The 9/11 Forum

In this thread I'd like to take a look at the mechanics of upper block tilting, approaching it from several directions:

- observational
- analytical
- computational

The focus initially (perhaps almost exclusively) will be on the tilting of the so-called upper block of WTC1. There is speculation as to how rigid this block was over time, and some observations to suggest it was mostly disintegrated by the time it had dropped its own height. The question of how long the block remained structurally sound following measurable initiation, or whether it was already unsound going into this interval, is naturally integral to this topic.

Of what value are idealized tilting models? The answer is the same for any other simplified modeling of the engineering mechanics: to discover basic mechanical principles relating to the problem and find any bounding conditions present. The simple models will not represent the actual but, if you don't know what a simple model can do, what can you say about the real thing?

Though this is a narrow topic compared to the whole of collapse, it's quite complex in its own right. There are many aspects to explore over time, and it will be pure stream of conciousness. Others are welcome to contribute.

The generally accepted extreme idealization of the tilt of WTC1 can be described as follows:

- upper block tilts as rigid body about a fixed axis (one degree of freedom)
- hinge is located on the plane of the north wall at or about story 98
- tilting is directly southward

None of these are expected to be true and indeed all have been reasonably demonstrated to be false. Yet it is in practice a pretty good first approximation and a starting point for understanding more complex scenarios. Here the motion of any point of the upper block is contained in a plane and the problem may be considered in a 2D space.

This graphic illustrates an example of 7 degree tilt under these assumptions, along with rays (originating off to the right) from the estimated Sauret camera location to measurement locations on the tower.




Some recent high resolution measurements indicate the upper block did not behave as a rigid body during the early phase of tilting. Personally, I currently take the evidence as provisional, but not refuted.

On one hand, the non-uniform and dynamically decreasing capacity of the supporting lower section could be expected to lead to deformation of the upper block, of some degree. If this deviation from true rigidity could be mapped with any accuracy, it may provide clues as to the load redistribution and failure sequence over time.

On the other hand, some support the idea that the upper block rotated more or less as a rigid body, regardless of whether the other provisions above hold true. The hinge (if it can be called that) location was probably a dynamic quantity and the rotation in two degrees of freedom, and these are used to suggest that measurements consistent with deformation can be explained via these other means. Maybe a slight amount of deformation is allowable in such a scheme, but seemingly not to extent of slumping in the core - for some reason.

The proper starting point for a very simple model would be an analytic treatment which develops an equation of motion for a simple planar system, using area moment of inertia and a pair of forces acting on a rectangle with one corner constrained to a point. I don't feel like doing that now but I will reserve this space to revisit it later in case the motivation strikes.

The subject of reaction force came up recently in relation to the (early) tilt of WTC1. The response of the lower section to the tilting of the upper block was mentioned as a possible explanation for, or influential factor in, the apparent deficit of motion observed in the NW corner as compared to the antenna in select measurements. Undoubtedly, there is a response in the lower section which will act to displace the hinge location towards the camera.

Intuitively, since the tower below is more massive on the whole and is coupled to ground, the amount of displacement towards the camera at the hinge would be less than that of the roofline away from the camera, regardless of cause, otherwise the roofline would stay in one horizontal location as the building buckled below, not observed. If the cause is force exerted on the hinge by the upper block in opposition to the horizontal component of tilt motion, even more so.

Still, it's an interesting question to ponder, so I did. This is the first subject to be examined.

The first experiment is to examine the action/reaction influence on hinge horizontal position. It uses a 2D mass-spring simulation with a grid of equal masses arranged in a rectangular lattice as a crude approximation to the upper block. The masses are connected to nearest neighbors in both dimensions by translational springs and rotational springs are supplied at each mass node. No regard is given for the scale of mass, only for the length dimension and the distribution of mass. The lower left corner of the block (or sheet) is constrained against translation so that it acts as a hinge, while the sheet is allowed to rotate about the corner point under the influence of gravity and optional supporting force (ensemble). Time begins with a level upper block being released.


Two types of trials are conducted:

1) The hinge is constrained against translation in both the x and y directions
2) The hinge is constrained against translation in only the y direction

Both cases allow free rotation about the hinge point, any retarding moment is supplied by optional support forces, if any. In this model, there is no lower section. The attribute of providing a hinge, either fixed or free to move horizontally, is supplied by way of a constraint rule on the solver. The attribute of support or more generally resistance to motion is supplied by external forces defined to act on a point in a particular direction, which need not be constant.

The first set of experiments omit external forces except gravity. This is the most elementary starting point but also leads to large angular acceleration, so there's no need to run long to see what's happening. The gravity-only sims run for one second.

