The stiffness that belongs to the span

Some of a beam’s deflection is not bending, and for a rolled steel beam at ordinary proportions it is about one per cent — a correction, quoted once and then dropped.

For a sandwich panel it is a quarter of the answer, and the reason is a single ratio.

Two deflections with different powers #

A beam under a distributed load deflects by two mechanisms, and they scale differently.

$$\delta_b = \frac{5}{384}\frac{qL^4}{EI} \qquad \delta_s = \frac{1}{8}\frac{qL^2}{GA_s}$$

Bending goes as $L^4$ and shear as $L^2$, so the ratio goes as $L^{-2}$ — and the shear share of the total falls as the square of the span.

That single curve carries the whole of the argument. The shear share is not a property of the material or of the section; it is a property of the proportion, and a panel that is comfortable at one span is a different member at half of it.

Which free body produced the number #

Two free bodies, one for each mechanism, and they are cut differently.

For bending, cut the panel on a plane perpendicular to its axis and take the moment of the stresses on the cut. The faces are far from the neutral axis and carry a couple; the core is close to it, is soft, and carries almost nothing. The section’s rigidity is $E_f A_f d^2/2$ plus two negligible terms.

For shear, cut the same section and take the shear on it. Now the roles reverse: the faces are thin and carry almost no shear, and the core — the whole depth of it — carries essentially all of it, at a nearly uniform stress $V/bd$.

The two mechanisms use different parts of the same section. That is what makes the sandwich the extreme case: the material that supplies the bending stiffness supplies no shear stiffness, and the material that supplies the shear stiffness supplies no bending stiffness. There is no overlap at all.

For a solid beam the same two free bodies use the same material, its $E$ and its $G$ differ by a factor of 2.6, and the shear term is small. A section has two areas and the tables give one of them; a sandwich has two areas that share no material.

There is a second reading of the two powers that says which members are at risk without any numbers at all.

Bending deflection is an integral of curvature, and curvature accumulates twice along the member — once to a slope and once to a displacement. Shear deflection is an integral of shear strain, and shear strain accumulates once. Two integrations against one, which is where the $L^4$ against $L^2$ comes from and why it is a property of the mechanism rather than of any material.

So the shear share rises whenever the member gets shorter relative to its depth, and it does so for every member ever built. What differs between members is only where on that curve they sit, and the ratio $E/G$ of the two mechanisms’ materials is what places them.

That gives a one-line test. A member is shear-sensitive when its span-to-depth ratio is below about $\sqrt{E/G}$, which is 1.6 for steel — no beam is that short — and 53 for the sandwich here, which is longer than most panels are. The same criterion, two answers a factor of thirty apart, and the whole difference is in a material property nobody looks up.

The ratio that decides it #

The shear share at a given slenderness depends on one dimensionless group: $E_f/G_c$, the face modulus over the core’s shear modulus.

For the panel above that is 70,000 over 25 — 2,800. For a steel beam the equivalent ratio is $E/G = 2.6$. Three orders of magnitude, and the shear share moves with it.

A change in one material property that appears nowhere in the bending calculation has quadrupled a deflection. The core’s shear modulus is the least-specified number in a sandwich panel and one of the two that decide its serviceability, which is a poor arrangement and is the reason sandwich panel design is done from manufacturers’ tested values rather than from first principles.

The effective stiffness is not a section property #

Combining the two terms gives an effective flexural rigidity, and it has the span in it:

$$\left(EI\right)_{\text{eff}} = \frac{EI}{1 + \dfrac{k\,EI}{GA_s L^2}}$$

with $k$ a constant depending on the load pattern. That expression is not a section property, and it cannot be tabulated, because it contains $L$.

Three consequences follow and each is a habit that has to be unlearned.

A panel’s stiffness cannot be quoted. Manufacturers quote $EI$ and $GA_s$ separately for exactly this reason, and a designer combines them at the span in hand.

Superposition across spans fails. A continuous panel’s spans have different effective stiffnesses if they have different lengths, so the moment distribution between them is not the one $EI$ alone would give.

And the load pattern enters. The constant $k$ differs between a uniform load and a point load, so a panel has a different effective stiffness for each — which is a stranger statement than it sounds and is a straightforward consequence of the two terms having different shapes along the span.

