Foundations — Sandwich structures — face sheets, core

This page assumes you have seen none of the notation in the parent note. We build each symbol from a picture before it is ever used. If a term links to another vault note, that note goes deeper — but you will not need it to follow the line you are reading.

1. Length, thickness, width, depth — the shape words

Before any physics, we need words for how big the pieces are. A sandwich panel is a flat slab, so it has three sizes.

Reading Fig 1. The two magenta bars top and bottom are the face sheets, each of height $t_f$ — notice how thin they are. The hatched pink band between them is the core, of height $c$. The orange double arrow on the right measures the whole stack $h$; the violet arrow underneath measures the span $L$; and the small navy arrow at the left reminds you that the panel also extends into the page by its width $b$. Fix those five arrows in mind — every formula on this page is one of them squared, multiplied, or compared.

Because the two skins sit on the outside of the core, the depths add up:

Why the topic needs these. Every advantage of a sandwich is a story about $h$ being large while $t_f$ stays small. The whole subject lives in the comparison $t_f\ll h$ (read: "$t_f$ is much smaller than $h$"). The symbol $\ll$ is not "less than" — it means "so much less that we are allowed to ignore it next to the bigger thing."

2. Force, and force spread out: $F$, $w$, $p$

These connect by simple multiplication:

Why the topic needs these. During launch the panel is not pushed by one finger; it is pressed everywhere by its own weight times the rocket's acceleration. We turn that even pressure $p$ into a line load $w$ so we can treat the panel as a beam — the simplest object whose bending we know how to predict. (See Beam Bending Theory.)

3. The bending picture: neutral axis, stress, and $y$

Bend a beam and something beautiful happens: the top surface gets squeezed (compression) and the bottom gets stretched (tension). Somewhere in between is a line that is neither — the neutral axis.

Reading Fig 2. The violet dashed line through the middle is the neutral axis, where $y=0$. The magenta sloped line on the right is the stress: it leans away from the vertical, meaning stress grows the further you go from the middle. Above the axis the orange shading marks compression (top fibres squeezed); below, the magenta shading marks tension (bottom fibres stretched). The navy $y$ arrow shows the distance we measure. The key visual is that the stress line touches zero exactly at the neutral axis and fans out fastest at the surfaces.

Where $\sigma =My/I$ comes from

Rather than swallow the law whole, watch it build itself from two ideas you already own.

Idea 1 — strain grows straight-line with $y$. When the beam bends into a gentle arc, plane cross-sections stay flat and simply tilt. A fibre at height $y$ therefore stretches in proportion to $y$: fibres far from the neutral axis stretch most, fibres at $y=0$ not at all. Call the stretch-per-length (the strain) $\epsilon =\kappa y$, where $\kappa$ (Greek "kappa") is the curvature — how sharply the beam is arced. This is pure geometry of an arc; look again at the fanning stress line in Fig 2, which is really this straight-line strain.

Idea 2 — stress is stiffness times strain. For the elastic material, stress = $E$ (defined in §5) times strain, so $\sigma =E\kappa y$. Stress is therefore also a straight line in $y$.

Now the moment $M$ is just the total turning effect of all those little stresses about the neutral axis: each thin slice of area $\mathrm{d}A$ at height $y$ carries force $\sigma \mathrm{d}A$ and a lever arm $y$, so

$M=\int y\sigma \mathrm{d}A=\int y(E\kappa y)\mathrm{d}A=E\kappa \int y^2\mathrm{d}A.$

That leftover integral $\int y^2\mathrm{d}A$ is exactly the "distance-squared weighting" we meet in §4 — it is $I$. So $M=E\kappa I$, which rearranges to $E\kappa =M/I$. Substituting back into $\sigma =E\kappa y$ gives the law:

4. The second moment of area $I$ — "how spread out is the material?"

The integral $\int y^2\mathrm{d}A$ that fell out of the derivation above has a name and a picture. It is the single most important quantity in this topic, so we build it slowly.

Where the rectangle formula and its $1/12$ come from

Take a solid rectangle of width $b$ and some thickness ${\tau }_{\text{rect}}$ (a generic thickness — "whatever height that rectangle happens to have," not the face thickness $t_f$). Its neutral axis runs through the middle, so $y$ runs from $-{\tau }_{\text{rect}}/2$ to $+{\tau }_{\text{rect}}/2$. A horizontal strip at height $y$ has area $\mathrm{d}A=b\mathrm{d}y$, so:

$I_{\text{rect}}={\int }_{-\tau /2}^{+\tau /2}{y}^2(b\mathrm{d}y)=b{\left[\frac{y^3}{3}\right]}_{-\tau /2}^{+\tau /2}=\frac{b}{3}\left(\frac{{\tau }^3}{8}+\frac{{\tau }^3}{8}\right)=\frac{b{\tau }^3}{12}.$

When we apply this to a face sheet we substitute ${\tau }_{\text{rect}}=t_f$; when we apply it to a whole solid plate we substitute the plate's own thickness. The letter is a placeholder, not a new physical quantity.

