INTERNAL FORCES 71 



found to be 0.000534 the length. What was the modulus of 



elasticity? 



i fi onn 



= 30,000,000 Ibs. (in round numbers). 



In the example the bar was one inch square. It may make it 

 more clear if the modulus of elasticity is defined as the ratio found 

 by dividing the unit stress by the unit deformation, or unit strain. 

 Stress is a force and strain is the deformation produced by a force. 



The modulus of elasticity is used to compare the relative 

 deformation of materials which must act together. Steel may be 

 said to have a modulus of elasticity of 30,000,000. Concrete is 

 made of so many different mixtures and the workmanship varies 

 so greatly between specimens that an average value of the modulus 

 of elasticity must be taken for each mixture. The average value 

 of 1:2:4 concrete is 2,000,000 and the ratio between the moduli 

 of elasticity of structural grade steel and 1:2:4 concrete is taken 



to be - r - 15. The writer a few years ago called this 



4,000,000 



the " ratio of deformation " instead of the " ratio between moduli 

 of elasticity," and his term is very commonly used now. To give 

 a clear explanation of the matter assume a piece of concrete with 

 a unit cross-sectional area and beside it a piece of steel with the 

 same area. An equal load is placed on each piece and the short- 

 ening measured. It will be discovered that the steel shortened 

 one-fifteenth as much as the concrete, therefore the ratio of defor- 

 mation = 15. Concrete can really have no modulus of elasticity, for 

 it is a brittle material with an elastic limit very difficult to measure. 

 When, however, concrete is tested it shows enough consistency 

 in deformation to warrant the adoption of a value for the modulus 

 of elasticity by means of which a workable ratio of deformation 

 may be obtained. 



Reinforced Concrete Beams 



Whereas in beams of a uniform material there is a gradual 

 and uniform increase in stress from the neutral axis to the top 

 and bottom, in beams of reinforced concrete this is true only of 

 the upper portion of the beam. Roughly, the tensile strength of 

 concrete is about one-tenth the compressive strength. Since in a 

 beam the tensile and compressive forces must be equal, the neutral 

 axis in a beam of plain concrete under load will be very high. 



