A complete theory of a direct integration method for solving the steady-state integral transport equation in general geometry is presented together with special techniques for an accurate treatment of monoenergetic radiation source and for mitigation of the ray effect. Emphasis is on several characteristic features, which make the method well adapted to shielding calculations, such as an exact treatment of anisotropic scattering by applying the Klein-Nishina formula for Compton scattering and the differential scattering cross section itself for neutron elastic scattering, analytical integration of the flux term and also direct integration of the source term over the spatial variable in the radiation moving direction, the absence of iterative calculations for obtaining the group angular flux but, instead, applying the point-energy calculation, and optional use of an analytical unscattered flux calculation for mitigating the ray effects. For verifying the validity of the present method, several comparisons of the calculations are presented using the one- and the two-dimensional codes, PALLAS-PL, SP-Br and PALLAS-2DCY-FC, with the experiments adopted as shielding benchmark problems. Fairly good agreement is obtained between PALLAS calculations and experiments on the gamma-ray angular flux spectra at several angles as well as the energy spectra at two and three mean-free-paths in water. For neutron streaming through a cylindrical duct and also an annular duct, PALLAS calculations are in fairly good agreement with experiments in terms of the reaction rate except for thermal neutrons, where an obvious underestimation is obtained. For neutron deep penetration in an iron shield, selected for examining the weakest point of the method, a PALLAS calculation is found to be adequate for shielding design calculations, though some discrepancy is seen between calculation and experiment on neutron energy spectra at 20- and 30-in. depths.