The complementary functions method for the element stiffness matrix of arbitrary spatial bars of helicoidal axes

Abstract

Abstract The statical behaviour of a spatial bar of an elastic and isotropic material under arbitrary distributed loads having a non‐circular helicoidal axis and cross‐section supported elastically by single and/or continuous supports is studied by the stiffness matrix method based on the complementary functions approach. By considering the geometrical compatibility conditions together with the constitutive equations and equations of equilibrium, a set of 12 first‐order differential equations having variable coefficients is obtained for spatial elements of helicoidal axes. The stiffness matrix and the element load vector of a helicoidal bar with a non‐circular axis and arbitrary cross‐section are obtained taking into consideration both the presence of an elastic support and the effects of the axial and shear deformations. For helicoidal staircases, the significance of both axial and shear deformations and eccentricities existing in wide and shallow sections are also investigated. The developed model has been coded in Fortran‐77, which has been applied to various example problems available in the relevant literature, and the results have been compared.