Web Buckling - General Loading

Plate Buckling Method

Plate Buckling under Combined In-Plane Loads: Closed Form and Energy Method

Problem Statement

A rectangular plate 0 ≤ x ≤ a, 0 ≤ y ≤ b of thickness t carries in-plane loads per unit width: a uniform longitudinal load N_x on edges 1 (x = 0) and 2 (x = a), a uniform transverse load N_y on edges 3 (y = 0) and 4 (y = b), a uniform shear N_xy, and in-plane bending parts N_xb and N_yb that vary linearly across the width. Every load is tension positive; a compressive load is entered negative. The bending part N_xb is the value the linear distribution adds at edge 4, so N_x(y) = N_x + N_xb(2y/b − 1); N_yb is the value added at edge 2. In advanced mode the loads may instead be entered as the values at the two edges and the tool derives the mean and bending parts. Each edge is simply supported (S), clamped (C) or free (F).

The question is the factor λ by which all the entered loads must be multiplied together to make the plate buckle. The tool reports the ratio to requirement R_tot = 1/λ (applied over allowable along the load path) and the margin of safety M.S. = λ − 1. It does this twice, by two independent methods, so that one checks the other: the closed form of the handbooks (one buckling coefficient per load, an interaction equation for their combination) and the energy method (a Rayleigh-Ritz solution of the plate under the combined load, which needs no interaction equation). Both can include transverse shear deformation and plasticity. Buckling is a strength check, so both margins are written at ultimate load (ULT): the Load Level radio says whether the entered loads are ultimate (used as entered) or limit (multiplied by the factor of safety FS, default 1.5, before the ratios and the energy solution); no yield factor applies.

Plate Stiffness

An isotropic plate has the flexural stiffness D = E t³/[12(1 − ν²)]. An orthotropic plate (a laminate treated as homogeneous, or a corrugated or stiffened sheet smeared) has four: D₁₁ = E_x t³/[12(1 − ν_xyν_yx)], D₂₂ = E_y t³/[12(1 − ν_xyν_yx)], D₁₂ = ν_yx D₁₁ and D₆₆ = G_xy t³/12, with the reciprocity relation ν_yx = ν_xy E_y/E_x so that only ν_xy is entered. With E_y left blank the row is isotropic.

Closed-Form Method: Buckling Coefficients

Each load acting alone has a critical value N_cr = k π²D/b², where b is the width across the load and k the buckling coefficient for the support of the two edges parallel to the load (the unloaded edges). The coefficients come from Timoshenko and Gere and from NACA TN 3781 (Gerard and Becker):

• Compression, both unloaded edges simply supported: k = (m b/a + a/(m b))², minimised over the number of half-waves m along the load; it is 4.00 for a long plate and larger for a short one. For an orthotropic plate the same minimisation is done on N_cr = (π²/b²)[(m b/a)²D₁₁ + 2(D₁₂ + 2D₆₆) + (a/(m b))²D₂₂].
• Both unloaded edges clamped: k = 6.98 (long plate); orthotropic form with the coefficients 2.67 and 5.33 of the clamped-edge energy solution.
• One simply supported, one clamped: the geometric mean of the two, 5.28 against the exact long-plate value 5.42.
• One unloaded edge free (a flange): k = 6(1 − ν)/π² + (b/a)², i.e. 0.425 + (b/a)² at ν = 0.3, when the other edge is simply supported; 1.28 when it is clamped. The orthotropic flange is N_cr = [12D₆₆ + (π b/a)²D₁₁]/b².
• Both unloaded edges free: a wide column, N_cr = c π²D/a² with the end-fixity coefficient c of the loaded edges (1, 2.05 or 4).
• Shear: k_s = 5.35 + 4(b/a)² for four simply supported edges and 8.98 + 5.6(b/a)² for four clamped edges, b the shorter side. These are fits to the converged curves; Timoshenko's five-term series is not used because it is a first Ritz approximation that holds only near a square plate and grows without limit for a long one. With some edges clamped the allowable is interpolated geometrically, N_SS^{1 − n_C/4}N_CC^n_C/4 with n_C the number of clamped edges. No closed form exists for shear with a free edge.
• In-plane bending: k_b = 23.9 when the compressed edge is simply supported and 41.8 when it is clamped (long plate); the restraint of the compressed edge governs, the tension edge changes the coefficient by less than 1 %. The orthotropic coefficients are k_b = 13.4 + 10.45θ and 26.4 + 13.35θ − 0.25θ² with θ = (D₁₂ + 2D₆₆)/√(D₁₁D₂₂), from the converged energy solution referred to √(D₁₁D₂₂).

The closed form does not credit the fixity of the loaded edges: a clamped loaded edge raises the coefficient of a short plate, so the handbook value is conservative there. The energy method does credit it, which is one reason its allowables can come out higher.

