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angle of twist applying Eq assumed axial axis beam bending body forces boundary conditions buckling Castigliano's theorem centroid column compressive constant coordinates cross section curvature curve cylinder deflection deformation derived Determine disk displacement elastic equations of equilibrium Example expression factor of safety failure Figure finite difference finite element finite element method flange given by Eq Hooke's law inertia integration load material matrix maximum principal maximum shearing stress maximum stress membrane method midspan Mohr's circle nodal force normal stress obtained plane strain plane stress plastic plate principal stresses problem radial radius rectangular represents respectively rotation shear center shear force shear strain shearing stress theory shown in Fig simply supported solution static statically indeterminate strain energy stress components stress function stresses acting subjected Substituting surface tangential stress temperature tensile thickness torque torsion unit length values yield zero