كتاب Advanced Strength and Applied Stress Analysis
منتدى هندسة الإنتاج والتصميم الميكانيكى
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منتدى هندسة الإنتاج والتصميم الميكانيكى
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 كتاب Advanced Strength and Applied Stress Analysis

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Advanced Strength and Applied Stress Analysis
Second Edition
Richard G. Budynas  

كتاب Advanced Strength and Applied Stress Analysis  A_s_a_13
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LIST OF SYMBOLS
area, area of cross section, tight-wave vector amplitude, analyzer filter axis
area bounded by perimeter centerline of a thin-wallcd tube
dimension, crack width, varying amplitude of a light-wave vector
dimension, beam width
equivalent width of a composite beam section
material calibration constant for transmission photoelasticity
correction factor for photoelastic coatings
distance from the neutral axis to an outer beam fiber, speed of light, strain
pulse speed
flexural rigidity of a thin plate, diameter
diameter
directional numbers of a plane
modulus of elasticity, voltage
modulus of elasticity scale factor
eccentricity, distance &om the centtoidal axis to the neutral axis of a curved
beam
concentrated force
force scale factor
equivalent concentrated force
body forces per unit volume
material calibration constant for photoelastic coatings, finite element nodal
force, coefficient of friction
strength uncertainty factor
stress uncertainty factor
shear modulus, Griffith energy release rate
gravitational constant
dimension, depth of beam
depth of plastic region
second-area moments (area moments of inertia) of a cross section
principal second-area moments (area moments of inertia) of a cross section
stress invariants
unit vectors in the x, y, z directions respectively
polar second-area moment (polar moment of inertia) of a cross section
equivalent polar second-area moment (polar moment of inertia) of a cross
section
column support factor
fatigue stress concentration factor
stress intensity factor
critical stress intensity factor
static stress concentration factor, strain gage transverse sensitivity factor
spring constant, form correction factor for shear, stress optic coefficient
AAab CC
,
Dd
df , dy, dt
EEe FF F
1 F F
JV * y* * z
f UuG8h
Iyr /j, ly*
/1.4/3
l,J, k
Jh K ,K,Kc,Kk
xvfixvlit LIST OF SYMBOLS
length
length scale factor
directional cosines
strain gage length
strain pulse length
applied or reaction concentrated bending moment (couple)
moment scale factor
limit (plastic) moment
plate bending moments per unit length in polar coordinates
plate bending and twisting moments per unit length in rectangular
coordinates
yield moment
net internal bending moments about axes parallel to the y and z axes
respectively
mass, margin of safety
axes of the principal second-area moments
normal force, number of cycles, photoelastic isochromatic fringe order,
shape functions
photoelastic isochromatic fringe order at an angle of incidence 8
design factor, index of refraction, angular speed (rpm)
directional cosines
concentrated force, polarizer filter axis
pressure scale factor
critical buckling load
limit force
pressure, press-fit interference pressure
first-area moment of a partial area of a beam section
shear force per unit length, distributed load intensity, notch sensitivity
factor
reaction force, radius, crack resistance force, electrical resistance
strain gage nominal resistance
radius
cylindrical coordinates
the distance from the center of curvature to the centroidal axis of a curved
beam
radiusof gyration
the distance from the center of curvature to the neutral axis of a curved
beam
plastic zone radius
elastic section modulus
perimeter length of the centerline of a closed thin-walled tube
strain gage axial sensitivity
endurance strength
fatigue strength
strain gage factor
strain gage transverse sensitivity
yield strength
ultimate strength
position, curvilinear coordinate
LL,/
m, n
lI P'MM
MP
Mr, Me
Mx, My, Myy
M y
My, ML
mm
, n
N n„n
rty, n.
PP
Per
Pi
PQ f) RRr,r
8, z
rc
r*
ri,
r;
5SS0
SE
S F
s
*
S y
So
sLIST OF SYMBOLS xlx
beam shape factor
torsional moment (couple), temperature
thickness, time
pulse time
directional cosines of the net shear stress on an isolated surface
strain energy
strain energy per unit volume
displacements in the JC, y, t directions respectively
displacements in \htr, 8, z directions respectively
net internal shear force, input voltage
plate shear force per unit length in polar coordinates
plate shear forces per unit length in rectangular coordinates
beam shear forces in rectangular coordinates
displacement of beam centroida! axis
weight, work
complementary work
work potential
force per unit length, work per unit volume, width, weighting factor
rectangular coordinates
inelastic section modulus
SF
Tx uu,u
v, w
M,. «0.
V V
V
’ x> r y
V V
'y* ri
Vc
W w,x
y, z
Z
Greek
angle, coefficient of thermal expansion, angular location of the neutral
plane for unsyrnmetrical bending relative to the m axis
angle, angular location of the neutral plane for unsyrnmetrical bending
relative to the y axis
change, shift in phase of light waves
deflection, press-fit radial interference
deflection scale factor
Kronecker delta
normal strain
strain scale factor
axial strain
transverse strain
shear strain, weight density, temperature coefficient of resistivity
angular deflection
angle, angular deflection
angular deflection per unit length
wavelength, Lam£ constant
Lamfc constant
Poisson’s ratio
natural coordinates
potential energy
radius of curvature, mass density, resistivity
Neuber constant
normal stress
alternating and mean stresses in fatigue applications
stress scale factor
principal stresses
a 0 A88 ee ,ey0e'9A v
b y
np'p T< 'O ’G
li ^2» ^3LIST OF SYMBOLS
critical buckling stress
equivalent von Mises stress
shear stress
equivalent Tresca shear stress
Airy stress function, complementary strain energy
Prandtrs stress function
angle, helix angle of twist
angular speed (rad/s)
O’
er
T 4
>
4>
(t)
Mathematical
V2 Laplacian operator
Jacobian matrix
Transformation matrix
[J ]
[T]CONTENTS
Principal Strains 81
2.3 Generalized Stress-Strain Relations 82
2.4 The Equilibrium Equations 86
2.5 Compatibility 89
2.6 Summary of Important Equations 97
2.7 Problems 101
List of Symbols xvii 2.2.2
O N I
BASIC CONCEPTS OF FORCE, STRESS,
STRAIN, AND DISPLACEMENT 1
1.0 Introduction 1
1.1 Force Diagrams 2
1.2 Force Distributions 2
1.3 Stress 10
1.4 Strain, Stress-Strain Relations 20
1.4.1 Normal Strains 20
1.4.2 Shear Strains 23
1.4.3 Thermal Strains 24
1.5 Displacements, Strain-Displacement
Relations 24
1.5.1 Rectangular Coordinates 24
1.5.2 Cylindrical Coordinates 29
1.6 Summary of Important Relationships 33
1.7 Problems 37
C H
A REVIEW OF THE FUNDAMENTAL
FORMULATIONS OF STRESS, STRAIN,
AND DEFLECTION 113
Introduction 113
Assumptions and Limitations 114
Axial Loading 114
3.2.1 Axial Stresses 114
3.2.2 Axial Strains and Deflections 117
Torsion of Circular Shafts 129
3.3.1 Torsional Stresses 129
3.3.2 Torsional Strains and
Deflections 130
Beams in Bending 132
3.4.1 Shear Force and Bending Moment
Equations and Diagrams 132
3.4.2 Bending Stresses 138
3.4.3 Transverse Shear Stresses 143
3.4.4 Bending Strains and Deflections 152
Bending of Symmetric Beams in Two
Planes 158
Thin-Walled Pressure Vessels 162
3.6.1 Stresses 162
3.6.2 Strains and Deflections in a Circular
Cylinder 166
Superposition 166
Statically Indeterminate Problems 175
Stress and Strain Transformations 185
3.9.1 Plane Stress 185
3.9.2 Principal Stress 186
3.9.3 Maximum In-Plane Shear Stress - 189
3.9.4 Strain Transformations 191
3.0
3.1
3.2
3.3
( H A P T I R T W O
STRESS AND STRAIN.
