Mechanics of Materials

E. P. papav Professor of Civil Engineering University of California, Berkeley

Text in Collaboration with:

S. Nagarajan

SECOND EDITION

Associate Research Scientist L«kheed Missiles &: Space Company Sunnyvale, California

Problems with Assistance of

ZA Lu Associate, T. Y. Lin Internalional, Inc. Son Francisco, California

PRENTICE-HALL, INC., Upper Saddle River, New Je=y 07458

Contents

Abbreviations and Symbols: See Inside Front Cover

Preface xi

1. Stress-Axial Loads 1

I-I Introduction. 1 1-2 Method of Sections. 3 1·3 Stress. 4 1-4 Axial Load; Normal Stress. 7 1·5 Average Shearing Stress. 9 1-6 Problems in Normal and Shearing Stress. 11 )·7 Allowable Stresses ; Factor of Safety. 18 1-8 Design of Axially Loaded Members and Pins. 22 1·9 Basic Approach. ]5

2. Strain-Hooke's Law-Axial Load Problems 33

2·1 Introduction. 33 2·2 Strain. 33 2·3 Stress·Strain Diagram. 34 2-4 Hooke's Law. 35 2-5 Further Remarks on Stress·Strain Diagrams. 36 2-6 Deflection of Axially Loaded Rods. 37 2-7 Poisson's Ratio. 41

-2-8 Generalized Hooke's Law. 43 2-9 Shearing Stresses on Mutually Perpendicular Planes. 45 2·10 Hooke's Law for Shearing Stress and Strain. 46

-2·11 Stress Concentrations. 48

3. Torsion 57

3-1 Introduction. 57 3·2 Application of Method of Sections. 57 3-3 Basic Assumptions. 59 3-4 The Torsion Formula. 60 3-5 Remarks on the Torsion Formula. 6]

v

3·6 Design of Circular Members in Torsion. 65 3-7 Angle of Twist of Circular Members. 67

·3-8 Shearing Stresses and Deformations in Circular Shafts in the Inelastic Range. 70 ·3·9 Stress Concentrations. 75

·3·10 Solid Noncircular Members. 76 ·3·11 Thin-Walled Hollow Members. 80 ·3·12 Shaft Couplings. 82

4. Axial Force-Shear-and Bending Moment 91

4-1 Introduction. 91 4-2 Diagrammatic Conventions for Supports. 92 4-3 Diagrammatic Conventions for Loading. 94 4-4 Classification of Beams. 95 4-5 Calculation of Beam Reactions. 97 4·6 Application of Method of Sections. 100 4-7 Shear in Beams. 101 4-8 Axial Force in Beams. 103 4-9 Bending Moment in Beams. 103 4-10 Shear, Axial·Force, and Bending·Moment Diagrams. 105 4-11 Step-by·Step Procedure. lJI

5. PUre Bending of Beams 119

5·1 Introduction. 119 5·2 Some Important Limitations of the Theory. 119 5·3 Basic Assumptions. 120 5-4 The Flexure Formula. 123 5·5 Computation of the Moment of Inertia. 126 5·6 Remarks on the Flexure Formula. 130

·5-7 Pure Bending of Beams with Unsymmetrical Section. 134 ·5·8 Inelastic Bending of Beams. 135 ·5-9 Stress Concentrations. 142 ·5·10 Beams of Two Materials. 144 ·5-11 Curved Beams. 150

6. Shearing Stresses in Beams 163

6-1 Introduction. 163 6·2 Relation Between Shear and Bending Moment. 164 6·3 Shear Flow. 168 6-4 The Shearing Stress Formula for Beams. 174

·6-5 Limitations of the Shearing Stress Formula 182 ·6-6 Further Remarks on the Distribution of Shearing Stresses. 184 ·6·7 Shear Center. 186

7. Compound Stresses 199

7·1 Introduction. 199 7-2 Superposition and its Limitation. 200 ·7·3 Remarks on Problems Involving Axial Forces and Bending Moments:

The Dam Problem. 208

CONTENTS vi

·7-4 Special Limitation: The Chimney Problem. 2/0 7-5 A Force Applied to a Prismatic Member Anywhere Parallel

to Its Axis. 211 7-6 Unsymmetrical Bending. 216 7-7 Superposition of Shearing Stresses. 219

