Structural Engineering & Statics
where the load goes
Internal forces

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.

Assumes The slab that spans both ways, The hinge put in on purpose and The structure that survives losing a member.

20 min read 10 figures The load must go somewhereOne support too manyWhich failure arrives first

A yield-line calculation is an upper bound on the load a slab can carry, and it is usually a good one. It is also, for a slab whose edges are held by the rest of the floor, wrong by a factor of about four — in the safe direction, which is the reason it has stood for seventy years without anybody minding very much. What the calculation leaves out is not a subtlety about the reinforcement. It is a force in the plane of the slab that nobody applied and nobody drew.

The same restraint, twice, with opposite signsA 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.00.511.522.5020406080100120140deflection ÷ thicknessload the strip carriesarchingcatenaryyield line× 3.89 at 0.24tnothing to arch against
Fig. 1 A 4 m strip of 200 mm slab whose ends cannot move apart, against deflection in its own thicknesses. The flat line at 30.0 is the yield-line load, which is what the same strip would carry with free ends. The rising branch peaks at 116.6 — 3.89 times as much — at a deflection of 0.24 of the thickness, and then falls back as the arch runs out of depth before the reinforcement picks it up as a cable.

Two mechanisms are on that curve, they have opposite signs, and the same restraint produces both.

Which free body produced the number

Take half the strip, cut at mid-span, and draw what crosses the cut.

Before anything moves, the answer is a shear and a moment, and the mechanism analysis proceeds from there. Now let the strip deflect by ww at mid-span and ask a purely geometric question: how long is the strip’s own centreline now?

For a parabolic deflected shape it is longer than the straight chord by

Δ=∫0L12(v′)2 dx=8w23L\Delta = \int_0^L \tfrac{1}{2}(v')^2\,dx = \frac{8w^2}{3L}

At w=48w = 48 mm on a 4 m span that is 1.54 mm. The strip has to accommodate 1.54 mm of extra length between two supports that have not moved, so it must shorten by that amount — and a strip that is being shortened against its will is a strip in compression.

That compression is not applied by anybody. It is generated by the geometry, resisted by the strip’s own axial stiffness and by whatever the surrounding floor supplies, and it is exactly the same mechanism that makes an arch an arch: a member that cannot spread must thrust.

The arch inside the depth

Once the thrust exists, the free body at each hinge carries a compression as well as a moment, and the compression has a lever arm.

At mid-span the compression sits near the top of the section; at each end hinge it sits near the bottom. The vertical distance between the two thrust positions is the arch’s rise, and it is limited by the depth the strip has left after deflecting:

r=t−w2,N≤fc b (t−w)2r = \frac{t - w}{2}, \qquad N \le \frac{f_c\,b\,(t-w)}{2}

so the load the arch carries is 8Nr/L28Nr/L^2, which for a plain strip with no reinforcement at all is 2fcb(t−w)2/L22f_c b(t-w)^2/L^2. A restrained strip with no plastic moment whatsoever carries 86.6 per unit width, against a yield-line load of exactly zero. That single comparison is the cleanest statement of what membrane action is: it does not need the section to have any bending capacity, because an arch does not.

A three-pinned arch, rise 2.6 on span 9A three-pinned arch under a uniform load. One moment equation about the crown hinge gives a horizontal thrust of 23.37, with no stiffness and no assumption about the section. The thrust line lands on the axis everywhere, so there is no bending anywhere in the arch.crown hinge — no moment here, by constructionH = 23.4H = 23.427.027.0thrust line and axis coincide — the definition of funicular
Fig. 2 The arch this collection has already drawn, with its thrust computed rather than assumed. What is different inside a slab is only the scale: the rise is a fraction of the slab’s own thickness rather than a fraction of the span, and the thrust is generated by restraint rather than by shape.

