The cross-section coordinate $y$ is measured upward from the base (ski/board riding surface). The wood core sits between the below-core (tension) and above-core (compression) composite layers.
All thickness quantities are in mm. Moduli are in GPa. Stiffness quantities (EI, GJ) are computed per unit width (i.e. for a strip of width $b = 1\,\text{mm}$), so the unit is $\text{GPa·mm}^3/\text{mm}$. To obtain the stiffness of the full board, multiply by the board width in mm.
Each composite ply is characterised by the following properties:
| Property | Symbol | Unit | Description |
|---|---|---|---|
| Tensile modulus | $E_t$ | GPa | Young's modulus on the tension side of NA |
| Compressive modulus | $E_c$ | GPa | Young's modulus on the compression side of NA |
| Tensile strength | $\sigma_t$ | MPa | Failure stress in tension |
| Compressive strength | $\sigma_c$ | MPa | Failure stress in compression |
| In-plane shear modulus | $G_{xy}$ | GPa | Dominates torsion; highest for ±45° plies |
| Density | $\rho$ | g/cm³ | Layer density |
| Default thickness | $t$ | mm | Nominal laminate thickness |
Wood core types (paulownia, poplar, ash, beech) use the same property set with separate tension and compression moduli representing the anisotropy of wood along its grain direction.
Function analyzeSection(belowLayers, coreThickness, aboveLayers, coreType) analyses a single cross-section at a given position along the board. Layers are stacked from $y=0$ upward in the order: below-core plies → wood core → above-core plies.
The neutral axis $y_\text{NA}$ is the $y$-position where bending stress is zero. Because the tensile and compressive moduli of composite plies may differ ($E_t \neq E_c$), the neutral axis is found iteratively:
where for each layer $k$: $A_k = t_k \cdot b$ is the cross-sectional area (per unit width $b=1$), $y_{m,k}$ is the centroidal $y$-coordinate, and $E_k$ is assigned as $E_t$ for layers fully below $y_\text{NA}$ and $E_c$ for layers fully above it. Layers split by the neutral axis contribute both moduli proportionally. Iteration continues until $|y_\text{NA}^\text{new} - y_\text{NA}^\text{old}| < 10^{-8}\,\text{mm}$ (typically converges in < 10 steps).
The flexural rigidity per unit width is computed by summing the parallel-axis contributions of each layer about $y_\text{NA}$:
where $\bar{y}_k = (y_{\text{bot},k} + y_{\text{top},k})/2$ is the layer centroid. For layers split by $y_\text{NA}$, the tension and compression sub-portions are treated separately with their respective moduli.
Unit: $\text{GPa·mm}^3/\text{mm}$ (per unit width). Displayed scaled by $10^{-3}$ as $\times 10^3\,\text{GPa·mm}^3/\text{mm}$.
Torsional stiffness is computed using Classical Laminate Theory (CLT). The plate torsion stiffness $D_{66}$ per unit width is:
where $z = y - y_\text{geom}$ is measured from the geometric mid-plane $y_\text{geom} = t_\text{total}/2$ (not from the neutral axis). The factor-of-2 relationship between $D_{66}$ and the effective torsional stiffness for a plate gives:
Unit: $\text{GPa·mm}^3/\text{mm}$ (per unit width). To get the absolute torsional stiffness of the board, multiply by the board width: $GJ_\text{board} = GJ \times w$.
Stiffness balance measures the symmetry of bending stiffness between the tension and compression faces:
where the sums run over the portions of each layer on the respective side of $y_\text{NA}$. A value of 1.0 means both faces are equally stiff.
Analogous to stiffness balance, using strength instead of modulus:
A value of 1.0 means both faces accumulate the same "strength-thickness product". Values far from 1.0 indicate that one face will yield before the other under increasing bending load.
Identifies which side of the laminate has the layer that will fail first (the weakest link). The failure strain of each layer is:
The minimum failure strain on each side is found:
A value of 1.0 means both sides reach their failure strain simultaneously. Values significantly different from 1.0 indicate the lower-side face will fail first.
Function analyzeBoard(belowLayers, aboveLayers, coreProfile, coreType) applies analyzeSection at every point in the core thickness profile. The core profile is a list of $(x_i, t_{\text{core},i})$ pairs giving the core thickness at each position $x_i$ along the board length.
The result is a 1-D array of section analyses indexed by position. Charts show how EI, GJ, balance, and neutral axis vary from tip to tip. The midpoint section (at the position of maximum core thickness) is used for the headline metric values.
The single-number "Standard Hardness" answers: what constant EI would give the same mid-span deflection as the real, varying-stiffness board under a centred 3-point load?
For a simply-supported beam of length $L$ with a central point load $P$, the virtual-work deflection integral is:
where $m(x)$ is the bending moment diagram for a unit load ($P=1$):
The equivalent uniform stiffness that produces the same deflection is then:
The integral is evaluated numerically using the trapezoidal rule over the core profile points. The $m(x)^2/EI$ weighting means the stiff centre contributes more than the thin, flexible tips — soft tips pull the equivalent stiffness well below the midpoint value.
