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 stability of the pier on the right-hand side of the arch in Fig. 227, is determined on the assumption of the concentrated locomotive loading on the left-hand end of the next arch which is at the right of the given arch, and the uniform loading over the right-hand end of the given arch. We therefore draw through the point m' a line of force parallel to mk, and also produce the line In until it intersects the other line of force in the point s. A line from s parallel to R2, therefore, gives the line of action of the resultant of the forces passing down the pier, for this system of loading. Since this system of loading will give the most unfavorable condition, or the condition which will give a resultant with the greatest variation from the perpendicular, we shall consider this as the criterion for the stability of the pier. The piers were drawn with a batter of 1 in 12, and it should be noted that the resultant R, is practically parallel to the batter line. If the slope of R2 were greater than it is, the batter should then be increased. The value of R2 is scaled from the force diagram as 55,650 pounds. The force R2 is about 14 inches from the face of the pier, and this would indicate a maximum intensity of pressure of 221 pounds per square inch. This is a safe pressure for a good class of masonry work. The actual pressure on the top of the pier is somewhat in excess of this, on account of the weight of that portion of the arch between the virtual abutment at n and the top of the pier; and the total pressure at any lower horizontal section, of course, gradually increases; but on the other hand, the weight of the pier combines with the resultant thrust of the two arches to form a resultant which is more nearly vertical than R2, and the center of pressure therefore approaches more nearly to the axis of the pier. The effect of this is tc reduce the intensity of pressure on the outer edge of the pier; and since the numerical result obtained above is a safe value, the actual maximum intensity of pressure is certainly safe.