Structural Engineering & Statics
where the load goes
Concept

Progressive collapse — where it appears

Failure spreading from a local loss because what remained could not find another path, which is a robustness question and not a strength one. It is checked by removing a member and asking what is left, which is a question no strength calculation on the intact structure poses.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

Take that one away and the load finds another route. A 6-panel pratt truss under 20 kN at each top node, before and after member 2 is removed. The load redistributes. The worst-affected survivor now carries 2.03 times what it did, and four members that carried nothing before are now working. Whether that is survival depends on how much spare capacity was there, which is a different question from whether the frame was strong enough.

The structure that survives losing a member

Every check in this collection asks what a structure carries. None of them asks what is left when part of it is gone — and two frames with the same members, the same weight and the same factor of safety can answer that question completely differently.

structures · Robustness The same restraint, twice, with opposite signs. A 4 m strip of 200 mm slab whose ends cannot move apart, against deflection measured in its own thicknesses. The flat line is what a yield-line calculation gives, which is what the same strip would carry if its ends were free: 30.0 per unit width. The rising branch is compressive membrane action — the deflected strip is forced into an arch — and it peaks at 116.6, which is 3.89 times the yield-line load, at a deflection of 0.24 of the thickness. Past that the arch runs out of depth and the load falls back to the flexural one; past a deflection of one thickness there is no arch left and the reinforcement starts carrying the strip as a cable. It gets back to the arch's load at 2.17 thicknesses, which is one part in 9 of the span — a sag nobody would design for and exactly what a floor does instead of falling.

The force nobody put in the model

A slab strip whose ends cannot move apart is not the strip in the yield-line calculation. Deflecting shortens the chord between its ends, the ends do not come in, and the strip is forced into an arch — worth four times the load it was designed for, at a movement nobody would see.

internal-forces · Membrane action A check made on a perimeter, not on a section. One bay of a flat slab, 7.2 m square, on a 400 × 400 mm column. The heavy closed line is the control perimeter, 2d from the column face with its corners rounded at that radius — 4427 mm long against 1600 mm round the column itself. The shaded area inside it delivers no shear across it and is subtracted from the load; everything outside arrives through the perimeter. At 12 kN/m² that is 604 kN across 4427 × 225 mm, a shear stress of 0.606 N/mm² against a resistance of 0.658.

A check made on a perimeter, not on a section

Every shear check in this collection is made on a plane cut through a member. A slab sitting on a column has no such plane, because the shear leaves in every direction at once — so the check is made on a closed line, and a line grows with the column while the load grows with the square of the bay.

internal-forces · Punching shear A check made on a perimeter, not on a section. One bay of a flat slab, 7.2 m square, on a 400 × 400 mm column. The heavy closed line is the control perimeter, 2d from the column face with its corners rounded at that radius — 4427 mm long against 1600 mm round the column itself. The shaded area inside it delivers no shear across it and is subtracted from the load; everything outside arrives through the perimeter. At 12 kN/m² that is 604 kN across 4427 × 225 mm, a shear stress of 0.697 N/mm² against a resistance of 0.658.

Turn the column, and the slab passes

A flat slab that is comfortable under gravity fails its punching check the moment a moment arrives at the column, and nothing about the load has changed. The fix is not more concrete. It is the column's plan shape and, at equal area, which way round it is turned — worth more than adding half again as much column.

internal-forces · Punching shear Removing each member in turn. Every member of a 8-panel pratt truss removed one at a time, with the worst demand on the survivors plotted against the member removed. Four of the 35 leave a mechanism — the bars drawn to the top of the frame — and for those there is no redistribution to compute, because there is no structure left. The rest redistribute, and the worst of them asks a survivor for 2.04 times what it carried before. A single number for robustness does not exist: it depends on which member goes.

A determinate truss has no robustness at all

Remove any one of a Warren truss's thirty-one members and what is left is a mechanism. Not weakened — gone, with no set of forces that holds the load in any position. Robustness is not a property a structure has by degree; it is bought by adding members that carry nothing until something else stops carrying, and a truss without them has none of it to measure.

structures · Robustness The cable pays the factor the plateau saved. The force a beam of two 8.0 m spans over the column, 600 mm deep, plastic moment 800 kN·m, its far ends held against rotation and against spreading (plastic collapse load 400 kN) reaches after a sudden loss divided by the load it carries — the dynamic factor — against that load as a share of the plastic collapse load. Below the plateau the beam is a linear spring and the factor is 2. On the bending plateau it falls to 1.18 at 0.90 times the collapse load, because a flat resistance does its work at full strength from the start. Past it the beam becomes a cable, whose resistance is again straight, and the factor climbs back: 1.73 at 1.5 times, 1.88 at 2, 1.97 at 3.

The cable that pays the factor back

A frame that loses a column carries the floor across the gap first by bending and then, as the beam sags, as a cable. The hope is that a path which stiffens as it deflects needs less than the factor of two a suddenly loaded spring does. On the bending plateau it does — 1.18 at nine tenths of the collapse load. But past one depth of sag the cable is a straight line again, and the factor climbs straight back toward two: 1.73 at one and a half times the collapse load. And the connections are asked for a tenth of a radian on the way.

structures · Robustness

Named alongside it

The objects these essays reach for when they reach for this one.

RobustnessAlternative load pathCatenaryControl perimeterDeterminacyDuctilityEffective depthFlat slabLoad pathMechanismMembrane actionPunching shear

All concepts