The correct analysis of a loaded ring can be complicated enough to justify computer solutions using finite-element methods. However, with adequate accuracy for design of the cross section (ring) of a soil-loaded cast iron soil pipe, a few reasonable assumptions result in analysis that makes design simple. Computers are not required or justified.
SIMPLIFYING ASSUMPTIONS
1. The ring is thin walled, i.e., the ratio of wall thickness to diameter is less than ca, 1:10. If the ring is thin walled, mean diameter may be used for analysis without significant error.
2. The pipe material performs elastically.
3. Ring deformations are small. For example, stress analyses are sufficiently accurate even though the effect of ring deflection is neglected, provided that ring deflection is less than ten percent.
Ring deflection is the percent decrease in vertical diameter due to soil loading. It is essentially equal to the corresponding increase in horizontal diameter. In fact, a cast iron soil pipe with DmT=60 does not deflect six percent without exceeding the modulus of rupture. Moreover, the ring is so stiff that ring deflection is generally less than one- or two percent in typical installa-tions.
4. Loads and reactions are symmetrical around a vertical axis.
5. Loads and reactions are either concentrated loads (truck load per unit length of pipe) or uniform-ly distributed loads (constant pressures).
6. The three-edge bearing load is equivalent to the parallel-plate load for purposes of stress analy-sis. (See Figure 1.)
7. All loads and reactions are vertical. Radial pressures, such as internal pressure or external hydro-static pressure (including internal vacuum) are disregarded. In fact, a cast iron soil pipe with Dm/t = 60 can withstand over 100 psi of external hydrostatic pressure and vacuum. Clearly, internal hydrostatic pressure is of no concern in typical design. It is equivalent to a depth in water of over 230 feet.
HORIZONTAL SOIL SUPPORT
Horizontal soil support on the sides of the ring is disregarded. This is conservative because tal soil support decreases ring deflection and so decreases flexural stress in the ring. (Any horizon-tal support provides an additional safety factor.) Horizonhorizon-tal soil support is not always dependable in the case of relatively stiff rings, such as in cast iron soil pipe. Because it requires either excellent
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compaction of the sidefill soil against the pipe or enough horizontal ring expansion to develop hor-izontal soil support. Neither can be completely assured in typical stiff-ring installations. With Dm/T less than about 60, cast iron soil pipe has a pipe stiffness greater than 250 lb/in2. Because of the ring stiffness, ring deflection of cast iron soil pipe in typical installations is less than one or two percent.
PIPE STIFFNESS
Pipe stiffness is defined as F/∆, where F is a parallel plate load on a ring and ∆ is the deflection due to that load. The test procedure is essentially the same as the three-edge bearing test described in the next section. Because ring deflection is not a concern in the design of cast iron soil pipe, ring stiffness is not important and is not considered further.
DESIGN SOIL PRESSURE
By equating σ = R, the failure stress is a function of the three-edge bearing load W at failure. If an appropriate safety factor is included, allowable external soil pressure P can be written in terms of W.
P is the design soil pressure.
Values of P are listed in the last column of Table 1 for the three most common loadings in the design of cast iron soil pipe. The three loadings are called Installation Conditions 1, 2, and 3.
For design, the allowable P can be found from the design soil pressure equations of Table 1 in terms of three-edge bearing strength W and pipe diameter. For convenience, the values of P are found in Table 1.
SAFETY FACTOR
A safety factor is included in each equation for P. All safety factors include 1.25 to account for sta-tistical deviation of loads, geometry, and material properties. In addition, the idealized loads assumed for analysis are adjusted conservatively to reflect actual installation conditions.
Figure 1—Typical Loads on Rings Vertical and Symmetrical.
For the concentrated reaction of Installation Condition 1, the critical stress occurs at point A.
(See Table 1). The critical stress is the sum of the ring-compression stress plus the flexural stress Mc/I, where Mc is the moment that can be found by the area-moment method, virtual work, or Castigliano theorem; where c, the section modulus of the wall, is t2/6. Noting that the ring compres-sion stress is negligible for typical installations, the critical stress σ can be calculated and equated to the three-edge bearing stress R (45,000 psi) at failure (called modulus of rupture). The result is a crit-ical load of PDm = 1.084 W. Yet from experience, the concentrated reaction does not happen in the field. The actual distribution of the reaction justifies an increase of more than 15 or 20 percent in crit-ical load PD. If only 15 percent, the adjusted critcrit-ical load becomes PD = 1.25 W in units of pounds and feet. If the safety factor of 1.25 is included and Dm is inches, the design soil pressure is
P = 12W Dm
For the distribution reaction PD of Installation Condition 2, Table 1, the same reasoning applies to the safety factor except that the uniformly distributed pressure cannot be assured in the field. From experience, the actual distribution of pressure results in a slight pressure concentration that justifies a decrease of less than roughly 15 or 20 percent in the critical load PD. If 20 percent, the adjusted critical load becomes PD = 2.08 W in units of pounds and feet. If the safety factor of 1.25 is includ-ed and if D is inches, the design soil pressure is
P = 20W Dm
For Installation Condition 3, the same rationale for the safety factor applies as for Installation Condition 2.
It goes without saying that the margin of safety is increased significantly by such conditions as the arching action of the soil envelope, the horizontal support of the ring by sidefill soil, and the addi-tional strength of hubs or joints. All of these conditions were conservatively neglected in the safety-factor analysis. Of course, special installation conditions may require a modified safety safety-factor, depending on risk.
The strength of a pipe cross section (ring) is measured by the three-edge bearing test. A section of pipe barrel is positioned on two closely spaced, longitudinal quarter-round supports, as shown in Figure 2a.
Figure 2—(a) Three-Edge Bearing Test to Determine Failure Load W;
(b) Equivalent Free-Body Diagram for Analysis.
Loading Here units are adjusted such that: Load PDm at Failure
For σ = R **Adjusted PDm
Design Soil Pressure P Reduced by Safety Factor of 1.25
P = 20W
*R = Stress at failure.
***Because in practice reactions are not as ideal as assumed in diagrams, adjustment from experience is included in an adjusted load PDm. The concentrated reaction is, in fact, slightly distributed. Consequently, PDm is adjusted up by 15 percent. The distributed reaction is not really uniform, so PDm is adjusted downward about 20 percent.
***K = Stress Reduction Factor for Compressible Soil Envelope = 1⁄2