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  1. 1. Mechanics of Materials Second Edition Madhukar Vable Michigan Technological University
  2. 2. M. Vable II Mechanics of Materials: DEDICATED TO MY FATHER Professor Krishna Rao Vable (1911--2000) AND MY MOTHER Saudamini Gautam Vable (1921--2006)Printed from: January, 2010
  3. 3. M. Vable III Mechanics of Materials: Contents CONTENTS PREFACE XI ACKNOWLEDGEMENTS XII A NOTE TO STUDENTS XIV A NOTE TO THE INSTRUCTOR XVI CHAPTER ONE STRESS Section 1.1 Stress on a Surface 2 Section 1.1.1 Normal Stress 2 Section 1.1.2 Shear Stress 4 Section 1.1.3 Pins 5 Problem Set 1.1 9 MoM in Action: Pyramids 22 Section 1.1.4 Internally Distributed Force Systems 23 Quick Test 1.1 28 Problem Set 1.2 28 Section 1.2 Stress at a Point 30 Section 1.2.1 Sign convention 31 Section 1.3 Stress Elements 32 Section 1.3.1 Construction of a Stress Element for Axial Stress 32 Section 1.3.2 Construction of a Stress Element for Plane Stress 33 Section 1.4 Symmetric Shear Stresses 34 Section 1.5* Construction of a Stress Element in 3-dimension 36 Quick Test 1.2 39 Problem Set 1.3 39 Section 1.6* Concept Connector 43 History: The Concept of Stress 43 Section 1.7 Chapter Connector 44 Points and Formulas to Remember 46 CHAPTER TWO STRAIN Section 2.1 Displacement and Deformation 47 Section 2.2 Lagrangian and Eulerian Strain 48 Section 2.3 Average Strain 48 Section 2.3.1 Normal Strain 48 Section 2.3.2 Shear Strain 49 Section 2.3.3 Units of Average Strain 49 Problem Set 2.1 59Printed from: Section 2.4 Small-Strain Approximation 53 Section 2.4.1 Vector Approach to Small-Strain Approximation 57 MoM in Action: Challenger Disaster 70 Section 2.5 Strain Components 71 Section 2.5.1 Plane Strain 72 Quick Test 1.1 75 Problem Set 2.2 76 Section 2.6 Strain at a Point 73 Section 2.6.1 Strain at a Point on a Line 74 Section 2.7* Concept Connector 79 January, 2010
  4. 4. M. Vable IV Mechanics of Materials: Contents Section 2.7.1 History: The Concept of Strain 79 Section 2.7.2 Moiré Fringe Method 79 Section 2.8 Chapter Connector 81 Points and Formulas to Remember 82 CHAPTER THREE MECHANICAL PROPERTIES OF MATERIALS Section 3.1 Materials Characterization 83 Section 3.1.1 Tension Test 84 Section 3.1.2 Material Constants 86 Section 3.1.3 Compression Test 88 Section 3.1.4* Strain Energy 90 Section 3.2 The Logic of The Mechanics of Materials 93 Quick Test 3.1 98 Section 3.3 Failure and Factor of Safety 98 Problem Set 3.1 100 Section 3.4 Isotropy and Homogeneity 112 Section 3.5 Generalized Hooke’s Law for Isotropic Materials 113 Section 3.6 Plane Stress and Plane Strain 114 Quick Test 3.2 117 Problem Set 3.2 117 Section 3.7* Stress Concentration 122 Section 3.8* Saint-Venant’s Principle 122 Section 3.9* The Effect of Temperature 124 Problem Set 3.3 127 Section 3.10* Fatigue 129 MoM in Action: The Comet / High Speed Train Accident 131 Section 3.11* Nonlinear Material Models 132 Section 3.11.1 Elastic–Perfectly Plastic Material Model 132 Section 3.11.2 Linear Strain-Hardening Material Model 133 Section 3.11.3 Power-Law Model 133 Problem Set 3.4 139 Section 3.12* Concept Connector 141 Section 3.12.1 History: Material Constants 142 Section 3.12.2 Material Groups 143 Section 3.12.3 Composite Materials 143 Section 3.13 Chapter Connector 144 Points and Formulas to Remember 145 CHAPTER FOUR AXIAL MEMBERSPrinted from: Section 4.1 Prelude To Theory 146 Section 4.1.1 Internal Axial Force 148 Problem Set 4.1 150 Section 4.2 Theory of Axial Members 151 Section 4.2.1 Kinematics 152 Section 4.2.2 Strain Distribution 153 Section 4.2.3 Material Model 153 Section 4.2.4 Formulas for Axial Members 153 Section 4.2.5 Sign Convention for Internal Axial Force 154 Section 4.2.6 Location of Axial Force on the Cross Section 155 January, 2010
