The statical behaviour of a planar bar of an elastic and isotropic material having an arbitrary axis and a cross-section supported elastically by single and/or continuous supports is studied by the stiffness matrix method based on the complementary functions approach. The Timoshenko beam theory is extended for three-dimensional curvilinear bars taking into consideration the effects of axial and shear deformation. 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 the spatial elements. From expressions developed in such manner for three-dimensional bars, the governing equations for the special case of a planar bar loaded within or perpendicular to its plane are derived, which are next solved by the complementary functions method. The stiffness matrix and the element load vector of a planar bar with an arbitrary axis are obtained taking into consideration both the presence of an elastic support and the effects of axial and shear deformations. 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.