Environmental temperature changes produce cyclic thermal stresses in structures. Because temperature and hence thermal stresses are random and mechanical properties such as strength, relaxation modulus, and fracture toughness are time and temperature dependent as well as statistically variable quantities in rocket motor grains, crack growth will also be random. To include all these effects, the Forman crack growth rate relation will be used. The probability of failure, defined as the chance that a crack growing under the influence of such stresses exceeds the critical crack length, is calculated. A hazard function will be defined as the probability of failure during a single cycle after the cylinder has survived N previous cycles. The hazard function will then be integrated to yield the progressively increasing probability of fracture. The stress distribution through the cylindrical wall is not uniform; as a result, a step by step finite element analysis will be carried out to determine stress intensity and critical crack size as the crack travels through the varying stress field. The probability of failure is conditioned on the chance that a crack exists. / Ph. D.