Strength of Materials Textbook PDF

www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com om.c ts an ir sp ga.c w w w www.cgaspirant...

www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com om.c ts an ir sp ga.c w w w www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com MULTICOLOUR E DITION om.c ts (Mechanics of Solids) an [A Textbook for the students of B.E./B.Tech., A.M.I.E., U.P.S.C. (Engg. Services) and other Engineering Examinations] (SI UNITS) ir sp R.S. KHURMI ga.c w w S. CHAND & COMPANY LTD. w (AN ISO 9001 : 2000 COMPANY) RAM NAGAR, NEW DELHI - 110055 www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com S. CHAND & COMPANY LTD. (An ISO 9001 : 2000 Company) om Head Office : 7361, RAM NAGAR, NEW DELHI - 110 055 Phones : 23672080-81-82, 9899107446, 9911310888; Fax : 91-11-23677446 Shop at: schandgroup.com; E-mail: [email protected] Branches : 1st Floor, Heritage, Near Gujarat Vidhyapeeth, Ashram Road, Ahmedabad-380 014. Ph. 27541965, 27542369, [email protected] No. 6, Ahuja Chambers, 1st Cross, Kumara Krupa Road, Bangalore-560 001. Ph : 22268048, 22354008, [email protected] 238-A M.P. Nagar, Zone 1, Bhopal - 462 011. Ph : 4274723. [email protected] 152, Anna Salai, Chennai-600 002. Ph : 28460026, [email protected] S.C.O. 2419-20, First Floor, Sector- 22-C (Near Aroma Hotel), Chandigarh-160022, Ph-2725443, 2725446, [email protected] ts 1st Floor, Bhartia Tower, Badambadi, Cuttack-753 009, Ph-2332580; 2332581, [email protected] 1st Floor, 52-A, Rajpur Road, Dehradun-248 001. Ph : 2740889, 2740861, [email protected] Pan Bazar, Guwahati-781 001. Ph : 2514155, [email protected] Sultan Bazar, Hyderabad-500 195. Ph : 24651135, 24744815, [email protected] Mai Hiran Gate, Jalandhar - 144008. Ph. 2401630, 5000630, [email protected] an A-14 Janta Store Shopping Complex, University Marg, Bapu Nagar, Jaipur - 302 015, Phone : 2719126, [email protected] 613-7, M.G. Road, Ernakulam, Kochi-682 035. Ph : 2381740, [email protected] 285/J, Bipin Bihari Ganguli Street, Kolkata-700 012. Ph : 22367459, 22373914, [email protected] Mahabeer Market, 25 Gwynne Road, Aminabad, Lucknow-226 018. Ph : 2626801, 2284815, [email protected] Blackie House, 103/5, Walchand Hirachand Marg , Opp. G.P.O., Mumbai-400 001. ir Ph : 22690881, 22610885, [email protected] Karnal Bag, Model Mill Chowk, Umrer Road, Nagpur-440 032 Ph : 2723901, 2777666 [email protected] 104, Citicentre Ashok, Govind Mitra Road, Patna-800 004. Ph : 2300489, 2302100, [email protected] sp © 1968, R.S. Khurmi All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the Publishers. S. CHAND’S Seal of Trust ga In our endeavour to protect you against counterfeit/fake books we have put a Hologram Sticker on the cover of some of our fast moving titles. The hologram displays a unique 3D multi-level, multi-colour effect of our logo from different angles when tilted or properly illuminated under a single source of light. S. CHAND Background artwork seems to be “under” or “behind” the logo, giving the illusion of depth..c A fake hologram does not give any illusion of depth. Multicolour edition conceptualized by R.K. Gupta, CMD First Edition 1968 w Subsequent Editions and Reprints 1970, 71, 72, 73, 74, 75 (Twice), 76, 77 (Twice), 78 (Twice), 79 (Twice), 80, 81, 82 (Twice), 83, 84 (Twice), 85, 86, 87 (Twice), 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 2000, 2001, 2002, 2003, 2004, 2005, 2006 Multicolour Revised Edition 2007, Reprint 2007 w Reprint with Corrections 2008 Code: 10 320 ISBN : 81-219-2822-2 w PRINTED IN INDIA By Rajendra Ravindra Printers (Pvt.) Ltd., 7361, Ram Nagar, New Delhi-110 055 and published by S. Chand & Company Ltd., 7361, Ram Nagar, New Delhi-110 055. www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com om Preface To The Twenty-Third Edition It gives a great pleasure in presenting the new multicolour edition.c of this popular book to innumerable students and academic staff of the Universities in India and abroad. The favourable and warm reception, which the previous editions and reprints of this book have enjoyed all over ts India and abroad, has been a matter of great satisfaction. The present edition of this book is in S.I. Units. To make the book really useful at all levels, a number of articles as well as solved and an unsolved examples have been added. The mistakes, which had crept in, have been eliminated. Three new chapters of Thick Cylindrical and Spherical Shells, Bending of Curved Bars and Mechanical Properties of Materials have also been added. ir Any errors, omissions and suggestions for the improvement of this volume, will be thankfully acknowledged and incorporated in the next edition. sp E-mail : [email protected] Website : R.S. KHURMI N. KHURMI ga www.khurmis.com Address : B-510, New Friends Colony, New Delhi-110025 Mobile : 9810199785.c w w w (v) www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com om Preface To The First Edition I take an opportunity to present Strength of Materials to the students.c of Degree and Diploma, in general, and A.M.I.E (I) Section ‘A’ in particular. The object of this book is to present the subject matter in most concise, compact, to the point and lucid manner. ts While writing the book, I have always kept in view the examination requirements of the students and various difficulties and troubles, which they face, while studying the subject. I have also, constantly, kept in view the re- an quirements of those intelligent students, who are always keen to increase their knowledge. All along the approach to the subject matter, every care has been taken to deal with each and every topic as well as problem from the fundamentals and in the simplest possible manner, within the mathematical ir ability of an average student. The subject matter has been amply illustrated by incorporating a good number of solved, unsolved and well graded ex- amples of almost every variety. Most of these examples are taken from the sp recent examination papers of Indian as well as foreign Universities and pro- fessional examining bodies, to make the students, familiar with the types of questions, usually set in their examinations. At the end of each topic, a few exercises have been added, for the students to solve them independently. ga Answer to these problems have been provided, but it is too much to hope that these are entirely free from errors. At the end of each chapter, Highlights have been added, which summarise the main topics discussed in the chapter for quick revision before the examination. In short, it is earnestly hoped that.c the book will earn the appreciation of the teachers and students alike. Although every care has been taken to check mistakes and misprints, yet it is difficult to claim perfection. Any errors, omissions and suggestions for w the improvement of this volume, brought to my notice, will be thankfully ac- knowledged and incorporated in the next edition. R.S. KHURMI w w (vi) www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com Contents 1. Introduction 1 — 11 om 1. Definition. 