Members with Far End Hinged.............................................. 315

18.4 Analysis of Frames without Any Sidesway............................ 318

18.5 Analysis of Frames with Sidesway......................................... 323

Chapter 19 Moment Distribution Method............................................................ 327

19.1 Introduction............................................................................ 327

19.2 General Principles and Definitions......................................... 327

19.2.1 Sign Convention........................................................ 327

19.2.2 Fixed-End Moments (FEMs)..................................... 327

19.2.3 Member Stiffness Factors......................................... 328

19.2.4 Joint Stiffness Factor................................................. 329

19.2.5 Distribution Factor (DF)............................................ 331

19.2.6 Member Relative Stiffness Factor............................. 332

19.2.7 Carry over Factor...................................................... 333

19.3 Basic Concept of Moment Distribution Method.................... 333

19.4 Stiffness Factor Modifications................................................ 335

19.4.1 Member Pin Supported at Far End............................ 335

19.5 Analysis of Continuous Beams.............................................. 339

19.6 Analysis of Frames without Sidesway.................................... 341

19.7 Analysis of Frames with Sidesway.........................................344

19.7.1 Multistorey Frames................................................... 350

Chapter 20 Kani’s Method or Rotation Contribution Method............................. 353

20.1 Introduction............................................................................ 353

20.2 Basic Concept......................................................................... 353

20.2.1 Members without Relative Lateral

Displacement............................................................. 353

20.2.2 Members with Relative Lateral Displacement.......... 362

20.3 Analysis of Frames with Sidesway with Vertical

20.4 Analysis of Frames with Sidesway with Vertical

20.5 Analysis of Frames with Columns with Unequal Height....... 387

Chapter 21 Column Analogy Method..................................................................399

21.1 Introduction............................................................................399

21.2 Basic Concept.........................................................................399

21.3 Development of the Column Analogy Method......................402

21.4 Stiffness and Carry over Factors Determined by

Method of Column Analogy...................................................406

21.5 Fixed End Moments due to Support Settlement.....................407

21.6 Analysis of Portal Frames......................................................409

Chapter 22 Beams and Frames Having Nonprismatic Members......................... 413

22.1 Introduction............................................................................ 413

Members................................................................................. 413

22.3 Moment Distribution for Structures Having Nonprismatic

Members................................................................................. 414

22.3.1 Beam Pin Supported at Far End................................ 415

22.3.4 Support Settlement.................................................... 416

22.4 Slope Deflection Equation for Structures Having

Nonprismatic Members.......................................................... 417

22.4.2 Relative Joint Translation.......................................... 417

22.4.3 Rotation at A.............................................................. 418

22.4.4 Rotation at B.............................................................. 418

Chapter 23 Introduction to Matrix Structural Analysis....................................... 419

23.1 Introduction............................................................................ 419

23.2 Analytical Model.................................................................... 419

23.3 Member Stiffness Relations in Local Coordinates

for 2D Truss............................................................................ 420

23.4 Coordinate Transformation for 2D Truss............................... 422

23.5 Displacement Transformation Matrix for 2D Truss............... 423

23.6 Force Transformation Matrix.................................................424

23.7 Member Global Stiffness Matrix for 2D Truss...................... 425

23.8 Application of Stiffness Method for Truss Analysis.............. 428

23.9 Application of Stiffness Method for Space Truss Analysis... 430

23.10 Application of Stiffness Method for Beam Analysis............. 430

23.11 Beam Structure Complete Global Stiffness Matrix............... 432

23.12 Application of Stiffness Method for Frame Analysis............. 434

Chapter 24 Introduction to Plastic Analysis of Structure.................................... 437

24.1 Introduction............................................................................ 437

24.2 Stress-Strain Curve of a Ductile Material.............................. 437

24.3 Plastic Moment....................................................................... 438

24.4 Methods of Analysis...............................................................440

24.4.1 The Maximum Principle, Static Theorem, Lower

Bound Theorem, or Safe Theorem............................444

24.4.2 The Minimum Principle, Kinematc Theorem,

Upper Bound Theorem, or Unsafe Theorem.............444

24.4.3 The Uniqueness Theorem.........................................445

24.5 Static Method for Determining Collapse Load......................445

24.6 Kinematic Method for Determining Collapse Load..............449

24.7 Plastic Analysis of Portal Frames........................................... 452

Appendix A............................................................................................................ 457

Appendix B............................................................................................................ 459

Appendix C............................................................................................................465

Appendix D............................................................................................................ 467

Bibliography..........................................................................................................489

Index....................................................................................................................... 491

### Biography

Debabrata Podder is currently working as an Assistant Professor (CE) at National Institute of Technology Meghalaya, India. He has completed his Engineering education from Jadavpur University and Indian Institute of Technology Kharagpur. After a brief stint as Civil Engineer (S002) at Shapoorji Pallonji & Co. Ltd., Mumbai, he took to academics. He has published one book named as Residual stresses, distortions and their mitigation for fusion welding (ISBN: 978-3-659-94213-6) and papers in several peer reviewed journals. His research interests include welding induced deformations and residual stresses, FEA of structures, theoretical and computational solid mechanics, structural analysis and design. Santanu Chatterjee is a Bachelor of Engineering from Jadavpur University, Kolkata in Construction Engineering and professionally certified from IIT Kanpur on Finite Element Analysis (NPTEL – IIT Kanpur). He has research and industrial experience in the field of Civil/Structural Design Engineering for more than 14 years. He has been also involved in civil design of several solar power plants. He also takes interest in concepts of string theory and quantum mechanics and recently got his paper selected for International Conference on Mathematical Modeling in Physical Sciences (ICM-square 2019 and published in Journal of Physics: Conference Series 1391 (2019) doi:10.1088/1742-6596/1391/1/012066) and one of his papers got awarded as best scientific paper at National Institute of Technology Durgapur (Published in International Journal of Recent Technology and Engineering (IJRTE) ISSN: 2277-3878, Volume-8 Issue-2S7, July 2019), International Conference on renewable energy( ICCARE-2019), India. Presently he is working on development of a calculation tool for transmission line foundation system layout that would enable engineers to draw transmission tower foundation layout with minimal input and compute design quantitates for execution of work.