• DTDHandbook
• Contact
• Contributors
• 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.

 (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 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