2. Fundamental Units. 3. Derived Units. 4. Systems of Units. 5. S.I. Units (International Systems of Units). 6. Metre. 7. Kilogram. 8. Second. 9. Presentation of Units and Their Values..c 10. Rules for S.I. Units. 11. Useful Data. 12. Algebra. 13. Trigonometry. ts 14. Differential Calculus. 15. Integral Calculus. 16. Scalar Quantities. 17. Vector Quantities. 18. Force. an 19. Resultant Force. 20. Composition of Forces. 21. Parallelogram Law of Forces. 22. Triangle Law of Forces. 23. Polygon Law of Forces. 24. Moment of a Force. ir 2. Simple Stresses and Strains 12 — 24 1. Introduction. 2. Elasticity. 3. Stress. sp 4. Strain. 5. Types of Stresses. 6. Tensile Stress. 7. Compressive Stress. 8. Elastic Limit. 9. Hooke′s Law. ga 10. Modulus of Elasticity (or Young′s Modulus). 11. Deformation of a Body Due to Force Acting on it. 12. Deformation of a Body Due to Self Weight. 13. Principle of Superposition. 3. Stresses and Strains in Bars of Varying Sections 25 — 46.c 1. Introduction. 2. Types of Bars of Varying Sections. 3. Stresses in the Bars of Different Sections. 4. Stresses in the Bars of Uniformly Tapering Sections. 5. Stresses in the Bars of Uniformly Tapering Circular Sections. w 6. Stresses in the Bars of Uniformly Tapering Rectangular Sections. 7. Stresses in the Bars of Composite Sections. 4. Stresses and Strains in Statically Indeterminate Structures 47 — 71 1. Introduction. w 2. Types of Statically Indeterminate Structures. 3. Stresses in Simple Statically Indeterminate Structures. 4. Stresses in Indeterminate Structures Supporting a Load. 5. Stresses in Composite Structures of Equal Lengths. w 6. Stresses in Composite Structures of Unequal Lengths. 7. Stresses in Nuts and Bolts. (vii) www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com 5. Thermal Stresses and Strains 72 — 90 1. Introduction. 2. Thermal Stresses in Simple Bars. 3. Thermal Stresses in Bars of Circular Tapering Section. 4. Thermal Stresses in Bars of Varying Section. om 5. Thermal Stresses in Composite Bars. 6. Superposition of Thermal Stresses. 6. Elastic Constants 91 — 107 1. Introduction. 2. Primary or Linear Strain. 3. Secondary or Lateral Strain. 4. Poisson′s Ratio. 5. Volumetric Strain..c 6. Volumetric Strain of a Rectangular Body Subjected to an Axial Force. 7. Volumetric Strain of a Rectangular Body Subjected to Three Mutually Perpendicular Forces. 8. Bulk Modulus. ts 9. Relation Between Bulk Modulus and Young′s Modulus. 10. Shear Stress. 11. Principle of Shear Stress. 12. Shear Modulus or Modulus of Rigidity. 13. Relation Between Modulus of Elasticity and Modulus of Rigidity. an 7. Principal Stresses and Strains 108 — 147 1. Introduction. 2. Principal Planes. 3. Principal Stress. 4. Methods for the stresses on an Oblique Section of a Body. 5. Analytical Method for the Stresses on an oblique Section of a ir Body. 6. Sign conventions for Analytical Method. 7. Stresses on an Oblique Section of a Body subjected to a Direct Stress in One Plane. 8. Stresses on an oblique Section of a Body subjected to Direct sp Stresses in Two Mutually Perpendicular Directions. 9. Stresses on an Oblique Section of a Body subjected to a Simple Shear Stress. 10. Stresses on an Oblique Section of a Body Subjected to a Direct Stress in One Plane and Accompanied by a simple shear Stress. 11. Stresses on an oblique Section of a Body Subjected to Direct ga Stresses in Two Mutually Perpendicular Directions and Accompanied by a Simple Shear Stress. 12. Graphical Method for the Stresses on an Oblique Section of a Body 13. Sign Conventions for Graphical Method 14. Mohr′s Circle for Stresses on an Oblique Section of a Body.c Subjected to a Direct Stress in One Plane. 15. Mohr′s circle for Stresses on an Oblique Section of a Body Subjected to Direct Stresses in Two Mutually Perpendicular Direction. 16. Mohr′s Circle for Stresses on an Oblique Section of a Body Subjected to Direct Stresses in One Plane Accompanied by a w Simple shear Stress. 17. Mohr′s Circle for Stresses on an Oblique Section of Body Subjected to Direct Stresses in Two Mutually Perpendicular Directions Accompanied by Simple Shear Stress. w 8. Strain Energy and Impact Loading 148 — 161 1. Introduction. 2. Resilience. 3. Proof Resilience. w 4. Modulus of Resilience. 5. Types of Loading. (viii) www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com 6. Strain Energy Stored in a Body when the Load is Gradually Applied. 7. Strain Energy Stored in a Body, when the load is Suddenly Applied. 8. Strain Energy Stored in a Body, when the load is Applied with Impact. 9. Strain Energy Stored in a Body of varying section. 10. Strain Energy stored in a Body due to Shear Stress. om 9. Centre of Gravity 162 — 183 1. Introduction. 2. Centroid. 3. Methods for Centre of Gravity. 4. Centre of Gravity by Geometrical Considerations. 5. Centre of Gravity by Moments. 6. Axis of Reference. 7. Centre of Gravity of Plane Figures..c 8. Centre of Gravity of Symmetrical Sections. 9. Centre of Gravity of Unsymmetrical Sections. 10. Centre of Gravity of Solid Bodies. 11. Centre of Gravity of Sections with Cut out Holes. ts 10. Moment of Inertia 184 — 207 1. Introduction. 2. Moment of Inertia of a Plane Area. an 3. Units of Moment of Inertia. 4. Methods for Moment of Inertia. 5. Moment of Inertia by Routh′s Rule. 6. Moment of Inertia by Integration. 7. Moment of Inertia of a Rectangular Section. 