Mathematical Models of Progressive Collapse and the Question of How Did the World Trade Centers Perish

Abstract

We derive discrete and continuous class of mathematical models that describe a progressive collapse in a fictional one-dimensional structure, where we consider plastic and elastic types of collisions. We examine static (collapse initiation lines, derived from the ultimate yield strength of the structural steel) and dynamic (duration of collapse, computed using mathematical models) features of events that comprised the collapse in WTC 1 and 2. We show that $(a)$, the dynamic and static aspects of the collapse are mutually consistent and weakly dependent on the class or type of mathematical model used, and $(b)$, that the NIST scenario, in which the buildings collapse after a sequence of two damaging events (airplane impact and subsequent ambient fires), is inconsistent with respect to the structural strength of the buildings. Our analysis shows that the force that resisted the collapse in WTC 1 and 2 came from a single structural element, the weaker perimeter columns, while the second structural element, the stronger core columns, did not contribute. We discuss two non-obvious inconsistencies between the mathematical models of progressive collapse based on the NIST scenario, and the practical realizations of collapse in WTC 1 and 2: $(i)$, the average avalanche pressure is 3 orders of magnitude smaller than the pressure the vertical columns are able to withstand, and $(ii)$, the intact vertical columns can easily absorb through plastic deformation the energy of the falling top section of the WTCs. We propose collapse scenario that resolves these inconsistencies, and is in agreement with the observations and with the mathematical models.