Nonlinear dynamic analysis of a structure with local inelastic collapse

Figure 3 shows a series of contour plots of equivalent plastic strain (plotted on the deformed configuration) from the static analysis using element type CPEG8R, and Figure 4 shows the same plots for the analysis using element type CPEG4R. Figure 3 clearly shows that the discretization is too coarse or should be rezoned later in the deformation, but it is judged that this is not critical to the overall moment-rotation response prediction. Figure 5 shows the moment-curvature responses predicted by the analyses. The unstable nature of the response is clearly illustrated.

Figure 6 shows a series of deformed configuration plots from the dynamic analysis. After the shutdown of the force at 0.06 seconds, the momentum of the pipe is enough to cause localization of the deformation at the root of the cantilever as the section collapses there: the pipe whips around this hinge in a full circle and beyond its initial configuration. As well as this major hinge at the root, permanent plastic deformation develops throughout most of the pipe, leaving it bent into an arc. Time history plots of the tip displacement are shown in Figure 7 and of the curvature strain at the localized hinge in Figure 8. Figure 9 shows the moment-curvature response for the element at the support and shows the elastic unloading and reloading that takes place during and at the end of the event. Figure 10 shows the history of the energy content during the dynamic analysis and clearly shows the initial build-up of kinetic energy, which is then converted almost entirely to plastic dissipation.

This two-stage approach to the problem has the advantages of being simple and computationally inexpensive. It contains some obvious approximations. One is that interaction effects between bending, axial, and torsional behavior are neglected. This lack of interaction between the various modes of cross-sectional response is a basic approximation of the nonlinear beam general section option. In reality, axial or torsional strain will have the effect of reducing the strength of the section in bending. This effect is unlikely to be significant in a case that is dominated by bending, but it can be important if large axial or torsional loadings occur. The approach also neglects the effect of the axial gradient of the cross-sectional behavior on the response. This may be a significant error, but its evaluation would require a detailed, three-dimensional analysis for comparison; and that exercise is beyond the scope of this example. Another possibly significant error is the neglect of rate effects on the response. The cross-sectional collapse involves large strains, which occur in a very short time in the dynamic loadings, so high strain rates arise. It is likely that the material will exhibit strain rate dependence in its yield behavior and will, therefore, be rather stiffer than the static analysis predicts it to be. This should have the effect of spreading the hinge along the pipe and reducing the localization (because the strain rates increase at the section where most deformation is occurring, and that increased strain rate increases the resistance of the section). The magnitude of this effect can be estimated from the solution we have obtained. From Figure 3 we see that typical strains in the section are about 10–20% when the section is far into collapse; and Figure 8 shows that, in the dynamic event, it takes about 0.2 seconds for this to occur. This implies average gross strain rates of about 1.0 per second in that period of the response. In typical piping steels such a strain rate might raise the yield stress 5–10% above its static value. This is not a large effect, so the mitigation of localization by rate effects is probably not a major aspect of this event. Again, a more precise assessment of this error would require a fully three-dimensional analysis. Overall it seems likely that this simple and computationally inexpensive two-stage approach to the problem is providing results that are sufficiently realistic to be used in design, although it would be most desirable to compare these results with physical experimental data or data from a full, detailed, three-dimensional analysis to support that statement. Finally, it should be noted that the section considered here is relatively thick ($R/t=$3.5). In pipes with thin walls ($R/t>$20) it is to be expected that the behavior will be affected strongly by internal fluid pressure in the pipe and by the interaction between axial and bending forces. Such thin-walled pipes could be modeled at relatively low cost by using ELBOW elements directly in the dynamic analysis instead of this two-stage approach. An additional concern with very thin pipes is that they are more likely to tear and leak, rather than choke the flow.