In the life-cycle cost analysis of a structure, the total cost of damage caused by external hazards like earthquakes, wind storms and flood is an important but highly uncertain component. In the literature, the expected damage cost is typically analyzed under the assumption of either the homogeneous Poisson process or the renewal process in an infinite time horizon (i.e., asymptotic solution). The paper reformulates the damage cost estimation problem as a compound renewal process and derives general solutions for the mean and variance of total cost, with and without discounting, over the life cycle of the structure. The paper highlights a fundamental property of the renewal process, referred to as renewal decomposition, which is a key to solving a wide range of life cycle analysis problems. The proposed formulation generalizes the results given in the literature, and it can be used to optimize the design and life cycle performance of structures.

The delay time model is a practical way to model random occurrences of failures and the effect of inspection and maintenance actions on the reliability of a repairable system. The delay time model involves two random variables describing the time of initiation of defects and time to failure after the defect initiation. This article presents a clear and structured approach to the evaluation of maintenance cost using the theory of stochastic renewal processes. This article derives the mean, variance, skewness and kurtosis of the maintenance cost in a finite time horizon. Furthermore, the probability distribution of cost is accurately estimated using the Hermite polynomial model. Using the cost distribution, the value at risk is estimated and proposed as a measure to optimize the maintenance program.