Mechanics of Materials, Second Edition

Mechanics of Materials, Second Edition

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Andrew Pytel
The Pennsylvania State University

Jaan Kiusalaas
The Pennsylvania State University

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Book Details
 576 p
 File Size 
 7,622 KB
 File Type
 PDF format
 2012, 2003 Cengage Learning 

This textbook is intended for use in a first course in mechanics of materials.
Programs of instruction relating to the mechanical sciences, such as mechanical,
civil, and aerospace engineering, often require that students take this
course in the second or third year of studies. Because of the fundamental
nature of the subject matter, mechanics of materials is often a required course,
or an acceptable technical elective in many other curricula. Students must
have completed courses in statics of rigid bodies and mathematics through
integral calculus as prerequisites to the study of mechanics of materials.
This edition maintains the organization of the previous edition. The
first eight chapters are dedicated exclusively to elastic analysis, including
stress, strain, torsion, bending and combined loading. An instructor can
easily teach these topics within the time constraints of a two-or three-credit
course. The remaining five chapters of the text cover materials that can be
omitted from an introductory course. Because these more advanced topics
are not interwoven in the early chapters on the basic theory, the core material
can e‰ciently be taught without skipping over topics within chapters.
Once the instructor has covered the material on elastic analysis, he or she
can freely choose topics from the more advanced later chapters, as time
permits. Organizing the material in this manner has created a significant
savings in the number of pages without sacrificing topics that are usually
found in an introductory text.
The most notable features of the organization of this text include the
following: . Chapter 1 introduces the concept of stress (including stresses acting on
inclined planes). However, the general stress transformation equations
and Mohr’s circle are deferred until Chapter 8. Engineering instructors
often hold o¤ teaching the concept of state of stress at a point due to
combined loading until students have gained su‰cient experience analyzing
axial, torsional, and bending loads. However, if instructors wish
to teach the general transformation equations and Mohr’s circle at the
beginning of the course, they may go to the freestanding discussion in
Chapter 8 and use it whenever they see fit. . Advanced beam topics, such as composite and curved beams, unsymmetrical bending, and shear center, appear in chapters that are distinct
from the basic beam theory. This makes it convenient for instructors to
choose only those topics that they wish to present in their course. . Chapter 12, entitled ‘‘Special Topics,’’ consolidates topics that are
important but not essential to an introductory course, including energy
methods, theories of failure, stress concentrations, and fatigue. Some,
but not all, of this material is commonly covered in a three-credit
course at the discretion of the instructor.
. Chapter 13, the final chapter of the text, discusses the fundamentals of
inelastic analysis. Positioning this topic at the end of the book enables
the instructor to present an e‰cient and coordinated treatment of
elastoplastic deformation, residual stress, and limit analysis after
students have learned the basics of elastic analysis. . Following reviewers’ suggestions, we have included a discussion of the torsion of rectangular bars. In addition, we have updated our
discussions of the design of columns and reinforced concrete beams.
The text contains an equal number of problems using SI and U.S. Customary
units. Homework problems strive to present a balance between directly
relevant engineering-type problems and ‘‘teaching’’ problems that illustrate the
principles in a straightforward manner. An outline of the applicable problemsolving
procedure is included in the text to help students make the sometimes
di‰cult transition from theory to problem analysis. Throughout the text and
the sample problems, free-body diagrams are used to identify the unknown
quantities and to recognize the number of independent equations. The three
basic concepts of mechanics—equilibrium, compatibility, and constitutive
equations—are continually reinforced in statically indeterminate problems.
The problems are arranged in the following manner: . Virtually every section in the text is followed by sample problems and homework problems that illustrate the principles and the problemsolving
procedure introduced in the article. . Every chapter contains review problems, with the exception of optional topics. In this way, the review problems test the students’ comprehension
of the material presented in the entire chapter, since it is not
always obvious which of the principles presented in the chapter apply to
the problem at hand. . Most chapters conclude with computer problems, the majority of
which are design oriented. Students should solve these problems using
a high-level language, such as MATHCAD= or MATLAB=, which
minimizes the programming e¤ort and permits them to concentrate on
the organization and presentation of the solution.