TRANSFORMATIONS, EQUILIBRIUM,
AND COMPATIBILITY 46
3.4
2.0 Introduction 46
2.1 Stress Transformations 47
General Three-Dimensional Stress
Transformations 47
Plane Stress Transformations 56
Mohr’s Circle for Plane Stress 57
Three-Dimensional Stress
Transformation Simplified 62
Principal Stresses 66
Mohr’s Circles in Three
Dimensions 72
Maximum Shear Stress 75
2.2 Strain Transformations 78
Strain Transformations, General 78
3.5
2.1.1
3.6
2.1.2
11.3
11.4
3.7
11.5 3.8
11.6 3.9
11.7
12.1
xlxll CONTENTS
5.4.1 Shear Flow in Open Thin'Walled
Beams 280
5.4.2 Shear Center for Open Thin-Walled
Beams with One Axis of
Symmetry 283
5.4.3 Shear Center for Open Unsymmetric
Thin-Walled Beams 291
54.4 Shear in Closed Thin-Walled
Sections 296
5.5 Composite Beams in Bending 302
5.6 Curved Beams 309
5.6.1 Tangential Stresses 309
5.6.2 Approximate and Numerical
Calculations of e 317
5.6.3 Radial Stresses 322
5.7 Bending of Thin Flat Plates 324
5.7.1 Governing Equations in Rectangular
Coordinates 324
5.7.2 Tabulated Solutions of Uniformly
Loaded Rectangular Plates 329
5.7.3 Governing Equations for Axisymmetric
Circular Plates in Bending 330
5.7.4 Tabulated Solutions of Circular
Plates 336
5.7.5 Superposition 341
5.8 Thick-Walled Cylinders and Rotating
Disks 348
5.9 Contact Stresses 357
5.9.1 Distributed Contact Loading 357
5.9.2 Contact Between Curved
Surfaces 361
5.10 Stress Concentrations 364
5.11 Problems 371
5.12 References 403
3.10 Budding Instability of Columns in
Compression 192
3.11 Problems 199
3.12 References 219
C H A P T I R F O U R
CONCEPTS FROM THE THEORY OF
ELASTICITY 220
4,0 Introduction 220
4.1 Plane Elastic Problems 221
Definition 221
Governing Equations 222
Conversion between Plane Stress and
Plane Strain Problems 224
4.2 The Airy Stress Function 225
Rectangular Coordinates 225
Polar Coordinates 230
Curved Beam in Bending 233
Circular Hole in a Plate Loaded in
Tension 235
Concentrated Force on a Flat
Boundary (Flamant Solution ) 238
Disk with Opposing Concentrated
Forces 240
4.3 Prandtl’s Stress Function for Torsion 244
General Formulation 244
Torsion on a Rectangular Cross
Section 251
4.4 Discussion 255
4.5 Problems 255
4.6 References 260
4.1.1
4.1.2
4,1.3
4.2.1
4.2.2
4.2.3
4.2.4
4.2.5
4.2.6
4.3.1
4.3.2
C H A P T I R F I V I
TOPICS FROM ADVANCED MECHANICS
OF MATERIALS 261
C H A P T I R S I X
ENERGY TECHNIQUES IN STRESS
ANALYSIS 404
5.0 Introduction 261
5.1 Shear Flow 261 ,
5.2 Torsion of Closed Thin Walled Tubes 262
Single Cell Sections in Torsion 263
Multiple Cell Sections in
Torsion 267
5.3 Bending of Unsymmetrical Beams 272
Stresses 272
Deflections 279
5.4 Further Discussion of Transverse Shear
Stresses 280
6.0 Introduction 404
6.1 Work 409
5.2.1 6.2 Strain Energy 410
5.2.2 6.2.1 Uniaxial Case 410
Additional Normal Stresses 411
Shear Stress 411
General State of Stress 412
Plane Stress 413
6.2.2
6.2.3
5.3.1 6.2.4
5.3.2 6.2.5
6.3 Total Strain Energy in Bars with Simple
Loading Conditions 414CONTENTS Kill
7.3.5 Coulomb-Mohr Theory for Brittle
Materials 513
7.3.6 Design Equations 515
7.4 Fracture Mechanics 518
7.4.1 Introduction 518
7.4.2 Crack Modes and the Stress Intensity
Factor 520
7.4.3 The Plastic Zone Collection 526
7.5 Fatigue Analysis 532
7.5.1 Fatigue Strength and Endurance
Limit 532
7.5.2 Cyclic Stress with a Static
Component 534
7.5.3 Fatigue Strength Reduction
Factors 538
7.5.4 Equivalent Stresses (Plane Stress) 541
7.5.5 Estimating Ufe for Nonreveising or
Nonrepetitive Stress Cycles 543
7.6 Structural Stability 546
7.6.1 Column Buckling 547
7.6.2 Buckling of Plates 556
7.7 Inelastic Behavior 561
7.7.1 EPP Materials 561
7.7.2 Plastic Behavior of Straight Beams in
Bending 564
7.7.3 Depth of the Plastic Zone (Rectangular
Beam) 568
7.7.4 Residual Stresses (Rectangular
Beam) 569
7.7.5 Residual Stresses and Fatigue
(Rectangular Beam) 571
7.8 Engineering Approximations Used in
Statically Indeterminate Problems 573
7.8.1 Considering Deflections of Flexible
Elements Only 573
7.8.2 Limit Analysis 579
7.9 Problems 584
7.10 References 596
Axial Loading 414
Torsional Loading of a Solid
Circular Bar 414
Transverse Loading 415
6.4 Castigliano’s First Theorem 419
6.5 The Complementary-Energy Theorem 424
6.6 Castigliano’s Second Theorem 428
Deflections of Statically Determinate
Problems 428
Deflections Due to Temperature
Changes 445
6.7 Deflections of Thick-Walled Curved
Beams 446
6.8 Castigliano’s Second Theorem Applied to
Statically Indeterminate Problems 450
6.9 The Virtual Load Method 456
Axial Loading 458
Torsional Loading 459
Bending 460
6.10 The Virtual Load Method Applied to
Statically Indeterminate Problems 462
6.11 Rayleigh’s Method Applied to Beams in
Bending 464
6.12 The Rayleigh-Ritz Technique Applied to
Beams in Bending 469
6.13 Straight Beams Undergoing the Combined
Effects of Axial and Transverse
Loading 471
6.13.1 Unconstrained Beams 471
6.13.2 Constrained Beams 479
6.14 Problems 483
6.3.1
6.3,2
6.3,3
6.6.1
6,6.2
6.9.1
6.9.2
6.9.3
C H A P T I R S I V I N
STRENGTH, FAILURE MODES, AND
DESIGN CONSIDERATIONS 498
7.0 Introduction 498
7.1 “Strength” 498
7.2 The Design Factor 500
7.3 Strength Theories 505
Basis of Theories 505
Tresca (Maximum-Shear-Stress)
Theory for Ductile Materials 509
