ON STRESS DISTRIBUTION IN ENGINEERING MATERIALS. 495 



observation of phenomena that do not admit of exact treatment The 

 process of wire-drawing is a case in point. In drawing, wire is subjected 

 to three dimensional stresses in the dies, viz., axial tension and two-dimen- 

 sional compression — a combination which, in the model shown in fig. 20, is 

 represented by tlie region at the rear of the upper right-hand quadrant, 

 where the (vertical) tension is comparatively low. Approximate calcula- 

 tions, based on the hypothesis of constant limiting strain -energy, indicate 

 that the pull required to give 7h per cent, reduction in dies of ordinary 

 proportions should vary from 30 to 50 per cent, of the tensile strength ; 

 whereas, in practice, it is found that the actual pull required seldom 

 reaches half the tensile strength. Although the influence of friction is 

 too uncertain to admit of accurate calculation, the agreement is, at least, 

 within the limits of uncertainty. 



It may be noted, also, that strain-energy limits are already used in 

 practice, to a limited extent, in connection with calculations relating to 

 springs. The energy that can be stored per unit volume of solid round 

 wire, wound free in a closely-coiled helical spring, with maximum shear- 

 stress gr is W = iifl^), "where C is the modulus of rigidity. As q varies 

 from about 40 to 80,000 lbs. per sq. in. and C is approximately 12,000,000, 

 W runs from 330 to 1,300 inch -lbs. per cubic inch. In a spring required 

 to carry a concentrated load P with an elastic deflection 8, the total 

 stored-energy is (|P 0) ; hence the volume of metal required is simply 

 (ip Q _j- W). The hypothesis that tlie limiting strain-energy per unit 

 volume is the same, in torsion and bending, leads to the conclusion that 

 the helical spring should be capable of storing 50 per cent, more energy 

 than a well-designed coach-spring of equal volume (with constant maximum 

 stress along its many leaves) ; the expressions for the mean strain-energy 

 being respectively (j2*/C) and (^/^/B), which are in the ratio 6:4. It may 

 be observed that this conclusion is roughly endorsed by experience of the 

 endurance of such springs under comparable conditions of service. 



