**Fundamentals of Strength of Materials**

Dr. Debabrata Nag, Dr. Abhijit Chanda

**I****SBN: 13: 978-81-265-2286-6**

**Publication Date: July 2010/ ****Price: Rs 379/ ****Pages: 788/-**

**Key Features**

- Complete syllabi coverage of all leading universities of various engineering disciplines like mechanical, civil, electrical, aeronautical, chemical, metallurgy.
- Topics explored and elaborated for both elementary as well as advanced levels.
- Self-explanatory figures with liberal use of free-body diagrams to aid easy understanding.
- Well-graded solved examples from easy to difficult levels in each chapter to explain the subjective intricacies and problem-solving tactics.
- Last 5 years’ questions from various university examinations included at the end of all chapters.
- Model question papers for giving scope of mock tests appended at the end of the book.

**Appendices includes:**

– Deliberation on the topic of area moment of inertia.

– Summarised results of beam deflections for various beam configurations.

– Various symbols with their respective units and brief explanation on the various systems of units.

– Elaboration on the topic of pure bending and quick calculations for area under parabolas.

**Excellent pedagogy includes**:

ü 660+ illustrations.

ü 140+ review questions.

ü 230+ solved examples.

ü 260+ unsolved problems.

**CD contains:**

- Three useful chapters containing some special topics on leaf springs, beams of composite materials and continuous beams in form of Chapters 17, 18 and 19.
- History of the subject and its progress through various centuries.
- Lab manual containing some important experiments with detailed theory and illustrations.
- Last 10 years’ IES and GATE completely solved questions with explanatory answers.

**Uses of the Book**

- Helpful for the university students and also practicing engineers working in the industries for reference.
- Serves as a bridging subject for the applied subjects like Machine Design and Theory of Structures.
- Serves as the basic background for the more advanced-level subjects like Theory of Elasticity, Stress and Deformation Analysis or Advanced Mechanics of Solids.

**About the book **

This book covers one of the most fundamental subjects of Engineering discipline – Strength of Materials, also known as Mechanics of Materials, Mechanics of Deformable Bodies or Mechanics of Solids globally. The subject lays the ground for various Engineering subjects, ranging from Machine Design, Finite-Element Analysis, Theory of Structures, Bio-Mechanics, and Fracture Mechanics.

In this book, the topics are broadly divided into two parts: Elementary Strength of Materials and Advanced Strength of Materials, thereby progressing from basic fundamentals to detailed analysis. The first eight chapters deal with basic concepts of strengths of materials such as theories of stress and strain, torsion, deflection and buckling of columns. The remaining chapters deal with the advanced topics such as advanced theories of stress and strain, energy principles, failure theories, theories of curved and continuous beams, unsymmetric or asymmetric bending.

**Elementary Strength of Materials**

In this part, we describe the fundamentals of the subject and have carefully included those topics which are generally taught in the various engineering disciplines ranging from Mechanical, Civil to Aeronautical, Chemical, Metallurgical engineering departments in their sophomore levels. Chapters 1–8 form a part of this level.

It may seem that the above package of topics covers mostly the course curriculum requirements of the undergraduate Strength of Materials generally followed by the various institutes for various engineering disciplines across the country. However, disciplines like Mechanical, Electrical, Aeronautical, Civil, Metallurgy may have some additional topics in their respective curriculum. For example, certain institutes have Analysis of Trusses also in their syllabus of Strength of Materials. However, we feel that it would be more justified and logical to include it in the course curriculum of Engineering Mechanics as has been the age-old practice and followed by the various engineering institutes. Consequently, we have not included this topic in our present work.

**Advanced Strength of Materials**

In this part, we have endeavoured to include those topics which require a slight maturity in the subject. Chapters 9–19 form a part of this level.

The topics are meant for more serious and matured reading of the subject. Some of these topics can be well-included in the undergraduate curriculum of the subject for those engineering disciplines like Mechanical, Aeronautical, Civil, Chemical and Metallurgy. Also, the authors believe that the coverage of the above topics will be useful for those students who are willing to study this subject in more advanced level.

