The structure that survives losing a member

Every calculation in this collection has been about a structure that is intact. The load arrives, the members carry it, and the check is whether each of them is strong enough for the share it has been given.

There is a different question, and no ordinary check asks it: what happens when one member is not there. A vehicle hits a column, a weld was never made, a bolt group corroded, a member was left out during erection and nobody noticed. The structure that is left is a different structure, and whether it stands up is not a consequence of anything computed about the one that was designed.

That it stands is not obvious and it is not general. The same frame loses other members and does something entirely different.

Redundancy is necessary and it is not sufficient #

The frame above is four times redundant: twenty-five members and three reactions against twenty-four equations. Statics alone cannot find its forces, which is the condition that requires stiffness, and it is also the condition that makes an alternative path possible at all. A determinate truss has no spare paths by definition — remove any member and the count falls below the equations, and what is left is a mechanism.

But four redundancies do not mean four members can be lost.

The four critical members are the two end diagonals and the two end panels of the bottom chord, and the reason is local rather than global. Redundancy is a count over the whole structure and collapse is an event at one place. The counter-diagonals that made the frame redundant are all in the interior panels, so the interior is well provided with alternatives and the end panels have none. A structure can be redundant everywhere on average and determinate where it matters.

The demand depends on which member goes #

For the members whose loss is survivable, the interesting quantity is what the survivors are then asked for — the ratio of the force after to the force before.

Set that beside the 2.03 of the hero figure and the pattern is clear: losing a lightly loaded member costs little and losing a heavily loaded one costs a lot, which sounds obvious and has a non-obvious corollary. The members whose loss is survivable are exactly the members that were not doing much, and a structure optimised so that every member is fully stressed has removed all of them.

That is the uncomfortable finding of this field. Optimisation and robustness pull in opposite directions: the ideal fully stressed design has no member whose loss is cheap, and every deletion of an “unnecessary” member is a deletion of a reserve path. The economy is real and so is the cost, and neither shows up in a check on the intact structure.

The members that were doing nothing #

A truss solution routinely returns members carrying zero, and this collection has drawn them faintly and moved on.

The robustness reading is different and better. A zero-force member is a path that is available and unused: it costs material, it carries nothing today, and it is the thing that carries the load tomorrow when its neighbour is gone. In the damaged frame above, four members that were at less than a tenth of the peak force ended up carrying between 128 and 144 kN.

A structure’s reserve paths are exactly the members its analysis says are unnecessary, which is why an engineer’s instinct to delete them should be resisted precisely when the analysis is most confident. A triangle that cannot fold is the minimum arrangement; everything past it is either economy or insurance, and the two are the same members.

Two frames that pass every check and differ completely #

The comparison worth making is between a frame with reserves and one without, both of which satisfy every requirement anybody would write down.

Both trusses span the same distance, carry the same load, weigh about the same, and have the same factor of safety on every member. One of them survives losing twenty-one of its twenty-five members and the other survives losing none. Nothing in a member-by-member check distinguishes them, and nothing in the drawings distinguishes them except a set of diagonals that a value engineer would remove on the grounds that the analysis says they carry nothing.

The comparison also sets a boundary on how far this can be pushed. Counter-diagonals are cheap on a truss and there is no equivalent trick for a column: a building column that is lost has to be replaced by an entirely different mechanism, usually the floors above hanging in catenary. Robustness is easy where the structure is a network and hard where it is a tree, and a building is mostly a tree — loads run down columns to foundations with no lateral alternatives.

The tie is a robustness measure and not a strength one #

The provisions that came out of the failure at the end of this essay are almost all written as ties: a minimum tension capacity between every element and its neighbours, specified in kilonewtons and detailed rather than calculated.

That is a strange kind of requirement, because nothing in the intact structure carries the tie force. It is dormant in every load case the building is designed for, appears in no analysis output, and its size is not derived from any calculation of the damaged state — it is a stated number with an empirical origin.

The justification is the one this essay has been making. What decides whether a damaged structure survives is whether the load can find a route, and a route made of ties is a route made of things a detailer can install without an analysis. A tie is the physical embodiment of a load path that does not exist yet — and it is exactly the kind of provision that a project under pressure deletes, because nothing appears to need it.

Two related habits belong beside it. Connections stronger than members, so that when a member is overloaded it yields rather than tearing out of its joint — the capacity-design argument, arriving here from the direction of accidental damage rather than from earthquakes. And continuity, so that a beam over a lost support becomes a longer beam rather than two falling ones, which is the same continuity that was bought earlier for the reduction in peak moment and turns out to have been paying a second premium all along.

Why the count is not the criterion #

Counting redundancies is the first thing anybody does and it can be wrong in both directions. It can be too optimistic, as above, where four redundancies do not protect four members. It can also be wrong about determinacy itself.