The peak is where two opposing trends cross. At small ww the thrust is small because the geometric shortening is small; as ww grows the thrust grows quadratically but the lever arm falls linearly, and the crush limit takes over. The product peaks at w=48w = 48 mm — 0.24 of the thickness, one part in 83 of the span — where the thrust has reached 2,280 kN per metre of width and the lever arm is 76 mm.

The deflection is smaller than anybody would notice

That number deserves to be sat with, and it is worth reading the whole of the rising branch rather than only its top. A 4 m span is allowed about 16 mm at its serviceability limit, and at 16 mm the restrained strip is already carrying 1.39 times its yield-line load. At 24 mm it is at 1.84 times, at 32 mm at 2.43, and the peak of 3.89 arrives at 48 mm — three times the serviceability deflection, and still only a quarter of the slab’s own thickness.

deflection ÷ thickness load carried × yield line
8 mm 0.04 33.1 1.10
16 0.08 41.8 1.39
24 0.12 55.3 1.84
32 0.16 73.0 2.43
48 0.24 116.6 3.89

So the enhancement is not something a slab reaches only in extremis; a third of it is present before a floor has moved further than a code allows. That is one reason load tests on real floors so consistently outperform their own calculations — the same gap the slab essay records and attributes elsewhere. The other explanations — strain hardening, tensile strength ignored in the analysis, two-way action counted conservatively — are all real and none of them is worth a factor of four.

A two-way slab is a one-way slab as soon as it is not squareThe 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.11.522.530.40.50.60.70.80.91long span ÷ short spanshare taken by the short strips6 × 8 m: 76.0%by 2 : 1 it is a one-way slab
Fig. 3 The two-way calculation this argument sits on top of. Yield-line analysis gives a slab’s collapse load from a mechanism, and it is an upper bound on the load THAT MECHANISM can carry — which is a lower bound on nothing at all once the strip is restrained.

The restraint that has to exist

Everything above needs the ends held, and the honest question is what holds them.

In a real floor, the surrounding slab does. An interior panel is restrained by the panels around it, which are themselves restrained, and the whole plate is a self-equilibrating raft in which the thrust from one panel is the reaction of another. An edge panel has much less restraint and a corner panel less still, so the enhancement is largest exactly where the reinforcement is heaviest and smallest at the edges, which is the wrong way round from a designer’s point of view. It is the stiffest path taking the load applied to a floor plate rather than to a set of beams.

The generator refuses to draw the unrestrained case as anything but the flexural load: with the end springs made soft, the peak falls to 30.0, which is the yield-line value to the last figure. That is not a result — it is the check that the model has not put anything in by hand.

Two consequences follow. Membrane action is a property of a floor, not of a slab, so it cannot be computed from a panel in isolation. And it is destroyed by the things that destroy in-plane continuity: a movement joint, a large opening, a demolition, an edge beam that is itself free to spread.

Where the arch runs out

A line of thrust, and the masonry it has to stay insideAn arch ring of 9% of the span in thickness, rising 28% of the span, under its own weight as a uniform load. Any horizontal thrust between 3.85 and 5.23 puts a line of compression entirely inside the masonry, so the arch stands — and which of them it actually takes is not decided by statics. The two extremes are drawn: the minimum-thrust line, which rides high at the crown and low at the haunches, and the maximum-thrust line, which does the opposite.thrust anywhere from 3.85 to 5.23 fitsH = 3.85, leastH = 5.23, most
Fig. 4 The same limit stated for masonry: an arch stands while a thrust line can be drawn inside the material, and fails when the material runs out of depth to contain one. A restrained slab strip is an arch whose masonry is 200 mm deep, and the thrust line has the same job to do inside it.

Past the peak the lever arm shrinks faster than the thrust can grow, and at w=tw = t there is no rise left at all. The arch branch has fallen back to exactly the yield-line load, which is the check the model makes at that station: with r=0r = 0 the arching term is zero and only the plastic moments remain.

