Myriad
7th January 2008, 05:55 PM
In recent quasi-technical discussion threads in this forum, especially with Heiwa, realcddeal, and Sizzler, I've noticed that despite the frequent repetition of the phrase "progressive collapse" in discussion of the collapses of the World Trade Center towers, many people are not really clear on what the phrase really means.
The problem is, it appears obvious. "Progressive" means it happened in stages, right? The falling upper block destroys one floor, then the next floor down, and so forth. As the saying goes, clunkity-clunk, clunkity-clunk, clunkity-clunk.
And as far as it goes, that's true. A progressive collapse or -- as I'm going to generalize the concept here, a progressive failure -- happens in stages.
But that doesn't capture the full meaning of the phrase. If that's all it meant, just about every failure would be a progressive failure, and just about everything that happens would be "progressive." (The only exceptions I can think of are certain quantum state changes.) I'm progressively writing this post that you'll later be progressively reading, earlier I ate a progressive lunch, as my TV showed me progressive commercials. "Progressive collapse" would be redundant and we might as well just say "collapse."
What's missing from that partial definition is the nature of the causal connection between the successive stages of the event. So, here's my definition of what makes for a true "progressive" failure: a progressive failure is a failure that happens in stages, in which the effects of earlier stages are necessary to make the subsequent stages possible.
Let's look at some examples.
Example 1: Elevator
Imagine an elevator suspended from four cables. Each cable has a breaking stress of 5 tons.
Scenario A: A load of 30 tons is placed in the elevator. That load exceeds the total capacity of the cables, and the cables break. The elevator falls.
Scenario B: The elevator is carrying a load of 8 tons. An axle breaks in the winch mechanism over which the cables run, and the cable guide wheels fall out of place. This introduces different amounts of slack into three of the four cables. The cable that did not get additional slack is now carrying most or all of the load. As the load of 8 tons exceeds its capacity of 5 tons, it breaks. That shifts all or most of the load to the next cable, the one with the least slack. It breaks too, for the same reason. The process continues for the remaining two cables, and the elevator falls.
Example 2: The Titanic
The Titanic is divided by bulkheads into a series of watertight compartments from fore to aft. These bulkheads extend to well above the waterline (but not to the very top of the hull). As long as just some of the compartments remain filled with air, the ship has enough buoyancy to float. This makes the ship "unsinkable."
Scenario A: The Titanic collides with an iceberg, opening a crack along the entire length of its keel. Since there is no way to prevent every compartment in the ship from flooding, it will fill up with water and sink.
Scenario B: The Titanic collides with an iceberg, opening a crack along about the forward third of the length of the ship. This causes five of the watertight compartments in the bow portion of the ship to fill. Even with those compartments filled with water, the ship should have enough buoyancy to stay afloat (there were sixteen compartments in all). Unfortunately, all the buoyancy is at the stern end, so the ship cannot stay level. The flooded compartments weigh the bow of the ship down just enough that water can spill over the top of the bulkheads between the aftmost ruptured compartment and the foremost intact one. This causes that next compartment to fill with water, further weighing down the bow so that the water can now spill over into the next compartment back, and so on until Kate and Leo end up in the drink.
Example 3: Column Loads
We have a load-bearing wall made of 9 horizontal beams supported by 10 columns. A load of 100 tons is bearing on the horizontal beam(s) directly above each column (so the total load on the wall is 1000 tons). To break a column requires a load of 210 tons. The connections between the beams and the columns are diagonally braced at the corners, making them unusually rigid.
Scenario A: A sudden additional load of 2000 tons falls onto the wall. As this overloads every column, the whole wall collapses.
Scenario B: A truck crashes into, and breaks, one of the end columns (call it column 1). The load formerly supported by column 1 is now transferred to the adjacent column 2. Furthermore, the rigidity of the beam connections to column 2 means that the beams between column 1 and column 3 act as a lever; the load on the column 1 end acts on the lever with column 2 as a fulcrum, relieving part (let's say, half) of the load on column 3 and transferring that force to column 2 as well. Column 2's load is now greater than 210 tons, and it fails. Now column 3 is under the same kind of extra load, so it fails, and so forth until all the columns have failed and the whole wall collapses.
