An explicit matrix algorithm for solving three-dimensional elastic wedge under surface loads

Analytical solutions for three-dimensional wedge problems are difficult to obtain. It thus leads inevitably to the use of alternative methods such as the finite element method in practice. We here present an explicit matrix algorithm for solving 3D wedge problems under general surface loads: arbitrarily distributed normal and shear loads. The methodology is based on the concept of overlapping two half-spaces formed by the surfaces of a wedge and all calculations are based on half-space equivalent loads. We show that the equivalent loads on the two half-spaces are directly related to the product of the original loads on the wedge and transformation matrices. The transformation matrices are functions of wedge angle, mesh structure, and Poisson's ratio, but not the applied load. Hence, a 3D wedge problem can be solved using those classical solutions for half-spaces once the equivalent loads are obtained. Stress analyses of elastic wedge with different wedge angles are conducted. Results of three special cases: wedge angle of 170°, 90° and 60° are compared with half-space results, the published data of quarter-space and FEM, respectively. The new algorithm is validated by the good correlation shown in the comparison. The effect of wedge angle on internal stresses is also discussed.