The arrangement at t=t0 and t=1s is shown in the images below:



The model shown here is 'weak' in terms of ratio of spring constant to density. The purpose was to exaggerate the distribution of stress (color) experienced by the sheet as it rotated freely about the hinge and to be a graphic reminder that, in this first series of simulations, the upper block is pretty stiff, but it is by no means rigid. Subsequent models generating the data presented next were stiffened up considerably from this example but do display multi-modal oscillation. No damping is used, except as otherwise noted.

Future simulations will include truly rigid bodies, among other things.

The extreme hinge idealization has the hinge fixed in space. Realistically, the hinge can move in both the x and y dimensions as the building flexes below. The hinge is not free to move in an unconstrained fashion; by definition it is the attach point to the building therefore must be wherever the top of the lower section's north wall is. There may be unloading at the hinge plane resulting in a slight increase of hinge elevation, and there's bound to be some sway in the opposite direction from the tilt as a reaction.

What the actual motion would be over time would be very hard to determine by any means, but there is a way to place bounds on the action with a simple simulation. In the 2D simulation space, a lower corner can be constrained to remain in the same location as the block rotates under the influence of gravity, or can be free to move in the horizontal direction only. The first, XY-constrained, represents the idealization perfectly. The second, Y-constrained, represents the maximum possible reaction influence: the hinge is totally free to move in the direction opposite of the 'roofline' and center of mass motion.

This last is essentially a 'hinge on ice'. It can't rise or fall but it can and will slide side to side. In the real tower, there is (presumably) a very solid connection to a much greater mass stiffly coupled to ground. It will provide significant resistance to deformation and thus horizontal translation. The real situation must always be a reaction deflection less than the Y-constrained simulation because the latter is zero resistance.

Moreover, this first pair of runs uses no support force - the block swings around the hinge as if there were empty space underneath. Since the real tilt was more of a quasi-static affair, the free swing represents a bounding case for maximum impulse directed horizontally at the hinge. The real initiation which took place slowly over a much longer interval of time would necessarily generate less horizontal deflection than this sim.

Therefore it is seen that the models about to be presented are limiting cases for the reaction displacement in the simplified scenario.

Unsurprisingly, the hinge moved horizontally in reaction to the tilt, and this influenced the coordinate of the roofline corner, but not a lot.

Here is a plot comparing the horizontal deflections versus time. The horizontal axis runs 0 - 1 second of simulation time and the vertical axis is meters of displacement in the horizontal dimension. Blue is XY-constrained, red is Y-constrained. (I will only once in this thread beg pardon for using the default graph format of the spreadsheet program, which does no labeling of axes. I know labels are very very important to some people but life is short and those people are not paying me to do this, so I reserve the right to describe graphs in the accompanying text. Thank you.)



Thus we see that a roofline which is tilting in a given direction will in fact move in that direction even if the hinge is free to move. Extraordinary.*

The corner above the ideal fixed hinge moves 3.5m laterally in the direction of tilt where the corner above the sliding hinge only moves 2.5m, also in the direction of tilt. This is about a 30% reduction, but certainly represents the greatest possible deflection since nothing resists it. It also neglects lean in the direction of tilt which may develop in a very slow rotation, due to increasingly eccentric loading. Worst case - 30%. Realistically, who knows? I'd guess less than half that at the outside.

Reaction force cannot account for an apparently stationary corner as viewed from the north, in this grossly simplified model. Except for uniform density, there's every reason to expect a tower to exhibit similar properties under tilt.


* footnote for dumbshits who don't recognize sarcasm when they see it - you're looking at it; the results are ANYTHING BUT extraordinary.

Some context is required.

The simulations run for only 1 second but encompass about 3-4 degrees of tilt since there's nothing holding the block except the hinge. The vertical displacement in both cases is quite small, naturally, since the vertical motion is nil until an appreciable angle is obtained. The full range of tilt in the simulations are still effectively small angles. Maximum displacement is about 22cm for the sliding hinge:



Both drop a much smaller amount than move laterally, and take a considerable time to show any drop at all. The sliding hinge drops faster, though, and even this small displacement can be picked up on the Sauret video. So, while the sliding hinge will give less horizontal motion, it produces a correspondingly greater drop. Both dimensions project onto the image plane of the Sauret video. Later I'll map these roofline coordinates to apparent elevation angle as viewed by the Sauret camera.


Results for the two trials are still being processed. If anything particularly interesting comes up, I'll report it.

