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The
temperature stresses of a two-hinged concrete arch are less severe, while those
for a three-hinged concrete arch may be neglected; but the concrete construction
of hinges in concrete arch ribs adds considerably to the cost. In the following
demonstration, the concrete arch rib is considered as a single line OCB (Fig.
228), which is assumed to have the properties of a concrete arch rib namely,
the moment of inertia, modulus of elasticity of the material, and the
consequent resisting moment. The curved line PQR represents the special
equilibrium concrete polygon corresponding to someone condition of loading.
Although this line is drawn as a curved line, it is assumed to be a curve which
is made up of a large number of correspondingly short lines, each of which
corresponds to a section of an equilibrium concrete polygon similar to those
described under "Concrete Arches." This equilibrium concrete polygon is yet to be
determined. In Church's "Mechanics of Engineering," Chapter XI is
given the mathematical proof of three general equations which apply to this
problem. No demonstration will here be made of these three equations, which are
as follows; The practical meaning of the first of these equations may be
described as follows: the line represents one of an even number of very short
sections into which the length OCB of the concrete arch rib has been divided.

M
represents the transverse moment acting on the concrete arch rib at that
section under the particular condition of loading which is being considered. E
is the modulus of elasticity of the material, and is the moment of inertia of
the section. At some of the sections the moment is positive, and at some it is
negative. The product divided by the product of E and I, is therefore sometimes
positive and sometimes negative. According to this equation, the summation of
these various products for each short section of the rib equals zero; or, in
other words, the summation of the positive products will exactly equal
numerically the summation of the negative products. The other two parts of
Equation 48 must be interpreted similarly, the only difference being that in
each case the term is multiplied by the corresponding value of 1/ for one of
the equations, and by x for the other.

This group of three equations has
nothing to do with the form of the special equilibrium concrete polygon PQR. It
may also be proved by analytical mechanics, that if the curve PQR represents
the special equilibrium concrete polygon corresponding to some system of
loading, and z represents the vertical distance between the concrete arch rib
and the special equilibrium concrete polygon at any section, then the moment M
at that section a of the rib, equals Hz, in which H is a constant which may be
determined from the force diagram. The curve PQR represents a typical special
equilibrium concrete polygon which crosses the concrete arch rib at two points.
These points of intersection indicate points of contra flexure, where the
transverse moment changes its direction of rotation, and where it is therefore
zero. When the special equilibrium concrete polygon is above the rib curve, we
call the moment positive; and when it is below, we call it negative. When it is
positive, it means that there is tension in the lower part of the rib, and
compression in the upper part. The conditions are, of course, the reverse of
this when the curve is below the rib. We may therefore substitute Hz for the
value of H in the group of Equations 48; and since II and E are both constant
for all points, from the principle enunciated in Article 423, we may not only
place them outside of the sign of summation, but may even drop them altogether,
since the summation equals zero; and we may therefore transform Equations 48 to
the following formula.

**Are You in ****Gardner Massachusetts****? Do You
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Concrete Cutting Company**

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