Foundations — Classical laminate theory — ABD matrix

Before any equation makes sense, we have to earn every symbol. This page introduces them one at a time, each with a plain-words meaning, a picture, and the reason the topic cannot live without it.

1. The laminate itself — what are we even looking at?

Look at the figure. Each sheet is drawn separately, then pressed together. The whole point of building it this way: a single sheet is very stiff along its fibers but floppy across them, so we stack sheets at different angles to be stiff in every direction at once.

Why the topic needs this: the ABD matrix exists precisely because the stack has layers pointing different ways. If it were one uniform block, ordinary beam/plate formulas would do.

2. The coordinate frame — up, sideways, and through the thickness

Picture a deck of cards lying flat. $x$ and $y$ run along the face of the cards; $z$ is the direction you'd poke a pencil straight down through the deck. The middle card is $z=0$; cards below are negative $z$, cards above are positive $z$.

Why the topic needs this: every integral in the parent note ("$\int zdz$") is a sum over the thickness direction $z$. Without a fixed $z=0$ line to measure from, "distance from the mid-plane" has no meaning.

3. Ply boundary heights $h_k$ and thickness $t_k$

Read the figure like a ruler standing on its end. For a 4-ply stack that is $2$ mm total, the boundaries are $h_0=-1,h_1=-0.5,h_2=0,h_3=0.5,h_4=1$ (mm). Notice the bottom heights are negative — that sign is not decoration, it is what makes symmetric stacks cancel later.

Why the topic needs this: the parent's three matrices are built from $h_k-h_{k-1}$, $h_k^2-h_{k-1}^2$, and $h_k^3-h_{k-1}^3$. Each is just "top height minus bottom height" raised to a power — you cannot compute one without $h_k$.

4. Strain $ϵ$ and shear strain $\gamma$ — measuring "how much it deformed"

The left square in the figure stretches (normal strain). The right square tilts (shear strain). Together, $[{ϵ}_x,{ϵ}_y,{\gamma }_{xy}]$ fully describe any small in-plane deformation.

Why the topic needs this: the response the ABD matrix predicts is strain. Strain is the answer, so we must name it before writing the equation.

5. Mid-plane strain ${\mathbf{ϵ}}^0$ and curvature $\mathbf{\kappa }$

Why the topic needs this: these six numbers (${\mathbf{ϵ}}^0$ and $\mathbf{\kappa }$) are the unknowns the ABD matrix solves for.

6. Stress $\mathbf{\sigma }$ — the internal push inside each ply

Picture the fibers as springs: strain is how far you pulled them, stress is how hard they pull you back.

Why the topic needs this: the parent integrates stress over thickness to get the forces and moments. Stress is the bridge from deformation to load.

7. The reduced stiffness matrix $[Q{]}_k$ — each ply's personality

The subscript $k$ is doing heavy lifting: it says this table belongs to layer $k$ at its own angle. Where $[Q{]}_k$ actually comes from — building it from $E_1,E_2,G_{12},{\nu }_{12}$ and rotating it by the fiber angle — is its own story in Reduced Stiffness Matrix Q and Transformation of Stiffness.

Why the topic needs this: $[Q{]}_k$ is the only place material properties enter. Every entry of $[A]$, $[B]$, $[D]$ is a weighted sum of the $[Q{]}_k$ tables.

8. Force resultant $\mathbf{N}$ and moment resultant $\mathbf{M}$

Why the topic needs this: $\mathbf{N}$ and $\mathbf{M}$ are the inputs — what you apply. The whole ABD relation is "$\{\mathbf{N},\mathbf{M}\}$ in, $\{{\mathbf{ϵ}}^0,\mathbf{\kappa }\}$ out."

9. The integral $\int \dots dz$ — "add up through the thickness"

The three integrals the parent needs are elementary:

These are the powers $1$, $2$, $3$ that produce $[A]$, $[B]$, $[D]$ respectively. The why of each: $\int 1$ counts area (stretch), the extra $z$ in $\int z$ is the lever arm (coupling), and $\int z^2$ is lever-arm-squared (bending).

Why the topic needs this: the definitions of $[A]$, $[B]$, $[D]$ are nothing but $[Q{]}_k$ multiplied by these three integrals and summed over plies.

10. The result: $[A]$, $[B]$, $[D]$ in one sentence each

Whether $[B]$ vanishes depends on stacking symmetry — that is the whole subject of Symmetric and Balanced Laminates.

How the foundations feed the topic

Every arrow says "you need the left box before the right one makes sense." The final box is the parent topic.

Equipment checklist

Cover the right side and answer before revealing.