CONCRETE AND STEEL IN COMBINATION 33 



Suppose a load passes frequently over a reinforced concrete 

 structure, and let us consider a certain member in compression. 

 The first application, let us say, causes d deformation with a 

 permanent set d'. Then it should be clear, that the second appli- 

 cation causes an additional deformation of only d-d' , which is 

 the same as saying that both the concrete and the steel did not 

 return to their original position before the second application. 

 There must have been some compression still left in the steel, 

 with an equal amount of tension in the concrete, when the second 

 load was applied; in other words, the total deformation fixes 

 the stress in the steel. Thus, for the determination of the relative 

 stresses in the two materials for a working stress in the concrete, 



the modulus for the concrete should be expressed by ^~B (Fig. 10) 



and not by the ratio ^r=- 

 YD 



Let us take the case of a beam. The compressive stresses in 

 the concrete at any section will vary from zero at the neutral axis 

 to the value A B, for example, at the extreme fiber. At inter- 

 mediate points, the stresses and corresponding deformations follow 

 approximately the law of the curve OA. In this case the slope 

 of the chord OA does not exactly represent the facts, but for 

 working loads the error is small. 



Tests on prisms show that the modulus of elasticity as desig- 

 nated above for ordinary concrete, 30 days old and under work- 

 ing loads, ranges from 2,500,000 to 3,500,000 Ib. per square inch. 

 As a rule, the denser and older the concrete, the higher the 

 modulus. 



Now that the meaning of the term modulus of elasticity has 

 been made clear with reference to both steel and concrete, 

 the reason for the occurrence of a constant ratio of the moduli 

 in specifications and building codes will be discussed. 



Consider the stresses in the steel and concrete of a reinforced 

 concrete column, reinforced with longitudinal rods only. Let 



f a =unit stress in steel. 



f c =unit stress in concrete. 



Es = modulus of elasticity of steel. 



E c = modulus of elasticity of concrete. 



Since the modulus of elasticity of a material is the ratio of 

 stress to deformation, it follows that, for equal deformations, the 



