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DTD Handbook

Handbook for Damage Tolerant Design

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    • Sections
      • 1. Introduction
      • 2. Fundamentals of Damage Tolerance
      • 3. Damage Size Characterizations
      • 4. Residual Strength
      • 5. Analysis Of Damage Growth
      • 6. Examples of Damage Tolerant Analyses
      • 7. Damage Tolerance Testing
      • 8. Force Management and Sustainment Engineering
      • 9. Structural Repairs
      • 10. Guidelines for Damage Tolerance Design and Fracture Control Planning
      • 11. Summary of Stress Intensity Factor Information
        • 0. Summary of Stress Intensity Factor Information
        • 1. Background of Stress Intensity Factors
        • 2. Methodology For Determining Stress Intensity Factors
        • 3. Selected Stress Intensity Factor Cases
        • 5. Computer Codes
        • 6. References
    • Examples

Section 11.3. Selected Stress Intensity Factor Cases

This section will present a catalog of stress-intensity factor solutions for some typical crack geometries.  Many of these solutions are found in computer programs and handbooks.  Tables 11.3.1 through 11.3.5 summarize the solutions that are presented.  The solutions are categorized by the location of the crack, either embedded, in a plate (surface or edge), or at a hole, in Tables 11.3.1 through 11.3.3.  Solutions for cracks in a cylinders and sphere are summarized in Table 11.3.4, and the ASTM standard specimens are listed in Table 11.3.5.  Table 11.3.6 includes listings of the parameters used in the drawings and equations as well as their definitions.

Following these tables, the equations for the stress-intensity factor solutions are given.  The solutions are presented in the same order as listed in the tables.

The remote loading solutions are presented in the form:

(11.3.1)

where the coefficient Fi is expressed as a function of geometry, and i indicates the loading type.  Some of the cases considered can be used to develop more complex solutions through the methods of superposition and compounding

Table 11.3.1.  Embedded Cracks

Description

Illustration

References

Embedded Crack in a Plate

Newman & Raju [1984]

 

Forman, et al. [1998]

 


Table 11.3.2.  Cracks in a Plate

Description

Illustration

References

Surface Crack in Plate

Newman & Raju [1984]

Forman, et al. [1989]

Through Crack in the Center of a Plate

Fedderson [1966]

Paris & Sih [1964]

Roberts & Kibler [1971]

Forman, et al. [1998]

Corner Crack at the Edge of a Plate

Raju & Newman [1988]

Forman, et al. [1998]

Through Crack at the Edge of a Plate

Tada, et al. [1973]

Forman, et al. [1998]

 


Table 11.3.3.  Cracks from Holes

Description

Illustration

References

Radial Corner Crack from a Hole

Newman & Raju [1984]

Forman, et al. [1989]

Radial Through Crack from a Hole

Shivakumar & Hsu [1977]

Zatz, et al. [1981]

Isida [1973]

Forman, et al. [1989]

Corner Crack from a Hole in a Lug

Newman & Raju [1984]

Forman & Mettu [1992]

Forman, et al. [1998]

Through Crack from a Hole in a Lug

Shivakumar & Hsu [1977]

Zatz, et al. [1981]

Forman, et al. [1989]

 


Table 11.3.4.  Cracks in Cylinders and Spheres

Description

Illustration

References

Surface Crack in a Solid Cylinder

Forman & Shivakumar [1986]

Forman, et al. [1998]

Longitudinal Surface Crack in a Cylinder

Newman & Raju [1979]

Forman, et al. [1989]

Longitudinal Through Crack in a Cylinder

Newman [1976]

Forman, et al. [1998]

Thumbnail Crack on a Hollow Cylinder

Raju & Newman [1984]

Forman, et al. [1989]

Circular Through Crack in a Cylinder

Forman, et al. [1985]

Forman, et al. [1998]

Through Crack in a Sphere

Erdogan & Kibler [1969]

Forman, et al. [1998]

 


Table 11.3.5.  ASTM Standard Specimens

Description

Illustration

References

Standard Center-Cracked Tension Specimen

Fedderson [1966]

Standard Compact Specimen

ASTM E399 [2000]

Standard Round Compact Specimen

ASTM E399 [2000]

Standard Arc-Shaped Specimen

ASTM E399 [2000]

Standard Bend Specimen

ASTM E399 [2000]


Table 11.3.6.  Description of Parameters Used for SIF Solutions

Parameter

Description

a

Crack Depth

c

Crack Length

t

Thickness

W

Width

D

Hole diameter;
cylinder diameter

B

Distance from hole center to edge of plate
Thickness (ASTM standard solutions)

R

Cylinder radius

s0

Remote tension stress

s1  and s2

Bending stresses

s3

Bearing stress

 


