86 BROOMALL : 



horizontal shear to the neutral axis. Along with this longi- 

 tudinal shear in the flange there must also exist its right- 

 angled companion shear likewise with vertical shearing planes. 



As regards the variation in value of the stresses in a given 

 cross section of the Channel, it is evident that iu the web the 

 direct stress varies as the distance from neutral surface and 

 the horizontal and vertical shear vary inversely as the square 

 of the distance from the neutral surface. It is also evident 

 that in the flanges the direct stresses are independent of the 

 distance from web and are constant, while the longitudinal 

 and cross shears vary inversely as the distance from the web. 



Knowing the resultant forces acting upon the elements of 

 the body, it is easy to predict the characteristics of the 

 actual lines of maximum internal stress. In Figure 2 are 

 indicated the forces above enumerated which the elements 

 must resist, and the nature of the resulting lines of maximum 

 stress. The formulas for tracing these lines may be found in 

 any advanced work on the mechanics of materials, and are 



s 



Direct Stress : cot 26=^ (i) 



2 V 



s . ^ 



Shear : tan 2<^= — (2) 



2 V 



Where 6 = angle of direct stress with longitudinal direction 



cf> = angle of shearing plane with longitudinal direction 



s = direct unit stress, tension positive, compression 



negative 

 V ^ unit longitudinal shear 



The figure shows respectively top view, elevation and 

 bottom view of the beam. The various forces are represented 

 by arrows or by the letters T for tension, C for compression 

 and S for shear. 



So far the stress lines as determined are based upon stat- 

 ical principles. When the Channel deflects under its loads 

 these lines of stress will vary to a small extent. Under 



