52 BROOMALL : 



material. It will be seen that the effect is to produce counter- 

 clockwise shear Cfor sections on the left of the middle), and 

 that a rotation of the element would result were it not for the 

 vertical shear distributed over the vertical cross section of the 

 element. The total vertical shear furnishes the necessary- 

 vertical unit shear at each point to equilibrate the horizontal 

 unit shear and prevent rotation of the various elements. The 

 value of the horizontal unit shear, in a rectangular cross 

 section beam, varies according to the parabolic law, as sug- 

 gested by the dotted lines. Its value can be calculated for 

 any point according to the well known formula : 



V ■sr^ '^ 



v = — > da . z (3) 



I b -^^ z 



For rectangular beams there is little doubt the result so 

 obtained is quite close to the true value. In beams of other 

 cross section it is questionable whether the formula is reliable, 

 and as regards the I Beam it will be evident from subsequent 

 considerations that the formula does not represent at all the 

 true relationship between the forces, even though the results 

 may be approximately correct. 



Up to this point we have been confining ourselves mainly 

 to the relationship between the direct stresses and shears, the 

 resultants of the external loads, which act upon the elements 

 in a beam of rectangular cross section. By resolving these 

 forces for direct stress and shear, the direction and amount of 

 the maximum stresses may be found, as already mentioned. 

 So much for the stresses in the beam of rectangular section. 



Taking up the I Beam, it is evident that the first step in 

 the investigation is, as before, to ascertain the relation of 

 direct stress and shear in our second group of forces. As 

 stated, these forces are the resultants of the external loads 

 acting upon the elements of the body, and from them the 

 maximum stresses in these elements are found by resolution. 



In what follows we will consider only the solid, one-piece 