·7-8 Stresses in Closely Coiled Helical Springs. 221 *7-9 Deflection of Closely Coiled Helical Springs. 223

8. Analysis of Plane Stress and Strain 235

8-1 Introduction. 235 8-2 The Basic Problem. 236 8-3 Equations for the Transformation of Plane Stress. 238 8-4 Principal Stresses. 240 8-5 Maximum Shearing Stresses. 241 8-6 An Important Transformation of Stress. 245 8-7 Mohr's Circle of Stress. 246 8-8 Construction of Mohr's Circle of Stress. 248

·8-9 Mohr's Circle of Stress for the General State of Stress. 253 ·8-10 Analysis of Plane Strain: General Remarks. 255 ·8-11 Equations for the Transformation of Plane Strain. 256 ·8-12 Mom's Circle of Strain. 258 ·8-13 Strain Measurements; Rosettes. 261 ·8-14 Additional Linear Relations Between Stress and Strain and Among

E, G, and v. 263

Appendix to Chapter 8 Transformation of Moments of Inertia of Areas to Different Axes 266

8A-I Transformation Equations for Rotation of Axes. 266 8A-2 Principal Axes and Principal Moments of Inertia. 267

9. Combined Stresses-Pressure Vessels-Failure Theories 275

CONTENTS

9-1 Introduction. 275 9-2 Investigation of Stresses at a Point. 275 ·9-3 Members in a State of Two-Dimensional Stress. 282 ·9-4 The Photoelastic Method of Stress Analysis. 283 9-5 Thin-Walled Pressure Vessels. 288 9-6 Remarks on Thin-Walled Pressure Vessels. 291 9-7 Failure Theories: Preliminary Remarks. 293 9-8 Maximum Shearing Stress Theory. 294 9-9 Maximum Distortion Energy Theory. 295 9-10 Maximum Normal Stress Theory. 296 9-11 Comparison of Theories: Other Theories. 298

vii

10. Design of Members by Strength Criteria 311

10-1 Introduction. 311 10-2 Design of Axially Loaded Members. 311 10-3 Design of Torsion Members. 311 10-4 Design Criteria for Prismatic Beams. 311 10-5 Shear Diagrams by Summation. 316 10-6 Moment Diagrams by Summation. 318

*10-7 Further Remarks on the Construction of Shear and Moment Diagrams. 323

*10-8 Moment Diagram and the EJastic Curve. 327 10-9 Design of Prismatic Beams. 329

*10-10 Design of Nonprismatic Beams. 334 *10-11 Design of Complex Members. 336

11. Deflection of Beams 353

11-1 Introduction. 353 11-2 Strain-Curvature and Moment-Curvature Relations. 354 11-3 The Governing Differential Equation for Deflection of Elastic

Beams. 356 11-4 Alternative Differential Equations of Elastic Beams. 358 11 -5 Boundary Conditions. 359 11-6 Solution of Beam Deflection Problems by Direct Integration. 361 11-7 Statically Indeterminate Elastic Beam Problems. 370 11-8 Remarks on the Elastic Deflection of Beams. 374

*11-9 Elastic Deflection of Beams in Unsymmetrical Bending. 376 *11-10 Inelastic Deflection of Beams. 377. *11-11 Introduction to the Moment-Area Method. 380 *11-12 Derivation of the Moment-Area Theorems. 381

12. Statically I ndeterminate Problems 403

12-1 Introduction. 403 12-2 A General Approach. 404 12-3 Stresses Caused by Temperature. 411 12-4 Analysis of Indeterminate Systems Based on Superposition. 4/4

*12-5 Force Method. 414 *12-6 Displacement Method. 411 *12-7 Moment-Area Methods for Statically Indeterminate Beams. 428 * 12-8 The Three-Moment Equation. 435 *12-9 Constants for Special Load Cases. 437 *12-10 Limit Analysis of Beams. 440 *12-1 1 Concluding Remarks. 447

13. Columns 461

13-1 Introduction. 461 13-2 Stability of Equilibrium. 463 13-3 The Euler Formula for a Pin-Ended Column. 465

CONTENTS VIII

-13-4 Euler Formulas for Columns with Different End Restraints. 467 -13-5 Elastic Buckling of Columns Using Fourth-Order Differential

Equation. 470 -13-6 Analysis of Beam·Columns. 471

13-7 Limitations of the Euler Formulas. 473 13-8 Generalized Euler Buckling-Load Formulas. 475 13-9 The Secant Formula. 477 )3.-10 Design of Columns. 480 13-11 Column Formulas for Concentric Loading. 482

