This study concerns the formulation and validation of a 3d frame element that accounts for the inelastic response under the interaction of axial, flexural, shear and torsional effects. The proposed finite element addresses the need for a general beam element capable of accurately simulating the global as well as the local response of slender structural elements.
The new element accounts for the warping of arbitrary cross sections due to shear and torsion by introducing warping degrees of freedom at each section in order to accommodate higher order strain kinematics. The warping profile at a section uses the necessary number of Lagrange interpolation functions for the desired accuracy of local response. The warping distribution along the element axis is described by either Lagrange polynomials or spline interpolation functions. The number of interpolation parameters can be adjusted to control the accuracy of the local and global response.
The 3d beam element formulation is derived from a mixed Hu-Washizu variational potential, with the inclusion of the warping displacements as independent variables. In the proposed formulation the section response is coupled through the interpolation of the stress resultants and the warping displacements along the element axis. Because the stress resultants satisfy the element equilibrium exactly with the use of suitable force interpolation functions, the element is free of shear-locking. Non-uniform warping is accommodated by the warping displacement distributions, so that the proposed element can represent the stresses arising from local warping constraints. The element is incorporated in a general purpose finite element analysis program with the consistent linearization of the governing equations for warping force equilibrium and for element compatibility resulting in a robust algorithm for the element state determination.
The element is validated with several examples of linear and nonlinear material response of steel members. The linear elastic response is validated with analytical results, finite element models and some available experimental measurements. The inelastic response under monotonic and cyclic load conditions involves shear link specimens with wide flange and box section under high shear. The accuracy and computational efficiency of the proposed element is demonstrated by comparing the numerical results with available experimental measurements from eccentrically braced steel frames and with local response results from solid and shell finite element models.
The study concludes with the investigation of the axial, flexural and torsion interaction under large displacements. The corotational formulation is derived from a general hypothesis that establishes the accuracy and scope of application of this method. Several numerical examples are used for the illustration of key nonlinear geometry aspects of slender elements under the interaction of axial, flexural and torsional effects.
