Two skins and the space between them

Take two aluminium sheets seven tenths of a millimetre thick and set them 60 mm apart with foam between. The pair weighs a little over four kilograms per square metre, spans two and a half metres, and deflects twelve millimetres under a design load. Bonded face to face with no gap, the same two sheets deflect twenty-two thousand times as far.

That is not a composite material doing something clever. It is the second moment of area doing what it always does, taken to the limit: the faces carry the bending as a couple, the lever arm between them is the whole depth, and the material in between is there to hold them apart and to carry the shear.

What makes the section worth an essay of its own is that both of the usual approximations fail. The parallel-axis term is not a correction to be added — it is the section. And the shear deflection is not a correction either.

Which free body produced the number #

Cut the panel and take moments about the centroid of one face.

The other face’s force has a lever arm of $d = c + t_f$, the distance between the two face centroids. Everything else crossing the cut — the core’s own longitudinal stress — has a lever arm smaller than that and a stress smaller by a factor of a thousand, because the core’s modulus is 60 N/mm² against the faces’ 70,000. So the moment is carried by a pair of face forces at a known separation, and the face stress is

$$\sigma_f = \frac{M}{b\,d\,t_f}$$

with no second moment in it anywhere. That is the same arithmetic as a bolt group’s lever arm or a truss chord’s: a moment divided by a distance gives a force, and dividing that by an area gives a stress.

The bending stiffness follows from the parallel-axis theorem written out and then almost entirely thrown away:

$$D = \underbrace{\frac{E_f b t_f d^2}{2}}_{98.8\%} + \underbrace{\frac{2E_f b t_f^3}{12}}_{0.004\%} + \underbrace{\frac{E_c b c^3}{12}}_{1.2\%}$$

The separation term is the section. The faces’ own bending about their own centroids is four thousandths of one per cent of it, and the core’s own bending is one per cent.

The shear that cannot be neglected #

Every deflection on this site so far has been the second integral of a moment. That calculation drops a term, and for a solid beam the dropped term is worth about a per cent.

Here it is worth nine, and at shorter spans it is worth sixty.

The reason is a ratio. Bending deflection is $5qL^4/384D$ and shear deflection is $qL^2/8S$, so their ratio contains $L^2 \cdot S/D$ — and $S/D$ for a sandwich is a very small number, because $D$ contains the faces’ modulus and $S$ contains only the core’s shear modulus. On this panel the faces are 2,800 times stiffer in tension than the core is in shear.

$$\frac{\delta_{shear}}{\delta_{bend}} = \frac{48}{5}\cdot\frac{D}{S L^2}$$

At a span-to-depth ratio of 39 the shear term is 9 per cent of the total. At 20 it is 27 per cent. At 10 — a roof panel between closely spaced purlins, or a wall panel between rails — it is 60 per cent, and a deflection calculated from bending alone is out by a factor of two and a half.

The core is bought by thickness, not by material #

Thickening the core does two things at once and both of them are large.

The second moment goes as $d^2$, so doubling the core roughly quadruples the bending stiffness. The shear rigidity $S = G_c b d^2/c$ goes as $d^2/c$, which for a thick core is very nearly linear in $c$. And the weight goes up by the core’s own density, which for a structural foam is around three per cent of the faces’.

That combination is why sandwich panels exist. There is no other section in this collection where a designer can multiply the stiffness by four for a weight penalty of a few per cent, and the reason is that the material being added is doing almost nothing except occupying space.

The face that buckles into its own core #

A sandwich has a failure mode no other section here has, and it is the one that usually governs.

The compression face is a thin plate on an elastic foundation — the core. It buckles into that foundation in short waves, and the stress at which it does contains no length at all:

$$\sigma_{wr} = \tfrac{1}{2}\sqrt[3]{E_f E_c G_c}$$

For this panel that is 236 N/mm², against a face stress of 41. The cube root is what makes the number robust: a core three times softer lowers the wrinkling stress by only 44 per cent, so the failure mode is present in every sandwich and is rarely close in a well-made one. Where it does govern is where the core has been made very light, or where it has been damaged, or at a support where the face is in compression across a small local area.

The absence of a length in that expression is the same phenomenon as a plate’s local buckling stress depending on a width-to-thickness ratio rather than on a length, and for the same reason: the buckle chooses its own wavelength, so the member’s dimensions never enter.

Two materials, no transformed section #

The obvious tool for a section of two materials is the transformed section: scale one material’s width by the modular ratio and treat the result as homogeneous. It works here and it is a waste of effort.

The modular ratio between an aluminium face and a foam core is about 1,200. Transforming the core into an equivalent width of face material turns 1,000 mm of foam into 0.85 mm of aluminium, which is a strip narrower than the faces are thick. The transformed section is two flanges and a hairline, and the second moment of a hairline is nothing.

That is worth saying rather than skipping, because it is the general rule for any composite of very unequal materials: the transformed section is exact and the approximation of ignoring the weak material entirely is better than the effort of including it. What the weak material cannot be ignored for is shear, where it is the only participant.

What the interface has to do #

The faces and the core are bonded, and the bond carries the longitudinal shear flow between them — the same $VQ/Ib$ that holds any two-layer beam together.

