Analyses of plane parabolic arches using parabolic curved beam elements

Abstract

The stiffness matrix of a general planar six degrees of freedom curved beam element was derived utilizing Castigliano's second theorem. The theorem was also employed in the derivation of the equivalent loads due to uniformly distributed applied loads. The arch shape was defined by the parabolic equation in global coordinates. Through coordinate transformation, the element equation in local coordinates was obtained. The general curved beam stiffness matrix was then reduced to be the parabolic curved beam stiffness matrix. A finite element computer program was written utilizing this parabolic beam element, and the Gauss-Legendre quadrature was employed as the numerical integration technique. Example problems were solved utilizing the parabolic curved beam formulation. These solutions were compared to those obtained by simulating the parabolic arch as a series of straight beam elements. Results showed that the curved beam element was more economical and more reliable than the straight beam element.