8. Moment of Inertia of a Hollow Rectangular Section. ir 9. Theorem of Perpendicular Axis. 10. Moment of Inertia of a Circular Section. 11. Moment of Inertia of a Hollow Circular Section. 12. Theorem of Parallel Axis. 13. Moment of Inertia of a Triangular Section. sp 14. Moment of Inertia of a Semicircular Section. 15. Moment of Inertia of a Composite Section. 16. Moment of Inertia of a Built-up Section. 11. Analysis of Perfect Frames (Analytical Method) 208 — 252 ga 1. Introduction. 2. Types of Frames. 3. Perfect Frame. 4. Imperfect Frame. 5. Deficient Frame. 6. Redundant Frame..c 7. Stress. 8. Tensile Stress. 9. Compressive Stress. 10. Assumptions for Forces in the Members of a Perfect Frame. w 11. Analytical Methods for the Forces. 12. Method of Joints. 13. Method of Sections (or Method of Moments). 14. Force Table. w 15. Cantilever Trusses. 16. Structures with One End Hinged (or Pin-jointed) and the Other Freely Supported on Rollers and Carrying Horizontal Loads. 17. Structures with One End Hinged (or Pin-jointed) and the Other Freely Supported on Rollers and Carrying Inclined Loads. w 18. Miscellaneous Structures. (ix) www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com 12. Analysis of Perfect Frames (Graphical Method) 253 — 285 1. Introduction. 2. Construction of Space Diagram. 3. Construction of Vector Diagram. 4. Force Table. om 5. Magnitude of Force. 6. Nature of Force. 7. Cantilever Trusses. 8. Structures with One End Hinged (or Pin-jointed) and the Other Freely Supported on Rollers and Carrying Horizontal Loads. 9. Structures with One End Hinged (or Pin-jointed) and the Other Freely Supported on Rollers and Carrying Inclined Loads. 10. Frames with Both Ends Fixed..c 11. Method of Substitution. 13. Bending Moment and Shear Force 286 — 343 1. Introduction. 2. Types of Loading. ts 3. Shear Force. 4. Bending Moment. 5. Sign Conventions. 6. Shear force and Bending Moment Diagrams. an 7. Relation between Loading, Shear Force and Bending Moment. 8. Cantilever with a Point Load at its Free End. 9. Cantilever with a Uniformly Distributed Load. 10. Cantilever with a Gradually Varying Load. 11. Simply Supported Beam with a Point Load at its Mid-point. 12. Simply Supported Beam with a Uniformly Distributed Load. ir 13. Simply Supported Beam with a Triangular Load Varying Gradually from Zero at Both Ends to w per unit length at the Centre. 14. Simply Supported Beam with a Gradually Varying Load from Zero at One End to w per unit length at the other End. 15. Overhanging Beam. sp 16. Point of Contraflexure. 17. Load and Bending Moment Diagrams from a Shear Force Diagram. 18. Beams Subjected to a Moment. 19. Beams Subjected to Inclined Loads. 20. Shear Force and Bending Moment Diagrams for Inclined Beams. ga 14. Bending Stresses in Simple Beams 344 — 363 1. Introduction. 2. Assumptions in the Theory of Simple Bending. 3. Theory of Simple Bending. 4. Bending Stress..c 5. Position of Neutral Axis. 6. Moment of Resistance. 7. Distribution of Bending Stress Across the Section. 8. Modulus of Section. w 9. Strength of a Section. 10. Bending Stresses in Symmetrical Sections 11. Bending Stresses in Unsymmetrical Sections. 364 — 382 w 15. Bending Stresses in Composite Beams 1. Introduction. 2. Types of Composite Beams. 3. Beams of Unsymmetrical Sections. w 4. Beams of Uniform Strength. 5. Beams of Composite Sections (Flitched Beams). (x) www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com 16. Shearing Stresses in Beams 383 — 404 1. Introduction. 2. Shearing Stress at a Section in a Loaded Beam. 3. Distribution of Shearing Stress. 4. Distribution of Shearing Stress over a Rectangular Section. om 5. Distribution of Shearing Stress over a Triangular Section. 6. Distribution of Shearing Stress over a Circular Section. 7. Distribution of Shearing Stress over an I-section. 8. Distribution of Shear Stress over a T-section. 9. Distribution of Shearing Stress over a Miscellaneous Section. 17. Direct and Bending Stresses 405 — 421 1. Introduction. 2. Eccentric Loading..c 3. Columns with Eccentric Loading. 4. Symmetrical Columns with Eccentric Loading about One Axis. 5. Symmetrical Columns with Eccentric Loading about Two Axes. 6. Unsymmetrical Columns with Eccentric Loading. ts 7. Limit of Eccentricity. 18. Dams and Retaining Walls 422 — 462 1. Introduction. 2. Rectangular Dams. an 3. Trapezoidal Dams with Water Face Vertical. 4. Trapezoidal Dams with Water Face Inclined. 5. Conditions for the Stability of a Dam. 6. Condition to Avoid Tension in the Masonry of the Dam at its Base. 7. Condition to Prevent the Overturning of the Dam. 8. Condition to Prevent the Sliding of Dam. ir 9. Condition to Prevent the Crushing of Masonry at the Base of the Dam. 10. Minimum Base Width of a Dam. 11. Maximum Height of a Dam. sp 12. Retaining Walls. 13. Earth Pressure on a Retaining Wall. 14. Active Earth Pressure. 15. Passive Earth Pressure. 16. Theories of Active Earth Pressure. 17. Rankine′s Theory for Active Earth Pressure. ga 18. Coulomb′s Wedge Theory for Active Earth Pressure. 19. Conditions for the Stability of Retaining Wall. 19. Deflection of Beams 463 — 489 1. Introduction. 2. Curvature of the Bending Beam..c 3. Relation between Slope, Deflection and Radius of Curvature. 4. Methods for Slope and Deflection at a Section. 5. Double Integration Method for Slope and Deflection. 6. Simply Supported Beam with a Central Point Load. w 7. Simply Supported Beam with an Eccentric Point Load. 8. Simply Supported Beam with a Uniformly Distributed Load. 9. Simply Supported Beam with a Gradually Varying Load. 10. Macaulay′s Method for Slope and Deflection. 11. Beams of Composite Section. w 20. Deflection of Cantilevers 490 — 508 1. Introduction. 2. Methods for Slope and Deflection at a Section. w 3. Double Integration Method for Slope and Deflection. 4. Cantilever with a Point Load at the Free End. 5. Cantilever with a Point Load not at the Free End. (xi) www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com 6. Cantilever with a Uniformly Distributed Load. 7. Cantilever Partially Loaded with a Uniformly Distributed Load. 8. Cantilever Loaded from the Free End. 9. Cantilever with a gradually Varying Load. 10. Cantilever with Several Loads. 