🚀 RDO Solver

Solver Technical Capabilities

A guide to the mechanics formulations, statistical models, and structural engineering assumptions.

The Composite Mechanics & Failure Solver is built on classical closed-form mechanics formulations and statistical calibration routines. This guide details the capabilities and core mathematical assumptions behind the calculations to ensure structural engineers can trust and verify the results.

1. Classical Laminate Theory (CLT) Core

The primary solver computes the in-plane stiffness and coupled out-of-plane behavior of multi-directional composite stacks.

Stiffness Matrices ($A$, $B$, $D$)

Calculates the extensional, coupling, and bending matrices from the transformed reduced ply stiffnesses $[\bar{Q}]_k$ at coordinate angle $\theta_k$:

$$A_{ij} = \sum_{k=1}^{N} (\bar{Q}_{ij})_k (z_k - z_{k-1})$$ $$B_{ij} = \frac{1}{2} \sum_{k=1}^{N} (\bar{Q}_{ij})_k (z_k^2 - z_{k-1}^2)$$ $$D_{ij} = \frac{1}{3} \sum_{k=1}^{N} (\bar{Q}_{ij})_k (z_k^3 - z_{k-1}^3)$$

Hygrothermal Residual Stresses

Integrates thermal expansion and moisture swelling load vectors resulting from manufacturing curing cycles ($\Delta T$) and operating moisture states ($\Delta M$):

$$\begin{bmatrix} N_x^T \\ N_y^T \\ N_{xy}^T \end{bmatrix} = \sum_{k=1}^{N} [\bar{Q}]_k \{\bar{\alpha}\}_k \Delta T t_k, \quad \begin{bmatrix} M_x^T \\ M_y^T \\ M_{xy}^T \end{bmatrix} = \sum_{k=1}^{N} [\bar{Q}]_k \{\bar{\alpha}\}_k \Delta T t_k z_{mid,k}$$ $$\begin{bmatrix} N_x^H \\ N_y^H \\ N_{xy}^H \end{bmatrix} = \sum_{k=1}^{N} [\bar{Q}]_k \{\bar{\beta}\}_k \Delta M t_k, \quad \begin{bmatrix} M_x^H \\ M_y^H \\ M_{xy}^H \end{bmatrix} = \sum_{k=1}^{N} [\bar{Q}]_k \{\bar{\beta}\}_k \Delta M t_k z_{mid,k}$$

This allows the calculation of internal thermal-residual stresses within each ply, which are superimposed on the mechanical loads to check coupon margins.

Homogenized 2D Engineering Properties

Exposes the effective engineering constants of the combined multi-layer layup calculated from the compliance matrix $[a] = [A]^{-1}$:

$$E_x = \frac{1}{h \cdot a_{11}}, \quad E_y = \frac{1}{h \cdot a_{22}}$$ $$G_{xy} = \frac{1}{h \cdot a_{66}}, \quad \nu_{xy} = -\frac{a_{12}}{a_{11}}, \quad \nu_{yx} = -\frac{a_{21}}{a_{22}}$$ $$\eta_{xy,x} = \frac{a_{16}}{a_{11}}, \quad \eta_{xy,y} = \frac{a_{26}}{a_{22}}$$

2. Composite Failure Criteria Suite

Evaluates ply-by-ply stress states against standard strength limits to compute Margins of Safety (MoS) and Failure Indices (FI):

Criterion	Failure Mechanics & Characteristics
Maximum Stress	Checks $\sigma_1, \sigma_2, \tau_{12}$ independently against strength limits. Neglects stress interactions.
Maximum Strain	Checks strains $\varepsilon_1, \varepsilon_2, \gamma_{12}$ independently against strain limits. Good for fiber-dominated failure.
Tsai-Hill	An energy-based quadratic interaction model. Predicts yielding or failure of anisotropic materials.
Tsai-Wu	A tensor polynomial quadratic model including cross-term interaction ($F_{12} = -0.5\sqrt{F_{11}F_{22}}$). Highly sensitive to combined shear and tension.
Hashin	Separates fiber failure (tension vs. compression) from matrix failure modes. Used heavily in carbon-epoxy modeling.
Puck & Cuntze	Advanced inter-fiber failure (IFF) models tracking the physical wedge/fracture plane angle in matrix cracks. Highly trusted for thick laminates.

3. 2D Plate Mechanics & Stability

Solves rectangular plates (dimensions $a \times b$) under simply supported (SSSS) constraints using a coupled Ritz numerical method (Rayleigh-Ritz energy formulation). This method integrates the full $[A], [B], [D]$ stiffness matrices, capturing in-plane stretching (extension-bending coupling due to $B_{ij}$) and bending-twisting coupling ($D_{16}, D_{26}$):

Stability & Buckling

Determines the critical buckling load factor ($\lambda_{cr}$) under combined loading ($N_x, N_y, N_{xy}$) by condensing out the in-plane variables and solving the generalized eigenvalue problem:

$$\left( [K^{ww}] - [K^{coupling}]^T [K^{in}]^{-1} [K^{coupling}] \right) \mathbf{q}_w = \lambda [K_G^{ww}] \mathbf{q}_w$$

The solver outputs the critical load factor, dominant mode shape $(m, n)$, and a 3D coordinate deflection grid for visual verification of local skin buckling.

