Another useful property, which will be utilized later, and which may be readily verified from Figs. 215 and 216, is that, no matter what equilibrium polygon may be drawn, the two extreme lines of the equilibrium polygon, if produced, intersect in the resultant there fore, when it is desired to draw an equilibrium polygon which shall pass through any two abutment points, such as yz or yz', we may draw from these two abutment points, two lines which shall intersect at any point on the resultant R. We may then draw two lines which will be respectively parallel to these lines from the extremities p and q of the load lines, their intersection giving the pole of the correspond ing force diagram.
401. Equilibrium Polygon for Non=Vertical Forces. The above method is rendered especially simple, owing to the fact that the forces are all vertical. When the forces are not vertical, the method becomes more complicated. The principle will first be illustrated by the problem of drawing an equilibrium polygon which shall pass through the points y, z, and v in Fig. 217. We shall first draw the two non-vertical forces in the force diagram. The resultant R of the forces A and B is obtained as shown in Fig. 213. Utilizing the property referred to in the previous article, we may at once draw two lines through y and z which intersect at some assumed point e on the resultant R. Drawing lines from p and q parallel respectively to ez and ey, we determine the point o' as the trial pole for our force diagram. As a check on the drawing, the line joining the inter sections b and c should be parallel to the ray o's, thus again verifying one of the laws of Statics. If the line be is produced until it inter sects the line yz produced, and a line is drawn from the intersection x through the required point v, it will intersect the forces it and B in the points d and g. Then du will be one of the lines of the required equilibrium polygon. By drawing lines from q and p parallel to yd and zy, we find their intersection o", which is the pole of the required force diagram. There are two checks on this result: (1) the line
so" is parallel to dg; and (2) the line o'o" is horizontal.
If the line be is horizontal or nearly so, the intersection (x) of be and yz produced is at an infinite distance away, or is at least off the drawing. If be is actually horizontal, the line dg will also be a hori zontal line passing through v. When be is not horizontal, but is so nearly so that it will not intersect yz at a convenient point, the line du may be determined as is indicated by the dotted lines in the figure. Select any point on the line yz, such as the point a. Through the given point v, draw a vertical line which intersects the known line be in the point Jr. From some point in the line be (such as the point b), draw the horizontal line bh and the vertical line b,i. The line from o through k intersects the horizontal line from b in the point h. From the point h, drop a vertical; this intersects the line ov produced, in the point m. From m, draw a horizontal line which intersects the vertical line from b. This intersection is at the point n. The line .un forms part of the required line dy. As a check on the work, the lines zy and yd should intersect at some point f on the force R. Another check on the work, which the student should make, both as a demon stration of the law and as a proof of the accuracy of his work, is to select some other point on the line yz than the point o, and likewise some other point on the line be than the point b, and make another independent solution of the problem. It will be found that when the drawing is accurate, the new position for the point n will also be on the line dy.
In applying the above principle to the mechanics of an arch, the force ,1 represents the resultant of all the forces acting on the arch on one side of the point v through which the desired equilibrium polygon is required to pass; and the force B is the resultant of all the forces on the other side of that point. A practical illustration of thi6 method will be given later.