194 JOHN C. KOCH 



23. If II beam rests upon fixed supports which sustaiit' it and 

 its loads, the forces acting through the supports and rteisting 

 these weights are called reactions. The reactions upwari bal- 

 ance the loads downward. If the position and amount o'^- the 

 loads carried by a beam are known it is easy to computt,. tiie 

 amounts of the reactions from the principles of equ'librium 

 already considered. For equilibrium we must have: 



Algebraic sum of all forces =0 

 Algebraic sum of all moments = 



24. Vertical shear. A beam may fail by shearing in a vertical' 

 section as shown in figure 6 a for a cantilever beam, and in 

 figure 6 b for a beam on two supports. This shearing is caused 

 by two equal, parallel forces acting in opposite directions very 

 near the same section. 



25. The vertical shear varies considerably at different sec- 

 tions. It can be readily seen that the greatest tendency to shear 

 is at the supports where the loads are transferred from the beam 

 to the supports. Near the supports the vertical shear is equal 

 to the reactions. By simple additions and the application of 

 the laws of equilibrium the vertical shear at every part of a 

 beam may be readily determined. 



26. Beams. Beams usually fail by cross-breaking or rupture 

 transversely. Throughout Part II for simplicity, the weight of 

 the beam itself will not be considered in the analysis. In figures 

 7 a and 7 b the following notation is used: 



P = any concentrated load. 



m = distance from right support to line of action of P. 



Ri = reaction of cantilever, figure 7 a. 



Ri, Ri = reactions of simple beam, figure 7 b. 



n = distance from left support to line of action of P, in 

 figure 7 b. 

 Let figure 7 a represent a cantilever beam carrying a single con- 

 centrated load P, applied at any point along the beam at the 

 distance m from the support. The tendency of the load P to 

 cause rotation about the point of support of the beam is meas- 

 ured by the bending moment produced at that section. 