von Mises (Maximum-Energy-ofDistortion) Theory for Ductile
Materials 509
Comparison between the Tresca and
von MisesTheories (Plane Stress) 512
C H A P T I R I I G H T
EXPERIMENTAL STRESS ANALYSIS 597
7.3.1
8.0 Introduction 597
8.1 Dimensional Analysis 598
8.2 Analysis Techniques 601
8.2.1 Symmetry 601
When a Surface Perpendicular to a
Free Surface Exists Without Shear
Stress 602
7.3.2
7.3.3
8.2.2
7.3.4xhr CONTENTS
The One-Dimensional Truss
Element
—The Rayleigh-Ritz
Method 681
The Assembly Process 683
Distributed Loads 692
Thermal Stress 694
The Two-Dimensional Truss
Element 698
Skew Supports 708
The Three-Dimensional Truss
Element 711
9.3 Beam and Frame Elements 718
The Planar Beam Element 718
Distributed Loading 727
Pin Releases (Hinges) in Beam
Elements 736
Beams in Two-Plane Bending 741
The Frame Element 742
Three-Dimensional Transformation of
the Frame Element 743
Load-Stiffening and Buckling of
Beams 747
9.4 Two-Dimensional Elastic Elements 757
The Two-Dimensional Constant Strain
Triangle (CST) Element 758
The Two-Dimensional Isoparametric
Quadrilateral Element 767
9.5 Higher-Order and Three-Dimensional
Elastic Elements 777
9.6 Problems 778
9.7 References 788
8.3 Strain Gages, General 604 9.2.2
8.4 Strain Gage Configurations 607
8.5 Strain Gage Instrumentation 610
The Wheatstone Bridge 610
Commercial Strain Gage Indicator
Systems 613
8.6 Characteristics of Strain Gage
Measurements 616
Linearity of the Grid Material 617
Transverse Sensitivity of the
Gage 617
Temperature Effects 620
Lead-Wire Connection 623
Strain Gradient 624
Zero Shift and Hysteresis Effects 624
Dynamic Response 624
Gage-Current Heating Effects 625
Noise from Electric and/or Magnetic
Fields 626
8.7 The Theory of Photoelasticity 626
Electromagnetic Wave Representation
of Light 626
Polarization 628
Refraction 630
9.2.3
8.5.1
8.5.2 9.2.4
9.2.5
9.2.6
9.2.7
8.6.1
9.2.8
8.6.2
8.6.3
9.3.1
8.6.4
9,3.2
8.6.5
9.3.3
8.6.6
8.6.7
9.3.4
8.6.8
9.3.5
8.6.9
9.3.6
9.3.7
8.7,1
8.7.2
8.7.3 9.4.1
8.7.4 Birefringence 631
8.7.5 Stress and Birefringence 634
Isoclinic Fringe Analysis 637
8.7,7 Isochromatic Fringe Analysis 642
8.8 Techniques Used in Photoelastic
Applications 645
8.8.1 Photoelastic Material Calibration 645
8.8.2 Fractional Fringe Orders 647
8.8.3 Separation of the Principal Stresses, cT\
and 8.8.4 Reflection Photoelasticity 654
Stress Freezing in Three-Dimensional
Photoeiasticity 660
9.4.2
8.7.6
C H A P T E R T I N
FINITE ELEMENT MODELING
8.8.5 TECHNIQUES 789
10.0 Introduction 789
10.1 Planning and Creating the Finite Element
Model (Preprocessing) 791
10.2 Element Selection and Mesh
Strategy 792
10.2.1 Introductory Remarks 792
10.2.2 Element Selection 796
10.2.3 Element Input Information 802
10.2.4 Mesh Generation 803
10.2.5 Two-Dimensional Meshing
Strategies 808
10.2.6 Submodeling 811
10.2.7 Symmetry 812
8.9 Problems 662
8.10 References 672
C H A P T I R M I N I
INTRODUCTION TO THE FINITE
ELEMENT METHOD 673
9.0 Introduction 673
9.1 Node and Element Subscript Notation 677
9.2 The Truss Element 677
The One-Dimensional Truss ElementDirect Stiffness Method 678
9.2.1I N D E X
A
Abrupt change in cross section, 115-116, 235, 365-371, 873-878
Adjoint matrix, 894
Airy stress function:
polar coordinates, 230-244
rectangular coordinates, 225-229
Alternating stress component, 535
Analysis techniques, experimental, 601-604
Analyzer filter, 629
Angle of twist:
circular shafts, 130-131
closed thin-walled tubes, 264-265, 268
noncircular shafts, 245-249
Area moment of inertia (see Second-area moment)
Area properties, tables, 845-847
Assumptions and limitations of theory, 114
Axial and transverse loading of beams, 471-482
Axial deflection of beam due to lateral loading, 473-474
Axial loading of bars:
combined with bending, 471-482
composite materials, (example 3,2-1) 118
deflections, 118
effects of loading conditions, 115-116
restrictions to basic formulation, 115
strain energy, 414
strains, 117
stresses, 114-117
with sudden change in section, 115-116, 235, 365-371, 873-877
using finite elements, 678-717
913•14 INDEX
B
Babinet-Soldi compensator, 645, 647, 648-650
Bathe, K. J„ 788, 842
Bazant, Z. P., 596
Beams in bending (see Bending)
Belegundu, A. D., 788
Belyayev, N. M., 403
Bending:
composite beams, 302-309
carved beams
Airy’s stress function, 233-235
deflections:
thick-walled, 446-449
thin-walled, 441-145
eccentricity, e, 311
approximate, 317-318
numerical determination, 318-321
neutral axis, 310
radial stresses, 322-324
strain energy, 446-448
tangential stresses, 309-317
straight beams:
axially constrained, 479-482
circular, in wo planes, 158-162
combined with axial loading, 471 482
composite, 302-309
deflections:
Castigliano’s method, 432-439
compatibility, (example 2.5-1 ) 90
correction for wide beams, (problem 4.4) 256
direct integration, 152-157
Rayleigh-Ritz method, 469-471
Rayleigh’s method:
combined with axial loading, 473-482
simple bending, 464-469
singularity functions, (example 3.4-7) 154, (example D.2-3) 865
superposition, 166-172
tables, 849-857
virtual load method, 460
integral relations, 859-860
neutral axis, 140, 276, 278
singularity functions, 858-867
strain energy, 417
strains, 152
