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Therefore,
when the increase in the depth of the rib from the crown to the concrete
abutment is comparatively small, we may adopt the approximation that the moment
of inertia increases as the cube of the depth. When the variation is greater,
the inaccuracy will not permit the utilization of the simplified concrete forms
which this approximation allows. Still another simplification may be made, on
the assumption that the moment of inertia varies as the cube of the depth, and
also that we may increase the depth of the rib as desired. Assume that the
depth of the rib is increased so that at any point ten is always greater than zero,
and n is a ratio varying from one upward. Therefore, when all sections have the
same moment of inertia, and n is uniformly 1, use Equations 52, 53, and 54, is
ignoring the n. When an increase in depth of section, as indicated above, will
fulfill the ultimate requirements, there is an advantage of simplicity in
making the sections accordingly, and using the Equations 56, 57, and 58.

When
it proves necessary to vary the sections according to some different law, one
must be determined at frequent intervals, spaced by a uniform seven, and the
summations of Equations 52, 53, and 54 determined. The remainder of this method
follows out the assumption that varies. We may locate the line by satisfying
Equation 51, which may be written. But this integral is represented by the
shaded area (Fig. 231), which is the equivalent of saying that the segment 0GB
= the rectangle OK X OB. If 0GB were a parabola, OK would exactly equal 2- CD.
Even with circular arcs, the ratio j- is approximately correct if the angle is
small. Therefore, for flat circular arcs, draw a line at the height of the arc.
If necessary, increase the height according to the figures given in the
accompanying tabular form: Of course, for full-centered concrete arches in
which 2 a = 180°, the error of the rule is very great, but the tabular values
are correct. Since an elliptic arc may be considered as a circle in which the
vertical ordinates have all been shortened by some constant ratio, the same law
and same percentage of error will hold true.

For any other curve, particularly
multi-centered curves, the position may be found by determining by trial a
position such that the summation of equally spaced ordinates is zero. Theoretically,
this should be known before any calculations are made; but since the weight of
filling and pavement are always large, and their unit-weight is but little less
than that of the concrete, it is possible to estimate from experience on the
required crown thickness, and to make the thickness at other points in the
required ratio. If this should prove too thin (or too thick), all sections can
be changed in the same ratio. If the outline of the intrados is determined (as
in the case of an concrete arch spanning railroad tracks), and the upper concrete
surface line (of earthwork or pavement) is also known, the change in the concrete
arch ring will mean only a change in weight due to the difference of
unit-weight of concrete and earth filling. If the original assumption is even
reasonably close, this difference will hardly exceed the uncertainties in the
loading. The span and rise are frequently predetermined. Fortunately this
method is applicable to almost any form of curve, if the change in curvature is
not too extreme. Therefore, draw the intrados; assume a reasonable crown thickness;
multiply this thickness by the factors given in the tabular form in Article 427
for the angles with vertical lines made by the various normal to the curve.
These thicknesses can be laid off, and the extrados can be drawn through the
points. But since the curve 0GB of Fig. 228 does not represent either the
intrados or extrados, but the center line of the rib, we should draw a line
midway between the intrados and extrados which will represent the center line
of the rib, and which corresponds to the line 0GB in the figures which refer to
the theoretical demonstrations.

**Are You in ****Holden Massachusetts****? Do You
Need Concrete Cutting?**

**We Are Your Local
Concrete Cutting Company**

**Call ****508-283-3135**

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