Axisymmetric stresses and displacements in a finite circular bar

A class of axisymmetric boundary value problems for a finite solid circular bar is considered. The lateral surface is assumed to be traction free, whereas its end sections are subjected to prescribed tractions and/or displacements. The solution utilizes Love's stress function to generate a family of biorthogonal eigenfunctions on the interval [0,1]. The problem is formally reduced to an infinite system of linear algebraic equations, explicit expressions being given in the case of mixed boundary conditions on both ends, or all boundary conditions prescribed on a single end. Three example problems are solved: first, a bar with no body forces, a set of self-equilibrating normal stresses prescribed on one end, and no tractions on the other end; second, a bar with no body forces, a set of axisymmetric shear stresses prescribed on one end, and no tractions on the other end; third, a vertical cylinder resting on a rigid, frictionless surface, loaded by its own weight. Selected numerical results are presented in graphical form.