**Common Elements in the Two Parts**

In both these levels we placed a common chapter – Stress–Strain analysis. This topic has been discussed from its fundamental level in the most elementary way and from the standpoint of more advanced tensorial theory. We sincerely believe that after going through this chapter a student wishing to pursue the subject in his/her engineering courses will have a modest introduction to the more advanced level subjects such as Theory of Elasticity, Theory of Stress and Deformation Analysis, etc. We have intentionally kept the topic of strain energy in the advanced level as we integrated many energy-related theorems along with it. This topic, we believe, shall certainly help students when they study Finite Element Methods or Matrix approach of Structural Analysis.

In both levels, all the chapters have quite a large number of numerical examples completely solved to bring out the intricacies of the relevant topics.

** **

**Bibliography and Appendices**

At the end of the book, we have appended Bibliography section, where we have included a list of references. Also, we have compiled a set of appendices containing:

- Short deliberation on the topic of area moment of inertia as we frequently need this concept in our present studies. The topic in its complete detail can be found in any standard book on Engineering Mechanics.
- Summarised results of beam deflections for various beam configurations for easy reference.
- Summarised table containing the different symbols and their units used in the book supported by a brief discussion on various systems of units.
- Quick calculation of area under parabola and deliberation on the topic of pure bending.

**To the Readers of the Book**

For the benefit of the students and in order to test their understanding of the subject, we have added a series of Model Question Papers at the end of the book. Each setup is a complete one and consists of standard questions from the relevant chapters. Attempting to answer these questions as mock tests will definitely help them in preparing themselves for any examination.

Moreover, the book is accompanied by a CD which contains:

- Some special topics covering a short deliberation on the stress analysis of leaf springs, beams of composite materials and continuous beams provided in form of Chapters 17, 18 and 19.
- History of the subject and its progress through various centuries.
- Lab manual containing some important experiments with detailed theory and illustrations:
- Tensile test for mild steel rod
- Torsion test for mild steel solid shaft
- Hardness test
- Izod impact test for metals and alloys
- Last 10 year’s questions of Indian Engineering Services (IES) examination and GATE including their solutions, completely explained. The authors sincerely believe that this additional study material will help the readers to get acquainted with the standards of questions that are usually followed in such examinations and the model answers for them.

**About the Authors**

**Dr. Debabrata Nag**, a graduate in Mechanical Engineering from Jadavpur University, is presently designated as the Reader in the Department of Mechanical Engineering in Applied Mechanics specialisation of his alma mater. He has over 7 years of teaching experience both in Undergraduate and Postgraduate levels and over 12 years of industrial experience in finite element stress analysis of industrial piping systems. Credited with a number of research papers in various International journals, his research interest includes areas of numerical modeling of non-Newtonian fluids, biological fluids, mathematical theories of mechanical vibration, theory of elasticity and dynamics of engineering systems. Dr. Nag has also co-authored the book “Fundamentals of Engineering Mechanics”, published by Scholar Books, Kolkata with Dr. Abhijit Chanda.

**Dr. Abhijit Chanda**, a graduate in Mechanical Engineering from Jadavpur University, is presently designated as the Reader in the Department of Mechanical Engineering in Applied Mechanics specialization of his alma mater. His teaching experience spreads over 9 years both in Undergraduate and Postgraduate levels. Dr. Chanda was previously associated with Research Institute and had a brief industrial experience also. Having been a “Young-Scientist” award winner of DST, Dr. Chanda handled number of research projects and papers in various International and National level journals. His research interest includes Material Science, Mechanical Behaviour of Materials, Bio-Materials, etc. He is also the Joint Director of the School of Bio-Engineering of Jadavpur University. Dr. Chanda has co-authored the book “Fundamentals of Engineering Mechanics”, published by Scholar Books, Kolkata with Dr. Debabrata Nag.

