We consider a post-critical deformation of an elastic rod with an extensible axis. The problem is considered in the geometrically nonlinear statement with account for the shear. To take into account the compressibility of the axis of a rod, we propose the variant of the model considering shear deformations. According to this model, the compressibility of the axis enters into the equations governing both the elongation-compression and bending. The shear was taking into account by several methods. As a result, both numerical and analytical solutions were obtained. The results are compared with the results obtained from a finite-element analysis when the rod was considered as a two-dimensional medium.