Formulação variacional de elementos finitos estabilizados aplicado ao problema de viga de Timoshenko com geometria arbitrária no espaço tridimensional

Abstract

We present a mathematical model for flexible structures with arbitrary geome- tries in three-dimensional space, governed by beam theory that accounts for bending,

shear, and torsion, within the framework of small displacements and strains. We be- gin by revisiting the classical theory of straight beams, founded on the Euler–Bernoulli

and Timoshenko models, and derive the governing equations for deformation under

distributed loading. These concepts are then extended to spatially curved beams, re- sulting in the derivation of constitutive and equilibrium equations that form a system

of ordinary differential equations. To solve this problem, an abstract variational formu- lation founded on the Hellinger–Reissner principle is adopted, which enables general- ized displacements and internal forces to be treated as independent variables. In this

context, the displacement and internal force fields are approximated by interpolation

functions of the same polynomial degree, continuous for displacements and discon- tinuous for internal forces, thereby allowing greater flexibility in accurately capturing

their internal variations. To guarantee the numerical stability of the approximations, least-squares residual terms are directly incorporated into the equilibrium equations at the element level, which prevents locking phenomena in the solution. Numerical results are presented to demonstrate the potential of the formulation.