Geometric Stiffness of Space Frames Using Symbolic Algebra

Abstract

This paper is concerned with the derivation of the geometric stiffness matrix for space frames. Symbolic algebra is used to calculate the gradient of the member force vector that defines, in global coordinates, the geometric stiffness matrix. Members of solid cross-sections with no warping are considered. The independent nodal variables are the common position coordinates of the member ends. An additional independent variable, which cannot be related to the position coordinates, is the angle of twist at one end of each member. Since the angle of twist is defined locally, its effect on the stiffness matrix is found separately and then transformed to the global coordinates. The advantages of the present approach are its explicitness in derivation and clear physical insight. Finally, the geometric stiffness matrix is implanted into an existing 3D nonlinear frame analysis program. For the analytical, numerical and experimental benchmark examples studied, the present results appear to be in good agreement with the results available in the literature.