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Concrete Cutting One
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Concrete Cutting Coring Gardner MA Mass Massachusetts

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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 Need Concrete Cutting?

We Are Your Local Concrete Cutting Company

Call 508-283-3135

We Service all surrounding Cities & Towns.