A Simple One-dimensional Model of Collapse of Some Tall Buildings

We develop one-dimensional model of an avalanche to examine the destruction of World Trade Center (WTC) 2 on September 11, 2001, after the building was hit by an airplane. Its main ingredient is the way the building opposes the progress of the avalanche, which is expressed in terms of a reaction force R at the location of the collapse front (a moving interface between the building and the avalanche). We consider three realistic functional forms of R: (i), the uniform reaction where R = r ·M · g, with M being the total mass of the building, g being the Earth’s gravity, and r being a constant independent of position; (ii), the uniform safety, where R = s · m(Y ) · g, with m(Y ) being the mass of the avalanche, the bottom of which is at the position Y = Y (t), and s being a constant independent of position; (iii), the uniform collapse where R = r ·M · g + s ·m(Y ) · g. In each case the collapse model results in an ordinary differential equation (ODE) of the second degree. We choose the necessary initial and final conditions for the ODE as follows: as the position where collapse starts we take the top floor of the impact area, while the initial velocity of the avalanche is zero. To simplify our analysis we assume that the impacted floors have sustained maximal damage so we set R ≡ 0 therein, while for the floors below, the functional form of R is one of the above three. We numerically solve the models to find for what values of parameters r or s, or both, the total collapse times are equal to the measured total collapse time of WTC 2 of Tc = 11 sec: r ≃ 0.1 in the uniform reaction; s ≃ 0.22 in the uniform safety; and r/0.1 + s/0.22 ≃ 1 in the uniform collapse model. For comparison, r = 1, s = 1, and r + s ≥ 1, are the minimal values that building must have in order to be standing, absent any other strain, in the uniform reaction, the uniform safety, and the uniform collapse models, respectively. The realistic estimate for WTC 2 is (rWTC , sWTC) ≃ (0.1, 1.4), in the uniform collapse model. Based on the magnitude of this discrepancy we surmise that the collapse of WTC 2 was inconsequential to the damage sustained by the airplane impact. Rather, the collapse of the building was caused by a wave of massive destruction (WMD) which propagated through the building moments before the avalanche started, and in the wake of which the avalanche actually developed.