But the strip has not been sitting still while that happened. The same geometric extension that generated the thrust is still growing, and once the section has cracked through, the material available to resist it is the reinforcement. At w=0.6tw = 0.6t the demand has exceeded what 800 mm²/m of 500 N/mm² bar can supply, the tie has yielded, and it is carrying its full 400 kN per metre.

So the two branches overlap. The catenary overtakes the arch at 0.6 of the thickness, when the arch still has 40% of its lever arm; switching between them at w=tw = t would put a step in the curve, and the honest picture is where the two cross. The minimum on the curve is 54.0 — still 1.8 times the yield-line load, and the strip never falls back to it.

The cable that a floor becomes

The cable and the arch are the same curveThe shape a cable takes under a uniform load is a parabola, and it carries that load in pure tension. Reflected, the identical curve carries the same load in pure compression, which is what an arch is.cable: pure tensionreflected herearch: pure compression
Fig. 5 The pairing this collection keeps returning to: a cable and its reflected arch, carrying the same load by the same shape with the sign reversed. A restrained slab does both, one after the other, in the same 200 mm of concrete.

Beyond the crossing the strip is a cable. Its capacity is 8Tw/L28Tw/L^2 with TT the tie’s yield force, so it rises linearly with deflection and has no maximum at all — it fails when the reinforcement fractures or when its anchorage pulls out, whichever comes first.

That is a robustness argument rather than a design one, and the number that makes it so is where the catenary gets back to the arch’s peak: 2.17 thicknesses of deflection, which is 434 mm, one part in 9 of the span. Nobody designs for that. A floor that has dropped a ninth of its span is a floor that has visibly failed, and it is still holding.

Take that one away and the load finds another routeA 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.intactmember 2 removedworst demand 2.03×
Fig. 6 Why that matters. When a structure loses a member, the question is whether an alternative path exists — and for a floor above a lost column the alternative is precisely this: the slab hanging in catenary, at deflections no serviceability calculation would tolerate, carried by reinforcement and anchorages nobody sized for tension.

The essay on robustness names catenary action as the mechanism a building falls back on when a column goes, and this is that mechanism computed. Its two requirements are both about detailing rather than about capacity: continuity of reinforcement through supports, so that a tie exists at all, and anchorage able to develop it, so that the tie can reach yield rather than pulling out of a lap.

The same curve, elsewhere

A load with a maximum in it, and nothing bifurcatesLoad against apex movement for a two-bar frame of half-span 1000 mm and rise 150 mm. The load rises to 133.4 kN at a movement of 64 mm — well short of the 150 mm that would bring the apex level — and then falls. Past that point the frame can only be held by taking load away, so under a dead weight it goes: 260 mm of movement at constant load, arriving inverted and in tension. The minimum on the path is -133.4 kN, the exact negative of the maximum, because the geometry is symmetric about the flat position and the arithmetic knows it.050100150200250300350-150-100-50050100150movement of the apex (mm)load (kN)limit point: 133.4 kNit jumps 260 mmas builtat the limitafter it goes
Fig. 7 The snapping frame this family was written for, and the reason the membrane curve is drawn on the same axes: both are a load-deflection path with a limit point on it, reached by a geometry that stiffens and then runs out. The difference is what is left afterwards — the frame has nothing and the strip has a second mechanism of the opposite sign.

The comparison is exact in form and instructive in its difference. A shallow frame that snaps through has a maximum load, a falling branch and a region where it can carry nothing; a restrained slab strip has a maximum load, a falling branch, and a rising branch beyond. Both are geometric-nonlinear problems in which the load path changes kind as the structure moves, and neither is visible to any analysis that assumes small displacements.

That is the general lesson. A structure whose geometry is part of its load path has a capacity that depends on how far it has moved, and the linear analysis that computes forces from an undeformed shape has no way to represent it — in either direction.

How much of it survives being asked for

Membrane action is largest on short spans and shallowest on long ones, and the trend is worth tabulating because it is the reverse of what a designer would guess.

