## Description

This book is specially designed for the course on “Strength of Materials II” for Civil engineering. All the concepts have been explained in a simple manner and problems are solved in a step-by-step procedure. A large number of illustrative examples have been solved, for better understanding of basic principles and fundamental concepts. This book is written in a simple and logical sequence.

## Table of Content

**Chapter 1 Energy Principles**

1.1 Strain Energy

1.2 Resilience

1.3 Proof Resilience

1.4 Modulus of Resilience

1.5 Stresses Due to Different Types of Loads

1.5.1 Expression for Strain Energy Stored in a Body, When the Load is Applied Gradually

1.5.2 Expression for Strain Energy Stored in the Body, When the Load is Applied Suddenly

1.5.3 Expression for Strain Energy Stored in the Body, When the Load is

Applied with Impact

1.6 Strain Energy Due to Shear

1.7 Strain Energy Due to Torsion

1.8 Strain Energy Due to Bending

1.9 Strain Energy Due to Shear in a Beam

1.10 Castigliano’s Theorems

1.10.1 Determination of Deflection and Slope in Beams Using Castigliano’s

Theorems

1.10.2 Determination of Deflection in Trusses

1.11 Maxwell’s Reciprocal Theorem or Law of Reciprocal Deflection

1.12 Principe of Virtual Work

1.12.1 Virtual Work and Complementary Virtual Work

1.12.2 Principle of Virtual Work and Complementary Virtual Work

1.12.3 Virtual Work-Unit Load Method

1.12.4 Determination of Deflection and Rotations in Beams

1.12.5 Determination of Deflection and Rotation in Trusses

1.13 Williot Mohr’s Diagram

**Chapter 2 Indeterminate Beams**

2.1 Beams

2.2 Types of Supports

2.3 Classification of Beams

2.3.1 Equations of Equillibrium

2.3.2 Statically Determinate Beams

2.3.3 Statically Indeterminate Beams

2.3.4 Degree of Static Indeterminacy

2.4 Classification of Structures

2.5 Types of Loads

2.6 Shear Force

2.7 Bending Moment

2.8 Analysis of Indeterminate Beams

2.8.1 Propped Cantilever Beam

2.8.2 Analysis of Propped Cantilever Beam

2.8.3 Maximum Deflection Due to Different Loading Conditions in Cantilever

Beams

2.8.4 Propped Cantilever Beam with Central Load

2.8.5 Propped Cantilever Beam with Uniformly Distributed Load Over the

Entire Span

2.9 Fixed Beam

2.9.1 Methods of Analysis of a Fixed Beam

2.9.2 Determination of Deflection

2.9.3 Fixed Beam Carrying a Point Load at Mid span

2.9.4 Fixed Beam Carrying a Non-Central Concentrated Load

2.9.5 Fixed Beam Carrying Uniformly Distributed Load

2.9.6 Fixed Beam Subjected to a Non-Central Couple M

2.9.7 Fixed Beam Carrying Uniformly Varying Load, Whose Intensity Varies

from Zero at one End to w Per Unit Run at the Other End

2.9.8 Fixed Beam Carrying Uniformly Varying Load for a Given Distance

From One End

2.9.9 Fixed Beam with Various Loading Conditions

2.9.10 Fixed Beams with Sinking of Support

2.10 Continuous Beam

2.10.1 Clapeyron’s Theorem of Three Moments

2.10.2 Continuous Beams with Overhangs

2.10.3 Continuous Beams with Varying Cross Section

2.10.4 Continuous Beams with Sinking of Supports

2.10.5 Propped Cantilever Beams-Theorem of Three Moments

**Chapter 3 Columns and Cylinder**

3.1 Column or Strut

3.1.1 Slenderness Ratio

3.1.2 Buckling Load

3.1.3 Buckling Load Factor

3.1.4 Classification of Columns

3.1.5 Strength of Columns

3.1.6 End Conditions

3.1.7 Equivalent Length (le)

3.1.8 Failure of a Column

3.1.9 Critical Load/Crippling Load/Buckling Load (Pcr)

3.2 Euler’s Column Theory for Long Column

3.2.1 Assumption in the Euler’s Column Theory

3.2.2 Sign Convention for Bending Moment

3.2.3 Euler’s Formula

3.2.4 Limitations for the Use of Euler’s Formula

3.2.5 Applicability of Euler’s Theory

3.3 Derivation of Euler’s Formula (For Different End Conditions)

3.3.1 When Both Ends of the Column are Hinged or Pinned

3.3.2 When One End of the Column is Fixed and Other End is Free

3.3.3 When Both Ends of the Column are Fixed

3.3.4 When One End of the Column is Fixed and the Other End is Hinged

3.3.5 Buckling Load for Various Cases

3.4 Rankine’s Hypothesis

3.5 Long Column Subjected to Eccentric Loading

3.6 Short Column

3.6.1 Centrally Loaded Short Column

3.6.2 Eccentrically Loaded Short Column

3.6.3 Condition for No Tension in the Section

3.6.4 Middle Third Rule [Core or Kernel of the Section]

