Laminate Bending Stiffness

Laminate Bending Stiffness (structures/composites/laminate-bending-stiffness)

Use when the task is the bending side of classical lamination theory: the bending-extension coupling matrix B and the bending matrix D of a composite laminate stack, together with the full ABD assembly for symmetric and unsymmetric layups. It pairs with structures/composites/laminate-stiffness, which owns the in-plane symmetric-laminate A-matrix synthesis, and feeds structures/composites/laminate-plate-buckling, which consumes the D11, D22, D12 and D66 bending terms as given inputs to its critical load analysis.

Domain quick reference

• A ply has orthotropic stiffness Q11, Q12, Q22, Q66 in material axes from the engineering constants E1, E2, nu12, G12; rotation to the laminate axes gives the Qbar terms, the integrand of every integral below.
• The stack is a bottom-to-top list of (theta_deg, thickness_m) plies; the interface z coordinates put the mid-plane at z = 0.
• A_ij = sum_k Qbar_ij,k (z_k - z_k-1), the in-plane extensional stiffness (N/m).
• B_ij = (1/2) sum_k Qbar_ij,k (z_k^2 - z_k-1^2), the bending-extension coupling matrix (N). B is identically zero to machine precision for a mirror-symmetric stack and real, nonzero for an unsymmetric stack.
• D_ij = (1/3) sum_k Qbar_ij,k (z_k^3 - z_k-1^3), the bending matrix (N m). A balanced symmetric angle-ply has A16 = A26 = 0 but D16 and D26 are NOT zero, so the D terms must come from the z-cubed integral and cannot be read off the in-plane A pattern.
• A single isotropic layer gives D11 = E h^3 / (12 (1 - nu^2)), the classic plate result and a closed-form check on the model.
• Equivalent flexural constants of a symmetric stack with D16 = D26 = 0: E_bx = 12 (D11 D22 - D12^2) / (D22 h^3), E_by analogous, nu_bxy = D12 / D22, G_bxy = 12 D66 / h^3.
• FAR-25 and CS-25 frame the transport-aeroplane composite airframe certification context; the CLT math is common mechanics, summary-only.

Workflow

1. Collect the ply engineering constants E1, E2, nu12, G12 and the bottom-to-top ply stack as (theta_deg, thickness_m) tuples.
2. Rotate each ply to the laminate axes with ply_rotated_stiffness, the integrand of the z-moment integrals.
3. Assemble the bending-extension coupling matrix with b_coupling_matrix over the z-squared thickness integral.
4. Assemble the bending matrix with d_bending_matrix over the z-cubed thickness integral.
5. Get the full ABD assembly, including the A sub-block, with abd_stiffness_matrices for symmetric or unsymmetric stacks.
6. Pull the four bending terms D11, D22, D12, D66 with laminate_bending_terms in the order the laminate plate buckling analysis fixes as its given inputs.
7. Recover the equivalent flexural engineering constants with equivalent_flexural_constants for symmetric stacks with vanishing D16 and D26.
8. Or run the one-shot laminate_bending_report for the full a, b, d dict plus the scalar bending terms and the b_max_abs coupling magnitude.
9. Confirm the deterministic checks with the contract test scripts/test_laminate_bending_stiffness.py.

Worked example

AS4/3501-6 carbon/epoxy: E1 = 181.0 GPa, E2 = 10.3 GPa, G12 = 7.17 GPa, nu12 = 0.28, ply thickness t = 0.125 mm.

• Symmetric [0/90]s, plies (0, 90, 90, 0) x 0.125 mm, h = 0.5 mm: A11 = 48.0393243936 MN/m, A22 = 48.0393243936 MN/m, A12 = 1.44846222217 MN/m, A66 = 3.585 MN/m. B is zero to machine precision, max |B_ij| = 8.52651282912e-14 N m. D11 = 1.67060433677, D22 = 0.331034179627, D12 = 0.0301762962953, D66 = 0.0746875 (N m), the four terms laminate_bending_terms returns for the buckling hand-off; D16 and D26 vanish below 1e-9 N m. Equivalent flexural constants (D16 = D26 = 0): E_bx = 160.113939519 GPa, E_by = 31.7269538028 GPa, G_bxy = 7.17 GPa, nu_bxy = 0.0911576451995.
• Unsymmetric [0/90]2T, plies (0, 90, 0, 90) x 0.125 mm: the A block matches the [0/90]s case exactly (same ply set), but B11 = -2679.14031429 N m and B22 = +2679.14031429 N m, three orders above the D scale, a real bending-extension coupling no symmetric stack can produce. D11 = D22 = 1.0008192582 N m, softer than the [0/90]s stack because the stiff 0-degree plies are not at the faces.
• Symmetric balanced angle-ply [+45/-45]s: A16 = A26 = 0 exactly (balance removes the in-plane coupling) while max |B_ij| = 2.27373675443e-13 N m (the B == 0 identity holds again) and D16 = D26 = 0.334892539286 N m, NONZERO: balance removes the A coupling but not the D coupling, the reason the z-cubed integral is required.
• Isotropic reduction, E = 70 GPa, nu = 0.3, h = 2 mm: D11 = 51.2820512821 N m, matching E h^3 / (12 (1 - nu^2)) within 1e-9 relative; equivalent_flexural_constants recovers E_bx = 70 GPa, G_bxy = 26.9230769231 GPa and nu_bxy = 0.3 within 1e-9 relative.

