Local Buckling Coefficients: Method

Local Buckling Coefficients of Flat Plate Elements

What local buckling is

A thin-walled compression member, a stiffened skin or a shear web is an assembly of flat plate elements joined along their edges. Each element can buckle on its own between those edges while the junction lines stay straight: the web of a channel bows between its flanges, a flange waves between the web and its free edge, a skin panel buckles between stringers and frames. This is local buckling, as opposed to the primary (Euler, torsional) buckling of the member as a whole. It is not failure, since a plate carries load beyond its buckling stress, but it changes the stiffness and the stress distribution and is normally required not to occur below limit load in skins, webs and stiffeners.

The critical stress of every case has the same form (NACA TN 3781, eq. 1):

σ_cr = η η̄ k π² E / [12 (1 − ν²)] (t/b)²,

with b the width of the element, t its thickness, E and ν the elastic constants, k the buckling coefficient that carries the loading, the aspect ratio a/b and the edge conditions, η the plasticity-reduction factor above the proportional limit and η̄ the cladding reduction factor of an alclad sheet. The whole difficulty is in k, η and η̄, and that is what Gerard and Becker's Handbook of Structural Stability Part I (1957) catalogues. This tool evaluates them for one element and, when asked, for a whole section.

Conventions

• The plate is a × b: a along the direction of the compressive stress (or along the bending stress), b across it. The loaded edges x = 0 and x = a carry the compression; the unloaded edges y = 0 and y = b are the ones whose support condition matters most. For shear, where all four edges carry stress, the coefficient is referred to the shorter side.
• Edge conditions: SS simply supported (no deflection, free to rotate), C clamped (no deflection, no rotation), F free, and elastically restrained: a rotational spring of stiffness S (moment per unit length per radian) along the edge, measured by the restraint coefficient ε = S b / D with D = E t³/[12(1 − ν²)]. TN 3781 defines ε as the ratio of the rotational rigidity of the edge stiffener to that of the plate; Lundquist and Stowell (NACA Report 733) wrote it as 4 S₀ b/D with S₀ the edge moment per quarter radian, which is the same number. ε = 0 is simply supported, ε → ∞ clamped; k is insensitive to ε beyond about 50 for a plate and 20 for a flange (TN 3781 fig. 17).
• A plate has both unloaded edges supported; a flange has one unloaded edge free. Widths are centre-line widths from a junction to the free edge or to the next junction.
• Stresses are compressive and positive; F_cy is the compressive yield strength and n the Ramberg–Osgood exponent of ε = σ/E + 0.002 (σ/F_cy)ⁿ.

Compression: the exact solution

With the loaded edges simply supported, the plate equation D ∇⁴w + N_x ∂²w/∂x² = 0 separates: w = Y(y) sin(mπx/a) with m half-waves along the length, and Y satisfies Y′′′′ − 2α² Y′′ + (α⁴ − α² N_x/D) Y = 0, α = mπ/a, whose solution is a combination of cosh, sinh, cos and sin. The four conditions at the two unloaded edges (two per edge: deflection and slope, deflection and moment, moment and Kirchhoff shear, or deflection and the spring relation D ∂²w/∂y² = ±S ∂w/∂y) give a 4 × 4 determinant whose lowest root in k = N_x b²/(π² D) is the buckling coefficient for that m. The tool finds the root by scanning and bisection and takes the minimum over m; for a long plate it minimises over the half-wave length λ = a/m treated as continuous. This is the method of Lundquist and Stowell (Reports 733 and 734), of which TN 3781's figures 14 to 17, 19 and 20 are plots, and it reproduces the report's long-plate values: 4.00 (SS–SS), 6.97 (C–C; Table 7 prints 6.98, fig. 17 prints 6.96), 5.42 (SS–C), 0.4256 + (b/a)² (SS–free at ν = 0.3; the report's 0.43 is the limit a → ∞) and 1.280 (C–free). Bryan's formula k_c = (mb/a + a/(mb))² for the simply supported plate, with minima of 4 at a/b = m and cusps at a/b = √(m(m+1)), is the special case.

