The paper presents a new stress monitoring method for cylindrical pressure elements of the power unit. It can be used to monitor a power unit's start-up, shutdown, and load change to reduce the unit's connection time to the power grid. The outer surface of the component is thermally insulated. Circumferential thermal stress at the point of its concentration at the hole edge is determined based on the stress determined at a greater distance from the hole. The circumferential thermal and pressure stresses at the hole edge are calculated by multiplying the stress values determined for the non-weakened wall by the respective concentration factors. In addition, the heat transfer coefficient (HTC) at the inner surface of the pressure element is necessary to determine the thermal stress concentration factor. Heat transfer correlations available in the literature are used to calculate HTC. The transient element wall temperature is measured five to 12 mm from the inner surface of the cylindrical component. An analysis of the measurement point locations on the accuracy of temperature and stress determining at the inner component surface was investigated. A practical application of the proposed method using real measurement data is presented.