Classical Laminate Theory: Method

Classical Laminate Theory: conventions, equations and the failure criteria

This page states what the Classical Laminate Theory tool computes and in which conventions, so a result can be checked by hand. The formulas are those of Jones, Mechanics of Composite Materials (2nd ed., chapter 4) and MIL-HDBK-17-3F chapter 5; the symbols are the handbook's. Every step of the tool's code (library composites/_shared/clt.php) is commented with the same equation numbers.

1. Conventions

• Axes. x and y lie in the plane of the laminate, z is normal to it. The plies are listed from the bottom surface (z = −h/2) upward; ply 1 is at the bottom, z₀ = −h/2 and z_k is the top of ply k. This is Jones' numbering (his layer 1 sits at z₀ = −h/2).
• Ply angle θ. From the laminate x axis to the fibre (1) direction, positive counter-clockwise looking down the +z axis (x toward y). The 2 direction is in the plane at 90° from the fibre.
• Stress resultants. N_x, N_y, N_xy are forces per unit width (tension positive). M_x, M_y, M_xy are moments per unit width defined by M_x = ∫σ_x z dz, so a positive M_x stretches the top (+z) surface and compresses the bottom.
• Strains. Engineering shear strain γ_xy = 2ε_xy everywhere. The through-thickness strain field is ε(z) = ε⁰ + z κ with ε⁰ the mid-plane strains and κ the curvatures (κ_xy the twist).
• Units. The tool accepts either unit system; internally every quantity is in SI base units and the page shows the display units (Msi or GPa for moduli, ksi or MPa for stresses, lb/in or kN/m for A, lb or kN for B, lb·in or N·m for D, με for strains).

2. Lamina stiffness

A unidirectional ply under plane stress is orthotropic with four independent constants E₁, E₂, G₁₂, ν₁₂; the minor Poisson's ratio follows from reciprocity, ν₂₁ = ν₁₂ E₂ / E₁. The reduced stiffness in the fibre axes (Jones eq. 2.61) is

Q₁₁ = E₁ / (1 − ν₁₂ ν₂₁),   Q₂₂ = E₂ / (1 − ν₁₂ ν₂₁),   Q₁₂ = ν₁₂ E₂ / (1 − ν₁₂ ν₂₁),   Q₆₆ = G₁₂,

and σ = Q ε in those axes. Rotated to the laminate axes by θ (m = cos θ, n = sin θ) it becomes the transformed reduced stiffness Q̄ (Jones eq. 2.85):

Q̄₁₁ = Q₁₁ m⁴ + 2 (Q₁₂ + 2 Q₆₆) m² n² + Q₂₂ n⁴
Q̄₁₂ = (Q₁₁ + Q₂₂ − 4 Q₆₆) m² n² + Q₁₂ (m⁴ + n⁴)
Q̄₂₂ = Q₁₁ n⁴ + 2 (Q₁₂ + 2 Q₆₆) m² n² + Q₂₂ m⁴
Q̄₁₆ = (Q₁₁ − Q₁₂ − 2 Q₆₆) m³ n + (Q₁₂ − Q₂₂ + 2 Q₆₆) m n³
Q̄₂₆ = (Q₁₁ − Q₁₂ − 2 Q₆₆) m n³ + (Q₁₂ − Q₂₂ + 2 Q₆₆) m³ n
Q̄₆₆ = (Q₁₁ + Q₂₂ − 2 Q₁₂ − 2 Q₆₆) m² n² + Q₆₆ (m⁴ + n⁴)

The strain and stress transformations between the two axis systems (with engineering shear strain) are

ε₁ = m² ε_x + n² ε_y + m n γ_xy,   ε₂ = n² ε_x + m² ε_y − m n γ_xy,   γ₁₂ = −2 m n ε_x + 2 m n ε_y + (m² − n²) γ_xy;
σ₁ = m² σ_x + n² σ_y + 2 m n τ_xy,   σ₂ = n² σ_x + m² σ_y − 2 m n τ_xy,   τ₁₂ = −m n σ_x + m n σ_y + (m² − n²) τ_xy.

A ply's thermal expansion coefficients rotate as α_x = α₁ m² + α₂ n², α_y = α₁ n² + α₂ m², α_xy = 2 m n (α₁ − α₂) (Jones eq. 4.70).

3. Laminate stiffness

Integrating the ply stresses through the thickness with the linear strain field gives the laminate constitutive relation [N; M] = [A B; B D] [ε⁰; κ] (Jones eq. 4.22 to 4.24):

A_ij = Σ_k Q̄_ij,k (z_k − z_k−1),   B_ij = ½ Σ_k Q̄_ij,k (z_k² − z_k−1²),   D_ij = ⅓ Σ_k Q̄_ij,k (z_k³ − z_k−1³).

