Large-Deflection Plates

Large-Deflection Plates Method

Large-Deflection Plates: the von Kármán Equations, Levy's Fourier Solution, the Clamped Plate by Edge Moments and the Circular Plate

What the tool computes

The deflection and the stresses of a thin flat plate under uniform pressure when the deflection is of the order of the thickness, so that the stretching of the middle surface carries a part of the load that small-deflection (Kirchhoff) theory ignores. For a rectangular plate with all edges simply supported or all clamped, and with the edges either held against in-plane displacement or free to slide, the tool solves von Kármán's equations by Levy's Fourier method; for a circular plate, clamped or simply supported, it solves the axisymmetric equations on a radial grid. It reports the centre deflection against the load parameter p a⁴/(E t⁴), the small-deflection value and the deflection ratio, the membrane, bending and total stresses at the centre and at the edge midpoints, the average edge tensions an immovable frame must react, Levy's tabulated values beside the series result, a convergence record, and charts of the load–deflection and load–stress curves with the design point.

Why small-deflection theory fails

Kirchhoff theory assumes the middle surface does not stretch, so the deflection is proportional to the load and the stresses are pure bending. When the deflection reaches a fraction of the thickness the middle surface must stretch to follow the curved shape: a plate whose edges are held develops membrane tensions that grow with the square of the deflection and resist the pressure directly, like a drum skin; even a plate whose edges can slide inward develops a self-equilibrating membrane field (tension in the middle, compression near the corners). The load–deflection curve therefore bends over, the deflection grows eventually with the cube root of the pressure, and the extreme-fibre stress is the sum of a membrane and a bending part. For Levy's simply supported square plate the small-deflection formula overstates the centre deflection by a factor of four at w/t = 1.4 (the tool's default case), and the bending stress at the centre is about half of the small-deflection value while a membrane stress of nearly the same size has appeared.

The von Kármán equations

With w the deflection and F the Airy stress function of the median-fibre stresses (σ′_x = F_,yy, σ′_y = F_,xx, τ′_xy = −F_,xy), von Kármán (1910) coupled the compatibility of the membrane strains with the equilibrium of the bent plate:

∇⁴F = E [ (w_,xy)² − w_,xx w_,yy ],     D ∇⁴w = p + t ( F_,yy w_,xx + F_,xx w_,yy − 2 F_,xy w_,xy ),

with D = E t³/[12(1 − ν²)]. The first equation says that the Gaussian curvature of the deflected surface forces the middle surface to stretch; the second adds to the plate equation the transverse component of the membrane forces acting on the curved surface. The extreme-fibre bending stresses are σ″_x = −[E t/(2(1 − ν²))](w_,xx + ν w_,yy) and likewise for y, and the total extreme-fibre stress is σ′ + |σ″| on the face where the two add. The equations are nondimensional in W = w/t and the load parameter λ = p a⁴/(E t⁴), and the stresses scale as σ a²/(E t²): the tool prints both the physical and the dimensionless values, so a result can be compared directly with Levy's tables and figures.

Levy's Fourier solution (NACA Report 737)

Levy (1942) expanded the deflection of a simply supported rectangular plate in the double sine series w = ΣΣ w_mn sin(mπx/a) sin(nπy/b), m, n odd for a symmetric load, which satisfies the simply supported conditions term by term. The right-hand side of the compatibility equation is then a sum of products of sines and cosines, that is a cosine series, and the stress function follows term by term as F = −p̄_x y²/2 − p̄_y x²/2 + ΣΣ b_pq cos(pπx/a) cos(qπy/b), each b_pq a quadratic form in the w_mn (his equation (8)). Substituting w and F in the equilibrium equation and matching the sine coefficients of the pressure gives one cubic equation per deflection coefficient (his equation (9) and Tables 1 and 2), which he solved by successive approximation for a square plate with six coefficients. The tool does exactly this with the products of trigonometric functions worked out exactly and the cubic system solved by Newton's method with load continuation; the user chooses 3, 4 or 5 odd harmonics per direction (9, 16 or 25 coefficients), and the page prints the neighbouring counts as a convergence record. With Levy's own six coefficients the tool reproduces every entry of his Table 9 to the printed digits (w₁₁/h = 0.5, 1.0, 1.5, 2.0 at λ = 14.78, 51.4, 132.0 and 278.5, centre deflections 0.485, 0.952, 1.402, 1.846) and the stress-function coefficients of his Table 10.

