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The
moment of resistance of a concrete beam equals the total tension in the steel,
or the total compression in the concrete (which are equal), times (d - x).
Therefore we have the choice of two values (as before): = cbkd
(d —x). If the economical percentage p has already been determined from Equation
28, then either equation may be used, as most convenient, since they will give
identical results. If the percentage has been arbitrarily chosen, then the
least value must be determined, as-was described in Article 267. For 1:3: 5
concrete, using as before r = 12, and with a working value for c = 500, and s =
16,000, we find from Equation 28 that the economical percentage of steel
equals: 1 500500X12 = 2 16,000 (500) < 12 + 16,000) = .0043 From Table XV we
find by interpolation that, for r = 12, and p = .0043, k = .273. Then X = -Icc1
= .091d; and (d—x) = .909d. Substituting these values in either formula of
Equation 29, we have: M 62 bd2. The percentage of steel computed from Equation
28 has been called the most economical percentage, because it is the percentage
which will develop the maximum allowed stress in the concrete and the steel at
the same time, or by the loading of the concrete beam to some definite maximum
loading. The real meaning of this is best illustrated by a numerical example
using another percentage. Assume that the percentage of steel is exactly
doubled, or that p = 2 >< .0043 = .0086. From Table XV, for r = 12, and p
= .0086, we find k=.362; x =.121d; and (d—x) = .879d. Substituting these values
in both forms of Equation 29, we have: M0 = 80 bd2; and, M= 121 W2. The
interpretation of these two equations, and also of the equation found above (M
= 62 bd2), is as follows: Assume a concrete beam of definite dimensions b and
d, and made of concrete whose modulus of elasticity is -- that of the modulus
of elasticity of the reinforcing steel; assume that it is reinforced with steel
having a cross-sectional area = .0043 bd. Then, when it is loaded with a load
which will develop a moment of 62 bd2, the tension in the steel will equal
16,000 pounds per square inch, and the compression in the concrete will equal
500 pounds per square inch at the outer fiber. Assume that the area of the
steel is exactly doubled. One effect of this is to lower the neutral axis (k is
increased from .273 to .362), and more of the concrete is available for
compression. The load may be increased about 29 per cent, or until the moment
equals 80 bd2, before the compression in the concrete reaches 500 pounds per
square inch. Under these conditions the steel has a tension of about 10,600
pounds per square inch, and its full strength is not utilized. If the load were
increased until the moment was 121 bcl2, then the steel would be stressed to
16,000 pounds per square inch, but the concrete would be compressed to over 750
pounds, which would of course be unsafe with such a grade of concrete. If the
compression in the concrete is to be limited to 500 pounds per square inch,
then the load must be limited to that which will give a moment of 80 bd2. Even
for this the steel is doubled in- order to increase the load 29 per cent.
Whether this is justifiable, depends on several circumstances the relative cost
of steel and concrete, the possible necessity for keeping the dimensions of the
concrete beam within certain limits, etc. Usually a much larger ratio of steel
than 0.43 per cent is used; 1.0 per cent is far more common; but when such is
used, it means that the strength of the steel cannot be fully utilized unless
the concrete can stand high compression. A larger value of r will indicate
higher values of k, which will indicate higher moments; but r cannot be
selected at pleasure. It depends on the character of the concrete used; and,
with E, constant, a large value of .r means a small value for E0, which also
means a small value for c, the permissible compression stress.

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

**We Are Your Local
Concrete Cutter**

**Call 603-622-4441**

**We Service Newington
NH and all surrounding Cities & Towns**