The solution of an axisymmetric problem of thermoelasticity is considered and possibility of rigid and soft destruction when on internal and external surfaces of area there is a heat exchange with environment under Newton’s law is analyzed. Considered tasks have important practical value as allow to estimate thermodurability of many details of the metallurgical equipment working at increased temperatures. For the long hollow cylinder at an axisymmetric temperature field fields of temperatures and tension are defined, the coefficient characterizing, according to Davidenkov-Friedman’s integrated durability and nature of destruction – fragile or viscous destruction is calculated. The system analysis of influence of coefficients of a thermolysis, the geometrical sizes, ways of fixing of end faces of the cylinder at a size of temperature tension and nature of destruction is made.