Beer F.P., Johnston E.R., DeWolf J.T., Mazurek D.F. Mechanics of Materials

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Fundamentals of Engineering

Examination

Appendix E

Engineers are required to be licensed when their work directly

affects the public health, safety, and welfare. The intent is to ensure

that engineers have met minimum qualifications, involving compe-

tence, ability, experience, and character. The licensing process

involves an initial exam, called the Fundamentals of Engineering

Examination, professional experience, and a second exam, called the

Principles and Practice of Engineering. Those who successfully com-

plete these requirements are licensed as a Professional Engineer. The

exams are developed under the auspices of the National Council of

Examiners for Engineering and Surveying.

The first exam, the Fundamentals of Engineering Examination,

can be taken just before or after graduation from a four-year accred-

ited engineering program. The exam stresses subject material in a

typical undergraduate engineering program, including Mechanics of

Materials. The topics included in the exam cover much of the mate-

rial in this book. The following is a list of the main topic areas, with

references to the appropriate sections in this book. Also included are

problems that can be solved to review this material.

Stresses (1.3–1.8; 1.11–1.12)

Problems: 1.1, 1.7, 1.31, 1.37

Strains (2.2–2.3; 2.5–2.6; 2.8–2.11; 2.14–2.15)

Problems: 2.7, 2.19, 2.41, 2.49, 2.63, 2.68

Torsion (3.2–3.6; 3.13)

Problems: 3.6, 3.28, 3.35, 3.51, 3.132, 3.138

Bending (4.2–4.6; 4.12)

Problems: 4.11, 4.23, 4.34, 4.47, 4.104, 4.109

A29

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Shear and Bending-Moment Diagrams (5.2–5.3)

Problems: 5.6, 5.9, 5.42, 5.48

Normal Stresses in Beams (5.1–5.3)

Problems: 5.18, 5.21, 5.55, 5.61

Shear (6.2–6.4; 6.6–6.7)

Problems: 6.2, 6.12, 6.32, 6.36

Transformation of Stresses and Strains (7.2–7.4; 7.7–7.9)

Problems: 7.6, 7.13, 7.33, 7.41, 7.81, 7.87, 7.102, 7.109

Deflection of Beams (9.2–9.4; 9.7)

Problems: 9.6, 9.10, 9.72, 9.75

Columns (10.2–10.4)

Problems: 10.11, 10.21, 10.28

Strain Energy (11.2–11.4)

Problems: 11.10, 11.14, 11.19

A30

Appendix E

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Photo Credits

CHAPTER 1

Opener: © Construction Photography/CORBIS RF; 1.1: © Vince

Streano/CORBIS; 1.2: © John DeWolf.

CHAPTER 2

Opener: © Construction Photography/CORBIS; 2.1: © John DeWolf;

2.2: Courtesy of Tinius Olsen Testing Machine Co., Inc.; 2.3, 2.4,

2.5: © John DeWolf.

CHAPTER 3

CHAPTER 4

Opener: © Lawrence Manning/CORBIS; 4.1: Courtesy of Flexifoil;

4.2: © Tony Freeman/Photo Edit; 4.3: © Hisham Ibrahim/Getty

Images RF; 4.4: © Kevin R. Morris/CORBIS; 4.5: © Tony Freeman/

Photo Edit; 4.6: © John DeWolf.

CHAPTER 5

Opener: © Mark Segal/Digital Vision/Getty Images RF; 5.1: © David

Papazian/CORBIS RF; 5.2: © Godden Collection, National Information

Service for Earthquake Engineering, University of California, Berkeley.

CHAPTER 6

Opener: © Godden Collection, National Information Service for

Earthquake Engineering, University of California, Berkeley; 6.1:

shutterstock.com.

CHAPTER 7

Opener: NASA; 7.1: © Radlund & Associates/Getty Images RF;

7.2: © Spencer C. Grant/Photo Edit; 7.3: © Clair Dunn/Alamy;

7.4: © Spencer C. Grant/Photo Edit.

CHAPTER 8

Opener: © Mark Read.

CHAPTER 9

Opener: © Construction Photography/CORBIS; 9.1: Royalty-

Free/CORBIS; 9.2 and 9.3: © John DeWolf; 9.4: Courtesy of Aztec

Galvenizing Services; 9.5: Royalty-Free/CORBIS.

