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.

Horizontal thrust, H = (W
2)n∑zy − ∑y·∑z
n∑y2 − (∑y)2 (1)
Moment at Crown, M0 = ½W∑z − H∑y
n (2)
Shear at Crown, V0 = ½W∑zx
∑x2 (3)

For symmetrical loading such as W on the left and W on the right the horizontal thrust and crown moment due to both loads are double those found by the above formulas, while the crown shear V0 is zero. For several loads unsymmetrically placed the formulas are to be applied to each in succession and the results added algebraically, the value of V0 being taken as negative for the left semi-arch and positive for the right semi-arch.

For any joint whose middle point is at a distance x from the crown

M = M0 + Hy + V0x − ∑Wz,
V = V0 − ∑W,

where ∑W is the sum of all the loads between the joint and the crown and ∑Wz is the sum of the moments of those loads with respect to the middle of the joint. The components of the resultant thrust normal and parallel to the joints are,

N = H cos θ − V sin θ,
F = H sin θ + V cos θ,

in which θ is the angle which the plane of the joint makes with the vertical.

The distances from the neutral axis to the resistance line are,

at the crown, e0 = M0
H,
at the joint, e = M
N.

The resistance line should be located as in the vouissoir method and if not within the middle third a new design should be studied.

105. Reinforced Concrete Sewer Design.—The method to be followed in the design of reinforced concrete arches is similar except that the moment of inertia should include both the concrete and the steel, that is,

I = Ic + nIs,

in which I is the moment of inertia to be employed, Ic is the moment of inertia of the concrete, Is is the moment of inertia of the steel, and n is the ratio of their moduli of elasticity, generally taken as 15. All of the moments of inertia are referred to the neutral axis of the beam. The reinforcement called for in precast circular pipes is given in Table 39. Sewers cast in place are ordinarily designed to avoid reinforcement, except where the depth of cover is small and the sewer may be subjected to superimposed loads.

Concrete sewers are sometimes reinforced longitudinally, with expansion joints from 30 to 50 feet apart. This reinforcement is to reduce the size of expansion and contraction cracks by distributing them over the length of a section. The pipe is divided into sections to concentrate motion due to expansion or contraction at definite points where it can be cared for.

The amount of longitudinal reinforcement to be used is a matter of judgment. It varies in practice from 0.1 to 0.4 per cent of the area of the section. Since the coefficients of expansion of concrete and of steel are nearly the same, movements of the structure are as important as the stresses due to changes in temperature.

Because of the uncertain and difficult conditions under which concrete sewers are frequently constructed it is advisable to specify the best grade of concrete and not to stress the concrete over 450 pounds per square inch in compression, with no allowable stress in tension. The concrete covering of reinforcing steel should be thicker than is ordinarily used for concrete building design, because of the possibility of poor concrete allowing the sewage to gain access to the steel, resulting in more rapid deterioration than would be caused by exposure to the atmosphere. A minimum covering of about 2 inches is advisable, except in very thin sections not in contact with the sewage. A minimum thickness of concrete of about 9 inches is frequently used in design, although crown thicknesses of 4½ inches have been used with success. Greater thicknesses should be used near the surface, particularly in locations subjected to heavy or moving loads.

Brick linings are often provided for the invert where moderately high velocities of about 10 feet per second when flowing full are to be expected. For velocities in the neighborhood of 20 feet per second the invert should be lined with the best quality vitrified brick. Although concrete may erode no faster than brick under the same conditions, brick linings are more easily replaced and at a smaller expense.