Orthotropic Plate Bending: Method

Orthotropic Plate Bending: Huber's Equation, the Rigidities of Stiffened and Corrugated Plates, the Lévy Series and the Affine Reduction

What the tool computes

A thin rectangular plate whose bending stiffness differs in two perpendicular directions parallel to its edges is called orthotropic (orthogonally anisotropic). The stiffness may come from the material itself (plywood, a unidirectional or cross-ply laminate, rolled sheet with a texture), or from the construction: ribs or stringers smeared over their pitch, a gridwork of beams, a corrugated sheet, a reinforced-concrete slab with unequal reinforcement. The tool takes the plate's four rigidities, built from one of these descriptions or entered directly, and solves the small-deflection bending problem under a uniform, hydrostatic, patch or point load for a plate simply supported on two opposite edges, the other two simply supported, clamped or free. It reports the deflection, the bending and twisting moments, the curvatures, the plate and rib stresses, the edge reactions and corner forces, the equivalent isotropic plate, charts along the centre lines, a fringe of the deflection and margins of safety.

Huber's equation

With the Kirchhoff hypothesis (normals stay straight and normal, no transverse shear deformation) the moment–curvature relations of an orthotropic plate are

M_x = −(D_x ∂²w/∂x² + D₁ ∂²w/∂y²),   M_y = −(D_y ∂²w/∂y² + D₁ ∂²w/∂x²),   M_xy = 2 D_xy ∂²w/∂x∂y,

and the equilibrium of a plate element gives Huber's equation (Huber 1914, 1923; Timoshenko and Woinowsky-Krieger 1959, Chapter 11, eq. 212)

D_x ∂⁴w/∂x⁴ + 2 H ∂⁴w/∂x²∂y² + D_y ∂⁴w/∂y⁴ = q,    H = D₁ + 2 D_xy.

D_x and D_y are the bending rigidities for curvature along x and along y, D₁ the Poisson coupling rigidity and D_xy the torsional rigidity. The four are not independent of the plate's Poisson ratios: Betti's reciprocal theorem gives μ_x D_y = μ_y D_x (Libove and Batdorf, NACA Report 899, eq. 8 and Appendix B), which in the present notation says that the single coupling rigidity D₁ = μ_y D_x = μ_x D_y serves both moment equations. For an isotropic plate D_x = D_y = H = D, D₁ = νD and D_xy = (1 − ν)D/2, and the equation reduces to the biharmonic equation D∇⁴w = q of the Plate Bending tool, whose results this tool reproduces to machine precision in that limit. The shears and the Kirchhoff edge reactions follow from the moments: Q_x = −(D_x w_,xxx + H w_,xyy), V_x = −(D_x w_,xxx + (D₁ + 4D_xy) w_,xyy), and likewise for y; at a corner where neither edge is clamped a concentrated force R = 2M_xy appears.

The rigidities

Homogeneous orthotropic material. With the engineering constants E_x, E_y, ν_xy, G_xy and the reciprocity ν_yx = ν_xy E_y/E_x (Timoshenko Art. 86, eq. a–c): D_x = E_x t³/[12(1 − ν_xyν_yx)], D_y = E_y t³/[12(1 − ν_xyν_yx)], D₁ = ν_yx D_x, D_xy = G_xy t³/12. These are the D₁₁, D₂₂, D₁₂, D₆₆ of a single-ply laminate, which the tool's test checks against the Classical Laminate Theory library.

Plate stiffened by ribs. Timoshenko's statement for a plate reinforced by equidistant stiffeners in one direction (Art. 86, case e, Fig. 185) is D_x = H = E t³/[12(1 − ν²)] and D_y = E t³/[12(1 − ν²)] + E′I/a₁, "in which E and ν are the elastic constants of the material of the plating, E′ the Young modulus and I the moment of inertia of a stiffener, taken with respect to the middle axis of the cross section of the plate"; for two sets of stiffeners (case f) each adds its E′I per pitch to its own rigidity. Taking I about the plate middle plane, I_r = I_own + A e² with e = t/2 + h_r/2, is how the eccentricity of the rib enters: the rib is assumed to bend about the plate middle plane, which is exact only when the plating is much stiffer in stretching than the ribs (the usual case for closely spaced light ribs) and otherwise overestimates D by the amount the neutral surface moves toward the ribs. The tool follows Timoshenko and prints the assumption. Rib torsion is neglected in Timoshenko's stiffened-plate statement, but in his gridwork case (g) the twisting rigidity is the mean of the two sets' torsional rigidities per pitch, H = (C₁/b₁ + C₂/a₁)/2 with C = GJ; the tool offers that addition, G_rJ_r/(2s) per set, as an option (on by default). With the plating thickness set to zero the same inputs describe a gridwork of beams: D₁ = 0 and H comes from the beam torsion only, so the plate is of Timoshenko's third kind (complex roots) unless the beams are stiff in torsion. J of the rectangular rib is Saint-Venant's series for the rectangle, exact to better than 10⁻⁶.