CHAPTER 10

Opener: © Jose Manuel/Photographer’s Choice RF/Getty Images; 10.1:

Engineering, University of California, Berkeley; 10.2b: © Peter Marlow/

Magnum Photos.

CHAPTER 11

Opener: © Corbis Super RF/Alamy; 11.1: © Daniel Schwen; 11.2:

Sec Envel.

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Index

Accuracy, numerical, 17, 44

Actual deformation, 95, 99

Allowable load and allowable stress, 4

factor of safety, 31–32, 44

shearing stresses, 156–158

Allowable-stress method, 235

design of columns under an eccentric load, 675–676, 685–686

Aluminum

design of columns under a centric load, 664–665

properties of, 58, 60, 129, A12–A13

structural tubing, 202

American Forest & Paper Association, 665

American Institute of Steel Construction, 662, 667

American standard beams (S-beams), 231, 388

American standard channel steel (C shapes), properties of,

A22–A23

American standard shape steel (S shapes), properties of,

A20–A21

American wide-flange beam (W-beam), 231, 388

Analysis and design of beams for bending, 314–379

computer problems, 378–379

design of prismatic beams for bending, 339–349, 370,

371–372

introduction, 316–319

nonprismatic beams, 361–369, 373

relations among load, shear, and bending moment,

329–339, 371

review problems, 374–377

shear and bending-moment diagrams, 319–328, 370–371

summary, 370–373

using singularity functions to determine shear and bending

moment in a beam, 350–361, 372–373

Analysis and design of simple structures, 14–16

determining bearing stresses, 16

determining normal stress, 14–15

determining shearing stress, 15–16

Angle of twist, 143, 145–147, 189

adding algebraically, 161

in elastic range, 159–163, 165, 211

Angle steel

equal legs, A24–A25

properties of, A24–A27

unequal legs, A26–A27

Anisotropic materials, 63, 130

Anticlastic curvature, 234, 306

Areas. See Moments of areas

Average value, of stresses, 9, 42

Axes

centroidal, A6, A9–A10

of symmetry, A3

Axial loading

bearing stress in connections, 13, 43

centric, 42

deformations under, 67–71, 101–103

eccentric, 42, 284–293, 308

normal stress, 9–11, 42

shearing stress, 11–13, 43

slowly increasing, 694

stress and strain distribution under, 52–139

stress and strain in, 138–139

Axisymmetry, of circular shafts, 146, 197

Bauschinger effect, 65

Beam deflections and slopes, 585, 720–721, 725, A28

Beam elements

of arbitrary curved surface, longitudinal shear on, 428

of arbitrary shape, longitudinal shear on, 399–400

shear on the horizontal face of, 384–386, 427

Beams. See also Analysis and design of beams for bending

of constant strength, 373

nonprismatic, 318, 361–369, 373

overhanging, 554

simply-supported, 554

statically indeterminate, 561–571, 620

of variable cross section, 551

Bearing stresses, 4, 13, 16, 18, 43

in connections, 13, 43

determination of, 16

Bearing surfaces, 13, 43

Bend and twist, 415, 420

Bending. See also Pure bending

analysis and design of beams for, 314–379

of curved members, 294–304, 308

of members made of several materials, 242–245, 306

stresses due to, 419, 531, 679

Bending moment, 225, 235, 263

relation to shear, 330–335

Bending-moment diagrams, 318–328, 333–335, 370–371

by parts, 551, 597–604, 623

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Boundary conditions, 554, 564–565, 574–576, 619