Closed-Form Method: Transverse Shear, Plasticity and Yield Cutoff

Transverse shear. A thick plate, or a sandwich-like one, also deforms in transverse shear, which lowers the buckling load. With the Transverse Shear box checked the tool enters the moduli G_xz and G_yz (a blank G_yz is taken equal to G_xz; both blank on an isotropic row gives E/[2(1 + ν)]; both blank on an orthotropic row leaves the row thin and says so). The closed-form allowable is reduced in the series-spring form N_cr = N_t/(1 + N_t/K_s), with N_t the thin-plate value and K_s = κ G_s t, κ = 5/6 and G_s the smaller modulus: the bending and shear flexibilities add. This is exact for cylindrical bending and conservative for a plate mode, whose exact reduction is smaller.

Plasticity. Above the proportional limit the plate resists the bending of buckling with the moduli of its stress-strain curve, not with E. Stowell's unified theory (NACA Report 898) gives the plasticity reduction factor η = σ_cr/σ_e for each structure case in terms of the secant and tangent moduli E_sec, E_tan at the critical stress and the ratio r = √(1/4 + 3/4 E_tan/E_sec):

• A, long flange, unloaded edge simply supported: E_sec/E;
• B, long flange, unloaded edge clamped: (E_sec/E)(0.330 + 0.670r);
• C, long plate, both unloaded edges simply supported: (E_sec/E)(1/2 + r/2);
• D, long plate, both unloaded edges clamped: (E_sec/E)(0.352 + 0.648r); one edge of each kind takes the geometric mean of C and D;
• E, F, G, short, square and long plates loaded as columns: 1/4 E_sec/E + 3/4 E_tan/E, 0.114 E_sec/E + 0.886 E_tan/E, and E_tan/E;
• shear: E_sec/E (Gerard and Becker, confirmed by the tube tests of NACA TN 3184) with the curve entered at the equivalent compressive stress √3 τ; bending: the compression case of the compressed edge, entered at the extreme-fibre stress.

The moduli come from the Ramberg-Osgood curve ε = σ/E + 0.002(σ/F_cy)^n_c: E_sec = E/[1 + 0.002(E/σ)(σ/F_cy)^n_c] and E_tan = E/[1 + 0.002n_c(E/F_cy)(σ/F_cy)^{n_c − 1}]. Because η depends on the stress it reduces, the critical stress is the solution of σ_cr = η(σ_cr)σ_e, found by bisection. Stowell takes Poisson's ratio as 1/2 in both the elastic and the plastic critical stress, so no (1 − ν_e²)/(1 − ν²) factor appears. The correction applies to isotropic rows with F_cy and n_c entered.

Yield cutoff. The Ramberg-Osgood curve has no ultimate, so a corrected critical stress above F_cy (above F_cy/√3 in shear, the von Mises shear yield) means the plate yields before it buckles: the allowable is then the yield load F_cy t (or F_cy t/√3), a yield load and not a buckling load. The cutoff belongs to the plastic-correction option; elastic allowables with the box off are not capped.

Closed-Form Method: Utilization Ratios and Interaction

Each load gives a utilization ratio R = applied/allowable with compression positive: R_x = −N_x/N_x,cr and R_y = −N_y/N_y,cr, so a tensile load gives a negative ratio and relieves the interaction; shear and bending enter by magnitude. The loads are combined with the interaction equation of NACA TN 3781 and Bruhn:

R_x + R_y + R_xb^p + R_yb^p + R_xy² = 1

which reduces to the published pairs R_c + R_s² = 1 (compression and shear), R_c + R_b^1.75 = 1 (compression and bending), R_b² + R_s² = 1 (bending and shear, so p = 2 when bending is combined with shear only, else 1.75) and R_x + R_y = 1 (biaxial compression) when the other loads vanish, and is conservative with more loads present. The transverse load and bending enter by analogy. The load factor λ is the value at which the equation with every R replaced by λR equals 1 (solved numerically); then R_tot = 1/λ and M.S. = λ − 1. When every term is tensile the equation never reaches 1 and the plate does not buckle. When a load has no closed-form allowable for the fixity (a free loaded edge, shear or bending with a free edge) the closed form cannot form the ratio and the energy method must be used.

Energy Method: the Rayleigh-Ritz Solution

The total potential of the buckled plate is Π = U + λV, with the bending strain energy and the work of the in-plane loads

U = ½ ∫ [D₁₁ w_xx² + 2D₁₂ w_xxw_yy + D₂₂ w_yy² + 4D₆₆ w_xy²] dA,    V = ½ ∫ [N_x(y) w_x² + N_y(x) w_y² + 2N_xy w_xw_y] dA.