A free edge brings Poisson's ratio into the coefficient itself: the long flange has k_c = 6(1 − ν)/π², 0.4256 at ν = 0.3 and 0.4073 at ν = 0.33 (TN 3781 fig. 2). The tool solves with the entered ν; the report's charts use 0.3.

Clamped loaded edges raise k_c for short plates (the dashed curves of fig. 14) and have no effect above a/b ≈ 4. No separable solution exists for them; the tool uses converged Ritz tables (polynomial boundary functions × Legendre polynomials, 14 × 14 terms, Gauss integration) for the five fig. 14 cases, computed for the tool and checked against the figure and against Timoshenko's 10.07 for the clamped square plate.

Unequal restraint. For a plate whose two unloaded edges have different ε, TN 3781 eq. (49) takes the geometric mean of the k values for each ε applied to both edges; the tool's exact solution needs no such rule for the listed cases, and uses the geometric mean only for the SS–C pair in shear and bending, where the tables are for equal restraint on both edges.

Shear and bending

Shear and in-plane bending do not separate, so the coefficients come from energy solutions. The tool carries tables in (b/a, ε) for shear and (a/b, ε) for bending, computed by a Ritz solution with the edge springs as additional energy (converged to four figures), and interpolates them. Their checks: shear k_s = 5.35 (SS, long) and 8.98 (C, long) of TN 3781 Table 7, 9.34 for the simply supported square plate (Stein and Neff, NACA TN 1222), 14.6 to 14.7 for the clamped square (Budiansky and Connor, NACA TN 1559); bending k_b = 23.9 for the simply supported long plate and Timoshenko's values at finite a/b. The widely used approximations k_s = 5.35 + 4 (b/a)² and 8.98 + 5.6 (b/a)² lie within a few percent of the tables. For long plates with elastic restraint the report's fig. 23(a) gives k_s∞ against ε; the tool's table agrees with it within reading accuracy.

One printed value differs: TN 3781 Table 7 gives k_b = 41.8 for a long plate with clamped unloaded edges, taken from the approximate energy solution of Schuette and McCulloch (NACA TN 1323). The converged value is 39.6 (Timoshenko and Gere; Bulson), five percent lower, and the tool uses it; the report's figure is printed beside it on the results page. For the simply supported plate the two agree (23.9).

For shear the coefficient is referred to the shorter side of the panel; the tool swaps the sides when a < b. A free edge carries no shear, and in-plane bending of a flange with its free edge in compression is not a case the tables cover; the tool says so and shows the SS–SS value for information only.

Plasticity

Above the proportional limit the plate is stiffer in some directions than in others and the elastic coefficient overestimates the stress. TN 3781 (Table 2) recommends Stowell's deformation-theory factors (NACA Report 898, table on p. 128), each a function of the secant and tangent moduli E_sec, E_tan at the buckling stress:

• long flange, SS edge: η = E_sec/E;
• long flange, clamped edge: η = (E_sec/E)[0.330 + 0.670 √(¼ + ¾ E_tan/E_sec)];
• long plate, SS edges: η = (E_sec/E)[½ + ½ √(¼ + ¾ E_tan/E_sec)];
• long plate, clamped edges: η = (E_sec/E)[0.352 + 0.648 √(¼ + ¾ E_tan/E_sec)];
• shear: η = E_sec/E (TN 3781 eq. A5, the curve entered at the equivalent compressive stress √3 τ);
• bending: the hinged-flange factor E_sec/E for a simply supported plate and the restrained-flange factors for restrained edges (TN 3781 p. 52).