Compare Standard Hardness to EI midpoint: the ratio reveals how much the tip taper softens the overall board response. For a board with no taper, both values are equal.
The analogous question for torsion: what constant GJ would give the same total twist angle under an applied end torque as the real board?
For a shaft of varying torsional stiffness $GJ(x)$ subjected to a torque $T$, torsional compliance adds in series. The total twist angle is:
The equivalent uniform stiffness that gives the same total twist is:
This is the harmonic-mean-like average of $GJ$ along the board. Because compliance sums in series, the softest sections dominate — the thin, low-$GJ$ tips have a disproportionately large influence. This is the opposite weighting from the bending case, where the stiff centre dominates.
Compare Standard Torsion Stiffness to GJ midpoint: a large difference means the tips are torsionally much softer than the centre and significantly limit the whole-board torsional rigidity.
Uniformity measures how consistently the Stiffness Balance is maintained along the entire board length:
where $\text{Balance}_\text{centre}$ is the balance value at the 50th percentile position along the board, and $\text{Balance}_\text{tips}$ is the average of the first and last three profile points.
At the thin tips, the wood core contributes very little stiffness, so the composite layers on both faces dominate the balance ratio. Any asymmetry between below-core and above-core layups that is masked by the thick core in the centre becomes fully exposed at the tips.
The flex test panel simulates a symmetric 3-point bend with supports placed symmetrically about the board centre. Given the input parameters (applied moment $M$, board width $w$, and distance between supports $s$):
where $L$ is the board length from the core profile.
The same virtual-work integral as for Standard Hardness is applied, but restricted to the span $[x_L, x_R]$:
where $m_s(x)$ is the moment diagram for a unit load at the centre of the span.
The moment $M$ is applied at the span centre. The equivalent point force and mid-span deflection are:
where $M$ is in N·mm, $s$ in mm, $EI_\text{span}$ in GPa·mm³/mm, and $w$ is the board width in mm. The factor $10^3$ converts GPa to N/mm².
If a measured deflection $\delta_\text{meas}$ is entered, the actual EI is back-calculated by inverting the deflection formula:
The calibration factor $EI_\text{measured} / EI_\text{span}$ indicates whether the real board is stiffer (> 1) or softer (< 1) than the model predicts.
| Metric | Formula / Definition | Unit | Target / Good range |
|---|---|---|---|
| Stiffness Balance | $(E{\times}t)_\text{tension}\ /\ (E{\times}t)_\text{compression}$ | — | 0.85 – 1.15 |
| Uniformity | $|$Balance$_\text{centre}$ − Balance$_\text{tips}|$ | — | < 0.05 |
| Strength Balance | $(\sigma{\times}t)_\text{tension}\ /\ (\sigma{\times}t)_\text{compression}$ | — | ≈ 1.0 |
| Strain Balance | $\varepsilon^\text{min}_\text{tension}\ /\ \varepsilon^\text{min}_\text{compression}$ | — | ≈ 1.0 |
| Neutral Axis | $y_\text{NA}$ from iterative $\sum E_k A_k y_k / \sum E_k A_k$ | mm from base | — |
| E×t Tension | $\sum_{k\,\in\,\text{tension}} E_{t,k}\,t_k$ | GPa·mm | — |
| E×t Compression | $\sum_{k\,\in\,\text{compression}} E_{c,k}\,t_k$ | GPa·mm | — |
| EI midpoint | $\sum_k E_k [b t_k^3/12 + b t_k d_k^2]$ at thickest section | ×10³ GPa·mm³/mm | — |
| EI mean | $\int_0^L EI(x)\,dx\ /\ L$ | ×10³ GPa·mm³/mm | — |
| Standard Hardness | $L^3 \Big/ \Big(48 \int_0^L m(x)^2/EI(x)\,dx\Big)$ | ×10³ GPa·mm³/mm | Compare boards at equal width |
| GJ midpoint | $2D_{66} = \frac{2}{3}\sum_k G_{xy,k}(z_\text{top}^3 - z_\text{bot}^3)$ at thickest section | ×10³ GPa·mm³/mm | — |
| Standard Torsion Stiffness | $L \Big/ \int_0^L 1/GJ(x)\,dx$ | ×10³ GPa·mm³/mm | Compare boards at equal width |
| Flex Test EI span | $s^3 \Big/ \Big(48 \int_{x_L}^{x_R} m_s^2/EI\,dx\Big)$ | ×10³ GPa·mm³/mm | — |
| Predicted deflection δ | $M s^2 / (12 \cdot EI_\text{span} \cdot w \cdot 10^3)$ | mm | — |
| Spring rate k | $P / \delta = 4M / (s \cdot \delta)$ | N/mm | — |
Laminate Calculator v2 · Oxess / C-Sink Sports · Documentation generated from source code