An apparatus for stabilizing a core plate against lateral deflection. A plurality of core plate wedge assemblies (e.g., four) are installed in the circumferential gap between the circular outer peripheral edge of the core plate and the circular cylindrical wall of the middle shell of the shroud at respective azimuthal positions. These core plate wedge assemblies are wedged into place with preload to maintain the spacing between the core plate and the shroud, thereby maintaining the alignment of the fuel assemblies. The core plate wedge assemblies further serve to transmit loads from the nuclear fuel core to the core shroud at azimuthal locations where lateral shroud restraints outside the shroud contact the reactor pressure vessel.