Role of Compaction Ratio in the Mathematical Model of Progressive Collapse

where x 2 is the position of the avalanche front, ρ 0 = M/H with M the total mass and H the total height of the building and g is the gravity, while R = R(x 2 ) is a local resistive force. Here, the dot above the quantity indicates its differentiation with respect to time.

The compaction ratio κ is defined as

where ρ is a density of the “compacted” section of the building, cf. Fig. 1.

We reexamine the steps that lead to Eq. ( 1) from the point of view of classical mechanics.

As can be seen from Fig. 1 we can choose between two generalized coordinates. The first choice is x 1 = x 1 (t) which for t > 0 well describes the motion of the avalanche. The second choice is x 2 = x 2 (t), which represents the position of the avalanche front -an idealized point-like boundary between the stationary and the moving part of the building, at which the compaction takes place. Motion of the avalanche front is more complex than the motion of the avalanche as it combines the motion of the avalanche with its spatial growth due to non-zero compaction ratio. While care must be exercised when deriving an equation of motion for each of them, the final result may not depend on the choice of generalized coordinate.

For simplicity, we assume that the total energy and the total mass in the system buildingavalanche is conserved. We note that in that case we obtain the fastest avalanche, as then there are no conversion losses of the potential energy of the building into the kinetic and then into the crushing energy of the avalanche. Also, the conservation of energy allows us to use Lagrangian formalism to derive the equation of motion.

First, we state the two constraints of descent,

Here, the top section stretches from x 0 = x 0 (t) down to x 1 = x 1 (t), while the compacted building occupies from x 1 down to x 2 = x 2 (t). The point x 2 is the avalanche front. With

Eq. (3a) we state that the length of the top section h does not change in descent, while with Eq. (3b) we express the conservation of the mass of the building. Differentiation of Eq. ( 3) with respect to time yields dynamical constraints, ẋ1 = ẋ0 and ẋ1 = (1 -κ) ẋ2 .

Second, we find kinetic, potential and latent energy necessary for the Lagrangian formulation. The kinetic energy K is given by K = 1 2 dx ρ(x) v 2 (x), where the velocity distribution in the avalanche is v(x) = ẋ1 for x ∈ [x 0 , x 2 . The kinetic energy of the avalanche is thus

The potential energy U of the whole building is U = -dx ρ(x) x g, yielding

The latent energy

Given a Lagrangian L = K -U -L, which is a function of a generalized coordinate x and its generalized velocity ẋ, the equation of motion follows from d dt ∂L/∂ ẋ = ∂L/∂x. We recall that we have two choices for the generalized coordinate: x ≡ x 1 for the motion of avalanche, or x ≡ x 2 for the avalanche front. If x 2 is chosen as a generalized coordinate, we simply obtain,

With x 1 as a generalized coordinate we note that

As expected, the two equations of motion are identical. In Eqs. ( 6) and ( 7) we introduced an additional parameter η which may take values 1, if the total energy is conserved, or 0, if this is not the case. A distinction between the two cases is discussed in [2]. Comparison between Eq. ( 7) and Eq. (1) of Bažant and Verdure’s shows that due to non-zero compaction the avalanche propagates through the building faster than it travels, which leads to an amplification of the building’s resistive force by the factor (1 -κ) -1 > 1. In other words, the avalanche front in Bažant and Verdure’s model, Eq. ( 1), is faster than the one proposed in Eq. ( 7). Eq. ( 7) as a proposed correction to Eq. ( 1) is a major result of this technical note.

We next discuss the estimates for the resistive force R that can be found in the literature.

subdivision after the statement from the NIST report that the damage to the buildings was concentrated in their primary zones while leaving the secondary zones intact. We found that to reach the collapse time 2 T the initial estimate of R had to be reduced by 75%. In our report we dubbed the 75%-strength-reducing wave that preceeded the avalanche the wave of massive destruction (WMD). As the avalanche was not supplying the energy to the WMD, and the WMD propagated before the avalanche, we concluded that the WMD caused the avalanche.

Currently, we can only speculate about the source of the 60-75%-strength-reducing wave and its coupling to the avalanche. However, a piece of information that would provide an important insight is the descent curve, which describes position as a function of time of some visible part of the building, say its top, x 0 = x 0 (t). Once the descent curve is known it is the acceleration, ẍ0 = ẍ0 (t), that can be directly connected to R through a mathematical model.

In fact, in [5] we examine the descent curve

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