Art. 68. 



THEORY OF FLEXURE. 



95 



limits. It has been found, by careful experiments, that original- 

 ly plane cross sections remain plane and perpendicular to the 

 neutral surface, and that the length of the neutral surface does 

 not change ; and this is practically true for stone, which does not 

 follow Hooke's law, even when it is, at the same time, also sub- 

 ject to shearing stresses. Stone, concrete, and cast iron do not 

 meet the requirements of the theory of flexure because they do 

 not follow Hooke's law; the discrepancy is particularly large 

 for cast iron, but the error is on the side of safety. 



The influence of shearing stresses which are usually com- 

 bined with bending stresses is discussed in Art. 87. 



The third assumption can, of course, not be in accord with 

 the facts. Since there is tension on one side of the neutral axis 

 and compression on the other, there will be transverse contrac- 

 tion and expansion (9), and the interaction of the fibers will be 

 different from that in pure tension and compression, because the 

 the stress on the inner fibers is less than on the outer ones. Since 

 the contraction and expansion decrease to zero at the neutral 

 axis, the inner fibers interfere with both the transverse and longi- 

 tudinal deformations of the outer fibers (13). 



The effect of the interactions of the fibers depends upon the 

 form of the cross section, but the assumption is on the side of 

 safety ; the interference with the deformations of the outer fibers 

 makes the maximum stress less than the theory of flexure indi- 

 cates. The more the area of the cross section is concentrated 

 into two narrow strips parallel to the neutral axis, the less the 

 discrepancy. The error is greater with a rectangular than with 

 an I-shaped cross section. With a cross section, unsymmetrical 

 about the neutral axis, this axis would have a sligthly different 

 location from that indicated by the theory of flexure; if this 

 were not so, the "sum of the horizontal components" would not 

 be zero, because there would be a greater resistance on that side 

 of the neutral axis where there is a greater proportion of fibers 

 vertically over each other (Above NN, Fig. 67). 



Under the usual working stresses, the assumptions in the 

 theory of flexure are practically correct for all materials of con- 

 struction and their adoption has been all but universal. It 

 should be remembered, however, that long compression flanges 

 of beams should be stayed against transverse buckling, and that 

 when bending and shearing stresses are combined, the limita- 

 tions pointed out in Arts. 87 and 67 must also be observed. 



