The Crush-Down Equation for Non-Constant Velocity Profiles

Ba\v{z}ant et al. have proposed a model for a gravity-driven collapse of a tall building that collapses after column failure in a single storey. Therein the collapsing building is described by three distinct sections. The top section which consists of the part above the first failing storey, the middle section which is pushed from above by the top section and consists of compacted building material, and the part of the building below which is still undamaged. The middle part is gaining height during the collapse, the lower section is loosing height. The resulting equation of motion is called Crush-Down equation. In a first approach Ba\v{z}ant and Verdure used a constant velocity profile for the middle section, namely the top section and the middle section are assumed to have the same velocity. In a second approach by Ba\v{z}ant, Le, Greening and Benson this assumption is dropped and the model is slightly modified. However, their modifications are based on unphysical assumptions and lead to an erroneous version of the Crush-Down equation. We give a detailed account of how to implement a non-trivial velocity profile for the middle section and thereby derive a more accurate version of the Crush-Down equation.