ABSTRACT
Most solutions in two dimensions, for other than linear stress fields where direct integration is possible, can be obtained most easily by the semi-inverse method using stress functions. Usually Airy’s stress function is employed which, in two dimensions, reduces the three field equations to one fourth-order partial differential equation.* This “biharmonic equation” is exact for plane strain and nearly so for plane stress if the plate is reasonably thin.**
For simple geometry and loadings, a wide range of practical problems in various coordinate systems have been solved since elasticity was formalized. These results are used extensively for design since many structures can be idealized to plane stress or plane strain without serious error. Moreover, such two-dimensional solutions are essential benchmarks to, on the one hand, establish the range where the simpler strength-of-materials type analysis is adequate and, on the other, to validate the more complicated numerical or experimental models necessary to determine fields in specific structures that
cannot be safely idealized to a shape or loading pattern where a closed-form analytic solution is feasible.
