Second-moment methods are widely applied in structural reliability. Recently, so-called first-order reliability methods have been developed that are capable of producing reliable estimates of the failure probability for arbitrary design situations and distributional assumptions for the uncertainity vector. In essence, nonlinear functional relationships or probability distribution transformations are approximated by linear Taylor expansions so that the simple second-moment calculus is retained. Failure probabilities are obtained by evaluating the standard normal integral, which is the probability content of a circular normal distribution in a domain bounded by a hyperplane. In this paper second-order expansions are studied to approximate the failure surface and some results of the statistical theory of quadratic forms in normal variates are used to calculate improved estimates of the failure probability.