A new method for deriving rigorous bounds on the effective elastic constants of a composite material is presented and used to derive a number of known as well as some new bounds. The new approach is based on a presentation of those constants as a sum of simple poles. The locations and strengths of the poles are treated as variational parameters, while different kinds of available information are translated into constraints on these parameters. Our new results include an extension of the range of validity of the Hashin-Shtrikman bounds to the case of composites made of isotropic materials but with an arbitrary microgeometry. We also use information on the effective elastic constants of one composite in order to obtain improved bounds on the effective elastic constants of another composite with the same or a similar microgeometry.