General finite element description for non-uniform and discontinuous beam elements

The theory of generalized functions is used to address the static equilibrium problem of
Euler–Bernoulli non-uniform and discontinuous 2-D beams. It is shown that if simple integration rules are
applied, the full set of response variables due to end nodal displacements and to in-span loads can be derived,
in a closed form, for most common beam profiles and arbitrary discontinuity parameters. On this basis, for
finite element analysis purposes, a non-uniform and discontinuous beam element is implemented, for which
the exact stiffness matrix and the fixed-end load vector are derived. Upon computing the nodal response, no
numerical integration is required to build the response variables along the beam element