The shape transformations of red blood cells stuck in capillary narrowings with radii close to the critical radius where the maximum deformations occur are analyzed. The membrane skeleton deformations are studied within the effective network model and the continuum elastic model, whereas the area-difference elasticity model is applied to describe the phospholipid bilayer. A minimization of the total free energy is performed to determine the cell shapes in a stopped flow, which are calculated by a triangulated representation of the membrane surface. The shapes are asymmetric, characterized by a single invagination, which decreases with decreasing radii of the narrowing and vanishes at its critical radius. The largest stretching deformations of the skeleton are at the ends of the elongated shape, and remarkable shear deformations appear around the invagination. The membrane's mechanical energy increases with the decreasing radius of the narrowing, predominantly due to the deformation of membrane skeleton. The increase in the shear energy is significantly larger than any other energy contribution within both models. The pressure differences needed for the penetration into the narrowing are strongly coupled with the membrane's mechanical energy. Their values were found to be of the order of 10 Pa. Both models correspond well.