**Table of Contents **

Preface

**Part A Elementary Strength of Materials**

**1. Stress and Strain**

1.1 Stress

Average Normal Stress

Average Shear Stress

Stresses on Inclined Plane

1.2 Strain

Normal Strain

Shear Strain

1.3 Relationship between Stress and Strain

Generalised Hooke’s Law

Relationship between Different Elastic Moduli

Working Stress and Factor of Safety

1.4 Statically Indeterminate Systems

1.5 Thermal Stress

1.6 Stress Concentration

Summary

Key Terms

Review Questions

Numerical Problems

Answers

**2. Torsion**

2.1 Basic Equations

Torsion of Thin Tubes

Torsion of Solid Non-circular Shafts

2.2 Power Transmission

2.3 Failure due to Torsion

2.4 Close-Coiled Helical Spring

Summary

Key Terms

Review Questions

Numerical Problems

Answers

**3. Thin-Walled Pressure Vessels**

3.1 Governing Equation

3.2 Special Cases

Cylindrical Pressure Vessel

Spherical Pressure Vessel

Conical Pressure Vessel

3.3 Deformation Analysis of Thin-Walled Pressure Vessels

Summary

Key Terms

Review Questions

Numerical Problems

Answers

**4. Biaxial Stresses**

4.1 Fundamental Equations: Derivation and Discussion

4.2 Mohr’s Circle for Biaxial Stress

4.3 General Biaxial Stress Situation

4.4 Graphical Representation by Mohr’s Circle of Stresses

4.5 Principal Stresses and Principal Planes

Maximum Shear Stress

4.6 Steps for Drawing the Mohr’s Circle

Summary

Key Terms

Review Questions

Numerical Problems

Answers

**5. Shear Force and Bending Moment of Beams**

5.1 Relationship between Shear Force and Bending Moment

5.2 Fundamental Equations of Shear Force and Bending Moment

5.3 Alternate Method for Finding Shear Force and Bending Moment

Summary

Key Terms

Review Questions

Numerical Problems

Answers

**6. Stresses in Beams**

6.1 Bending of Beams

6.2 Governing Equations for Bending Stress

6.3 Governing Equation for Shear Stress

Summary

Key Terms

Review Questions

Numerical Problems

Answers

**7. Deflection of Beams**

7.1 Derivation of Differential Equation of Elastic Line or Elastica

7.2 Methods for Solving Differential Equation of Elastic Line

Double Integration Method

Another Form of Deflection Equation

7.3 Moment-Area Method or Mohr’s Theorems

7.4 Discontinuity Functions

Macaulay’s Function

7.5 Effect of Shear Force on Beam Deflection

Summary

Key Terms

Review Questions

Numerical Problems

Answers

**8. Buckling of Columns**

8.1 Buckling: Elastic Instability

8.2 Derivation of Expressions for Critical Load

Pinned–Pinned or Pin–Ended Column

Fixed–Free Column

Fixed–Fixed Column

8.3 Euler’s Curve

8.4 Eccentric Loading: The Secant Formula

8.5 Columns with Initial Curvature

8.6 Empirical Column Formulas

Summary

Key Terms

Review Questions

Numerical Problems

Answers

Part B Advanced Strength of Materials

**9. Analysis of Stress and Strain**

9.1 Ideas of Stress at a Point

9.2 Equations of Equilibrium and Symmetry of Stress Matrix

9.3 Stress Transformation Equation

Principal Stresses – Diagonalisation of Stress Matrix, [sij ]

