Therefore, if is the true closing line, we have the propor tion: trial is to true Ti as is to or The value of v,m, and v„m„ having been found, the true closing line is obtained by drawing a line from m, to The lines and m,m18 should intersect at the middle of the span—a check always wise to note. Notice that by the above method,* the magnitude of neither trial T,. and true T,. nor trial and true are necessary, and also that the positions of trial and trial T are the same for all systems of loads, both of which facts constitute an advantage of this method over the one ordinarily used.
If the construction has been correctly made, the summation of the vertical intercepts above the closing line between it and the trial equilibrium polygon is equal to the summation of the ordinates below that line—a test easy to apply. In the drawing of which Fig. 233, facing page 688, is a photographic reduction and which had a scale of 1 inch = 3 feet, the sums of the bm intercepts were: above, 27.37 ft.; below 27.29 ft.
If in the force diagram, a line be drawn from the trial pole, P', to the load line parallel to the closing line the inter section Q will divide the load line into the true reactions at the right and the left abutments. The true pole is at some point, as yet undetermined, on a horizontal line through Q.
It is a principle of the equilibrium polygon that moving the pole vertically does not alter either the magnitude or the position of the intercepts, but does change the direction of the closing line. There fore if the trial pole is moved vertically to the horizontal line through Q, and a new equilibrium polygon be drawn, the closing line of the new equilibrium polygon will be horizontal; but the intercepts will not have changed either their magnitudes or their positions hori zontally. (This equilibrium polygon is not drawn in Fig. 233, since it is of no special advantage and would therefore only encumber the drawing).
Since the span of the trial equilibrium polygon is equal to the span of the arch ring, and since moving the pole horizontally does not alter the position horizontally of the several ordinates of the trial equilib rium polygon, if verticals be drawn through q,. and q,, the points
in which the closing line and the trial equilibrium polygon intersects, the intersections of these verticals with the reference line and respectively, will be points on the true equilibrium polygon that is to be constructed upon the line True Pole Distance. The moment at any point is equal to the intercepts in the equilibrium polygon multiplied by the pole distance; and hence increasing the pole distance decreases the intercepts in the equilibrium polygon, and vice versa. The true equilibrium polygon must give I ck .y = s ak. y (see equation 12, page 675); and hence the trial pole must be moved accordingly. If the trial pole is moved vertically to the horizontal line through Q, the closing line will be horizontal (§ 1324); and if then the trial pole is moved along the horizontal line through Q so as to change I bm . y to I ak. y, the new position will be the true pole for the given loading. Therefore the true pole distance = the trial pole distance To solve equation 17 proceed as follows: In the trial equilibrium polygon, measure the several intercepts bm, and also measure the several ordinates, ad (=y), from the neutral line to the span line, AB; and compute The several values of bm and of y are given in Table 94, page 683. In the example in hand, I bm. y = — 229.78. On the line of action of each load, measure the several intercepts, ak, from the neutral line to the reference line and compute The -values of ak and of y are given in Table 94, page 683. In the example in hand, I ak.y = —192.72. Equation 17 then becomes: True Equilibrium Polygon. Locate the true pole by measur ing the true pole distance from Q; and then beginning at, say, draw the equilibrium polygon c„; .... which should pass through The graphical construction of the equilibrium polygon can be checked as follows: Multiplying the pole distance is the same as dividing the intercepts of the equilibrium polygon, and hence we may compute the intercepts ck at once by dividing each bm inter cept by the ratio I bm. y _ I ak. y, and lay off these quantities from the line vertically on the lines through the centers of the several sections into which the neutral line is divided, having regard to the sign of bm.
The equilibrium polygon constructed as above is the true equilibrium polygon for the given loads. The proof is as follows: By construction, s ck. y = I ak. y, which satisfies equation 12, page 675.
By construction, each ck has been made equal to the correspond ing bm divided by a constant ratio, and in 4 1323 it was shown that I bm = 0; and hence I ck = 0, which satisfies equation 14, page 675 Each intercept ck is vertically over the corresponding intercept bm, and in magnitude each ck is equal to the corresponding bm divided by a constant ratio; and in § 1321 it was shown that bm. x = 0, and therefore I ck.x = 0, which satisfies equation 16, page 675.