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The
area of a curve bounded by a straight line at the bottom, and which has
vertical and equally spaced ordinates, may be computed with very close accuracy
by the adoption of Simpson's rule. If y° represents the ordinate at the
beginning of the curve (and in this case y. = 0), while y1 up to n represent
the lengths of the several ordinates (being the last ordinate, and, in this
case, also equal to 0; and n being equal to the even number of divisions, in
this case 20), then the area may be expressed by the formula: Applying this
rule, we find that the area will be 640.50 square feet. Dividing this by the
span, 61.4, we find that the height above the line OB will be 10.59 feet. The
approximate two-thirds rule would give us 10.17. Making a rough interpolation
in the tabular form of Article 428, we could say that for an angle 2 a equal to
106° 16', the quantity to be added to the result by the two-thirds rule would
be approximately 5 percent. Adding 5 percent to 10.17, we would have 10.68,
which gives a rough check with the far more accurate value just found. We shall
assume that the concrete arch carries a filling of earth or cinders weighing
100 pounds per cubic foot, that the top of this filling is level, and that it
has a thickness of one foot above the crown. Since concrete weighs about 150
pounds per cubic foot, we shall assume this weight of 150 pounds as the unit of
measurement, and therefore reduce the ordinates of earthwork to the load line
for the top of the earth, as shown in Fig. 232.

We shall assume as an
additional dead load a pavement weighing 80 pounds per square foot, and shall
therefore lay off an ordinate of a1 foot above the ordinates for the
earth-filling load. For this particular problem, we shall only investigate a
live load of 200 pounds per square foot, extending over one-half of the spans
from the concrete abutment to the center. From our previous work in concrete
arches, we know that such a loading will test the concrete arch more severely
than a similar unit live load extending over the entire concrete arch; and
therefore, if the concrete arch proves safe for this eccentric load, we may certainly
assume that it will be safe for a full load. These load lines are laid off
similarly to the method elaborated in Article 409. The concrete arch has
already been laid off. in equal horizontal sections, each having a width of
3.07 feet. The two end sections are slightly longer if we consider the entire
load which is vertically over the extreme ends of the extrados of the concrete
arch. The 18 sections lying between the end sections have a width of 3.07 feet,
and a variable height which may be considered as extending from the top of the
load line down to the intrados. We may therefore multiply the widths of these
sections by their various heights, and by 150, and obtain the number of pounds
weight on each section, and we find the loads as follows: The sum total of this
loading, which represents the total dead and live load on a section of the concrete
arch one foot wide (in the direction of the axis of the concrete arch), is
63,814 pounds. We lay off these various loads on the right-hand side of the
drawing in a vertical line, using a scale of 5,000 pounds per inch. Selecting a
pole 01 at random, we draw rays to the various points in the load line. Commencing
at the point 0, we draw the segments of the trial equilibrium concrete polygon
parallel with the rays in the force diagram which run from the point 01, to the
load line, and obtain the trial equilibrium concrete polygon 0B1. By drawing
from o the line o1m parallel to the line 0B1, we obtain the point n on the load
line, from which we draw an indefinite horizontal line which intersects the
concrete.

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

**We Are Your Local
Concrete Cutting Company**

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

**We Service all
surrounding Cities & Towns.**