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.
Forces Nos. 1 to 17 are drawn in the force diagram of Fig. 227 at the scale of 4,000 pounds per inch. Forces 1 to 8, inclusive, have a resultant whose direction is given by the line marked R1" which joins the extremities of forces 1 to 8. Similarly, the direction of the resultant (7?/ or R2') of forces 9 to 17, inclusive, is given by the line which joins the extremities of this group. The direction of the resultant of all the forces Nos. 1 to 17, is given by the line joining the extremities of these forces in the force diagram, this resultant being marked R2, By choosing a pole at random (the point o2' in the force diagram), drawing rays to the forces, and beginning at the left-hand abutment, we may draw the trial equilibrium polygon, which passes through the point a on force No. 17. The line through a parallel to the last ray, has the direction ab. Producing the section of the polygon which is between forces 8 and 9 (and which is parallel to the ray which reaches the load line between forces 8 and 9), it intersects the first and last lines of the trial equilibrium polygon at the points b and d. The point b is therefore a point on the resultant R2' of forces Nos. 9 to 17 inclusive; and by drawing a line parallel to the force R2' in the force diagram, we have the actual line of action of the resultant.
Similarly, the line of action of the force R2" is determined by drawing from the point d a line parallel to R2" in the force diagram. Their intersection at the point e gives a point in the line of action of the resultant of the whole system of forces, R2; and by drawing from the point e a line parallel to R2 of the force diagram, we have the line of action of R2. We select a point (f) at random on the resultant R2, and join the point f with the center of each abutment. By drawing lines from the extremities of the load line parallel to these two lines from f, they intersect at the point o2". A horizontal line through o2" is therefore the locus of the pole of the true equilibrium polygon passing through the center of both abutments. The line fn intersects R2' in the point g, and the line fm intersects the force R2" in the point h. The intersection of gh with the vertical through the center (the point i) is the trial point which must be raised up to the point c, which is done by the method illustrated in Article 401. The application of this method gives the line kl, passing through c; and the line ln is therefore the first line of the special equilibrium polygon for the complete system of forces from No. 1 to No. 17; and the line km is similarly the last line of that polygon. By drawing lines from the extremities of the load line, parallel to ln and km, we find that they intersect at the point o2", which is the pole of the special equilibrium polygon passing through n, c, and m, for the complete system of forces Nos. 1 to 17.
Fig. 227. Pressures on Voussoirs of a Full-Centered Arch.
As a check on the work, the intersection of these lines from the ends of the load line, parallel to ln and fan, must be on the horizontal line passing through o2". By drawing rays from the new pole o2'" to the load line, and completing the special equilibrium polygon, we should find as a double check on the work, that both of these partial polygons starting from m and n should pass through the point c; and also that the section of the polygon between forces Nos. 8 and 9 lies on the line kl. This gives the special equilibrium polygon for the system of forces Nos. 1 to 17, which corresponds with the second condition of loading, as specified above.
The first condition of loading is given by duplicating about the center, in the force diagram, the system of forces from No. 17 to No.9 inclusive. Since this system of forces is symmetrical about the center, we know that its resultant R1 passes through the center of the arch, and that it must be a vertical force. We may draw from the middle of force No. 9 a horizontal line, and also draw a vertical from the lower end of the load line. Their intersection is evidently at the center of the resultant R1, which is therefore carried above this horizontal line for an equal amount. Joining the upper end of R1 with the upper end of force No. 9, we have the direction and amount of the force R1". The intersection of ng with the force R1 at the point j, gives a point which, when joined with the point m, gives one line of a trial equilibrium polygon passing through the required points m and n, but which does not pass through the required point c. The intersection of jm with the force R1", at the point p, gives us the line pg, which is the same kind of line for this trial polygon as the line hg was for the other.
By a similar method to that used before and as described in detail in Article 401, we obtain the line qr passing through c, which gives us also the section of our true equilibrium polygon between forces Nos. 8 and 9. The line rn also gives us that portion of the true equilibrium polygon for this system of loading, from the point n up to the force No. 17.
By drawing a line from the lower end of the load line, parallel to nr, until it intersects the horizontal line through the middle of force No. 9 at the point o1', we have the pole of the special equilibrium polygon for this system of loading, which is the first condition of loading. The rays are drawn from o1' only to the forces from No. 9 to No. 17 inclusive, and the special equilibrium polygon is completed between n and c by drawing them parallel to these rays.
On account of the symmetry of loading, we know that the equilibrium polygon would be exactly similar on the left-hand side of the arch. In discussing these equilibrium polygons, we must therefore remember that of the two equilibrium polygons lying between the extrados and intrados on the right-hand side of the arch, the upper line represents the line of pressure for a uniform loading over the whole arch (the first condition of loading), while the lower line on the right-hand side, and also the one equilibrium polygon which is shown on the left-hand side of the arch, represent the special equilibrium polygon for the second condition of loading.