Stiffness Formulation for Nonprismatic Beam Elements

Abstract

This paper presents a method to define two-dimensional (2D) and three-dimensional (3D) elastic-stiffness matrices for nonprismatic elements (tapered or haunched), based on traditional beam theory and the flexibility method. The proposed formulation includes deformations and the shape of the cross section but neglects warping deformations. Although more rigorous formulations for tapered elements have already been addressed, the proposed procedure is presented so its direct application or implementation in computer programs for structural analysis is straightforward. The procedure is compared against the design tables of the Portland Cement Association (PCA). It is demonstrated that the PCA tables are obsolete for today's state-of-the-knowledge on nonprismatic members because they can lead to significant errors. A new set of design aids for most common cross sections used in building structures have been developed to substitute the PCA handbook of frame constants. Closed-form solutions for linearly tapered elements of rectangular, square, and circular cross sections are provided. Finally, it is demonstrated that the stiffness factors for nonprismatic elements depend on the span-to-depth ratio of the element ( L /h). To the writer's knowledge, no one has proved this fact before.