We analyze the behavior of a dense granular aggregate made by identical, elastic spheres, uni-axially compressed at constant pressure. Our goal is to predict the evolution of the effective moduli along the loading path when small perturbations are applied to stressed states. The analytical model is based upon the average strain theory. We show that the moduli in the anisotropic state normalized with the corresponding initial isotropic value, are captured by a so crude model. Numerical simulations support this result.