Nonlinear Generalized Beam Theory for open thin-walled members

Abstract

In the framework of the Generalized Beam Theory (GBT) a new cross-section analysis is proposed, specifically suited for nonlinear elastic thin-walled beams (TWB). The approach is developed according to the nonlinear Galerkin method (NGM), which calls for the evaluation of nonlinear (passive) trial functions, to be used in conjunction with linear (active) trial functions, in describing the displacement field. The set of (quadratic) trial functions is determined here by requiring that the classic Vlasov's kinematic hypotheses of the linear theory (i.e. (a) transverse inextensibility and (b) unshearability) are satisfied also in the nonlinear sense. The linear field is described by the so-called conventional displacements, by neglecting non-conventional displacements, which violate Vlasov's hypotheses. The nonlinear trial functions thus generated are innovative deformation fields, which describe extensional and shear displacements in a different way from that of the non-conventional fields of the linear theory. In particular, they consist of non-constant tangential and out-of-plane displacements of the cross-section profile, able to ensure inextensibility and unshearability of all the plate elements, by balancing the second-order strains induced by the conventional displacements. Since nonlinear trial functions do not increase the number of the unknowns, the GBT spirit, as a reduction method, is preserved. A very promising example is discussed, which shows that equilibrium paths can be determined by using few linear trial functions in conjunction with the corresponding nonlinear trial functions, supplying good results when compared with burdensome finite-element solutions

Convergence of Hencky-Type Discrete Beam Model to Euler Inextensible Elastica in Large Deformation: Rigorous Proof

“…We proved a Γ-convergence result for a Hencky-type discretization of an inextensible Euler beam in large deformation regime. Future investigations should generalize the result (in a suitable form) for extensible beam models; moreover, it will be interesting to extend the convergence argument to Generalized Beam Models ( [32,31,30,25]) and also to the dynamics of the dicrete system, which should of course take into account the possibility of various kinds of dynamic instabilities ( [43,27,26]). Finally, it has to be remarked that Hencky-type discretization for Elastica has proven to be very effective, and is in fact used by several computational software packages (as for instance by MATLAB ).…”

Large deformations of 1D microstructured systems modeled as generalized Timoshenko beams

“…by means of 3D-printing) makes it now possible to produce slender objects which display a richer behavior than what can be captured by Euler beam model (see e.g. [11][12][13] for interesting examples, [14][15][16] for cases in which dynamical/instability problems are addressed and [17][18][19][20] for an approach using asymptotic justification; a review of complex structures employing fibers that can be modeled as generalized beams is [21]). The original model from Timoshenko was established in a linear framework.…”

Generalized Beam Theory for Thin-Walled Beams with Curvilinear Open Cross-Sections

“…In the above the membrane components of strain in tangential direction s and shear mid-surface indeformability are omitted according to Vlasov's conditions Equation (22). After expressing the stress and strain components in Equation (25) in terms of the unknown amplitudes a j (x), by means of the relevant equations from Sections 2.1 and 2.2, the elastic potential energy becomes:…”