  5. 5. M. Vable V Mechanics of Materials: Contents Section 4.2.7 Axial Stresses and Strains 155 Section 4.2.8 Axial Force Diagram 157 Section 4.2.9* General Approach to Distributed Axial Forces 162 Quick Test 4.1 164 Problem Set 4.2 164 Section 4.3 Structural Analysis 171 Section 4.3.1 Statically Indeterminate Structures 171 Section 4.3.2 Force Method, or Flexibility Method 172 Section 4.3.3 Displacement Method, or Stiffness Method 172 Section 4.3.4 General Procedure for Indeterminate Structure 172 Problem Set 4.3 178 MoM in Action: Kansas City Walkway Disaster 187 Section 4.4* Initial Stress or Strain 188 Section 4.5* Temperature Effects 190 Problem Set 4.4 193 Section 4.6* Stress Approximation 194 Section 4.6.1 Free Surface 195 Section 4.6.2 Thin Bodies 195 Section 4.6.3 Axisymmetric Bodies 196 Section 4.6.4 Limitations 196 Section 4.7* Thin-Walled Pressure Vessels 197 Section 4.7.1 Cylindrical Vessels 197 Section 4.7.2 Spherical Vessels 199 Problem Set 4.5 200 Section 4.8* Concept Connector 202 Section 4.9 Chapter Connector 203 Points and Formulas to Remember 204 CHAPTER FIVE TORSION OF SHAFTS Section 5.1 Prelude to Theory 205 Section 5.1.1 Internal Torque 209 Problem Set 5.1 211 Section 5.2 Theory of torsion of Circular shafts 214 Section 5.2.1 Kinematics 215 Section 5.2.2 Material Model 216 Section 5.2.3 Torsion Formulas 217 Section 5.2.4 Sign Convention for Internal Torque 218 Section 5.2.5 Direction of Torsional Stresses by Inspection. 219 Section 5.2.6 Torque Diagram 222 Section 5.2.7* General Approach to Distributed Torque 228 Quick Test 5.1 238Printed from: MoM in Action: Drill, the Incredible Tool 230 Problem Set 5.2 231 Section 5.3 Statically Indeterminate Shafts 239 Problem Set 5.3 243 Section 5.4* Torsion of Thin-Walled Tubes 247 Problem Set 5.4 249 Section 5.5* Concept Connector 251 Section 5.5.1 History: Torsion of Shafts 251 Section 5.6 Chapter Connector 252 Points and Formulas to Remember 253 January, 2010
  6. 6. M. Vable VI Mechanics of Materials: Contents CHAPTER SIX SYMMETRIC BENDING OF BEAMS Section 6.1 Prelude to Theory 254 Section 6.1.1 Internal Bending Moment 258 Problem Set 6.1 260 Section 6.2 Theory of Symmetric Beam Bending 264 Section 6.2.1 Kinematics 265 Section 6.2.2 Strain Distribution 266 Section 6.2.3 Material Model 267 Section 6.2.4 Location of Neutral Axis 267 Section 6.2.5 Flexure Formulas 269 Section 6.2.6 Sign Conventions for Internal Moment and Shear Force 270 MoM in Action: Suspension Bridges 275 Problem Set 6.2 276 Section 6.3 Shear and Moment by Equilibrium 282 Section 6.4 Shear and Moment Diagrams 286 Section 6.4.1 Distributed Force 286 Section 6.4.2 Point Force and Moments 288 Section 6.4.3 Construction of Shear and Moment Diagrams 288 Section 6.5 Strength Beam Design 290 Section 6.5.1 Section Modulus 290 Section 6.5.2 Maximum Tensile and Compressive Bending Normal Stresses 291 Quick Test 6.1 295 Problem Set 6.3 295 Section 6.6 Shear Stress In Thin Symmetric Beams 301 Section 6.6.1 Shear Stress Direction 302 Section 6.6.2 Shear Flow Direction by Inspection 303 Section 6.6.3 Bending Shear Stress Formula 305 Section 6.6.4 Calculating Qz 306 Section 6.6.5 Shear Flow Formula 307 Section 6.6.6 Bending Stresses and Strains 308 Problem Set 6.4 315 Section 6.7* Concept Connector 321 Section 6.7.1 History: Stresses in Beam Bending 322 Section 6.8 Chapter Connector 323 Points and Formulas to Remember 324 CHAPTER SEVEN DEFLECTION OF SYMMETRIC BEAMS Section 7.1 Second-Order Boundary-Value Problem 325 Section 7.1.1 Boundary Conditions 326 Section 7.1.2 Continuity Conditions 326Printed from: MoM In Action: Leaf Springs 334 Problem Set 7.1 335 Section 7.2 Fourth-Order Boundary-Value Problem 339 Section 7.2.3 Boundary Conditions 340 Section 7.2.4 Continuity and Jump Conditions 341 Section 7.2.5 Use of Template in Boundary Conditions or Jump Conditions 341 Problem Set 7.2 348 MoM in Action: Skyscrapers 353 Section 7.3* Superposition 354 Section 7.4* Deflection by Discontinuity Functions 357 January, 2010
