413. Correcting a Design. The above general method of testing an arch consists of first designing the arch, and then testing it to see whether it will satisfy all the required conditions. • In case some condition of loading is found which will cause the line of pres sure to pass outside of the middle third or to introduce an excessive unit-pressure in the stones, it is theoretically necessary to begin anew with another design, and to make all the tests again on the basis of a new design; but it is usually possible to determine with sufficient closeness just what alterations should be made in the design so that the modified design will certainly satisfy the required conditions. For example, if the line of pressure passes on the upper side of the middle third at the haunches of the arch; a thickening of the arch at that point until the line of pressure is within the middle third of the revised thickness, will usually solve the difficulty. The effect of the added weight on the haunch of the arch will be to make the line of pressure move upward slightly; but the added thickness can allow for this. As another illustration, the unit-pressure, as determined for the crown of the arch, might be considerably in excess of a safe pres sure for the _arch, and it might indicate a necessity to thicken, the arch, not only at the center, but also throughout the length of the arch.
For example, in the above numerical case, although it is probably not really necessary to alter the design, the 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 71 inches (or say V, inches, to allow generously for the slight raising of the line of pres sure) from the extrados, while the thickness of the arch is increased from 19 inches to 22 inches. But the line of pressure would now be within the middle third.
414. Location of True Equilibrium Polygon. In the above demonstration, it is assumed that the true equilibrium polygon will pass through the center of each abutment, and also through the center of the keystone; and the test then consists in determining whether the equilibrium 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 maxi mum 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 an arch, it is possible to draw an infinite number of equilib rium polygons; and then the question arises, which polygon, among the infinite number that ca-n be drawn, represents the true equilibrium 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 voussoir would tend to pass as nearly as possible through the center of the voussoir; but since the forces of an equilibrium polygon, which rep resent a combination of lines of pressure, must all act simultaneously, it is evident that the line of pressure will pass through the voussoirs by a course which will make the summation of the intensity of pres sures at the various joints a minimum. It is not only possible but probable that the true equilibrium polygon does u-ot pass through the center of the keystone, but at some point a little above or below, through which a polygon may be drawn which will give a less sum mation of pressures than those for a 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 polygon always passes through the voussoirs in such a way that the summation of the intensities of the pressures is the least possible combination of pressures; and therefore any polygon which can be drawn through the Youssoirs in such a way that the pressures at all the joints are safe, merely indicates that the arch will be safe, siilec 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 polygon which is assumed to be the true polygon.
When an equilibrium polygon for eccentric loading passes through the arch at some distance from the center of the joint at one part of the arch, and very near the center of the joint in all other sections, it can be safely counted on, that the true 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. For example, the true equilibrium polygon for the third condition of loading (see Fig. 223) probably passes a little nearer the center on the left-hand haunch, and a little farther away from the center on the right-hand haunch, where there is a larger margin; in other words, the whole equilibrium polygon is slightly lowered throughout the arch. No definite reliance should lie placed on this allowance of safety; but it is advantageous to know that the margin exists, even though the margin is very small. The margin, of course, would reduce to zero in case the equilibrium polygon chosen actually represented the true equilibrium polygon. While it would be convenient and very satisfactory to be able to obtain always the true equilibrium polygon, it is sufficient for the purpose to obtain a polygon which indicates a safe condition when we know that the true polygon is still safer.