# A Timoshenko Beam Formulation with 3D Response for Linear and Nonlinear Materials

## Palavras-chave:

Timoshenko beam, three-dimensional response, nonlinear analysis## Resumo

Beam finite elements are computationally efficient, widely used, and sufficiently accurate for many

applications. The intrinsic a-priori assumptions and simplifications of classical beam theories do not always

represent the actual three-dimensional stress-strain distributions. The cross-section of the Euler-Bernoulli beam

remains both plane and orthogonal to the beam axis. This theory does not account for shear deformation, which is

considered by Saint-Venant’s solution. Timoshenko beam theory includes the rotation of the cross-section, which

remains plane. This assumption yields an average shear strain, which does not correspond with the actual three-

dimensional distribution of shear strains. Shear coefficients can be adopted as a correction for beams made of

linear materials. However, the analysis of beams of nonlinear materials must consider the three-dimensional stress-

strain relationships at each point. An integrated formulation of a beam element with three-dimensional response is

discussed. The arbitrary cross-section of the corresponding Timoshenko beam element remains neither plane nor

orthogonal to the beam axis. The element kinematics is defined by two fields: the displacement shapes of the cross-

sections and the axial functions of their corresponding averaged movements. The deformed displacement shapes

are obtained by minimizing the potential energy of a beam slice submitted to the compatibility constraints of the

kinematics framework. Higher order models reproduce the three-dimensional distribution of stresses and strains

of Saint-Venant’s solution for concentrated loads and Michell’s solution for uniformly loaded beams. The solution

also agrees with three-dimensional finite element models with idealized boundary conditions. This paper presents

improvements that simplify the theory, focuses on first-order linear and nonlinear analyses, and discusses some

examples.