Response nonnormality is investigated for a yielding primary structure and a linear secondary system (P-S) subjected to a normally distributed ground acceleration. The nonlinearity considered is bilinear hysteretic (BLH) yielding in the primary structure. The coefficient of excess (COE), which is a normalized fourth cumulant function, is used as a measure of the nonnormality in the current study. An initial effort focuses on the nonnormality of primary absolute acceleration, since this is the base excitation of a light secondary system. Analytical and numerical results for a nonlinear but nonhysteretic substitute structure are shown to be in good agreement with those from simulation for both mean squared levels and COE of response. It is shown that the acceleration of the primary system can be significantly nonnormal in some situations. Linear substitute methods are used for analytically evaluating the nonnormality of secondary response. The basic concept is to use a linear model with nonnormal excitation to replace the nonlinear primary element with normal excitation, with the goal of matching the trispectrum for the acceleration of these two systems. The trispectrum is the frequency decomposition of the fourth cumulant function. Periodogram analysis (a special FFT technique for obtaining polyspectra) is developed for evaluating the trispectral function of BLH primary acceleration. A two filters model (with a more narrowband fourth cumulant filter) gives good approximations for the COE values of secondary response in most cases including both cascade and noncascade analysis. The probability of failure of secondary response affected by nonnormality due to nonlinearity in the primary is investigated. A nonnormality correction factor (NCF) which is equal to the ratio of the expected life for a Gaussian process to the expected life for the non-Gaussian process is used as an index of the nonnormality effect. Analytical approaches based on knowledge of the first four response cumulants are developed to approximate the NCF values. It is shown that the NCF for first-passage failure generally is more significant than for fatigue failure based on the cases in this study, and both failure modes can be significantly affected by the nonnormality in some situations.