stress functions, 226, (example 4.2-2) 226
stresses:
from equilibrium equations, 87-89
flexural, 138-143
lateral, (example 2.4-1) 87INDEX 915
Bending: (continued )
transverse shear, solid beams, 143-151
compound, 151-152
thin-walled, 280-302
unsymmetric, 272-279
thin flat plates:
axisymmetric circular, 330-348
rectangular, 324-330
Bending moment equations and diagrams:
by sections, 133-138
sign convention:
single-plane, 133
two-plane, 158
singularity functions, 137-138, 860-867
superposition, 166-168
tables, 849-857
Bending moment relation to:
deflection, 153
strain energy, 417
transverse shear force, 136, 859-860
Bending test specimen, fatigue test, 532
Biharmonic equation:
Cartesian coordinates, 225
polar coordinates, 231
Birefringence, 631-637
BLH Electronics, Inc., 607,621
Body forces, 18, 87
Brittle materials, strength theories, 507, 513-515, 516
Broek, D„ 596
Buckling ( see Columns in compression; Stability)
C
Calibration of photoelastic materials, 645-647, (example R.R-1) 657
Canale, R. P„ 219
Castigliano’s theorems ( see Energy methods)
Cedolin, L., 596
Centroid of rivet areas, 574
Centroids of plane areas, 845-847
Chandmpatla, T. R., 788
Chapra, S. C„ 219
Circular plates in bending, 330-348
Circular shafts in:
torsion, 129-132
two-plane bending, 158-162
Coefficient of thermal expansion:
definition, 24
effect on strain gages, 620
Cofactor, 893916 INDEX
Cofactor matrix, 893-894
Collapse modes, 580-584
Columns in compression, 192-19B, 547-555
critical force, 115, 193, 548
critical stress, 195, 549
design equations, 554-555
eccentric loading, 552-554
Euler buckling theory, 192-198
Euler load, 193-194
Euler’s curve, 195, 551-554
inelastic, 549
modes, 548-549
parabolic equation, 551
secant formula, 554
tangent-modulus method, 549-551
Compatibility:
correction of an incompatible stress field, (example 2.5-3) 95
discussion, 89-97
equations, 96-97, 100
Compensating gage, 612-613
Complementary energy, 424
Complementary energy theorem:
Castigliano's second theorem, 428
discussion, 424
—428
virtual load method, 456-457
Complementary virtual energy, 457
axial loading, 458
bending, 460-462
temperature changes, 461
torsional loading, 459
Complementary virtual work, 457
Complementary work, 424
Composite beams:
axially loaded, (example 3.2-1) 118
bending, 302-309
Compound beams, 151-152
Compressive stress, 10-11
Concentrated force acting on a flat boundary, 238-240
Considfcre, A. G., 549
Constantan, 605
Contact between curved surfaces, 361-364
Contact stresses, 357-364
Conversion of isoclinic fringes to stress trajectories, 637-641
Conversion factors, USCU to SI units, 843-844
Cook, R. D., 260, 788, 842
Correction factors for photoelastic coatings, 656-657
Coulomb-Mohi theory of failure, 507, 513-515, 516
Crack growth, 518-520
Crack modes, 520
Crack resistance force, 520INDEX 917
Critical buckling force, 115, 548
Critical stress intensity factor, 524-526
Cubic structure, 83
Cumulative damage, fatigue, 545
Curved beams in bending:
Airy’s stress function, 233-235
deflections:
thick-walled, 446-449
thin-walled, 441-445
eccentricity, e, 311
approximation, 317-318
numerical determination, 318-321
neutral axis, 310
radial stresses, 322-324
strain energy, 446-448
tangential stresses, 309-317
Cyclic loading ( see Fatigue)
Cylinders, pressurized:
thick-walled, 348-357
thin-walled, 162-166
Cylindrical coordinates:
displacements:
definition, 30-32
relation to strain, 30-32
stress components, 29-30
stress-strain relations, 29-30
Cylindrical surfaces in contact, 364
D
Dally, J. W., 672
Deflections:
axial loading, 118, 120—128
large deflections, 125-128
bending, 152-157
axially constraint
Castigliano’s first th<
Castigliano’s second
combined bending and axial loading, 471-482
composite beams, 304
complementary energy theorem, 408, 424-428
curved beams:
thick-walled, 446-449
thin-walled, 441-445
Rayleigh-Ritz method, 408, 469-471
Rayleigh’s method, 408, 464-469, 473-482
relation to strain:
cylindrical coordinates, 30-32, 35-36
rectangular coordinates, 24-27, 34-35
singularity functions, (example 3.4-7) 154, (example D.2-3) 86591» INDEX
Deflections: (continued)
superposition, 166-162
thick.-walled curved beams, 446-449
thin-walled pressurized circular cylinder, 166
torsion:
circular shafts, 130-132
closed thin-walled tubes, 264—266, 268
unsymmetric bending, 279
virtual load method, 408, 456-464
Delta rosette, 607, 881-882, 887-888
Depth of plastic zone in beams loaded inelastically, 564, 568-569
Design factor, 500
rationalization, 500-504
used in design equations, 515-516
Determinant of a matrix, 892-893
Dimensional analysis, 598-601
Diphase behavior of polymers, stress freezing, 660-662
Directional cosines:
net shear stress on an oblique surface, 64
normal to an oblique surface, 48-50
principal stresses, 67, 99
Directional numbers, 48
Disk:
rotating, 348-350, 355-357
subjected to two opposing concentrated forces, 240-244
photoelastic fringe patterns, 638-639, 643, 644, 648
stress trajectories, 637-641
Distortion, energy of, 507, 510
Ductile materials:
limit analysis, 579-584
plastic behavior, 561-573
strength theories, 509-513
Dynamic response of strain gages, 624-625
iimaataH
