AC 33.70-5 Equation A-1. Test Case Cyclic Crack Growth Rate 𝑑𝑎 𝑑𝑁 = exp(𝐵) (𝐾𝑒𝑓𝑓𝐾𝑡ℎ ) 𝑃 [ln (𝐾𝑒𝑓𝑓𝐾𝑡ℎ )]𝑄 [ln ( 𝐾𝑐𝑟𝐾𝑒𝑓𝑓)]𝐷 A.2.5.4 Crack Grow th Ra tes.
The crack growth rates described in paragraph A.2.5.3 are displayed in tables A-3 and A-4. For reference:
• The cyclic crack growth threshold is denoted as 𝐾𝐾𝐾𝐾ℎ.
• The material fracture toughness is denoted as 𝐾𝐾𝐾𝐾𝐾𝐾.
A.2.5.5 R Ratio Corrections and Walker Model.
R ratio corrections are applied using the Walker model, where the effective stress intensity Keff, is defined in equation A-2.
𝐾𝐾𝐾𝐾𝐾𝐾𝐾𝐾 = 𝐾𝐾𝐾𝐾𝑑𝑑𝐾𝐾(1 − 𝑅𝑅)𝑚𝑚 Equation A-2. Walker model Where:
• 𝑅𝑅 is the stre ss intensity ratio, R=Kmin/Kmax
• 𝐾𝐾𝐾𝐾𝑑𝑑𝐾𝐾 𝑑𝑑𝑎𝑎𝑑𝑑 𝐾𝐾𝐾𝐾𝐾𝐾𝑎𝑎 = stress intensities at max and min stress points of the cycle pair.
• 𝐾𝐾 is the Walker exponent.
A.2.5.6 Temperature-Dependent Walker Exponent.
The Walker exponent m varies with temperature, and its values are provided in tables A-3 and A-4. Depending on the computed R ratio:
• Use the m- column if R is negative.
• Use the m+ column if R is positive or zero.
Table A-3. Test Case Cyclic Crack Growth Rate Parameters (deg F) Temperature (deg F) Kth (ksi*sqrt(in)) Kcr (ksi*sqrt(in)) B P
0.517 Q D m+ m-
0.0976 32 5.0025 67.0 -15.062 1.668 -2.523 0.6845 75 5.0025 67.0 -15.062 0.517 1.668 -2.523 0.6845 0.0976 500 7.00 67.0 -14.000 0.900 1.200 -1.800 0.6462 0.1401 700 7.94 67.0 -13.500 1.080 0.980 -1.460 0.6282 0.1601 900 8.88 67.0 -13.000 1.260 0.760 -1.120 0.6102 0.1801 1000 9.35 67.0 -12.750 1.350 0.650 -0.950 0.6012 0.1901 1100 9.82 67.0 -12.501 1.440 0.540 -0.780 0.5922 0.2001 1150 9.82 67.0 -12.501 1.440 0.540 -0.780 0.5922 0.2001 A–13
AC 33.70-5
Table A-4. Test Case Cyclic Crack Growth Rate Parameters (deg C) Temperature (deg C)
0.0 Kth (MPa*sqrt(m))
5.4970 Kcr (MPa*sqrt(m))
73.6 B -18.735 P
0.517 Q
1.668 D -2.523 m+
0.6845 m-
0.0976
23.9 5.4970 73.6 -18.735 0.517 1.668 -2.523 0.6845 0.0976
260.0 7.69 73.6 -17.673 0.900 1.200 -1.800 0.6462 0.1401
371.1 8.72 73.6 -17.173 1.080 0.980 -1.460 0.6282 0.1601
482.2 9.76 73.6 -16.673 1.260 0.760 -1.120 0.6102 0.1801
537.8 10.27 73.6 -16.423 1.350 0.650 -0.950 0.6012 0.1901
593.3 10.79 73.6 -16.174 1.440 0.540 -0.780 0.5922 0.2001
621.1 10.79 73.6 -16.174 1.440 0.540 -0.780 0.5922 0.2001 A.2.5.7 Elastic-Plastic Finite Element Analysis.
For notched locations where local plasticity may occur, it is common to “shake down” elastic stresses before performing crack growth analyses.
