Preface ...................................................... XIII
List of Contributors ........................................... XV
Chapter 1. Probabilistic approaches to the material-related
reliability of fracture-sensitive structures
J.W.Provan
1.1. Introduction. .............................................. 1
1.1.1. Introductory remarks ................................ 1
1.1.2. Reliability: general considerations ................. 3
1.1.3. Review of fatigue reliability models ................ 4
1.1.3.1. The exponential distribution ............... 4
1.1.3.2. The normal of Gaussian distribution ........ 5
1.1.3.3. The log-normal distribution ................ 6
1.1.3.4. The gamma distribution ..................... 7
1.1.3.5. The Weibull distribution ................... 8
1.1.3.6. The Gumbel (extreme-value)
distributions ............................. 10
1.1.3.7. The Birnbaum-Saunders distribution ........ 12
1.1.3.8. Other reliability distributions ........... 14
1.1.4. The hazard rate concept ............................ 14
1.2. P-S-N Analysis ............................................ 15
1.1.2 Introductory remarks ................................ 15
1.2.2. P-S-N diagram ...................................... 15
1.2.3. Reliability when the cycles-to-failure
are dependent on the initial strength
of the component ................................... 17
1.2.4. Time dependent stress-strength ..................... 19
1.2.5. Further considerations ............................. 19
1.3. Stochastic crack growth ................................... 22
1.3.1. Introduction ....................................... 22
1.3.2. Stochastic crack propagation ....................... 22
1.3.3. Significantly weaker spot stochastic crack
growth ............................................. 26
1.4. The micromechanics approach to fatigue failure ............ 27
1.4.1. Introduction ....................................... 27
1.4.2. The foundation of micromechanics ................... 27
1.4.2.1. The micromechanic axioms pertaining
to fatigue failure ........................ 27
1.4.2.2. Experimental and theoretical studies
on the elastic response of metals ......... 29
1.4.3. Fatigue crack initiation ........................... 31
1.4.4. Fatigue crack propagation .......................... 32
1.4.5. Fatigue crack experimental investigations .......... 35
1.4.6. Comparison of theoretical and experimental
results ............................................ 36
1.5. A fatigue reliability law based on probabilistic
micromechanics ............................................ 37
1.5.1. Introductory remarks ............................... 37
1.5.2. The micromechanics fatigue reliability
relation ........................................... 38
1.5.3. An experimental investigation of fatigue
reliability laws ................................... 39
1.6. Concluding remarks ........................................ 44
Chapter 2. Probabilistic damage tolerance analysis of
aircraft structures
B.Palmberg, A.F.Blom and S.Eggwertz
2.1. Introduction .............................................. 47
2.2. Basic assumptions ......................................... 48
2.3. Load spectra .............................................. 49
2.3.1. Specification of loads and environment ............. 49
2.3.2. Acquisition of load spectra in service ............. 50
2.3.3. Load sequences, truncations ........................ 52
2.3.4. Scatter considerations ............................. 53
2.4. Stress and stress intensity analysis ...................... 54
2.4.1. Structural models .................................. 54
2.4.2. Available handbook and literature results .......... 58
2.4.3. Disturbances such as residual stresses,
contact stresses, friction ......................... 60
2.5. Imperfections ............................................. 62
2.5.1. Classification of imperfections .................... 62
2.5.2. Quality control .................................... 62
2.5.3. Equivalent initial flaw sizes ...................... 63
2.5.4. US military damage tolerance requirements .......... 67
2.6. Crack growth .............................................. 68
2.6.1. Constant amplitude crack growth rate ............... 68
2.6.2. Variable amplitude crack growth rate ............... 70
2.6.3. Variability in crack growth rate ................... 72
2.6.4. Predicting fatigue crack growth .................... 73
2.6.5. Stochastic modelling of crack growth ............... 77
2.7. Fracture mechanics and residual strength .................. 86
2.7.1 Material behaviour .................................. 86
2.7.2 Linear elastic fracture mechanics ................... 89
2.7.3 Nonlinear fracture mechanics ........................ 92
2.7.4 Scatter in fracture toughness ....................... 95
2.7.5 Probability of failure .............................. 96
Inspection during service life .................... 102
2.8.1 Damage sources and inspection procedures ........... 102
2.8.2 Probability of crack detection by NDI methods ...... 102
2.8.3 Length of intervals ................................ 106
2.8.4 US military damage tolerance requirements .......... 107
2.8.5 Crack size distribution after multiple
inspections ........................................ 108
2.9. Structural safety ........................................ 113
2.9.1 Discussion of USAF damage tolerance requirements ... 113
