We present a generalization of Signorini's method to the case of live loads which allows us to derive approximate solutions to some pure traction-value problems in finite elastostatics. The boundary-value problems and the corresponding compatibility conditions are formulated in order to determine the displacement of the system up to the second-order of approximation. In particular, we consider the case of homogeneous and isotropic elastic bodies and we solve the following two traction-value problems with live loads:(i) a sphere subjected to the action of a uniform pressure field;(ii)a hollow circular cylinder whose inner and outer surfaces are subjected to uniform pressures. Then, starting from these solutions, we suggest experiments to determine the second-order constitutive constants of the elastic body. Expressions of the second-order material constants in terms of displacements and Lame' coefficients are determined.