I have a pipe holding liquid at 80°C. The outside atmosphere is -2°C. The pipe gets a temperature gradient over the wall thickness. The outside fibers will thus restrain the inner fibers from expanding. I would like to know the increase in the inner dia of the pipe. I have found temperature gradient dependent formulas in Roark’s formulas for the stress in the casing wall but I can’t find any analytical formulas for the diametrical change on the inside (or outside) diameter.