Elastic constants for an isotropic material -ta, 21 .5,-23 896 , (example 2.2-1) 80, 84
Elastic limit, 505
Elastic
-plastic hehavior, 561-573
Elasticity:
mathematical theory, 220-255
modulus, 21
E-LP material, 562
Endurance limit, 533
Energy methods, 404-482
Castigliano’s first theorem, 408, 419-424
Castigliano’s second theorem, 408
deflections due to temperature changes, 445
statically determinate problems, 428-449INDEX #1#
Energy methods: {continued )
statically indeterminate problems, 450-456
complementary energy theorem, 408, 424-428
Rayleigh-Ritz method, 408, 469-471
Rayleigh’s method, 408, 464-469, 473-482
virtual load method, 408, 456-464
Engesser, E, 549
E-PP material, 561-564
Equilibrium equations, 86-89, 100
Equivalent force, 5
Equivalent section of a composite beam, 302
area properties, 304
Euler buckling theory, 192-198
Euler load, 193-194
Experimental methods, 597-662
P
Factor of safety (see Design factor)
Failure of materials {see Fatigue; Fracture mechanics; Strength theories)
Fatigue:
alternating stress component, 535
bending test, 532
combined with static failure theories, 541-542
cyclic stress with a static component, 534-535
cumulative damage, 545
discussion, 532-534
endurance limit, 533
equivalent alternating stress, 543
equivalent von Mises stress, 541-542
Gerber line, 535
Goodman line, 535
mean stress component, 535
Miner’s method, 545
Neuber’s constant, 539-540
notch radius, 539
number of cycles to failure, 533-534
residual stresses, improvements by, 571-573
S~N diagram, 534
Sodeiberg line, 535
strength, 532
reduction factors, 538
Stress concentrations, effects of, 539-540
surface effects, 538
Finite element method:
assembly process, 683,
axisymmetric element, 795, 816-817
beam element:
offsets, 800-801
planar, 718,»20 INDEX
Finite element method: (continued)
load stiffening and buckling, 747-757
pin (hinge) releases for beams, 736
two-plane, 741
boundary elements, 708, 818, 820-822
consistent load vector, 729
degrees of freedom, 673, 793, 796-801
direct stiffness method, 678
discussion, 673
distributed loading, 692, 727
element library, 674
errors, 674-676
force-stress matrix, 679
the frame element, 742, 743
Gaiiss-Legendre quadrature, 772
Gauss points, 772
global coordinate system, 698
global transformation matrix, 700, 713, 746
higher-order elements, 777
inverse of reduced stiffne^matrix, 685
interpolation functions (see Shape functions)
isoparametric element, 768
isoparametric quadrilateral element, 767
local coordinate system, 698
modeling techniques, 789-835
multipoint constraints, 826-829
nodal loads, 679, 718
node, 673
numerical integration of [k], 772
parasitic shear, 777
partitioning, 685
the penalty method, 818
the Rayleigh-Ritz method, 681
reduced stiffness matrix, 685
rigid elements, 826-829
shape functions, 679, 720, 760, 768, 769
skew supports, 708
static and work equivalent (consistent) load vector, 729
static equivalent (lumped) load vector, 727
stiffness matrix, 680, 682, 722, 746, 749, 762, 772, 774
strain-deflection matrix, 679
stress-strain matrix:
plane strain, 766
plane stress, 761
subparametric element, 768
subscript notation, 677
superpardmetric element, 7
system stiffness matrix, 684INDEX »21
Finite element method: (continued)
thermal stress, 694
transformed stiffness matrix, 698, 711, 743
truss element:
one dimensional, 677-698
three-dimensional, 711
two dimensional, 698
two-dimensional triangular element, 758
Finite element modeling techniques, 789-835
aspect ratio, 795
boundary conditions (see constraints below )
connecting different element types, 796-801
constraints, 823-829
contour plots, 831-832
smoothing, 832
degrees of freedom, 793, 796-801
discussion, 789-791
echo of the input, 791 , 830
dement input information:
geometric properties, 802
material properties, 802
element loads, 820
element selection, 792-801
elements suited to structural characteristics, 797-798
load application, 818-823
load scale factors, 823
mesh density, 803
mesh generation, 803
automated, 807-808
manual, 803
semiautomatic, 803-807, 808-810
mesh refinement, 803
mesh strategy, 792-796, 808-811
multiple load cases, 818, 822-823
nodal loads. 679, 718, 819-820
offset beams to model plate ribs, 800-801
planning, 791
preprocessing, 790, 791-830
postprocessing, 790, 830-834
processing, 790, 830
skew, 795
submodeling, 811-812
symmetry, SI2-818
axisymmetry, 816
cyclic symmetry, 815-816
load antisymmetry,814-815
reflective symmetry, 812-815
superposition, 816922 INDEX
Finite element modeling techniques:{continued )
text output, 834
transition region, 796
units, 792
warp, 798
First-area moment, Q, 145
Flamant solution, 238-240
Flexural rigidity of a thin flat plate, 327, 332
Flexure formula, bending:
in one plane, 138
in two planes, 158, 272, 278
Flow analogy, 367-369
Force diagrams, 2
Force distributions, 2-9
Forces, body, 18, 87
Form correction factor for transverse shear,418
Fracture mechanics, 518-532
Fracture toughness, 524-526
Free-body diagrams, 2
Free surface conditions, 280, 602
Fringe order, 636.643-645
Fully plastic conditions, 564
Gage:
compensating, 612-613
strain, 604-626
Gage factor, 617, 883
Gauss-Legendre quadrature, 772
Gauss points, 772
Generalized Hooke’s law, 82-85
