The decomposed form and boundary conditions of elastic beams with free faces

From the decomposition theorem of elastic beams, two classes of exact stress states are investigated for the equations of three dimensional elasticity governing elastic beams in bending deformations with free faces. One of these is the analogue of the Levy solution for elastic plates and is designated as the interior state. The other complementary class corresponds to a decaying state and is designated as the Papkovich-Fadle state. The appropriate boundary conditions have been established recently for the prescribed data at the end edge of beams to induce only an exponentially decaying elastostatic state. The present paper describes how these conditions may be used to determine the boundary conditions of these two states. The decomposition theorem of beams effectively allows us to split the prescribed edge-data correctly into two parts, one for the interior solution components and the other for the decaying solution components. An analytical solution of the decaying state is formulated to verify the validity of our boundary conditions. The results in turn show that the necessary conditions for the Papkovich-Fadle state are also sufficient conditions. The boundary conditions obtained for the interior state show that the interior solution determined by these conditions is the correct solution in the beam interior up to exponentially small terms. Moreover, with the separate consideration of the interior and decaying solution components, a relatively simple analytical solution is often practical and desirable, and the numerical computation process is essentially simplified. As an illustrative example, the present results are applied to the end-loaded cantilever beam.