  7. 7. M. Vable VII Mechanics of Materials: Contents Section 7.4.1 Discontinuity Functions 357 Section 7.4.2 Use of Discontinuity Functions 359 Section 7.5* Area-Moment Method 364 Problem Set 7.3 367 Section *7.6 Concept Connector 369 Section 7.6.1 History: Beam Deflection 370 Section 7.7 Chapter Connector 371 Points and Formulas to remember 373 CHAPTER EIGHT STRESS TRANSFORMATION Section 8.1 Prelude to Theory: The Wedge Method 375 Section 8.1.1 Wedge Method Procedure 375 Problem Set 8.1 379 Section 8.2 Stress Transformation by Method of Equations 383 Section 8.2.1 Maximum Normal Stress 384 Section 8.2.2 Procedure for determining principal angle and stresses 384 Section 8.2.3 In-Plane Maximum Shear Stress 386 Section 8.2.4 Maximum Shear Stress 386 Quick Test 8.1 389 Section 8.3 Stress Transformation by Mohr’s Circle 389 Section 8.3.1 Construction of Mohr’s Circle 390 Section 8.3.2 Principal Stresses from Mohr’s Circle 391 Section 8.3.3 Maximum In-Plane Shear Stress 391 Section 8.3.4 Maximum Shear Stress 392 Section 8.3.5 Principal Stress Element 392 Section 8.3.6 Stresses on an Inclined Plane 393 Quick Test 8.2 400 MoM in Action: Sinking of Titanic 401 Problem Set 8.2 402 Quick Test 8.3 408 Section *8.4 Concept Connector 408 Section 8.4.1 Photoelasticity 409 Section 8.5 Chapter Connector 410 Points and Formulas to Remember 411 CHAPTER NINE STRAIN TRANSFORMATION Section 9.1 Prelude to Theory: The Line Method 412 Section 9.1.1 Line Method Procedure 413 Section 9.2.2 Visualizing Principal Strain Directions 419Printed from: Problem Set 9.1 414 Section 9.2 Method of Equations 415 Section 9.2.1 Principal Strains 413 Section 9.2.2 Visualizing Principal Strain Directions 419 Section 9.2.3 Maximum Shear Strain 420 Section 9.3 Mohr’s Circle 423 Section 9.3.1 Construction of Mohr’s Circle for Strains 424 Section 9.3.2 Strains in a Specified Coordinate System 425 Quick Test 9.1 428 Section 9.4 Generalized Hooke’s Law in Principal Coordinates 429 Problem Set 9.2 433 January, 2010
  8. 8. M. Vable VIII Mechanics of Materials: Contents Section 9.5 Strain Gages 436 Quick Test 9.2 446 MoM in Action: Load Cells 447 Problem Set 9.3 442 Section *9.6 Concept Connector 448 Section 9.6.1 History: Strain Gages 448 Section 9.7 Chapter Connector 449 Points and Formulas to Remember 450 CHAPTER TEN DESIGN AND FAILURE Section 10.1 Combined Loading 451 Section 10.1.1 Combined Axial and Torsional Loading 454 Section 10.1.2 Combined Axial, Torsional, and Bending Loads about z Axis 454 Section 10.1.3 Extension to Symmetric Bending about y Axis 454 Section 10.1.4 Combined Axial, Torsional, and Bending Loads about y and z Axes 455 Section 10.1.5 Stress and Strain Transformation 455 Section 10.1.6 Summary of Important Points in Combined Loading 456 Section 10.1.7 General Procedure for Combined Loading 456 Problem Set 10.1 468 Section 10.2 Analysis and Design of Structures 473 Section 10.2.1 Failure Envelope 473 Problem Set 10.2 480 MoM in Action: Biomimetics 485 Section 10.3 Failure Theories 486 Section 10.3.1 Maximum Shear Stress Theory 486 Section 10.3.2 Maximum Octahedral Shear Stress Theory 487 Section 10.3.3 Maximum Normal Stress Theory 488 Section 10.3.4 Mohr’s Failure Theory 488 Problem Set 10.3 491 Section 10.4 Concept Connector 492 Section 10.4.1 Reliability 492 Section 10.4.2 Load and Resistance Factor Design (LRFD) 493 Section 10.5 Chapter Connector 494 Points and Formulas to Remember 495 CHAPTER ELEVEN STABILITY OF COLUMNS Section 11.1 Buckling Phenomenon 496 Section 11.1.1 Energy Approach 496 Section 11.1.2 Eigenvalue Approach 497Printed from: Section 11.1.3 Bifurcation Problem 498 Section 11.1.4 Snap Buckling 498 Section 11.1.5 Local Buckling 499 Section 11.2 Euler Buckling 502 Section 11.2.1 Effects of End Conditions 504 Section 11.3* Imperfect Columns 518 Quick Test 11.1 511 Problem Set 11.2 511 MoM in Action: Collapse of World Trade Center 525 Section *11.4 Concept Connector 526 Section 11.4.1 History: Buckling 526 January, 2010
