Limiting ourselves to only loading and system parameter uncertainties, the dynamic loading to a structure can be modeled as a random function while the system or material parameters can be modeled as random fields. Through a process of discretization, these random functions and random fields can be lumped into a vector of random variables that completely describe the uncertainties in loading and material parameters. These uncertainties result in a finite probability of failure or of the structural system not performing as intended. A powerful method to compute the failure probability is via the time history of the mean rate of out-crossing the "safe domain" by the structural system. Computing the mean out-crossing rate at any instant of time amounts to solving iteratively a constrained optimization problem. Each iteration of the constrained optimization problem requires the gradients of the constraints with respect to the vector of uncertain parameters and hence the response sensitivities with respect to loading and material parameters since the constraints are constituted in terms of response quantities. (Abstract shortened by UMI.)