144 



HOMOGENEOUS BEAMS 



measure of this tendency is the product of the weight W 

 multiplied by its distance from cd, and, since this tendency 

 is the moment of a force that tends to bend the beam, it is 

 called the bending moment. 



A further inspection of Fig. 3 will show that through the 

 bending action of the load the upper part of the beam is 

 subjected to a tensile stress, while the lower part is sub- 

 jected to a compressive stress. 



Fig. 3 also shows that the greater the distance of the par- 

 ticles in the assumed section above or below the center #, 

 the greater will be their displacement. In other words 



1 . I 



I I 



i *-r 



A 

 ?-*' 



D' 



FIG. 4 



since the stress in a loaded body is directly proportional to 

 the strain, or relative displacement of the particles, it fol- 

 lows that the stress in a particle of any section is propor- 

 tional to its distance from the center line ab, and that the 

 greatest stress is in the particles composing the upper and 

 lower surfaces of the beam. 



In Fig. 4, let ABCD represent a cantilever. A force F 

 acts on it at its extremity A . This force will tend to bend 

 the beam into something like the shape shown by A'BCD'. 

 It is evident from what has preceded that the effect of the 

 force F in bending the beam is to lengthen the upper fibers 

 and to shorten the lower ones. Hence, the upper part A 'B 

 is now longer than it was before the force was applied ; that 

 is, A'B is longer than AB. It is also evident that D'C is 

 shorter than DC. Further consideration will show that there 

 must be a fiber, as SS", that is neither lengthened nor short- 



