Structural Engineering & Statics
where the load goes
Concept

Collapse mechanism — where it appears

An arrangement of hinges that turns a structure into a linkage, giving an upper bound on the load at which it falls down. Every assumed mechanism gives a load that is never too low, so the search is a minimisation and the mechanism nobody thought of is the dangerous one.

Named by 15 essays across 6 fields — each of them below, with the objects they name alongside it.

The collapse mechanism of a propped cantilever. A collapse mechanism, with the hinge position found by searching rather than quoted. Every position gives an upper bound on the collapse load; the lowest is 7.29, at a hinge 58.6 per cent along, which is a coefficient of 11.657 times Mp over the square of the span.

After the first yield, which is not the end

A steel beam whose extreme fibre has reached yield has not failed. It has started forming a hinge, and collapse waits until there are enough hinges to make a mechanism.

internal-forces · Plastic hinge Prying action in a tee stub. A tee stub pulled by its web with 100 kN per bolt. The 20 mm flange is in the one-hinge regime, so the prying force at the flange tip is 50.63 kN and the bolt carries 150.63 kN — 1.51 times what was applied. The flange stops prying entirely at 26.97 mm thick, and collapses on its own at 110 kN.

The force the bolt never saw applied

Pull a tee stub with a hundred kilonewtons and its bolt carries a hundred and fifty. The extra comes from the flange bending and pressing its own edge against the thing it is bolted to, and no free body of the connection as a point contains it.

connections · Prying A two-way slab is a one-way slab as soon as it is not square. The share of the load carried by the strips spanning the short way, against the ratio of the sides. The two families of strips cross at the centre and must deflect equally there, and a strip's deflection goes as the fourth power of its span — so at a ratio of 1.33 the short strips already take 76% and at 2 they take 94%. The panel drawn here is 6 × 8 m, a ratio of 1.33, and its short strips take 76.0%. Two-way action is worth having at a ratio of one and worth almost nothing by two.

The slab that spans both ways

A panel supported on four sides sends its load in two directions at once, and the share is decided by a fourth power — so a panel a third longer than it is wide has already stopped being a two-way slab in any useful sense. What it does at collapse is a different calculation with a different answer.

structures · Two-way spanning 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 The same load, two diagrams, both in equilibrium. One span of a pair of 7 m spans under 5 kN/m, drawn twice. The elastic solution puts 31 kNm over the support and 17 in the span. Reducing the support moment by 30% and taking what statics then gives leaves 21 and 21: the section the beam needs falls from 31 kNm to 21, a saving of 30%. Both curves are in equilibrium with the same load — the mid-span ordinate plus half the support moment is the free moment 31 kNm for either — and the second is legitimate for that reason alone. What it costs is 1.0 milliradians of rotation at the support, which the section has to be able to deliver.

The moment that was moved on purpose

The elastic analysis of a continuous beam gives one set of moments. It is not the only set the beam is allowed to have, and taking a smaller one at the support is legal, cheaper, and paid for in a rotation that has to be delivered before the design exists.

internal-forces · Moment redistribution The two theorems close on the answer from opposite sides. A pinned-base portal frame under 200 kN at mid-span and 80 kN at the eaves, with a plastic moment of 200 kNm. The three horizontal lines are mechanisms — work equations, each an upper bound: the beam mechanism gives 1.000, sway 1.250 and the combined one 0.714. The curve is the lower bound: for each value of the one redundant, the largest load factor whose moment field stays inside the plastic moment everywhere. Its peak is 0.714, and it touches the lowest mechanism exactly — so the collapse load is known rather than bracketed. Using the beam mechanism instead would have claimed 1.40 times the real capacity, and the sway one 1.75 times, both of them on the wrong side.