The collapse mechanism of a propped cantileverA 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.sagging hinge at 4.69hinge at the fixed endlowest upper bound: 7.29every hinge position gives an upper bound on the collapse loadassumed position of the sagging hingecoefficient 11.66 Mp ÷ L²
Fig. 8 The mechanism the enhancement is measured against. A yield-line load is a hinge count and a work equation, and it contains no geometry beyond the hinge positions — which is exactly why it cannot see a thrust that only exists once the mechanism has moved.
span yield-line with restraint enhancement peak at
3 m 53.3 228.3 4.28× 0.19 t
4 m 30.0 116.6 3.89× 0.24 t
6 m 13.3 42.4 3.18× 0.34 t
8 m 7.5 20.1 2.68× 0.42 t

The arching term goes as t2/L2t^2/L^2 and the flexural term as 1/L21/L^2, so the ratio would be independent of span if the thrust were crush-limited throughout — and it is not, because on a longer span the geometric shortening at a given fraction of the thickness is smaller, so the thrust is demand-limited over more of the range and the peak moves outward. The effect is worth most where a slab is thick relative to its span, which is to say in exactly the flat-slab and transfer-slab construction where nobody is short of capacity.

It moves, or it pushes. Never both, and never neitherA 30 m steel member 30 °C warmer than it was built, in three conditions. Free, it grows 10.8 mm and carries nothing. Held, it moves nothing and carries 75.6 MPa — which is E·α·ΔT and contains neither the length nor the area of the member, so the identical stress arises in a two-metre strut. Held by a spring it does some of each: 3.2 mm of movement and 53.2 MPa, and the split is decided by the spring rather than by the member.free at one end10.8 mmno stressheld at both ends75.6 MPaheld by a spring of 100 kN/mm53.2 MPaE·α·ΔT = 210000 × 12×10⁻⁶ × 30 = 75.6 MPa, at every length
Fig. 9 The general form of the argument, which this collection has met before. A member prevented from changing length develops force instead of movement, and the force depends on the restraint rather than on the member. Membrane action is that statement with the length change caused by deflection rather than by temperature.

The parallel with thermal restraint is exact and is the most useful way to file this. In both cases a length change is demanded by something other than a load, the restraint refuses it, and a force appears whose magnitude is set by the restraint’s stiffness rather than by anything in the loading. Where they differ is the sign of the consequence: a restrained thermal expansion is a nuisance, and a restrained deflection is a gift.

Where the model stops

The strip is rigid-plastic and one-dimensional. A real slab spans two ways, its membrane forces are a field rather than a line, and the compressive arch in one direction interacts with the tensile ring around the outside. The two-way case has a compression zone in the middle and a tension ring at the edges, and its enhancement is larger than a one-way analysis gives.

The restraint is taken as rigid. With finite restraint the thrust is smaller, the peak lower and further out, and the whole curve flattens. Real restraint is neither rigid nor knowable, and this is the largest uncertainty in the subject by a wide margin.

No concrete crushing softening is included. The arch is capped by a rectangular stress block at fcf_c, and a real compression zone at that strain is already spalling. The peak is therefore optimistic and the falling branch steeper than drawn.

Nothing here is a design method. Codes generally ignore membrane action for exactly the reasons above, and the honest position is that it is a large, real, unquantifiable reserve rather than a capacity to be spent.

And the enhancement disappears at fire. A heated slab expands, which relieves the very geometric shortening that generated the thrust, and then loses stiffness so that the deflection grows — so a fire moves a floor along this curve from the arching branch to the catenary one, which is why a floor in a fire hangs rather than arching and why the tensile branch is the one fire engineering is interested in.