A is the extensional stiffness, D the bending stiffness and B the membrane-bending coupling, which vanishes for a laminate symmetric about its mid-plane. A balanced laminate (every +θ ply matched by a −θ ply of the same thickness) has A₁₆ = A₂₆ = 0; D₁₆ and D₂₆ are generally not zero even then, because the ±θ plies sit at different distances from the mid-plane. The tool reports B relative to √(max A · max D), and A₁₆/A₂₆ and D₁₆/D₂₆ relative to the largest term of their matrix, so the size of each coupling can be judged.

The 6 × 6 inverse [ε⁰; κ] = [a b; bᵀ d] [N; M] is computed by Gauss-Jordan elimination with pivoting. For a symmetric laminate b = 0, a = A⁻¹ and d = D⁻¹.

4. Engineering constants

In-plane (from the a block, so a coupled laminate gets the values of a plate free to curve; for a symmetric laminate these are Jones eq. 4.62): E_x = 1/(h a₁₁), E_y = 1/(h a₂₂), G_xy = 1/(h a₆₆), ν_xy = −a₁₂/a₁₁, ν_yx = −a₁₂/a₂₂, and the shear-coupling ratios η_xs = a₁₆/a₁₁, η_ys = a₂₆/a₂₂ (zero for a balanced laminate). For an unsymmetric laminate the page also prints the values from A alone, which apply when the plate is held flat.

Flexural (from the d block): E_xf = 12/(h³ d₁₁), E_yf = 12/(h³ d₂₂), G_xyf = 12/(h³ d₆₆), ν_xyf = −d₁₂/d₁₁, ν_yxf = −d₁₂/d₂₂. E_xf is the modulus a bending test of a strip gives: the strip is free to take the anticlastic curvature.

Partial inversion, D*. With the in-plane forces zero, N = 0 = A ε⁰ + B κ gives ε⁰ = −A⁻¹ B κ and M = (D − B A⁻¹ B) κ: the reduced bending stiffness D* = D − B A⁻¹ B (MIL-HDBK-17-3F eq. 5.3.3.3(d)); likewise A* = A − B D⁻¹ B with M = 0. For a symmetric laminate D* = D. From D* the tool reports two stiffnesses per direction:

• cylindrical (κ_y = κ_xy = 0, the anticlastic curvature suppressed as in a wide plate): D*₁₁, written as the modulus 12 D*₁₁/h³;
• free (M_y = M_xy = 0, the anticlastic curvature allowed as in a narrow strip): 1/d*₁₁ with d* = (D*)⁻¹, written as 12/(h³ d*₁₁), which is E_xf.

The cylindrical value exceeds the free one by the Poisson restraint, 1/(1 − ν_xyf ν_yxf) for an uncoupled laminate. Plate bending and buckling equations use the cylindrical stiffness (their D is D*); beams and strips use the free one.

Through-thickness Poisson's ratios. Classical laminate theory has no thickness direction, so ν_xz and ν_yz are obtained from the plies' 3-D compliance under plane stress: each ply strains through its thickness by ε₃ = −(ν₁₃/E₁) σ₁ − (ν₂₃/E₂) σ₂, with ν₁₃ = ν₁₂ (transverse isotropy about the fibre) and ν₂₃ entered per material. The plies expand through the thickness independently, so the laminate's thickness strain is the thickness-weighted mean of the ply values, and ν_xz = −ε_z/ε_x under N_x alone, ν_yz likewise under N_y. The flexural versions use the first moment of ε₃: κ_z = (12/h³) ∫ ε₃ z dz is the slope of the linear part of the thickness strain, and ν_xzf = −κ_z/κ_x under M_x alone. For a unidirectional laminate these reduce to ν₁₂ and ν₂₃.

Directional stiffness. The last figure shows E_x(φ) and G_xy(φ), the in-plane moduli of the laminate seen in axes rotated by φ: every ply angle is reduced by φ, A is rebuilt and inverted. A quasi-isotropic laminate gives a flat line; the tool's test script checks the quasi-isotropic laminate against the invariants of Tsai and Pagano, E = (U₁² − U₄²)/U₁, ν = U₄/U₁, G = U₅.

5. Response to the loads

A temperature change ΔT from the stress-free (cure) temperature enters through the thermal stress resultants (Jones eq. 4.71 to 4.74)

N^T = Σ_k Q̄_k α_k ΔT (z_k − z_k−1),   M^T = ½ Σ_k Q̄_k α_k ΔT (z_k² − z_k−1²),

which are added to the mechanical loads before the inverse is applied: [ε⁰; κ] = [a b; bᵀ d] [N + N^T; M + M^T]. In each ply the total strain is ε(z) = ε⁰ + z κ; the mechanical (stress-producing) strain is ε − α ΔT and the stress is σ = Q̄ (ε − α ΔT). Both are rotated to the lamina axes. The failure criteria use the mechanical lamina strains (which equal S σ) and the lamina stresses; the page reports both the total and the mechanical fibre strains when ΔT ≠ 0.