Deviatoric Stress Matrix, Hydrostatic Stress

Octahedral Plane – Octahedral Shear Stress

9.4 Plane-Stress Formulation

9.5 Graphical Representation

9.6 Analysis of Strain

Strain at a Point

Shear Strain

9.7 Deformation Geometry

9.8 Plane-Strain Condition

9.9 Strain-Compatibility Relations

9.10 Strain Components in x–y Plane

9.11 Stress–Strain Relationship based on Material Behaviours

9.12 Different Material Behaviours

9.13 Stress–Strain Relations of Hookean Materials

9.14 Other Stress–Strain Relations

Perfectly Elastic Material

Rigid Perfectly Plastic Material

Perfectly Elastic–Plastic Material

Viscoelastic (Linear) Materials

Summary

Key Terms

Review Questions

Numerical Problems

Answers

**10. Energy Principles**

10.1 Concept of Strain Energy

Strain Energy due to Uniaxial Tension/Compression

Strain Energy due to Shear

Strain Energy due to Bending

10.2 Complementary Strain Energy

10.3 Energy-Related Theorems

Virtual Work Theorem

Total Potential Energy Theorem

Castigliano’s Theorem I

Complementary Virtual Work Theorem

Total Complementary Potential Energy Theorem

Crotti–Engesser and Castigliano’s Second Theorem

Maxwell–Betti’s Reciprocity Theorem

10.4 Closely Coiled Helical Spring – Revisited

10.5 Open-Coiled Spring

Summary

Key Terms

Review Questions

Numerical Problems

Answers

**11. Theories of Failure**

11.1 Failure of Materials

11.2 Failure Theories of Ductile Materials

Maximum Principal Stress Criterion (Rankine, Lame’s Theory)

Maximum Principal Strain Theory (Saint Venant’s Theory)

Maximum Strain Energy Theory (Beltrami–Haigh’s Theory)

11.3 More Accurate Yielding Criteria of a Ductile Material

Maximum Shear Stress Theory (Tresca–Guest and Coulomb’s Theory)

Maximum Distortional Energy Density Theory or Maximum Octahedral Shear Stress Theory (von-Mises–Maxwell–Huber–Henky’s Theory)

11.4 Failure Theories of Brittle Materials

Mohr–Coulomb Theory

Modified Mohr–Coulomb Theory

11.5 Concluding Remarks

11.6 Assumptions

Summary

Key Terms

Review Questions

Numerical Problems

Answers

**12. Combined Loadings**

12.1 Axial Load and Torsion

12.2 Axial Load and Bending

12.3 Bending and Twisting

Summary

Key Terms

Review Questions

Numerical Problems

Answers

**13. Unsymmetric Bending of Beam**

13.1 Unsymmetric Bending

Bending about a Principal Axis

13.2 Bending about Arbitrary Axis

13.3 Concluding Remarks

Summary

Key Terms

Review Questions

Numerical Problems

Answers

**14. Shear Stresses in Thin-walled beams**

14.1 Shear Stress in Symmetric Beams with Thin-Walled Open Sections

14.2 Shear Stress Distribution in Thin-Walled Asymmetric Open Sections

Summary

Key Terms

Review Questions

Numerical Problems

Answers

**15. Axisymmetric Problems in Strength of Materials**

15.1 Mathematical Preliminaries

15.2 Thick Cylinder Pressure Vessels

Stress Equations for Thick Cylinder

Special Cases

Compound Cylinder

15.3 Rotating Disc with Constant Thickness

15.4 Rotating Disc with Variable Thickness

Rotating Disc of Uniform Strength

15.5 Concluding Remarks

Summary

Key Terms

Review Questions

Numerical Problems

Answers

**16. Curved Beam Theory**

16.1 Theory of Curved Beams

16.2 Radial Stresses in Curved Beam

16.3 Concluding Remarks

Summary

Key Terms

Review Questions

Numerical Problems

Answers

**17. Leaf Springs**

17.1 Beams of Uniform Strength

17.2 Deflection of Beam of Uniform Strength

17.3 Leaf Spring

Stress Deformation Analysis for Leaf Springs

Summary

Key Terms

Review Questions

Numerical Problems

Answers

**18. Beams of Composite Materials**

18.1 Bending Stress in a Composite Beam

18.2 Reinforced Concrete Beam

Summary

Key Terms

Review Questions

Numerical Problems

Answers

**19. Statically Indeterminate Beams – Continuous Beams**

19.1 Analysis of Continuous Beams

Second Area-Moment Theorem

19.2 Three-Moment Equation

Summary

Key Terms

Review Questions

Numerical Problems

Answers

Model Question Paper 1

Model Question Paper 2

Model Question Paper 3

Model Question Paper 4

**Appendix A**

A.1 Area Moment of Inertia

A.2 Product Area Moment of Inertia

A.3 Parallel-Axis Theorem

**Appendix B**

B.1 Deflection and Elastic Equations of Some Common Beams

B.2 Area, Centroid and Area Moment of Inertia for Some Common Sections

**Appendix C**

C.1 Symbols and Units

C.2 System of Units

C.3 Area under Parabola

**Appendix D**

D.1 Pure Bending

Bibliography

Index