1:4 THEORY OF FLEXURE. Art, 68. 



The assumptions in tJic theory of flexure are: 



1. Originally plane cross sections will be planes after bend- 

 ing takes place. 



2. The modulus of elasticity is constant throughout the 

 beam. It follows that it is the same for tension and compression 

 (for there will evidently be both kinds of stresses), and the 

 stress must nowhere exceed the elastic limit (11). 



3. The fibers do not act upon each other but independently. 

 When bending- takes place, the fibers between any two cross 



sections are compressed above the neutral surface (Fig. 07) and 

 elongated below it. According to the first assumption the change 

 of length of the fibers is directly proportional to their distances 

 from the neutral axis, as shown in Fig. 68, DD being originally 

 parallel to BB. 



Under the second assumption, Hooke's law (8) must hold 

 for all parts; it therefore follows that the unit stresses vary 

 directly as the deformations, that is, the intensity of the bending 

 stress increases directly as the distance from the neutral axis as 

 shown in Fig. 67. This simply means that if the stress per 

 square inch is 150 Ibs. at a distance of one inch from the neutral 

 axis, at a distance of two inches it will be 300 Ibs., at a distance 

 of 10 inches, 1500 Ibs., etc. The total stress at a certain distance 

 from the neutral axis will, of course, depend upon the width ot 

 the crovss section. 



Since the moment of resistance must be equivalent to the 

 moment of a couple, the resultant of the tensile stresses must 

 equal the resultant of the compressive stresses, which makes 

 "the sum of the horizontal components" zero because all other 

 forces are vertical. 



It is evident that the maximum intensity of stress will occur 

 in the fiber which is most remote from the neutral axis. The 

 stress per square inch upon this fiber must not exceed the work- 

 ing stress, and if the cross section is made of such dimensions and 

 such shape that the unit stress in the extreme fiber is equal to th-? 

 working stress, the moment of resistance will equal the bending 

 moment, and the problem is solved. 



The first assumption upon which the theory of flexure is 

 based has the merit of being the simplest which can be made 

 under the given circumstances. The second assumption limits its 

 application to certain materials, and to stresses within the elastic 



