Errors in Residual Stress Measurements due to Random Counting Statistics

Abstract

AbstractThe measurement of residual stress, using X-ray diffraction techniques, is based on the change in diffraction angle determined for the Intensity maximum of some suitable reflection from the sample when this is placed consecutively with its surface at two different angles to the diffracting planes. These diffraction angles may be obtained in a variety of ways, but are most often calculated from measurements of three X-ray diffraction intensities at angles selected in the immediate vicinity of the peak maximum at each sample angle and fitting each set of data to a parabolic curve. A simple mathematical expression may be derived relating the diffraction angles, and hence the residual stress, to the measured X-ray intensities; there will, however, be statistical errors in the calculated diffraction angles due to random counting errors in the measurement of the X-ray diffraction intensities. From the expression relating the residual stress to the X-ray intensities an equation has been derived giving the standard deviation in the residual stress due to random counting errors. In addition, a simple approximation has been obtained from this equation showing that the standard deviation is decreased by increasing the number of counts accumulated for each X-ray intensity measurement and by increasing the size of the angular increments between the data points. It will also be shown that, using the approximation, it is possible to estimate in advance the number of accumulated counts at each point necessary to attain a desired standard deviation in a residual stress measurement.