In the meantime, the initial conditions for the next series of simulations are being 'cooked'. These will have external forces applied to the bottom elements of the block, directed upward, intended to function as support or resistance to tilting. Because the structure is stiff but not exactly rigid, it oscillates when released. The bounciness on release may as well be eliminated for the next round. Better to start with a model which has settled with damping to stable static equilibrium with the entire bottom row XY-constrained. It's running now, a little over five seconds in.

That's all for now.

Jumping ahead...

The next configuration is with a set of forces added to the bottom to give resistance to tilting. These are shown graphically:



These forces will remain constant throughout the trial, both magnitude and direction. This is not very realistic but one step at a time. It's more realistic than a "follower" force which maintains its orthogonality with the bottom row as tilt progresses. In that scheme, the resistive force against angular motion is constant. With this, the component of force acting about the hinge decreases as the tilt increases. Gravitational force acting on the body will have an increasing component as tilt increases, in these ranges.

If the sum of forces equals 32/33 the total load (one corner, 1/33th of the mass, is supported by the fixed hinge), the block stays in place as it should. Reduce these forces and it tilts.

This is tilt angle versus time for initial "capacity" slightly greater than 95% of load. That is, all 32 forces were reduced to 0.95 times the value necessary to maintain static equilibrium with the hinge.



I'm surprised how fast the tilt is. If the vertical pixel displacement in Sauret were interpreted as tilt angle, this is much faster by comparison. The next run, at 99% of DCR 1, is in progress.

It is curious. The component of resistive force decreases as the cosine of the tilt angle, so does not diminish appreciably in the early tilt. The torque induced by gravitation increases (roughly) 5% max. How does it fall so fast? Why am I surprised? Analytical treatment required.

The previous graph with zero support shows 15cm y deflection at 1s, the 95% run has less than a millimeter at 1s and 31cm at 5s. That is a lot faster...

There are two torques acting on the block.


Due to gravity:
τ_g = mgrcos(θ - ψ)
where ψ = atan(24/32) = 36.87 degrees = 0.6435 rad
and r = 40 meters
=>
τ_g = 40mgcos(θ - ψ)


Due to support:
τ_s = -(m - dm)g(r + dr)cos(θ)
=>
τ_s = -(32/33)mg(33)cos(θ) = -32mgcos(θ)


Sum of torques:
τ = τ_s + τ_g = -32mgcos(θ) + 40mgcos(θ - ψ)
= 8mg(5cos(θ - ψ) - 4cos(θ))

This must be zero at θ = 0 and it is.

However, the imbalance between the two climbs sharply as theta increases.

τ = mg(40cos(θ - 0.6435)-32cos(θ))

at 5 degrees, the net torque is about (2 meters)*mg. The initial supporting torque was only (32 meters)*mg. This is 1/16th the supporting torque before the tilt, acting to make it tilt - with a support torque at 100% capacity with no tilt.

Obviously, this angular dependent loss of support in my scheme is on top of any reduction in capacity specified. So, the 95% capacity is somewhat misleading. At exactly 100% capacity, it will keep tilting once started and accelerate due to geometric considerations until bottom dead center (hanging diagonal).

That's why it seemed a little too fast as full capacity was approached. Longer lever arm for gravity acting on the center of mass.

The 99% initial DCR is done. Rotates about 2.6 degrees in 5 seconds. Getting there. It's a little hard to get a slow tilt...

...without just contriving a tangential force ever so slightly less than that required to maintain equilibrium.

This shows the geometry and how the support forces go non-orthogonal as tilt progresses:

OneWhiteEye wrote:That's why it seemed a little too fast as full capacity was approached. Longer lever arm for gravity acting on the center of mass.

This seems to be the first fundamental result, if a bit obvious. A block will not tilt unless the support torque drops below supporting capacity. A block which has tilted will display increasing imbalance of torque as the tilt progresses, all things being equal. Since the primary supports are designed to handle load while plumb, their load carrying capacity must decrease in early tilt. Therefore, small tilts rapidly turn into larger tilts if the hinge holds.

All in all, tilting on a wall hinge is a snowballing process. Once begun, it's going to accelerate unless some new force acts to retard it. In the actual collapse, this could be load distribution to intact floors below, lateral force from the core, whatever. Even at 0.99 DCR, barely enough to get the tilt started, the block tilts more than 2.5 degrees in 5 seconds.

Of course, this does not account for dynamic redistribution of loads and local overloading, nor does it factor in a distribution of load-displacements with peaks above and below demand. It's just a global average of sorts. The next pass may do a better job of simulating the characteristics of a failing lower structure.