A continuum mechanics based derivation of Reissner’s large-displacement finite-strain beam theory: the case of plane deformations of originally straight Bernoulli–Euler beams

In the present paper, we present a continuum mechanics based derivation of Reissner’s equations for large-displacements and finite-strains of beams, where we restrict ourselves to the case of plane deformations of originally straight Bernoulli–Euler beams. For the latter case of extensible elastica, we succeed in attaching a continuum mechanics meaning to the stress resultants and to all of the generalized strains, which were originally introduced by Reissner at the beam-theory level. Our derivations thus circumvent the problem of needing to determine constitutive relations between stress resultants and generalized strains by physical experiments. Instead, constitutive relations at the stress–strain level can be utilized. Subsequently, this is exemplarily shown for a linear relation between Biot stress and Biot strain, which leads to linear constitutive relations at the beam-theory level, and for a linear relation between the second Piola–Kirchhoff stress and the Green strain, which gives non-linear constitutive relations at the beam theory level. A simple inverse method for analytically constructing solutions of Reissner’s non-linear relations is shortly pointed out in Appendix I.