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.