Second-order two-cycle analysis of frames based on interpolation functions from the solution of the beam-column differential equation

Abstract

In geometrically nonlinear problems solved using the Finite Element Method (FEM), the structure response is directly influenced by the level of discretization and the nonlinear solution algorithm used. To reduce the discretization dependence, exact solutions are developed based on the deformed infinitesimal element equilibrium. To deal with the nonlinear solution problem, the two-cycle method can be used, since it is not dependent on load or displacement steps. The two-cycle method developed by Chen & Lui (1991) uses the classical geometric matrix and is not accurate for high axial loads. This happens because the geometric matrix is obtained using Hermitian polynomials which are approximate solutions. To circumvent this issue, the frame element's tangent matrix is obtained using interpolation functions that match the homogeneous solution of the differential equation of the beam-column problem. The main objective of this study is to carry out a second order analysis of the frames and obtain equilibrium paths using the two-cycle method and the tangent stiffness matrix based on solutions of the differential equations obtained from the element's deformed configuration. The results in terms of displacements and rotations for the examples studied are identical to the analytical solutions, showing that the combination of the two-cycle method with the exact element formulation is promising and can diminish the need for discretization.

Evaluation of the P-Δ (P-Delta) effect in columns and frames using the two-cycle method based on the solution of the beam-column differential equation

“…Therefore, the resulting matrix coefficients depend on the axial force in the element. The complete formulation and derivation of the tangent stiffness matrix is given in [5 , 14] . The main difference now is that there are no separate components of the tangent stiffness matrix, i.e., it presents a closed form for its coefficients, which depend on elastic parameters and axial loads, as shown in Eq.…”