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The general method of determining this center of gravity requires the use of differential calculus, and is a very long and tedious calculation. But the final result may be reduced to a surprisingly simple form, as expressed in the following equation: x = kd 12-4q - Assuming, as explained above, the value of q = -, this reduces to: x=.357kd - When q equals zero, the value of x equals .333 k-d; and, at the other extreme, when q = 1, x .375 kd. There is, therefore, a very small range of inaccuracy in adopting the value of q = * for all computations. 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. 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 module of elasticity (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 module of elasticity 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 =, as explained above. Substituting q = - in Equation 16, we have: The various values for the ratio of the module 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 =, 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. Theoretically there is an indefinite number of values of r, the ratio of the module of elasticity of the steel and the concrete.. The modulus for steel is fairly constant at about 29,000,000 or 30,000,000. The value of the initial modulus for concrete varies according to the quality of the concrete, from 1,500,000 to 3,000,000, for stone concrete. An average value for cinder concrete is about 750,000. Some experimental values for stone concrete have fallen somewhat lower than 1,500,000, while others have reached 4,000,000 and even more. We may probably- use the following values with the constant value of 29,000,000 for the steel.            The value given above for 1:6:12 concrete is mentioned only because the value r 20 is sometimes used with the weaker grades of concrete, and the value of approximately 1,450,000 for the elasticity of such concrete has been found by experimenters. The use of such a lean' concrete is hardly to be recommended, because of its unreliability. Considering the variability in cinder concrete, the even value of r = 40 is justifiable, rather than the precise value 38.67. The previous calculations have been made as if the percentage of the steel might be varied almost indefinitely. While there is considerable freedom of choice, there are limitations beyond which it is useless to pass; and there is always a most economical percentage, depending on the conditions.

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