Gerber line, 535
Gere, J. M„ 596
Goodier, J. N„ 219, 260, 403
Goodman line, 535
Griffith, 518
Griffith criterion. 519-520
Griffith energy release rate, 520, 522
H
Hardrath, H. F., 596
Hertz, H„ 361, 403
Hertzian stresses, 361
Hoersch, V. A., 403
Hooke’s law:
generalized, 82-85
isotropic material, 21—24, 29-30, 33-34, 84INDEX 923
Hoop stress in a pressurized cylinder, 164
Hydrostatic state of stress, 66, 509-510
Hysteresis in strain gages, 624
I
Indicial notation, 84
Inelastic behavior:
beam of rectangular cross section in bending:
depth of plastic zone, 564, 568-569
elastic limit moment, 564
maximum strain, 569
plastic limit moment, 564
plastic section modulus, 566
residual strains and stresses (rectangular beam), 569-573
plate with hole, loaded in tension, 563
relation to fatigue, 571-573
shape factor, 565-566
shifting of neutral axis, 566-568
Inglis, 518
Integral relations for beanos in bending, 859-860
Internal forces, 10
Interpolation functions ( see Shape functions)
Invariants, stress, 69
Inverse of matrix, 894-895
Irwin, C. R., 528, 596
Isochromatic fringes, 636
Isoclinic fringes, 636
Isoelastic, 605
Isoparametric element, 774
Isotropic material, 84
Isotropic point (in plane of analysis), 71, 187, 637
J
Jacobian matrix, 770
Johnson, B. G., 596
Johnson, J. E., 596
Johnson, R. C., 596
Juvinall, R. C., 5%
K
Karma, 605
Kela, A., 842
Kirsch, G„ 235
Kirsch’s problem, 235
Kobayashi, A. S., 672
Kronecker delta, 85
Kuhn, P., 596934 INDEX
L
Lamp’s constants, 84
Laplacian operator,— 2, 223
Lateral contraction, 118
Light, behavior, 626-628
Light field, 648
Limit analysis, 579-584
Limit load, 579
Limit moment, 564
Limitations of theory, 114
Localized yielding, 366, 526
Logan, D. L„ 788
Lohner, R., 842
Longitudinal stress in a pressurized cylinder, 164
Love, A. E. H., 260
M
Macaulay’s functions, 860
Malleus, D. S„ 788, 842
Margin of safety, 500
Mathematical theory of elasticity, 220-255
Matrix:
stiffness (see Stiffness matrix)
strain, 46-47
stress, 19, 46
stress*strain, 761, 767
system stiffness, 684
transformation, 55-56, 98, 896-899
Matrix algebra, 889-896
Matrix transformation, 55-56, 98, 896-899
Maximum shear stress, 75-78, 99
Maxwell’s reciprocity theorem, 83
Maxwell’s wave theory, 626-628
Mean stress component, 535
Measurements Group, Inc., 614, 615, 616, 646, 650, 654, 658, 660, 662
Membrane equation, 162-163
Membrane stresses, 163
Micro-Measurements Division of Measurements Group, Inc., 606, 607
Miner, M. A., 596
Miner’s method, 545
Minor (of a matrix), 893
Mixed second-area moment, 272, 868
Modeting load conditions, 818-823
Modulus:
elasticity, 21
section:
elastic, 143
plastic, 566INDEX 915
Modulus: (continued)
shear, 23
Young’s, 21
Mohr’s circle:
strain gage rosette, 608
stress:
three-dimensional, 72—75
two-dimensional, 57-62
Moment of inertia ( see Second-area moment)
Monochromatic light, 626
Murakami, Y., 596
N
Neuber, H„ 539, 596
Neuber's constant, 539-540
Neutral axis in bending:
curved beams, 310
straight beams;
symmetric, 140
unsymmetric, 276, 278
Neutral axis shift due to plastic behavior, 566-568
Newton-Raphson method, 125, (example 3.2-5) 126
Nichrome V, 605
Nodal loads, 679, 718, 819-820
Node, 673
Normal stress, 11, 14
Norton, R. L., 596
Notch radius, 539
Null-balance, 613
O
Oblique incidence method,651-652, 658-660
Octahedral shear stress, (prob. 2.8) 103, 507, (prob. 7.32) 586
Orthotropic material, 83
Osgood. C. C., 596
Parallel-axis theorem, 868
Parikh, R, 842
Paris, P, C., 596
Peterson, R. E-, 219, 403
Perucchio, R., 842
Phase shift, 632
Photoelastic coating, 654
Photoelasticity, 626-662
analyzer filter, 629926 INDEX
Photoelasticity: (continued )
Babinet-Soleil compensator, 645, 647, 648-650
birefringence, 631-634
due to stress, 634-637
coating used in reflection method, 654
correction factors for photoelastic coatings, 656-657
fast axis, 631
fractional fringe orders, 647-650
fringe order.636, 643-645
isochromatic fringe, 636
isochromatic fringe analysis, 642-645
isoclinic fringe, 636
isoclinic fringe analysis, 637-642
light field, 648
material calibration:
reflection, (example 8.8-1) 657
transmission, 645-647
material properties, 646
Maxwell’s wave theory, 626-628
oblique incidence method, 651-652, 658-660
phase shift, 632
polariscope:
circular, 642
plane, 635
reflection, 655
polarization:
circular, 633-634
linear, 628-629
polarizer filter, 628
principal axes of stress, location, 636
principal strain difference equation, 656
principal stress difference equation, 635, 646
principal stresses, separation, 650-654
quarter-wave plate, 632-634
reflection, 654-660
refraction:
index, 630
principle, 630
shear difference method, 652-654
slow axis, 631
stress concentration, example, 365-366
stress freezing, 660-662
stress-optic coefficient, 635
stress separation techniques, 650-654, 658-660
stress trajectories, construction, 637-642
Tardy method, 647-648
three-dimensional methods, 660-662
tint of passage, 645INDEX 917
Photoelasticity: (continued )
transmission methods, 626-654
wavelength, light, 626
Pilkey, W. D., 403, 596
Plane elastic problems:
conversions between plane stress and plane strain problems, 224
definition, 221-222
governing equations, 222-224
Plane strain, 222, 223, 766
