Stiffness Analysis of Grids Including Warping

Two methods of including warping effects in the stiffness method of analysis are presented. Method B seems to be superior to Method A for cases where the warping constant is not large. In the limiting case where I_w=0, the warping effects disappear and leave only the familiar GI_x/L. When I_w is small relative to I_x (approximately pL>5 for each element) computational errors grow, because the stiffness matrix tends to become singular as the elements on the main diagonal associated with warping approach zero. This is not a serious practical problem, as good solutions can be obtained using an ordinary grid analysis neglecting warping for structures where warping stiffness is small and p is large. Composite bridges seem to fall near the borderline where warping can be neglected. The bridge used in the example above was noncomposite so that warping would be significant. Bimoment and warping torsion are obtained for grid structures. Results of computer programs based upon these methods are shown to agree closely with published solutions for straight beams, a curved beam and a curved highway bridge.