In this study, two failure modes, yield buckling of the compression ring section and strength failure in the roof-to-shell of the tank, have been proposed for a vertical vaulted tank. The failure criteria of the two failure modes in the roof-to-shell of vault tanks are established via finite element analysis of three tanks of 640 m3, 3200 m3, and 6800 m3 in volume. The finite element models are built with axisymmetric elements and spatial multi-elements. Based on the strength failure criterion, the failure pressure formula in the vaulted tank roof-to-shell is derived. The maximum relative error between the theoretical calculation and numerical simulation is 9.7%. Finally, we verify the strength failure criterion through a tank failure test; the maximum relative error between the test and theoretical calculation is 9.6%. The failure pressure of both failure modes has been compared and analyzed. The failure pressure of the yield buckling in the compression ring section is about 1.65 times that of the strength failure in the roof-to-shell of the tank.