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# How is a progressive-collapse analysis performed in which a column is removed to evaluate dynamic response?

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1. Before doing any analysis, be clear on the purpose. For many structures, the purpose of the analysis is to show that if one column is suddenly removed, there is at most a small amount of damage. This is relatively easy, and may require only linear static analysis. For some structures actual collapse may be a concern. In this case the analysis is much more complex. Also, the calculated response is likely to be so sensitive to the modeling assumptions that the analysis may be little more than an academic exercise. The attached paper by Graham Powell can provide some guidance.
2. Before running a dynamic analysis, start with a static analysis (remove the column, analyze for static gravity loads plus static loads equal to the column forces (upwards load), then add static downwards load equal to two times the column forces). Use this to get an upper bound estimate of the vertical deflection (the factor of two is the dynamic amplification factor for an undamped elastic structure when the column is suddenly removed), and also to see if there is any substantial inelastic behavior.
3. If a dynamic analysis is needed, it is probably a good idea to start with an elastic analysis. For zero damping the results should be close to those for a static analysis with an impact factor of 2.0. Be sure to use a time step that is short enough to capture the vertical vibrations, which are likely to have short periods. The analysis results may be sensitive to the assumed distribution of vertical masses, and to the assumed damping. Since there is only one substantial displacement cycle when a column is suddenly removed, it can be argued that there is less effective damping than for earthquake response, with many cycles. For damping, be very careful with the beta-K part of Rayleigh damping. It may be wise to specify only alpha-M damping, or to assume zero damping.
4. If an elastic analysis indicates that there may be significant inelastic behavior, run an inelastic static analysis. If the amount of inelastic behavior is small, this should run OK. Among other things, look at the energy balance (of the type considered in the attached paper).
5. If an inelastic dynamic analysis is still needed, then run one, but be aware that the calculated response is likely to be very sensitive to modeling assumptions such as strengths, strain hardening ratios, damping, mass distribution, and whether the catenary effect is considered.
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