-----
In each case, Scenario A represents a failure that I would not consider "progressive" -- even though if we observed them closely, we would see for instance that in scenario 2A some of the compartments on the Titanic fill faster than others and (if we had high-speed film of the other two A scenarios) some cables and some columns fail before others do. But in all those cases, each "stage" of failure individually does not depend on other failures having already occurred.
In the B scenarios, the situation is quite different. These are what I would consider truly progressive failures. Each starts with only partial failure, and total failure only occurs because each increment of partial failure leads to further partial failure. They all have a perverse "chain reaction" or "oh, crap" quality to them. They all look, at first glance, as though they should not have happened. Four cables "should" be able to support a load that normally any 2 of them could support. A ship that still has eleven of sixteen intact compartments "should not" sink. A wall with 9 fully intact columns, in which each column can support more than one fifth of the total load, "should not" collapse. When you look at the whole event, it's tempting to think that there must be some point where the cascade of events toward disaster could have, and should have, just stopped happening. The nature of progressive failure situations is that no such stopping point exists.
Though I'm mostly discussing engineering failures here, the principles apply more generally than that. Some market crashes are progressive collapses of a sort. One small sale or small piece of bad news "should not" cause the value of a commodity to plummet. But if the traders in that commodity are excessively focussed one another's actions, so that each sale that takes place convinces another trader to also sell, it might.
Here are some observations about progressive failures:
1. (To repeat the previous paragraph): Progressive failures, when not thoroughly analyzed or understood, look like they should not have happened.
2. Progressive failure scenarios often involve unbalanced forces. This, to a large extent, is the reason behind observation 1. If we only look a the average forces, we can see no sensible reason for failure. The average buoyancy of the Titanic is more than adequate to keep her afloat. The average tension on the elevator cables is well within their capacity. The average load on the columns is only slightly greater than normal, and well within their "safety margins." What those simplistic assessments overlook is that in a progressive failure scenario, stress way above the average gets concentrated on the next point of failure. Averages go out the window.
Another example outside the realm of engineering: many battlefield tactics are based on attempting to cause a progressive failure of the enemy's positions. A smaller force "should not" be able to rout a larger stronger one, and would not, if they met head-on. But a smaller weaker force that turns an enemy's flank and rolls up their line can do so, by concentrating force in an unbalanced way that makes the "average" strengths of the two forces irrelevant.
3. Progressive failures can happen very quickly -- except in the movies. In the movies, progressive failure is almost always depicted as happening slowly, with long time intervals between each stage. The only reason for that is that it's more dramatic that way. The classic example is the rope that slowly breaks, one strand parting at a time, as the hero hanging from it races to get to the top of the cliff. Eventually, the rope is down to a single thin strand -- but that holds just long enough (sometimes, to compound the silliness, not breaking until a moment after the hero lets go of it). In reality, if that last strand was enough to hold the hero's weight for twenty seconds, the rope would not have broken in the first place.
In reality, the elevator cables in Scenario 1 would not pause for dramatic effect after each one failed. They would break in very rapid succession. To the unaided eye they would look like they all broke at once. The same for the columns in example 3: the force of the unbalanced overload transfers through the structure at the speed of sound in whatever material they're made of. It's quite possible that the columns would all fail within a fraction of a second of the vehicle impact that took out the first one, and the whole wall could collapse nearly straight down.
The only exception in the examples is the Titanic scenario -- and note that in that case, the non-progressive "A" scenario would also happen gradually. In general, a progressive failure is likely to happen at about the same speed as a straightforward failure of the same system caused by a massive overload.
4. Progressive failures can sometimes be prevented, and not just through brute-force means. In the elevator example, the progressive failure scenario would be prevented by making each cable strong enough to bear the entire load, but also by making the cables more elastic. In the wall example, the progressive failure scenario would be prevented by making each column stronger, but also by making the horizontal beams weaker. Higher watertight compartments would have prevented the Titanic sinking. (I've sometimes wondered if the Titanic's sinking could have been arrested or slowed by deliberately flooding the two compartments farthest aft.) Of course, these solutions might have drawbacks as well (such as bouncy elevator rides, reduced nominal load-bearing capacity for the wall, and cramped grand promenade decks). Progressive failure is only one of many kinds of failure possibility that engineers have to weigh when designing safety factors. Also, the failure scenario must be anticipated in the first place. That can be the hard part.
