1. A technique was developed for calculating temperature fields and thermal stresses by the method of finite elements. This technique was approved for calculating the thermal stress state of cylindrical specimens during heating by high-frequency current. The calculations of temperature fields were verified experimentally. 2. A comparison of results of calculations of thermal stresses by the MKE and by formulas for the plane axisymmetric problem of thermoelasticity showed that, with a sufficiently large temperature range constant along the specimen axis for a distance of not less than 1R from the central section, there is satisfactory agreement between the results obtained by both methods. With a decrease in the section of constant temperature along the oz axis, the difference between stresses calculated by methods I and II increases, reaching 80%, if the stresses calculated according to the MKE are taken for 100%. 3. In evaluating the limiting stress state of cylindrical specimens, it is necessary to consider the set of all principal stresses, since the equivalent stresses computed according to different theories of strength — including theories 3 and 4, usually used in calculating the strength of structural members made of similar materials — may substantially exceed the value of stress σz, used in a number of situations for evaluating the stress state of structural members under conditions of abrupt heat transfers.