In designing some types of fuel elements, it is necessary to find the thermal stresses in a continuous medium performated by cylindrical channels of circular cross-section. In order to set up the problem, assume that the strength and heat transfer calculation are based on the phenomena that occur in a typical unit lattice cell, and that the heat generation is uniform over the whole section: Therefore, the problems reduce to solve the temperature and thermal stresees in a polygonal prism with a central circular hole under uniform heat generation and for insulated outer boundary. In out previous papers we obtained the steady thermal stress distribution in such formed regions by use of the five elementary functions method.In starting up or shutting down of the reactor, however, the problem becomes one of the unsteady state. It is felt that no analysis has been given to the transient thermoelastic problem of the multiply-connected region. The problems considered here are concerned with unsteady thermal stresses in a polygonal prism with a circular hole under a uniform heat generation. The solving process is divided into two basic steps. First, the heat conduction equation is solved by the technique of Laplace transform. Secondly, the plane thermoelastic problem is solved by use of the five elementary functions method. Throughout the treatment of both problems, in order to satisfy the boundary conditions, we use the point-matching technique in conjunction with the least square method.