The majority of studies of the heat transfer inside the hot box treats the heat transfer as a steady-state process. This paper demonstrates that this approach cannot be applied to the most dangerous cases of the cyclic thermal stress. The significant thermal gradients may occur in the skirt to shell junction of a high-temperature vessel and set up critical thermal stresses. It is a common practice to use a hot box to equalize temperatures of a shell and a skirt support. Reduction of thermal gradient results from a radiative heat transfer inside this hot box. Where a heating/cooling rate is high enough, as in coke drums, for example, the accounting of transient alters radically the distribution of a thermal stress state, and allows us to reconsider the mechanics of the fracture growth in the skirt to shell weld. This paper proves that during the cooling of coke drums some parts of the skirt support have higher temperatures than the shell, which causes tensile circumferential stresses in the weld. The intensity of the radiative heat transfer falls rapidly, when cooling a shell down to 247 °C, which leads to the increase of thermal gradients in the weld zone. This paper proposes a solution to the thermal problem in 2D, and strain-state analysis — in 3D, due to the presence of skirt slots equally spaced around skirt circumference, which increases the circumferential flexibility. The two-dimensional thermal field has been interpolated to a three-dimensional hexagonal grid for solving the thermo-strength transient problem.