Computations have been made for the effect of a nonlinearity of the hard spring type upon the fatigue damage ensuing on excitation of a resonator by a stationary, broad‐band, random Gaussian force. The fatigue theory used assumes simple accumulation of damage in proportion to a power of the peak values of response strain, without interactions between peaks. Two cases are distinguished: (a) strain proportional to displacement; (b) strain nonlinearly dependent on displacement, as is the surface fiber strain in a bar with pinned ends, responding in a single mode.Numerical results are presented that indicate for case (a) a continuous decrease, with increasing nonlinearity, of the rate of accumulation of damage, while for case (b) this damage rate increases to a maximum as large as 3.4 times the linear estimate, and then falls below the value for linear response. Probability density (in response amplitude) of damage is greater at small amplitudes, and less at large, than for the linear case. The value of response amplitude of maximum damage density is also affected by nonlinearity.