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Under
this condition, the average pressure on the concrete of the concrete slab i8
always greater than c, or at least it is never less than e. As previously
explained, the average pressure just equals 1c when the neutral axis is at the
bottom of the concrete slab. We may therefore say that the total pressure on
the concrete slab is always greater than I- c b t. We therefore write the
approximate equation: M=cbt > (dt).
As before, the values obtained from this equation are safe, but are
unnecessarily so. Applying them to Example 2, Article 291, by substituting M =
1,350,000, b' = 60, t = 4, and (d - t) = 24.5, we compute c = 459. But we know
that this approximate value of c is greater than the true value; and if this
value is safe, then the true value is certainly safe. The more accurate value
of c, computed in Article 291, is 352. If the value of c in Equation 38 is
assumed, and the value of d is computed, the result is a depth of concrete beam
unnecessarily great.
If the concrete beam is so shallow that we may know, even
without the test of Equation 36, that the neutral axis is certainly within the concrete
slab, then we may know that the center of pressure is certainly less than - I
from the top of the concrete slab, and that the lever-arm is certainly less
than (d - I); and we may therefore modify Equation 37 to read: A (d—t).
Applying this to Example 1 of Article 291, and substituting = 900,000, s =
16,000, (d - - I) = (13.75 - 1.67) = 12.08, we find that A = 4.65, instead of
the 4.59 previously computed. This again illustrates that the formula gives an
excessively safe value, although in this case the difference is small. Equations
37 and 38 should be considered as a pair which is applied according as the
steel or the concrete is the determining feature. When the percentage of steel
is assumed (as is usual), both equations should be used to test whether the
unit-stresses in both the steel and the concrete are safe. It is impracticable
to form a simple approximate equation corresponding to Equation 39, which
will-express the moment as a function of the compression in the concrete.
Fortunately it is unnecessary, since, when the neutral axis is within the concrete
slab, there is always an abundance of compressive strength.
Every solution for concrete
beam construction should be tested at least to the extent of knowing that there
is no danger of failure on account of the shear between the concrete beam and
the concrete slab, either on the horizontal plane at the lower edge of the concrete
slab, or in the two vertical planes along the two sides of the concrete beam.
Let us consider a T-concrete beam such as is illustrated in Fig. 106. In the
lower part of the figure is represented one-half of the length of the flange,
which is considered to have been separated from the rib. Following the usual method
of considering this as a free body in space, acted on by external forces and by
such internal forces as are necessary to produce equilibrium, we find that it
is acted on at the left end by the abutment reaction, which is a vertical
force, and also by a vertical load on top. We may consider F' to represent the
summation of all compressive forces acting on the flanges at the center of the concrete
beam. In order to produce equilibrium, there must be shearing force acting on
the underside of the flange. We represent this force by Sli.
Since these two forces are the only horizontal forces, or forces with
horizontal components, which are acting on this free body in space, F' must
equal S. Let us consider z to represent the shearing force per unit of area. We
know from the laws of mechanics that, with a uniformly distributed load on the concrete
beam, the shearing force is at the ends of the concrete beam, and diminishes
uniformly towards the center, where it is zero. Therefore the average value of
the unit-shear for the half-length of the concrete beam must equal z. As
before, we represent the width of the rib by b.
Are You in Seabrook New Hampshire? Do You
Need Concrete Cutting?
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Call 603-622-4441
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