The methodology to produce five concrete specimens with high density has been conducted, incorporating natural aggregates such as sand and basalt as well as commercial aggregates like lead and copper. These specimens have been designed for use in structural and radiological shielding purposes. The five specimens exhibit exceptional mechanical features, characterized by superior performance and remarkable stability. The specimen labeled heavyweight concrete with 36% Pb (HWC36Pb) has been found to have a compressive strength of 28 MPa and a tensile strength of 2.76 MPa. Furthermore, the current work specimens show excellent potential for attenuating gamma, neutron, proton, and alpha radiation, particularly at low energies. This supports their potential use in medical and nuclear facilities. Radiation attenuation results show that the specimen with the ID HWC36Pb exhibits the maximum attenuation coefficient for gamma and neutron radiation among the other investigated specimens, which were either doped with a small quantity of lead or doped with copper. The mass attenuation coefficient of HWC36Pb at a gamma energy of 662 keV was determined to be 0.0905 cm2/g. Furthermore, it demonstrates the highest fast neutron removal cross-section, measuring 0.0944 cm−1.