The conventional method of fracture probability calculations such as that adopted by the NRC-sponsored PRAISE CODE and the FAVOR CODE developed in this laboratory are both based on Monte Carlo simulation. Heavy computations are required. A new method of fracture probability calculation is developed by direct probability integration. The preliminary version of the development was published in an earlier paper. More detailed development of the method is presented here. The present approach offers simple and expedient method to obtain numerical values of fracture probability. This method can be applied to problems as general as the method of Monte Carlo simulation. This approach also provides a clear physical picture on the meaning of the probability of fracture. Parametric studies are made in this paper to show the variation of the numerical values of the probabilities of fracture as a result of the change of the standard deviation of either fracture toughness or the radiation-induced temperature shift. Also, it is shown numerically that a limiting probability can be obtained if the standard deviation of the fracture toughness approaches zero that implies a deterministic fracture toughness. It confirms the theoretical proof shown in Eq. (11). The limiting probability is the simplistic probability of crack count used by this author where both toughness and temperature shift are assumed to be deterministic values. The general probability of fracture developed here is simply a generalization of the crack count, except the crack count is selected with the appropriate fracture toughness in the toughness distribution. The toughness for the problem considered here is then multiplied by the appropriate temperature shift in the distribution function of the temperature shift. Although the present development is based on linear fracture mechanics assumption and applied to the radiated reactor vessel steel, there is no difficulty in viewing the present development as a general formulation that is capable of handling as many random variables as required by the fracture model. The multiplicity of the integration corresponds to the number of random variables. The probability integral is applied in this paper to calculate the probability of fracture for the high flux isotope reactor (HFIR) vessel that has been weakened due to the radiation embrittlement. The random variables used here are the crack length, the fracture toughness, and the radiation-induced temperature shift that is needed in the parametric representation of the radiated vessel steel.

1.
Chang
,
S. J.
,
1998
, “
Probability of Fracture and Life Extension Estimate of the High Flux Isotope Reactor Vessel
,”
ASME J. Pressure Vessel Technol.
,
120
, pp.
209
296
;
2.
also, 1997, ed., K. K. Panahi, ASME PVP-Vol. 355, pp. 207–214.
1.
Chang, S. J., 1998, “Fracture Probability and Leak Before Break Analysis for the Cold Neutron Source Moderator Vessel,” eds., S. J. Chang and D. Brochard, ASME PVP-Vol. 366, pp. 35–44.
2.
Chang
,
S. J.
,
1994
, “
Probability of Fracture for HFIR Pressure Vessel Caused by Random Crack Size or by Random Toughness
,”
ASME J. Pressure Vessel Technol.
,
116
, pp.
24
29
.
3.
Cheverton, R. D., Merkle, J. G., and Nanstat, R. K., 1988, “Evaluation of HFIR Pressure Vessel Integrity Considering Radiation Embrittlement,” ORNL/TM-10444, Oak Ridge National Laboratory, Oak Ridge, TN.
4.
Cheverton, R. D., and Dickson, T. L., 1998, “HFIR Vessel Life Extension With Enlarged HB-2 and HB-4 Beam Tubes,” ORNL/TM-13698, Oak Ridge National Laboratory, Oak Ridge, TN.
5.
Hirsch
,
P. B.
,
Roberts
,
S. G.
, and
Sanmuels
,
J.
,
1988
, “
The Brittle-Ductile Transition in Silicon
,”
Proc. R. Soc. London, Ser. A
,
A421
, pp.
25
53
.
6.
Chang
,
S. J.
, and
Ohr
,
S. M.
,
1981
, “
Dislocation Free Zone Model of Fracture
,”
J. Appl. Phys.
,
52
, pp.
7174
7181
.
7.
Gates
,
R. A.
,
1983
, “
The Relationship Between Load Factors and Failure Probabilities Determined From a Full Elastic-Plastic Probability Fracture Mechanics Analysis
,”
Int. J. Pressure Vessels Piping
,
13
, pp.
155
167
.
8.
Wilson, R., Mitchell, B. J. and Ainsworth, R. A., 1996, “Relationship Between Conditional Failure Probabilities and Corresponding Reserve Factors Derived From R6 Failure Assessment Diagram,” ASME PVP-Vol. 323, pp. 401–404.
9.
Remec
,
I.
,
Wang
,
J. A.
,
Kam
,
F. B.
, and
Farrel
,
K.
,
1994
, “
Effect of Gamma Induced Displacements on HFIR Pressure Vessel Materials
,”
J. Nucl. Mater.
,
217
, pp.
258
268
.
10.
Marshall, W., 1982, “An Assessment of the Integrity of PWR Pressure Vessels,” United Kingdom Atomic Energy Authority.