To access additional course materials, please visit www.cengagebrain.com
At the cengagebrain.com home page, search for the ISBN of your title (from the back cover of your book) using the search box at the top of the page, where these resources can be found, for instructors and students. The following ancillaries are available at www.cengagebrain.com. . Study Guide to Accompany Pytel and Kiusalaas Mechanics of Materials, Second Edition, J. L Pytel and A. Pytel, 2012. The goals of the Study Guide are twofold. First, self-tests are included to help the student focus on the salient features of the assigned reading. Second, the study guide uses ‘‘guided’’ problems which give the student an opportunity to work through representative problems before attempting to solve the
problems in the text. The Study Guide is provided free of charge. . The Instructor’s Solution Manual and PowerPoint slides of all figures and tables in the text are available to instructors through

We would like to thank the following reviewers for their
valuable suggestions and comments:
Roxann M. Hayes, Colorado School of Mines Daniel C. Jansen, California Polytechnic State University, San Luis Obispo Ghyslaine McClure, McGill University J.P. Mohsen, University of Louisville Hassan Rejali, California Polytechnic State University, Pomona In addition, we are indebted to Professor Thomas Gavigan, Berks
Campus, The Pennsylvania State University, for his diligent proofreading.
Andrew Pytel
Jaan Kiusalaas

Table of Contents
Stress 1
1.1 Introduction 1
1.2 Analysis of Internal Forces; Stress 2
1.3 Axially Loaded Bars 4
a. Centroidal (axial) loading 4
b. Saint Venant’s principle 5
c. Stresses on inclined planes 6
d. Procedure for stress analysis 7
1.4 Shear Stress 18
1.5 Bearing Stress 19

Strain 31
2.1 Introduction 31
2.2 Axial Deformation; Stress-Strain
Diagram 32
a. Normal (axial) strain 32
b. Tension test 33
c. Working stress and factor of safety 36
2.3 Axially Loaded Bars 36
2.4 Generalized Hooke’s Law 47
a. Uniaxial loading; Poisson’s ratio 47
b. Multiaxial loading 47
c. Shear loading 48
2.5 Statically Indeterminate Problems 54
2.6 Thermal Stresses 63

Torsion 75
3.1 Introduction 75
3.2 Torsion of Circular Shafts 76
a. Simplifying assumptions 76
b. Compatibility 77
c. Equilibrium 77
d. Torsion formulas 78
e. Power transmission 79
f. Statically indeterminate problems 80
3.3 Torsion of Thin-Walled Tubes 91
*3.4 Torsion of Rectangular Bars 99

Shear and Moment in Beams 107
4.1 Introduction 107
4.2 Supports and Loads 108
4.3 Shear-Moment Equations and
Shear-Moment Diagrams 109
a. Sign conventions 109
b. Procedure for determining shear
force and bending moment diagrams 110
4.4 Area Method for Drawing Shear-Moment Diagrams 122
a. Distributed loading 122
b. Concentrated forces and couples 124
c. Summary 126

Stresses in Beams 139
5.1 Introduction 139
5.2 Bending Stress 140
a. Simplifying assumptions 140
b. Compatibility 141
c. Equilibrium 142
d. Flexure formula; section modulus 143
e. Procedures for determining bending stresses 144
5.3 Economic Sections 158
a. Standard structural shapes 159
b. Procedure for selecting standard shapes 160
5.4 Shear Stress in Beams 164
a. Analysis of flexure action 164
b. Horizontal shear stress 165
c. Vertical shear stress 167
d. Discussion and limitations of the shear stress formula 167
e. Rectangular and wide-flange sections 168
f. Procedure for analysis of shear stress 169
5.5 Design for Flexure and Shear 177
5.6 Design of Fasteners in Built-Up Beams 184

Deflection of Beams 195
6.1 Introduction 195
6.2 Double-Integration Method 196
a. Di¤erential equation of the elastic curve 196
b. Double integration of the di¤erential equation 198
c. Procedure for double integration 199
6.3 Double Integration Using Bracket Functions 209
*6.4 Moment-Area Method 219
a. Moment-area theorems 220
b. Bending moment diagrams by parts 222
c. Application of the moment-area method 225
6.5 Method of Superposition 235

Statically Indeterminate Beams 249
7.1 Introduction 249
7.2 Double-Integration Method 250
7.3 Double Integration Using Bracket Functions 256
*7.4 Moment-Area Method 260
7.5 Method of Superposition 266

