Composite Overwrapped
Pressure Vessel Design Suite

All internal computation is SI (Pa, m, kg, K); the UI's unit toggle converts only at the display/input edge. Stress resultants (force per unit length) carry units N/m = Pa·m.

Stored-gas mass uses the Redlich-Kwong equation of state, not the ideal gas law — at 20–70 MPa pressurant conditions the real-gas deviation is non-negligible:

$$ P = \frac{RT}{V_m - b} - \frac{a}{\sqrt{T}\, V_m (V_m + b)} $$

with constants from each gas's critical temperature/pressure:

$$ a = \frac{0.42748\, R^2 T_c^{2.5}}{P_c}, \qquad b = \frac{0.08664\, R\, T_c}{P_c} $$

Given P, T, the molar volume is solved implicitly via Brent's method (scipy.optimize.brentq), then density and compressibility follow directly:

$$ \rho = \frac{M}{V_m}, \qquad Z = \frac{P V_m}{R T} $$

Volume/area from numerical revolution of the meridian (Pappus's theorems in integral form). For a fiber path that is a geodesic on a surface of revolution, Clairaut's relation holds along the entire path:

$$ r(s)\sin\alpha(s) = \text{const} = r_\text{polar}, \qquad \sin\alpha_\text{cyl} = \frac{r_\text{polar}}{R} $$

The dome meridian is generated from the local slope equal to the local fiber angle, with a closure factor enforcing exact tangency at the cylinder junction:

$$ \frac{dz}{dr} = \frac{\tan\alpha(r)}{\sqrt{1-(r/R)^2}} $$

Substituting $r=R\sin\theta$ removes both endpoint singularities analytically:

$$ \frac{dz}{d\theta} = R\tan\alpha(R\sin\theta), \qquad \theta \in \left[\arcsin\tfrac{r_\text{polar}}{R},\ \tfrac{\pi}{2}\right] $$

solved via scipy.integrate.solve_ivp (RK45), cached per r_polar/R ratio.

Fiber-only membrane equilibrium (matrix neglected — conservative first-pass sizing). Thin-cylinder resultants:

$$ N_\text{hoop} = PR, \qquad N_\text{axial} = \frac{PR}{2} $$

Solved for helical (±α) and hoop ply thickness at design fiber stress $\sigma_f = f_\text{design}\cdot X_t$:

$$ t_h = \frac{N_\text{axial}}{\sigma_f \cos^2\alpha}, \qquad t_\theta = \frac{N_\text{hoop} - \sigma_f\, t_h \sin^2\alpha}{\sigma_f} $$

Total thickness $t_h+t_\theta=(N_\text{axial}+N_\text{hoop})/\sigma_f$ is independent of α up to the angle where $t_\theta\to0$:

$$ \tan^2\alpha_\text{ideal} = \frac{N_\text{hoop}}{N_\text{axial}} = 2 \quad\Rightarrow\quad \alpha_\text{ideal} = \arctan\sqrt{2} \approx 54.7356^\circ $$

Beyond it, $t_\theta$ clamps to zero and $t_h$ alone must grow — validation case 2 confirms this transition numerically.

Reduced ply stiffness in material axes:

$$ Q_{11}=\frac{E_1}{1-\nu_{12}\nu_{21}},\ \ Q_{22}=\frac{E_2}{1-\nu_{12}\nu_{21}},\ \ Q_{12}=\frac{\nu_{12}E_2}{1-\nu_{12}\nu_{21}},\ \ Q_{66}=G_{12} $$

transformed to laminate axes via $[T]$ ($m=\cos\theta,\ n=\sin\theta$) and the Reuter matrix $[R]=\text{diag}(1,1,2)$:

$$ [\bar Q]_k = [T]^{-1}[Q]_k [R][T][R]^{-1} $$

ABD assembly over ply interface coordinates $z_{k-1},z_k$:

$$ A_{ij}=\sum_k [\bar Q_{ij}]_k(z_k-z_{k-1}),\quad B_{ij}=\frac12\sum_k [\bar Q_{ij}]_k(z_k^2-z_{k-1}^2),\quad D_{ij}=\frac13\sum_k [\bar Q_{ij}]_k(z_k^3-z_{k-1}^3) $$

Membrane solve for the pressure-vessel case ($M=0$):

$$ N = [A]\,\varepsilon^0, \qquad N = [N_\text{axial},\ N_\text{hoop},\ 0]^T $$

inverted for midplane strain, then per-ply stress from $[\sigma]_k=[Q]_k[\varepsilon]_k$ after rotating strain into material axes. Effective laminate moduli, back-calculated from A:

$$ E_x = \frac{A_{11}A_{22}-A_{12}^2}{A_{22}\, t}, \qquad E_y = \frac{A_{11}A_{22}-A_{12}^2}{A_{11}\, t} $$

