60 



STRENGTH OF MATERIALS 



or, since - + h 2 = 



In this equation q is proportional to 6, and hence the maximum 

 value of q_ is at the center where b = 2 r. Hence 



The maximum unit shear on a circular cross section is therefore 



equal to f of its average value. 



57. Cases in which shear is of especial importance. In Article 53 



it was shown that at points where the normal bending stress is a 



maximum the shear is zero. 

 I MI this reason it is usu- 

 ally Mltlicient to dime! 



a beam so as to carry the 



maximum landing Bl 

 sal eh \vitln.ut regard to 



the >hear. 1 1. \vever. ill 



certain cases. ..f which the 



X 



FIG. 45 



arc examples, it 

 is necessary to calculate 

 the shear also, and combine it with the bending stress. 



For an I-beam the static moment J //'//' is nearly as great dire. -th- 

 under the flange as for a section through the neutral axis ; ami t 

 fore, by formula (28), the shear is very large at this point, as shown 

 on the shear diagram in Fig. 45. Hence the shear and landing 

 stress are both large under the flange, and the resultant stress at 

 this point may, in some cases, exceed that at the outer 1 



Again, if a beam is very short in comparison with its depth. r if 

 the material of which it is made offers small resistance to shear in 

 certain directions, as in the case of a wooden beam parallel to the 

 grain, a special investigation of the shear must be made. F r instance, 

 consider a rectangular wooden beam of length /, breadth 6, and depth //, 

 bearing a single concentrated load P at its center. Then the total 