11. Cantilever of Composite Section. om 21. Deflection by Moment Area Method 509 — 526 1. Introduction. 2. Mohr′s Theorems. 3. Area and Position of the Centre of Gravity of Parabolas. 4. Simply Supported Beam with a Central Point Load. 5. Simply Supported Beam with an Eccentric Point Load. 6. Simply Supported Beam with a Uniformly Distributed Load. 7. Simply Supported Beam with a Gradually Varying Load..c 8. Cantilever with a Point Load at the Free end. 9. Cantilever with a Point Load at any Point. 10. Cantilever with a Uniformly Distributed Load. 11. Cantilever with a Gradually Varying Load. ts 22. Deflection by Conjugate Beam Method 527 — 547 1. Introduction. 2. Conjugate Beam. 3. Relation between an Actual Beam and the Conjugate Beam. an 4. Cantilever with a Point Load at the Free End. 5. Cantilever with a Uniformly Distributed Load. 6. Cantilever with a Gradually Varying Load. 7. Simply Supported Beam with Central Point Load. 8. Simply Supported Beam with an Eccentric Point Load. 9. Simply Supported Beam with a Uniformly Distributed Load. 10. Simply Supported Beam with a Gradually Varying Load. ir 23. Propped Cantilevers and Beams 548 — 569 1. Introduction. 2. Reaction of a Prop. 3. Cantilever with a Uniformly Distributed Load. sp 4. Cantilever Propped at an Intermediate Point. 5. Simply Supported Beam with a Uniformly Distributed Load and Propped at the Centre. 6. Sinking of the Prop. 24. Fixed Beams 570 — 597 ga 1. Introduction. 2. Advantages of Fixed Beams. 3. Bending Moment Diagrams for Fixed Beams. 4. Fixing Moments of a Fixed Beam. 5. Fixing Moments of a Fixed Beam Carrying a Central Point Load. 6. Fixing Moments of a Fixed Beam Carrying an Eccentric Point Load..c 7. Fixing Moments of a Fixed Beam Carrying a Uniformly Distributed Load. 8. Fixing Moments of a Fixed Beam Carrying a Gradually Varying Load from Zero at One End to w per unit length at the Other. 9. Fixing Moments of a Fixed Beam due to Sinking of a Support. w 25. Theorem of Three Moments 598 — 623 1. Introduction. 2. Bending Moment Diagrams for Continuous Beams. w 3. Claypeyron′s Theorem of Three Moments. 4. Application of Clapeyron′s Theorem of Three Moments to Various Types of Continuous Beams. 5. Continuous Beams with Simply Supported Ends. 6. Continuous Beams with Fixed End Supports. w 7. Continuous Beams with End Span Overhanging. 8. Continuous Beams with a Sinking Support. 9. Continuous Beams Subjected to a Couple. (xii) www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com 26. Moment Distribution Method 624 — 652 1. Introduction. 2. Sign Conventions. 3. Carry Over Factor. 4. Carry Over Factor for a Beam Fixed at One End and Simply Supported at the Other. om 5. Carry Over Factor for a Beam, Simply Supported at Both Ends. 6. Stiffness Factor. 7. Distribution Factors. 8. Application of Moment Distribution Method to Various Types of Continuous Beams. 9. Beams with Fixed End Supports. 10. Beams with Simply Supported Ends. 11. Beams with End Span Overhanging..c 12. Beams With a Sinking Support. 27. Torsion of Circular Shafts 653 — 678 1. Introduction. 2. Assumptions for Shear Stress in a Circular Shaft Subjected to ts Torsion. 3. Torsional Stresses and Strains. 4. Strength of a Solid Shaft. 5. Strength of hollow shaft. an 6. Power Transmitted by a Shaft. 7. Polar Moment of Inertia. 8. Replacing a Shaft. 9. Shaft of Varying Section. 10. Composite Shaft. 11. Strain Energy due to Torsion. 12. Shaft Couplings. ir 13. Design of Bolts. 14. Design of Keys. 28. Springs 679 — 694 1. Introduction. sp 2. Stiffness of a Spring. 3. Types of Springs. 4. Bending Springs. 5. Torsion Springs. 6. Forms of Springs. 7. Carriage Springs or Leaf Springs (Semi-elliptical Type). ga 8. Quarter-Elliptical Type Leaf Springs. 9. Helical Springs. 10. Closely-coiled Helical Springs. 11. Closely-coiled Helical Springs Subjected to an Axial Load. 12. Closely-coiled Springs Subjected to an Axial Twist. 13. Open-coiled Helical Springs..c 14. Springs in Series and Parallel. 29. Riveted Joints 695 — 721 1. Introduction. w 2. Types of Riveted Joints. 3. Lap Joint. 4. Butt Joint. 5. Single Cover Butt Joint. w 6. Double Cover Butt Joint. 7. Single Riveted Joint. 8. Double Riveted Joint. 9. Multiple Riveted Joint. 10. Chain Riveted Joint. w 11. Zig-zag Riveted Joint. 12. Diamond Riveted joint. 13. Pitch of Rivets. (xiii) www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com 14. Failure of a Joint. 15. Failure of the Rivets. 16. Shearing of the Rivets. 17. Crushing of the Rivets. 18. Failure of the Plates. 19. Tearing off the Plate across a Row of Joints. om 20. Tearing off the Plate at an Edge. 21. Strength of a Rivet. 22. Strength of the Plate. 23. Strength of a Riveted Joint. 24. Efficiency of a Riveted Joint. 25. Design of a Riveted Joint. 26. Eccentric Riveted Connections. 27. Transmission of Load Through Rods. 28. Types of Rod Joints..c 29. Knuckle Joint. 30. Cotter Joint. 30. Welded Joints 722 — 741 ts 1. Introduction. 2. Advantages and Disadvantages of Welded Joints. 3. Type of Welded Joints. 4. Butt Weld Joint. 5. Fillet Weld Joint. an 6. Plug or Slot Weld Joint. 7. Technical Terms. 8. Strength of a Welded Joint. 9. Unsymmetrical Section Subjected to an Axial Load. 10. Eccentric Welded Joints. 11. Eccentric Welded Joint Subjected to Moment. 12. Eccentric Welded Joint Subjected to Torsion. ir 31. Thin Cylindrical and Spherical Shells 742 — 754 1. Introduction. 2. Failure of a Thin Cylindrical Shell due to an Internal Pressure. sp 3. Stresses in a Thin Cylindrical Shell. 4. Circumferential Stress. 5. Longitudinal Stress. 6. Design of Thin Cylindrical Shells. 7. Change in Dimensions of a Thin Cylindrical Shell due to an Internal Pressure. ga 8. Change in Volume of a Thin Cylindrical Shell due to an Internal Pressure. 9. Thin Spherical Shells. 10. Change in Diameter and Volume of a Thin Spherical Shell due an Internal Pressure. 11. Riveted Cylindrical Shells. 12. Wire-bound Thin Cylindrical Shells..c 32. Thick Cylindrical and Spherical Shells 755 — 772 1. Introduction. 2. Lame′s Theory. w 3. Stresses in a Thick Cylindrical Shell. 4. Stresses in Compound Thick Cylindrical Shells. 5. Difference of Radii for Shrinkage. 