Liner and composite as two bonded concentric thin cylinders, equal strain compatibility:

$$ \varepsilon = \frac{N}{E_\text{liner}t_\text{liner} + E_\text{composite}t_\text{composite}}, \qquad \sigma_\text{liner}=E_\text{liner}\varepsilon,\quad \sigma_\text{composite}=E_\text{composite}\varepsilon $$

In the liner-only limit this collapses exactly to Barlow's formula $\sigma_\text{hoop}=PR/t$ (validation case 1). Lamé thick-wall cross-check (inner radius a, outer b), maximized at the bore:

$$ \sigma_{\theta,\max} = \frac{P(a^2+b^2)}{b^2-a^2} $$

flagged once t/R ≥ 0.1 makes the thin-wall estimate meaningfully non-conservative (validation case 11 confirms convergence as t/R→0). Von Mises biaxial yield — the liner sees hoop and axial stress at once:

$$ \sigma_\text{vM} = \sqrt{\sigma_\text{hoop}^2 - \sigma_\text{hoop}\sigma_\text{axial} + \sigma_\text{axial}^2} $$

(equal-biaxial identity $\sigma_\text{vM}=\sigma$ exactly — validation case 10). External-pressure buckling, classical thin-shell (long-cylinder, n=2 lobe):

$$ P_{cr,\text{classical}} = \frac{E_\text{liner}}{4(1-\nu^2)}\left(\frac{t}{R}\right)^3, \qquad P_{cr} = \gamma_\text{ext}\, P_{cr,\text{classical}},\ \ \gamma_\text{ext}=0.8\ (\text{default}) $$

During proof pressure the liner can yield in hoop tension while the composite stays elastic; on depressurization the elastic composite springs back further than the now permanently-stretched liner, leaving the liner in residual compression (validation case 6 confirms the sign). Elastic-plastic proof strain:

$$ \varepsilon_\text{proof} = \frac{P_\text{proof}R - \sigma_y t_\text{liner} + E_h \varepsilon_y t_\text{liner}}{E_h t_\text{liner} + E_\text{composite}t_\text{composite}}, \qquad \varepsilon_y = \frac{\sigma_y}{E_\text{liner}} $$

Elastic unload to zero, then reload elastically to MEOP:

$$ \sigma_{\text{liner,res}} = E_\text{liner}(\varepsilon_\text{after unload}-\varepsilon_p), \qquad \sigma_\text{liner}(\text{MEOP}) = \sigma_{\text{liner,res}} + E_\text{liner}\cdot \frac{P_\text{MEOP}R}{E_\text{liner}t_\text{liner}+E_\text{composite}t_\text{composite}} $$

Netting-scale:

$$ P_{burst,\text{netting}} = P_\text{design}\cdot\frac{X_t}{\sigma_{f,\text{design}}} $$

First-ply-failure: since the CLT membrane response is linear in pressure for a fixed stiffness state, each ply's strength ratio at a reference pressure scales directly to its failure pressure:

$$ P_{burst,\text{FPF}} = P_\text{ref}\cdot\min_k R_k $$

Last-ply-failure (progressive ply-discount): after each matrix failure, that ply's transverse/shear stiffness is discounted (E₂, G₁₂→×10^-6), the ABD matrix rebuilt, repeated until a fiber-dominated failure occurs — the true burst point.

PV/W historical correlation (sanity check, not predictive):

$$ \frac{PV}{W} = \frac{P_{burst}\cdot V_\text{internal}}{W_\text{composite}}, \qquad \left[\frac{PV}{W}\right]_\text{m} = \frac{PV/W}{g} $$

Typical flight-qualified bands: carbon/epoxy Type IV ≈ 25–45 km; aramid/epoxy ≈ 15–25 km.

