In the operation of rotary kilns the losses of heat to the atmosphere through the shell are up to 20% of the total consumption. A reduction in these heat losses is possible by adding additional thermal insulation to the lining as the result of changing the form of the refractory with the formation on it of cells in which is laid a material with good thermal insulation properties. However, the change in configuration must be coordinated with the amount of thermomechanical stresses existing in the refractory material.In solving the problem of determination of the temperature stresses in the refractory of a rotary kiln let us consider the lining as a series of isotropic rings with a thickness equal to the height of the refractory. The action of these rings on one another is small because of the small slope of the furnace and it may be neglected.In considering the stressed state of a refractory part located in such a ring let us assume that the circumferential stresses caused by thermal expansion are insignificant as the result of the presence of thermal expansion joints in the lining. Taking into consideration that the ring cross section in the radial direction and the conditions of fastening of the refractory are the same for all portions of the refractory and the distribution of thermal loads does not change around the ring, let us consider the refractory as a plate of a single thickness in the plane strained condition. Since during operation the refractory is in the elastic state [I] and the temperature loads may be considered separately from the mechanical [2], the thermoelastic stresses in the refractory are described by the equation [3] with the boundary conditions [TI (x, y) ] ----0 AxyAxy [@ (x, y) ] +aEAxv