Table 11.3.7.  Stress Intensity Solutions for Embedded Cracks

Embedded Crack in a Plate

See Tables 11.3.11 for ff and fx equations

 


Table 11.3.8.  Stress Intensity Solutions for Cracks in a Plate

Surface Crack in a Plate

Tension

Bending

See Table 11.3.12 for M0, g1, ff , and fx  equations

Through Crack in the Center of a Plate

Tension

Bending

Corner Crack at the Edge of a Plate

Tension

Bending

See Tables 11.3.12 for ff , fa and fx  equations and
Table 11.3.9 for fi

Through Crack at the Edge of a Plate

Tension

Bending


Table 11.3.9.  Calculation of fi for Corner Crack Solution

 obtained from interpolating in f0, f1, f2 tables as follows

1

Four data points, fI{(a/c)j, a/t, c/W}|j=1,2,3,4, are calculated using cubic spline interpolation, where (a/c)j are listed tabular values of 0.2, 0.4, 0.5, 1.0, 2.0, 2.5, and 5.0, and, in general, (a/c)j=1,2 < a/c and (a/c)j=3,4 > a/c.

2

fi(a/c) are then calculated from the above four data points using piecewise Hermite polynomial interpolation.

Table of F0 Values

 

 

c/W  =  0.0

c/W  =  0.1

c/W  =  0.2

c/W  =  0.5

c/W  =  0.8

c/W  =  1.0

a/c

a/t

a-tip

c-tip

a-tip

c-tip

a-tip

c-tip

a-tip

c-tip

a-tip

c-tip

a-tip

c-tip

0.2

0.0

1.037

1.280

1.041

1.285

1.043

1.291

1.070

1.330

1.102

1.390

1.128

1.441

 

0.1

1.078

1.311

1.083

1.318

1.087

1.322

1.116

1.355

1.145

1.408

1.169

1.452

 

0.2

1.157

1.374

1.161

1.380

1.169

1.388

1.207

1.420

1.240

1.470

1.268

1.513

 

0.5

1.515

1.752

1.536

1.787

1.571

1.833

1.732

1.993

1.944

2.243

2.124

2.448

 

0.8

2.031

2.498

2.098

2.663

2.196

2.832

2.749

3.528

3.623

4.603

4.378

5.491

 

1.0

2.475

3.286

2.578

3.585

2.749

3.931

3.790

5.340

5.523

7.514

7.026

9.311

0.4

0.0

1.073

1.173

1.077

1.177

1.082

1.183

1.130

1.244

1.201

1.314

1.254

1.365

 

0.1

1.094

1.198

1.097

1.201

1.104

1.206

1.161

1.267

1.233

1.343

1.289

1.398

 

0.2

1.131

1.241

1.135

1.246

1.147

1.257

1.227

1.337

1.306

1.417

1.375

1.488

 

0.5

1.317

1.488

1.339

1.521

1.378

1.567

1.577

1.749

1.865

2.072

2.117

2.349

 

0.8

1.636

1.985

1.691

2.069

1.780

2.198

2.318

2.781

3.239

3.816

4.066

4.723

 

1.0

1.941

2.504

2.015

2.638

2.167

2.861

3.111

3.972

4.813

5.875

6.355

7.559

0.5

0.0

1.086

1.158

1.090

1.160

1.097

1.165

1.150

1.220

1.235

1.302

1.308

1.381

 

0.1

1.102

1.179

1.106

1.180

1.113

1.185

1.178

1.245

1.271

1.339

1.350

1.424

 

0.2

1.130

1.211

1.134

1.217

1.147

1.228

1.238

1.310

1.345

1.417

1.439

1.511

 

0.5

1.272

1.414

1.294

1.446

1.335

1.492

1.550

1.684

1.879

2.045

2.161

2.355

 

0.8

1.546

1.827

1.596

1.899

1.684

2.018

2.224

2.574

3.169

3.609

4.010

4.516

 

1.0

1.801

2.260

1.871

2.368

2.021

2.558

2.931

3.568

4.595

5.380

6.163

7.059

1.0

0.0

1.138

1.138

1.142

1.141

1.145

1.144

1.236

1.192

1.416

1.343

1.601

1.523

 

0.1

1.141

1.142

1.144

1.144

1.154

1.152

1.261

1.220

1.470

1.399

1.683

1.609

 

0.2

1.144

1.145

1.152

1.154

1.172

1.172

1.309

1.267

1.565

1.486

1.801

1.685

 

0.5

1.198

1.232

1.220

1.251

1.267

1.309

1.547

1.547

2.075

2.056

2.555

2.514

 