  9. 9. M. Vable IX Mechanics of Materials: Contents Section 11.5 Chapter Connector 527 Points and Formulas to Remember 528 APPENDIX A STATICS REVIEW Section A.1 Types of Forces and Moments 529 Section A.1.1 External Forces and Moments 529 Section A.1.2 Reaction Forces and Moments 529 Section A.1.3 Internal Forces and Moments 529 Section A.2 Free-Body Diagrams 530 Section A.3 Trusses 531 Section A.4 Centroids 532 Section A.5 Area Moments of Inertia 532 Section A.6 Statically Equivalent Load Systems 533 Section A.6.1 Distributed Force on a Line 533 Section A.6.2 Distributed Force on a Surface 534 Quick Test A.1 535 Static Review Exam 1 536 Static Review Exam 2 537 Points to Remember 538 APPENDIX B ALGORITHMS FOR NUMERICAL METHODS Section B.1 Numerical Integration 539 Section B.1.1 Algorithm for Numerical Integration 539 Section B.1.2 Use of a Spreadsheet for Numerical Integration 540 Section B.2 Root of a Function 540 Section B.2.1 Algorithm for Finding the Root of an Equation 541 Section B.2.2 Use of a Spreadsheet for Finding the Root of a Function 541 Section B.3 Determining Coefficients of a Polynomial 542 Section B.3.1 Algorithm for Finding Polynomial Coefficients 543 Section B.3.2 Use of a Spreadsheet for Finding Polynomial Coefficients 544 APPENDIX C REFERENCE INFORMATION Section C.1 Support Reactions 545 Table C.1 Reactions at the support 545 Section C.2 Geometric Properties of Common Shapes 546 Table C.2 Areas, centroids, and second area moments of inertia 546 Section C.3 Formulas For Deflection And Slopes Of Beams 547Printed from: Table C.3 Deflections and slopes of beams 547 Section C.4 Charts of Stress Concentration Factors 547 Figure C.4.1 Finite Plate with a Central Hole 548 Figure C.4.2 Stepped axial circular bars with shoulder fillet 548 Figure C.4.3 Stepped circular shafts with shoulder fillet in torsion 549 Figure C.4.4 Stepped circular beam with shoulder fillet in bending 549 Section C.5 Properties Of Selected Materials 550 Table C.4 Material properties in U.S. customary units 550 Table C.5 Material properties in metric units 550 Section C.6 Geometric Properties Of Structural Steel Members 551 Table C.6 Wide-flange sections (FPS units) 551 January, 2010
  10. 10. M. Vable X Mechanics of Materials: Contents Table C.7 Wide-flange sections (metric units) 551 Table C.8 S shapes (FPS units) 551 Table C.9 S shapes (metric units) 552 Section C.7 Glossary 552 Section C.8 Conversion Factors Between U.S. Customary System (USCS) and the Standard Interna- tional (SI) System 558 Section C.9 SI Prefixes 558 Section C.10 Greek Alphabet 558 APPENDIX D SOLUTIONS TO STATIC REVIEW EXAM 559 APPENDIX E ANSWERS TO QUICK TESTS 562 APPENDIX H ANSWERS TO SELECTED PROBLEMS 569 FORMULA SHEET 578Printed from: January, 2010
  11. 11. M. Vable XI Mechanics of Materials: Preface PREFACE Mechanics is the body of knowledge that deals with the relationships between forces and the motion of points through space, including the material space. Material science is the body of knowledge that deals with the properties of materials, including their mechanical properties. Mechanics is very deductive—having defined some variables and given some basic premises, one can logically deduce relationships between the variables. Material science is very empirical—having defined some variables one establishes the relationships between the variables experimentally. Mechanics of materials synthesizes the empirical relationships of materials into the logical framework of mechanics, to produce formulas for use in the design of structures and other solid bodies. There has been, and continues to be, a tremendous growth in mechanics, material science, and in new applications of mechanics of materials. Techniques such as the finite-element method and Moiré interferometry were research topics in mechanics, but today these techniques are used routinely in engineering design and analysis. Wood and metal were the pre- ferred materials in engineering design, but today machine components and structures may be made of plastics, ceramics, poly- mer composites, and metal-matrix composites. Mechanics of materials was primarily used for structural analysis in aerospace, civil, and mechanical engineering, but today mechanics of materials is used in electronic packaging, medical implants, the explanation of geological movements, and the manufacturing of wood products to meet specific strength requirements. Though the principles in mechanics of materials have not changed in the past hundred years, the presentation of these