Load level. The allowable strains and strengths are ultimate values, so every margin is written at ultimate load and its box says ULT whatever level the loads were entered at. With "Limit" chosen as the applied load type, the strains and stresses are shown at the entered level and the criteria are evaluated with the mechanical loads multiplied by the factor of safety FS (1.5 by default; the thermal part is not factored, since it does not grow with the applied load). With "Ultimate" chosen the loads are used as entered, never multiplied by FS, and the factor field is greyed. The tool has no yield check, so it carries no yield factor.

6. Failure criteria and first-ply failure

Maximum strain (the default, always applied): each mechanical lamina strain is compared with its own allowable, the tension value when the strain is positive and the compression value (entered as a magnitude) when negative, and |γ₁₂| with the shear allowable. The strain ratio R = strain / allowable is the fraction of the allowable used; the margin of each mode is M.S. = 1/R − 1. Both surfaces of every ply are checked, because the strain field is linear and the extremes of a ply are at its faces. The first ply to fail is the one with the largest R over all plies, surfaces and modes; the laminate's first-ply-failure load factor is λ = 1/R_max (the entered loads can grow by λ before that ply reaches its allowable) and the tool's margin is M.S. = λ − 1.

Maximum stress (when the five strengths of the material are entered): the same bookkeeping with σ₁, σ₂, τ₁₂ and F₁^t, F₁^c, F₂^t, F₂^c, F₁₂.

Tsai-Wu (Tsai and Wu 1971; Jones section 2.9.4), plane stress: F₁σ₁ + F₂σ₂ + F₁₁σ₁² + F₂₂σ₂² + F₆₆τ₁₂² + 2F₁₂σ₁σ₂ = 1 with F₁ = 1/F₁^t − 1/F₁^c, F₂ = 1/F₂^t − 1/F₂^c, F₁₁ = 1/(F₁^t F₁^c), F₂₂ = 1/(F₂^t F₂^c), F₆₆ = 1/F₁₂² and the interaction coefficient F₁₂ = −½ √(F₁₁ F₂₂), the generalized von Mises value used when no biaxial test is available. Scaling the stresses by a factor R gives a R² + b R = 1, a = F₁₁σ₁² + F₂₂σ₂² + F₆₆τ₁₂² + 2F₁₂σ₁σ₂, b = F₁σ₁ + F₂σ₂; the positive root R = [−b + √(b² + 4a)]/(2a) is the strength ratio (the factor the ply stresses can grow by), M.S. = R − 1, and the failure index at the applied stresses is a + b (1 = failure).

First ply failure is usually a matrix mode (transverse tension or shear in the off-axis plies). A laminate normally carries load beyond it; the fibre-direction margin is the one that governs last-ply failure, so the page prints the maximum and minimum fibre strains beside the first-ply margin. Progressive failure (ply discount) is not implemented.

7. Ply families and rules of thumb

The page gives the thickness fractions of the 0°, ±45° (both +45° and −45°) and 90° families and of any other angle, and checks the 10 % rule of MIL-HDBK-17-3F section 5.6.2 (at least 10 % of the plies in each of the four directions 0°, +45°, −45°, 90°), which protects the laminate against secondary loads the analysis did not consider. Symmetry and balance are checked ply by ply from the stacking sequence, not from the size of B and A₁₆.

8. Stacking code

Instead of the ply table a stacking code can be typed, read from the bottom ply up: angles separated by "/", optional square brackets; "±45" or "+-45" for the pair +45/−45 ("∓45" for −45/+45); "0_2" for two plies at 0°; "(0/90)3" for a group repeated three times and "(45/−45)s" for a mirrored group; the suffix "]s" mirrors the whole sequence (symmetric), "]2s" repeats it twice and mirrors, "]so" is symmetric with a single mid-ply (the last ply listed is not repeated), "]T" marks a total sequence. A material letter before an angle, "B:45", takes that ply from material slot B (default A). Every ply takes the ply thickness of its material.

9. Checks

The library is checked by test-scripts/test_clt.php (127 checks) against identities and closed forms: the Q̄ invariants and closed forms at 45° and 90°, the strain and stress transformations, a unidirectional laminate (E_x = E₁, the strain N_x/(E₁h), free thermal expansion without stress, beam curvature 12 M_x/(E₁h³) with κ_y = −ν₁₂ κ_x, surface stress 6 M_x/h²), the quasi-isotropic [0/±45/90]_s laminate (A from the Tsai-Pagano invariants, E = (U₁² − U₄²)/U₁, G = E/(2(1 + ν)), direction independence), the unsymmetric [0/90]_T laminate (B₁₁ = (Q₂₂ − Q₁₁) t²/2, saddle curvature on cooling), the [0/90]_s closed forms for A₁₁ and D₁₁, the stress integrals returning N and M, the thermal residual stresses self-equilibrating, the criteria at their limits (a stress equal to its strength gives a strength ratio of 1) and the stacking-code parser.