Plane stress, 19, 22, 33, 56-62, 70-72, 87, 98-100, 185, 221, 222, 224, 761
Plastic hinges in beams, 579
Plastic limit moment, 564, 568
Plastic section modulus, 566
Plastic zone correction, 526-530
Plate in tension with central circular hole, 235-238, 365-369
Plates (thin, flat):
buckling, 556-561
coefficient, 560-561
circular, 330-348
flexural rigidity, 327, 332
rectangular, 324-330
superposition of axisymmetric circular plates, 341-348
tables:
axisymmetric circular, 336-340
rectangular, 329-330
Plesha, M. E„ 788, 842
Poisson’s ratio, 21
Polar coordinates:
Airy stress function, 230-244
axisymmetric problems, 232
Polar moment of inertia ( see Polar second-area moment)
Polar second-area moment:
circular section, 130
effective, noncircular sections, 253-254, 265
Polariseope:
circular, 642
plane, 634
reflection, 655
Polarization of light:
circular, 633-634
plane, 628-629
Polarizer filter, 628
Prandtl's stress function, 244-247
Press fits, 352-355
Pressure vessels, thin-walled, 162-166
Pressurized cylinders:
thick-walled, 348-357
thin-walled, 162-166
*92* INDEX
Principal axes for second-area moments:
directions, 868-872
restrictions, beams in bending. 140
Principal strains, 81-82
Principal stresses:
determination:
three-dimensional, 66-69, 99
two-dimensional, 70-72, 99, 186-189
difference, by photoelastic method, 635, 646
directions, by photoelastic method, 636
separation, by photoelastic method, 650-654
Product of inertia ( see Mixed second-area moment)
Properties of areas, 845-848
Q
Quarter-wave plate:
behavior, 632-634
in a circular polariscope, 642
R
Radius of:
curvature, 153
gyration, 195
Rayleigh-Ritz method, 408, 469-471
Rayleigh’s method, 408, 464-469, 473-482
Rectangular plates in bending, 324-330
Rectangular rosette, 607-610, 879-881, 885-887
Reddy, J. N„ 788
Redundant supports (see Statically indeterminate problems)
Reflection photoelasticity, 654-660
Reflection polariscope, 655
Refraction:
basic principles, 630
relative index, 630
Residual strains, 569
Residual stresses:
fatigue, related to, 571-573
rectangular beam in bending, 569-573
Rigid elements, 826-829
Riley, W. F„ 672
Rosettes, strain gage, 607-610, 619, 879-888
Rotating disks, 348-350, 355-357
Rotation of an elastic element, 26-27, 31-32
9
Saint-Venant’s principle, 4, 114
Salmon, C. G, 596INDEX 929
Saxena, M., 842
Scaling factors, 599-600
Second-area moment:
mixed, 272, 868
parallel-axis theorem for, 868
plane areas, 138, 868
polar, 253-254, 265
principal axes, 871
tables, 845-847
Section modulus:
elastic, 143
plastic, 566
Shanley, F. R., 549
Shape factor, 565-566
Shape functions, 679, 720, 760, 768, 769
Shear center:
closed thin-walled sections, 296-302
open thin-walled beams with one axis of symmetry, 283-291
open unsymmetric thin-walled beams, 291-295
Shear difference method, 652-654
Shear flow, 129, 261-262, 263, 280-283
Shear force equations and diagrams {see entries for bending moment equations and diagrams )
Shear modulus:
definition, 23
relationship with E and v, 23, (example 2.2-1) 80
Shear strain:
elasticity definition, 47
engineering definition, 23-24, 46
Hooke’s law:
cartesian coordinates, 23-24, 33-34
cylindrical coordinates, 29-30, 35
generalized, 82-85
relation to deflection:
cylindrical coordinates, 30-32, 35-36
rectangular coordinates, 26-27, 34
Shear stress:
absence, on lines of symmetry, 601-602
components, 15-17
definition, 12, 15
direction, 64
maximum, 75-78, 99
in-plane, 189-191
net:
definition, 16
determination, 62-65
octahedral, (prob. 2.8) 103
relation between cross shear stresses, 19
torsional:
circular shafts in torsion, 129-130930 INDEX
Shear stress; {continued )
closed thin-walled tubes in torsion, 262-271
noncircular shafts in torsion, 244-255
open thin-walled beams in torsion, 253-255
transverse, in bending of beams, 280-302
Shear yield strength, 506
SI units, 843-844
Sih, G. C., 596
Singular matrix. 895
Singularity functions, 858-867
Skew supports, 708
Slenderness ratio, 195
S' N diagram, 534
Soderberg line, 535
Sokolnikoff, I, S„ 260
Spherical pressure vessel, 164
Spherical surfaces in contact, 363-364
Stability (structural), 546-561
Statically indeterminate problems:
approximations used in design, 573-584
energy methods, 450-456, 462-464
geometry of deformation, 175-185
Stiffness matrix;
bending element:
combined with axial loading, 749
single-plane bending, 722
two-plane bending, 741
frame element, 743, 745-746
isoparametric quadrilateral element, 774
system, 684
two-dimensional, 701
three-dimensional, 713
triangular element, 762
Strain:
compatibility, equations, 96-97, 100
displacement relations:
cartesian coordinates, 24-29, 34-35
cylindrical coordinates, 30-32, 35-36
inelastic distribution in rectangular beam, 569
normal
, 20-22
relations with displacements:
cylindrical coordinates, 30-32, 35-36
rectangular coordinates, 24-27, 34-35
shear:
engineering definition, 23-24, 46
theory of elasticity definition, 47
thermal, 25INDEX 9*1
Strain energy:
axial loading. 414
definition, 410-411
form correction factor for transverse shear, 418
per unit volume, 411
general state of stress, 412-413
shear stress, 411-412
state of plane stress, 413
triaxial stress, 411
