The determination of the maximum pressure that a pressure vessel can withstand is of considerable importance in design. This paper represents a third of a series dealing with the prediction of the maximum internal pressure which can be applied to thick-walled cylindrical vessels.The report gives results for three pressure vessel steels designated as; A212 Grade B Firebox Quality, A285 Grade C Firebox Quality and T304 Stainless.Thirty-nine specimens of these steels, with rati's of outside to inside diameter from 1.2 to 4.0, were tested to fracture.From these tests, the pressure-expansion relations were determined for these specimens and the maximum pressures applied were found. New theories were developed for predicting the pressure-expansion relations and maximum pressure for these vessels based upon the simple tension true stress-strain relations of the material.A comparison of these theoretical predictions with the experimental results shows good agreement for the A285 Grade C Firebox Steel and the T304 Stainless Steel.For the A212 Grade B Firebox Steel the agreement was fairly good.A simpler approximate theory based upon the exact theory was developed.This approximate theory was found to be satisfactory for the A285 Grade C Firebox Steel but not satisfactory for the A212 Grade B Firebox Steel.