The harmonics beyond the retained set are not dropped: they respond almost linearly to the pressure (and, for the clamped plate, to the edge moments) and are carried as a tail of a few hundred terms whose stiffness includes the average membrane tensions of the retained solution. This tail removes the slow convergence of the bending stresses that a bare truncated Navier series shows, and it is what lets the clamped plate below be treated with a few edge-moment coefficients.

In-plane edge conditions

The constants p̄_x and p̄_y are the average membrane stresses on the edges. Levy showed (his equations (10) to (12)) that with this stress function every edge remains straight and free of shear stress, and that the in-plane displacement of an edge is uniform along it: u(a) − u(0) = −(p̄_x − ν p̄_y) a/E − (1/2b) ∫∫ (w_,x)² dA. Two limiting conditions are offered. Immovable edges (Levy's "edge displacement zero"): the displacement is set to zero, which fixes the average tensions, (σ̄_x − ν σ̄_y) a/E = (1/2b) ∫∫ (w_,x)² dA; the tool reports them as stresses and as the edge tensions N = σ̄ t per unit length that the supporting frame must carry. Free to slide: the average edge stress is zero, the edges move inward as rigid lines, and only the self-equilibrating part of the membrane field remains; the tool reports the approach of the opposite edges. A real panel lies between the two. A riveted sheet–stringer panel whose edges are also held along their length (u = v = 0 on the edge) is in principle stiffer than either; Wang (NACA TN 1425 and 1462, 1948) solved that case by finite differences for the square and for b/a = 1.5 and 2, and the tool prints his centre deflections beside the series result where they apply (for the square his riveted plate deflects 11 % less than Levy's at λ = 132; at b/a = 1.5 and 2 his coarse grid gives values within a percent of the series).

The clamped plate (NACA Report 740)

For a plate with clamped edges Levy kept the simply supported sine series and added the unknown edge bending moments, expanded as m_x(y) = Σ k_s sin(sπy/b) on the x-edges and likewise on the y-edges, replacing them by an equivalent pressure distribution; the condition of zero slope at every edge, Σ_m m w_mn = 0 for every n (his equations (18) to (20)), then determines the moment coefficients. The tool does the same in the weak (Galerkin) form of the plate equation: the edge moments enter through the boundary term ∮ m (∂φ/∂n) ds of every sine test function, and the zero-slope conditions are the constraints whose Lagrange multipliers the edge moments are. The edge bending stress is then read directly from the moment coefficients, 6m/t², which the truncated sine series itself could never supply (its second derivatives vanish on the edge). Because the slope sums converge only like 1/r², the tail of high harmonics described above is essential here; with it the small-deflection limit of the tool gives the exact clamped-square coefficients 0.0012653 for the centre deflection and −0.0513 for the edge moment, and in the large-deflection range it reproduces Levy's 22-term Table 5 for the centre deflection within 2 % and for the edge bending stress within 3.5 % up to his last row (λ = 402, w/t = 1.9), against the 2 % he claims for his own solution.

One correction to Report 740 has to be recorded. Its equation (16) prints the first stress-function coefficient as b_0,2 = (E/32)(w₁₁² + 9w₁₃² + 2w₁₁w₁₃ − 18w₁₃w₃₃ + …), whereas the exact product algebra, and Report 737's own Table 10 (its value 1.174 at w₁₁/h = 1 is reproduced only with −2w₁₁w₁₃, the printed sign would give 1.310), require the term −2w₁₁w₁₃. The slip propagated into the stress expressions (26) and (27) of Report 740 and into the membrane and total stress columns of its Table 5, which run some 15 % below the correct values (at λ = 63.4 the table gives 1.06 and 1.33 for the membrane stress at the edge midpoint and at the centre; the series and an independent quadrature solution on a different basis both give about 1.25 and 1.57). The deflections, the edge-moment coefficients and the edge bending stresses of Report 740 are unaffected and are reproduced here. The tool prints Table 5 beside its own values with this note.