Verification

• Confirm abd_stiffness_matrices on the [0/90]s stack reproduces the D11, D22, D12, D66 and A11, A22, A12, A66 values above within 1e-6 relative.
• Confirm max |b_coupling_matrix| is below 1e-9 N m for the [0/90]s and [+45/-45]s stacks (the symmetric B == 0 identity, never asserted as exact equality to zero) and near 2679.14 N m for the unsymmetric [0/90]2T stack.
• Confirm the leading principal minors of the symmetric 3x3 D matrix are all positive for the [0/90]s, [0/90]2T and [+45/-45]s stacks.
• Confirm the isotropic reduction and the equivalent flexural constant recovery within 1e-9 relative.
• Confirm every non-positive or boolean modulus, thickness or nu12 outside [0, 1), an empty ply list, and a non-positive-definite D pair each raise ValueError.
• Run the contract test offline: python3 scripts/test_laminate_bending_stiffness.py (31 tests, deterministic).

Related leaves

• structures/composites/laminate-stiffness: the in-plane symmetric laminate A-matrix synthesis; this leaf's A sub-block is produced only inside the full ABD assembly and matches that synthesis for symmetric stacks.
• structures/composites/laminate-plate-buckling: the consumer that takes the D11, D22, D12 and D66 terms this leaf produces as its workflow step 1 inputs for the critical buckling load.
• structures/composites/laminate-first-ply-failure: mid-plane strain recovery of a symmetric balanced laminate from its in-plane compliance, a separate in-plane failure surface.
• structures/composites/unidirectional-lamina-micromechanics: the constituent-to-lamina prediction of E1, E2, nu12 and G12 that feed this leaf as inputs.

Pitfalls

• Reading the D16 and D26 terms off the in-plane A pattern: a balanced angle-ply has A16 = A26 = 0 exactly but D16 = D26 = 0.334892539286 N m nonzero in the worked [+45/-45]s example, so the bending coupling must be computed by the z-cubed integral, never inferred.
• Asserting B == 0 as exact float equality: the symmetric-stack residue is a floating-point cancellation artifact (8.5265e-14 and 2.2737e-13 N m in the worked examples), not a literal zero; use an absolute tolerance.
• Treating the stacking order as free for the B and D matrices: the A block only depends on the ply set, but [0/90]s and [0/90]2T share the same A block while their B and D matrices differ completely (B == 0 against B11 = -2679.14 N m).
• Feeding the equivalent flexural constants a stack with nonzero D16 or D26: the function assumes a cross-ply or specially orthotropic symmetric stack; report the full D matrix instead for a balanced angle-ply.

Behavior contract (gate 3)

Run the deterministic contract test (stdlib unittest, offline):

python3 scripts/test_laminate_bending_stiffness.py

The test covers the [0/90]s worked example (A, B, D blocks and the four bending terms), the symmetric B == 0 identity on the [0/90]s and [+45/-45]s stacks, the unsymmetric B coupling and D terms of the [0/90]2T stack, positive definiteness of the D matrix on all three worked stacks, the balanced-stack A16/A26 zero versus D16/D26 nonzero split, the A-consistency of the ABD assembly against the closed form re-derived in-test, the isotropic reduction and equivalent flexural constant recovery, the one-shot report keys and determinism, and ValueError rejection of every non-physical input.

Compliance

• Standards referenced, not reproduced: FAR-25 is US government work (public domain) and CS-25 is a free EASA download; the CLT math is common mechanics, summary-only per standards-map.yaml.
• compliance: STANDARDS-REF, gated: false.