The Classic Stress-Function Approach

The Classic Stress-Function Approach

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 var-ious coordinate systems have been solved since elasticity was formalized. These results are used extensively for design since many structures can be ide-alized 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 ana-lytic solution is feasible.

Hundreds of solutions are available and there is space in this and the next two chapters to present only a limited number* selected on the basis of one or more of the following criteria:

a.  to best demonstrate the stress-function method,

b.  to give insight as to the fundamental behavior of isotropic and devi-atoric stress fields,

c.  to present questions, perhaps unanswered, concerning the theory,

d.  to suggest a new problem not yet solved that might excite a student’s interest, or

e.  to display particular utility for design in themselves or by superpo-sition.

The presentation will be in a somewhat diagrammatic form with frequent ref-erence to figures augmented by sketches and notes. This approach is intended to emphasize the rational aspects of the semi-inverse method based on physi-cal reasoning.