RECTANGULAR BEAMS 49 



at the point where it cuts this cross-section is proportional directly 

 to the distance of this fiber from the neutral axis of the cross-section. 



Assumption 2. As to the evidence in favor of this law, experi- 

 ment shows that as long as a material such as steel is not strained 

 beyond safe limits, this law holds. However, wrought iron and 

 steel are the only important structural materials which closely 

 follow this law, and they only within their elastic limits. But 

 under working conditions, these materials are not stressed beyond 

 their elastic limit and so the formulas ordinarily hold. Timber, 

 stone, and cast iron can hardly be said to obey Hooke's law, yet 

 for working conditions the common flexure formulas for these 

 materials are roughly correct and they are in general use. 



The important question to be decided is does concrete 

 follow the laws stated above? In regard to the first assumption, 

 it can be said that careful measurements show some deviation 

 from a plane, but in general this assumption seems to be war- 

 ranted by the results of observed deformations under working 

 loads. Concerning the second law, as regards the concrete in a 

 reinforced concrete beam, it should be clear that the assumption 

 will not strictly apply except for low stresses. 



31. Plain Concrete Beams. OS in Fig. 23 is the stress-defor- 

 mation diagram for concrete in compression, with which the 

 student is already familiar. The curve shown here is identical 

 with Fig. 10, the only difference being that the deformations are 

 represented vertically instead of horizontally. This change has 

 been made in order that we may apply the curve directly to the 

 cross-section of a beam. The curve OT is the stress-deformation 

 diagram for concrete in tension and is of use only in the analysis 

 of plain concrete beams. 



We have just seen that Assumption 1 may be applied to beams 

 of concrete. This assumption leads us to the conclusion that 

 deformations of the fibers are proportional to the distances of the 

 fibers from the neutral axis. Figs. 24 and 25 show the curves 

 applied to the vertical cross-section AX of a beam. For example, 

 suppose the deformation at point a, Fig. 25, is the same as rep- 

 resented by Oa in Fig. 23. The corresponding stress given by 

 the compression curve in Fig. 23 is aa' '. Lay off the distance 

 ao! in Fig. 25 to represent that stress. Proceeding similarly for 

 all points and connecting, we have the stress curve A"OX" ', which 

 is nothing more than a portion of the stress-deformation diagrams 

 in Fig. 23 plotted to a different scale. 



