Introducing Keith Seffen's "Progressive Collapse Of The WTC: A Simple Analysis"

The existence of the paper, "Progressive Collapse of the World Trade Centre: a Simple Analysis", by University of Cambridge senior lecturer Keith Seffen, is no longer in doubt.

Dr. Seffen has claimed his paper shows the collapse of the WTC towers was "destined to be rapid and total" once it began, and the rapid and total destruction of the towers was "an ordinary thing to have happened."

The paper was cited on September 11, 2007, by the BBC and others, who described it as having been published by the Journal of Engineering Mechanics (JEM), a monthly publication of the American Society of Civil Engineers (ASCE). But a search of the ASCE archives revealed that the paper had not in fact been published.

In a recent series of articles I mentioned this discrepancy and documented some of the efforts made by myself and others to obtain further information on this story -- all of which had been ignored.

A few people who read those stories had academic credentials and were able to provoke a response where humble bloggers had failed: the questions they sent to Ross Corotis, the editor of the JEM, were answered, and yesterday I was able to report that Keith Seffen's paper is scheduled to be published in the February, 2008 issue of the Journal of Engineering Mechanics.

If that was good news, this is even better: one of the other people who read those stories had even more impressive credentials, and he went digging in a different place. And you'll never guess what he managed to unearth ... or will you?

Listen: It's a lot like blowing up a balloon -- a long, thin one. It takes a lot of pressure to get it started, but once it gets going, the rest is easy.

Progressive Collapse of the World Trade Centre: a Simple Analysis

K. A. Seffen

T: +44 1223 764137; F: +44 1223 332662; E: kas14@cam.ac.uk

Abstract

The collapse behaviour of the World Trade Centre (WTC) towers is considered formally as a propagating instability phenomenon. The application of associated concepts enables the residual capacities of both towers after the onset of collapse to be formally estimated. This information is combined into a simplified variable-mass collapse model of the overall dynamical behaviour. The resulting, non-linear governing equation of motion can be solved in closed form, to yield compact information about the overall collapse conditions.

Keywords: progressive failure, residual strength, dynamic analysis

Introduction

The collapse of the World Trade Centre (WTC) towers on 11 September 2001 was a devastating, catastrophic event. Those factors responsible for the onset of collapse are now well established. Despite localised and substantial horizontal impacts by fuel-laden aircraft, both towers survived until the intense fire compromised the ability of the remaining, in-tact columns close to the aircraft impact zones to sustain the weight of the buildings above them. The subsequent near free-falling of these upper parts over the height of just one storey resulted in dynamical “over-loading” of the relatively undamaged lower columns by a factor of 30 compared to their static load capacity, according to Bazant and Zhou (2002). They argue that the storey immediately below bears the brunt in terms of a localised, plastic buckling of its columns, and they show that the commensurate dissipation cannot arrest the motion of the falling part, leading to a sustained collapse.

This paper examines the collapse sequence by referring the behaviour to concepts familiar in studies on propagating instabilities. Such studies usually deal with progressive collapse of structures, where damage accrues in a prescribed fashion following an initiation phase. Depending on the local collapse behaviour inveigled by the instability sweeping through the initially undamaged structure, it becomes possible to ascertain the level of loading required to sustain its propagation, or conversely, to quantify the ability of the structure to resist or comply with collapse, thereby defining its “residual capacity”.

In the case of the WTC towers, it is clear that the initial loads imposed by both parts falling onto the undamaged buildings beneath were exceptionally high due to the unforeseen preceding events, and that damage was bound to propagate into the floors below: this is the initiation phase. It is also clear that both collapse modes were progressive, as indicated by film footage: there was the sound of each successive impact of floor upon floor and a matching sequence of lateral ejection of debris. Therefore, it is valid to consider the behaviour formally in the proposed terms, and in doing so, the aims of this paper are twofold.

First, this paper aims to show that progressive collapse confers a substantial reduction in the performance of the undamaged building compared to its static strength after the onset of failure. Other, insightful studies on the WTC towers have estimated this capacity by informal methods, usually by comparing the rate of energy dissipated by the collapsing members to the rates at which the falling parts acquire kinetic energy and lose gravitational potential energy. In terms of progressive behaviour, the attention is confined to the localised collapse of a given storey, which confirms for both WTC towers that there was insufficient residual capacity to arrest this particular type of collapse mode (Bazant and Zhou 2002). However, the link to progressive collapse is improperly asserted by claiming that, because each storey locally collapses in an unstable manner, successive storeys are bound to fail sequentially (Zhou and Yu 2004). This claim is partially true, but it must also account for the transfer of loading between storeys, which is defined by the final stages of storey collapse. By implementing a simple interpretation here, it becomes rather straightforward to compute the residual capacity of the undamaged building in the proper sense of progressive collapse.

The second aim is to formulate a compact dynamical model of the progressive collapse of the overall building. Even though at any time, the building falls by the columns failing discretely but uniformly within a single storey, a propagating instability viewpoint ensures that the behaviour is independent of the particular snap-shot of current deformation. Accordingly, the assumption of progressive collapse enables a continuum viewpoint, which permits a simpler formulation compared to, say, a finite element analysis. Moreover, a closed-form solution becomes available here, which imparts essential transparency to the conditions governing collapse progression: conversely, it is possible to elucidate the conditions required for arresting dynamical collapse, in view of the design and/or retro-fitting (Newland and Cebon 2002) of safer multi-storey buildings.

Propagating Instabilities

Propagating instabilities feature in structures that are failing progressively. A highly deformed, localised region —the instability— is driven along the structure from the site of first damage, often at a load value below the damaging threshold. They are observed, for example, in the plastic collapse of long pipelines (Kyriakides 1994), where a discrete indentation can spread along the entire length. A frivolous but useful analogue is the inflation of a rubber party balloon: personal experience suggests that a higher lung pressure is required to motivate inflation than that needed to sustain it; and that the latter pressure is approximately constant and steady for a long, uniform balloon, irrespective of the inflated volume before becoming fully inflated. These two pressures can be identified in Fig. 1 from the typical inflation response for the overall balloon. The initial over-pressure is denoted as Pmax, and the steady-state propagation pressure is labelled P∗. The peak pressure is identically and more simply calculated from the uniform expansion of a cylindrical segment of balloon. The complete response of this segment is also shown in Fig. 1 for a typical non-linear constitutive behaviour belonging to rubber materials. The large changes in geometry also ensure that, beyond the peak value, the overall response is highly non-linear. The precise variation does not matter, but it must exhibit a characteristic up-down-up profile, for this enables two areas, A1 and A2, to be respectively enclosed above and below the horizontal line of P∗, as illustrated. The purpose in doing so, as detailed originally by Chater and Hutchinson (1984), leads to the Maxwell Construction, where the exact value of P∗ equates A1 and A2. The underlying hypothesis expresses the equality of two energetic processes: the work done by the pressure in expanding the segment of balloon from a volume VU to VD (equal to the area under the local pressure curve between the same limits) and the steady-state effort of the propagating front absorbing the same expansion under constant P∗ as the bulge front moves along the balloon.

There's a lot more; it's very technical and doesn't render well in HTML. So I won't post it here.

But I have the entire paper (in PDF format), and I'm willing to share.

Click here to download Keith Seffen's paper, "Progressive Collapse of the World Trade Centre: a Simple Analysis" (PDF)