thick-walled curved beams, 446-448
torsional loading of circular shafts, 414-415
transverse loading of beams, bending and direct shear, 415-419
Strain gage instrumentation, 610-616
commercial strain indicators, 613-616
switching and balancing units, 615
Wheatstone bridge circuit, 610-612
Strain gages, 604-626
adhesive, 606
compensating (temperature), 612, 620-623
configurations, 605, 607
corrections for transverse sensitivity, 619, 883-888
data sheet, 621
delta rosette, 607, 881-882, 887-888
dynamic response, 624-625
gage current heating effects, 625-626
gage factor, 617, 883
variation due to temperature change, 621-622
hysteresis, 624
installations, 606
lead-wire connections, 623-624
linearity of grid material, 617
materials, 605-606
measurement characteristics, 616-626
noise from electric and/or magnetic fields, 626
resistance change due to:
strain, 604
temperature change, 612, 620-623
resistivity, 604 <
rectangular rosette:
three-element, 607-610, 879-881, 885-887
two-element, 607, 619, 884-885
self-temperature-compensating gages, 620-621
sensitivity, 604
stain gradient, effects, 624
temperature effects, methods of correction, 612, 620-623
transverse sensitivity, 617-619, 883-888
corrections, 619, 883-888
transverse sensitivity coefficient, 618, 883
zero shift, 624922 INDEX
Strain indicators, 613-616
Strain matrix, 46-47
Strain-stress relations (see Stress-strain relations)
Strain transformations, 78-82, 100, 191-192
Strength;
definition, 498-500
fatigue, 532
offset yield, 505
ultimate, 499
yield, 505
Strength theories;
basis, 505-508
Coulomb-Mohr theory for brittle materials, 507, 513-515, 516
maximum distortion energy ( see von Mises theory)
maximum octahedral shear stress, 507, (prob. 7.32) 586
maximum principal strain, 506
maximum principal stress, 506
maximum shear stress (see Tresca theory)
maximum strain energy, 506
Tresca theory, 506, 509, 512, 515
von Mises, 507, 509-511, 512, 515
Strength uncertainty factor, 500-501
Stress:
alternating component, 535
axial, 114
bearing, 115
bending;
curved beams, 309-317, 322-324
straight beams, 138-143
thin flat plates, 324-348
compressive, 10-11
concentrations, 364-371
factor:
charts, 873-878
defined, 117, 365
plate in tension with central circular hole, 116-117, 235-238, 365-369
contact, 357-364
definition, 10-20
equilibrium equations for, 86-89, 100
flexure formula:
bending:
single-plane, 138
two-plane, 158, 272, 278
freezing, 660-662
function:
Airy’s, 225-244
PrandU’s, 244-247
hoop, 164
hydrostatic, 66, 509-510INDEX 933
Stress: (continued )
inelastic, 568
intensity factor. 520, 522-525
invariants, 69
longitudinal, 164
matrix, 19, 46
mean component, 538
normal, 10-12, 14-15
plane, 19, 221
principal:
three-dimensional analysis, 66-70, 99
two-dimensional analysis, 70-72, 99, 186-189
radial, (fig. 1.5-3a) 30
residual, 569-571
shear:
definition, 12, 15-16
net, 16
octahedral, (problem 2.8) 103
superposition, 172-175
tangential, (fig. 1.5-3a) 30
tensile, 11
torsional shear, 129-130
trajectories, 637
transformation simplified, 62-65, 98-99
transformations:
three dimensional analysis, 47-51, 62-70, 72-78, 97-99
two-dimensional analysis, 56-57, 70-72, 98-99
transverse (straight beams), 143-151
Tresca (equivalent), 509
uncertainty factor, 501-502
von Mises (equivalent), 511, 512
Stress-optic coefficient, 635
Stress-optic equation, 635
Stress-strain relations, 21-24, 29-30, 33-34, 35, 82-85 *
Stress-strain matrix:
plane strain, 766
plane stress, 761
Superposition, principle, 166-175
Switching and balancing units, 615
T
Tada, H., 596
Tangential stress, (fig. 1.5-3a) 30
Tardy method, 647-648
Taylor, R. L„ 788, 842
Temperature compensation of strain gages, 612-613, 620-623
Tensile stress, 11
Tensor, 896-899934 INDEX
Theories of failure (.see Strength theories)
Thermal coefficient of resistivity, 604
Thermal expansion, coefficient of, 24
Thermal strains, 24
Thick-walled cylinders, 348-357
Thin-walled cylinders, 162-166
Thin-walled tubes in torsion:
closed sections, 262-271
open sections, 253-255
Thomas, H. R., 403
Three-dimensional pfiotoelasticity, 660-662
Three-element rosettes:
delta, 607, 881-882, 887-888
rectangular, 607-610, 879-881, 885-887
Timoshenko, S. P., 219, 260, 403, 596
Tint of passage, 645
Toroidal shell, (example 3.6-1) 164
Torsion:
angle of twist, 130-131
circular shaft, 129-130
closed thin-walled tubes, 262-271
multiple cell, 267-271, 296-302
single cell, 263-267
elliptical cross section, (problem 4.15) 258
helix angle, 130
open thin-walled sections, 253-255
PrandtTs stress function, 244-247
rectangular cross section, 251-253
strain energy, 414-415
triangular cross section, (example 4.3-2) 249
Trajectories, stress, 637
Transformation:
coordinate, 46
directional cosines, 48-50 •
second-area moments, 871
strain, 78-82, 100, 191-192
stress, 47-78, 97-99, 185-191
stress tensor, 896-899
Transpose matrix, 892
Transverse sensitivity of strain gages, 617-619, 883-888
Transmission photoelasticity, 626-654
Transverse shear stresses in straight beams, 143-151
Tresca stress, 509
Tresca theory, 506, 509, 512, 515
Triangular finite element, 758-767
U
Ugural, A. C., 403
Ultimate strength, 499


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