Determination of the Influence of Internal Stresses on the Strength of Materials.—­We call internal stresses those which exist within the mass of any hollow cylinder or other body, when it appears to be in a state of repose, or not under the influence of external forces.  When pressure is applied to a hollow cylinder, either externally or internally, the interior layers into which its walls may be conceived to be divided are subjected to a new series of stresses, the magnitude of which is independent of those already existing.  These additional stresses combine with the former in such a manner that at every point of the thickness of the cylinder they have common resultants acting in various directions.  Thus, if we call t the internal stress existing at a distance r_x from the axis of the cylinder, and in a direction tangential to its cross section, and T the additional stress due to pressure inside the cylinder acting at the same point and in the same direction, then the newly developed stress will be t + T.

If R and r0 be the external and internal radii of the cylinder, and if we suppose the external pressure nil, then, if the pressure inside the bore be P0, the stress on the radius r_x is determined by the following expression deduced from the well-known fundamental formulae of Lame:[1]

r0 squared      R squared + (r_x) squared
T = P0 ------- . -------------
R squared-r0 squared      (r_x) squared

From which we see that T is a maximum when r_x = r0, i.e., for the layer immediately next to the bore of the cylinder.  Calling t0 the internal stress in this layer, and T0 the stress resulting from the action inside the bore of the pressure P0, and allowing that the sum of both these quantities must not exceed the elastic limit U of the material, we have—­T0 = U — t0.  And for this value of T0, the corresponding pressure inside the bore will be

R squared — r0 squared
P = (U - t0) ----------. 
R squared + r0 squared

This pressure increases with the term (U — t0).  With t0 positive, i.e., when the internal stresses in the thickness of the hollow cylinder are such that the metal of the layers nearest to the bore is in a state of tension and that of the outer layers in a state of compression, then the cylinder will have the least strength when t0 has the greatest numerical value.  Such stresses are termed injurious or detrimental stresses.  With t0 negative, the strength of the cylinder increases with the numerical value of t0, and those stresses which cause compression in the layers nearest to the bore of the cylinder and tension in the outer layers are termed beneficial or useful stresses.