-13-12 Column Formulas for Eccentric Loading. 486 -13-13 Beams without Lateral Supports. 492

-14. Structural Connections 501

14-1 Introduction. 501 14-2 Riveted and Bolted Connections. 501 14-3 Methods of Failure of a Riveted or Bolted Ioint. 503 14-4 Eccentric Riveted and Bolted Connections. 512 14-5 Welded Connections. 516 14-6 Eccentric Welded Connections. 517

15. The Energy Methods 525

15-1 Introduction. 525 15-2 Elastic Strain Energy for Uniaxial Stress. 525 15-3 Elastic Strain Energy in Pure Bending. 528 15-4 Elastic Strain Energy for Shearing Stresses. 530

-15-5 Strain Energy for Multiaxial States of Stress. 532 -15-6 Design of Members for Energy Loads. 532

15-7 Deflections by the Energy Method. 534 -15-8 Impact Loads. 536 -15-9 Virtual Work Method for Deflections. 540 -15-10 Virtual Work Equations for Elastic Systems. 542 -15-11 Statical1y Indeterminate Problems. 548

'16. Thick-Walled Cylinders 557

16-1 Introduction. 557 16-2 Solution of the General Problem. 557 16-3 Special Cases. 562 16-4 Behavior of Ideally Plastic Thick·Walled Cylinders. 564

Appendix Tables 569

Index 583

CONTENTS llRC

MCAST

ix

Index

A

Abbreviations and symbols, front endpaper

Allowable stress: definition of, 19 in bending, 331. 332, 570 on bolts, 507 in riveted joints. 506 in torsion, 65, 67 table of. 570

Angle of twist: circular shaft, 67 hollow shaft, 82 rectangular shaft, 77

Angle sections, propenies for designing, 576, S77

Area-moment method (see Momenl-orea

method) Areas, useful properties of, 57\ Axes:

principal. 134, 216, 267 neutral, 122

Axial force diagrams, 105 Axial loads (also see Columns):

definition of, 7

B

deformation due to, 37 stresses due to, 7

Beam-columns, 471

Beams: bending moments in, 103 bending stresses in , 125, 329

Beams (Conl'd.) built-in (see Bearns, find) cantilever, 96 center of twist (see Shear Center) classification. 95 connections for, 509 constant strength, 335 continuous, 96 cover plated, 174,336 crippling of, 333 curved, 150 definitions of, 91 deflection of (see Deflection) design of, 130. 312 elastic curve for. 327,355 elastic section modulus of, 130 elastic strain energy in, 528 fixed, 95, 419, 431 flexure fonnula for. 125 inelastic bending of, 135 lateral instability of. 119, limit analysis of, 440 maximum bending stresses in, 125.

492 neuual axis in, 122 of two materials. 144 of variable cross·section. 143,335 overttanging.96 plastic analysis of, 138.440 plastic section modulus of. 140 prismatic, 142, 312 reactions of, 97. 370, 417 radius of curvature of. 355 reinforced concrete, 147 restrained, 95, 428 section modulus of. 130 , 138

583

INDEX

Beams (Cont'd.j shear in, 101.316 shearing stresses in, 163, 174 stresses in, 123, 174,312,539 simple, 95 simply supported, 95 statically indeterminate, 11.370,404.

409,414,428,435 uniform strength (see COlIStant

str~ngth)

unsymmetrical bending of. 134,216 Bearing stress. 11,504, S06 Bending:

combined with axial loads, 200-216 of beams (see B~ams) pure, 107 , 119,312 inelastic, 135 deflections due to, 353-393 skew (see unsymmetrical) strain energy in, 528 stresses due to, 125,312 unsymmetrical. 216

Bending moment: and elastic curve, relation between,

327, 355 and shear, relation between, 164,318 definition of. 103 diagrams. 10:5,318,429 diagrams by summation method, 318 sign convention for. 104

Biaxial stress, 275, 288 Boilers:

design of, 288 Bolted joints (see Riveud joiflls) Bolts, 507 Bredt's formula , 81 Buckling:

of beams, 492 of columns, 461-492 of vacuum chambers. 292

Bulk: modulus of elasticity, 265 Bult joints. 292, S02, 510

c

Cantilever. 96 Center of twist (see Shear cerller) Centroids of areas. 571 Channel sections. properties for

designing, 575

Clapeyron's equation, 436 Circ::umferential stress, 290 Coefficient of thermal expansion, 411,