Alternatively, a full elastic-plastic finite element analysis can be conducted. If plasticity is considered, the material’s stress-strain properties (listed in tables A-5 and A-6) should be referenced and used along with the Ramberg-Osgood or Ludwick’s model given in equation A-3.
Equation A-3. Ramberg-Osgood Stress Strain Relationship
Table A-5. Test Case Material Constants to be Used for Considerations of Plasticity (deg F) Temperature (deg F) E (ksi) K (ksi) n 32 29,000 205.00 0.0500 68 29,000 205.00 0.0500 500 27,600 202.00 0.0500 700 26,600 200.00 0.0500 900 25,700 195.00 0.0500 950 25,600 200.00 0.0580 1000 25,500 205.00 0.0660 1100 24,400 210.00 0.0750 1150 24,400 210.00 0.0750 A–14
AC 33.70-5
Table A-6. Test Case Material Constants to be Used for Considerations of Plasticity (deg C) Temperature (deg C) E (MPa) K (MPa) n
0.0 199,955 1413.43 0.0500
20.0 199,955 1413.43 0.0500
260.0 190,302 1392.74 0.0500
371.1 183,407 1378.95 0.0500
482.2 177,202 1344.48 0.0500
510.0 176,512 1378.95 0.0580
537.8 175,823 1413.43 0.0660
593.3 168,238 1447.90 0.0750
621.1 168,238 1447.90 0.0750 A.2.6 Anomaly Distribution Curve.
The anomaly distribution, provided in appendix B, defines the probability of occurrence as a function of anomaly depth. No POF benefit is assumed from manufacturing credits for the component in the test case.
A.2.7 Inspection POD.
The POD curve used to determine the effect of an in-service inspection is shown in table D-1. This table is consistent with figure A4-1 of AC 33.70-2. The test case assumes one directed fluorescent penetrant inspection (FPI) at the component's mid-life (1500 flight cycles). The POD relates the probability of detecting an anomaly to flaw length, using the convention shown in figure A-11.
A–15
AC 33.70-5
Figure A-11. Test Case Conventions Used for Test Case Inspection Scenario A.3 Test Case Analysis Guidelines.
Analytical guidelines for probabilistic assessments are provided with the intent to minimize the variations of the manufacturer’s results due to analytical assumptions. The test case presented is based on a typical surface anomaly probabilistic fracturemechanics approach. The blade slot test case considers an anomaly located on the slot surface and on the slot edge (figure A-12). The POF is to be computed for each location. This analytical approach can be broken down into the following steps:
• Stress analysis.
• Slot location definition (maximum p rincipal stress location).
• Crack growth model definition.
• Crack growth calculations.
• Relative risk calculation.
A–16
AC 33.70-5 A.3.1 Stress Analysis.
The applicant should follow their own best practices for performing stress analysis for each time point of the mission profile. This includes:
• Selecting the level of mesh refinement.
• Handling blade attachment (assume frictionless contact).
• Accounting for material and geometric non-linearities.
Acceptable results have been obtained using various mesh refinement levels and blade attachment interface methods.
A.3.2 Slot Location Definition (Max Principal Stress Location).
The risk assessment should focus on locations where maximum principal stress occurs during the mission. These locations include:
• Slot surface.
• Slot edge (figure A-12).
The crack’s center should be placed at the maximum principal stress location.
Figure A-12. Test Case Anomaly Placement within Test Case A–17
AC 33.70-5 A.3.3 Crack Growth Model Definition.
For each location defined in figure A-12, develop a crack growth model using the following assumptions:
• Surface Cracks: Use a 2:1 aspect ratio for a semi-circular crack, with surface length (2c) equal to twice the depth (a).
• Corner Cracks: Use a 1:1 aspect ratio for a quarter-circle crack, with surface length
(c) equal to the depth (a).
• Orientation of the crack plane should be normal to the maximum principal stress.
• Ensure that plate geometry does not extend outside the physical boundaries of the component.
A.3.4 Crack Growth Calculations.
