We use the replica trick and a variational method to determine the effective elastic coefficients of a disordered composite. We obtain for them a self-consistent formula, which is satisfactory from the points of view of low disorder and low dilution expansions. When the bulk moduli K and the shear moduli mu are such that K>~2mu, it satisfies Hashin-Shtrikman bounds and is close to the usual effective-medium approximation. In the case K>~2mu, we observe a deviation of Hashin-Shtrikman bounds which can be understood by analogy with an equivalent one-dimensional problem. Finally, this calculation allows us to derive the rigidity threshold pr for any dimension d.