princi- ples must evolve to provide the students with a foundation that will permit them to readily incorporate the growing body of knowledge as an extension of the fundamental principles and not as something added on, and vaguely connected to what they already know. This has been my primary motivation for writing this book. Often one hears arguments that seem to suggest that intuitive development comes at the cost of mathematical logic and rigor, or the generalization of a mathematical approach comes at the expense of intuitive understanding. Yet the icons in the field of mechanics of materials, such as Cauchy, Euler, and Saint-Venant, were individuals who successfully gave physical meaning to the mathematics they used. Accounting of shear stress in the bending of beams is a beautiful demonstration of how the combination of intuition and experimental observations can point the way when self-consistent logic does not. Intui- tive understanding is a must—not only for creative engineering design but also for choosing the marching path of a mathemat- ical development. By the same token, it is not the heuristic-based arguments of the older books, but the logical development of arguments and ideas that provides students with the skills and principles necessary to organize the deluge of information in modern engineering. Building a complementary connection between intuition, experimental observations, and mathematical generalization is central to the design of this book. Learning the course content is not an end in itself, but a part of an educational process. Some of the serendipitous devel- opment of theories in mechanics of materials, the mistakes made and the controversies that arose from these mistakes, are all part of the human drama that has many educational values, including learning from others’ mistakes, the struggle in under- standing difficult concepts, and the fruits of perseverance. The connection of ideas and concepts discussed in a chapter to advanced modern techniques also has educational value, including continuity and integration of subject material, a starting reference point in a literature search, an alternative perspective, and an application of the subject material. Triumphs and trag- edies in engineering that arose from proper or improper applications of mechanics of materials concepts have emotive impactPrinted from: that helps in learning and retention of concepts according to neuroscience and education research. Incorporating educational values from history, advanced topics, and mechanics of materials in action or inaction, without distracting the student from the central ideas and concepts is an important complementary objective of this book. The achievement of these educational objectives is intricately tied to the degree to which the book satisfies the pedagogi- cal needs of the students. The Note to Students describes some of the features that address their pedagogical needs. The Note to the Instructor outlines the design and format of the book to meet the described objectives. I welcome any comments, suggestions, concerns, or corrections you may have that will help me improve the book. My e- mail address is January, 2010
  12. 12. M. Vable XII Mechanics of Materials: Acknowledgments ACKNOWLEDGMENTS A book, online or on in print, is shaped by many ideas, events, and people who have influenced an author. The first edition of this book was published by Oxford University Press. This second on-line edition was initially planned to be published also on paper and several professionals of Oxford University Press helped in its development to whom I am indebted. I am very grateful to Ms. Danielle Christensen who initiated this project, brought together lot of outstanding people, and contin- ued to support and advise me even when it was no longer her responsibility. The tremendous effort of Mr. John Haber is deeply appreciated who edited the entire book and oversaw reviews and checking of all the numerical examples. My thanks to Ms. Lauren Mine for the preliminary research on the modules