6. Thick spherical shells. w 33. Bending of Curved Bars 773 — 794 1. Introduction. 2. Assumptions for the Stresses in the Bending of Curved Bars. 3. Types of Curved Bars on the Basis of initial Curvature. w 4. Bars with a Small Initial Curvature. 5. Bars with a Large Initial Curvature. 6. Link Radius for Standard Sections. (xiv) www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com 7. Value of Link Radius for a Rectangular Section. 8. Value of Link Radius for a Triangular Section. 9. Value of Link Radius for a Trapezoidal Section. 10. Value of Link Radius for a Circular Section. 11. Crane Hooks. 12. Rings. om 13. Chain Links. 34. Columns and Struts 795 — 820 1. Introduction. 2. Failure of a Column or Strut. 3. Euler′s Column Theory. 4. Assumptions in the Euler′s Column Theory. 5. Sign Conventions. 6. Types of End Conditions of Columns..c 7. Columns with Both Ends Hinged. 8. Columns with One End Fixed and the Other Free. 9. Columns with Both Ends Fixed. 10. Columns with One End Fixed and the Other Hinged. 11. Euler′s Formula and Equivalent Length of a Column. ts 12. Slenderness Ratio. 13. Limitations of Euler′s Formula. 14. Empirical Formulae for Columns. 15. Rankine′s Formula for Columns. an 16. Johnson′s Formula for Columns. 17. Johnson′s Straight Line Formula for Columns. 18. Johnson′s Parabolic Formula for Columns. 19. Indian Standard Code for Columns. 20. Long Columns subjected to Eccentric Loading. 35. Introduction to Reinforced Concrete 821 — 834 ir 1. Introduction. 2. Advantage of R.C.C. Structures. 3. Assumptions in the Theory of R.C.C. 4. Neutral Axis. 5. Types of Neutral Axes. sp 6. Critical Neutral Axis. 7. Actual Neutral Axis. 8. Moment of Resistance. 9. Types of Beam Sections. 10. Under-reinforced Sections. 11. Balanced Sections. ga 12. Over-reinforced Sections. 13. Design of Beams and Slabs. 36. Mechanical Properties of Materials 835 — 843 1. Introduction. 2. Classification of Materials. 3. Elastic Materials..c 4. Plastic Materials. 5. Ductile Materials. 6. Brittle Materials. 7. Classification of Tests. w 8. Actual Tests for the Mechanical properties of Materials. 9. Tensile Test of a Mild Steel Specimen. 10. Working Stress. 11. Factor of Safety. 12. Barba′s Law and Unwin′s Formula. w 13. Compression Test. 14. Impact Test. 15. Fatigue Test. Appendix 845 — 852 w Index 853 — 862 (xv) www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com List of Symbols om A = Area of cross-section W = Load or Weight (N) a = Rankine’s constant w = Load per unit length (N/m) B, b = Width w = Specific weight (kN/m3) C = Shear modulus of rigidity x, y, z = Cartesian co-ordinates (N/mm2) y = Distance.c D, d = Depth = Deflection = Diameter Z = Section modulus E = Young’s modulus of elasticity ts (N/mm2) r, θ = Polar co-ordinates e = Linear strain α = Co-efficient of linear expansion (/ °C) = Eccentricity α, θ, β = an Angle (rad) G = Centre of Gravity ⎛ 1⎞ = Centroid of area or lamina μ = Poisson’s ratio or ⎜⎝ ⎟⎠ m g = Acceleration due to gravity η = Efficiency (9.81 m/s2) ε = Strain ir H, h = Height (m) ρ = Density (kg/m3) I = Moment of inertia (mm4) φ = Shear strain J = Polar moment of inertia (mm4) i = Slope sp K = Bulk modulus of elasticity (N/mm2) δ = Deflection k = Radius of Gyration Δ = Deflection k = Stiffness of Spring (N/mm) δl = Change in length ω = ga L, l = Length (m) Angular velocity (rad/s) M, m = Mass (kg) µ = Co-efficient of friction M = Bending moment (N-m) σ = Normal stress (N/mm2) N = Speed (r.p.m.) τ = Shear stress (N/mm2).c n = Number σc = Circumferential (or hoop) stress P = Force (N) σl = Longitudinal stress p = Pressure (N/mm2) σr = Radical stress w R, r = Radius σt = Tangential stress T, t = Time (s) = Tearing stress T = Torque (N-m) σb = Bending stress w = Twisting Moment = Bearing stress U = Strain Energy σ1, σ2, σ3 = Principal streses V = Volume (m3) Le w = Slenderness ratio k (xvi) www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com Chapter 1 om.c Introduction ts Contents 1. Definition. 2. Fundamental Units. an ir 3. Derived Units. 4. Systems of Units. 5. S.I. Units (International Systems of sp Units). 6. Metre. 7. Kilogram. 8. Second. 9. Presentation of Units and Their Values. ga 10. Rules for S.I. Units. 11. Useful Data. 12. Algebra. 13. Trigonometry. 1.1. Definition 14. Differential Calculus. In day-to-day work, an engineer comes.c 15. Integral Calculus. across certain materials, i.e., steel girders, angle 16. Scalar Quantities. irons, circular bars, cement etc., which are used 17. Vector Quantities. in his projects. While selecting a suitable 18. Force. material, for his project, an engineer is always w 19. Resultant Force. interested to know its strength. The strength of 20. Composition of Forces. a material may be defined as ability, to resist its 21. Parallelogram Law of Forces. failure and behaviour, under the action of 22. Triangle Law of Forces. w external forces. It has been observed that, under 23. Polygon Law of Forces. the action of these forces, the material is first 24. Moment of a Force. deformed and then its failure takes place. A detailed study of forces and their effects, w alongwith some suitable protective measures for the safe working conditions, is known as Strength of Materials. As a matter of fact, such www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com 2  Strength of Materials a knowledge is very essential, for an engineer, to enable him, in designing all types of structures and machines. om 1.2. Fundamental Units The measurements of physical quantities is one of the most important operations in engineering. Every quantity is measured in terms of some arbitrary, but internationally accepted units, called fundamental units. All the physical quantities, met with in Strength of Materials, are expressed in terms of the following three fundamental quantities : 1. Length, 2. Mass and 3. Time..c 1.3. Derived Units Sometimes, physical quantities are expressed in other units, which are derived from fundamental units, known as derived units, e.g., units of area, velocity, acceleration, pressure, etc. ts 1.4. Systems of Units Following are only four systems of units, which are commonly used and universally recognised. an 1. C.G.S. units, 2. F.P.S. units, 