Neutron spectra in the energy range above 10-4 eV transmitted through iron shields succeedingly are measured with a BC501A liquid scintillation detector and the Bonner Ball detector using quasi-monoenergetic neutron sources generated via the 7Li(p,n) reaction by 43- and 68-MeV protons. Using the collimated source neutrons, the spectra are obtained on the neutron beam axis and at off-center positions. The calculations using the MORSE-CG and DOT3.5 codes with the DLC119 group cross-section data sets as well as the HETC-KFA2 code are carried out and compared with the measurements. The spectra calculated with the Monte Carlo code MORSE-CG and the DLC119 data agree with the measurements on the whole in the energy region above a few mega-electron-volts. It, however, is found that a few orders of Legendre expansion fail to reproduce the measurements at the position dominated by the neutrons scattered at a large angle. The calculations with the two-dimensional discrete ordinates code DOT3.5 and the DLC119 data also agree with the measurements in the overall energy region, though the restricted numbers of angular quadrature and a few orders of Legendre expansion led to the disagreement with the measurements at the off-center positions or in the neutron peak energy region. The comparison among the calculations with the Monte Carlo code HETC-KFA2 and the measurements show that the contribution of elastic scattering is very important in the intermediate energy region, and the treatment of the angular distribution of the elastic scattering reaction in the HETC-KFA2 code should be modified.