Please keep these points in mind if you're asking, or are being asked, questions like these:
If all the damage was on one side why did the building fall straight down?
The mass of the upper structure was X, the total capacity of the remaining undamaged columns was Y (>X), so how could collapse initiation have occurred?
How likely is it that all the columns along a tower wall could have failed at once?
Why didn't safety factors in the strength of the structure cause the collapse to stop after a few floors?
Why don't we see the collapse happening in stages? (That is, why didn't the process pause for dramatic effect after each major support column gave way, like in the movies?)
If collapse was inevitable why do you say engineers study it to improve future buildings?
----------
I have one question left to explore, which is were the collapses of the WTC towers really progressive collapses by the more specific (non-redundant) definition I've applied here? It's pretty academic, but I'm curious.
It's clear that collapse initiation for both towers, and for WTC7 as well, were true progressive failures. In those cases the "progression" of the progressive failure was horizontal, just like in my example 3B. But what about the other "progression," of the towers to the ground?
The answer appears to be, we don't know for sure. If at the start of the collapse there was enough potential energy in the moving upper blocks to destroy all of the floors below, without taking into account the mass of the floors destroyed along the way, then I would actually not call the collapses truly progressive. In that case the destruction of, say, floor 10 was not dependent on the destruction of the floors above it, except in the trivial sense that the floors above it were in the way and would inevitably have been destroyed first. (An analogy would be, in example 3A, a train moving parallel to the wall crashing through all the columns, taking them out one at a time in a definite sequence, but not really representing a progressive failure by my more specific definition).
If, however, the increase in falling mass from the floors destroyed along the way is necessary to explain the destruction of the lower floors all the way to the ground, then the collapses were progressive by any definition. In that case it was the effects of the earlier stages of the collapse itself that created the condition that allowed the collapse to continue -- specifically, the increased falling mass.
The various calculations published by both sides in the dispute are not (and probably cannot be) precise enough to settle this (admittedly minor) issue decisively. It's even possible that the collapse of tower 1 was progressive while tower 2, with its larger initial falling mass, was more like one big drawn-out sledgehammer blow.
In any case, I hope I've made my case that there's more to the concept of progressive collapse than the trivial observation that a building collapsed one floor at a time.
Respectfully,
Myriad
The problem is, it appears obvious. "Progressive" means it happened in stages, right? The falling upper block destroys one floor, then the next floor down, and so forth. As the saying goes, clunkity-clunk, clunkity-clunk, clunkity-clunk.
And as far as it goes, that's true. A progressive collapse or -- as I'm going to generalize the concept here, a progressive failure -- happens in stages.
But that doesn't capture the full meaning of the phrase. If that's all it meant, just about every failure would be a progressive failure, and just about everything that happens would be "progressive." (The only exceptions I can think of are certain quantum state changes.) I'm progressively writing this post that you'll later be progressively reading, earlier I ate a progressive lunch, as my TV showed me progressive commercials. "Progressive collapse" would be redundant and we might as well just say "collapse."
What's missing from that partial definition is the nature of the causal connection between the successive stages of the event. So, here's my definition of what makes for a true "progressive" failure: a progressive failure is a failure that happens in stages, in which the effects of earlier stages are necessary to make the subsequent stages possible.
Let's look at some examples.
Example 1: Elevator
Imagine an elevator suspended from four cables. Each cable has a breaking stress of 5 tons.
Scenario A: A load of 30 tons is placed in the elevator. That load exceeds the total capacity of the cables, and the cables break. The elevator falls.
Scenario B: The elevator is carrying a load of 8 tons. An axle breaks in the winch mechanism over which the cables run, and the cable guide wheels fall out of place. This introduces different amounts of slack into three of the four cables. The cable that did not get additional slack is now carrying most or all of the load. As the load of 8 tons exceeds its capacity of 5 tons, it breaks. That shifts all or most of the load to the next cable, the one with the least slack. It breaks too, for the same reason. The process continues for the remaining two cables, and the elevator falls.
Example 2: The Titanic
The Titanic is divided by bulkheads into a series of watertight compartments from fore to aft. These bulkheads extend to well above the waterline (but not to the very top of the hull). As long as just some of the compartments remain filled with air, the ship has enough buoyancy to float. This makes the ship "unsinkable."