Two ways of being wrong

Plastic analysis has two theorems and they point in opposite directions. Any equilibrium field that nowhere exceeds the plastic moment gives a load at or below the collapse load; any mechanism gives one at or above it. Only one of those errors is safe.

equilibrium · Bound theorems A bearing capacity is a mechanism, and here it is. Prandtl's collapse mechanism under a 3.0 m footing in a soil of 32° friction. A rigid wedge is driven down with the footing at 61° to the horizontal; a fan of radial shear turns the stress through exactly ninety degrees on a logarithmic spiral whose growth rate is tanφ; and a passive wedge at 29° has to be pushed up and out of the way. Nothing here is empirical — every angle is a function of φ alone — and the mechanism reaches 15.9 m from the centre, which is 10.6 times the footing's half width. That is why two footings closer together than about four widths do not have separate bearing capacities.

The ground is a mechanism

Bearing capacity is met as a formula with three terms and a table of coefficients, and that presentation hides what it is. Underneath is a plastic collapse mechanism — a rigid wedge, a fan of radial shear on a logarithmic spiral, and a passive wedge that has to be pushed up and out of the way — and every coefficient in the table is a property of that one drawing.

equilibrium · Bearing capacity The circle is searched for, and the first guess is 39 per cent optimistic. The same slope with 81 trial circles evaluated, each one through the toe and each one giving its own factor of safety. There is no equation whose solution is the answer: the slip surface is a shape the ground chooses, so the calculation is a search over shapes and the answer is the smallest number found — 1.191 against 1.650 for the circle a first guess puts through the toe from above the middle of the slope, which is 39 per cent optimistic. A slope analysis that reports one circle has reported nothing.

The surface that has to be searched for

Every other check in this collection is made at a section somebody drew. A slope has no section — the failure surface is a shape the ground chooses, so the calculation is a search over shapes, and the answer is the smallest number found rather than the solution of anything.

equilibrium · Slope stability Where the drift went. Storey drift at the target displacement, for the same frame with and without a soft ground storey at 50 per cent of the others' stiffness and strength. The regular frame spreads 219 mm over every storey; the soft one reaches 266 mm and puts 5.58 per cent of it into the ground storey against 0.53 next to it — a concentration of 5.9 against 1.7. The roof goes 22 per cent further, and where that extra displacement lands is the whole of the difference between the two buildings.

Weaker in one place, and better on every average

Take an eight-storey frame and make its ground storey half as stiff and half as strong. Its ductility demand falls, its first mode carries more of the mass, and its period lengthens into a gentler part of the spectrum. Three global numbers all improve, and the building is the one that collapses.

dynamics · Pushover Where two buildings touch, and what is there when they do. Closure between an 8-storey building 24 m tall and a 4-storey one 15 m tall, drawn against height, with the 50 mm gap between them. The two swaying out of phase close on each other more the higher up they are, so they first touch at 7.6 m and are in contact above it. The dots are the taller building's floors: its storeys are 3.00 m and the other's are 3.75, so two of them arrive part way up a column rather than at a slab. A blow at mid-height of a column asks it for a shear of half the impact and an end moment of Fh/8, neither of which is a demand any part of the design contains.

The floor that arrives at a column

The gap between two buildings is computed from their roof displacements, which is where each of them moves most. It is not where they touch, and it is not what is there when they do — a slab edge meeting a column part way up its height is a different event from two slabs meeting, and it is the one that appears in the photographs.

dynamics · Pounding Two ways for the same plate to fold. An end plate 200 mm wide with a bolt 45 mm from the web face and 55 mm from the edge, and the two families of yield line it can collapse along. The circle closes round the bolt and is 283 mm of hinge — a circle round the bolt. The straight pattern runs out to the plate's free edges and is 249 mm — hinges to the plate edges. The plate folds along whichever is cheaper, which here is the fan, and the 200 mm that comes out is the length of the equivalent tee stub — a dimension that is nowhere on the plate and is 100 per cent of its width.