2.9.2 Residual strength of structures .................... 115
2.9.3 Safety analysis .................................... 116
2.9.4 Numerical example .................................. 118
Concluding remarks ....................................... 128
Chapter 3. Aircraft structural reliability and risk analysis
F.H.Hooke
3.1. Introduction ............................................. 132
3.1.1 The reliability concept ............................ 133
3.2. Basic reliability and risk mathematics ................... 135
3.2.1 Mixed population with different risks .............. 137
3.3. Physical aspects of structural failure ................... 139
3.3.1. Loading actions ................................... 139
3.3.2. Structural behaviour .............................. 140
3.4. Mathematical-statistical model representing a real
structural situation ..................................... 142
3.5. Reliability without inspections - the safe life
situation ................................................ 146
3.5.1. Risk of static ultimate load failure .............. 147
3.5.2. Risk of failure with deteriorating strength ....... 148
3.5.3. Partitioning the risk ............................. 149
3.5.4. Averaging the instantaneous risk .................. 152
3.5.5. Truncation of the strength - time curve ........... 152
3.5.6. Structures with initial cracks .................... 154
3.5.7. Structures with multiple failure modes and
locations ......................................... 155
3.6. Risk of failure with inspectable structures ......... 156
3.7. Illustrative examples ............................... 158
3.7.1. Safe life situation - ultra high strength
material .......................................... 167
3.7.2. Safety-by-inspection situation: typical
aluminium alloy material .......................... 160
3.7.2.1. Distribution of strength with virgin
strength preserved ....................... 160
3.7.2.2. Change of strength distribution with
time-deteriorating strength .............. 161
3.7.2.3. Safety by inspection ..................... 162
3.8. Acceptable risk .......................................... 164
3.9. Reliability of reliability estimates ..................... 165
3.9.1. Confidence intervals related to the sampling
of H .............................................. 166
3.9.2. Confidence regions related to the sampling
of δlogH ........................................... 166
3.9.3. Confidence intervals related to the sampling
of U0 ............................................. 167
3.9.4. Confidence in relation to extrapolation of
the spectrum ...................................... 167
3.9.5. Confidence related to safe-by-inspection
structures ........................................ 168
3.10. General discussion ...................................... 168
Chapter 4. Stochastic crack growth models for applications
to aircraft structures
V.J.N.Yang, W.H.Hsi, S.D.Manning,
and J.L.Rudd
4.1. Introduction ............................................. 171
4.2. Stochastic models for fatigue crack propagation .......... 173
4.2.1. Stochastic crack propagation model ................ 173
4.2.2. Fatigue crack growth data in fastener holes ....... 174
4.2.3. Lognormal crack growth rate model and
analysis procedures ............................... 176
4.2.4. Lognormal random process model .................... 181
4.2.5. Lognormal white noise model ....................... 182
4.2.6. Lognormal random variable model ................... 183
4.2.7. Correlation with experimental test results ........ 186
4.2.7.1. General lognormal random process
model .................................... 186
4.7.2.1. Lognormal random variable model .......... 189
4.3. Second moment approximation .............................. 191
4.3.1. Mean and standard deviation of W{t) ............... 194
4.3.2. Weibull approximation ............................. 195
4.3.3. Gamma and other approximations .................... 196
4.3.4. Correlation between second moment
approximations and experimental test results ...... 197
4.4. Fatigue crack propagation in center-cracked specimens .... 198
4.4.1. Synergistic sine hyperbolic crack growth
rate function ..................................... 201
4.4.2. Stochastic models and second moment
approximations .................................... 202
4.4.3. Correlations with experimental test results ....... 203
4.5. Factors affecting probabilistic prediction of fatigue
crack propagation ........................................ 206
4.5.1. Fatigue crack growth analysis procedures .......... 206
4.5.2. Equal number of data points for each specimen ..... 207
4.5.3. Data processing procedures ........................ 208
4.6. Conclusions and discussions .............................. 209
Chapter 5. Durability of aircraft structures
S.D.Manning, J.N.Yang, and J.L.Rudd
5.1. Introduction ............................................. 213
5.2. Durability design requirements ........................... 214
5.2.1. Analytical requirements ........................... 215
5.2.2. Experimental requirements ......................... 216
5.3. Durability analysis criteria ............................. 216
5.3.1. Durability critical parts criteria ................ 216
5.3.2. Economic life criteria/guidelines ................. 217
5.4. Durability analysis methodology .......................... 219