How the two members differ, in one comparison #

Setting the steel beam and the sandwich panel side by side makes the mechanism visible without any algebra.

The steel beam’s flanges carry the moment and its web carries the shear, and the two are welded or rolled from one billet. The web is 57 per cent of the area and carries 97 per cent of the shear; the flanges are 43 per cent and carry most of the moment. Different parts, same material, moduli differing by 2.6.

The sandwich’s faces carry the moment and its core carries the shear, and the two are different materials chosen independently. The faces are aluminium at 70 GPa; the core is foam at 25 N/mm² in shear. Different parts, different materials, moduli differing by 2,800.

The structural arrangement is identical and the numbers are three orders of magnitude apart. That is the whole of why one member’s shear deflection is a footnote and the other’s is a design case, and it says where else to expect trouble: any member whose shear is carried by something softer than what carries its moment.

What the core is actually for #

The core’s job is separation, and the arithmetic of that is worth putting beside the shear argument.

The stiffness goes as the square of a dimension that is almost free to buy, which is what two skins and the space between them are for and is the whole economic case for the form.

But thickening the core does not help the shear term in proportion. The shear stiffness $G_c b d$ grows linearly with the separation while the bending stiffness grows as its square, so deepening a sandwich makes it relatively more shear-flexible — and the share of shear deflection rises as the panel is improved. That is the same trade the previous rung found on a truss, with the web taking over from the chords as the depth grows.

Where it stops being a sandwich #

Every argument here depends on the faces being much stiffer than the core, and the generator refuses a section where they are not — faces at least twenty times stiffer, or it is not a sandwich.

Past that the section is a composite of two materials and is analysed as one, with a transformed section and a stress distribution that is continuous rather than a pair of blocks. The sandwich is a limiting case rather than a category, and the interesting members are near the boundary: a plywood-webbed beam, a concrete slab on a light frame, a laminated glass panel with a thick interlayer.

The two failure modes that belong specifically to the sandwich are also boundary effects. Core shear failure is the shear stress in the core reaching its own strength, which is the same free body as the shear deflection and the same number in the denominator. Face wrinkling is a face buckling into the core as a strut on an elastic foundation, at a stress that contains no length at all — 236 N/mm² for this panel, whatever its span.

That second one is worth pausing on. A buckling stress with no length in it is unlike every other stability result in this collection, and the reason is that the face’s buckling half-wavelength is set by the core’s stiffness rather than by the member’s ends — the same structure as a strut on an elastic foundation, which forgets its own length for the same reason.

What it does to a design decision #

The span-dependence has a practical edge, and it is about which panel to specify rather than about how to analyse one.

A manufacturer’s table lists panels by core thickness and quotes an allowable span for a deflection limit. Those spans are computed with both terms, so they are correct — and the ratio between two panels’ allowable spans is not what a bending calculation would give.

Doubling the core thickness quadruples the bending rigidity and doubles the shear rigidity. If bending alone governed, the allowable span would rise by $4^{1/4} = 1.41$; because the shear term is present and grows relatively worse, the real gain is less, and on a short-span panel it can be substantially less.

So the returns on core thickness diminish faster than the bending arithmetic suggests, and the diminishing is worst exactly where a designer is most tempted to reach for it — a short heavily loaded panel, where the shear share is largest.

The opposite decision has the opposite shape. Improving the core’s shear modulus — a denser foam, a honeycomb instead of a foam — does nothing at all for the bending term and everything for the shear one. On a long panel it is wasted; on a short one it is the only move that works. Two specifications, two spans, and the choice between them inverts across the range, which is not something a single stiffness number could ever have expressed.

The same effect where nobody expects it #

Shear flexibility is not confined to sandwiches, and three other members have it for the same structural reason.

A truss’s web deflection is shear deflection, term by term: the diagonals and verticals extend under the panel shear, and their contribution to the movement is exactly the $\int V v/GA_s$ of a solid beam with the integral replaced by a sum.

A Vierendeel is worse, because it carries its shear by bending its chords in double curvature, which is a much softer mechanism than an axial diagonal.

And a laced or battened column is a shear-flexible member in compression rather than in bending, which is why its buckling load is a harmonic sum of a bending term and a shear term rather than the Euler load alone.

The unifying statement is that a member built out of pieces has a shear stiffness supplied by the connection between the pieces, and the connection is always softer than the pieces. A solid beam is the only case where it is not, and it is the case every intuition is trained on.

