A third of the load crosses sideways

A panel supported on four sides sends its load in two directions at once, and the share is decided by a fourth power. That account is correct as far as it goes, and it stops one mechanism short.

A plate has three internal moments per point, not two. There is a bending moment about $x$, a bending moment about $y$, and a twisting moment $M_{xy}$ — and the equilibrium equation that carries the load contains all three:

$$\frac{\partial^2 M_x}{\partial x^2} + 2\frac{\partial^2 M_{xy}}{\partial x \partial y} + \frac{\partial^2 M_y}{\partial y^2} = -w$$

The middle term is the one no beam strip has. A strip can bend and it cannot twist relative to its neighbour, because it has no neighbour — it is a strip.

Which free body produced the number #

An element of the plate, with three moments on it, solved by Navier’s double sine series so that all three come out together rather than one being assumed.

Take a 6 by 6 m panel, 200 mm thick, simply supported on four sides under 10 kN/m². Evaluate each of the three terms above at a point and each is that mechanism’s contribution to carrying the load there; they must add to $w$.

At the centre the two bending terms supply 22.9% each and the twisting term supplies 54.1%. More than half the load at the middle of the panel is carried by a moment nobody draws.

Averaged over the panel, weighted by the load, the split is 34.6% / 34.6% / 30.7%.

And the deflection coefficient comes out $\alpha = 0.00406$ in $\delta = \alpha w L^4 / D$ — which is the number every plate table has printed since Timoshenko, arrived at here from the series rather than looked up.

Where the twisting goes, which is nowhere and everywhere #

$M_{xy}$ is zero on both centrelines and largest at the corners. That distribution is the exact opposite of the bending moments’, which peak at the centre and vanish at the edges, and the complementarity is not a coincidence: it is what allows the three terms to sum to a constant load across a panel where each of them varies enormously.

It also means the twisting is invisible at the one point everybody checks. A designer computing the mid-span moment finds a number that the strip method reproduces reasonably well, sees agreement, and concludes the strip model is adequate — while the mechanism the strip model is missing is doing its work in the corners, where nobody looked.

The corner, which has to be held down #

Here is the consequence that shows up in a real building.

At a free corner of a simply-supported panel, the twisting moment on the two edges meeting there is statically equivalent to a downward point force of $2M_{xy}$. That is Kirchhoff’s result and it follows from replacing a distributed twisting moment along an edge by a shear flow — the shear flows from the two edges do not cancel at the corner, and what is left is a concentrated force.

For this panel it is 26.7 kN, at a point, which is 7.4% of the entire load on the slab.

So the slab has to be held down at each corner with 26.7 kN, or it lifts. A panel whose corners are free does lift there, deflects more than the calculation says, and cracks diagonally across the corner on its top face — a crack running at forty-five degrees, in a region where a strip-method design has put no top reinforcement at all.

That top steel across the corners of a two-way slab is the physical trace of this whole essay. It exists for a moment that appears in no strip calculation, in a direction no span suggests, and it is left out precisely when the designer has reasoned about the slab as two beams.

What the strip method gets wrong, and in which direction #

Give the whole load to two strips in bending and design each for what it carries. What has that produced?

A set of internal forces in equilibrium with the load, and reinforcement provided to carry them. That is the lower-bound theorem’s hypothesis, so the slab can carry at least that load: the strip method is safe by construction, and this is why it is a legitimate design method rather than a rough one.

What it costs is weight. The strip solution here deflects 4.05 mm against the plate’s 2.53 — 60% more — because it is doing without a mechanism that was available. The reinforcement follows.

And what it gets wrong is the corner. Safety in the lower-bound sense is about the collapse load, not about behaviour at working load, and a slab designed as strips will still lift and crack at its corners because the real plate insists on twisting whether the design contemplated it or not. A lower-bound design is safe against collapse and is not a description of what the structure does.

Why the plate is stiffer than its strip #

There is a second, smaller consequence of the same mechanism, and it is the one that shows up in a deflection check rather than in a reinforcement drawing.

Take the fourth-power split at face value: a square panel gives half its load to each direction, so each strip carries $w/2$ and deflects $5(w/2)L^4/384D$. That is 4.05 mm here. The plate deflects 2.53 mm — 38% less.

The extra stiffness has two sources and only one of them is the twist. A plate strip is also stiffer than an isolated beam strip of the same section by $1/(1-\nu^2)$, because the material either side of it stops it curving anticlastically. At $\nu = 0.2$ that is 4%, which is real and small. The other 34% is the twisting mechanism carrying load the strips never had to.

That distinction matters when a slab’s deflection is the governing check, which for a lightly loaded floor it usually is. Designing the reinforcement by strips and then checking the deflection by strips compounds a conservatism into a thicker slab than the panel needs, and on a repeated floor plate the thickness is the cost of the building — and the deflection that arrives three years late is sitting on top of whatever the elastic calculation gave.

It is not a Poisson effect #

There is a tempting misreading, which is that the twisting is a consequence of Poisson coupling — that the plate is bending one way, contracting the other way as a result, and that the twist is what comes of the two interfering.

Set $\nu = 0$ and check. The twist share goes up, from 30.7% to 38.4%, and the deflection coefficient does not move at all.

The reason is in the constitutive relation:

$$M_{xy} = D(1-\nu)\frac{\partial^2 w}{\partial x \partial y}$$

The twisting moment is proportional to $(1-\nu)$, so removing the Poisson coupling increases it. What produces it is the cross-derivative of the deflected surface — a purely geometric quantity, which is non-zero wherever the surface is not a cylinder or a sphere. A doubly curved surface whose two curvatures are not aligned with the axes has twist in it, and that is all $M_{xy}$ is.

