## Description

This book is specially designed for the undergraduate students of civil engineering and AMIE. The text covers the syllabi requirements of almost all technical universities in India and abroad. A lucid pattern, both in terms of language and content, has been adopted throughout the text. This book will prove to be a boon to the students preparing for engineering, AMIE and other competitive examinations.

## Table of Content

**Chapter 1 OVERVIEW OF STRUCTURAL ANALYSIS**

1.1 Introduction

1.2 Components of Structural Analysis

1.2.1 The Structure

1.2.2 The Method of Analysis

1.2.3 The Forces

1.2.4 The Deformations

1.3 Main Elements of a Structure According to their Form and Function

1.3.1 Tie Rods

1.3.2 Beams

1.3.3 Columns

1.4 Structural System

1.4.1 Trusses

1.4.2 Cables

1.4.3 Arches

1.4.4 Frames

1.4.5 Surface Structures

1.5 Joints

1.5.1 Pin Joint

1.5.2 Rigid Joint

1.6 Loads on Structures

1.6.1 Types of Loads

1.6.2 IS:875, Code of Practice for Structural Safety of Buildings

1.7 Stability of Structures

1.7.1 Types of Instability

1.8 Approaches to Structural Analysis

1.9 Principle of Superposition

1.10 Free Body Diagrams (F.B.D.)

1.11 Linear and Non-Linear Systems

1.12 Frequently Used Terms in Structural Analysis

1.12.1 Degrees of Freedom (DOF)

1.12.2 Statically Determinate Structures

1.12.3 Statically Indeterminate or Hyperstatic or Redundant Structures

1.12.4 Externally Indeterminate Structures

1.12.5 Internally Indeterminate Structures

1.12.6 Structure with Combined Indeterminacies

1.12.7 Kinematic Indeterminacy

1.13 Basic Tools of Analysis

1.13.1 Stress–Strain Relationship of the Material in the Structure

1.13.2 Equations of Static Equilibrium

1.13.3 Conditions of Compatibility

**Chapter 2 EQUATIONS OF STATIC EQUILIBRIUM**

2.1 Introduction

2.2 Notations and Sign Conventions

2.3 Loads

2.3.1 Concentrated Load

2.3.2 Distributed Load

2.3.3 Couple and Moment

2.4 Supports

2.4.1 Supports Providing Reaction with known

Line of Action (Roller Supports)

2.4.2 Supports Providing Reactions with

Unknown Direction (Hinged Supports)

2.4.3 Supports Providing Reactions and a

Couple (Fixed Supports)

