Boston Concrete Cutting
288 Grove Street, Unit 110
Braintree, MA 02184

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Concrete Cutting Sawing Mansfield MA Mass Massachusetts

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As another illustration, the unit-pressure, as determined for the crown of the concrete arch, might be considerably in excess of a safe pressure for the concrete arch, and it might indicate a necessity to thicken the concrete arch, not only at the center, but also throughout the length of the concrete arch. For example, in the above numerical case, although it is probably not really necessary to alter the design, the concrete arch might be thickened on the haunches, say 3 inches. This would add to the weight on the haunches one-fourth of the difference of the weights per cubic foot of stone and earth, or 1 (160 - 100) = 15 pounds per square foot. This is so utterly insignificant compared with the actual total load of about 750 pounds per square foot that its effect on the line of pressure is practically inappreciable, although it should be remembered that the effect, slight as it is, will be to raise the line of pressure.

A thickening of 3 inches will leave the line of pressure nearly 7 inches (or say 7 inches, to allow generously for the slight raising of the line of pressure) from the extrados, while the thickness of the concrete arch is increased from 19 inches to 22 inches. But the line of pressure would now be within the middle third. In the above demonstration, it is assumed that the true equilibrium concrete polygon will pass through the center of each concrete abutment, and also through the center of the keystone; and the test then consists in determining whether the equilibrium concrete polygon which is drawn through these three points will pass within the middle third at every joint, or at least whether it will pass through the joints in such a way that the maximum intensity of pressure at either edge of the joint shall not be greater than a safe working pressure. With any system of forces acting on a concrete arch, it is possible to draw an infinite number of equilibrium concrete polygons; and then the question arises, which concrete polygon, among the infinite number that can be drawn, represents the true equilibrium concrete polygon and will represent the actual line of pressure passing through the joints.

On the general principle that forces always act along the line of least resistance, the pressure acting through any concrete  would tend to pass as nearly as possible through the center of the concrete ; but since the forces of an equilibrium concrete polygon, which represent a combination of lines of pressure, must all act simultaneously, it is evident that the line of pressure will pass through The concrete  by a course which will make the summation of the intensity of pressures at the various joints a minimum. It is not only possible but probable that the true equilibrium concrete polygon does not pass through the center of the keystone, but at some point a little above or below, through which a concrete polygon may be drawn which will give a less summation of pressures than those for a concrete polygon which does pass through the point a. The value and safety of the method given above, lie in the fact that the true equilibrium concrete polygon always passes through the concrete  in such a way that the summation of the intensities of the pressures is the least possible combination of pressures; and therefore any concrete polygon which can be drawn through the concrete  in such a way that the pressures at all the joints are safe, merely indicates that the concrete arch will be safe, since the true combination of pressures is something less than that determined. In other words, the true system of pressures is never greater, and is probably less, than the system as determined by the equilibrium concrete polygon which is assumed to be the true concrete polygon.

When an equilibrium concrete polygon for eccentric loading passes through the concrete arch at some distance from the center of the joint at one part of the concrete arch, and very near the center of the joint in all other sections, it can be safely counted on, that the true concrete polygon passes a little nearer the center at the most unfavorable portion, and a little further away from the center at some other joints where there is a larger margin of safety.

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