called MoM in Action used in this book and to Ms. Adri- ana Hurtado for taking care of all the loose ends. I am also thankful to Mr. John Challice and Oxford University Press for their permissions to use the rendered art from my first edition of the book and for the use of some of the material that over- laps with my Intermediate Mechanics of Materials book (ISBN: 978-0-19-518855-4). Thirty reviewers looked at my manuscript and checked the numerical examples. Thanks to the following and anonymous reviewers whose constructive criticisms have significantly improved this book. Professor Berger of Colorado School of Mines. Professor Devries of University Of Utah. Professor, Leland of Oral Roberts University Professor Liao of Arizona State University Professor Rasty of Texas Tech University Professor Bernheisel of Union University Professor Capaldi of Drexel University Professor James of Texas A&M University Professor Jamil of University of Massachusetts, Lowell Professor Likos of University of Missouri Professor Manoogian of Loyola Marymount University Professor Miskioglu of Michigan Technological University Professor Rad of Washington State University Professor Rudnicki of Northwestern University Professor Spangler of Virginia TechPrinted from: Professor Subhash of University of Florida Professor Thompson of University of Georgia Professor Tomar of Purdue University Professor Tsai of Florida Atlantic University Professor Vallee of Western New England College January, 2010
  13. 13. M. Vable XIII Mechanics of Materials: Acknowledgments The photographs on Wikimedia Commons is an invaluable resource in constructing this online version of the book. There are variety of permissions that owners of photographs give for downloading, though there is no restriction for printing a copy for personal use. Photographs can be obtained from the web addresses below. Figure Description Web Address Number 1.1 S.S. Schenectady 1.36a Navier 1.36b Augustin Cauchy 2.1a Belt Drives 2.21a Challenger explosion 2.21b Shuttle Atlantis 3.51 Thomas Young 4.33a Kansas City Hyatt Regency walkway 5.42a Pierre Fauchard drill 5.42b Tunnel boring machine 5.55 Charles-Augustin Coulomb 6.33a Golden Gate bridge 6.33c Inca’s rope bridge. 6.128 Galileo’s beam experiment 6.72 Galileo Galilei. 7.1a Diving board. 7.14a Cart leaf springs 7.14b Leaf spring in cars 7.25a Empire State Building. 7.25b Taipei 101 7.25c Joint construction. 7.47 Daniel Bernoulli 8.33a RMS Titanic 8.33b Titanic bow at bottom of ocean. bow_seen_from_MIR_I_submersible.jpeg 8.33c Sliver Bridge.,_Ohio_side.jpg 10.42b Montreal bio-sphere. 11.20 World Trade Center Tower 11_Statue_of_Liberty_and_WTC_fire.jpg 11.21 Leonard Euler. 11.21 Joseph-Louis Lagrange. from: January, 2010
  14. 14. M. Vable XIV Mechanics of Materials: A note to students A NOTE TO STUDENTS Some of the features that should help you meet the learning objectives of this book are summarized here briefly. • A course in statics is a prerequisite for this course. Appendix A reviews the concepts of statics from the perspective of this course. If you had statics a few terms ago, then you may need to review your statics textbook before the brevity of presentation in Appendix A serves you adequately. If you feel comfortable with your knowledge of statics, then you can assess for yourself what you need to review by using the Statics Review Exams given in Appendix A. • All internal forces and moments are printed in bold italics. This is to emphasize that the internal forces and moments must be determined by making an imaginary cut, drawing a free-body diagram, and using equilibrium equations or by using methods that are derived from this approach. • Every chapter starts by listing the major learning objective(s) and a brief description of the motivation for studying the chapter. • Every chapter ends with Points and Formulas to Remember, a one-page synopsis of non-optional topics. This brings greater focus to the material that must be learned. • Every Example problem starts with a Plan and ends with Comments, both of which are