MECHANICS OF MATERIALS

Mechanics of Materials
Second Edition

Madhukar Vable
Michigan Technological University

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: http://www.me.mtu.edu/~mavable/MoM2nd.htm

January, 2010

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 59

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

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M. Vable IV

Section 2.7.1 History: The Concept of Strain 79
Section 2.7.2 Moiré Fringe Method 79

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 3.12.1 History: Material Constants 142
Section 3.12.2 Material Groups 143
Section 3.12.3 Composite Materials 143

CHAPTER FOUR AXIAL MEMBERS

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

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M. Vable V

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

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 5.2.3 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 5.1 238

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 5.5.1 History: Torsion of Shafts 251

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M. Vable VI

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 6.2.2 Strain Distribution 266
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 6.7.1 History: Stresses in Beam Bending 322

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 326

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

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M. Vable VII

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
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 8.4.1 Photoelasticity 409

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 419

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

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M. Vable VIII

Section 9.5 Strain Gages 436
Quick Test 9.2 446
MoM in Action: Load Cells 447
Problem Set 9.3 442
Section 9.6.1 History: Strain Gages 448

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
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
Section 10.4 Concept Connector 492
Section 10.4.1 Reliability 492
Section 10.4.2 Load and Resistance Factor Design (LRFD) 493

CHAPTER ELEVEN STABILITY OF COLUMNS

Section 11.1 Buckling Phenomenon 496
Section 11.1.1 Energy Approach 496
Section 11.1.2 Eigenvalue Approach 497

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
MoM in Action: Collapse of World Trade Center 525
Section 11.4.1 History: Buckling 526

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M. Vable IX


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 547

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

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M. Vable X

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 578

January, 2010

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 impact

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 mavable@mtu.edu.

January, 2010

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 Tech

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

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 http://en.wikipedia.org/wiki/File:TankerSchenectady.jpg
1.36a Navier http://commons.wikimedia.org/wiki/File:Claude-Louis_Navier.jpg
1.36b Augustin Cauchy http://commons.wikimedia.org/wiki/File:Augustin_Louis_Cauchy.JPG
2.1a Belt Drives http://commons.wikimedia.org/wiki/File:MG_0913_dreikrempelsatz.jpg
2.21a Challenger explosion http://commons.wikimedia.org/wiki/File:Challenger_explosion.jpg
2.21b Shuttle Atlantis http://commons.wikimedia.org/wiki/File:AtlantisLP39A_STS_125.jpg
3.51 Thomas Young http://commons.wikimedia.org/wiki/File:Thomas_Young_(scientist).jpg#filehistory
4.33a Kansas City Hyatt Regency walkway http://commons.wikimedia.org/wiki/File:Kansas_City_Hyatt_Regency_Walkways_Collapse_11.gif
5.42a Pierre Fauchard drill http://en.wikipedia.org/wiki/File:Fauchard-drill.jpg
5.42b Tunnel boring machine http://commons.wikimedia.org/wiki/File:Matilda_TBM.jpg
5.55 Charles-Augustin Coulomb http://commons.wikimedia.org/wiki/File:Coulomb.jpg
6.33a Golden Gate bridge http://commons.wikimedia.org/wiki/File:GoldenGateBridge-001.jpg
6.33c Inca’s rope bridge. http://commons.wikimedia.org/wiki/File:Inca_bridge.jpg
6.128 Galileo’s beam experiment http://commons.wikimedia.org/wiki/File:Discorsi_Festigkeitsdiskussion.jpg
6.72 Galileo Galilei. http://commons.wikimedia.org/wiki/File:Galileo_Galilei_3.jpg
7.1a Diving board. http://commons.wikimedia.org/wiki/File:Diving.jpg
7.14a Cart leaf springs http://en.wikipedia.org/wiki/File:Red_Brougham_Profile_view.jpg
7.14b Leaf spring in cars http://en.wikipedia.org/wiki/File:Leafs1.jpg
7.25a Empire State Building. http://upload.wikimedia.org/wikipedia/commons/f/fb/EPS_in_NYC_2006.jpg
7.25b Taipei 101 http://commons.wikimedia.org/wiki/File:31-January-2004-Taipei101-Complete.jpg
7.25c Joint construction. http://commons.wikimedia.org/wiki/File:Old_timer_structural_worker2.jpg
7.47 Daniel Bernoulli http://commons.wikimedia.org/wiki/File:Daniel_Bernoulli_001.jpg
8.33a RMS Titanic http://commons.wikimedia.org/wiki/File:RMS_Titanic_3.jpg
8.33b Titanic bow at bottom of ocean. http://commons.wikimedia.org/wiki/File:Titanic bow_seen_from_MIR_I_submersible.jpeg
8.33c Sliver Bridge. http://commons.wikimedia.org/wiki/File:Silver_Bridge_collapsed,_Ohio_side.jpg
10.42b Montreal bio-sphere. http://commons.wikimedia.org/wiki/File:Biosphere_montreal.JPG
11.20 World Trade Center Tower http://en.wikipedia.org/wiki/File:National_Park_Service_9-
11_Statue_of_Liberty_and_WTC_fire.jpg
11.21 Leonard Euler. http://commons.wikimedia.org/wiki/File:Leonhard_Euler_2.jpg
11.21 Joseph-Louis Lagrange. http://commons.wikimedia.org/wiki/File:Joseph_Louis_Lagrange.jpg