170 Columns:

critical load, 467 critical Stress, 473 design of. 480-492 double-modulus theory for, 476 eccentrically loaded, 477 .486 Euler 's formula, 467 fonnulas forconcentrically loaded,

482 long, 474, 475 parabolic formula for, 483 secant formula for, 477 short , 475 slenderness flltio of, 474 straight-line formulas for, 484 tangent-modulus formula for. 476

Combined stresses. 235-255 Compound stresses, 199-224 Complementary energy, 526 Concentflltion of stress (see Stress) Connections. 292. 501-520 Constant strength beams, 335 Consistent deformations, method of, 403 Continuous beams:

analysis of, 403, 414, 421, 428, 43:5, 440

definition of, 96 Contraflexure, 328 Couplings, shaft, 82 Creep, 19 Crippling of web, 333 Critical sections, I I . :59, 311, 313, 314 Curvature, 355 Curvature-area method, 377,393 Curved beams:

stresses in, ISO deflection of, 548

Cylinden: thick-walled,557-567 thin-walled,288-292

D

d'Alembert's principle. 4 Deftection:

of axially loaded rods, 37

584

INDEX

Deflection (Com'd .) of beams:

dummy-load method. 542 due to impact, 400 due to shear, 535 integration methods for, 358 moment·area method for. 380-393, 428-435

strain energy method for, 535, 540 statically indeterminate, 370, 428 successive integration method for. 361

superposition method for, 374 table of. 580 unsymmetrical bending, 216 yinual work method for, 543

of frameworks. 408, 416. 544. 549 of helical springs. 223

Degrees of freedom, 421 Design:

of axially loaded members, 22, 311 of beams, 130, 312. 329-336 of columns, 480-492 of complex members. 336 of connections, 501-520 oftonoion members. 65. 312

Deyiation. tangential. 383 Differential equation of elastic curve, 357,

358 Displacement method. 401. 421 Double·modulus theory, 476 Dynamic loads. 536

E

Eccentric loading: of columns, 477, 486 of riyeted joints. 512 ofshon blocks. 160--166 of welded connections. 517

Effectiye column length, 469 Efficiency of a joint, 292, 512 Elastic curve, 327, 355 Elastic limit. 37 Elastic modulus, 36 Elastic strain energy:

in bendins, 528 in shear. 530 in tension or compression, 527 in torsion, 531

Elastic strain energy (Com' d.) for mulliuial stresses. 532 for uniuial stress, 525

Elasticity: definition of. 37 modulus of. 36

Endurance limit. 20 Equivalent section in bending. 145 Euler'$ formula. 467

F

Factor of safety. 21 Factors of SlTeu·concentration:

for helical springs, 222 in bending, 142 in tension or compression. 48 in torsion, 75

Failure theories, 227 Fatigue. 20 Fiber stress. definition of, 21 Fillet weld, 516 Flange, definition of. 143 Flexibility method, 401. 414 Flexure fonnula:

for curved beams. 152 for straight beams. 125

F1exural rigidity, 359 Force method, 401,414 Formulas:

for centroids and moments of inertia of an:as.571

for deflection of beams. 580 for fixed·end actions of beams, 581

Fracture cri teria. 293-300 Frameworks. deflection of, 408, 416,

S44. 549 Free·body. definition of. 3 Fringe, 286

H

Helical springs. 221-224 Hinge, plastic, 441 Hooke's law:

for shearing stress and strain, 46 for uniaxial stress. 35 generalized, 43

585

INDEX

Hook.s, stresses in, 150,204 Hoop strc:ss, 290 Horizontal shearing stress, 175 Horsepower and torque relation, 66

I

I-shape beams: crippling in, 333 shearing stresses in, 180.333 table of properties for designing, 572,

'" Impact: dellection due to, 538 faaor, 538 loading, 536

Inelastic behavior: of beams, 135, 377 of tors ion members, 70

Inertia, moment of, 125,266 Inflection, point of. 328 Interaaion fonnula for columns, 487 Internal forces, 3, 91 Internal work., 525 Isoclinic, 282 Isotropy. 36

J

Joints:

K

bolted, 501, S03, 512 riveted. 10, SOl, 503, 512 welded, 292, 516

Kern, 215 Keyways,76 Kinemllic indetenninates, 421

L

Lame's problem. 557 Lap joint, S02 Lateral inslability of beams, 119,492 Leaf springs, 533

Limit analysis and design. 440 Line of zero stress, 213 Load factor, 22. 442 Loads:

axial,7 concentrated,94 distributed, 95 impaa, 536 live, 334

Localized stress (see Factors of Srrtss-concullralion)

longitUdinal stress in cylinder, 290

Materials, table of physil;al properties, 57.

