278 Cilley — Fundamental Propositions in the 



layer would be contracted twice as much as the side layers were 

 extended. It follows that the compression in the center layer 

 would be two-thirds its original compression, and the compres- 

 sive force and intensity of compression in it therefore two- 

 thirds that to which it was originally subjected. A little 

 further consideration will show that, except near the ends, the 

 binding material is subject to no appreciable shearing stress. 

 Thus we have created a fairly definite state of primary stress 

 and strain. 



]N T ow this state of primary stress and strain was derived from 

 a certain known state of stress and strain, — that in which the 

 middle layer was under a perfectly definite compressive stress, 

 and the side layers wholly free from stress, — by the removal of 

 the applied forces, — the compressive forces at the ends of the 

 middle thirds of the composite bar. The resulting changes in 

 stresses and strains would be those determined by the ordinary 

 theory of elasticity for this case, regardless of the compound 

 nature of the bar. Now, if the compressive forces had been 

 uniformly distributed over the entire ends of the compound 

 bar, their removal would simply have resulted in a uniform 

 change in stress over all sections of the compound bar, equal 

 in intensity (but opposite in kind) to the total compressive 

 force divided by the total area. When the end pressures are 

 concentrated on limited areas however, their removal does not 

 result in such a uniform change in stress. But, as Saint 

 Yenant long since pointed out, in a body with relatively great 

 dimensions in other directions, as in the present case, the effect 

 of the local distribution of a force is purely local, and is inap- 

 preciable at a distance from it which is relatively considerable. 

 So in this case, except near the ends, the consequences of the 

 removal of the end pressures will be the same as though they 

 had been uniformly distributed over the ends. This enables 

 us to perceive very clearly what the primary stresses and 

 strains must be and even to follow them to some extent where 

 they lose their uniformity near the ends, and the binding 

 material becomes subject to considerable shearing stress. 



We might easily conceive much more complex but definitely 

 strained built-up bodies, formed of numerous rods or of not 

 quite flat discs or of rings. The latter construction is fre- 

 quently employed in gun construction and its theory has 

 received considerable attention. Let us examine an allied 

 problem, a cylinder of uniform thickness, unlimited length and 

 subject to a given internal pressure, but no (sensible) external 

 pressure. We will suppose that the circular tension in the 

 cylinder is uniform from inside to outside, that the axial stress 

 is the same as if the tube were without stress when under no 

 pressure, and that there are no shearing stresses on the princi- 