What to carry away #

The shear share falls as the square of the span. It is a proportion rather than a property, so the same section is a different member at every length.

The deciding ratio is $E/G$ of whatever carries each mechanism. 2.6 for steel, 2,800 for a rigid-foam sandwich, and 11,667 for a light one.

The effective stiffness contains the span, so it cannot be tabulated, superposed across unequal spans, or quoted independently of the load pattern.

And deepening the section makes it relatively more shear-flexible, because the bending stiffness goes as the square of the separation and the shear stiffness only as the first power.

The measurement that settles it #

There is a clean experiment behind all of this, and it is worth knowing because it is how the two rigidities are actually obtained.

Test the same panel at two different spans and measure the mid-span deflection under a known load in each. Two equations, two unknowns — $EI$ and $GA_s$ — and both fall out with no assumption about the section’s geometry, the core’s modulus or the faces’ thickness.

That is the standard method and it exists because the alternative does not work. Computing $GA_s$ from a core’s quoted shear modulus requires a number that varies with density, temperature, age and manufacturing batch, and which is measured on a coupon rather than on a panel. Testing two spans measures the member instead of the material, which is the right thing to measure when the member is a laminate.

It also produces the check that catches a bad panel. A three-span test is over-determined: it gives two estimates of each rigidity, and a disagreement between them means the panel is not behaving as a sandwich — usually because the bond between a face and the core has failed somewhere, which is the failure mode that nothing on the outside shows.

A measured stiffness describes the structure that exists rather than the one that was specified, and for a manufactured laminate the gap between the two is wider than for anything else in this collection.

Where the model stops #

The core is treated as carrying uniform shear and no bending. It carries a little of both, and for a thick core the “little” is a few per cent of the second moment — which the figures report and which is small rather than absent.

The faces are membranes. Their own bending about their own centroids is 0.003 per cent of the section’s second moment, which justifies the treatment and stops doing so for thick faces.

The core is linear and elastic. Foam cores creep, and a sandwich panel’s long-term deflection under sustained load is governed by a shear creep nobody measures.

The panel is prismatic and simply supported. A continuous panel has different effective stiffnesses in different spans, and the moment distribution between them is not what $EI$ alone gives.

And the two mechanisms are added. That is exact for the deflection and not for the strength, where the interaction between face stress and core shear is a genuine interaction rather than a sum.

What to carry away, once more #

Two mechanisms, two powers of the span. Bending as the fourth, shear as the second, so the shear share falls as the square and a member’s stiffness is a function of how long it is.

Look for members whose shear is carried by something softer. A sandwich core, a truss web, a Vierendeel chord in double curvature, a battened column’s lacing — every one of them is a shear stiffness supplied by a connection rather than by a section.

And the test is $L/d$ against $\sqrt{E/G}$, which is 1.6 for solid steel and 53 for the panel here. Everything on this page follows from where a member sits relative to that number.

Three other flexibilities are usually left out of a deflection for the same reason this one is — because the beam formula does not contain them. The support that moves adds a rigid-body term; a truss’s member extensions are the axial equivalent of this one; and where a deflection actually comes from is the accounting that puts all of them in one column. A deflection quoted to three figures from a formula containing one of the four is precise about the wrong thing.

The ladder from here #

Later rungs on this anchor: the shear coefficient κ derived by strain energy for the standard families of section. Timoshenko beam theory set out properly, with the rotation of the section as an independent variable rather than the slope of the deflection. Shear deflection in redundant structures, where softening one member in shear redistributes force to another. Deep beam design by strut-and-tie, which abandons the section entirely and is the right answer at slendernesses below about two. Shear deformation in coupled shear walls, where it governs the coupling beam. Shear lag, which is the same failure of the plane-section assumption seen along a flange instead of through a depth. And core creep, which is a sandwich’s serviceability limit and is not in any of the equations here.

Sandwich construction was worked out during the Second World War for aircraft — the Mosquito’s fuselage is a balsa-cored sandwich — and the theory arrived with it, because nothing in existing beam theory covered a member whose core was three orders of magnitude softer than its faces. Everything on this page is in Allen’s 1969 monograph, and the shape of the argument has not changed since: two mechanisms, two different pieces of the section, and a ratio that decides which one a designer is actually dealing with.