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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. Of

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.

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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-

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

M. Vable XVI
Mechanics of Materials: A note to the instructor

itive ability. (ii) Computer programs, such as the finite-element method or those that reduce full-field experimental data, pro-
duce stress and strain values in a specific coordinate system that must be properly interpreted, which is possible if students
know how to use subscripts in determining the direction of stress on a surface. (iii) It is consistent with what the student will
see in more advanced courses such as those on composites, where the material behavior can challenge many intuitive expecta-
tions.
But it must be emphasized that the use of subscripts is to complement not substitute an intuitive determination of stress
direction. Procedures for determining the direction of a stress component by inspection and by subscripts are briefly described
at the end of each theory section of structural elements. Examples such as 4.3 on axial members, 5.6 and 5.9 on torsional shear
stress, and 6.8 on bending normal stress emphasize both approaches. Similarly there are sets of problems in which the stress
direction must be determined by inspection as there are no numbers given—problems such as 5.23 through 5.26 on the direc-
tion of torsional shear stress; 6.35 through 6.40 on the tensile and compressive nature of bending normal stress; and 8.1
through 8.9 on the direction of normal and shear stresses on an inclined plane. If subscripts are to be used successfully in
determining the direction of a stress component obtained from a formula, then the sign conventions for drawing internal
forces and moments on free-body diagrams must be followed. Hence there are examples (such as 6.6) and problems (such as
6.32 to 6.34) in which the signs of internal quantities are to be determined by sign conventions. Thus, once more, the comple-
mentary connection between intuition and mathematical generalization is enhanced by using double subscripts for stresses
and strains.
Other features that you may find useful are described briefly.
All optional topics and examples are marked by an asterisk (*) to account for instructor interest and pace. Skipping these
topics can at most affect the student’s ability to solve some post-text problems in subsequent chapters, and these problems are
easily identifiable.
Concept Connector is an optional section in all chapters. In some examples and post-text problems, reference is made to
a topic that is described under concept connector. The only purpose of this reference is to draw attention to the topic, but
knowledge about the topic is not needed for solving the problem.
The topics of stress and strain transformation can be moved before the discussion of structural elements (Chapter 4). I
strived to eliminate confusion regarding maximum normal and shear stress at a point with the maximum values of stress com-
ponents calculated from the formulas developed for structural elements.
The post-text problems are categorized for ease of selection for discussion and assignments. Generally speaking, the
starting problems in each problem set are single-concept problems. This is particularly true in the later chapters, where prob-
lems are designed to be solved by inspection to encourage the development of intuitive ability. Design problems involve the
sizing of members, selection of materials (later chapters) to minimize weight, determination of maximum allowable load to
fulfill one or more limitations on stress or deformation, and construction and use of failure envelopes in optimum design
(Chapter 10)—and are in color. “Stretch yourself ” problems are optional problems for motivating and challenging students
who have spent time and effort understanding the theory. These problems often involve an extension of the theory to include
added complexities. “Computer” problems are also optional problems and require a knowledge of spreadsheets, or of simple
numerical methods such as numerical integration, roots of a nonlinear equation in some design variable, or use of the least-
squares method. Additional categories such as “Stress concentration factor,” “Fatigue,” and “Transmission of power” prob-
lems are chapter-specific optional problems associated with optional text sections.

January, 2010

MECHANICS OF MATERIALS

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