Scenario A: The Titanic collides with an iceberg, opening a crack along the entire length of its keel. Since there is no way to prevent every compartment in the ship from flooding, it will fill up with water and sink.
Scenario B: The Titanic collides with an iceberg, opening a crack along about the forward third of the length of the ship. This causes five of the watertight compartments in the bow portion of the ship to fill. Even with those compartments filled with water, the ship should have enough buoyancy to stay afloat (there were sixteen compartments in all). Unfortunately, all the buoyancy is at the stern end, so the ship cannot stay level. The flooded compartments weigh the bow of the ship down just enough that water can spill over the top of the bulkheads between the aftmost ruptured compartment and the foremost intact one. This causes that next compartment to fill with water, further weighing down the bow so that the water can now spill over into the next compartment back, and so on until Kate and Leo end up in the drink.
Example 3: Column Loads
We have a load-bearing wall made of 9 horizontal beams supported by 10 columns. A load of 100 tons is bearing on the horizontal beam(s) directly above each column (so the total load on the wall is 1000 tons). To break a column requires a load of 210 tons. The connections between the beams and the columns are diagonally braced at the corners, making them unusually rigid.
Scenario A: A sudden additional load of 2000 tons falls onto the wall. As this overloads every column, the whole wall collapses.
Scenario B: A truck crashes into, and breaks, one of the end columns (call it column 1). The load formerly supported by column 1 is now transferred to the adjacent column 2. Furthermore, the rigidity of the beam connections to column 2 means that the beams between column 1 and column 3 act as a lever; the load on the column 1 end acts on the lever with column 2 as a fulcrum, relieving part (let's say, half) of the load on column 3 and transferring that force to column 2 as well. Column 2's load is now greater than 210 tons, and it fails. Now column 3 is under the same kind of extra load, so it fails, and so forth until all the columns have failed and the whole wall collapses.
-----
In each case, Scenario A represents a failure that I would not consider "progressive" -- even though if we observed them closely, we would see for instance that in scenario 2A some of the compartments on the Titanic fill faster than others and (if we had high-speed film of the other two A scenarios) some cables and some columns fail before others do. But in all those cases, each "stage" of failure individually does not depend on other failures having already occurred.
In the B scenarios, the situation is quite different. These are what I would consider truly progressive failures. Each starts with only partial failure, and total failure only occurs because each increment of partial failure leads to further partial failure. They all have a perverse "chain reaction" or "oh, crap" quality to them. They all look, at first glance, as though they should not have happened. Four cables "should" be able to support a load that normally any 2 of them could support. A ship that still has eleven of sixteen intact compartments "should not" sink. A wall with 9 fully intact columns, in which each column can support more than one fifth of the total load, "should not" collapse. When you look at the whole event, it's tempting to think that there must be some point where the cascade of events toward disaster could have, and should have, just stopped happening. The nature of progressive failure situations is that no such stopping point exists.
Though I'm mostly discussing engineering failures here, the principles apply more generally than that. Some market crashes are progressive collapses of a sort. One small sale or small piece of bad news "should not" cause the value of a commodity to plummet. But if the traders in that commodity are excessively focussed one another's actions, so that each sale that takes place convinces another trader to also sell, it might.
Here are some observations about progressive failures:
1. (To repeat the previous paragraph): Progressive failures, when not thoroughly analyzed or understood, look like they should not have happened.
2. Progressive failure scenarios often involve unbalanced forces. This, to a large extent, is the reason behind observation 1. If we only look a the average forces, we can see no sensible reason for failure. The average buoyancy of the Titanic is more than adequate to keep her afloat. The average tension on the elevator cables is well within their capacity. The average load on the columns is only slightly greater than normal, and well within their "safety margins." What those simplistic assessments overlook is that in a progressive failure scenario, stress way above the average gets concentrated on the next point of failure. Averages go out the window.
Another example outside the realm of engineering: many battlefield tactics are based on attempting to cause a progressive failure of the enemy's positions. A smaller force "should not" be able to rout a larger stronger one, and would not, if they met head-on. But a smaller weaker force that turns an enemy's flank and rolls up their line can do so, by concentrating force in an unbalanced way that makes the "average" strengths of the two forces irrelevant.