3. M.K.S. units and 4. S.I. units. In this book, we shall use only the S.I. system of units, as the future courses of studies are conducted in this system of units only. 1.5. S.I. Units (International System of Units) ir The eleventh General Conference* of Weights and Measures has recommended a unified and systematically constituted system of fundamental and derived units for international use. This system of units is now being used in many countries. In India, the Standards of Weights and Measures Act of 1956 (vide which we switched over to M.K.S. units) has been revised to recognise all the S.I. units in sp industry and commerce. In this system of units, the †fundamental units are metre (m), kilogram (kg) and second (s) respectively. But there is a slight variation in their derived units. The following derived units will be used in this book : 3 ga Density (or Mass density) kg/m 2 Force (in Newtons) N (= kg.m/s ) 2 –6 2 Pressure (in Pascals) Pa (= N/m = 10 N/mm ) 2 –6 2 Stress (in Pascals) Pa (=N/m = 10 N/mm ) Work done (in Joules) J (= N-m).c Power (in Watts) W (= J/s) International metre, kilogram and second are discussed here. 1.6. Metre w The international metre may be defined as the shortest distance (at 0°C) between two parallel lines engraved upon the polished surface of the Platinum-Iridium bar, kept at the International Bureau of Weights and Measures at Sevres near Paris. w * It is known as General Conference of Weights and Measures (G.C.W.M.). It is an international organisation of which most of the advanced and developing countries (including India) are members. This conference has been ensured the task of prescribing definitions of various units of weights and measures, which are w the very basis of science and technology today. † The other fundamental units are electric current, ampere (A), thermodynamic temperature, kelvin (K) and luminous intensity, candela (cd). These three units will not be used in this book. www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com Chapter 1 : Introduction  3 om.c A bar of platinum - iridium metre kept at a temperature of 0º C. 1.7. Kilogram ts The international kilogram may be defined as the mass of the Platinum-Iridium cylinder, which is also kept at the The standard platinum - kilogram is kept International Bureau of Weights and Measures at Sevres near at the International Bureau of Weights an Paris. and Measures at Serves in France. 1.8. Second The fundamental unit of time for all the four systems is second, which is 1/(24 × 60 × 60) = 1/86 400th of the mean solar day. A solar day may be defined as the interval of time between the ir instants at which the sun crosses the meridian on two consecutive days. This value varies throughout the year. The average of all the solar days, of one year, is called the mean solar day. 1.9. Presentation of Units and Their Values sp The frequent changes in the present day life are facilitated by an international body known as International Standard Organisation (ISO). The main function of this body is to make recommendations regarding international procedures. The implementation of ISO recommendations in a country is assisted by an organisation appointed for the purpose. In India, Bureau of Indian Standard formerly ga known as Indian Standards Institution (ISI) has been created for this purpose. We have already discussed in the previous articles the units of length, mass and time. It is always necessary to express all lengths in metres, all masses in kilograms and all times in seconds. According to convenience, we also use larger multiples or smaller fractions of these units. As a typical example, although metre is the unit of length, yet a smaller length equal to one-thousandth of a metre proves to.c be more convenient unit especially in the dimensioning of drawings. Such convenient units are formed by using a prefix in front of the basic units to indicate the multiplier. The full list of these prefixes is given in Table 1.1 w TABLE 1.1. Factor by which the Standard form Prefix Abbreviation unit is multiplied w 12 1 000 000 000 000 10 Tera T 9 1 000 000 000 10 giga G 106 w 1 000 000 mega M 3 1 000 10 kilo k 2 100 10 hecto* h www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com 4  Strength of Materials 1 10 10 deca* da –1 0.1 10 deci* d om 0.01 10–2 centi* c –3 0.001 10 milli m –6 0.000 001 10 micro μ 0.000 000 001 10–9 nano n –12 0.000 000 000 001 10 pico p.c 1.10. Rules for S.I. Units The Eleventh General Conference of Weights and Measures recommended only the fundamental and derived units of S.I. system. But it did not elaborate the rules for the usage of these units. Later ts on, many scientists and engineers held a number of meetings for the style and usage of S.I. units. Some of the decisions of these meetings are : 1. A dash is to be used to separate units, which are multiplied together. For example, a newton- meter is written as N-m. It should not be confused with mN, which stands for millinewton. an 2. For numbers having 5 or more digits, the digits should be placed in groups of three separated by spaces (instead of ††commas) counting both to the left and right of the decimal point. 3. In a †††four digit number, the space is not required unless the four digit number is used in a column of numbers with 5 or more digits. At the time of revising this book, the author sought the advice of various international authorities ir regarding the use of units and their values, keeping in view the global reputation of the author as well as his books. It was then decided to ††††present the units and their values as per the recommendations of ISO and ISI. It was decided to use : sp 4500 not 4 500 or 4,500 7 589 000 not 7589000 or 7,589,000 0.012 55 not 0.01255 or.012,55 6 7 30 × 10 not 3 × 10 or 3,00,00,000 ga The above mentioned figures are meant for numerical values only. Now we shall discuss about the units. We know that the fundamental units in S.I. system for length, mass and time are metre, kilogram and second respectively. While expressing these quantities, we find it time-consuming to write these units such as metres, kilograms and seconds, in full, every time we use them. As a result of this, we find it