Sandwich Panel

Sandwich Panel Method

Sandwich Panel Method: conventions, equations and the failure modes

The Sandwich Panel tool checks a flat panel of two thin facings on a honeycomb or foam core by the classical sandwich theory of MIL-HDBK-23A (chapters 3 and 4) and Allen, Analysis and Design of Structural Sandwich Panels (1969), with the core shear deformation included in the plate solution and in the buckling load. It is the companion of the Classical Laminate Theory tool: that tool gives a sandwich its A, B, D and facing stresses when the core is entered as a thick soft ply (the "core ply" shortcut), but it has no transverse shear, no core support of the facings and no core strengths. This tool takes the facings (equal or unequal) and a core from My Cores and writes the margins laminate theory cannot.

1. Conventions

• z up. Facing 1 is the top facing (+z), facing 2 the bottom. d is the distance between the facing mid-planes, h_c the core height, t_f the facing thickness.
• M_x = ∫σ_x z dz per unit width (the laminate tool's convention): positive stretches facing 1. N_x, N_y tension positive. V_x, V_y are transverse shear forces per unit width in the x-z and y-z planes.
• The pressure p is positive when it pushes down on facing 1 (toward the core). A downward pressure bends the panel so that facing 1 is in compression at the centre.
• The panel edges are a (along x) and b (along y). The honeycomb ribbon (L) direction runs along x or along y; the core shear in the x-z plane uses G_L and F_sL when the ribbon runs along x, else G_W and F_sW.
• The allowables are ultimate values, so every margin is written at ultimate load and its box says ULT whatever level the loads were entered at: limit loads are multiplied by the factor of safety FS (1.5 by default) before the margins are written, ultimate loads are used as entered and FS is greyed. The tool has no yield check and carries no yield factor.

2. Section properties

The facings carry the membrane forces and the bending couple; the core, whose in-plane stiffness is neglected, carries the transverse shear and holds the facings apart (Allen section 2.2). With the facing plate moduli Ē_x = E_x/(1 − ν_xy ν_yx), ν_yx = ν_xy E_y/E_x:

d = h_c + (t₁ + t₂)/2
D_x = Ē_x1 t₁ Ē_x2 t₂ d² / (Ē_x1 t₁ + Ē_x2 t₂) + (Ē_x1 t₁³ + Ē_x2 t₂³)/12   (Allen eq. 2.6 generalized to unequal facings; the first term is the couple, the second the facings' own bending, which the handbook's thin-facing formula Ē t d²/2 omits)
D_y likewise with the y moduli; D_xy with the facing shear moduli G_xy; D₁₂ = ν_yx D_x (facing-stiffness-weighted ν_yx)
A_x = E_x1 t₁ + E_x2 t₂ (membrane stiffness), A_y likewise
U_x = G_c,x d²/h_c, U_y = G_c,y d²/h_c   (transverse shear stiffness per unit width, Allen eq. 2.12)
neutral axis for x bending at z̄ = Ē_x1 t₁ d / (Ē_x1 t₁ + Ē_x2 t₂) above the centroid of facing 2.

The test script checks D_x, D_xy and A against the classical laminate theory library with the same sandwich entered as three plies and a soft core: they agree to 0.01 %.

3. Panel under pressure

For a simply supported rectangular panel under a uniform load q (= −p, the load in +z) the Navier double sine series solves the first-order shear deformation (Reissner-Mindlin) plate equations exactly mode by mode (Plantema 1966 chapter 5; Allen chapter 4). With w = W sin αx sin βy, φ_x = X cos αx sin βy, φ_y = Y sin αx cos βy (α = mπ/a, β = nπ/b, m and n odd), the constitutive relations M_x = D_x φ_x,x + D₁₂ φ_y,y, M_y = D₁₂ φ_x,x + D_y φ_y,y, M_xy = D_xy(φ_x,y + φ_y,x), Q_x = U_x(w_,x + φ_x), Q_y = U_y(w_,y + φ_y) and the three equilibrium equations Q_x,x + Q_y,y + q = 0, M_x,x + M_xy,y = Q_x, M_xy,x + M_y,y = Q_y give for every mode the 3 × 3 system

[U_xα² + U_yβ²] W + U_xα X + U_yβ Y = q_mn
U_xα W + [D_xα² + D_xyβ² + U_x] X + (D₁₂ + D_xy)αβ Y = 0
U_yβ W + (D₁₂ + D_xy)αβ X + [D_xyα² + D_yβ² + U_y] Y = 0,   q_mn = 16q/(π² m n).

The centre deflection, the centre moments and the shear forces at the middle of the edges are summed over 25 odd terms per direction. The Kirchhoff deflection (no shear deformation) is the U → ∞ limit, W = q_mn/[D_xα⁴ + 2(D₁₂ + 2D_xy)α²β² + D_yβ⁴]; the difference between the two is the core shear share of the deflection. Checks: the isotropic square plate gives w = 0.00406 qa⁴/D and M = 0.0479 qa² at ν = 0.3 (Timoshenko), the long plate the beam-strip values 5qa⁴/(384D) + qa²/(8U), M = qa²/8 and Q = qa/2.

