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Whenever
we are investigating the mechanics of a concrete arch rib which has a constant
moment of inertia, we may simplify Equations 49 by dropping out altogether the
denominators of those equations; but since concrete arch ribs are usually made
with deeper sections near the concrete abutments, the result will be greater
near the concrete abutments. Calling the line at the center I, then I equals itself,
in which n is a variable. If we substitute this value of I in the denominators
of Equations 49, then, since I. is a constant quantity, it may be placed
outside of the summation sign, and even dropped altogether, which practically
means that we substitute n for I in Equations 49. We shall also substitute for
z its value z" - z' (see Fig. 228), and shall rewrite Equations 49 as
follows, by making the substitutions. It will later be shown how we can draw a
line which will satisfy the following equations: Since the concrete arch rib
(represented by the curve OCB) is assumed to be symmetrical about its center C,
and since the variable is horizontal, any position of which will satisfy the
first of Equations 51 will also satisfy the second.

It is another principle of
the science of summations, that if we have a series of terms whose summation
equals zero, and also have another series of terms whose summation equals zero,
but whose terms are made up of the difference of two terms, one of which
corresponds in each case to the terms of the first summation, then we may say
that the summation of the other corresponding terms is likewise zero. For
example, the first one of Equations 50 consists of a series of terms which may
be rewritten: The first one of Equations 51 is the summation of a series of
terms, each with the form In each of these summations the different terms
corresponding to the variable values of z' exactly correspond. We may therefore
say that the summation of a series of corresponding terms, each one of the
form, will exactly equal zero; and we may therefore write Equation 52 as given below.
We may also combine the second part of Equation 50 with the second part of Equation
51 in a similar manner, and obtain Equation 53 as given below It will be found
more convenient to separate the third part of Equation 50 into two summations,
One of which consists of a series of terms and the other of a series of terms
consisting of yards of concrete and since the difference of these summations
equals zero, then the summations must equal each other, and we may therefore
write Equation 54.

An infinite number of equilibrium concrete polygons may be
drawn which will satisfy Equation 52 and 53. An equilibrium concrete polygon
may be drawn by trial, and the values of the summations for each side of
Equation 54 may be determined. But since the position of the line is definitely
determined by Equation 51, then the value of the right side of Equation 54 is
fixed, and we only need to alter the pole distance of the trial equilibrium concrete
polygon in the inverse ratio of the required change in z", and then
Equation 54 will be satisfied. Since changing all the values of z" in the
same ratio does not alter the satisfaction-of Equations 52 and 53, the changing
of the pole distance does not vitiate the previous work. The value for the true
pole distance is thus obtained, by which the true curve PQR may be graphically
drawn out. We may then determine the moment at any point, which is the product
of Hz for any point of the curve in which z is the vertical distance between
the center line of the concrete arch rib and the finally determined equilibrium
concrete polygon, and H is the pole distance corresponding to that concrete
polygon. It will be shown later that the thrust and the shear for any point of
the curve equal the projection onto the tangent and normal respectively of the
proper ray of the force diagram.

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

**We Are Your Local
Concrete Cutting Company**

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

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