------------
R squared - (r0) squared (r_x) squared
Therefore, to obtain the maximum resistance in the cylinder, the value
t_x of the initial stress will be determined by the difference (T - t'_x),[*need to check the prime with library or work out the equations]
and since P0 is given by Equation (1), then
/ r0 (r_x) squared + R squared \
t_x = T ( 1 - ---------- . ------------- ) (2)
\ R0 + r0 (r_x) squared /
The greatest value t_x = t0 corresponds to the surface of the bore
and must be t0 = -T, therefore
r0 squared + R squared
--------------- = 2
r0 (R + r0)
whence P0 = T sqrt(2) = 1.41 T.
From the whole of the preceding, it follows that in a homogeneous
cylinder under fire we can only attain simultaneous expansion of all the
layers when certain relations between the radii obtain, and on the
assumption that the maximum pressure admissible in the bore does not
exceed 1.41 U.
Equation (2) may be written thus--
R r_x - Rr
t_x = T -------- . ---------- (3)
R + r0 (r_x) squared
Substituting successively r_x = r0 and r_x = R, we obtain
expressions for the stresses on the external and internal radii--
R - r0 R R - r0
t_R = T -------- and t_r0 = -T ---- --------
R + r0 r0 R + r0
Therefore, in a homogeneous hollow cylinder, in which the internal
stresses are theoretically most advantageous, the layer situated next to
the bore must be in a state of compression, and the amount of
compression relative to the tension in the external layer is measured by
the inverse ratio of the radii of these layers.
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