3. Progressive failures can happen very quickly -- except in the movies. In the movies, progressive failure is almost always depicted as happening slowly, with long time intervals between each stage. The only reason for that is that it's more dramatic that way. The classic example is the rope that slowly breaks, one strand parting at a time, as the hero hanging from it races to get to the top of the cliff. Eventually, the rope is down to a single thin strand -- but that holds just long enough (sometimes, to compound the silliness, not breaking until a moment after the hero lets go of it). In reality, if that last strand was enough to hold the hero's weight for twenty seconds, the rope would not have broken in the first place.
In reality, the elevator cables in Scenario 1 would not pause for dramatic effect after each one failed. They would break in very rapid succession. To the unaided eye they would look like they all broke at once. The same for the columns in example 3: the force of the unbalanced overload transfers through the structure at the speed of sound in whatever material they're made of. It's quite possible that the columns would all fail within a fraction of a second of the vehicle impact that took out the first one, and the whole wall could collapse nearly straight down.
The only exception in the examples is the Titanic scenario -- and note that in that case, the non-progressive "A" scenario would also happen gradually. In general, a progressive failure is likely to happen at about the same speed as a straightforward failure of the same system caused by a massive overload.
4. Progressive failures can sometimes be prevented, and not just through brute-force means. In the elevator example, the progressive failure scenario would be prevented by making each cable strong enough to bear the entire load, but also by making the cables more elastic. In the wall example, the progressive failure scenario would be prevented by making each column stronger, but also by making the horizontal beams weaker. Higher watertight compartments would have prevented the Titanic sinking. (I've sometimes wondered if the Titanic's sinking could have been arrested or slowed by deliberately flooding the two compartments farthest aft.) Of course, these solutions might have drawbacks as well (such as bouncy elevator rides, reduced nominal load-bearing capacity for the wall, and cramped grand promenade decks). Progressive failure is only one of many kinds of failure possibility that engineers have to weigh when designing safety factors. Also, the failure scenario must be anticipated in the first place. That can be the hard part.
Please keep these points in mind if you're asking, or are being asked, questions like these:
If all the damage was on one side why did the building fall straight down?
The mass of the upper structure was X, the total capacity of the remaining undamaged columns was Y (>X), so how could collapse initiation have occurred?
How likely is it that all the columns along a tower wall could have failed at once?
Why didn't safety factors in the strength of the structure cause the collapse to stop after a few floors?
Why don't we see the collapse happening in stages? (That is, why didn't the process pause for dramatic effect after each major support column gave way, like in the movies?)
If collapse was inevitable why do you say engineers study it to improve future buildings?
----------
I have one question left to explore, which is were the collapses of the WTC towers really progressive collapses by the more specific (non-redundant) definition I've applied here? It's pretty academic, but I'm curious.
It's clear that collapse initiation for both towers, and for WTC7 as well, were true progressive failures. In those cases the "progression" of the progressive failure was horizontal, just like in my example 3B. But what about the other "progression," of the towers to the ground?
The answer appears to be, we don't know for sure. If at the start of the collapse there was enough potential energy in the moving upper blocks to destroy all of the floors below, without taking into account the mass of the floors destroyed along the way, then I would actually not call the collapses truly progressive. In that case the destruction of, say, floor 10 was not dependent on the destruction of the floors above it, except in the trivial sense that the floors above it were in the way and would inevitably have been destroyed first. (An analogy would be, in example 3A, a train moving parallel to the wall crashing through all the columns, taking them out one at a time in a definite sequence, but not really representing a progressive failure by my more specific definition).
If, however, the increase in falling mass from the floors destroyed along the way is necessary to explain the destruction of the lower floors all the way to the ground, then the collapses were progressive by any definition. In that case it was the effects of the earlier stages of the collapse itself that created the condition that allowed the collapse to continue -- specifically, the increased falling mass.
The various calculations published by both sides in the dispute are not (and probably cannot be) precise enough to settle this (admittedly minor) issue decisively. It's even possible that the collapse of tower 1 was progressive while tower 2, with its larger initial falling mass, was more like one big drawn-out sledgehammer blow.
In any case, I hope I've made my case that there's more to the concept of progressive collapse than the trivial observation that a building collapsed one floor at a time.
Respectfully,
Myriad