Breaking strength, 59

Brittle materials, 54, 58–61, 129

under plane stress, fracture criteria for, 469–477, 505

sudden failure of, 32, 151

Bulk modulus, 55, 96–98, 132

C shapes. See Standard shape steel channels

Cantilever beams, 554, 595, 623

and beams with symmetric loadings, 595–596, 623

Cast iron, properties of, A12–A13

Castigliano, Alberto, 735

Castigliano’s theorem, 694, 734–735, 753

deflections by, 736–739

Center of symmetry, 421, A3

Centric loading, 10, 223, 270

axial, 42

design of columns under, 660–674, 686

Centroid, 236

of an area, A2–A4

of a composite area, A4–A6

Centroidal axis, A6, A9–A10

Centroidal moment of inertia, 236, 400, 407, 515

Circular shafts

as axisymmetric, 146, 197

deformations in, 144–148, 184–186, 210, 212

made of an elastoplastic material, 186–189, 212–213

Clebsch, A., 354

Coefficients

influence, 732

of thermal expansion, 82, 131

Columns, 630–691

computer problems, 690–691

critical load, 684

design of under a centric load, 660–674, 686

design of under an eccentric load, 675–683, 685–686

eccentric loading, 649–660, 685–686

effective length, 632, 685

Euler’s formula for pin-ended columns, 635–638,

684–685

extension of Euler’s formula to columns with other end

conditions, 638–649

introduction, 632

review problems, 687–689

the secant formula, 632, 649–660, 685–686

slenderness ratio, 685

stability of structures, 632–635

summary, 684–686

Combined loadings, stresses under, 527–539, 613

Combined stresses, 419

Components of stress, 4, 27–30

Composite materials, 224

fiber-reinforced, stress-strain relationships

for, 103–107, 134

Compression, 227

modulus of, 97

Computations, 17

errors in, 17

Computer problems

analysis and design of beams for bending, 378–379

applying singularity functions to determine shear and

bending moment in a beam, 355

axial loading, 138–139

columns, 690–691

concept of stress, 49–51

deflection of beams, 627–629

energy methods, 757–758

principal stresses under a given loading, 545–547

pure bending, 312–313

shearing stresses in beams and thin-walled members,

434–435

torsion, 218–219

transformations of stress and strain, 510–511

Concentrated loads, 316

single, 720

Concentric stress, 679

Concept of stress, 2–51

computer problems, 49–51

Concrete

maximum stress in, 249

properties of, 129, A14–A15

renforced beams of, 245

stress-strain diagram for, 61

Constant strength, 319, 362, 373

Constants of integration, determination of, 558

Copper, properties of, A12–A13

Coulomb, Charles Augustin de, 469–470

Coulomb’s criterion, 469

Creep, 64

Critical load, 634

on columns, 684

Critical stress, 636

Cupronickel, properties of, A14–A15

Curvature, 232

anticlastic, 234, 306

radius of, 224, 235, 263

Curved members, bending of, 294–304, 308

Cylindrical thin-walled pressure vessels, stresses in, 505

Dead load, 33

Deflection of beams, 70, 86–87, 548–629

applying cantilever beams and beams with symmetric

loadings, 595–596, 623

applying moment-area theorems to beams with unsymmetric

loadings, 605–606, 625–626

applying superposition to statically indeterminate beams,

582–592, 621

bending-moment diagrams by parts, 597–604, 623

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Deflection of beams—Cont.