The further it deflects, the harder it pulls backTotal load against midspan sag for a 30 m cable of 1000 mm² prestressed to 500 kN, carrying 5 kN/m. The cubic H³ − T₀H² − w²L²EA/24 = 0 was bisected at every point of the curve, so the sag at the full 150 kN is 0.740 m rather than the 1.125 m the flat-cable formula WL/8T₀ gives — the straight dashed line, which is the tangent to this curve at the origin and nothing more. Its slope is the initial stiffness 8T₀/L = 133.3 kN/m; at the marked point the tangent has reached 341.2 kN/m, 2.56 times as stiff, and the horizontal component of the tension has risen from 500 kN to 760 kN. Nothing about the steel changed. The geometry got better at the job.00.20.40.60.811.2020406080100120140160midspan sag (m)total load on the cable (kN)the design load, 150 kNsolved 0.740 m1.125 mtangent here 341.2 kN/mk₀ = 8T₀/L = 133.3 kN/mthe flat-cable law
Fig. 10 The catenary branch’s own stiffness, in the field it belongs to. A cable resists load by changing shape, so its stiffness depends on how far it has already sagged — which is why the tensile branch of a restrained slab rises with deflection rather than reaching a plateau, and why nothing about it looks like a strength limit.

What the pictures cannot show

The load-deflection curve is drawn as a single line, and a real restrained slab does not follow one: it cracks, and each crack is a small drop and a redistribution. The curve is an envelope of a jagged process and its smoothness is an artefact of a rigid-plastic idealisation.

Nothing in the figure shows the in-plane force directly, which is the whole subject. The thrust is 2,280 kN per metre of width at the peak — a force comparable with what a column carries — travelling horizontally through a slab that is drawn as a line. There is no axis on the figure it could be plotted against, and it is invisible for the same reason a bimoment is: the drawing shows a mechanism, and this is a stress resultant the mechanism did not ask for.

And the horizontal axis is a deflection in thicknesses, which flatters the picture. Reading it in spans instead: the peak is at L/83L/83, the crossing at L/33L/33, and the recovery at L/9L/9. Only the first of those three is a structure anybody would still be standing under.

The ladder from here

Later rungs on this anchor: the two-way case, with its compression dome and its peripheral tension ring, and the Bailey method that computes it. The restraint stiffness as a design variable, and what an edge beam has to be to supply it. Membrane action in composite steel-concrete floors at fire, which is where the whole subject became a design method rather than a curiosity. Tie force requirements in codes, and the fact that they are specified as forces with no analysis behind them precisely because the analysis is intractable. The dynamic version, where a column is removed suddenly and the floor has to reach its catenary state through an impact rather than a slow deflection. And the negative case: the slab that is restrained on three sides and free on the fourth, where the thrust has nowhere to go and the whole enhancement disappears.

The finding worth carrying furthest is not the factor of four. It is that a structure’s real capacity can depend on a force that appears in no drawing, is applied by nobody, is generated by the structure’s own movement, and is destroyed by anybody who cuts a joint through it.

Where this goes next

11.522.530.40.50.60.70.80.91long span ÷ short spanshare taken by the short strips6 × 8 m: 76.0%by 2 : 1 it is a one-way slab 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. intactmember 2 removedworst demand 2.03× 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. crown hinge — no moment here, by constructionH = 23.4H = 23.427.027.0thrust line and axis coincide — the definition of funicular The hinge put in on purpose An arch with two pinned feet cannot be solved by statics. Add a third hinge at the crown — deliberately weakening it — and the whole structure falls out of one moment equation. 050100150200250300350-150-100-50050100150movement of the apex (mm)load (kN)limit point: 133.4 kNit jumps 260 mmas builtat the limitafter it goes The roof that jumps Every stability failure in this collection so far has been a bifurcation — a straight thing discovering it can be bent. A shallow frame does something else entirely. It stays perfectly symmetric, deforms steadily, and at some point the load it can carry starts to fall while it is still moving in the direction it was pushed.

Named alongside this one

Essays reaching for the same objects. Nobody chose these; they are what the concept index makes visible.

The objects this essay names

Each one links to every other essay that touches it.

Alternative load pathCatenaryCollapse mechanismCompatibilityHorizontal thrustLimit pointMembrane actionNo tension materialProgressive collapseRestraintRobustnessYield line