Large Deflections of Beams with an Unknown Length of the Reference Configuration

Abstract

The present paper deals with large deflections and deformations of beams. As a novel aspect in the non-linear theory of beams, we study a case, in which the length of the reference configuration is not known in advance, namely a one span beam, which is axially fixed at one end, but can be pulled out axially from a compartment at the other end by transverse external forces, the place of this end in the reference configuration thus being unknown. We do not adopt the assumption of axial inextensibility. For the sake of simplicity, we assume the beam to be slender, neglecting the influence of shear upon the deformation, and we restrict to the case of a plane static deformation. Without loss of generality, the ends of the beam are taken as clamped against bending, and the normal force is set to zero at that compartment, from which the beam is pulled out. A continuum mechanics based derivation of the field equations for large displacements of slender beams is presented, which allows us to determine constitutive relations between certain generalized structural quantities on the basis of a constitutive modeling at the stress-strain level. Boundary conditions are formulated for the unknown place of the beam end in the reference configuration, and an additional relation for determining the latter place is derived. We then present a coordinate transformation, which enables us to write the resulting boundary value-problem with respect to a domain of fixed length. The transformed boundary value problem together with the additional relation for the unknown place of the boundary in the reference configuration is solved with conventional numerical methods applied to classical problems in non-linear structural mechanics. The solutions are presented in a non-dimensional framework and a critical non-dimensional load factor is determined, above which equilibrium cannot be reached, i.e. about which the beam would be pulled out indefinitely.