The mass attenuation of high energy X or γ-ray photons is governed primarily by the atomic number and density of the concrete as a shielding material. Mass attenuation coefficients depend on pair production, photoelectric effect and Compton scattering. It is indicated that the effective atomic number Zeff and the effective electron number Neff are useful parameter for radiation shielding. The effective atomic number Zeff and the effective electron number Neff depend on the elemental compositions of the heavy-weight and normal-weight concretes.The iron, barium and calcium concentration of the concretes is more effective than the other elemental concentrations according to the results analyzed. The Zeff and Neff values of heavy-weight concretes are found to be higher than the Zeff and Neff values of the normal-weight concretes.