That is why the study is done by removal rather than by counting. Twenty-five analyses of twenty-five damaged frames answer a question that no property of the intact frame can, and each of them is a full solve — which is the reason this kind of check was impractical before it was cheap, and is now merely unusual.

Redistribution needs ductility to be real #

The elastic redistribution drawn in the figures assumes every surviving member can carry what it is now given. Where the demand is 2.03 and the design margin was 1.5, it cannot — and what happens next is decided by whether the overloaded member yields or breaks.

So robustness is not only a matter of geometry. It requires ductile members and ductile connections — the property that appears in none of the equations — and a frame with a brittle joint has no redistribution available at all: the overloaded member’s connection fractures, that member is now also gone, and the calculation restarts on a frame missing two. That is progressive collapse, and it is a sequence of the analysis in this essay run repeatedly with the damage accumulating.

The design responses follow directly and none of them is a strength requirement. Tie the structure together so that loads can find new routes; make the connections stronger than the members they join, so that yielding happens where it can be absorbed; provide alternative paths where the removal study says there are none; and where an element is genuinely irreplaceable, protect it — a bollard in front of a column is a structural measure.

The building this became a rule for #

Ronan Point was a twenty-two-storey precast panel block in east London. In May 1968 a gas explosion in a corner flat on the eighteenth floor blew out the load-bearing wall panels of that flat, and the corner of the building above it — having nothing left to stand on — fell, and the impact of it demolished the corner of every storey below. Four people died in a collapse that involved a small fraction of the building’s material and none of its strength.

The structural lesson was not that the panels were weak. They were adequate for the loads they carried, and the calculation of them was correct. What was missing was any means for the load of the flats above to reach the ground once one storey’s wall was gone — no route, at any stress — and no check anybody performed asked whether such a means existed.

Every disproportionate-collapse provision in every code descends from that morning, and their common form is the study above: remove each element in turn and require what is left to stand, or where it cannot, show that the resulting collapse is limited in extent. The provisions are usually written as tying requirements rather than as analyses, because a tie is something a detailer can draw and an analysis is something somebody has to do.

The word covers two different questions #

It is worth separating two things the literature calls robustness, because they need different work.

The first is structural: given that an element is gone, does what is left stand? That is the question the figures above answer, it is computable, and its answer is a set of demand ratios and a list of critical members. Nothing about probability enters, and nothing about the cause of the damage matters at all — a column lost to a lorry and a column lost to a missing weld leave the same frame.

The second is systemic: how much damage is proportionate to how much cause? That is the question the codes actually ask — a collapse must not be disproportionate to the event that started it — and it has no purely structural answer, because “disproportionate” compares a consequence with a cause that structural mechanics has no units for.

The two are usually run together and the confusion has a cost. A design can be excellent at the first and poor at the second: a frame in which every member’s loss is survivable, tied into a floor system that peels away one bay at a time, is robust member by member and fragile as a building. Conversely a structure with several critical members can be perfectly acceptable if those members are protected, inspected and unlikely to be lost — which is how every bridge with a fracture-critical member is treated.

The removal study answers the first question completely and the second not at all, and knowing which one a requirement is about decides whether the answer is an analysis or a judgement.

What the picture cannot show #

The removal is instantaneous in the drawings and gradual in the arithmetic. A member that disappears suddenly delivers its force to its neighbours dynamically, and the peak force is up to twice the static redistribution. Every demand ratio quoted above should be roughly doubled for a sudden loss, which turns a comfortable 1.07 into 2.14.

Nothing here checks the survivors against anything. The demand is a ratio, and whether it is acceptable depends on the margin the members were designed with — a demand of 2.03 is survivable if the members were at half capacity and fatal if they were at 80%. The study identifies the question; it does not answer it.

Only single removals are considered. Real damage takes out whatever is near it, which is usually several members at one place, and the number of double-removal cases is the square of the number of single ones. The practical alternative is to define a damage zone rather than a member, which is what the codes do.

Where the ladder goes #

The first rung is the dynamic factor: what a sudden removal does that a gradual one does not, and how the factor of two arises from an energy balance with no equation of motion in it.

The second is the same study applied to a frame rather than a truss, where the redistribution is of moment rather than of axial force and where the alternative path is often catenary action — the beams above a lost column hanging as cables, which requires connections able to carry a tension nobody designed them for.

The third is the awkward one. Everything above computes a consequence for a stated cause. The cause is a member being gone, and the probability of that happening is not a structural quantity at all — it is a question about vehicles, gas, corrosion, workmanship and malice, and the decision about how much robustness to buy cannot be made inside this subject.