quite convenient to use the following standard abbreviations, which are internationally.c recognised. We shall use : m for metre or metres km for kilometre or kilometres w kg for kilogram or kilograms * The prefixes are generally becoming obsolete probably due to possible confusion. Moreover, it is becoming w 3n a conventional practice to use only those powers of ten which confirm to 10 where n is a positive or negative whole number. † In certain countries, comma is still used as the decimal marker. ††† In certain countries, space is used even in a four digit number. w †††† In some question papers, standard values are not used. The author has tried to avoid such questions in the text of the book, in order to avoid possible confusion. But at certain places, such questions have been included keeping in view the importance of question from the reader’s angle. www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com Chapter 1 : Introduction  5 t for tonne or tonnes s for second or seconds min for minute or minutes om N for newton or newtons N-m for newton × metres (i.e., work done) kN-m for kilonewton × metres rad for radian or radians rev for revolution or revolutions.c 1.11. Useful Data The following data summarises the previous memory and formulae, the knowledge of which is very essential at this stage. ts 1.12. Algebra 0 0 1. a = 1 ; x = 1 an (i.e., Anything raised to the power zero is one.) m n m+n 2. x × x = x (i.e., If the bases are same, in multiplication, the powers are added.) xm ir m–n 3. xn = x (i.e., If the bases are same, in division, the powers are subtracted.) 2 sp 4. If ax + bx + c = 0 −b ± b 2 − 4ac then x= 2a where a is the coefficient of x2, ga b is the coefficient of x and c is the constant term. 1.13. Trigonometry In a right-angled triangle ABC as shown in Fig. 1.1..c b = sin θ 1. c A c = cos θ 2. a w b sin θ 3. a = cos θ = tan θ c b c 1 4. b = sin θ = cosec θ w 90º c 1 q 5. a = cos θ = sec θ B a C w a = cos θ = 1 = cot θ Fig. 1.1 6. b sin θ tan θ www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com 6  Strength of Materials 7. The following table shows the values of trigonometrical functions for some typical angles: angle 0° 30° 45° 60° 90° om 1 1 3 sin 0 2 1 2 2 3 1 1 cos 1 2 0 2 2 1 tan 0 3 1 3 ∞.c or in other words, for sin write. 0° 30° 45° 60° 90° ts 0 1 2 3 4 2 2 2 2 2 1 1 3 an 0 2 1 2 2 for cos write the values in reverse order; for tan divide the value of sin by cos for the respective angle. 8. In the first quadrant (i.e., 0° to 90°) all the trigonometrical ratios are positive. ir 9. In the second quadrant (i.e., 90° to 180°) only sin θ and cosec θ are positive. 10. In the third quadrant (i.e., 180° to 270°) only tan θ and cot θ are positive. 11. In the fourth quadrant (i.e., 270° to 360°) only cos θ and sec θ are positive. sp 12. In any triangle ABC, a b = c = sin A sin B sin C where a, b and c are the lengths of the three sides of a triangle. A, B and C are opposite angles of the sides a, b and c respectively. ga 13. sin (A + B) =sin A cos B + cos A sin B. 14. sin (A – B) =sin A cos B – cos A sin B. 15. cos (A + B) = cos A cos B – sin A sin B. 16. cos (A – B) = cos A cos B + sin A sin B..c tan A + tan B 17. tan (A + B) = 1 − tan A. tan B w tan A − tan B 18. tan (A – B) = 1 + tan A. tan B 19. sin 2A = 2 sin A cos A. w 2 2 20. sin θ + cos θ = 1. 2 2 21. 1 + tan θ = sec θ. 2 2 22. 1 + cot θ = cosec θ. w 1 − cos 2 A 23. sin A = 2 2 1 − cos 2 A 24. cos2 A = 2 www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com Chapter 1 : Introduction  7 25. 2 cos A sin B = sin (A + B) – sin (A – B). 26. Rules for the change of trigonometrical ratios: om sin (– θ) = – sin θ cos (– θ) = cos θ (A) tan (– θ) = – tan θ cot (– θ) = – cot θ sec (– θ) = sec θ cosec (– θ) – cosec θ.c = sin (90° – θ) = cos θ cos (90° – θ) = sin θ ts tan (90° – θ) = cot θ (B) cot (90° – θ) = tan θ sec (90° – θ) = cosec θ an cosec (90° – θ) = sec θ sin (90° + θ) = cos θ cos (90° + θ) = – sin θ (C) tan (90° + θ) = – cot θ ir cot (90° + θ) = – tan θ sec (90° + θ) = – cosec θ cosec (90° + θ) = sec θ sp sin (180° – θ) = sin θ cos (180° – θ) = – cos θ tan (180° – θ) = – tan θ (D) ga cot (180° – θ) = – cot θ sec (180° – θ) = – sec θ cosec (180° – θ) = cosec θ sin (180° + θ) = – sin θ.c cos (180° + θ) = – cos θ tan (180° + θ) = tan θ (E) cot (180° + θ) = cot θ w sec (180° + θ) = – sec θ cosec (180° + θ) = – cosec θ Following are the rules to remember the above 30 formulae : w Rule 1. Trigonometrical ratio changes only when the angle is (90° – θ) or (90° + θ). In all other cases, trigonometrical ratio remains the same. Following is the law of change: sin changes into cos and cos changes into sin, w tan changes into cot and cot changes into tan, sec changes into cosec and cosec changes into sec. Rule 2. Consider the angle θ to be a small angle and write the proper sign as per formulae 8 to 11 above. www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com 8  Strength of Materials 1.14. Differential Calculus d om 1. is the sign of differentiation. dx d d (x)8 = 8x7, d 2. (x)n = nxn–1 ; (x) = 1 dx dx dx (i.e., to differentiate any power of x, write the power before x and subtract one from the power). 3. d (C) = 0 ; d (7) = 0.c dx dx (i.e., differential coefficient of a constant is zero). 4. d (u. v) = u. dv + v. du dx dx dx ts ⎡ i.e., Differential ⎤ ⎡(Ist function×Differential ⎤ ⎢ coefficient of ⎥ ⎢coefficient of second function) ⎥ ⎢ product of any ⎥ = ⎢ + (2nd function×Differential ⎥ ⎢⎣ two functions ⎥⎦ ⎢⎣coefficient of first function) ⎥⎦ an v. du − u. dv 5. () d u = dx dx v v2 dx ⎡(Denominator × Differential ⎤ ⎡ i.e., Differential ⎤ ⎢coefficient of numerator) ⎥ ir ⎢ coefficient of two ⎥ ⎢ – (Numerator × Differential ⎥ ⎢ functions when one ⎥ = ⎢ coefficient of denominator ⎥ ⎢⎣ is divided by the other ⎥⎦ ⎢ Square of denominator ⎥ ⎣ ⎦ sp 6. Differential coefficient of trigonometrical functions d d (cos x) = – sin x (sin x) = cos x ; dx dx d d (tan x) = sec2 x ; (cot x) = – cosec2 x ga dx dx d d (sec x) = sec x. tan x ; (cosec x) = – cosec x. cot x dx dx Note. The differential coefficient, whose trigonometrical function begins with co, is negative..c 7. If the differential coefficient of a function is zero, the function is either maximum or minimum. Conversely, if the maximum or minimum value of a function is required, then differentiate the function and equate it to zero. w 1.15. Integral Calculus 1. ∫ dx is the sign of integration. w n +1 7 ∫ x dx = n +1 ; ∫ x dx = 7 n x 6 x 2. (i.e., to integrate any power of x, add one to the power and divide by the new power). w 3. ∫ 7dx = 7x ; ∫ C dx = Cx (i.e., to integrate any constant, multiply the constant by x). www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com Chapter 1 : Introduction  9 n +1 (ax + b) 4. ∫ (ax + b) n dx = (n + 1) × a (i.e., to integrate any bracket with power, add one to the power and divide by the new power om and also divide by the coefficient of x within the bracket). 