Max stress/strain (no mode interaction):

$$ R = \min\left(\frac{X_{t/c}}{\sigma_1},\ \frac{Y_{t/c}}{\sigma_2},\ \frac{S}{|\tau_{12}|}\right) $$

Tsai-Wu (quadratic interaction surface):

$$ F_1\sigma_1+F_2\sigma_2+F_{11}\sigma_1^2+F_{22}\sigma_2^2+F_{66}\tau_{12}^2+ 2F_{12}\sigma_1\sigma_2 = 1, \qquad R = \frac{-b+\sqrt{b^2+4a}}{2a} $$

Tsai-Hill (anisotropic Hill-yield adaptation):

$$ \left(\frac{\sigma_1}{X}\right)^2 - \frac{\sigma_1\sigma_2}{X^2} + \left(\frac{\sigma_2}{Y}\right)^2 + \left(\frac{\tau_{12}}{S}\right)^2 = \frac1{R^2} $$

Puck (2D action-plane, mode-resolved fiber/inter-fiber failure) — Mode A (transverse tension + shear), Mode B (shear-dominated compression), Mode C (compression-dominated):

$$f_{E,A} = \sqrt{\left(\frac{\tau_{21}}{S}\right)^2 + \left(1-p_{\perp\parallel}^{+}\frac{Y_t}{S} \right)^2\left(\frac{\sigma_2}{Y_t}\right)^2} + p_{\perp\parallel}^{+}\frac{\sigma_2}{S}$$

$$f_{E,C} = \left[\left(\frac{\tau_{21}}{2(1+p_{\perp\perp}^{-})S}\right)^2 + \left(\frac{\sigma_2}{Y_c}\right)^2\right]\frac{Y_c}{-\sigma_2}, \qquad R_\text{Puck}=\min(R_{FF},R_{IFF})$$

Liner metal fatigue (Basquin):

$$ \sigma_a = \sigma_f'(2N)^b \quad\Rightarrow\quad N = \frac12\left(\frac{\sigma_a}{\sigma_f'}\right)^{1/b} $$

Mean-stress correction (Goodman or Walker):

$$ \sigma_{a,\text{eq}} = \frac{\sigma_a}{1-\sigma_m/\sigma_u}, \qquad \sigma_{a,\text{eq}} = \sigma_{\max}\left(\frac{1-R_\sigma}2\right)^{1-\gamma} $$

Composite residual-strength wear-out:

$$ \sigma_\text{residual}(n) = \sigma_\text{static} - (\sigma_\text{static}-\sigma_a) \left(\frac{n}{N}\right)^c $$

Miner's cumulative linear damage:

$$ \sum_i \frac{n_i}{N_i} \le 1 $$

Classical Volkersen shear-lag bondline load transfer near the boss:

$$ \tau(x) = \tau_{\max} e^{-x/\lambda}, \qquad \lambda = \left[\frac{G_a}{t_a}\left(\frac1{E_\text{boss}t_\text{boss}} + \frac1{E_\text{composite}t_\text{composite}}\right)\right]^{-1/2}, \qquad \tau_{\max} = \frac{N_\text{axial}}{\lambda} $$

Boss/composite CTE mismatch on cooldown (same stiffness-weighted compatibility structure as liner load-sharing, applied to free thermal strain):

$$ \varepsilon_\text{common} = \frac{(E_\text{boss}t_\text{boss}\alpha_\text{boss} + E_\text{composite}t_\text{composite}\alpha_\text{composite})\,\Delta T} {E_\text{boss}t_\text{boss}+E_\text{composite}t_\text{composite}} $$

CTE mismatch between differently-oriented plies generates residual stress on cooldown, handled by a thermal force/moment term added to the CLT balance:

$$ N^T = \sum_k [\bar Q]_k\,\vec\alpha_k\,\Delta T\,(z_k-z_{k-1}) $$

solved at $[N,M]=[N^T,M^T]$; mechanical (stress-producing) strain is total strain minus each ply's own free thermal strain. Permeation (informational, Type IV/V) via steady-state Fick's law:

$$ \dot n = \frac{\Phi \cdot A \cdot \Delta P}{t_\text{liner}} $$

Minimize vessel mass over winding angle, aspect ratio, and hoop:helical ply ratio, subject to burst-margin and liner-stress-margin constraints:

$$ \min_{\alpha,\ AR,\ r_\text{ply}}\ m(\alpha, AR, r_\text{ply}) \quad\text{s.t.}\quad \frac{P_{burst}}{P_{burst,\text{req}}}-1 \ge 0,\quad \frac{\sigma_y}{\sigma_\text{liner}}-1\ge0 $$

via scipy.optimize.minimize (SLSQP) or differential_evolution (global, penalty-constrained). The search uses fast netting-scale/strain-compatibility proxies for responsiveness; a one-shot verification pass then re-solves the real netting→CLT→first-ply-failure chain at the reported optimum, surfacing any proxy/reality mismatch rather than hiding it (see the Tab 12 screenshot above and the Engineering Rigor section).