The deformation of the elastic transversally isotropic layer under normal pressure in a three-dimensional formulation is analyzed. The asymptotic formulas that describe the behavior of the soft layer, the rigidity of which in the transversal direction is much lower compared with the tangential direction are obtained. A comparison has been carried out of the exact and asymptotic solutions. The influence of the parameters on the value in relation to a change in the layer thickness is examined.