Beam and Ring Failures of Pipes
In the failure of pipes, it has for a long time been recognized that the apparent strength is different for the transverse bending that leads to ring-type failure (Fig. 10.5.3a) and for longitudinal bending of the whole pipe that leads to beam-type failure (Fig. 10.5.3b). Gustafsson and Hillerborg (1985) analyzed such failures using the fictitious (cohesive) crack model. A plot of the size effect that they obtained is shown in Fig. 10.5.3c. In this plot, <t. is defined as the maximum elastic stress according to mechanics of materials theory. Thus, for the ring failure, we have
where Fu is the maximum force per unit length of pipe, Di and D„ the inner and outer diameters, respectively, and f the pipe thickness. The nominal stress for the beam failure is
I = (10.5.10)
Figure 10.5.3 Size effect in unreinforced concrete pipes according to the computations of Gustafsson and Hilierborg (1985): (a) scheme of ring (crushing) failure; (b) scheme of bending (beam) failure; (c) strength vs. size for the two types of failure. – is the maximum elastically-computed stress at peak load, seethe text.)
in which Mu is the ultimate bending moment (at the failure cross section), and I is the centroidal moment of inertia of the ring cross section. It is apparent that the size effect displays the same general trends as the size effect for the modulus of rupture (Section 9.3), and, thus, is expected to have similar properties. Indeed, the plot in Fig. 10.5.3c is a modification of Gustafsson and Hillerborg’s results which uses the property that, for unnotched specimens, the size effect curve is independent of the softening when plotted as a function of D/i where t is the characteristic size associated to the initial linear softening. Gustafsson and Hilierborg performed the computations using Petersson’s bilinear softening curve (Section 7.2.1), and produced plots of оууц/ft vs. D/£ch the plot in Fig. 10.5.3c has been rescaled by taking into account that for such softening i — 0.6Ісіг and in the given form can be applied to any softening curve with initially linear softening.
From the foregoing results, it follows that smaller pipes are seen to be stronger and more ductile in their postpeak response than larger pipes (JDj in the figure is the inner pipe diameter). It follows from this analysts that a size independent ‘‘modulus of rupture” currently used in design (ACI Committee 318, 1989) is unconservative for large pipes. Gustafsson and Hilierborg also observed that the ring-type failure is more size sensitive than the beam-type failure (Fig. 10.5.3c).
The failure of pipes was also studied by Bazant and Cao (1986), who considered the test results from Gustafsson (1985) and Brennan (1978). They compared the available test results to Bazant’s size effect law and concluded that the size effect is strong and that Bazant’s size effect law could be used. However, it must also be cautioned that the size effect law should not be fully applicable in this case, because the pipes reach their maximum load after only a small crack growth (that is, a large crack does not develop before failure). Thus, the size effect is primarily due to the formation of the fracture process zone, as characterized, for example, by the cohesive crack model or crack band model. Therefore, Bazant’s size effect law might not work well if the range of sizes is increased or the scatter of measurements reduced.