specially set off to emphasize the importance of these two features. Developing a plan before solving a problem is essential for the development of analysis skills. Comments are observations deduced from the example, highlighting concepts discussed in the text pre- ceding the example, or observations that suggest the direction of development of concepts in the text following the example. • Quick Tests with solutions are designed to help you diagnose your understanding of the text material. To get the maxi- mum benefit from these tests, take them only after you feel comfortable with your understanding of the text material. • After a major topic you will see a box called Consolidate Your Knowledge. It will suggest that you either write a synopsis or derive a formula. Consolidate Your Knowledge is a learning device that is based on the observation that it is easy to follow someone else’s reasoning but significantly more difficult to develop one’s own reasoning. By deriving a formula with the book closed or by writing a synopsis of the text, you force yourself to think of details you would not otherwise. When you know your material well, writing will be easy and will not take much time. • Every chapter has at least one module called MoM in Action, describing a triumph or a tragedy in engineering or nature. These modules describe briefly the social impact and the phenomenological explanation of the triumph or trag- edy using mechanics of materials concept. • Every chapter has a section called Concept Connector, where connections of the chapter material to historical develop- ment and advanced topics are made. History shows that concepts are not an outcome of linear logical thinking, but rather a struggle in the dark in which mistakes were often made but the perseverance of pioneers has left us with a rich inheritance. Connection to advanced topics is an extrapolation of the concepts studied. Other reference material that may be helpful in the future can be found in problems labeled “Stretch yourself.” • Every chapter ends with Chapter Connector, which serves as a connecting link to the topics in subsequent chapters. OfPrinted from: particular importance are chapter connector sections in Chapters 3 and 7, as these are the two links connecting together three major parts of the book. • A glossary of all the important concepts is given in Appendix C.7 for easy reference.Chapters number are identified and in the chapter the corresponding word is highlighted in bold. • At the end is a Formula Sheet for easy reference. Only equations of non-optional topics are listed. There are no expla- nations of the variables or the equations in order to give your instructor the option of permitting the use of the formula sheet in an exam. January, 2010
  15. 15. M. Vable XV Mechanics of Materials: A note to the instructor A NOTE TO THE INSTRUCTOR The best way I can show you how the presentation of this book meets the objectives stated in the Preface is by drawing your attention to certain specific features. Described hereafter are the underlying design and motivation of presentation in the context of the development of theories of one-dimensional structural elements and the concept of stress. The same design philosophy and motivation permeate the rest of the book. Figure 3.15 (page 93) depicts the logic relating displacements—strains—stresses—internal forces and moments—exter- nal forces and moments. The logic is intrinsically very modular—equations relating the fundamental variables are indepen- dent of each other. Hence, complexity can be added at any point without affecting the other equations. This is brought to the attention of the reader in Example 3.5, where the stated problem is to determine the force exerted on a car carrier by a stretch cord holding a canoe in place. The problem is first solved as a straightforward application of the logic shown in Figure 3.15. Then, in comments following the example, it is shown how different complexities (in this case nonlinearities) can be added to improve the accuracy of the analysis. Associated