Two things the smeared model leaves out are estimated or flagged on the results page. The plating between the ribs bends locally under the pressure as a strip continuous over the ribs, with a moment of about q s²/12 at the ribs; this stress adds to the smeared one and is printed. And the smeared idealization needs several ribs across the span: the tool warns when the pitch exceeds a quarter of the plate width across the ribs.

Corrugated sheet. For a sheet of thickness h corrugated to the sinusoid z = f sin(πx/l) with the corrugations along y, Timoshenko (Art. 86, case c, after Seydel) gives D_x = (l/s) E h³/[12(1 − ν²)], D_y = E I, H = (s/l) E h³/[12(1 + ν)], where s is the developed length of one half-wave and I the second moment of the profile per unit projected width about its middle line. Across the corrugations the sheet bends about its own thickness over the longer developed path, which divides the flat-sheet rigidity by s/l; along them the profile acts as a beam; the twisting rigidity 2 D_xy = (s/l) G h³/6 carries the extra material, and there is no Poisson coupling (D₁ = 0). Timoshenko quotes the first-order developed length s = l (1 + π²f²/(4l²)) and an empirical expression for I; the tool integrates s and I exactly over the sinusoid instead (Gauss–Legendre quadrature), prints the first-order s beside the exact one, and its test reproduces Seydel's own example (NACA TM 705, 1933, example 3: a duralumin sheet 2l = 3.0 cm, 2f = 0.915 cm, δ = 0.039 cm, E = 750,000 kg/cm², ν = 0.25, for which Seydel computed D₁ = 3.27, D₂ = 3,790 and D₃ = 3.585 kg·cm) within 1 % for D_x and H; the exact sinusoid gives D_y 11 % below Seydel's value, whose profile was not sinusoidal. Seydel's D₁, D₂, D₃ are this tool's D_x, D_y, H.

Laminates. A symmetric cross-ply or balanced symmetric laminate with many plies has a bending-stiffness matrix with D₁₆ = D₂₆ ≈ 0, and then D_x = D₁₁, D_y = D₂₂, D₁ = D₁₂, D_xy = D₆₆ (Jones 1999, Chapter 5, "specially orthotropic" plates). Take the four numbers from the Classical Laminate Theory tool's D matrix, enter them here, and carry the moments and curvatures at the peak back to that tool for the ply stresses and the first-ply-failure margin. Angle-ply laminates with few plies have D₁₆, D₂₆ of the order of the other terms and are not orthotropic in bending; Huber's equation then underestimates the deflection.

The Lévy solution and its three cases

With edges 1 (x = 0) and 2 (x = a) simply supported the solution is the single series w = Σ Y_m(y) sin(mπx/a), and each Y_m satisfies D_y Y⁗ − 2Hα² Y″ + D_xα⁴ Y = q_m(y), α = mπ/a, with q_m the Fourier coefficient of the load. The characteristic roots s² = α² [H ± √(H² − D_xD_y)]/D_y give Timoshenko's three cases (Art. 91):

• Case 1, H² > D_xD_y: two real roots s₁, s₂; Y_m is a combination of e^−s₁y, e^−s₂y and their mirror images from the other edge. A plate with stiff ribs and ordinary plating torsion is usually here.
• Case 2, H² = D_xD_y: a double root s = α (D_x/D_y)^1/4; Y_m = (A + By) e^−sy + (C + D(b − y)) e^−s(b−y). The isotropic plate is here, and so is every "reducible" plate (below).
• Case 3, H² < D_xD_y: complex roots p ± iq; Y_m combines e^−py cos qy and e^−py sin qy with their mirror images. Corrugated sheets and gridworks with little torsional stiffness are here; the deflection then shows the "oscillating" decay away from a loaded line that Timoshenko describes.

The tool writes the homogeneous part on functions that decay from each edge (rather than cosh and sinh), so the 4 × 4 system of each harmonic stays well conditioned at any harmonic number, and it handles the boundary conditions on edges 3 and 4 as Y = Y″ = 0 (simply supported), Y = Y′ = 0 (clamped), or M_y = 0 and V_y = 0 (free: D_yY″ − D₁α²Y = 0 and D_yY‴ − (D₁ + 4D_xy)α²Y′ = 0). For a uniform or hydrostatic load the particular solution is the cylindrical-bending strip of rigidity D_x, summed in closed form, so the series converges exponentially at interior points; for a patch, line or point load the particular solution is the Green's function of the infinite strip, integrated over the loaded band in closed form, and the moments at the load converge like 1/m² (240 harmonics are used). The Navier double series for the all-simply-supported plate, w_mn = p_mn/{π⁴[D_x(m/a)⁴ + 2H(m/a)²(n/b)² + D_y(n/b)⁴]}, is printed as a convergence record and agrees with the Lévy series to the digits shown. Free or clamped conditions on edges 1 and 2 are not offered: rotate the plate so that the simply supported pair runs along y, remembering that D_x and D_y swap with it.

