Formulation for second-order inelastic analysis of steel frames including shear deformation effect

Finite elements based on the Euler-Bernoulli beam theory differed from elements that consider Timoshenko theory, once the first theory neglects the deformation due to shear and hence it is not suitable for thick and short beams. In Timoshenko beam formulation, the cross-sections remain plane but not perpendicular to the neutral axis after deformation due the effects of shear strains. This paper presents the development of a finite element model to be used in the inelastic second-order analysis of planar steel frames. The finite element model considers the spread of plastification within the cross-section and along the member length, several residual stress distributions, members shear deformations based on the Timoshenko theory, P-Δ and P-δ second order effects. The proposed theoretical development considers the updated Lagrangian formulation and the corotational technique for the consistent deduction of the element tangent stiffness matrix. The theory predicts that nodes will suffer large displacements and rotations, and the elements of the structure, large stretches and curvatures. A computer program capable of performing advanced inelastic analysis is developed and numerical examples are presented in order to prove the efficiency of the proposed formulation. It is shown that the Timoshenko beam model is clearly superior to Euler-Bernoulli model in precisely predicting the structural response.