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Mr.
A. L. Johnson, another noted authority, bases his computation of concrete formulae
on the assumption that the ultimate compressive strength of the concrete is
two-thirds of the value which would be required to produce that amount of
compression in case the initial modulus of elasticity were the true value for
all compressions. In other words, looking at Fig. 93, if o c is a line
representing the initial modulus of elasticity, then, if the elasticity were
uniform throughout, it would require a force of about 2,340 pounds (or d J) to
produce a proportionate compression of .00132 of the length (represented by o
d). Actually that compression will be produced when the pressure equals d e,
which is - of d J. It should not be forgotten that the above numerical values
are given merely for illustrative purposes. They would, if true, represent a
rather weak concrete. The following theory is therefore based on the assumption
that the stress-strain curve is represented by the parabolic curve o e (see Fig
93); and that the ultimate stress per square inch in the concrete c' is
represented by d e, which is of the compressive stress that would be required
to produce that proportionate compression if the modulus of elasticity of the
concrete were uniformly maintained at the value it has for very low pressures. The
theory of reinforced-concrete concrete beams is based on the usual assumptions
that:

(a)
The loads are applied at right angles to the axis of the concrete beam. The
usual vertical gravity loads supported by a horizontal concrete beam fulfill
this condition.

(b)
There is no resistance to free horizontal motion. This condition is seldom, if
ever, exactly fulfilled in practice. The more rigidly the bam is held at the
ends, the greater will be its strength above that computed by the simple
theory. Under ordinary conditions the added strength is quite indeterminate;
and is not allowed for, except in the appreciation that it adds indefinitely to
the safety.

(c)
The concrete and steel stretch together without breaking the bond between them.
This is absolutely essential.

(d)
Any section of the concrete beam which is plane before bending is plane after
bending.

In
Fig. 94 is shown, in a much exaggerated form, the essential meaning of
assumption d. The section a b c d in the unstrained condition, is changed to
the plane a' b' c' d' when the load is applied. The compression at the top = a
a' = b b'. The neutral axis is unchanged. The concrete at the bottom is
stretched an amount = c c' = d d', while the stretch in the steel equals -:
c>g g'. The compression in the concrete between the neutral axis and the top
is proportional to the distance from the neutral axis.

In
Fig. 95a is given aside d -view of the concrete beam, with section of the
fibers. Since neutral the fibers between the neutral axis and the compressive
face are compressed proportionally, then, if a' represents the linear
compression of the outer fiber, the shaded lines represent, at the same scale,
the compression of the intermediate fibers.

In
Fig. 95b, m it indicates the stress there would be in the outer fiber if the
initial modulus of elasticity applied to all stresses. But since the force
required to produce the compression a a' is pro proportionately so much less
than that required for the lesser compressions, the actual pressure in pounds
on the outer fiber may be represented by a concrete beam. This is also called
the cancroids of compression. The theoretical determination of this center of
gravity is virtually the same as the determination of the center of gravity of
the shaded area shown in the video above.

**Are You in Nashua ****New Hampshire****? Do You
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Concrete Cutter**

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