We study in detail the influence of deviations from the molecular cylindrical symmetry on the fundamental elastic properties of uniaxial nematic liquid crystals. Results for the elastic constants are obtained for a range of molecular length-width ratio, temperature, density and molecular parameters. We compare calculated values with the experimental data of 8 OCB. It is observed that the effect of non-cylindrical molecular symmetry on the values of elastic constants of uniaxial of a nematic phase is small.