3.6.5 Determination of Limit of Eccentricity for No Tension

3.7 Column Subjected to Biaxial Eccentric Loading

3.7.1 Symmetrical Column with Eccentric Loading About Both XX and Y Y

Axis

3.8 Thin Cylindrical and Spherical Shells

3.8.1 Stresses in Thin Cylinder

3.8.2 Change in Dimension of a Thin Cylinder Subjected to Internal Pressure

3.8.3 Maximum Shear Stress

3.8.4 A Thin Cylinder Subjected to Internal Fluid Pressure and Torque

3.8.5 Efficiency of a Joint

3.9 Thin Spherical Shells

3.9.1 Change in Dimensions of a Thin Spherical Shell Subjected to Internal

Pressure

3.10 Wire Wound Thin Cylindrical Shells

3.11 Thick Cylindrical and Spherical Shells

3.11.1 Thick Cylindrical Shells

3.11.2 Assumptions in Lame’s Theory

3.11.3 Stresses in a Cylindrical Shell

3.12 Compound Cylinders

3.12.1 Difference in Radii Due to Shrink Fitting

3.13 Thick Spherical Shells

**Chapter 4 State of Stress in Three Dimensions**

4.1 Stress

4.1.1 Types of Stresses

4.1.2 Uniaxial State of Stress

4.1.3 Biaxial State of Stress

4.1.4 Triaxial State of Stress

4.1.5 Direct and Shear Stresses

4.1.6 Notation of Stresses

4.1.7 Sign Convention

4.2 Stress Tensor

4.2.1 Symmetry of Stress Tensor

4.2.2 Spherical and Deviatoric Stress Tensor

4.3 Strain Tensor

4.3.1 Spherical and Deviatoric Strain Tensor

4.3.2 Volumetric Strain

4.4 The State of Stress at a Point

4.4.1 Stress Transformation

4.5 Principal Stresses and Principal Planes

4.5.1 Invariants of Stress Tensor

4.5.2 Determination of Principal Stresses and Principal Planes

4.6 Principal Planes

4.6.1 Determination of Direction Cosines

4.6.2 Determination of Direction Cosines using Matrix Method

4.7 Theories of Failure

4.7.1 Maximum Principal Stress Theory (Rankine’s Theory)

4.7.2 Maximum shear stress theory (Coulomb’s Theory or Guest’s Theory)

4.7.3 Maximum Principal Strain Theory (St. Venant’s Theory)

4.7.4 Maximum Strain Energy Theory (Haigh’s Theory)

4.7.5 Maximum Shear Strain Energy Theory or Distortion Energy Theory

(Von Mises−HenckyTheory)

4.8 Octahedral Plane

4.8.1 Octahedral Stresses

4.8.2 Octahedral Shearing Stress Theory

4.9 Significance of Theories of Failure

**Chapter 5 Advanced Topics in Bending of Beams**

5.1 Unsymmetrical Bending

5.2 Moment of Inertia

5.3 Product of Inertia

5.3.1 Sign Convention

5.4 Principal Axes

5.4.1 Principal Moment of Inertia

5.4.2 Determination of Principal Moment of Inertia

5.5 Stresses Due to Unsymmetrical Bending

5.6 Deflection of Beam Due to Unsymmetrical Bending

5.7 Shear Centre

5.7.1 Shear Flow

5.7.2 Bending Axis and Shear Centre

5.7.3 Shear Centre for Channel Section

5.7.4 Shear Centre for Unequal I-section

5.8 Bending of Curved Beams

5.8.1 Stresses in Curved Beams

5.9 Position of Neutral Axis for Various Sections

5.9.1 Rectangular Section

5.9.2 Trapezoidal Section

5.9.3 Triangular Section

5.9.4 Inverted T−Section

5.9.5 I−Section

5.9.6 Circular Section

5.10 Stresses in Crane Hooks

## About The Authors

**Dr. Binu Sukumar** is Professor & Head, Department of Civil Engineering, R.M.K. Engineering College, Chennai. She has more than 25 years of experience in teaching. She obtained her B.Tech in Civil Engineering from Government College of Engineering, Trivandrum, M.Tech in Structural Engineering from Indian Institute of Technology, Madras and Ph.D. from Anna University, Chennai. She is a member of the Editorial Board for International Journal of Advances in Engineering Research & International Journal of Research in Science and Technology and Reviewer for many international journals published by Elsevier and Springer. She is a recognized Research Supervisor of Anna University.

**P Joyson Silva **is Assistant Professor, Department of Civil Engineering, R.M.K. Engineering College, Chennai. He has obtained his M.Tech in Structural Engineering from NIT Trichy. He has published many papers in International and National Journals. He is currently pursuing Ph.D program at Anna University, Chennai

**V Sherin **is Assistant Professor, Department of Civil Engineering, KCG College of Technology, Chennai. She obtained her Masters in Structural Engineering securing the Gold Medal from Mepco Schlenk Engineering College (Autonomous), Tamil Nadu.

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