The pressure's centre moments and edge shear forces are added to the entered N, M and V (the shear forces as magnitudes) and the whole is taken to ultimate level. Combining the centre moment with the edge shear force is conservative: they do not occur at the same point.

4. Facing stresses, core shear and crushing

σ_f1x = N_x E_x1/A_x + M_x/(d t₁), σ_f2x = N_x E_x2/A_x − M_x/(d t₂) and likewise in y (MIL-HDBK-23A eq. 3-1 and 3-2: the membrane force shared by stiffness, the moment carried as a couple M/d by the facings). Each facing's stress in x and y is compared with its tensile or compressive allowable by sign; the margin is on the worse direction. Core shear τ_c = V/d in each plane against the core shear strength of that direction. The pressure acts on the core as a flatwise stress σ_z = p: compared with F_c (crushing) when it pushes on the facing, with the flatwise tensile strength when it pulls.

5. Facing wrinkling, intracell dimpling, shear crimping

A compressed facing can buckle locally into the core. On a honeycomb core the facing is a plate on an elastic foundation of stiffness E_c/h_c, which gives σ_wr = 0.82 √(E_f E_c t_f/h_c) (the Hexcel design value; the beam-on-foundation derivation gives the coefficient 0.577, the 0.82 includes the plate factor and the usual knockdown, MIL-HDBK-23A section 3.3). On a continuous (foam) core Hoff and Mautner's wrinkling stress is σ_wr = 0.5 (E_f E_c G_c)^1/3. In both, E_f is the facing modulus in the direction of the compression. Intracell dimpling of the facing into a honeycomb cell of size s: σ_d = 2 E_f/(1 − ν²) (t_f/s)² (Hexcel). Shear crimping is the limit of general buckling for a weak core: the panel fails by core shear at N = U = G_c d²/h_c, independent of the panel size (Allen section 8.3). The wrinkling and dimpling margins are on the worst compressive facing stress over both facings and both directions.

6. General buckling

For the simply supported panel under uniaxial compression N_x, each half-wave pair (m, n) has the Kirchhoff load N_K = [D_xα⁴ + 2(D₁₂ + 2D_xy)α²β² + D_yβ⁴]/α², reduced by the core shear deformation. The mode's buckling load is the eigenvalue of its shear-deformable (Reissner-Mindlin) 3 × 3 stiffness, the same matrix as in section 3, with the compressive force acting on the deflection term: condensing the two rotations gives N = [K_ww − k K_φφ⁻¹ kᵀ]/α², which for isotropic stiffness and U_x = U_y is Reddy's exact first-order shear deformation value N_K/(1 + D(α² + β²)/U) (Theory and Analysis of Elastic Plates); Allen's N_K/(1 + N_K/U) (eq. 8.15; MIL-HDBK-23A section 4.2) is its long-plate limit, 5.8 % low for a square plate, and a direction-weighted mean of U_x and U_y in that form was 3.7 % low on a panel with G_L = 2 G_W; the tool uses the condensed eigenvalue. The tool prints the effective shear stiffness N N_K/(N_K − N) of the critical mode. The minimum over m (n = 1 governs) is the buckling load; the tool prints it with the Kirchhoff minimum for comparison. Checks: 4π²D/a² for the isotropic square plate, Reddy's N_K/(1 + D(α² + β²)/U) with shear on the square plate and Allen's form on a long plate, the value tending to U as the core softens, and the energy-route cross-check (section 8).

7. What is not covered

Clamped or free edges (the pressure solution and the buckling are for four simply supported edges; for other edges use the moments and shear forces of a beam or plate solution as the entered M and V), combined in-plane shear buckling, curved panels and the core pressure they generate, inserts and concentrated loads, thermal gradients between the facings, and the edge-close-out details. The facing allowables are taken the same in x and y. A facing chosen from the Saved Laminate dropdown is run through the laminate library: E_x, E_y, G_xy, ν_xy from the inverse of its ABD matrix, the thickness as the sum of the plies, and the allowables as the laminate's first-ply-failure stresses under uniaxial tension and compression (maximum strain criterion on the lamina allowable strains, the smaller of the x and y values of each sign; a sign whose allowable strains are missing on the lamina is left blank). For the sandwich solution the facing is a homogeneous sheet (its own bending stiffness is the smeared E t³/12 rather than the laminate's D, negligible for thin facings). Its plies are then checked individually: the facing's ultimate membrane stresses σ_fx, σ_fy become the forces per unit width N = σ t_f on the facing's stack, laminate theory gives the mid-plane strains (uniform through the facing, which is a membrane in sandwich theory), each ply's strains are rotated to its fibre axes and compared with the lamina allowable strains by the maximum strain criterion, and the facing's first-ply-failure load factor λ on its membrane stresses gives a further margin M.S. = λ − 1 (section "Facing Ply Strains").

8. Checks

test-scripts/test_sandwich.php (37 checks): the section formulas against hand values for equal and unequal facings and against the classical laminate theory library, the Navier series against Timoshenko's square-plate coefficients and the beam-strip closed forms with shear, the buckling closed forms and the crimping limit, the wrinkling and dimpling formulas, and the deflection profile.