A semidestructive method has been developed for determining the principal residual stresses and directions in rectangularly orthotropic materials. The reduction equations are based upon a set of functions that describe the surface strain-relaxation field about a hole drilled to a limited depth into the material. Three constants contained in the strain functions have to be determined by calibration tests; they are related to three general constants and the elastic material constants to establish applicability to an orthotropic material. Expressions for the planar residual-stress components in the material-symmetry directions are then devel oped, and from Mohr's stress circle, the principal residual stresses and directions are determined.