5.4.1. General description ............................... 220
5.4.2. Assumptions and limitations ....................... 220
5.4.3. Initial fatigue quality (IFQ) model ............... 221
5.4.3.1. IFQ model equations for Case I (b > 1) ... 224
5.4.3.2. IFQ model equations for Case
II (b = 1) ............................... 225
5.4.4. Durability analysis procedures .................... 226
5.5. Durability analysis details .............................. 227
5.5.1. EIFS distribution ................................. 228
5.5.2. Test/fractographic guidelines ..................... 229
5.5.2.1. Test guidelines .......................... 230
5.5.2.2. Guidelines for fractographic data ........ 232
5.5.3. Fractographic data pooling concepts ............... 232
5.5.4. Determination of EIFSD parameters ................. 234
5.5.4.1. General concepts and guidelines .......... 235
5.5.4.2. Estimation of Qi* ......................... 236
5.5.4.3. Estimation of ai, βi, and ei .............. 237
5.5.4.4. Determination of a and Qβ ................ 238
5.5.4.5. Estimation of EIFSD parameters
and evaluation of goodness-of-fit ........ 239
5.5.5. Statistical scaling of ft parameter ............... 245
5.5.6. Probability of crack exceedance ................... 247
5.5.6.1. Service crack growth master curve ........ 247
5.5.6.2. Crack exceedance predictions ............. 249
5.5.7. Formats for presenting durability analysis
results ........................................... 251
5.6. Durability analysis demonstration ........................ 251
5.6.1. Fighter lower wing skins .......................... 251
5.6.2. Complex-splice specimens subjected to bomber
load spectrum ..................................... 259
5.7. Comparison of deterministic and probabilistic
approaches for durability analysis ....................... 263
5.7.1. Deterministic crack growth approach ............... 263
5.7.2. Probabilistic approach ............................ 264
5.7.3. Conceptual comparisons ............................ 264
5.7.3.1. Durability analysis based on DCGA ........ 265
5.7.3.2. Durability analysis based on
probabilistic approach ................... 265
5.7.4. Conclusions ....................................... 266
5.8. Summary and concluding remarks ........................... 267
Chapter 6. The reliability of pressurized water reactor vessek
R.F.Cameron, G.O.Johnston, and A.B.Lidiard
6.1 Introduction ........................................... 269
6.2. Statistics of pressure vessel failure .................... 272
6.2.1. Nuclear primary circuit ........................... 274
6.2.2. Non-nuclear vessels ............................... 275
6.3. General physical aspects ................................. 277
6.3.1. Cracks - their causes, detection and repair ....... 281
6.3.1.1. Causes and incidence of cracks ........... 282
6.3.1.2. Detection of cracks by non-destructive
examination .............................. 283
6.3.1.3. Repair of cracks ......................... 283
6.3.2. Material toughness and failure mechanisms ......... 283
6.3.3. Crack growth by fatigue ........................... 287
6.3.4. Transient loadings ................................ 290
6.4. Mathematical formulation ................................. 291
6.4.1. General expressions for the failure integral ...... 293
6.4.2. Deterministic crack growth ........................ 296
6.4.3. Effect of previous loading: the
'cold hydrotest' .................................. 298
6.5. The distribution functions of physical quantities ........ 299
6.5.1. The initial crack-size distribution function ...... 299
6.5.1.1. Cracks arising in manufacture ............ 300
6.5.1.2. Efficiency of the detection of cracks .... 302
6.5.2. Fracture toughness and flow stress ................ 304
6.5.3. Fatigue crack growth and the transfer function .... 306
6.6. Applications ............................................. 307
6.6.1. U.S. and European L.W.R. pressure vessel
calculations ...................................... 308
6.6.2. Survey of results and conclusions ................. 309
6.6.2.1. Absolute failure rates ................... 309
6.6.2.2. Dependence of failure rate upon time
in service ............................... 311
6.6.2.3. Sensitivity to the crack incidence
function ................................. 312
6.6.2.4. Sensitivity to fracture toughness ........ 314
6.6.2.5. Sensitivity to stress-intensity
function and failure condition ........... 315
6.6.2.6. Sensitivity to crack growth rates ........ 316
6.6.2.7. Relative effects of different
transients ............................... 317
6.6.2.8. Relative contributions of different
regions of the vessel .................... 317
6.6.2.9. In-service inspection .................... 318
6.7. Conclusion ............................................... 318
Appendix 6.1 .................................................. 320
Appendix 6.2 .................................................. 320
Appendix 6.3 .................................................. 321
Chapter 7. Applications of PFM in the nuclear industry to
reactor pressure vessel, main coolant piping
and steel containment
R.Wellein
7.1 Introduction .............................................. 325
7.2. Reactor pressure vessel .................................. 326
7.2.1. Brittle fracture of the RPV ....................... 327
7.2.2. Defect distributions .............................. 327
