The maximum resistance of a tube or hollow cylinder to external stresses will be attained when all the layers are expanded simultaneously to the elastic limit of the material employed. In that case, observing the same notation as that already adopted, we have—
| P0 = T | R - r0 ——— r0 | (1) |
But since the initial internal stresses before firing, that is previous to the action of the pressure inside the bore, should not exceed the elastic limit,[2] the value of R will depend upon this condition.
In a hollow cylinder which in a state of rest is free from initial stresses, the fiber of which, under fire, will undergo the maximum extension, will be that nearest to the internal surface, and the amount of extension of all the remaining layers will decrease with the increase of the radius. This extension is thus represented—
| tx1 = P0 | r02 ———— R2 - r02 | · | rx2 + R2 ———— rx2 |
Therefore, to obtain the maximum resistance in the cylinder, the value tx of the initial stress will be determined by the difference T - t'x, and since P0 is given by Equation (1), then
| tx = T | ( | 1 - | r0 ——— R0 + r0 | · | rx2 + R2 ———— rx2 | ) | (2) |
The greatest value tx = t0 corresponds to the surface of the bore and must be t0 =-T, therefore
| r02 + R2 ———— r0 (R + r0) | = 2 |
whence P0 = T √2 = 1.41 T.