Effective elastic moduli and ultrasonic velocity of materials having aligned cylindrical pores have been derived using a series expansion in terms of the difference between the upper and lower bounds of elastic moduli obtained by the variational method. The theoretical results for polycrystalline alumina agree well with the experimental data, confirming the suggestion of previous researchers that a matrix containing parallel cylindrical pores orientated perpendicular to the applied stress, provide a better model than a spherical one in describing the porosity-dependence of elastic moduli in sintered specimens.