boundary conditions, 619

by Castigliano’s theorem, 736–739, 753

computer problems, 627–629

direct determination of the elastic curve from the load

distribution, 559–560

equation of the elastic curve, 553–558, 619

introduction, 550–552

maximum, 607–608, 624, 694, 722, 725, A28

method of superposition, 580–582, 585–587, 621

moment-area theorems, 592–595, 621–622

review problems, 625–626

under a single load, 722–732

statically indeterminate beams, 561–571, 620

summary, 618–624

under transverse loading, 552–553, 618

using moment-area theorems with statically indeterminate

beams, 609–617, 624

using singularity functions to determine, 571–580,

620–621

by the work-energy method, 722–732

Deformations, 54, 86–87, 113, 167, 225, 561, 610. See also

Elastic deformations; Plastic deformations

actual, 95, 99

under axial loading, 67–71, 101–103

of a beam under transverse loading, 552–553, 618

in a circular shaft, 144–148, 210

computing, 17

maximum, 716

permanent, 224

in a symmetric member in pure bending, 226–228

in a transverse cross section, 233–241, 306

Design considerations, 30–35. See also Analysis and design

allowable load and allowable stress, 31–32, 44

determination of the ultimate strength of a material,

30–31

factor of safety, 44

for impact loads, 718–719

load and resistance factors, 33, 44, 341–343

for loads, 31

of prismatic beams for bending, 339–349, 370–372

selection of an appropriate factor of safety, 31–33

specifications of, 33

of transmission shafts, 143, 176–178, 518–527, 541

*Design considerations, of transmission shafts, 211

Design of columns

allowable-stress method, 662–664, 675–676, 686

aluminum, 664–665

under a centric load, 660–674, 686

under an eccentric load, 675–683, 686

for greatest efficiency, 643

interaction method, 676–677, 686

with load and resistance factor design, 667–669

structural steel, 662–664, 667–669

wood, 665–667

Deterioration, 32

Determination

of the bearing stresses, 16

of constants of integration, 558

of elastic curve, 559–560

of first moment, A4–A6

of forces, 113, 441

of the moment of inertia of a composite area, A10–A11

of the normal stress, 14–15

of the shearing stress, 15–16

of the shearing stresses in a beam, 386–387, 428

of the ultimate strength of a material, 30–31, 44

Deviation, tangential, 594

Diagonal stays, 52–53

Diagrams

free-body, 4, 17–18, 34–35, 42, 70–71

loading, 357

of shear, 319–328, 333–335, 342–343, 370–371

of shear and bending-moment, 319–328, 370–371,

597–604, 623

of stress-strain relationships, 54, 56–61, 129, 186, 716

Dilatation, 97, 132

bulk modulus, 96–98, 132

Dimensionless quantities, 56

Discontinuity, 350

Displacement, relative, 69

Distributed loading, 316, 613

Distribution of stresses

in a narrow rectangular beam, 390–399, 428

over the section, 418–419

statically indeterminate, 10

Double shear, 13

Ductile materials, 54, 58–60, 129, 151

under plane stress, yield criteria for, 467–469, 504

Eccentric axial loading, 42, 224

general case of, 284–293, 308

in a plane of symmetry, 270–278, 307

Eccentric loading, 223, 270

columns under, 649–660, 686

design of columns under, 675–683, 686

Effective length, of columns, 632, 685

Efficient design, for columns, 643

Elastic action, 123

Elastic core, radius of, 189

Elastic curve

direct determination from the load distribution, 559–560

equation of, 553–558, 563–565, 574–576, 619, A28

Elastic deformations, 229–232, 305

under axial loading, 130

Elastic flexure formula, 230, 305

Elastic limit, 63–64, 130

Elastic range, 229

angle of twist in, 159–163, 211

shearing stresses within, 210

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Elastic section modulus, 230, 259, 306

