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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.

**Are You in New Boston ****New Hampshire****? Do You
Need Concrete Cutting?**

**We Are Your Local
Concrete Cutter**

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**We Service New Boston
NH and all surrounding Cities & Towns**