HOOKS, LINKS, AND SPRINGS 



157 



The maximum moment, therefore, occurs at the points of application 

 of the forces. From formula (102), the direct stress at these points 

 is zero. 



Having determined the position and amount of the maximum 

 bending moment, the maximum bending stress can be calculated by 

 the graphical method explained in Article 125, or, if the piece is rec- 

 tangular in section, by formulas (99) and (100) or (101) in Article 126. 



Problem 152. A wrought-iron anchor ring is 6 in. in inside diameter and 2 in. in 

 sectional diameter. With a factor of safety of 4, find by the graphical method of 

 Article 125 the maximum pull which the ring can 

 withstand. 



Problem 153. A cast-iron pipe 18 in. in in- 

 ternal diameter and 1 in. thick is subjected to 

 a pressure of 150 Ib. /linear foot at the highest 

 point of the pipe. Find the maximum stress in 

 the pipe. 



HINT. Use formula (101), Article 126. 



129. Plane spiral springs. Consider a 

 plane spiral spring, such as the spring of 

 a clock or watch. Let P denote the force 

 tending to wind up the spring, and c the 



perpendicular distance of P from the spindle on which the spring is 

 wound (Fig. 111). Also, let dx denote a small portion of the spring 

 at any point A distant y from P. Then the moment at A is M = Py ; 

 and hence, from Article 6 6, the angular deformation dfi for the portion 

 dx is given by the formula 



, ~ Mdx Pydx 



dp = = - 



FIG. Ill 



El 



El 



Therefore the total angular deformation of the spring is 



Since the average value of y is c, and the integral of dx is the length 

 of the spring I, ~ t 



I ydx = cl, 



and hence 



T> / 





 El 



