Using a general nonlinear elasticity law for small strain, solutions of some elastomechanic problems are stated depending on only one coordinate. If Cartesian coordinates are applied, the equations of the simple tensile test are immediately given, by which the two arbitrary functions of the nonlinear elasticity law can easily be determined. In cylindrical and polar coordinates, the stresses are calculated in thick tubes and thick hollow spheres subjected to a uniform internal pressure. Numerical examples demonstrate a considerable deviation of stress distribution from linear theory, especially at the boundary, so that the decrease of stress maximum known in technical practice can be explained by small deviations from linearity.