Enhancement of elementary beam theories in order to obtain exact solutions for elastic rectangular beams

Boley's method is utilized in order to show that the elementary Bernoulli–Euler beam theory can be enhanced such that exact solutions of the plane-stress theory of linear elasticity are obtained for force loaded rectangular beams. An equivalent enhancement is derived for the elementary Timoshenko theory of beams. The enhancement terms act analogous to thermal loadings; they follow from the force loading of the rectangular beam in an explicit form. The resulting boundary value problem of fourth order can be efficiently solved by means of symbolic computer codes. As an illustrative example, a redundant beam is studied, which is simply supported at one end, and which is clamped at the other end. Outcomes for three alternative clamped end boundary conditions are compared.