With tension positive a tensile load raises Π and stiffens the plate, a compressive one lowers it. The deflection is written as a finite double series w = ∑_i∑_j a_ij X_i(x) Y_j(y) whose functions satisfy the geometric boundary conditions: X_i(x) = (1 + ξ)^p(1 − ξ)^q P_i(ξ), with ξ = 2x/a − 1, P_i the Legendre polynomial and the boundary function enforcing the support of edge 1 by p and of edge 2 by q: 0 for a free edge, 1 for a simply supported edge (w = 0), 2 for a clamped edge (w = 0 and zero slope). Y_j(y) is built the same way from edges 3 and 4. This is the boundary characteristic polynomial basis of Bhat (1985) with Legendre rather than Gram-Schmidt polynomials; the Ritz method needs only the geometric conditions, and a free edge's zero moment and shear are approached as the series grows. Every integrand is a polynomial, so Gauss-Legendre quadrature integrates it exactly.

Making Π stationary with respect to every a_ij gives the symmetric generalized eigenproblem K a = λ M a, K from U (positive definite whenever the supports prevent rigid-body motion) and M from −V (indefinite in general). With the Cholesky factor K = L L^T and C = L⁻¹M L^−T, the eigenvalues μ of the symmetric matrix C are 1/λ: the largest positive μ gives the buckling load factor λ₁, the next ones the higher modes, and no positive μ means the plate does not buckle under that combination (every term tensile). The eigenvector gives the buckled shape, drawn in the two figures with its node lines; the half-waves are counted along the row and column of the mode through its largest lobe. The tool solves the whole problem in plain PHP (Householder tridiagonalisation and the implicit QL algorithm), so the combined load is solved directly and no interaction equation is involved: R_tot = 1/λ₁ and M.S. = λ₁ − 1 as before.

Energy Method: Series Size, Convergence and the Error Estimate

A finite series can only make the plate stiffer than it is, so the Ritz load factor is an upper bound that falls as terms are added. The basis needs about two polynomial terms per half-wave of the buckled shape plus a few for the boundary function, and the number of half-waves grows with the longer side over the shorter one, r = max(a/b, b/a). The Series Terms setting Auto therefore picks n = 2 round(3 + r) terms per direction, between 6 and 16 (14 with transverse shear), which keeps the load factor within 0.1 % of the converged value for square to moderately long plates; a square plate is converged at 8 terms, a plate of r = 6 under shear needs 14. A number chosen instead is used for every row. The limit of 16 is run time, not accuracy: the eigenproblem has n² unknowns and its cost grows about as n⁶ (3 s per solve at 16 terms on the development server, four times that with transverse shear), and the page carries a time budget after which remaining rows are marked as not solved.

For the example row the problem is solved again at n − 4 and n − 2 terms. Once the series resolves every half-wave the error falls geometrically, so from the last two steps d₁ = λ(n − 4) − λ(n − 2) and d₂ = λ(n − 2) − λ(n) the step ratio q = d₂/d₁ is known and the steps still to come add up to d₂ q/(1 − q), Aitken's extrapolation. The page prints this estimate of the error left. When the steps are not yet shrinking the series is still finding the mode's half-waves and no estimate is possible: a larger series can then find a shorter half-wave, and the last step is neither a bound nor an estimate. A last step above 0.1 % prints a warning; the results are kept as computed, with the reminder that an upper bound on λ makes R_tot low and M.S. high. Long plates with more than about seven half-waves (a simply supported plate beyond a/b of 8, a clamped one beyond 5, or a plate much stiffer across the load than along it) are the usual case; there the closed-form coefficient has reached its long-plate value and is the better answer.

Energy Method: Transverse Shear (Mindlin-Reissner Plate)

With the Transverse Shear box checked the plate is a first-order shear-deformable plate. The normal to the middle surface rotates by φ_x and φ_y, which differ from the slopes by the transverse shear strains γ_xz = w_x − φ_x and γ_yz = w_y − φ_y. The strain energy is the bending energy written in the rotations plus the shear energy ½ ∫ [κ G_xz t (w_x − φ_x)² + κ G_yz t (w_y − φ_y)²] dA, with the shear correction factor κ = 5/6; the load work keeps its form in w. Each rotation has its own series: a clamped edge fixes w and both rotations, a simply supported edge fixes w and the rotation about the normal to the edge (the hard condition, which returns the thin-plate result as the shear stiffness grows), a free edge fixes nothing. The rotation unknowns are condensed out exactly before the eigenproblem is solved (the load matrix acts on w only), so the eigenproblem keeps n² unknowns; the summary table shows the condensed count in parentheses. For a simply supported isotropic plate the theory gives N_cr = N_K/(1 + D s/(κ G t)) with s = (mπ/a)² + (nπ/b)² and N_K the thin-plate value of the same mode, which the solution reproduces to four figures; the closed form's series-spring reduction is the cylindrical-bending limit of this and is conservative for a plate mode.

