54 REINFORCED CONCRETE CONSTRUCTION 



C'C represent the unit deformation, or deformation per 

 unit length, in the metal which is stressed to the 

 amount f s . 



C = total compression in concrete at a section of the beam. 



T = total tension in steel at a section of the beam. 



EC represent the modulus of elasticity of concrete in 

 compression. 



E 8 represent the modulus of elasticity of steel in tension. 



n = ratio j^- 



&C 



d = distance from compression surface to axis of 



reinforcement. 

 k = proportionate depth of neutral axis from below the 



compression surface. 

 a s = area of cross-section of steel. 

 b = breadth of a rectangular beam. 

 p = "steel ratio "= the ratio of the area of steel to 



f a* 



area of concrete = , -,. 

 oa 



M c = resisting moment as determined by concrete. 



M 8 = resisting moment as determined by steel. 



M = bending moment or resisting moment in general. 



Now the total compressive resistance is equal to the area of the 

 triangular figure AOA" multiplied by 6, the breadth of the beam. 

 But the area AOA" = 1/2 (AA")kd = l/2f c kd. Hence, the total 

 compressive resistance C is equal to 1/2 f c kbd. 



The total tensile resistance T is evidently the cross-sectional 

 area of the metal multiplied by the uniform intensity of stress 

 thereon =a s f s . 



Since the total compressive resistance above the neutral axis 

 must be equal to the total tensile resistance below the same, we 

 have 



l/2f c kbd = a s f s (I) 



From the assumption that deformations vafy as the distances 

 of the fibers from the neutral axis, 



AA' CC' AA' CC' 

 or 



OA OC kd d(l-k) 

 But by definition of the values represented by AA' and CC' , we 

 have 



