Art. 76. 



PRINCIPAL STRESSES IN BEAMS. 



113 



been shown that, in general, both occur at any point in a beam. 

 The question arises, why is the resultant stress the principal 

 stress (12) not calculated ? 



According to Art. 75, the shearing stresses increase from 

 the neutral axis outward, and according to Art. 68, the 

 bending stresses increase in an opposite direction. In the outer 

 fibers, the bending stresses are a maximum and the shearing 

 stresses zero; at the neutral axis the reverse is the case. The 

 maximum resultant stress occurs somewhere between the 

 neutral axis and outer fiber, and its intensity might be 



than the maximum intensity of the bending stress, depend in-- 

 upon the form of cross section. The common I section is an un- 

 favorable one in this respect. Since it is unknown how shearing 

 stresses are distributed over an I section, calculations must al- 

 ways be made upon certain assumptions; such calculations in- 

 dicate that, within certain limitations, the bending and shearing 

 stresses may be considered separately, and, in simple beams, th'j 

 shearing stresses need not be considered at all (See example Art. 

 70). 



The resultant stress is evidently less in a beam uniformly 

 loaded (supported both ends) than in one carrying a concen- 

 trated load, because in the one case the maximum shear comes at 

 the supports, and the maximum moment at the center section, 

 while in the other these sections coincide. (See Figs. 101 and 

 96. Also compare shear and moment diagrams for other cases.) 

 Suppose a beam to be designed to resist the maximum moment 

 caused by a single load; if now the span be decreased, the moment 

 becomes smaller, and the shear remains the same. It is evident 

 that in very short beams, the bending stresses are negligible as 

 compared with the shearing stresses, and the usual procedure is 

 not applicable. This is especially true for timber beams. The 

 resistance to shear in sections parallel to the Fibers being the 

 least, short timber beams sometimes fail (at the neutral axis) in 

 horizontal shear. A similar failure might occur in very short 1 

 beams heavily loaded, because the web is thin. 



For example, if a timber beam 12"X12" has a span of 4 ft. 

 and carries a load of 24000 Ibs. at its middle, its extreme fiber 



BtreBB iB = ^ = ^|^=1000 lbB.perBq.in.(Eq. 10). The 



maximum unit shearing stress is -|x ^jj^= 125 lbs - P er S( l- in - 

 (Eq. 23). Now if the working stresses are 1000 lbs. in the ex- 