Energy Method: Plasticity by Deformation Theory

With the Plastic Correction box checked the stiffness inside the energy functional is the plastic one. Hencky's deformation theory, the theory behind Stowell's Report 898 and his shear and combined-load notes (NACA TN 1681, TN 1990), gives the plate stiffness from the membrane stresses σ_x, σ_y, τ at each point: with the stress intensity σ_i = √(σ_x² − σ_xσ_y + σ_y² + 3τ²) and the Ramberg-Osgood moduli E_sec, E_tan at σ_i, the incremental compliance is

C = (1/E_sec) [[1, −½, 0], [−½, 1, 0], [0, 0, 3]] + (1/E_tan − 1/E_sec) q q^T,    q = (3/(2σ_i)) (s_x, s_y, 2τ),

with the deviatoric stresses s_x, s_y and Poisson's ratio 1/2 in the plastic range, as Stowell takes it. The plate stiffness is D = C⁻¹t³/12 at every Gauss point, including the bending-twisting couplings a shear stress brings in. In uniaxial compression this is exactly Stowell's energy expression; in pure shear the twisting stiffness carries E_tan, which is why his TN 1681 finds the shear factor a little below the secant value E_sec/E that the closed form uses and that Peters' tube tests follow: the energy method is slightly conservative in shear (about 0.88 to 0.97 of E_sec/E). Flow (incremental) theory is not used; it overpredicts plastic buckling loads, the known plastic-buckling paradox.

Because the stiffness depends on the stress, which is λ times the applied load, the eigenproblem is nonlinear in λ and is solved as a fixed point: the stiffness is frozen at a trial λ, the linear problem solved, and the iteration closed (regula falsi) when the two agree. As in Stowell's report, the plastic factor is divided by the elastic factor of the ν = 1/2 plate solved the same way, η = λ_p/λ_e(ν = 1/2), so that the error of taking ν = 1/2 cancels, and η multiplies the elastic factor computed with the entered ν. For one load on a long plate this reproduces his table; for any fixity, combination or distribution it is the same procedure carried through by the series solution, so mixed cases need no judgement about which of his cases applies. The yield cutoff is kept as one criterion: λ may not exceed the value at which the largest stress intensity over the plate reaches F_cy (which is F_cy t for uniform compression and F_cy t/√3 for shear, the closed form's two caps). With transverse shear on, the shear stiffness is reduced by E_sec/E at each point. Orthotropic rows and rows without F_cy or n_c stay elastic.

Comparing the Two Methods

The two results should be close but are not expected to be equal. The energy method gives a higher load factor where the closed form is conservative by construction: it credits clamped loaded edges, it solves the combined load directly instead of through the interaction equation (on a long simply supported plate under biaxial compression and shear the interaction gives a R_tot about a third higher), and it carries both transverse shear moduli exactly. It gives a lower one in shear with plasticity, by the deformation-theory twisting stiffness above. The single-load table of the example row compares the two methods load by load: a difference of a few percent is the closed form's rounding of the handbook coefficient; a large positive one points at the loaded-edge fixity. A free edge in shear or bending, where the closed form has no coefficient, is solved by the energy method alone.

Reading the Output

The closed-form section works through the example row chosen in the Example Row dropdown: stiffnesses, one allowable per load with its coefficient, the transverse-shear and plastic corrections when on, the utilization ratios, the interaction equation and its load factor, R_tot and the margin box. The energy-method section first tabulates every row of the plate table (fixity, aspect ratio, loads, the elastic factor, η and the cutoff flag when plasticity is on, λ₁, R_tot, M.S., the closed-form R_tot of the example row, the second and third modes, the half-waves, the series size and the solve time), then works through the example row: the three lowest load factors with their half-wave counts, the convergence with the series size and the error estimate, each load alone against the closed-form allowable, the plastic correction with η and the cutoff check, R_tot and the margin box, and the first two buckled shapes as colour plots of w/w_max with their node lines.

Limits

Flat rectangular plates only, with uniform or linearly varying edge loads and uniform shear; no lateral pressure, no initial imperfection, no post-buckling strength (the closed-form coefficients and the energy method both give the bifurcation load). The orthotropic plate is homogeneous through the thickness, so laminates with bending-extension coupling or large bending-twisting coupling terms are outside the model. Plasticity is for isotropic rows only. Elastic restraint of an edge between simply supported and clamped is not offered; a free edge in shear or bending, and clamped loaded edges, are covered by the energy method only. For plates whose mode has more than about seven half-waves the energy series is not converged at 16 terms and the closed-form long-plate coefficient should be used.