120 STRENGTH OF MATERIALS 



two forces P acting in the direction of the axis of the cylinder (Fi^ 

 Then the bending moment at any point of the spring is M = 7V. It 

 the radius r of the coil is large in comparison with the diameter of 

 the wire, and if the spring is closely wound, the plane of the external 

 moment M is very nearly perpendicular to the axis of the helix, and 

 consequently the bending strain can be assumed to be zero in com- 

 parison with f he torsional strain. Under this assumption the maxi- 

 mum stress is found, from equation (55), to be 



Similarly, the maximum stress in a spring of square or rectangular 

 cross section can be found by substituting M = 7V in equations (61) 



and (62). 



To find the amount by which the spring 

 is extended or compressed, let dO be the 

 angle of twist for an element of the helix 

 of length dl. Then (Fig. 88). tl,,. 



axis of the spring, a point M in this 

 FIG. 88 " l tne same horizontal plane with the ele- 



ment dl is displaced vertically an amount 



MN=rdd in the direction of the axis. Therefore the total axial 

 compression or extension D of the spring is the sum of all the infini- 

 tesimal displacements rdd for every element dl ; whence 



V = CrdO. 



From equation (57), 6 = 2 Ml = 



Tra*G 7ra*G 

 f) -p 



Therefore dd = dl, and consequently 



7ra 4 G 



in which I is the length of the helix. 



If n denotes the number of turns of the helix, then, under the 

 above assumption that the slope of the helix is small, l = 

 approximately, and hence 



