arXiv subjects

Charles M. Beck

Publications and source records attributed to Charles M. Beck.

3 recordsLinked to original sources

arXiv2009

Descent curve and the phases of collapse of WTC 7

We examine four WTC 7 descent curves, labeled "C," "E," "N," and "O," either anonymously published, or confidentially communicated to us. Descent curve describes apparent height of a collapsing building as a function of time. The set "C" suggests that there are three active phases of collapse. Phase I is a free fall for the first $H_1\simeq28$m or $T_1\simeq2.3$s, during which the acceleration $a$ is that of the gravity, $a=g=9.8$m/s$^2$. In Phase II, which continues until drop $H_2\simeq68$m, or $T_2\simeq3.8$s, the acceleration is $a\simeq5$m/s$^2$, while in Phase III which continues for the remaining of the data set, $a \simeq -1$m/s$^2$. We propose that the collapse of WTC 7 is initiated by a total and sudden annihilation of the base (section of the building from the ground level to $H_1$), which then allows the top section (building above $H_1$) to free fall during Phase I, and then collide with the ground in Phase II and III. The total duration of the collapse, assuming that Phase III continues to the end, is in the range $7.8-8.6$s. We derive a physical model for collision of the building with the ground, in which we correct the "crush-up" model of Bažant and Verdure, J. Engr. Mech. ASCE, {\bf 133} (2006) 308, and estimate the magnitude of the resistive force in the top section. We compare our findings to those of NIST investigators and find an agreement with respect to the distribution of damage in the primary zone. We conclude that the building was destroyed in a highly controlled fashion.

physics.pop-ph↗

arXiv2008

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 Bažant 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 Bažant 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 Bažant 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.

physics.pop-ph↗

arXiv2007

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

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.

physics.pop-ph↗