8 MODULUS OF ELASTICITY. Art. 11, 



In members composed of two different materials, if the 

 deformations are equal, from Hooke's law the unit stresses in 

 the materials will be directly proportional to their moduli of 

 -elasticity. This gives us a means of estimating the stresses in 

 the two materials of a reinforced concrete member due to shrinkage 

 in setting or to temperature changes. 



Suppose a bar of steel one inch square were embedded in 

 the center of a concrete member 6 in. square, and the whole 

 subjected to a fall in temperature of 40 Fahr. If the coefficient 

 of expansion of the steel is .0000065, and of the concrete 

 .0000055, the drop in temperature will cause tension in the steel 

 and compression in the concrete, the bond between the two being 

 assumed to be perfect. 



If the two materials could contract independently, the 

 steel would contract 40 X .0000065 = .00026 in. per inch of 

 length, and the concrete would contract 40 X .0000055 = .00022 

 in. per inch; but the combination being firmly bonded, will 

 contract some amount between these two values, causing ten- 

 sion in the steel and compression in the concrete. As equilib- 

 rium exists in the member the total tension in the steel must 

 equal the total compression in the concrete, or 



s s A s =s c A c , from which s s =35s c . ... (a) 



The total deformation in each material will be the algebraic 

 sum of the deformations due to temperature and to stress, and 

 these total deformations must be equal, as the bond is not broken. 



Let d st = deformation of steel due to temperature ; 



d ss = deformation of steel due to stress; 



d a = deformation of concrete due to temperature; 



d c ,= deformation of concrete due to stress; 



#,=30,000,000; 



E c = 2,000,000. 

 ; Then d n "d u ^d et +d etj or 



.00026 = .00022 + 8c .. . . (&) 



30000000 2000000 



