a = y − W2x2
H.[[72]]
A resistance line should be drawn with this new horizontal thrust. If no resistance line can be found lying wholly within the middle third, new sections should be designed until a resistance line can be drawn lying wholly within the middle third—unless the arch is to be reinforced. A number of satisfactory arches should be designed and the easiest one to build should be selected. This method is limited in its application to sewer arches with rigid side walls and it cannot be extended to include the invert. Although an approximate method it is accurate within less than 10 per cent of the true stresses and is usually quite close.
Fig. 84.—Method for Dividing Arch into Proportion I
S.
The elastic method for the design of arches locates the true line of resistance without approximations and is more accurate though not so simple to apply as the static or vouissoir method. In this method a desired form of arch is drawn as in the static method and subdivided into vouissoirs so that the distance S along the neutral axis between joints is such that the ratio I
S shall be the same for all vouissoirs. I is the average of the moments of inertia of the surfaces of the two limiting joints about the neutral axis. If the thickness of the arch is constant the distance between joints will be the same. The method for dividing the arch into sections such that the ratio I
S shall be a constant[[73]] is as follows: divide the half arch axis into any number of equal parts; measure the radial depth at each point of division; lay off the length of the arch axis to scale on a straight line; divide this line into the same number of equal parts as the half arch, as shown in Fig. 84; at each point erect a perpendicular equal in length by scale to the moment of inertia at the corresponding point on the arch section; draw a smooth curve through the tops of these lines; draw a line ab at any slope from the center of the original straight line to the curve, and then a line bc back to the straight line to form an isosceles triangle abc; continue forming these triangles in a similar manner thus dividing the original straight line in the required ratio. The distance between joints is represented by the bases of the triangles. By construction the altitude of the triangle represents the average moment of inertia between the two limiting joints. The base of each isosceles triangle is S, and I
S = ½ tan α in which α is the base angle of all the isosceles triangles.
Fig. 85.—Elastic Arch Analysis.
The following steps in the procedure are taken from the second edition of the American Civil Engineers Pocket Book, p. 634:
In Fig. 85 let the middle points of the joints be marked 1, 2, 3, etc. and the coordinates x and y from the crown be found for each by computation or measurement. For a load W placed at one of these points, let z denote the distance from it, toward the nearest skewback, to another middle point. Let ∑zx be the sum of the products of all the values of z by the corresponding x, and ∑zy be the sum of all the products of z by the corresponding y; that is, each z in the last two summations is multiplied by the x or y of the point back of W which corresponds to z.
For a single load W on the left semi-arch of Fig. 85 the following formulas are deduced from the elastic theory, n being the number of parts into which the semi-arch is divided.