The face that dents and the core that crushes

Two skins and the space between treated a sandwich panel as a section: two thin stiff faces carrying the bending as a couple, a thick light core holding them apart and carrying the shear. It made the faces membranes, with no bending stiffness of their own, and said where that stops being right: “a face under a point load bends about its own centroid”, and local indentation under a fixing “is the failure mode that governs most real panels and is not a section calculation at all.”

This essay does that calculation. Put a load on one face of a panel, over a width small beside the panel, and the panel’s bending hardly enters: what matters is the face alone, bending as a thin plate, and the core under it, compressed through its thickness. The face spreads the load; the core pushes back; and one of the two reaches its limit first.

The separation is sharper than it sounds. The panel’s bending stiffness comes from its faces held 80 mm apart, and it is more than a hundred thousand times the stiffness of one face bending about its own centroid. A load that bends the whole panel is carried by that couple; a load concentrated on one face is carried first by the face alone, because the core under it is too soft to hand the load to the far face until it has been compressed. Everything local happens in a strip a few centimetres wide, inside which the panel as a section does not yet exist.

A thin beam on a soft bed #

Take a strip of face of unit width across the load. It bends with a stiffness $EI = Et^3/12$ per unit width — tiny, because the face is thin — and it rests on the core, which resists being compressed through its thickness like a bed of springs of modulus $k = E_c/t_c$, the core’s modulus over its thickness. That is exactly the beam that sits on the ground, with the face as the beam and the core as the ground, and it has the same characteristic length:

$$\frac1\beta = \left(\frac{4EI}{k}\right)^{1/4}.$$

For a steel-faced polyurethane cladding panel — faces of 0.5 mm steel with a yield stress of 320 N/mm², an 80 mm core of modulus 4 N/mm² that crushes at 0.12 N/mm² — the spreading length is 20.5 mm. A load on the face is shared over a strip about four times that wide, and the face lifts off its core again beyond 48 mm, where the beam-on-a-bed solution changes sign.

The figure shows the two limits being approached together, at different rates. Under 3 N per mm the core is at six tenths of its crushing strength and the face at nine tenths of its yield stress. The face, not the core, is the part in danger — which is the opposite of what the panel’s usual description, a strong face on a weak core, suggests.

The reason is the face’s thickness entering twice. Its bending stress is its moment over $t^2/6$, and half a millimetre squared is a quarter of a square millimetre: a moment of 13 N·mm per mm of width, a few newtons on a lever a few millimetres long, is enough to yield it. A strong material in a thin sheet is a weak plate. The core, meanwhile, is weak in stress but has 20 mm of spreading to work with, and the load it sees under the middle of the strip is less than a third of what the strip applies.

This is the same arithmetic as a crane wheel on a girder, where the top flange bends under the wheel and spreads it along the web, and the length it spreads over is an output of the flange’s stiffness against the web’s. There the flange is thick and the web is strong, and the question is whether the web cripples. Here the face is thin and the core is weak, and the question is which of the two gives first — but in both the load decides its own bearing length, and in neither can anyone type that length into a calculation before the stiffnesses have been compared.

Two thresholds from one solution #

While nothing has yielded or crushed the whole problem is linear, so both thresholds come from one elastic solution. For a load narrow against the spreading length, the core’s peak stress is $P\beta/2$ and the face’s peak moment $P/4\beta$, so

$$P_{crush} = \frac{2\sigma_c}{\beta}, \qquad P_{yield} = 4\beta M_y = \frac{2\beta f_y t^2}{3}.$$

The first grows as $1/\beta$, which grows as the face’s thickness to the three quarters: a thicker face spreads the load further and the core under it is asked for less. The second grows as $\beta t^2$, as the face’s thickness to the five quarters: a thicker face is stronger in bending faster than it spreads. The two meet at a crossover thickness

$$t^* = \frac{3\sigma_c^2 E}{k\,f_y^2},$$

which is 1.77 mm for this panel under a narrow load. Below it the face yields before the core crushes; above it, the core crushes first. A load wider than the spreading length is carried more by the core directly and moves the crossover to thinner faces.

The exponents are the whole story. Every thickening of the face buys strength in bending faster than it buys spreading, so a thick enough face will always outlast its core, and a thin enough face will always yield over it. Nothing about the materials can change that ordering except by moving $t^*$: the crossover rises as the square of the core’s strength, because a stronger core outlasts more faces, and falls as the square of the face’s yield stress, because a stronger face needs less thickness to outlast the core.

