A variational formulation of classical nonlinear beam theories

This article intends to present a concise theory of spatial nonlinear classical beams followed by a special treatment of the planar case. Hereby the considered classical beams are understood as generalized one-dimensional continua that
model the mechanical behavior of three-dimensional beam-like objects. While a onedimensional continuum corresponds to a deformable curve in space, parametrized by a single material coordinate and time, a generalized continuum is augmented by further kinematical quantities depending on the very same parameters. We introduce the following three nonlinear spatial beams: The Timoshenko beam, the Euler–Bernoulli beam and the inextensible Euler–Bernoulli beam. In the spatial theory, the Euler–Bernoulli beam and its inextensible companion are presented as constrained theories. In the planar case, both constrained theories are additionally described using an alternative kinematics that intrinsically satisfies the defining constraints of these theories.