Margin of safety, 21 Maximum distort ion energy theory, 295 Maximum nonnal scress theory, 296 Maximum shearing stress theory, 294 Members of two materials, 144 Membrane analogy for torsion, 79 Method of sections:

definition of. 3 for axially loaded members, 6 for beams, 100 for torsjon members, 57

Middle-third rule, 161 Modulus:

bulk,265 of elast icity, 36 of resilience, 527 ofrigidily,47 ofrup'ure in bending, 138 ofNp'ure in torsion. 72

Modulus of elasticity related to modulus of rigidity, 47,

Mohr's circle: for moments of inenia, 268 for strain, 258 for stress, 248

Moment-area method: for detenninale beams. 380-388 for indetenninate beams. 428-435

Moment (also see Btnding nwtnt'nt): definition of. 103 diagrams, 105, 318

586

INDEX

Moment (Com'd.)

sign convention for. 104 MomenT of ineni.·

Motlr's circle for. 268 of plane areas, 125 , 126 parallel-axis theorem for, 127 polar, 61 principal axes of. 134.267 table of, 571

Moving bodies, stresses due to, 538

Necking. 34 Neuual a",is, 122 Neutral surface, 122 Normal stress:

o

combined with shearing stress. 235. 240

definition of, 4 in uially loaded members. 8 in bending, 123 muimum and minimum. 241

Octohedral shearing stress theory. 296

p

Parabolic column fonnula. 483 Parallel-uis theorem. 127 Plane strain, 560 Plane stress, 263, 561 Plastic analysis:

of beams, 138, 440 of torsional members, 70

Pitch, in riveted joints. 502 Photoelutic method of stress analysis,

283 Point of inf!.ection, 328 Poisson's ratio, 41 Polar moment of inertia, 61 Pressure vessels, thin-walled, 288 Principal axes, of inertia, 127, 267 Principal planes. of bending, 134.216 Principal shearing stress. 241

Principal strain, 259 Principal stress. 240 Properties:

of angles, 576. 577 of standard steel beams, 572 of channel sections, 575 of pipe, 578 of rectangular timber, 579 of wide·f!.ange steel beams, 573, 574

Proportional limit, 36

R

Radius: of curvature, 355 of gyration, 473

Reactions, calculation of, 97 Redundant reactions, 370, 414 Reinforced concrete beams, 147 Relation among E, G, and V, 47,264 Relation between shear and bending

moment, 164,318 Relation among E, G, and v , 47 , 264

moment, 357 Repeated loading, 19,51 Residual stress, 20 Resilience, modulus of, 527 Restrained beams, 95, 428 Riaidity, fl.e",ural. 361 Rigidity, modulus of, 47 Riveted joints:

concentrically loaded, 10, SOI-512 eccentrically loaded. 512-517 methods of failure, S03 structural. 506-517

Rosettes, strain. 261 Rupture, modulus of, 72, 138

s

St. Venant's principle, 49 Secant fonnula for columns, 477 Section modulus:

elastic, 130 plastic. 140 tables. 572-579

Shaft (see Torsion)

587

INDEX

Shape factor, 140 Shear:

and bending momen!' relation between, 164,318

definition of, 101 diagrams, !OS diagrams by summation method, 316 sign convention for, 102

Shear cenler, 186 Shear diagrams (see Shear) Shear flow, 81,168 Shearing deflections of beams, 535 Shearing deformation, 46 Shearing force in beams (see Shear) Shearing modulus of elasticity, 47 Shearing strain, 47 Shearing stress:

definition of, 5 due to tension or compression, 276 in beams, 175 in circular shafts, 61, 75 in non-circular shafts, 78, 81 in rivets and bolts, 9, 504, 506 maximum, 178,241 on perpendicular planes, 45 principal,241