Perform crack growth calculations based on the predicted stresses and crack growth rate data. These calculations determine the residual life (life to failure from an assumed crack size) for each location. Conduct the calculations for a range of initial crack sizes to ensure that the component service life is covered. These are the guidelines for crack growth calculations:
• Assume all anomalies act as sharp propagating cracks, with no incubation/nucleation life.
• The crack plane should be oriented perpendicular to the maximum principal stress at the crack origin. Once defined the plane should remain fixed.
• Consider the impact of stress gradients.
• Use crack growth rate data provided for the test case.
• The effect of beneficial surface residual stresses, such as shot peening, should not be considered in the test case.
• Use a representative stress intensity factor solution for surface and corner cracks.
• A TMF method is prescribed to ensure consistency. The TMF method is used to consider how fast a crack will grow when the temperature and stress profiles are out of phase i.e., the timepoints associated with the peak stress and temperature in the loading cycle are not the same.
Note: The TMF method described in this AC consists of a stress rainflow (based on absolute stress) to determine the mission point’s pairs (peaks & valleys). Then, for each pair, the higher temperature of the pair’s points is used for the crack growth calculation. Figure A-13 shows the normalized (value/maximum) stress and temperature excursions at the slot bottom of the AC test case, where useful mission points were identified using letters A through F. Applying the prescribed TMF method to this mission profile results in the stress pairs and crack growth rate evaluation temperatures given in table A-7.
A–18
AC 33.70-5 A.3.5 Relative Risk Calculation.
A.3.5.1 POF for Limiting Locations.
The POF for both limiting locations (slot surface and slot edge) should be calculated based on the following factors:
• The provided anomaly distribution.
• Predicted crack growth life.
• Component service life.
• POD (see paragraph A.2.7).
A.3.5.2 Analysis Assumptions.
For this analysis:
• Consider anomalies as sharp cracks with no incubation life.
• Assume a single FPI at half the component service life.
• Assume 100% of the fleet is inspected.
A.3.5.3 Calculate Relative Risk or POF.
To calculate the relative risk or POF for the component, adjust the POF prediction for a single slot using equation A-4. Assume:
• One anomaly per slot
• 72 slots on the disk.
• Service life of 3000 flight cycles.
• POFslot is the probability of fracture of a single slot at or before the component service life.
• No manufacturing credits (value = 1.0).
POFslotPOFcom pone nt = 1 −(1 − ) Nef fManufacturing Credits Equation A-4. Test Case Component Level POF Calculation A–19
AC 33.70-5
Figure A-13. Test Case TMF Mission Example
Table A-7. Test Case Prescribed TMF Method Results Pair Max Stress Min Stress Temperature 1 Point B Point A Point A 2 Point C Point D Point C 3 Point E Point F Point F A–20
AC 33.70-5 A.4 Test Case Results.
A.4.1 OEM Results.
This test case has been analyzed independently by several OEMs, and a statistical analysis of their results, given in events per service life, is provided in table A-8 (where m = mean value; s = sample standard deviation).
Table A-8. Test Case Acceptable Component POF Results for the Slot Locations Location Failure Risk Events/Service Life Mean value: m m – 1.65 s m + 1.65 s Slot Surface Without in-service inspection With an in-service directed FPI inspection
9.15E-05
6.10E-05
8.14E-05
4.95E-05
1.03E-04
7.52E-05 Slot Edge Without in-service inspection With an in-service directed FPI inspection
4.74E-05
1.74E-05
3.74E-05
1.06E-06
6.02E-05
2.86E-05 A.4.2 Acceptance Criteria.
All results within the range (m-1.65s, m+1.65s) for both conditions are considered acceptable. This range is the interval centered on the mean value covering 90 percent of the result population assuming a log-normal distribution.
A.4.3 Geometry Verification.
Due to the complexity of axial blade slots (figures A-3 through A-8), verifying the geometry used in the finite element analysis is crucial. Based on results from seven independent analyses, the peak surface stress in the axial blade slot should fall within the range of 172 ksi to 177.1 ksi. Paragraph 1.3.3 provides neutral files that contain the test case geometry and may be used to generate the finite element model. Most commercial finite element software packages can read these files directly. If any conflicts arise between the geometry shown in figures A-3 through A-8 and the neutral files, use the neutral file.
