A non-stationary ground motion is assumed to be expressed as the product of an envelope function representing the temporal amplitude modulation and another function representing the frequency content of a stationary random process. In most of the previous investigations on probabilistic critical excitation problems, the envelope functions of non-stationary ground motions were fixed and the critical frequency contents were found. In contrast to the previous studies, the critical envelope functions are investigated here with the frequency contents fixed to the critical one found in the previous study. The mean total energy of the ground motions is constrained and the mean-square drift of a single-degree-of-freedom system is maximized in the present problem. It is shown that the order interchange of the double maximization procedure with respect to time and to the envelope function can be an efficient and powerful solution algorithm for finding the critical envelope function. An upper bound of the mean-square drift enabling the efficient and rather accurate evaluation of the mean-square drift is also derived by the use of the Cauchy-Schwarz inequality. (C) 2004 Elsevier Ltd. All rights reserved.