Percentiles of von Mises Stress from Combined Random Vibration and Static Loading by Approximate Noncentral Chi Square Distribution

Firstly, a calculation for percentiles of von Mises stress in linear structures subjected to Gaussian random loads is extended to the case of Gaussian random loads having nonzero mean values, i.e.,the inclusion of static loads. The development is restricted to the case of plane stress. The method includes calculation of a given percentile of von Mises stress to any desired accuracy, a rapid estimate of the percentile, and upper and lower bounds on the von Mises stress. The calculation expands the cumulative distribution function of the von Mises stress as a series of noncentral chi-square distributions. Summation of a sufficient number of terms of the series calculates the percentile to the desired accuracy. The rapid estimate of the percentile interpolates the distribution of the von Mises stress in a small number of inverse noncentral chi-square² distribution functions. The upper and lower bounds on the percentiles take advantage of the noncentral chi-square distribution of summations of normally distributed stress components. Second and third calculation methods arise from approximations of the distribution of quadratic forms of noncentral normal variables, or equivalently, linear combinations of noncentral chi-square variables. These methods provide rapid estimates of percentiles of von Mises stress in linear structures under random loads having nonzero mean values. The accuracy and computational efficiency of the methods are reviewed and compared. The methods are expected to have wide application in design of and prognostics for components subjected to constant structural loads coupled with random loading arising from vibrations caused by wind, waves, seismic events, engines, turbulence, acoustic noise, etc.