The thermal behavior of a horizontal steel structure loaded with an overlying layer of corium is investigated numerically. The objective is to determine the temperature transients of the corium/steel system and melt-front propagation (if any) in the steel structure subjected to different accident conditions. Water is assumed to be present below the steel structure so that cooling is available on the bottom side of the steel structure by downfacing boiling of water. A numerical model is developed which accounts for transient heating of the corium and transient conduction in the steel structure. Finite difference equations are derived by discretizing the governing equations using an implicit scheme. To facilitate the numerical solution procedure, the Landau Transformation technique is employed to immobilize the solid-liquid melting interface for the case with steel melting. The governing finite-difference equations are cost into a tridiagonal matrix form and then solved by a general Thomas algorithm along with an iterative updating procedure. An integral technique is employed to calculate the bulk temperature of the corium and the melt-front location is determined by integrating the interfacial energy balance equation using the fourth order Runge Kutta method and the Secant shooting method. Numerical results are obtained showing the conditions for incipient melting and the transient thermal responses of the corium/steel system as functions of the steel structure thickness, the inventory of the corium, the decay heating level, and the thermal boundary conditions.