0.8

1.364

1.413

1.399

1.470

1.486

1.565

2.056

2.075

3.171

3.171

4.196

4.162

 

1.0

1.481

1.615

1.545

1.686

1.685

1.801

2.514

2.555

4.162

4.190

5.977

5.977

Table of F1 Values

 

 

c/W  =  0.0

c/W  =  0.1

c/W  =  0.2

c/W  =  0.5

c/W  =  0.8

c/W  =  1.0

a/c

a/t

a-tip

c-tip

a-tip

c-tip

a-tip

c-tip

a-tip

c-tip

a-tip

c-tip

a-tip

c-tip

0.2

0.0

1.037

1.280

1.041

1.285

1.043

1.291

1.070

1.330

1.102

1.390

1.128

1.441

 

0.1

0.939

1.287

0.940

1.289

0.945

1.294

0.975

1.336

1.029

1.400

1.077

1.458

 

0.2

0.855

1.295

0.862

1.296

0.870

1.302

0.910

1.360

0.972

1.435

1.025

1.510

 

0.5

0.683

1.475

0.689

1.486

0.706

1.520

0.820

1.632

0.956

1.829

1.070

1.990

 

0.8

0.392

1.762

0.428

1.811

0.469

1.898

0.730

2.231

1.135

2.811

1.494

3.204

 

1.0

0.056

2.050

0.093

2.129

0.165

2.266

0.572

2.793

1.264

3.745

1.883

4.577

0.4

0.0

1.073

1.173

1.077

1.177

1.082

1.183

1.130

1.244

1.201

1.314

1.254

1.365

 

0.1

0.941

1.152

0.943

1.160

0.956

1.170

1.015

1.214

1.087

1.307

1.188

1.396

 

0.2

0.820

1.148

0.828

1.157

0.842

1.168

0.911

1.212

0.997

1.333

1.124

1.455

 

0.5

0.515

1.195

0.538

1.210

0.562

1.236

0.694

1.378

0.877

1.603

1.027

1.807

 

0.8

0.194

1.340

0.217

1.360

0.247

1.400

0.488

1.705

0.903

2.243

1.255

2.739

 

1.0

-0.026

1.490

-0.018

1.503

0.035

1.573

0.357

2.044

1.028

2.857

1.698

3.599

0.5

0.0

1.086

1.158

1.090

1.160

1.097

1.165

1.150

1.220

1.235

1.302

1.308

1.381

 

0.1

0.946

1.130

0.952

1.139

0.965

1.148

1.027

1.192

1.117

1.297

1.233

1.417

 

0.2

0.808

1.114

0.820

1.126

0.840

1.140

0.915

1.183

1.019

1.320

1.167

1.482

 

0.5

0.475

1.124

0.490

1.140

0.526

1.164

0.660

1.313

0.873

1.573

1.055

1.831

 

0.8

0.129

1.223

0.150

1.243

0.184

1.281

0.422

1.570

0.838

2.099

1.197

2.654

 

1.0

-0.094

1.334

-0.079

1.343

-0.032

1.407

0.274

1.854

0.934

2.594

1.600

3.375

1.0

0.0

1.138

1.138

1.142

1.141

1.145

1.144

1.236

1.192

1.416

1.343

1.601

1.523

 

0.1

0.965

1.087

0.977

1.097

0.993

1.111

1.094

1.176

1.288

1.348

1.488

1.573

 

0.2

0.785

1.047

0.810

1.060

0.838

1.080

0.960

1.167

1.180

1.368

1.408

1.650

 

0.5

0.345

0.982

0.375

1.000

0.419

1.033

0.590

1.194

0.942

1.574

1.270

2.012

 

0.8

-0.070

0.961

-0.043

0.983

-0.006

1.031

0.228

1.280

0.698

1.831

1.189

2.551

 

1.0

-0.352

0.964

-0.323

0.990

-0.279

1.043

-.005

1.407

0.637

2.028

1.154

2.992

Table of F0 Values

 

 

c/W  =  0.0

c/W  =  0.1

c/W  =  0.2

c/W  =  0.5

c/W  =  0.8

c/W  =  1.0

a/c

a/t

a-tip

c-tip

a-tip

c-tip

a-tip

c-tip

a-tip

c-tip

a-tip

c-tip

a-tip

c-tip

0.2

0.0

1.037

1.280

1.006

1.054

0.976

0.822

0.900

0.138

0.800

-0.566

0.740

-1.033

 

0.1

1.078

1.311

1.050

1.080

1.020

0.848

0.955

0.150

0.866

-0.550

0.805

-1.018

 

0.2

1.157

1.374

1.119

1.123

1.090

0.896

1.039

0.190

0.952

-0.522

0.885

-0.996

 

