This section is from the book "Cyclopedia Of Architecture, Carpentry, And Building", by James C. et al. Also available from Amazon: Cyclopedia Of Architecture, Carpentry And Building.
The method of making this test is exactly similar to that previously given; but on account of a load eccentrically placed, the force diagram will be more distorted than in either of the cases previously given, and there is greater danger that the arch will prove to be unstable on such a test. An inspection of the equilibrium polygon for this case shows that the critical point is the joint between vous-soirs Nos. 3 and 4. This is what might be expected, since it is the joint under the heavy concentrated load. The ray in the force diagram which is parallel to the section of the equilibrium polygon passing through this joint, is the ray which reaches the load line between loads 3 and 4. This ray, measured at the scale of 1,500 pounds per square inch, indicates a pressure of 15,625 pounds on the joint. The line of pressure is 4f inches from the upper edge of the joint; it is outside of the middle third; and therefore the joint will probably open somewhere under this loading. According to the theory of the distribution of pressure over a stone joint, the pressure will be maximum on the upper edge of this joint, and will be zero at three times 4| inches, or 14.25 inches, from the upper edge. The area of pressure for a joint 12 inches wide will be 14.25 X 12 = 171 square inches. Dividing 171 into 15,625, we have an average pressure of 91 pounds, or a maximum pressure of twice this, or 182 pounds, per square inch at the edge of the joint. But this is such a safe working pressure for such a class of masonry as cut-stone vous-soirs, that the arch certainly would not fail, even though the elasticity of the stone caused the joint to open slightly at the intrados during the passage of the steam roller.
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 pressure 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 pressure 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 \ (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 1/2 inches (or say 7 1/2, inches, to allow generously for the slight raising of the line of pressure) 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.