**Chapter 3 ANALYSIS OF PLANE TRUSSES UNDER STATIC LOAD **

3.1 Introduction

3.2 Elements of a Truss

3.3 Classification of Coplanar Trusses

3.3.1 Simple Truss

3.3.2 Compound Truss

3.3.3 Complex Truss

3.4 Determinacy and Stability

3.5 Analysis of Simple Trusses

3.6 Method of Joints

3.7 Method of Sections (Ritter’s method)

3.8 Method of Tension Coefficients

3.8.1 Analysis of Plane Truss by the Method of Tension Coefficients

**Chapter 4 STRESS RESULTANT DEVELOPED IN STRUCTURAL MEMBERS**

4.1 Introduction

4.2 Definitions

4.3 Shear Force and Bending Moment

4.4 Relationship Between Load Intensity, Shear Force and Bending Moment

4.4.1 When Concentrated Load Acts on the Beam

4.4.2 When Couple Acts

4.5 Procedure for Analysis

4.6 Shear Force and Bending Moment Diagrams

4.6.1 Cantilever Beam with an End Load

4.6.2 Cantilever Beam of Length L Carrying a Uniformly Distributed Load

of W Per Unit Run Over the Entire Length

4.6.3 Cantilever Beam of Length L Carrying Uniformly Varying Load

4.6.4 Cantilever Beam of Length L Carrying a Uniformly Distributed Load

of W Per Unit Run for a Distance of a from Free End

4.6.5 Simply Supported Beam of Span L Carrying a Concentrated Load

W at Mid-Span

4.6.6 Simply Supported Beam with Concentrated Load Acting at

a Distance from a Left Support

4.6.7 Simply Supported Beam Subjected to Uniformly Distributed Load

4.6.8 Simply Supported Beam Subjected to Several Point Load

4.6.9 Simply Supported Beam Partly Loaded with

Uniformly Distributed Load

4.6.10 Simply Supported Beam with a Triangular Load

4.6.11 Simply Supported Beam Loaded with Gradually Varied Load

4.7 Load and Bending Moment Diagram from a Shear Force Diagram

4.8 Beams Subjected to Couples

4.8.1 Simply Supported Beam Subjected to a Couple at its Mid-Span

**Chapter 5 ANALYSIS OF BEAMS UNDER ROLLING LOADS**

5.1 Introduction

5.2 Classification of Moving Loads

5.3 Sign Convention

5.3.1 Shear Force

5.3.2 Bending Moment

5.4 A Single Concentrated or Point Load W Moving from

Left End to the Right End of the Beam

5.4.1 To Find Support Reactions, RA and RB

5.4.2 Negative Shear Force

5.4.3 Positive Shear Force

5.4.4 Bending Moment

5.5 Uniformly Distributed Rolling Load Longer than the

Span of the Girder Moving from Left End to Right End

5.5.1 Maximum negative shear force at section X

5.5.2 Maximum positive shear force at section X

5.5.3 Maximum Bending Moment

5.5.4 Absolute Maximum Bending Moment

5.6 Uniformly Distributed Rolling Load Shorter than the Span of the

Simply Supported Beam Moving from Left End to Right End

5.6.1 Maximum Negative Shear Force

5.6.2 Maximum Positive Shear Force

5.6.3 Bending Moment

5.6.4 Absolute Maximum Bending Moment

5.7 Two Concentrated Loads with a Specified Distance Between Them 406

5.7.1 Maximum Negative Shear Force

5.7.2 Maximum Positive Shear Force

5.7.3 Bending Moment

5.7.4 Absolute Maximum Bending Moment

5.8 Multiple Concentrated Loads / Several Concentrated

Loads Rolling from Left End to Right End

5.8.1 Proposition for the Maximum Bending Moment Under any given Load

5.8.2 Proposition for the Maximum Bending Moment

at any given Section on the span

5.9 Equivalent Uniformly Distributed Load (EUDL)

**Chapter 6 INFLUENCE LINES FOR BEAMS **

6.1 Introduction

6.2 Influence Lines for Beam Reactions

6.2.1 Influence Lines for Simple Beam Reactions

6.2.2 Influence Line for Reaction in a Cantilever Beam

6.2.3 Influence Lines for Reactions for a Beam with Overhang on One Side

6.2.4 Influence Lines for Reactions for a Beam with Overhang on Both Sides

6.3 Influence Lines for Internal Shears

6.3.1 Influence Lines for Simply Supported Beam Shear

6.3.2 Influence Lines for Cantilever Beam Shear

6.3.3 Influence Line for Shear Force for a Beam with Overhang

6.4 I.L.D. for Bending Moment at a Particular Section in a Simply Supported Beam

6.5 Series of Concentrated Loads

6.6 U.D.L. Shorter than the Span

6.6.1 Bending Moment

6.7 U.D.L. Longer than the Span of the Girder

6.7.1 Bending Moment

6.8 Influence Lines for Two Concentrated Loads

