734 BELL SYSTEM TECHNICAL JOURNAL 



existed. As will be shown, such stretching will cause the radius to 

 increase at those portions of the spring where the applied stresses are 

 highest, that is, for those turns near the dividing line of the arbor. 



Substituting for y, in equation (10), the distances from the neutral 

 axis to the extreme inner and outer fibers will give the strain in these 

 fibers. Provided Ri is considerably larger than /?/2, half the thickness 

 of the material, the distance to these extreme fibers becomes hjl and 

 the y in the denominator can be neglected in comparison to i?i. The 

 maximum fiber stress due to placing the spring on the arbor is this 

 value of strain multiplied by Young's modulus. Then 



This stress is in the form of compression for the outer fibers and tension 

 for the inner. Since a load on the clutch results in a tension in the 

 spring, the stress given by equation (20) must be added to the load 

 tension stress to get the total stress on the inner fibers of the spring. 

 This initial stress therefore reduces the load-carrying capacity of the 

 clutch. 



Residual Spring Stresses 

 When a straight wire is wound upon an arbor to form a coil spring 

 the strain on the inner and outer fibers must exceed that corresponding 

 to the yield-point and plastic-flow results. To simplify the discussion 

 assume the idealized stress-strain curve shown by the heavy lines of 

 Fig. 7(a). The stress distribution across any section of the wire while 

 wound on the winding arbor will be as shown by the heavy lines of 

 Fig. 7(b) where Syp is the yield-point stress, the maximum stress that 

 the material will sustain. The moment, across the section, required 

 to produce this bending is 



fft/2 

 Shydy (21) 



h/2 



where h is the width of the wire at any point of y distance from the 

 neutral axis and 5 is the stress at the same point. If the spring is 

 released, it expands to a radius R\ in which condition the external and 

 the internal moments are both zero. It is now possible by applying 

 the same moment as was given by equation (21) to reduce the radius 

 of curvature again to Rq but without causing additional plastic flow. 

 The added stress distribution produced by this second bending must 

 therefore follow a straight line as shown by 52-52, which together 

 with the residual stresses in the relaxed condition (radius R^ must 



