To Find the True Equilibrium Polygon

TO FIND THE TRUE EQUILIBRIUM POLYGON. The third step (§ 1310) is to construct on the line kk, Fig. 230, page 675, an equilibrium polygon which will satisfy the conditions: I ck = 0, r ck. x = 0, and F ck. y = I ak. y. The equilibrium polygon which satisfies these conditions is the true equilibrium polygon, and having this the stresses in the arch can readily be found. The method of finding the true equilibrium polygon can best be explained in con nection with the working of an example.

Tha Data We will use

for the illustration a segmental circular plain-concrete arch of the following dimensions: Span of the neutral line, 50 ft.; rise of the neutral line, 10 ft.; thickness at the crown, 2.0 ft.; thickness at the springing in the line of the radius of the neutral line, 5 ft.; depth of earth over the crown of the extrados, 2 ft.; live load over half the span, 100 lb. per sq. ft. The live load is taken over only half the span to illustrate a method of procedure for unsymmetrical loads. We will assume concrete to weigh 150, and earth 100 lb. per cu. ft. We will consider only a section of the arch ring 1 ft. long, i.e., 1 ft. perpendicular to the plane of the draw ing. Fig. 232 shows the dimensions of the arch.

The first step is to divide the neutral line of the semi-arch into a number of parts such that the length of each part divided by the moment of inertia of the cross section shall be constant. By the method of § 1311, the neutral line of the semi arch is divided into eight parts, as shown in Fig. 232. Table 93, page 680, shows the data employed in making this division, and also shows the values of 48 _ I. Of course, if the work were accurate, all of the quantities in the last column of Table 93 would be the same. The values of ds - 1 were de termined from a drawing like Fig. 231 having a scale of 2 inches to 1 foot.

The next step is to divide the arch into sections and find the dead and live load for each. In the solution of the problem to follow, we shall be required to measure the vertical intercepts, at the center of each division of the arch ring found in § 1311-12, of the equilib rium polygon and also be tween the equilibrium polygon and the neutral line of the arch ring; and the more nearly the equi librium polygon conforms to the curve of pressures, i.e., the line of resistance

or the linear arch, the more accurate the results. The line of resistance is a curve inscribed in the equilibrium polygon (see §1195); and hence we may either make the equilib rium polygon so many sided that it conforms closely to the pressure curve, or so construct the equilibrium polygon that its sides shall be tangent to the pressure curve at the points where the • intercepts are to be measured. The first necessitates the dealing with numerous loads, and hence entails considerable work; while the second can be done without unneces sary labor.

Therefore, if we draw vertical lines through a„ a,, etc., Fig. 232, page 679, the middle of the several divisions of the arch ring, and find the dead and live loads for each section, the resulting equilibrium polygon will have its sides almost exactly tangent to the pressure curve at the verticals through a„ a,, etc., the points at which the intercepts are to be measured. Since the end section of the arch ring is rather long, it was divided into three portions and the load found for each. The loads for each of the several sections are stated in the upper portion of Fig. 232. These loads act at the center of gravity of each vertical slice, which may usually be taken midway between the vertical sides except possibly for a few slices near the springing. For the latter portions, the center of gravity may be found as in § 935.

Trial Equilibrium Polygon.

The first step toward finding the true equilibrium polygon is to draw a trial equilibrium polygon. The arch is loaded as shown in Fig. 232. The arch is re-drawn to a larger scale in Fig. 233—a folding sheet facing page 688.