Role of Compaction Ratio in the Mathematical Model of Progressive Collapse

We derive a mathematical model of progressive collapse and examine role of compaction. Contrary to a previous result by Ba\v{z}ant and Verdure, J. Engr. Mech. ASCE 133 (2006) 308, we find that compaction slows down the avalanche by effectively increasing the resistive force. We compare currently available estimates of the resistive force, that of Ba\v{z}ant and Verdure (2006) corrected for compaction for World Trade Center (WTC) 2, and of Beck, www.arxiv.org:physics/0609105, for WTC 1 and 2. We concentrate on a damage wave propagating through the building before the avalanche that figures in both models: an implicit heat wave that reduces the resistive force of the building by 60% in Ba\v{z}ant and Verdure (2006), or a wave of massive destruction that reduces the resistive force by 75% in Beck (2006). We show that the avalanche cannot supply the energy to the heat wave as this increases the resistive force by two orders of magnitude. We thus reaffirm the conclusion of Beck (2006) that the avalanche is initiated in the wake of the damage wave.