The parallel-axis theorem — and why you add $A{d}^2$

We rarely have material sitting at the neutral axis. A sandwich skin sits far from it. To handle that we shift a shape away from the axis:

Reading Fig 3. The violet dashed line is again the neutral axis. The magenta bar high above it is a single face sheet, its own centre marked by the navy dot. The orange double arrow is the distance $d$ from that dot down to the neutral axis — for a real skin this is about $h/2$.

Why you add $A{d}^2$. Put the shape's own centre at height $d$. A patch that sits at height $y^{\prime }$ measured from the shape's own centre now sits at height $y=d+y^{\prime }$ from the neutral axis. Feed that into the definition:

$I=\int (d+y^{\prime }{)}^2\mathrm{d}A=\int (d^2+2d{y}^{\prime }+y^{\prime 2})\mathrm{d}A=d^2\int \mathrm{d}A+2d\int y^{\prime }\mathrm{d}A+\int y^{\prime 2}\mathrm{d}A.$

Read the three pieces: the first is $d^2$ times the total area $A$; the middle integral $\int y^{\prime }\mathrm{d}A$ is zero because $y^{\prime }$ is measured from the shape's own balance point (its centroid); the last is just the shape's own $I_{\text{own}}$. So the cross-term vanishes and you are left with a clean sum:

For a sandwich skin, $d\approx h/2$, so its contribution is $A{d}^2=b{t}_f(h/2{)}^2$. Since $t_f\ll h$, this outboard term utterly dwarfs the skin's own tiny $b{t}_f^3/12$ — which is exactly why the parent note throws the small term away.

5. Stiffness properties: $E$, $G$, and stiffness $EI$

The derivation in §3 already borrowed $E$; here is its full picture.

The subscripts tell you which part a symbol belongs to. Keep two families straight:

• Material properties of the faces vs. core: $E_f$ (faces) and $E_c,G_c$ (core).
• Geometry of the faces vs. core: $t_f$ (face thickness) and $c$ (core thickness). These are lengths, not material properties — the same $f$/$c$ subscript just tells you which part it measures.

Why two different moduli? Because the two jobs are different. The faces get stretched and squeezed by bending → we care about $E_f$. The core gets skewed sideways as the panel shears → we care about $G_c$. Using $E$ where you need $G$ (or vice versa) is a category error the topic warns against.

From stiffness to sag: the deflection $\delta$

Bigger $EI$ means less droop. For a simply-supported beam (resting on a support at each end) under a uniform line load $w$, the peak sag at mid-span is:

Reading Fig 4. The navy line is the unloaded straight beam. The magenta curve is the same beam sagging under the orange downward arrows (the line load $w$). The violet double arrow at mid-span marks $\delta$. The caption reminds you that doubling $EI$ halves that arrow — that is the entire promise of a sandwich, drawn as a picture.

6. Density $\rho$, yield stress ${\sigma }_y$, and the "specific" idea

In space, mass is money — every kilogram costs a fortune to launch. So engineers do not ask "how stiff?" but "how stiff per kilogram?"

Why the topic needs this. A sandwich can be $5$–$10\times$ stiffer per kilogram than a solid plate, even though a solid plate is stiffer per cubic metre. The winning number is always the "per kilogram" one, so $E/\rho$, ${\sigma }_y/\rho$ and $\rho$ appear everywhere.

7. Shear, core stress, and section modulus: $V$, $\tau$, $S$

Two last families of symbols close the loop.

Why dividing $I$ by $c_{\text{edge}}$ answers "how much moment can this survive." Rearrange the bending law from §3: $\sigma =My/I$. The material first reaches its limit stress ${\sigma }_y$ at the most stressed point, which is the outermost fibre where $y=c_{\text{edge}}$ is largest. Set $\sigma ={\sigma }_y$ there and solve for the moment: $M_{\max }={\sigma }_y\cdot \frac{I}{c_{\text{edge}}}={\sigma }_{y}S.$ So $S=I/c_{\text{edge}}$ is precisely the geometry factor that converts a material limit ${\sigma }_y$ into a survivable moment — bigger $S$, more moment before yield. That is why it bundles $I$ and $c_{\text{edge}}$ into one number.

Why the topic needs these. Bending stiffness $EI$ tells you how much the panel sags ($\delta$); the core shear ${\tau }_c$ tells you whether the light filling tears; and $S$ tells you the moment the faces can survive. A real design must pass all three — a stiff panel with a weak core still fails. (Shear failure and modal effects connect to Buckling and Instability and Vibration and Modal Analysis.)

How the foundations chain into one number

The map below is causal, not a table of contents: each arrow is an operation ("integrate", "multiply", "divide") that transforms the quantity before it into the quantity after it, ending at the two numbers a designer actually reports — the sag $\delta$ and the core margin.

Follow one path end to end: arc geometry gives a straight-line strain, times $E$ gives stress, summing lever arms ($\int y^2\mathrm{d}A$) gives $I$, times $E$ gives $EI$, and the sag formula turns $EI$ into the droop $\delta$. Two side branches turn $V$ into a core-stress check and ${\sigma }_y$ into a survivable moment. All three feed the single verdict: pass or fail.

Equipment checklist

Cover the right side and test yourself — you are ready for the parent note only if each answer is instant.

Ready? Continue to the parent topic.