This section is from the book "Cyclopedia Of Architecture, Carpentry, And Building", by James C. et al. Also available from Amazon: Cyclopedia Of Architecture, Carpentry And Building.
According to one of the fundamental laws of mechanics, the sum of the horizontal tensile forces must be equal and opposite to the sum of the compressive forces. Ignoring the very small amount of tension furnished by the concrete below the neutral axis, the tension in the steel =As = pbds = the total compression in the concrete. Therefore, applying Equation 11, pb d s = 1/2 (1 -1/3q) Ec εc k b d But s = Es εs; therefore, p Es εs = 1/2 (1 - 1/3 q) Ec εck
But Es/Ec = r; and by proportional triangles, as shown in Fig. 96,
εc /kd = εs /d- kd; or εs k / 1-k
Making these substitutions, we have: pr=1/2 (1-1/3q)k2/1-k ..(15)
Solving this quadratic for k, we have:
Equation 16 is a perfectly general equation which depends for its accuracy only on the assumption that the law of compressive stress to compressive strain is represented by a parabola. The equation shows that k, the ratio determining the position of the neutral axis, depends on three variables - namely, the percentage of the steel (p), the ratio of the moduli of elasticities (r), and the ratio of the deformations in the concrete (q). These must all be determined more or less accurately before we can know the position of the neutral axis.
On the other hand, if it were necessary to work out Equation 16, as well as many others, for every computation in reinforced concrete, the calculations would be impracticably tedious. Fortunately the extreme range in k for any one ratio of moduli of elasticities, is only a few per cent, even when q varies from 0 to 1. We shall therefore simplify the calculations by using the constant value q = 2/3, as explained above.
Substituting q = 2/3 in Equation 16, we have:
The various values tor the ratio of the moduli of elasticity (r) are discussed in the succeeding section. The values of k for various values of r and p, and for the uniform value of q = 2/3, have been computed in the following tabular form. Five values have been chosen for r, in conjunction with nine values of p, varying by 0.2 per cent and covering the entire practicable range of p, on the basis of which values k has been worked out in the tabular form. Usually the value of k can be determined directly from the table. By interpolating between two values in the table, any required value within the limits of ordinary practice can be determined with all necessary accuracy.