with each complexity are post-text problems (numbers written in parenthe- ses) under the headings “Stretch yourself ” or “Computer problems,” which are well within the scope of students willing to stretch themselves. Thus the central focus in Example 3.5 is on learning the logic of Figure 3.15, which is fundamental to mechanics of materials. But the student can appreciate how complexities can be added to simplified analysis, even if no “Stretch yourself ” problems are solved. This philosophy, used in Example 3.5, is also used in developing the simplified theories of axial members, torsion of shafts, and bending of beams. The development of the theory for structural elements is done rigorously, with assumptions identified at each step. Footnotes and comments associated with an assumption directs the reader to examples, optional sec- tions, and “Stretch yourself ” problems, where the specific assumption is violated. Thus in Section 5.2 on the theory of the tor- sion of shafts, Assumption 5 of linearly elastic material has a footnote directing the reader to see “Stretch yourself ” problem 5.52 for nonlinear material behavior; Assumption 7 of material homogeneity across a cross section has a footnote directing the reader to see the optional “Stretch yourself ” problem 5.49 on composite shafts; and Assumption 9 of untapered shafts is fol- lowed by statements directing the reader to Example 5.9 on tapered shafts. Table 7.1 gives a synopsis of all three theories (axial, torsion, and bending) on a single page to show the underlying pattern in all theories in mechanics of materials that the students have seen three times. The central focus in all three cases remains the simplified basic theory, but the presentation in this book should help the students develop an appreciation of how different complexities can be added to the theory, even if no “Stretch yourself ” problems are solved or optional topics covered in class. Compact organization of information seems to some engineering students like an abstract reason for learning theory. Some students have difficulty visualizing a continuum as an assembly of infinitesimal elements whose behavior can be approximated or deduced. There are two features in the book that address these difficulties. I have included sections called Prelude to Theory in ‘Axial Members’, ‘Torsion of Circular Shafts’ and ‘Symmetric Bending of Beams.’ Here numerical problems are presented in which discrete bars welded to rigid plates are considered. The rigid plates are subjected to displace- ments that simulate the kinematic behavior of cross sections in axial, torsion or bending. Using the logic of Figure 3.15, the problems are solved—effectively developing the theory in a very intuitive manner. Then the section on theory consists essen- tially of formalizing the observations of the numerical problems in the prelude to theory. The second feature are actual photo-Printed from: graphs showing nondeformed and deformed grids due to axial, torsion, and bending loads. Seeing is believing is better than accepting on faith that a drawn deformed geometry represents an actual situation. In this manner the complementary connec- tion between intuition, observations, and mathematical generalization is achieved in the context of one-dimensional structural elements. Double subscripts1 are used with all stresses and strains. The use of double subscripts has three distinct benefits. (i) It pro- vides students with a procedural way to compute the direction of a stress component which they calculate from a stress for- mula. The procedure of using subscripts is explained in Section 1.3 and elaborated in Example 1.8. This procedural determination of the direction of a stress component on a surface can help many students overcome any shortcomings in intu- 1 Many authors use double subscripts with shear stress but not for normal stress. Hence they do not adequately elaborate the use of these sub- scripts when determining the direction of stress on a surface from the sign of the stress components. January, 2010