On the application of complex potentials and photoelastic data to the solution of plane problems

Abstract

This article demonstrates the accuracy and convenience of a recently developed analytical technique for determining the stress and displacement fields for plane problems from point‐wise photoelastic data and finite order, polynomial approximations of complex potentials of the Kolosoff type. The order of the approximations is consistent with the assumption of a stress field which varies essentially in a quadratic manner within each region or subregion in which the approximations are fitted to the photoelastic data.The theoretical basis of the technique is reviewed briefly and the method is then applied to the problems of an end‐loaded cantilever, a ring and a disc under diametral compression, and a vertical point load on a half‐space. Uncertainties associated with inevitable errors in experimental data are avoided by generating isoclinic and isochromatic data from the theoretical solutions of the four problems, these ‘photoelastic results’ then forming the basic initial data upon which the technique operates.Computations of the stresses, rotations and displacements indicate that the complex potentials method is equally applicable along lines of symmetry, on contours around point loads, in regions where the stress is predominantly uniaxial in character, and also where the stress field is completely two‐dimensional. The computed and theoretical values are virtually in exact agreement whenever the density of the randomly distributed data points is such that the stress field is accurately approximated by a quadratic function within the region spanned by the data points. The local approximations of the potentials are fitted to the experimental results at these data points.