50 



RESISTANCE OF MATERIALS 



From this illustration it is evident that the resisting stress in a 

 beam required to equilibrate any system of external forces is of 

 two kinds: 



1. A compressive stress on one side, normal (that is, perpen- 

 .dicular) to the plane of the cross section. 



A tensile stress on the opposite side, also normal to the plane of 

 the cross section. 



2. A vertical shearing stress in the plane of the cross section. 

 33. Distribution of stress. The effect of the external bending 



moment on a beam originally straight is to cause its axis to become 

 bent into a curve, called the elastic curve. 

 Considering the beam to be composed of 

 single fibers parallel to its axis, it is found 

 by experiment that when a beam is bent, the 

 fibers on one side are lengthened and those 

 on the other side are shortened. Between 

 these there must evidently be a layer of fibers 

 which are neither lengthened nor shortened, 

 but retain their original length. The line in 

 which this unstrained layer of fibers inter- 

 sects any cross section is called the neutral 

 axis (Fig. 47). 



It is also found by experiment that a cross 

 section of the beam which was plane before flexure (bending) is 

 plane after flexure. This is known as Bernoulli's assumption.* As 

 a consequence of Bernoulli's assumption it is evident from Fig. 47 

 that the lengthening or shortening of any longitudinal fiber is pro- 

 portional to its distance from the neutral axis. But by Hooke's 

 law the stress is proportional to the deformation produced. There- 

 fore the normal stress at any point in the cross section is likewise 

 proportional to the distance of this point from the neutral axis. If, 

 then, the normal stresses are plotted for every point of any vertical 

 strip MN (Fig. 48), their ends will all lie in a straight line. This 

 distribution of stress is therefore called the straight-line law. 



* St. Venant has shown that Bernoulli's assumption is rigorously true only for certain 

 forms of cross section. If the bending is slight, however, as is the case in all structural 

 work, no appreciable error is introduced by assuming it to be true whatever the form 

 of cross section, 



FIG. 47 



