RECTANGULAR BEAMS 



47 



30. Assumptions in Common Theory of Beams. In order to 

 derive a formula by whic,h we can design concrete beams rein- 

 forced with steel, it will first be necessary to overhaul the assump- 

 tions on which the common theory of flexure is founded and see 

 if we can derive a formula, or formulas, by which we can deter- 

 mine stresses in both the steel and concrete of such beams. 



The two main assumptions in the common theory of beams 

 may be stated as follows: 



1. If, when a beam is not loaded, a plane cross-section be made, 

 this cross-section will still be a plane after the load is put on and 

 bending takes place. (Navier's hypothesis.) 



2. The stress is proportional to the deformation namely, to 

 the elongation or compression per unit of length. (Hooke's Law.) 



From the first assumption the following principle is obtained : 

 the unit deformations of the fibers at any section of a beam are 



D H 



E 6 



(a) 



FIG. 21. 



proportional to their distances from the neutral axis. From the 

 second assumption: the unit stresses in the fibers at any section 

 of a beam are also proportional to the distances of the fibers from 

 the neutral axis. 



It may be well to explain more at length what is meant by the 

 two assumptions and the results derived from them. 



Assumption 1. Imagine two originally parallel cross-sections, 

 ED and GH, Fig. 21a or Fig. 21b, so near to each other that the 

 curve taken after bending by that part of the neutral plane between 

 these sections may, without appreciable error, be accounted circu- 

 lar. Let ED and GH, Fig. 22a or 22b, be the lines in the loaded 

 beam in which these cross-sections cut the plane of the paper, and 

 let be the point of intersection of the lines ED and GH. Let 

 OF= r, FL = y, FK = l, LM = l + al, in which a is the elongation 

 per unit of length of a fiber at a distance y from the neutral axis, 

 y being variable; then, because FK and LM are concentric arcs 