Because the faces carry essentially all the longitudinal force, $Q$ at the interface is the whole of one face’s first moment, and the flow is $V d /$ (something close to $d^2$), which comes out as $\tau = V/(b d)$: the core’s shear stress, uniform through the depth. Uniform, because the core carries no bending stress, so there is nothing for the parabolic distribution to arise from. Every layer of the core is transferring the same amount.

A debonded sandwich is two loose sheets, and the factor of 22,000 evaporates. That is the same sentence as the two planks, at a more dramatic ratio, and it is why the adhesive line is treated as a structural component with a strength and a durability rather than as an assembly detail.

The weight, which is the whole point #

Every other section in this collection is compared on stiffness for a given area of material. A sandwich is bought on stiffness for a given mass, and the two comparisons rank sections differently.

Take the panel drawn and ask what a solid aluminium plate of the same bending stiffness would weigh. The plate needs $t^3 = 12D/(Eb)$, which comes out at 24 mm — thirty-four times the sandwich’s face thickness, and 66 kg per square metre against the sandwich’s four. The sandwich is sixteen times lighter for the same stiffness, and the ratio grows as the square root of the depth ratio: a deeper core widens the gap.

That number is why the section exists at all, and it is also why the section is used in places where nothing else on this site is used. An aircraft floor, a refrigerated lorry’s wall, a curtain-wall spandrel, a wind-turbine blade shell: all of them are structures whose governing constraint is mass or thickness rather than material cost, and all of them arrive at the same answer independently.

It is worth noticing what the comparison assumes, though. Stiffness per unit mass is the right measure only when the panel is stiffness-governed, and the two failure modes above — wrinkling and local bearing — are strength ones. A sandwich optimised on mass is a sandwich pushed toward thinner faces and a lighter core, which is the direction both of those modes get worse in.

Where the model stops #

The core is treated as an isotropic continuum with one shear modulus. A honeycomb core is not: it has two shear moduli differing by a factor of two, depending on which way the ribbon runs, and a panel loaded across the ribbon is a different panel from one loaded along it. A foam core is closer to isotropic and is not linear at the strains a wrinkle produces.

The faces are membranes. Their own bending stiffness has been dropped as negligible, which is right for the panel as a whole and wrong locally: a face under a point load bends about its own centroid over a length of a few millimetres, and the indentation that follows is a real failure mode with a completely different arithmetic.

And the panel is one span, uniformly loaded. The support reaction on a sandwich is a local compression on the core across a narrow bearing, and the core’s compressive strength is a fraction of a newton per square millimetre. Nothing in the section calculation sees it, and it is what actually limits how a sandwich panel is fixed down.

The section that stops being a section #

There is a boundary in this subject at which a member stops being described by a cross-section and starts being described by a surface, and a sandwich panel sits right on it.

Everything above treats the panel as a beam: a moment, a shear, a second moment, a span. That is correct while the panel spans one way between two supports, which is how cladding and roof decking are used. Support it on four sides and it becomes a plate — two-way spanning, with the load shared according to a fourth power of the aspect ratio — and the shear term becomes two shear terms with two different core moduli behind them.

Push further and the same construction becomes a shell. A curved sandwich carries load in its faces as membrane forces rather than as a bending couple, the core’s job becomes preventing the faces from buckling rather than resisting shear, and the arithmetic on this page barely applies. The wind-turbine blade and the boat hull are both sandwiches and neither is a beam.

What makes the boundary worth naming is that the failure modes travel with the construction even when the analysis does not. Wrinkling, debonding and local indentation are properties of two thin faces separated by something weak, and they are present in the plate and the shell exactly as they are in the beam — usually as the governing check in all three.

What the pictures cannot show #

The section drawing is exaggerated in one direction. The faces are 0.7 mm on a 61.4 mm section, which at the scale of the printed figure would be a line a fifth of a millimetre wide, so they are drawn thicker than they are and the stress block is drawn at a scale of its own.

Nor can the figures show what the panel is usually being asked for, which is not stiffness or strength but thermal performance. The core’s thickness is very often chosen by a $U$-value and the structure gets whatever depth that decision produces — which is why sandwich panels are so often far stiffer than they need to be in bending and far closer to their limits in wrinkling and in local bearing.

The assumption the figure rests on #

The core’s shear modulus is taken as 25 N/mm² and held constant. It is neither: it falls with temperature — a dark cladding panel in summer sun is at 70 °C, where a polyurethane core has lost a third of it — and it falls under sustained load, where creep in the core takes the long-term shear stiffness to perhaps half the short-term value. Both act on the term that this essay has argued is not negligible, so the long-term deflection of a sandwich panel is not the short-term one multiplied by anything simple.

The ladder from here #

Later rungs on this anchor: the honeycomb core and its two shear moduli, where the panel’s properties depend on the direction the ribbon was expanded in. Local indentation under a fixing, which is the failure mode that governs most real panels and is not a section calculation at all. The sandwich in compression as a wall panel, where the wrinkling stress and the overall buckling load interact. Creep of the core, which turns the shear term into a function of time. And the same arithmetic at a completely different scale — a stressed-skin roof deck, an orthotropic bridge deck, a hollow-core slab — where the faces are steel or concrete and the “core” is air, ribs or webs.