A–21
AC 33.70-5 APPENDIX B. DEFAU LT ANOMALY DISTRIBUTION CURVES.
B.1 Manufacturing Anomaly Distribution Curve for Disk Axial Blade Slots.
B.1.1 Overview of Anomaly Distribution Curve.
This appendix provides the anomaly distribution curve for manufacturing-induced anomalies in axial blade slots of titanium and nickel engine rotors.
B.1.2 Explanation of Probability of Exceedance Curve.
Figure B-1 shows the probability of an exceedance curve for manufacturing-induced anomalies in axial blade slots. This curve represents the likelihood of having an anomaly greater than a given depth in a single disk slot.
• Horizontal Axis: Represents anomaly depth (in inches).
• Vertical Axis: Represents the probability of exceedance, or the chance that an anomaly larger than depth x exists in a single slot.
B.1.3 Tabular Data and Default Anomaly Exceedance Curve Equation.
Table B-1 provides the tabular data corresponding to the curve in figure B-1. The probability of exceedance for different anomaly depths is defined by the following equation:
• Imperial units: 𝐹𝐹(𝐾𝐾) = 1.97𝐸𝐸 − 06 ∗ 1.0635 ∗ 𝐸𝐸𝐸𝐸𝐸𝐸 (− 61.546 ∗ 𝐾𝐾)
• SI units: 𝐹𝐹(𝐾𝐾) = 1.97𝐸𝐸 − 06 ∗ 1.0635 ∗ 𝐸𝐸𝐸𝐸𝐸𝐸 (− 2.4230 ∗ 𝐾𝐾) Where:
• 𝐾𝐾 ≥ 0.001 in (or 𝐾𝐾 ≥ 0.0254 mm) is anomaly depth in inches (or millimeters)
• 𝐹𝐹(𝐾𝐾) is the exceedance probability per slot.
B.1.4 Application of the Exceedance Curve.
The exceedance curve shown in figure B-1 applies to both the slot bottom surface and the slot bottom edge. This distribution is based on a per-slot unit and does not account for size effects such as surface area or edge length at the slot bottom.
B–1
AC 33.70-5
Damage Depth (mm)
0.2540 2.54000.0254 Probability of Exceedance (-)
1.0E-09 1.0E-08 1.0E-07 1.0E-06 1.0E-05 0.001 0.010 0.100 Damage Depth (in) Figure B-1. Distribution for Manufacturing Induced Anomalies in Axial Blade Slots B–2
AC 33.70-5
Table B-1. Tabular Data for Figure B-1 Depth (in)
0.0010
0.0020
0.0030
0.0040
0.0050
0.0060
0.0070
0.0080
0.0090
0.0100
0.0110
0.0120
0.0130
0.0140
0.0150
0.0160
0.0170
0.0180
0.0190
0.0200
0.0210
0.0220
0.0230
0.0240
0.0250
0.0260
0.0270
0.0280
0.0290
0.0300
0.0350
0.0400
0.0500
0.0600
0.0700
0.0800
0.0900
0.1000 Depth (mm)
0.0254
0.0508
0.0762
0.1016
0.1270
0.1524
0.1778
0.2032
0.2286
0.2540
0.2794
0.3048
0.3302
0.3556
0.3810
0.4064
0.4318
0.4572
0.4826
0.5080
0.5334
0.5588
0.5842
0.6096
0.6350
0.6604
0.6858
0.7112
0.7366
0.7620
0.8890
1.0160
1.2700
1.5240
1.7780
2.0320
2.2860
2.5400 Exceedance (-)
1.970E-06
1.852E-06
1.742E-06
1.638E-06
1.540E-06
1.448E-06
1.362E-06
1.280E-06
1.204E-06
1.132E-06
1.065E-06
1.001E-06
9.413E-07
8.851E-07
8.323E-07
7.826E-07
7.359E-07
6.919E-07
6.506E-07
6.118E-07
5.753E-07
5.409E-07
5.087E-07
4.783E-07
4.497E-07
4.229E-07
3.977E-07
3.739E-07
3.516E-07
3.306E-07
2.430E-07
1.787E-07
9.655E-08
5.217E-08
2.819E-08
1.524E-08
8.233E-09
4.449E-09 B–3