Circular plates

For an axisymmetric plate the equations reduce to two ordinary ones in ρ = r/a for the slope Θ = dW/dρ and the radial stress resultant S = ρ σ_r a²/(E t²): D(Θ″ + Θ′/ρ − Θ/ρ²) = λ ρ/2 + S Θ/ρ and S″ + S′/ρ − S/ρ² = −Θ²/(2ρ), with Θ = S = 0 at the centre, Θ = 0 (clamped) or Θ′ + νΘ = 0 (simply supported, zero radial moment) at the edge, and S′ − νS = 0 (immovable, zero circumferential strain) or S = 0 (free) there (Timoshenko and Woinowsky-Krieger, Articles 97 to 99). The tool solves them by second-order finite differences on 400 radial intervals with Newton iteration and load continuation, and prints the solution on 200 intervals as a grid-convergence check; an independent scipy boundary-value solution agrees to five digits. For the clamped plate the one-term energy solution of Timoshenko's Article 97, w₀ + k w₀³/t² = p a⁴/(64D), is printed as a hand check, with k computed for the entered Poisson's ratio by the same minimisation (0.484 for an immovable edge at ν = 0.3, against the 0.488 Timoshenko prints; it overestimates the deflection by about 7 % at w₀/t = 3).

Reading the results

The load parameter λ = p a⁴/(E t⁴) tells at a glance where the plate sits: below about 10 the small-deflection theory is within a few percent; Levy's tables run to 278.5 (simply supported) and 402 (clamped), where the centre deflection is about two thicknesses; beyond a few thousand the plate is a membrane whose deflection grows with the cube root of the load. The deflection ratio w_c/w_lin is the fraction of the small-deflection value that remains. In the stress table the membrane stress is the mid-plane value, the bending stress the extreme-fibre value on the bottom face (the face away from the pressure, in tension at the centre), and the total their sum on the face where they add; for a simply supported plate the largest total stress is at the centre, for a clamped plate at the midpoint of an edge, where the clamping moment dominates. The edge tensions of an immovable plate are the forces the frame must resist in plane; a frame that cannot resist them makes the plate behave more like the free-to-slide case, with larger deflections and a smaller membrane share. The convergence table shows how far the series has settled: the centre deflection and the membrane stresses agree between harmonic counts to a fraction of a percent, the bending stresses of the clamped plate to a few percent.

Load level and factor of safety

The load-type radio says what the entered pressure is, ultimate or limit; the margin box names the level of its allowable, never of the entered pressure. The one allowable F_a is an ultimate strength (F_tu for the tension face, or F_cy for the compression face as a stable-element ultimate), so the box is written at ultimate load (ULT). Because the stresses are not proportional to the load in large-deflection theory, a limit pressure is not factored after the fact: the plate is re-solved at FS × p (default FS 1.5, the ultimate factor of 14 CFR 25.303) and the total stress of that solution is compared with F_a. With an ultimate pressure entered the stress of the solution above is used as it is, never multiplied by FS, and the factor is greyed on the form. The tool offers one solve and one allowable only; a yield check at limit load would need a second solution at a second pressure, and the comparison with the small-deflection values is information, not a margin.

Limits

• Thin, elastic, initially flat. Von Kármán's equations keep the squares of the slopes and nothing else; they hold for deflections up to a few thicknesses and thicknesses below about a twentieth of the span. The material is linearly elastic: compare the total stress with the yield strength, because a plate at w/t = 2 is often already yielding at a clamped edge.
• Idealised edges. Simply supported or clamped against rotation, with Levy's straight shear-free edges in plane. Wang observed (NACA TN 1425) that measured panels with clamped edges behave near the centre as if simply supported, the clamping being local, and that above λ ≈ 175 the measured deflections exceed every theoretical solution.
• Convergence. The bending stresses of the clamped plate are the slowest quantities; read the convergence table. Levy himself estimated 2 % for his 22-term edge stress.
• Uniform pressure only, no in-plane edge loads (Levy's combined compression and pressure, Report 737 Tables 12 to 14, is not implemented), no orthotropy, no imperfections.
• Compressive edge zones. A plate whose edges slide freely develops circumferential or tangential membrane compression near the edge, which in a very thin plate wrinkles; the tool reports the stress and does not check wrinkling.