Huber's affine reduction and the isotropic tables

With the substitution y′ = y (D_x/D_y)^1/4 Huber's equation becomes D_x[w_,xxxx + 2 (H/√(D_xD_y)) w_,xxy′y′ + w_{,y′y′y′y′}] = q. When H = √(D_xD_y) this is the isotropic equation for a plate of rigidity D_x and side b′ = b (D_x/D_y)^1/4 along y, so the deflection of such an orthotropic plate is read from Timoshenko's isotropic tables at the reduced aspect ratio. The tool prints the ratio r_H = H/√(D_xD_y), the reduced side, Table 8's coefficient α for the reduced plate and the deflection it gives, beside the exact series value; for the default stiffened plate r_H is small and the reduction underestimates the deflection by a fifth, which is the point of printing both. The strip value 5 q a⁴/(384 D_x), the limit in which the ribs carry everything, is printed too.

Stresses

For a homogeneous orthotropic plate the bending stresses are linear through the thickness and σ_x = ±6M_x/t², σ_y = ±6M_y/t², τ_xy = ±6M_xy/t² on the faces, exactly as for an isotropic plate. For the stiffened plate the smeared theory places the neutral surface at the plate middle plane; the plating faces carry σ = E z (κ + ν κ_other)/(1 − ν²) with z = t/2, and a rib tip carries σ_r = E_r κ z_tip with z_tip = t/2 + h_r from the plate middle plane, tension at mid-span for ribs on the loaded plate's tension face. For the corrugated sheet the crest stress along the corrugations is M_yc/I with c = f + h/2, and the sheet bending stress across them is 6M_x(l/s)/h² because the moment per unit width is carried on the developed length. For a laminate the stresses depend on the lay-up and are left to the Classical Laminate Theory tool, which takes the moments and curvatures printed here.

Checks

The library reproduces the Plate Bending tool's isotropic results (its own Lévy series and Timoshenko's Table 8) to better than 10⁻¹² in the isotropic limit for every edge combination and load type, satisfies Huber's affine identity and Maxwell's reciprocity for point loads to machine precision, recovers the strip 5qa⁴/(384D_x) when D_y and H vanish, agrees with an independent numpy Navier sum (2001 × 2001 terms), a 13-point finite-difference solution with Richardson extrapolation and a scipy boundary-value integration of the Lévy ordinary differential equation for cases 1, 2 and 3, reproduces Seydel's corrugated-sheet rigidities, and matches the Classical Laminate Theory library's D matrix for a single ply and a [0/90]_s laminate (test-scripts/test_orthotropic_plates.php).

Load level and factors of safety

The load-type radio says what the entered load is, ultimate or limit; the margin boxes name the level of their allowable, never of the entered load. The plating face stress (the smeared value plus the local bending between the ribs where it applies) is tension on one face and compression on the other and is checked four ways: against F_tu at ultimate load (ULT) and F_ty at limit load × FS_yld (LIM) on the tension face, and against F_cy at limit load × FS_yld (LIM) and, since a sheet supported between ribs or a corrugation crest is a stable element that cannot cripple, against F_cy as its compression ultimate at ultimate load (ULT) on the compression face. The rib tips are checked on their tension side against F_tu (ULT) and F_ty (LIM) and on their compression side against the entered crippling stress F_cc (ULT; a free outstand can cripple below F_cy, so obtain it from the Crippling Strength tool; F_cy stands in when it is blank, and the page says so) and F_cy (LIM); the two sides are the signed extremes of the moment along each rib set over the plan, so a clamped plate reports both. The ribs are taken as the plating's material. With a limit load entered the strength checks multiply the stress by FS (default 1.5, the ultimate factor of 14 CFR 25.303) and the yield checks by FS_yld (default 1.15); with an ultimate load entered the strength checks use the stress as entered, never multiplied by FS, and the yield checks divide it by FS and multiply by FS_yld, so FS stays active in both positions. The problem is linear, so both levels come from one solution. The deflection limit is a limit-load check: the computed deflection is divided by FS when an ultimate load was entered. A laminate entered by its rigidities gets no stress box; carry its moments and curvatures to the Classical Laminate Theory tool.

Limits

Small deflections (the page flags w above half the structural depth), thin plates (depth below a tenth of the shorter span; a sandwich or a deep gridwork needs the transverse-shear theory of NACA Report 899), orthotropy aligned with the edges (no D₁₆, D₂₆), symmetric construction (no membrane–bending coupling), smeared ribs (several ribs across the span, local plating bending estimated separately, the neutral surface at the plate middle plane, no shear lag), edges 1 and 2 simply supported, and no elastic restraints or elastic foundation. Buckling of orthotropic plates is in the Plate Buckling tool.