SI units,S, back endpaper Sign convention:

for moment, 104 for shear, 102 for stress, 6

Simple beam, definition of, 95 Skew bending (see unsyrnmetrical

bending) Slenderness ratio, 474 S-N diagrams, 20 Spherical pressure vessels, 291 Spring constant, 224 Springs, helical:

stresses in, 221 deflection of, 223

Stability, 119,461-493 Statical moment of area. 169 Static ally indeterminate beams:

analysis by: displacement method, 421 force method, 414 flexibility method. 414 general approach, 404

St3tically indetenninate beams (CQnt' d. ) integration of differential equations. 370

moment-area, 428 superposition, 414 three-moment equation. 435 virtual work, 548

definition of, 11.370,403 Slatically indeterminate members:

analysis by: displacement method, 421 force method. 414 flexibility method, 414 stiffness method, 421 virtual work, 548

axially loaded, 404, 415 in frames, 549 in lorsion, 406, 417

Steel beams, sizes. properties for designing, 572-578

Stiffness method, 421 Straight-line colUmn formula, 484 Strain:

definition of. 34, 47 extensional, 255. 256 maximum, 259 Mohr's circle for, 258 plane, 256, 560 principal, 259 shearing, 47 thermal, 411 transformation of. 256

Strain energy (see Elastic strain energy) Strain rosettes. 261 Strength, ultimate, 36 Stress:

allowable, 18 bending, 125 bearing, 9, 504, 506 biaxial,275. circumferential,29O combined (see Combined stress) compound, 199 compressive,4 concentration factor:

definition of, 49 for axi3Uy loaded members, 49 in bending, 142 in springs. 222

588

INDEX

Stress (ConI'd.) in torsion, 75

conditions for uniform, 8, 103 critical,473 definition of, 4 fiber, 21 fl exure, 125 hoop, 290 impact, 539 in curved bars, 152 maximum and minimum normal, 241 Mohr's circle of. 246 normal, 4. 7 on inclined planes, 238, 276 plane, 238, 561 principal,240 residual, 20, 72, 140 shearing,S, 9, 45, 61. 174,241 state of. 235 tangential,558 tensile, 4 torsional shearing. 61 three-dimensional, 6, 297 transformation of. 238 two-dimensional, 238 uniaxial,8 unit, 5 working, 22 yield point, 36

Stress-strain diagram. 34 Stress trajectories, 283 Structural riveted joints, 501-516 Suddenly applied loads. 538 Superposition:

of deflections, 374, 414 of stresses. 199-221 principle of. 43 statically indeterminate problems

solved by. 414 , 548 Suppons, diagrammatic conventions for.

92

T

Tables, index of. 569 Tangential deviation, 383 Tangent modlllus, definilion of, 476 Tangential stress, 558

TemperalUre or thermal stresses, 411 Tensile test, 33 Tensile stress, 4 Theories of failure, 293-300 Thick-walled cylinders, 557-567 Thin-walled cylinders, 288-292 Three-moment equation, 435 Thrust, 103 Timber, sizes, propenies for designing,

579 Torque, internal, 58 Torsion:

angle oftwisl due to, 67, 78, 82 assumptions of theory. 59 elastic energy in. 531 formula, elastic, 61 inelastic, 70 of circular shafts, 60 of hollow members, 62, 80 of non-circular solid bars, 76 of rectangular bars, 78

Torsional stiffness, 69 Toughness, 527 Trajectories, stress. 283 Transformation:

of moments of inertia, 266 of strain, 256 of stress, 238

Transformed sections, 145 Tresca yield condition, 295 Triaxial stress, 6, 44, 254, 532 Twist, angle of (see Torsion)

u

Ultimate strength, 18, 36 Uniaxial stress, 8 Unit strain. 34 Unil stress, 5 Unsymmetrical bending, 216 Unsymmetrical sections subjected 10

bending, 134, 186. 218

v

Variable cross-section, beams of, 334 Venical shear (sec: Shear)

589

INDEX

Virtual force, S41 Vinual work method for dcftcctions,

>4<>-'SO von Mises yield condition, 296

w

Wahl correction factor for helical springs, 223

WetH:Ietinition of, 143

Welded connections, 292, 516-520 Wide ftlnge beams, properties for

designing, 573-574 Working stress, 22

y

Yield point or strength, 36 Young's modulus, 36

590