Elastic strain energy

under axial loading, 699–700, 751

in bending, 700–701, 751

for normal stresses, 698

for shearing stresses, 701–703, 751

in torsion, 701–702, 751

under transverse loading, 703

Elastic torque

formulas for, 149

maximum, 187, 213

Elastic torsion, formulas for, 210

Elastic unloading, 193, 265

Elastic versus plastic behavior of a material, 64–65, 130

Elasticity, modulus of, 54, 62–64, 130

Elastoplastic materials, 117, 134, 224, 256–257, 307

circular shafts made of, 186–189, 212–213

members made of, 256–260

Elementary work, 695

Elongation

maximum, 119

percent, 61

Endurance limit, 66, 130

Energy methods, 692–758

Castigliano’s theorem, 734–735, 753

computer problems, 757–758

deflection under a single load by the work-energy method,

722–732

deflections by Castigliano’s theorem, 736–739, 753

design for impact loads, 718–719

elastic strain energy for normal stresses, 698

elastic strain energy for shearing stresses, 701–703, 751

equivalent static load, 752

impact loading, 716–718, 752

introduction, 694

modulus of resilience, 751

modulus of toughness, 750–751

review problems, 754–756

statically indeterminate structures, 740–749, 753

strain energy, 694–696, 750

strain-energy density, 696–698, 750

strain energy for a general state of stress, 704–715, 752

summary, 750–753

work and energy under a single load, 719–722, 752–753

work and energy under several loads, 732–734

Engineering strain, 62

Engineering stress, 61

Equal-leg angle steel, A24–A25

Equations

of the elastic curve, 553–558, 563–565, 574–576, 619, A28

equilibrium, 43

of statics, 152

Equilibrium equations, 43

Equivalent force-couple system, at shear center, 419

Equivalent open-ended loadings, 373

Equivalent static load, 721–722, 752

Euler, Leonhard, 636

Euler’s formula, 632, 636, 654

extension to columns with other end conditions, 638–649

for pin-ended columns, 635–638, 684–685

Experimental materials, 93

Factor of safety, 44, 707

selection of appropriate, 31–33

Failure, of shaft, 185

Fatigue

limit of, 67

from repeated loadings, 32, 54, 66–67, 130

Fiber-reinforced composite materials, 63–64

stress-strain relationships for, 103–107, 130, 133

First moment, 385, A2–A6

determination of, A4–A6

First moment-area theorem, 551, 593, 598–601, 606, 621–622

Flexural rigidity, 554, 596, 619

Flexural stress, 230

Force-couple system, at shear center, equivalent, 419

Forces

determination of, 113, 441

unknown, 43

Formulas

elastic flexure, 230, 305

elastic torsion, 149, 210

Euler’s, 632, 635–649, 654

interaction, 676–677

secant, 632, 649–660, 685–686

Fracture criteria for brittle materials under plane stress, 439,

469–477, 505

maximum-normal-stress criterion, 469–470

Mohr’s criterion, 470–471

Free-body diagrams, 4, 17–18, 34–35, 42, 70–71

Fundamentals of Engineering Examination, A29–A30

Gages

length, 57

pressure, 478, 496

strain, 440

Gyration, radius of, A7–A9

Hardening, strain, 64

Hertz (Hz), 177, 212

Homogeneous materials, 93

Hooke, Robert, 62

Hooke’s law, 107, 117, 133, 148, 184, 186

generalized, 93–96, 100, 104, 132

modulus of elasticity, 62–64, 67, 130

Hoop stress, 478

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Horizontal shear, 385

Horsepower (hp), 212

Hydrostatic pressure, 97

Hz. See Hertz

IF/THEN/ELSE statements, 355

Impact loading, 694, 716–718, 752

Inertia. See Moments of inertia

Influence coefficients, 732

Integration

constants of, 558

methods of, 628

Interaction formula, 676–677

Interaction method, design of columns under an eccentric load,

676–677, 686

Internal torques, 150, 163

Iron. See Cast iron

Isotropic materials, 63, 93, 113, 130

Joule (J), 695

Kinetic energy, 716

Lamina, 63

Laminates, 105

Lateral strain, 93, 132

Line of action, of loading, 113

Load and Resistance Factor Design (LRFD), 33, 44, 341–343.

See also Allowable load and allowable stress

Load distribution, direct determination of the elastic curve

from, 559–560

Loading diagram, modified, 357

Loadings. See also Unloading

axial, 9–13, 42–43, 52–139, 284–293, 308

centric, 10, 42, 223, 270, 660–674, 686

combined, 527–539, 613

concentrated, 316

dead, 33

distributed, 316, 613

eccentric, 223, 270–278, 284–293, 307–308, 649–660,

675–683, 686

general conditions of, 27–30, 44, 541

impact, 694, 716–718, 752

line of action of, 113

multiaxial, 94–96, 132

open-ended, 373

redundant reaction, 584, 613

relation to shear, 329–330

Loadings—Cont.

repeated, 66–67, 130

statically equivalent, 114

symmetric, 595–596, 623

torsional, 519

transverse, 223, 316, 552–553, 618

ultimate, 31, 33, 667

unknown, 79–80

unsymmetric, 414–426, 429, 605–606, 625–626

visualizing, 234

Longitudinal normal strain, 228

Longitudinal shear

on a beam element of arbitrary curved surface, 428

on a beam element of arbitrary shape, 399–400

Longitudinal stress, 478–479

Lower yield point, 60

LRFD. See Load and resistance factor design

Macaulay, W.H., 354

Macaulay’s brackets, 354

Macroscopic cracks or cavities, detected in a structural

component, 471

Magnesium alloys, properties of, A14–A15

Margin of safety, 31

Materials. See also Anisotropic materials; Brittle materials;

