A Finite Element Formulation for a Beam with Varying Cross-Section Geometry along Its Length: Flexibility- Based Approach

This study addresses a finite element formulation for a beam element with changing cross-section geometry along its length. The study utilizes a flexibility-based formulation. In this method, internal element forces (axial and bending) are used to derive exact form of element stiffness matrix. To this end, equilibrium equations are fully satisfied along the element, which makes it sufficient to use one element per member to capture accurate results. This is in contrast with conventional displacement-based formulation in which displacements fields are used to obtain element stiffness matrix and equailibrium equations are only satisfied in a weighted integral form. Therefore, one needs to use more than one element per member to capture accurate results.

In particular, a web-tapered beam element is chosen as an example but the formulation steps given in the study can be easily extended to beams with different cross-section variation. Due to varying cross-section geometric properties along the length, the centroid axis is not a straight line, but rather a curved line. Hence, the proposed element considers changes in the centroid axis and calculate the stiffness matrix that includes effects of the curved centroid axis.

An elastic material behavior is assumed and no geometric nonlinear effects are considered. In addition, it is assumed that deformations are small and section remains section after the deformation. This study also provides a method to include shear deformations within the proposed element. In addition, it is shown how one can include distributed load effects on the element response. Finally, two numerical examples are provided to show the merits of the proposed element.