6.8.1 Shear Force

6.8.2 Bending Moment

6.9 Several Point Loads (Wheel Load Train) Moving on the Girder

6.9.1 Maximum Bending Moment at a given Cross Section

6.9.2 Absolute Maximum Bending Moment

6.10 Muller–Breslau’s Principle

6.10.1 Statement

6.10.2 Simply Supported Beam

6.10.3 Infl uence Line for VA

6.10.4 Infl uence Line for VB

6.10.5 Infl uence Line for MX

6.10.6 Infl uence line for VX

**Chapter 7 INFLUENCE LINES FOR TRUSSES**

7.1 Introduction

7.2 Influence Line for a Pratt Truss with Parallel Chords

7.2.1 Influence Line for the Force in the Member L_1 U_2

7.2.2 Influence Line for the Force in the Member U_2 U_3

7.2.3 Influence Line for the Force in the Member U_3 U_4

7.2.4 Influence Line for the Force in the Member L_1 L_2

7.2.5 Influence Line for the Force in the Member L_2 L_3

7.2.6 Influence Line for the Force in the Member L_3 L_4

7.2.7 Influence Line for the Force in the Member U_2 L_2

7.2.8 Influence Line for the Force in the Member U_2 L_3

7.2.9 Influence Line for the Force in the Member U_3 L_4

7.2.10 Influence Line for the Force in the Member U_3 L_3

7.2.11 Influence Line for the Force in the Member U_4 L_4

7.3 Influence Line for a Deck Type Warren Truss

7.3.1 Influence Line for the Force in the Member U_1 U_2

7.3.2 Influence Line for the Force in the Member U_2 U_2

7.3.3 Influence Line for the Force in the Member U_1 L_1

7.3.4 Influence Line for the Force in the Member L_1 L_2

7.3.5 Influence Line for the Force in the Member L_2 L_3

7.3.6 Influence Line for the Force in the Member L_1 U_2

7.3.7 Influence Line for the Force in the Member L_2 U_3

7.3.8 Influence Line for the Force in the Member L_2 U_2

7.4 Influence Lines for an Inclined Pratt Truss

7.4.1 Influence Line for the Force in the Member L_1 U_2

7.4.2 Influence Line for the Force in the Member U_2 U_3

7.4.3 Influence Line for the Force in the Member L_1 L_2

7.4.4 Influence Line for the Force in the Member L_2 L_3

7.4.5 Influence Line for the Force in the Member U_2 L_2

7.4.6 Influence Line for the Force in the Member U_2 L_3

7.4.7 Influence Line for the Force in the Member U_3 L_3

**Chapter 8 THREE-HINGED ARCHES**

8.1 Introduction

8.2 Types of Arches

8.3 Linear Arch or Theoretical Arch (Eddy’s Theorem)

8.3.1 Statement of Eddy’s Theorem

8.3.2 Note on Use of Eddy’s Theorem

8.4 Analysis of Three-Hinged Arches

8.4.1 Advantage of Three-hinged Arch Over Simply Supported Beam

8.5 Normal Thrust and Radial Shear

8.6 Types of Three-Hinged Arches

8.6.1 Parabolic Arches

8.6.2 Segmental or Circular Arches

8.7 B.M.D. for Three-Hinged Arches

8.8 Arch Versus Beam

8.9 Straining Actions in a Three-Hinged Arch

8.10 Springing Supports at Different Levels

8.11 Effect of Temperature on a Three-Hinged Arch

8.12 Influence Lines for a Concentrated Load

Moving Over a Three-Hinged Circular Arch

8.12.1 Influence Line for Horizontal Thrust

8.12.2 Influence Lines for Bending Moment

8.12.3 Influence Lines for Normal Thrust

8.12.4 Influence Lines for Radial Shear Force

8.13 Influence Lines for a Concentrated Load

Moving Over a Three-Hinged Parabolic Arch

8.14 Influence Lines for a Uniformly Distributed Load Moving Over a

Three-Hinged Parabolic Arch

8.14.1 To Find the Distance AF

**Chapter 9 CABLES AND SUSPENSION BRIDGES**

9.1 Introduction—Cables

9.2 Elements of Suspension Cables

9.3 Cable with Uniformly Distributed Load

9.4 Tension in the Cable Carrying Uniformly Distributed Load

9.5 Length of the Cable when Supports are at the Same Level

9.6 Cables with Point (Concentrated Loads) Loads

9.7 Tension in the Cable Supported at Different Levels

9.8 Length of the Cable when the Supports are at Different Levels

9.9 The Catenary Cable

9.10 Introduction—Suspension Bridge

9.11 Forces on Anchor Cables and Towers

9.11.1 Roller Support

9.11.2 Guided Pulley Support

9.12 Suspension Bridges with Three-Hinged Stiffening Girder

9.12.1 Analysis of Three-hinged Stiffening Girder

9.13 Influence Lines for Moving Loads Over the

Suspension Bridge with Three-Hinged Stiffening Girder