The steel cladding panel sits at a third of its crossover thickness. It yields its face under 3.4 N per mm — 3.4 kN per metre of load line, about a person’s weight on a strip a quarter of a metre long — and a yielded face does not spring back: it is a dent. That is why steel-faced roof panels are walked on only along their supporting purlins, and why the dents they collect are in the face, over an intact core.

A stiffer core protects the face and crushes itself #

The obvious remedy for a denting face is a better core, and the figure shows what “better” has to mean. A stiffer core supports the face more closely, the load spreads over a shorter length, and the face’s moment falls: the load that yields it rises from 2.2 N/mm with a core of 1 N/mm² to 4.6 with one of 16. But the same shorter spreading concentrates the load on the core, and at a fixed crushing strength the core’s own threshold falls, from 7.0 to 3.6. A stiffer core protects the face and crushes sooner itself; the two thresholds cross, and above the crossing the panel’s local strength is the core’s.

So the core has to gain strength as well as stiffness to help. Real foams do: both grow with density, strength faster than stiffness in the lower range, and a denser core is better on both counts. The figure’s warning is for a stiffer core of the same strength — a change of foam chemistry or a fibre-reinforced core that raises the modulus and not the crushing stress — which trades one local failure for the other.

Three panels, and which part gives first #

The three panels span the range. The steel cladding panel dents its face. The aluminium honeycomb panel — 1 mm faces on a 20 mm core far stiffer than foam — spreads a load over only 6.3 mm, so a 10 mm load is a wide one for it, carried mostly by the core directly, and the honeycomb crushes at 36.5 N/mm before the face yields at 44. The glass-fibre panel has faces four times as thick as the steel panel’s on a core seven times as strong, and its core crushes at 26 N/mm, well before its face yields at 49.

The comparison with a small pad on concrete is instructive. A steel bearing plate on a large concrete block carries up to three times the cylinder strength, because the concrete under it cannot crush without spreading sideways and the concrete around it will not let it. A foam or honeycomb core gains almost nothing that way: it crushes by its cells collapsing in place, with next to no sideways expansion, so the material around a loaded patch has nothing to push back against. The core’s crushing strength under a face is very nearly its crushing strength in a test block, and the only enhancement it gets is the face’s spreading — which is the whole of what the elastic solution computes.

The order matters because the two failures are different kinds of damage. A dented face over a sound core is cosmetic until it collects water; a crushed core under a face that sprang back is invisible, and it has taken away the face’s support over a patch — the support the face needs against wrinkling the next time the panel is bent. A crushed core is a latent defect, found by tapping or by the next overload, and a panel that crushes first hides its damage where a panel that dents first shows it.

After the first crush #

A panel whose core crushes first does not fail there. The crushed foam goes on carrying its crushing stress, the face bridges over it as a beam supported by a constant pressure, and the load climbs as the crushed zone widens: from 26 N/mm at first crush to 49 when the face finally yields, by which point the core has crushed over 51 mm. The load–indentation curve softens rather than stops.

The end of it is a mechanism: plastic hinges in the face, one under the load and one at each edge of the crushed zone, with the crushed core pushing up at its crushing stress between them. The work balance gives the collapse load in one line,

$$P_u = 4\sqrt{\sigma_c M_p}, \qquad M_p = \frac{f_y t^2}{4},$$

a load set by the core’s strength and the face’s plastic moment and by nothing else — not the core’s stiffness, not the panel’s span. It is the plastic hinge’s arithmetic applied to a beam lying on a crushing bed, and for the glass-fibre panel it is 66 N/mm, a third above the load at which the face first yields.

The derivation is one line of work and one line of minimising. With the outer hinges a distance $L$ either side of the load and the middle one sinking by $\delta$, the face rotates by $\delta/L$ at each outer hinge and twice that under the load, so it absorbs $4M_p\delta/L$; the core, crushing over a triangle of depth $\delta$ and length $2L$, absorbs $\sigma_c L\delta$. The load does $P\delta$ of work, so any assumed $L$ gives $P = 4M_p/L + \sigma_c L$, and the true mechanism is the one that asks least: $L = 2\sqrt{M_p/\sigma_c}$. That is the upper-bound reasoning the ground’s bearing capacity is built from, done on a problem simple enough to finish. For the glass-fibre panel $L$ is 36.5 mm and for the steel panel 25.8 mm — both longer than the elastic spreading length, because a plastic face with a constant crushing pressure under it has no reason to stay as short as an elastic one.