Which is worth carrying, because it says the same thing is going on in a shell: a curved surface carries in-plane forces including a shear, a bent surface carries moments including a twist, and both third components come from the mixed second derivative. They are the same mathematical object in two different theories.

What happens to the twist as the panel stretches #

The twist is a square panel’s business and it goes away with the shape, which is worth putting numbers on because the transition is quicker than intuition suggests.

Two things stand out. The deflection coefficient nearly triples between a square panel and one of 2.5 to 1, which is the fourth power arriving — a long panel really is a one-way slab, and that part of the received account is right.

The twist does not fall nearly as fast. At 2 to 1, where the fourth-power split has already given 94% of the load to the short direction and the panel is being treated as one-way in every practical sense, the twist is still carrying 22% and the corner force is still 5.3% of the load. So the panel proportion at which the corner reinforcement can be honestly omitted is a good deal further out than the proportion at which the two-way action can be.

That is a specific and useful asymmetry: the two-way behaviour disappears faster than the corner problem does.

The history, which explains why it is taught the way it is #

Kirchhoff’s plate theory is 1850 and the corner force is in it from the start. The strip method is Hillerborg’s and it is 1956. So the profession had the exact answer for a century before it adopted an approximate one on purpose, which is worth explaining.

The exact solutions exist only for a handful of shapes and boundary conditions: rectangular and simply supported, circular and clamped, and not much else. Every real slab has an opening, a re-entrant corner, a column somewhere, or an edge that is neither free nor fixed — and for those the series is unavailable and the designer is left with tables that do not cover the case.

Hillerborg’s contribution was not a better analysis. It was the observation that the lower-bound theorem lets a designer invent a load path and be safe, so a slab can be designed by choosing how the load will travel rather than by discovering how it does. That is far more powerful than an exact solution for the cases where an exact solution does not exist, and it is why it took over — the same argument the bound theorems make for plastic analysis generally.

The cost is the one this essay is about: a method that lets the designer choose the load path will be used by designers who choose one with no twist in it, and then the corner detail has to be remembered separately rather than falling out of the calculation. It is a trade of an exact answer for a general method, and the corner steel is what is paid.

Where this model stops #

The plate is thin and elastic. Everything here is Kirchhoff theory: no shear deformation through the thickness, no cracking, no reinforcement. A real reinforced slab cracks at working load, and a cracked slab’s stiffness is direction-dependent in a way that changes the share.

The supports are simple and unyielding. Continuous edges change the moments and the twist together; supports that deflect — a slab on beams rather than on walls — redistribute the load along the edges, and the corner force can be substantially reduced by a beam that sags.

Concentrated loads are outside it. The Navier series converges slowly for a point load and the moment under it is singular in the theory, which is why punching is a check made on a perimeter rather than a moment calculation.

And the corner force is a theory artefact in one respect. $2M_{xy}$ is a concentrated force because the theory replaced a distributed twisting moment by an equivalent shear; a thick-plate theory spreads it over a real length. The total is right, the concentration is an idealisation, and it is why the top steel is specified as a mat over a region rather than a bar at a point.

What the picture cannot show #

The figure draws $M_{xy}$ as a field over a plan, which makes it look like a quantity that acts at a point in a direction. It is not: a twisting moment is a pair of moments on two perpendicular faces of an element, and its magnitude depends on which pair of axes the element is drawn against. Rotate the axes by forty-five degrees and $M_{xy}$ at the corner becomes a pure difference of bending moments — which is exactly why the corner crack runs diagonally.

That is the honest description of what the corner steel resists: not a twisting moment, but the principal moment that the twisting moment is a component of. The diagonal is where it lies, the top face is where the tension is, and the twist is the name of the same thing in the wrong coordinate system.

The reinforcement that follows, in three directions #

The design consequence of all this is a reinforcement layout, and it is worth setting out because it is where the argument becomes a drawing.

Bottom steel, both ways, over the middle. This is what the strip method gives and it is right. Its quantity follows from the fourth-power split, and the two directions are not equal unless the panel is square.

Top steel over any continuous edge. Ordinary continuity, and nothing in this essay changes it.

Top steel across each free corner, running diagonally — or a mat in both directions, which is the practical equivalent. This is what the twist demands, and it is the one a strip calculation never asks for. Codes specify it as an area over a region of about a fifth of the span, and the usual detail is an orthogonal mat rather than diagonal bars, because a mat at forty-five degrees to the principal moment carries it perfectly well and is far easier to fix.

And bottom steel across the corner too, on many specifications, because the corner’s principal moments have opposite signs on the two diagonals: the one running into the corner is hogging on top and the one across it is sagging beneath.

The corner detail is therefore four layers where the field has two, over a region where a designer reasoning from spans expects the least. That inversion — most steel where the moment diagram is smallest — is the visible signature of a mechanism that no moment diagram contains.

The generalisation #

The habit worth carrying is to count a structure’s mechanisms before assuming it has two.

A plate has three. A shell has three membrane resultants and three bending ones. A grid replacing either has a member for some of them and no path at all for the rest. In every case the missing one is a shear or a twist — the cross-term — and in every case it is missing because the model was built out of things that span rather than out of things that are a surface.

That is a specific and repeatable oversight, and it has a specific test. Write the equilibrium equation for the real object, count its terms, and ask which term each element of the model supplies. Anything with no supplier is a mechanism the model has thrown away — safely, if the model is a lower bound, and never harmlessly.