A closed-form Timoshenko beam element with multiple localized singularities for nonlinear material behavior using the strong discontinuity approach

Abstract

The development of a novel closed-form beam element for nonlinear material behavior in structures is described, which is formulated according to Timoshenko beam theory and the embedded strong discontinuity approach. Multiple strong transverse and rotation discontinuities are considered to model several dissipative mechanisms due to localized singularities in the shear strain and curvature fields. Issues of dependence on numerical techniques, as occurs in displacement- and forced-based beam elements with the finite element method, are overcome since the formulated element is derived from a pure mathematical treatment. Theoretical principles are stemming from a Lagrangian variational model, from which singularities are naturally derived. The proposed formulation is solved by the conventional methods of calculus of variations, resulting in the boundary value problem. Then, closed-form expressions for any boundary conditions are obtained. By particularizing closed-form expressions for a basic system, a symmetric closed-form flexibility matrix is obtained. This matrix naturally leads to a symmetric closed-form stiffness matrix for any type of loading and unloading process, in which the nodeless degrees of freedom of the multiple embedded strong discontinuities are naturally condensed. Since Gauss integration methods are not used, the multiple localized singularities are not influenced by predefined positions of the integration points. However, the accuracy depends on the chosen number of singularities and their positions. Then, the proposed element is mesh independent when sufficient discontinuities are considered, which does not imply an increase in the degrees of freedom. Softening discontinuities are modeled at arbitrary positions in a single element, which overcomes the drawbacks of other embedded beam theories where more than one element is necessary to identify failure since strong discontinuities are generally located at the midpoint of the beam element. The obtained closed-form stiffness matrix is not always positive definite for strain-softening problems. Nevertheless, there are no major convergence issues as it is a symmetric matrix that does not require a constraint equation, so even the complex snap-back behavior in structures is adequately modeled. The capability of the formulated beam element for nonlinear material behavior in structures is validated with representative examples.