The steel panel’s collapse load of 6.2 N/mm sits less than twice its first-yield load. A person standing on a steel-faced roof panel between purlins, weight concentrated under a heel, is inside that margin: the dent begins well before collapse, which is the one mercy of a panel on the face-first side of the crossover.

The same beam, twice in this collection #

The face on its core is the second time the beam on an elastic foundation has decided a local failure here. A pile in the ground is the same beam turned vertical, with the soil as the bed, and its characteristic length plays exactly the role the spreading length does here: it says how much of the member is working, and every answer is a multiple of it. The difference is only scale — a pile’s characteristic length is two metres, a steel panel face’s is two centimetres — and the fourth root that makes both of them insensitive to their inputs. The same fourth root also decides when a pile is too short to bend: a member shorter than a couple of characteristic lengths behaves as a rigid body on its bed. A panel face never is, because a panel is always hundreds of spreading lengths wide; the local problem is always the long-beam one. Doubling the core’s stiffness shortens the spreading length by only 16 per cent, which is why the panel’s local behaviour is decided far more by which thresholds cross than by the precise stiffness of anything.

The cladding panel by hand #

For the 0.5 mm steel face: $EI = 210{,}000 \times 0.5^3/12 = 2{,}188$ N·mm per mm; $k = 4/80 = 0.05$ N/mm³; $\beta = (0.05/(4 \times 2{,}188))^{1/4} = 0.0489$ per mm, a spreading length of 20.5 mm. Under a narrow load the core crushes at $2 \times 0.12/0.0489 = 4.9$ N/mm, and the face, whose yield moment is $320 \times 0.5^2/6 = 13.3$ N·mm per mm, yields at $4 \times 0.0489 \times 13.3 = 2.6$ N/mm. The crossover thickness is $3 \times 0.12^2 \times 210{,}000/(0.05 \times 320^2) = 1.77$ mm. The 10 mm load of the figures is not narrow against 20.5 mm, and spreading it raises the face’s threshold to 3.36 while barely changing the core’s; the order is the same.

A strip of face, a Winkler core and a rigid far face #

The load is a strip across the panel’s width, so the face bends in one direction. A foot or a fixing is a patch, and a face under a patch bends in two directions and spreads the load in both, which raises both thresholds; the strip is the conservative idealisation, and the order of the two failures is decided by the same comparison.

The core is a bed of independent springs, each compressing through the whole core thickness. A real core also shears, which spreads the load further than springs do and makes the effective bed stiffer near the load; two-parameter foundation models add that shear term, and they move the thresholds by tens of per cent without moving the crossover by much, because both thresholds move together.

The far face does not move. On a panel spanning between supports, the far face bends too, and a load near mid-span also bends the panel as a whole; the local problem sits on top of the global one, and a face near its yield stress from the panel’s bending has less left for the local moment.

What the pictures cannot show #

That a dent is permanent and a crush is hidden. The figures give the loads at which each begins; what they leave behind is a matter of plasticity in the face and densification in the core, and a dented steel face with a sound core behaves structurally almost as the undented one did, while a crushed core under an undented glass-fibre face may have lost the support the face needs against wrinkling.

Nor can they show the bond. The face and core are assumed to stay joined, and under a local load the interface directly beneath the load is compressed while the interface just beyond the spreading length is pulled apart — the face lifting off where the beam-on-a-bed solution changes sign. A weak bond fails there first, under loads well below either threshold, and a panel that has debonded around an old dent has a free face over an area the figures do not include.

Still open: the honeycomb that is two cores #

A foam core is the same in every direction. A honeycomb core is not: its cells are expanded from a ribbon, and its shear stiffness along the ribbon is about twice its stiffness across it, while its crushing strength through the thickness is the same both ways. A local load on a honeycomb panel spreads further along the ribbon than across it, and the patch it crushes is an ellipse rather than a circle. Whether that makes a honeycomb panel’s local strength depend on the orientation of the load — a fixing’s bearing along the ribbon against across it — and by how much, is the question the two shear moduli leave, and it is the one the panel’s own section calculation already cannot answer.