How much of the plate is bending

A tee stub is an object nobody builds, and the whole component method rests on replacing a real end plate with one. The length of the substitute is not a dimension of the plate — it is the length of the cheapest fold the plate can collapse along, and two families of fold compete for it on a criterion with no strength in it at all.

connections · Prying Two capacity curves that agree, from two answers that do not. Base shear against roof drift for the same eight-storey frame with a soft ground storey, pushed with a fixed triangular pattern and with one recomputed from the tangent stiffness at every step. At a roof displacement of 426 mm they differ by 8.7 per cent — which is the number a capacity-spectrum procedure reads, and it is the number that is nearly the same. The storey drifts underneath these two curves differ by a factor of 3.8 at the fourth floor, and the drift is what the analysis was run for.

The pattern that stopped describing the building

A pushover analysis pushes with a load pattern chosen to resemble the first mode, and by the time the structure has done anything worth analysing it no longer has that mode. Recomputing the pattern as the frame softens changes the base shear by nine per cent and the drift at the fourth floor by a factor of four.

dynamics · Pushover Two collapses for the same plate, and only one of them fits. The same three bolt rows at 90 mm pitch, folding two ways. On the left each row folds on its own pattern — 249 mm of hinge line round each bolt — and the patterns overlap, because 249 mm of fold cannot fit in a 90 mm pitch. On the right the plate does what it can actually do: the hinges run straight from one row to the next, and the whole group folds on 429 mm rather than 746. The arithmetic follows the drawing — 377 kN for the group against 657 for the rows added up.

The rows have to share one fold

Each bolt row of an end plate is checked on its own, and the answers are added up. Two rows ninety millimetres apart cannot each fold the plate on a pattern two hundred and fifty millimetres long, because there is only one plate — so the group folds on one shorter pattern, and the sum was never available.

connections · Prying Two flanges, one bolt, one prying force. An end plate 18 mm thick with its bolt 45 mm from the beam web, bolted to a column flange 14 mm thick with the same bolt 30 mm from the column web; their tips are 35 mm beyond the bolt and bear on each other, so the prying force there is one force acting on both. Checked as two separate tee stubs on rigid bases, the end plate carries 109.3 kN per bolt (mode 2, web hinge and bolt) and the column flange 104.4 (mode 1, both hinges), so the component method gives the joint 104.4. The pair carries 92.3 kN, 12 per cent less, with the prying force at 44.7 kN and hinges in the end plate at its web and the column flange at the bolt line — the end plate's web hinge working with the column flange's bolt-line hinge, a mechanism that exists in neither tee stub on its own.

The hinge that forms in the other flange

An end plate bolted to a column flange is two tee stubs sharing one bolt, and the force at their tips is one force pressing on both. Checked separately, as the component method checks them, the end plate carries 109 kN a bolt and the column flange 104. Together they carry 92 — by a mechanism with one hinge in each flange that neither tee stub has on its own.

connections · Prying How far a local load spreads, and what is nearest its limit. Along a steel-faced polyurethane cladding panel, 0.5 mm faces of 320 N/mm² steel on an 80 mm core crushing at 0.12 N/mm², from the middle of a strip load of 3.0 N per mm of width spread over 10 mm: the face's deflection over its value under the load, the core's compressive stress over its crushing strength, and the face's bending stress over its yield stress. The load spreads over a length set by the fourth root of the face's bending stiffness over the core's, 1/β = 20.5 mm; the deflection is 1.44 mm under the load and changes sign beyond 48 mm. The core is at 0.60 of its crushing strength and the face at 0.89 of its yield stress: this panel's face yields at 3.36 N/mm and its core crushes at 5.00.

The face that dents and the core that crushes

A sandwich panel carries bending as a couple between its faces, and that is a calculation about the whole panel. Put a local load on one face — a foot, a fixing, a dropped tool — and the face becomes a thin beam on a soft bed, spreading the load over a length the fourth root of their stiffnesses sets. Then either the face yields and dents, or the core crushes beneath it, and which comes first is decided by one thickness: below it the face gives, above it the core does, and a steel-faced roof panel is on the wrong side of it for anyone who walks on it.

sections · Sandwich section

Named alongside it

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

Plastic hingeFree bodyPlastic momentPryingYield-lineBolt tensionComponent methodDuctilityEnd plateEquilibriumLever armLower-bound theorem

All concepts