Conduits to be supported upon compressible soil are often designed to act as a whole, assuming that all parts of the structure, including the invert, are equally subject to distortion under the loads. Fig. 103 represents a half-section of a conduit of this character. If we assume the middle of the invert, m, to be fixed in position, the mo ments and thrusts in a slice of the conduit 1 foot thick may be found in the manner used for the elastic arch in Chapter X. The axis of the conduit ring is divided into lengths as shown. The lengths of the divisions, coordinates of the centers of divisions with reference to the crown, and thicknesses of concrete at centers of division, are given in Table XXXIII.
The loads given (Table XXXII), are those due to the pressure of 20 feet of earth above the crown of the arch. The weight of the earth is taken at 100 pounds per cubic foot, and the intensity of the horizontal earth pressure at one-third that of the vertical pressure at the same point. In computing the loads, the unit pressures at the middle of the extrados of the division are considered as acting upon areas equal to the horizontal and vertical projections of the extrados of the divisions. The upward pressures upon the base are considered as acting vertically and uniformly distributed horizontally. The computations of loads and their moments about the centers of divi sion are shown in Table XXXII.
The moment and thrust at the crown section, a, may be obtained by the use of the formulas of Section 176. As the loading is sym metrical about the crown, rnL and are equal, and Formulas (33) and (35) of Section 176 become Table XXXIII gives the computation of the terms needed in these formulas. As the sections are rectangular, no reinforcement being considered in the computations, the value may be used in the formulas in place of s,//.
The load diagram is now drawn as shown, and the equilibrium polygon (or line of resistance) constructed, beginning with IL at a distance, e= 1.46 feet, above the middle of the crown section.
The thrusts acting upon the ends of divisions as found from the load diagram may be resolved into normal thrusts and shears as shown by the broken lines. These are tabulated in Table XXXIV. The moments at the centers of sections at the end of divisions may be obtained by multiplying the normal thrust upon the section by the distance from the center of section to the point at which the equilib rium polygon cuts the section, or they may be computed by Formula 10 of Section 163, which becomes for symmetrical loading Table XXXIV, gives the thrusts and moments with the resulting stresses at the extrados and intrados of the sections. These results show that there are tensions at the intrados of the crown section and in the invert, and at the extrados of sections f, g, and h which must be cared for by reinforcement. This reinforcement should be sufficient to carry the tensions in the section without materially changing the position of its neutral axis or the compression upon the concrete. To do this, the stress in the steel should be limited to about fifteen times that shown for the rectangular section, or about. 6000 at sections a and g and 9000 at. m. Computing the total tension in these sections, we find that an area of about 2 of steel per foot of length is required at a and g and about 4 at m. One
inch square bars spaced 6 inches apart near the intrados at sections a and L, then crossing to the extrados at e and extending along the extrados to section i, with 1-inch square bars spa-ced 6 inches apart near the intrados of the invert would answer the requirement.
The maximum shear occurs at section j, the unit shear being about 50 which is not excessive.
It seems probable that this analysis represents the conditions giving the maximum stresses possible in the structure. For a depth as great as 20 feet, the full pressure of the earth would probably not be borne by the structure. For greater depths, these pressures need not he increased, unless the earth is unstable.
The deflection of the conduit under the loads is outward upon the sides, and, if the earth is well packed around the sides of the conduit the earth will resist that deflection and the horizontal earth pres sures will probably be greater than those used in the analysis. This will diminish the bending moments at all points and reduce the need for reinforcement.
Longitudinal reinforcement is needed to prevent cracking of the concrete. Usually 1-inch bars, 12 inches apart, are sufficient for this purpose. Where the support of the soil under the conduit may not be uniform, it is desirable to guard against longitudinal deflection by the use of heavier reinforcement near the bases of the side walls.
187. Pressure Conduits.—Conduits to carry water under pressure are usually made of circular or oval form. The stresses caused by internal pressure are all tensile and should be taken wholly by the steel reinforcement.
Let P = the internal pressure per square inch; internal diameter of the conduit in inches; stress in the steel; As= the area of steel per inch of length.
Then we have, As= Low values of are desirable in order to minimize the possibility of cracks in the concrete. Satis factory results have been obtained in a number of instances with stresses from 10,000 to 15,000 pounds per square inch. The likeli hood of cracks will be reduced by using reinforcement giving me chanical bond, such as expanded metal, diagonal mesh or deformed bar, rather close spaced.
The thickness of concrete, except for small conduits under light pressure, should be at least 6 inches. When the pressure is consider able, it may be possible to reduce the possible leakage by use of a greater thickness with double lines of reinforcement and low tension in the steel.
Pressure conduits must be capable, like gravity conduits, of carrying any exterior loads which may come upon them when empty. They may he analyzed in the same manner as pipes or gravity con duits for exterior loadings.
Longitudinal reinforcement is required in conduits to prevent cracking due to changes in temperature and shrinkage of the concrete. When the conduit is divided into sections by use of expansion joints, light reinforcement may be sufficient between joints, although closer spacing is desirable than is required for longitudinal reinforcement in bridges or culverts. When prevention of leakage is important, and the probable changes in temperature not too great, continuous closely spaced longitudinal reinforcement may give better results than the use of expansion joints.