STRENGTH OF MATERIALS 289 



single load concentrated at the center is equal to one-half 

 the load, and is uniform throughout the beam. Where a 

 beam supports several concentrated loads, changes in the 

 amount of shear occur only at the points where the loads are 

 applied. The external shear is resisted by the internal shear, 

 or shearing stress, of the beam, which is numerically equal to the 

 external shear. If the external shear is denoted by V, and the 

 area of the cross-section by A , the average intensity of shearing 



V 

 stress in the section is . This shearing stress is not uniformly 



distributed, and in beams of rectangular cross-section, the 

 maximum intensity of shearing stress is . Hence, a rect- 

 angular beam must be so designed that this value will not 

 exceed the working shearing strength of the material. In 

 metallic beams with thin webs (plate girders), the shearing 

 stress may be considered as uniformly distributed over the 

 cross-section of the web. There is, also, at every horizontal 

 or longitudinal section of the beam, a horizontal shearing 

 stress the intensity of which at any point is equal to the inten- 

 sity of the vertical shearing stress at that point. 



Although the maximum intensity of shearing stress, both 

 horizontal and vertical, in wooden beams is usually small, 

 the shearing strength of wood along the grain is also small. 

 As the horizontal external shear usually acts along the grain, 

 the safe load for a wooden beam may depend on its shearing 

 strength and not on its bending strength. For instance, the 

 safe load for a beam 4 in.X12 in. and 4 ft. long is 16,000 lb., 

 uniformly distributed, when based on a fiber strength of 1,000 lb. 

 per sq. in. Such a load will produce a shearing stress per 



3X8000 



unit of area equal to = 250 lb. per sq. in., which exceeds 



2 X48 



the working shearing stress for the wood along the grain by 

 about 100 lb. per sq. in. 



The bending moment at any section of a loaded beam is equal 

 to the algebraic sum of the moments of all the external forces 

 Goads and reactions) to the right or left of the section about 

 that section. For example, the bending moments at several 