0.5

1.515

1.752

1.469

1.492

1.440

1.259

1.400

0.530

1.313

-0.276

1.250

-0.814

 

0.8

2.031

2.498

1.997

2.282

2.009

2.081

2.124

1.447

2.200

0.614

2.300

0.058

 

1.0

2.475

3.286

2.470

3.085

2.558

2.967

2.873

2.536

3.320

1.821

3.700

1.347

0.4

0.0

1.070

1.175

1.050

1.000

1.010

0.796

0.940

0.215

0.845

-0.335

0.769

-0.714

 

0.1

1.095

1.198

1.070

1.015

1.037

0.812

0.970

0.242

0.875

-0.324

0.806

-0.700

 

0.2

1.131

1.241

1.100

1.039

1.074

0.852

1.010

0.276

0.922

-0.284

0.859

-0.658

 

0.5

1.317

1.488

1.281

1.288

1.271

1.112

1.250

0.563

1.196

-0.045

1.150

-0.419

 

0.8

1.630

1.985

1.629

1.798

1.652

1.635

1.772

1.199

1.912

0.649

1.998

0.282

 

1.0

1.941

2.504

1.970

2.318

2.044

2.167

2.376

1.861

2.778

1.548

3.177

1.194

0.5

0.0

1.086

1.158

1.055

0.989

1.020

0.789

0.942

0.244

0.854

-0.269

0.792

-0.625

 

0.1

1.102

1.179

1.074

1.000

1.040

0.809

0.968

0.272

0.884

-0.255

0.825

-0.603

 

0.2

1.130

1.211

1.100

1.025

1.070

0.846

1.004

0.310

0.930

-0.212

0.878

-0.561

 

0.5

1.272

1.414

1.241

1.230

1.234

1.067

1.216

0.566

1.187

0.025

1.157

-0.311

 

0.8

1.546

1.827

1.538

1.649

1.560

1.502

1.701

1.123

1.851

0.652

1.938

0.362

 

1.0

1.801

2.260

1.851

2.075

1.926

1.939

2.271

1.685

2.680

1.435

3.068

1.132

1.0

0.0

1.138

1.138

1.087

0.965

1.047

0.785

0.982

0.345

0.961

-0.070

0.964

-0.352

 

0.1

1.141

1.142

1.097

0.977

1.060

0.810

1.000

0.375

0.983

-0.043

0.990

-0.323

 

0.2

1.144

1.145

1.111

0.993

1.080

0.838

1.033

0.419

1.031

-0.006

1.043

-0.279

 

0.5

1.192

1.236

1.176

1.094

1.167

0.960

1.194

0.590

1.280

0.228

1.407

-0.005

 

0.8

1.343

1.416

1.348

1.288

1.368

1.280

1.574

0.942

1.831

0.698

2.028

0.637

 

1.0

1.523

1.601

1.573

1.488

1.650

1.408

2.012

1.270

2.551

1.189

2.992

1.154

 


Table 11.3.10.  Stress Intensity Solutions for Cracks from Holes

Radial Corner Crack from a Hole

Tension

Bending

Bearing

 

See Tables 11.3.11 and 11.3.12 for additional equations

Radial Through Crack from a Hole

Tension

Bearing

See Tables 11.3.11 and 11.3.12 for additional equations

Corner Crack from a Hole in a Lug

Bearing

See Tables 11.3.11 and 11.3.12 for additional equations

Through Crack from a Hole in a Lug

Bearing

See Tables 11.3.11 and 11.3.12 for additional equations

 

Table 11.3.11.  Additional Equations Used for Calculating SIF at Holes

 

Thru Cracks

Part-thru Cracks

G0

G1

G2

 

Gw

 

z

 

 

Table 11.3.12.  Additional Equations Used for Calculating SIF for Cracks In a Plate and Cracks at Holes

Parameter

Equation

 

fx

, M1

1

m1

m2

m3

g1

g3

g4

p

G11

G12

G13

G21

G22

G23

 


Table 11.3.13.  Cracks in Cylinders and Spheres

Surface Crack in a Solid Cylinder

Tension

Bending

Longitudinal Surface Crack in a Cylinder

See Table 11.3.12 for M0, g1, ff , and fx  equations

Longitudinal Through Crack in a Cylinder

Thumbnail Crack on a Hollow Cylinder

Tension

Bending

Circular Through Crack in a Cylinder

Tension

 

Bending

Through Crack in a Sphere


Table 11.3.14.  Stress Intensity Solutions for ASTM Standard Specimens

Standard Center-cracked Tension Specimen

Standard Compact Specimen

Standard Round Compact Specimen

Standard Arc-Shaped Specimen

Standard Bend Specimen