7.2.3. Influence of inspections and tests ................ 330
7.2.4. Results ........................................... 330
7.3. Main coolant piping ...................................... 332
7.3.1. Leakage of the MCP ................................ 332
7.3.2. Model of the defect distribution in welds ......... 333
7.3.3. Influence of vibrations ........................... 336
7.4. Steel containment ........................................ 337
7.4.1. Two-criteria-approach ............................. 337
7.4.2. Distributions of material properties .............. 338
7.4.3. Distributions of defect dimensions ................ 342
7.4.4. Results ........................................... 346
7.5. Conclusions .............................................. 350
Chapter 8. Numerical methods in probabilistic fracture
mechanics
A.Bruckner
8.1. Introduction ............................................. 351
8.2. Processing of input data ................................. 352
8.2.1. Curve fitting ..................................... 352
8.2.1.1. Classical statistics ..................... 352
8.2.1.2. Bayesian methods ......................... 355
8.2.2. Extrapolation from incomplete data ................ 359
8.2.3. Model calculations ................................ 361
8.2.3.1. Model of crack formation in welds ........ 361
8.2.3.2. Extreme value theory ..................... 361
8.3. Numerical methods for the calculation of failure
probabilities ............................................ 364
8.3.1. Monte Carlo methods ............................... 364
8.3.1.1. Simulation ............................... 364
8.3.1.2. Random number generators ................. 365
8.3.2. Variance reduction ................................ 365
8.3.2.1. Crude Monte Carlo ........................ 365
8.3.2.2. Stratified sampling ...................... 367
8.3.2.3. Importance sampling ...................... 369
8.3.3. Applications to probabilistic fracture
mechanics ......................................... 372
8.3.3.1. Applications of stratified sampling ...... 372
8.3.3.2. Applications of importance sampling ...... 372
8.4. Approximation methods .................................... 375
8.4.1. Basic ideas ....................................... 375
8.4.1.1. Linear problems with normal
distributions ............................ 377
8.4.1.2. Non-linear problems with normal
distributions ............................ 378
8.4.2. Non-linear functions g and non-normal
distributions ..................................... 380
8.4.2.1 Normal tail approximation ................. 380
8.4.2.2 Alternative iteration schemes ............. 381
8.4.3 Applications to probabilistic fracture mechanics ... 382
Chapter 9. Probabilistic fracture mechanics
P.M.Besuner
9.1. Introduction ............................................. 387
9.2. General discussion ....................................... 390
9.2.1. Background ........................................ 390
9.2.2. Deterministic foundations ......................... 390
9.3. Statistical basis for PFM ................................ 391
9.4. Engineering models and PFM ............................... 393
9.5. Numerical methods ........................................ 394
9.5.1. Discussion ........................................ 394
9.5.2. Monte-Carlo methodology ........................... 395
9.5.2.1. Outline of analysis procedures ........... 396
9.5.2.2. Using the MC program ..................... 398
9.5.2.3. The basic Monte-Carlo simulation
method ................................... 399
9.5.2.4. Minimum requirements to improve the MC
computer program ......................... 400
9.6. PFM application examples ................................. 401
9.6.1. Optimization of safety factors with pure PFM ...... 401
9.6.2. A failure rate prediction with pure PFM ........... 405
9.6.2.1. The deterministic fracture mechanics
problem .................................. 408
9.6.2.2. Distribution of the input random
variables C, Δδ, Ai ...................... 408
9.6.2.3. The probabilistic analysis ............... 410
9.6.2.4. A dependence effect on the
N distribution ........................... 412
9.6.2.5. Probabilistic addition of crack
initiation cycles ........................ 413
9.6.3. RFC of inspectable hardware with calibrated PFM ... 415
9.6.3.1. General discussion of turbine disk
problem .................................. 415
9.6.3.2. Simulation of stress uncertainty ......... 417
9.6.3.3. Simulation of inspection uncertainty ..... 419
9.6.3.4. Simulation of analyses uncertainties ..... 419
9.6.3.5. Results of cost analysis ................. 421
9.6.3.6. New developments in calibrated PFM/RFC ... 423
9.6.4. CPFM for hard-to-inspect turbine shaft bores ...... 423
9.6.5. Advanced computer application for general
calibrated PFM .................................... 427
9.6.5.1. Discussion of baseline RFC procedure
simulation ............................... 430
9.7. Concluding remarks ....................................... 432
9.7.1. Summary ........................................... 432
9.7.2. Future considerations ............................. 434
9.7.3. Conclusion ........................................ 436
References .................................................... 437
Index ......................................................... 461
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