Stresses Due to Combined Loads 277
8.1 Introduction 277
8.2 Thin-Walled Pressure Vessels 278
a. Cylindrical vessels 278
b. Spherical vessels 280
8.3 Combined Axial and Lateral Loads 284
8.4 State of Stress at a Point (Plane Stress) 293
a. Reference planes 293
b. State of stress at a point 294
c. Sign convention and subscript notation 294
8.5 Transformation of Plane Stress 295
a. Transformation equations 295
b. Principal stresses and principal planes 296
c. Maximum in-plane shear stress 298
d. Summary of stress transformation procedures 298
8.6 Mohr’s Circle for Plane Stress 305
a. Construction of Mohr’s circle 306
b. Properties of Mohr’s circle 307
c. Verification of Mohr’s circle 308
8.7 Absolute Maximum Shear Stress 314
a. Plane state of stress 315
b. General state of stress 316
8.8 Applications of Stress Transformation to Combined Loads 319
8.9 Transformation of Strain; Mohr’s Circle for Strain 331
a. Review of strain 331
b. Transformation equations for plane strain 332
c. Mohr’s circle for strain 333
8.10 The Strain Rosette 338
a. Strain gages 338
b. Strain rosette 339
c. The 45 strain rosette 340
d. The 60 strain rosette 340
8.11 Relationship between Shear Modulus and
Modulus of Elasticity 342

Composite Beams 349
9.1 Introduction 349
9.2 Flexure Formula for Composite Beams 350
9.3 Shear Stress and Deflection in Composite Beams 355
a. Shear stress 355
b. Deflection 356
9.4 Reinforced Concrete Beams 359
a. Elastic Analysis 360
b. Ultimate moment analysis 361

Columns 371
10.1 Introduction 371
10.2 Critical Load 372
a. Definition of critical load 372
b. Euler’s formula 373
10.3 Discussion of Critical Loads 375
10.4 Design Formulas for Intermediate Columns 380
a. Tangent modulus theory 380
b. AISC specifications for steel columns 381
10.5 Eccentric Loading: Secant Formula 387
a. Derivation of the secant formula 388
b. Application of the secant formula 389

Additional Beam Topics 397
11.1 Introduction 397
11.2 Shear Flow in Thin-Walled Beams 398
11.3 Shear Center 400
11.4 Unsymmetrical Bending 407
a. Review of symmetrical bending 407
b. Symmetrical sections 408
c. Inclination of the neutral axis 409
d. Unsymmetrical sections 410
11.5 Curved Beams 415
a. Background 415
b. Compatibility 416
c. Equilibrium 417
d. Curved beam formula 418

Special Topics 425
12.1 Introduction 425
12.2 Energy Methods 426
a. Work and strain energy 426
b. Strain energy of bars and beams 426
c. Deflections by Castigliano’s theorem 428
12.3 Dynamic Loading 437
a. Assumptions 437
b. Mass-spring model 438
c. Elastic bodies 439
d. Modulus of resilience; modulus of toughness 439
12.4 Theories of Failure 444
a. Brittle materials 445
b. Ductile materials 446
12.5 Stress Concentration 452
12.6 Fatigue Under Repeated Loading 458

Inelastic Action 463
13.1 Introduction 463
13.2 Limit Torque 464
13.3 Limit Moment 466
13.4 Residual Stresses 471
a. Loading-unloading cycle 471
b. Torsion 471
c. Bending 472
d. Elastic spring-back 473
13.5 Limit Analysis 477
a. Axial loading 477
b. Torsion 478
c. Bending 479

Review of Properties of Plane Areas 487
A.1 First Moments of Area; Centroid 487
A.2 Second Moments of Area 488
a. Moments and product of inertia 488
b. Parallel-axis theorems 489
c. Radii of gyration 491
d. Method of composite areas 491
A.3 Transformation of Second Moments of Area 500
a. Transformation equations for
moments and products ofinertia 500
b. Comparison with stress transformation equations 501
c. Principal moments of inertia and principal axes 501
d. Mohr’s circle for second moments of area 502

Tables 509
B.1 Average Physical Properties of Common Metals 510
B.2 Properties of Wide-Flange Sections (W-Shapes): SI Units 512
B.3 Properties of I-Beam Sections (S-Shapes): SI Units 518
B.4 Properties of Channel Sections: SI Units 519
B.5 Properties of Equal and Unequal Angle
Sections: SI Units 520
B.6 Properties of Wide-Flange
Sections (W-Shapes): U.S. Customary Units 524
B.7 Properties of I-Beam Sections (S-Shapes): U.S. Customary Units 532
B.8 Properties of Channel Sections: U.S. Customary Units 534
B.9 Properties of Equal and Unequal Angle
Sections: U.S. Customary Units 535
Answers to Even-Numbered
Problems 539
Index 547


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