9.13.1 Influence Lines for a Single Concentrated Load Rolling Over

the Suspension Bridges with Three-Hinged Girders

**Chapter 10 DEFLECTION—ELEMENTARY METHOD**

10.1 Introduction

10.2 Differential Equation of Flexure

10.2.1 Sign Convention

10.3 Assumptions made in Deriving Differential Equation

10.4 Relationship among Shear Force, Bending Moment and Deflection

10.5 Introduction—Double Integration Method

10.5.1 Cantilever Beam Subjected to a

Single Concentrated Load at its Free End

10.5.2 Cantilever Beam of Length L Carrying load P

at a Distance from the Fixed End

10.5.3 Cantilever Beam Loaded with Uniformly

Distributed Load Throughout the Length of the Beam

10.5.4 Cantilever Beam Subjected to Moment at Free End

10.5.5 Cantilever Beam Subjected to Load Varying Linearly from

Zero at Free End to w/unit Length at Fixed End

10.5.6 Simply Supported Beam Carrying a Point Load at the Centre

10.5.7 Simply Supported Beam Subjected to

Uniformly Distributed Load Throughout its Span

10.5.8 A Simply Supported Beam Subjected to a Load Varying

Linearly from Zero at One End to w/unit Run at the Other End

10.5.9 Cantilever Beam Subjected to Uniformly

Distributed Load for a Distance from Fixed End

10.5.10 Cantilever Beam Subjected to Uniformly

Distributed Load for a Distance a from Free End

10.6 Introduction— Macaulay’s Method

10.6.1 Deflection of a Simply Supported Beam

with Point Load at the Centre of the Beam

10.6.2 Simply Supported Beam Subjected to an Eccentric Point Load

**Chapter 11 DEFLECTION—GEOMETRICAL METHODS**

11.1 Introduction

11.2 Moment-Area Method

11.2.1 Moment-Area Theorem

11.2.2 Proof of the Moment-Area Theorems

11.3 Basic Properties of Parabolic Curves

11.4 Conjugate Beam Method

11.4.1 Conjugate–Beam Supports

11.4.2 Sign Convention

11.4.3 Properties of Conjugate Beam

11.4.4 Actual Beam Versus Conjugate Beam

11.4.5 Procedure for Analysis

11.5 Singularity-Function Method

11.5.1 Properties of Singularity-Function

11.5.2 Integration and Differentiation of Singularity-Function

**Chapter 12 DEFLECTION—ENERGY METHOD**

12.1 Introduction

12.2 Strain Energy in Linear and Non-Linear Elastic Systems

12.2.1 Expression for Strain Energy and

Complimentary Strain Energy in Linear System

12.2.2 Expression for Strain Energy and Complimentary Strain

Energy in Non-Linear Elastic Material

12.3 Expression for Strain Energy Due to Axial Force

12.4 Strain Energy Due to Bending

12.4.1 Strain Energy Stored by a Beam Subjected to a

Uniform Bending Moment

12.5 Maxwell’s Reciprocal Deflection Theorem

12.6 Betti’s Theorem

12.7 Theorem of Minimum Potential Energy

12.8 Principle of Virtual Work

12.8.1 Principle of Virtual Forces

12.8.2 Principle of Virtual Displacements

12.9 The Crotti-Engesser Theorem

12.10 Castigliano’s Theorem

12.10.1 Castigliano’s First Theorem

(For Forces in an Elastic Structure)

12.10.2 Castigliano’s Second Theorem

(For Displacements in a Linearly Elastic Structure)

12.10.3 Alternative Derivation for Castigliano’s Theorem

12.10.4 General Equation of Strain Energy

12.10.5 To Find the Deflection at a Point Where Load is Not Acting

**Chapter 13 UNIT -VIRTUAL LOAD METHOD FOR BEAMS AND FRAMES**

13.1 Introduction

13.2 Unit-Virtual Load Method

13.2.1 Application of Unit-Virtual Load Method to Beams and Frames

**Chapter 14 UNIT -LOAD METHOD FOR TRUSSES**

14.1 Introduction

14.2 Unit-Load Method

14.2.1 Solution by the Method of Joints

14.2.2 Following Examples are Worked Out

using Tension Coefficient Method

14.3 Deflection Due to Lack of Fit and Temperature Changes

## About The Author

**P N Chandramouli** is Professor, Department of Civil Engineering, The National Institute of Engineering, Mysore. He received his B.E in Civil Engineering from University of Mysore, M.E from Indian Institute of Science, Bangalore and Ph.D from Indian Institute of Technology, Roorkee. He has over 30 years of teaching experience at The National Institute of Engineering. He is a life member of ISTE and ACCE.

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