56 



RESISTANCE OF MATERIALS 



105. Derive a formula for the pitch of a cast-iron gear to carry safely a driving 

 force F. 



Solution. Circular pitch is defined as the distance between corresponding points 

 on two successive teeth, measured along the pitch circle. Let P denote the circular 



pitch for the case in question (Fig. 52). 

 Then, if h denotes the depth of the 

 tooth, b its breadth, and t its thickness 

 at the root, the relative proportions 



ordinarily used are 

 h = .7P, = .5P 



= 2Pto3P, 



FIG. 62 



Height above pitch circle (called addendum) = .3 P, 

 Depth within pitch circle = .4 P. 



The driving force F is ordinarily applied tangent to the pitch circle. Assume, how- 

 ever, that by reason of the gear being worn, or from some other cause, it reaches the 

 tip of the tooth, as shown in the figure. Then, considering the tooth as a cantilever 

 beam, the maximum moment is 



M=Fh, 



and its section modulus at the root is 



Z = . 



Therefore, assuming a working stress for cast iron of p = 4500 lb./in. 2 , we have 



4500 = Fh, 

 6 



and, inserting the values 



6 = 2P, =iP, 



this becomes 



h= .7P, 



106. Find the moment of resist- 

 ance for the section given in problem 

 53, assuming the working stress for 

 structural steel as 16,000 lb./in. 2 



107. Find the section modulus 

 and moment of resistance for the 

 section given in problem 55. 



108. Find the section modulus 

 and moment of resistance for the 

 section given in problem 56. 



109. Find the moment of resistance of a circular cast iron beam 6 in. in diameter. 



110. Find the moment of resistance of a 24-in. steel I-beam weighing 801b./ft. 



111. Compare the moments of resistance of a rectangular beam Sin. x 14 in. 

 in cross section, when placed on edge and when placed on its side. 



FIG. 53 