1.16. Scalar Quantities The scalar quantities (or sometimes known as scalars) are those quantities which have magnitude only such as length, mass, time, distance, volume, density, temperature, speed etc. 1.17. Vector Quantities.c The vector quantities (or sometimes known as vectors) are those quantities which have both magnitude and direction such as force, displacement, velocity, acceleration, momentum etc. Following are the important features of vector quantities : ts 1. Representation of a vector. A vector is represented by a directed line as shown in Fig. 1.2. P It may be noted that the length OA represents the O A an magnitude of the vector. The direction of the Fig. 1.2. Vector vector is is from O (i.e., starting point) to A (i.e., end point). It is also known as vector P. 2. Unit vector. A vector, whose magnitude is unity, is known as unit vector. ir 3. Equal vectors. The vectors, which are parallel to each other and have same direction (i.e., same sense) and equal magnitude are known as equal vectors. 4. Like vectors. The vectors, whch are parallel to each other and have same sense but unequal magnitude, are known as like vectors. sp 5. Addition of vectors. Consider two vectors PQ and RS, which are required to be added as shown in Fig. 1.3 (a). Take a point A, and draw line AB parallel and equal in ga magnitude to the vector PQ to some convenient scale. Through B, draw BC parallel and equal to vector RS to the same scale. Join AC which will give the required.c sum of vectors PQ and RS as shown in Fig. 1.3 (b). This method of adding the two vectors is called the Triangle w Law of Addition of Vectors. Similarly, if more than two vectors are to be added, the same may be w done first by adding the two vectors, and then by adding the third vector to the resultant of the The velocity of this cyclist is an example of a vector quantity. first two and so on. This method of w adding more than two vectors is called Polygon Law of Addition of Vectors. www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com 10  Strength of Materials om.c Fig. 1.3 6. Subtraction of vectors. Consider two vectors PQ and RS whose difference is required to be ts found out as shown in Fig. 1.4 (a). an ir Fig. 1.4 sp Take a point A, and draw line AB parallel and equal in magnitude to the vector PQ to some convenient scale. Through B, draw BC parallel and equal to the vector RS, but in opposite direction, to that of the vector RS to the same scale. Join AC, which will give the required difference of the vectors PQ and RS as shown in Fig. 1.4 (b). 1.18. Force ga It is an important factor in the field of Engineering-science, which may be defined as an agent which produces or tends to produce, destroys or tends to destroy motion. 1.19. Resultant Force.c If a number of forces P, Q, R......... etc., are acting simultaneously on a particle, then a single force, which will produce the same effect as that of all the given forces, is known as a resultant force. The forces P, Q, R.... etc., are called component forces. The resultant force of the component forces w or the point through which it acts may be found out either mathematically or graphically. 1.20. Composition of Forces w It means the process of finding out the resultant force of the given component forces. A resultant force may be found out analytically, graphically or by the following laws : 1.21. Parallelogram Law of Forces w It states, “If two forces acting simultaneously on a particle be represented, in magnitude and direction, by the two adjacent sides of a parallelogram, their resultant may be represented, in magnitude and direction, by the diagonal of the parallelogram passing through the point of their intersection.” www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com Chapter 1 : Introduction  11 1.22. Triangle Law of Forces It states, “If two forces acting simultaneously on a particle be represented in magnitude and om direction, by the two sides of a triangle taken in order, their resultant may be represented, in magnitude and direction, by the third side of the triangle taken in opposite order.” 1.23. Polygon Law of Forces It states, “If a number of forces acting simultaneously on a particle be represented in magnitude and direction by the sides of a polygon taken in order, their resultant may be represented, in magnitude and direction, by the closing side of the polygon taken in opposite order.”.c 1.24. Moment of a Force It is the turning effect, produced by the force, on a body on which it acts. It is mathematically ts equal to the product of the force and the perpendicular distance between the line of action of the force and the point about which the moment is required. an ir sp ga.c w w w www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com www.cgaspirants.com Chapter 23 om Propped Cantilevers and.c Beams ts Contents 1. Introduction. 2. Reaction of a Prop. an ir 3. Propped Cantilever with a Uniformly Distributed Load. 4. Cantilever Propped at an Interme- diate Point. sp 5. Simply Supported Beam with a Uniformly Distributed Load and Propped at the Centre. 6. Sinking of the Prop. ga 23.1. Introduction We have already discussed in chapters 19 and 20 that whenever a cantilever or a beam is loaded,.c it gets deflected. As a matter of fact, the amount by which a cantilever or a beam may deflect, is so small that it is hardly detected by the residents. But sometimes, due to inaccurate design or bad w workmanship, the deflection of