Composite materials; Ductile materials; Elastoplastic

materials; Experimental materials; Homogeneous

materials; Isotropic materials; Orthotropic materials

bending of members made of several, 242–245, 306

determining ultimate strength of, 30–31

elastic versus plastic behavior of, 64–65, 130

Materials used in engineering, A12–A15

aluminum, A12–A13

cast iron, A12–A13

concrete, A14–A15

copper, A12–A13

cupronickel, A14–A15

magnesium alloys, A14–A15

Monel alloy 400, A14–A15

plastics, A14–A15

steel, A12–A13

timber, A14–A15

titanium, A14–A15

Matrix, 63, 104

Maximum absolute strain, 228

Maximum absolute stress, 229

Maximum deflection, 552–553, 607–608, 624, 694, 725, A28

Maximum deformation, 716

Maximum-distortion-energy criterion, 439, 468–469, 694

Maximum elastic moment, 224

Maximum elastic torque, 187, 213

Maximum elongation, 119

Maximum-normal-stress criterion, 440, 469–470

Maximum shearing strain, 491, 494

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Maximum-shearing-stress criterion, 439, 445, 455–456,

467–469, 505

Maximum stress, 716, 722, 725

Maxwell, James Clerk, 734

Maxwell’s reciprocal theorem, 734

Measurements of strain, strain rosette, 494–501, 506

Members

curved, 294–304, 308

made of an elastoplastic material, 256–260

noncircular, 197–200, 214

with a single plane of symmetry, 260–261

stability of, 8

symmetric, 224–225

thin-walled, 414–426, 429

two-force, 4–6

Membrane analogy, 199–200

Methods

of integration, 628

of problem solution, 16–17, 43

of statics, review of, 4–6

of superposition, 551, 580–582, 585–587, 621

Microscopic cracks or cavities, detected in a structural

component, 471

Minimum shearing stresses, 150, 152

Mistakes, errors in, 17

Modulus

bulk, 55, 96–98, 132

of compression, 97

elastic section, 230, 259, 306

of elasticity, 54, 62–64, 130

plastic section, 259

of resilience, 694, 697–698, 751

of rigidity, 55, 100, 105, 133

of rupture, 185, 212, 256

of toughness, 694, 697, 750–751

Mohr, Otto, 452, 470

Mohr’s circle

application to the three-dimensional analysis of stress, 464–466

creating, 454, 457–458, 480, 493

for plane strain, 440, 506

for plane stress, 440, 452–462, 489–491, 503, 506

Mohr’s criterion, 440, 470–472, 505

Moment-area theorems, 592–595, 610, 618, 621–622

application to beams with unsymmetric loadings, 605–606,

625–626

using with statically indeterminate beams, 609–617, 624

Moments of areas, A2–A11

centroid of a composite area, A4–A6

centroid of an area, A2–A4

determination of the first moment, A4–A6

determination of the moment of inertia of a composite area,

A10–A11

first moment of an area, A2–A4

parallel-axis theorem, A9–A10

radius of gyration, A7–A9

second moment or moment of inertia of an area, A7–A9

Moments of inertia, 235. See also Bending moment

centroidal, 236, 400, 407, 515

of a composite area, determining, A10–A11

polar, 165, A7

Monel alloy 400, properties of, A14–A15

Multiaxial loading, 104

generalized Hooke’s law, 94–96, 132

National Council of Examiners for Engineering and

Surveying, A29

National Design Specification for Wood Construction, 666

Necking, 58–59

Neutral surface, 227–229, 295, 305

Noncircular sections, 200

Nonprismatic beams, 318, 361–369, 373

beams of constant strength, 373

Normal strains, 487

under axial loading, 55–57, 129

longitudinal, 228

Normal stresses, 4, 9–11, 18, 20, 26, 42, 224, 317, 462, 528–530,

532, 694, 723–724, 751. See also Maximum-normal-

stress criterion

determination of, 14–15

elastic strain energy for, 698

Numerical accuracy, 17, 44

Oblique parallelepipeds, 98–99

Oblique plane, stresses on, 4, 44

Offset method, for determination of yield strength, 60

Open-ended loadings, equivalent, 373

Orthotropic materials, 55, 105

Overhanging beams, 554

Pa. See Pascals

Parallel-axis theorem, A9–A10

Parallelepipeds

oblique, 98–99

rectangular, 94

Pascals (Pa), 7

Percent elongation, a measure of ductility, 60

Percent reduction in area, a measure of ductility, 60

Permanent deformations, 224

Permanent set, 64, 119, 130

Permanent twist, 190–191, 193

Plane of symmetry, plastic deformations of members with a

single, 260–261

Plane strain, 109

Plane stress, 110, 706

transformation of, 438, 486–488, 506

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