Bending Deflection & Out-of-Plane Coupling

Computes the deflection field $w(x, y)$, bending moments, and out-of-plane stresses under uniform distributed pressure, concentrated point loads, and sinusoidal loads. The in-plane displacements $u_0(x, y)$ and $v_0(x, y)$ are solved simultaneously:

$$[K_{condensed}] \mathbf{q}_w = \mathbf{F}_w, \quad \mathbf{q}_{in} = - [K^{in}]^{-1} [K^{coupling}] \mathbf{q}_w$$

This allows accurate superposition of coupling stresses with external loads for precise margin of safety evaluation.

Eigenfrequencies & Vibrations

Calculates the natural frequencies $\omega_{mn}$ (Hz) and fundamental coupled mode shapes:

$$\left( [K^{ww}] - [K^{coupling}]^T [K^{in}]^{-1} [K^{coupling}] \right) \mathbf{q}_w = \omega^2 [M^{ww}] \mathbf{q}_w$$

4. Stochastic & Weibull Statistical Engine

Instead of treating strengths as fixed constants, the engine supports a full probabilistic structural reliability analysis:

Coupon Fit: Calibrates scale ($\beta$) and shape ($m$) Weibull moduli from uploaded experimental data using Maximum Likelihood Estimation (MLE).
Goodness-of-Fit: Performs the Anderson-Darling (AD) test at a $95\%$ confidence level to verify if the dataset fits the Weibull distribution.
Monte Carlo Simulation: Performs stochastic loops (e.g. 1,000+ runs) applying random Gaussian variations to laminate layer thickness ($t \sim \mathcal{N}(t_0, \sigma_t)$), orientation angles ($\theta \sim \mathcal{N}(\theta_0, \sigma_\theta)$), and elastic moduli ($E_1, E_2 \sim \mathcal{N}(\mu, \sigma)$).
A & B-Basis Allowables: Computes design allowables (B-basis / $T_{90}$, A-basis / $T_{99}$) under Normal distribution, Weibull distribution, and non-parametric (rank-based) paths, using statistical methods consistent with CMH-17 conventions. This is not an independently certified CMH-17 conformance — material data is user-supplied, and allowables should be validated against your program's qualification requirements before use in a certification package.

Key Engineering Assumptions & Limitations

To maintain accurate expectations and safety margins, note the following boundary conditions:

Thin-Plate Limit (Kirchhoff Theory) — default, opt-out: The plate buckling, bending, and vibration solvers default to classical theory, which neglects out-of-plane transverse shear deformation ($\tau_{xz}, \tau_{yz}$). Highly accurate for thin skins ($span/thickness > 20$), but underestimates deflections on thick laminates and foam/honeycomb sandwich core panels. A shear-deformable Mindlin/FSDT option (theory="mindlin") is available for symmetric, balanced (specially-orthotropic) laminates — see below.
FSDT Transverse Shear (opt-in): Laminate analysis now supports an optional First-Order Shear Deformation Theory (FSDT) extension — supply ply $G_{13}/G_{23}$ and transverse loads $Q_{xz}/Q_{yz}$ to get transverse shear strains and per-ply interlaminar shear stresses ($\tau_{xz}, \tau_{yz}$). For laminates that are symmetric and balanced (specially orthotropic), a more accurate equilibrium-based through-thickness distribution is also computed automatically, correctly vanishing at the free surfaces; general/unsymmetric laminates keep the simpler constant-through-ply approximation. A bounded through-thickness normal stress ($\sigma_{zz}$) estimate is available for a known local applied pressure on a qualifying laminate — a simplified load-path approximation, not a general result. The plate buckling/bending/vibration solvers gained the same shear-deformable (Mindlin) option directly — pass theory="mindlin" for specially-orthotropic laminates.
Soft-Core Sandwich Trap Warning: For structural sandwich panels with thick, soft cores (e.g., Rohacell, Nomex honeycombs where $span/thickness < 20$ and core shear modulus $G$ is low), transverse shear compliance dominates. The default Kirchhoff plate solvers can underpredict out-of-plane bending deflection by a factor of 2x to 3x and completely neglect core shear stress in this regime, risking core shear failure in real prototypes. Use theory="mindlin" on the plate solvers (specially-orthotropic sandwich lay-ups only) or the FSDT extension above for a single load case; general/unsymmetric sandwich lay-ups still require 3D FEA.
First Ply Failure (FPF) Assumption: Structural safety margins are calculated based on when the first single ply cracks. The solver does not model progressive damage propagation (degrading cracked plies and continuing to load until Last Ply Failure). FPF calculations are safe and conservative, but do not capture post-cracking load redistribution.
No Stress Concentrations: Calculations assume uniform stress/strain fields. Local stress concentrations (such as open holes, fastener bypass bearing loads, or bolt joints) are not computed and must be calculated using external factors (OHT/OHC methods).

Back to Dashboard