Stowell takes ν = ½ in both the elastic and the plastic critical stress, so no (1 − ν_e²)/(1 − ν²) factor appears. For a pair of edges with different conditions, or an elastically restrained edge, the tool blends the two limiting factors by a weighted geometric mean, the weight being the position of the elastic coefficient between the SS and clamped values. Since η depends on the critical stress it multiplies, σ_cr = η(σ_cr) σ_e is solved as a fixed point by bisection, with E_sec and E_tan from the Ramberg–Osgood curve. A corrected stress above F_cy (F_cy/√3 in shear) is cut off: the element yields first.

Cladding

An alclad sheet has a soft cladding on each face, a fraction f of the total thickness (0.10 for alclad 2024-T3 up to 0.062 in, 0.08 for alclad 7075-T6; TN 3781 Table 4). Once the buckling stress exceeds the cladding yield σ_cl the cladding carries no more and the effective stiffness falls. TN 3781 eq. (42) gives η̄ = (1 + 3βf)/(1 + 3f) with β = σ_cl/σ_cr, which is 1 below the cladding yield and tends to 1/(1 + 3f) at high stress; the report's more elaborate eq. (40) for stresses above the proportional limit of the core differs little from it. The tool applies η̄ inside the same fixed point as η.

Combined loads

With R = applied stress / critical stress for each component, TN 3781 Table 5 gives R_c + R_s² = 1 for compression and shear on long plates (and the same form inelastically, eq. 60) and R_b² + R_s² = 1 for bending and shear; for compression with bending it gives curves but no equation. The tool writes R_c + R_b^1.75 + R_s² = 1, which reduces to the report's two forms when one component is absent and takes the bending exponent from Bruhn C5, and scales all applied stresses together by the load factor λ_L that reaches the curve; the margin is λ_L − 1.

Whole-section local buckling

In a real section no element is simply supported or clamped: each is restrained by its neighbours, which are themselves under the same stress and trying to buckle. TN 3781 treats this through ε (fig. 17: the restraint a stiffener supplies), and the NACA charts for I, Z, channel and tube sections (Stowell and Lundquist, TN 743; Kroll, Fisher and Heimerl, ARR 3K04) solved it by the moment-distribution method for plates: the stiffness of each element against an edge rotation, at the common stress and half-wave length, is assembled at the junctions, and the section buckles when the assembled stiffness vanishes.

The tool does exactly that in closed form. Under uniform compression every element has the same stress and the same half-wave length λ; with the junction lines held straight, the Lévy solution for an element with unit rotations imposed at its junction edges gives its exact 2 × 2 (or 1 × 1 for a flange) edge-moment stiffness, M₀ = −D Y″(0), M₁ = +D Y″(b). These are assembled over the junction rotations; the lowest stress at which the determinant vanishes is bracketed between the lowest simply supported element value (below it every element is stable) and the lowest clamped element value (the first pole), found by bisection and minimised over λ. The results page lists, for each element, the coefficient it would have alone with simply supported junctions (TN 3781's restraint-free lower bound: the lowest of them is a safe estimate of the section), with clamped junctions, and the coefficient it actually carries in the section; the governing element is the one with the lowest lower bound, and the equivalent restraint coefficient ε its neighbours supply is given so that the report's figures 16 or 20 can be entered with it. The test script reproduces the moment-distribution tables of ARR 3K04 (k_W and k_F for I, channel, Z and tube sections) to within one percent.

Limits: the junction lines are assumed not to deflect, so the distortional mode of lipped sections with slender lips (flange and lip rotating about the web junction) is outside the analysis; so are primary buckling, curved or tapered elements, and the post-buckling strength (see the Crippling Strength and Effective Sheet Width tools). The elements are taken as long unless a member length is entered, in which case the half-wave length is capped at that length.

How to read the results

The coefficient k is printed with its source (exact solution, table) and, where TN 3781 Table 7 has a long-plate value, that value beside it. σ_e is the elastic critical stress; η, E_sec, E_tan and η̄ follow at the solved stress, then σ_cr and its ratio to F_cy. The margin compares σ_cr, a strength allowable, with the applied stresses at ultimate load (ULT): ultimate stresses are used as entered, limit stresses are multiplied by the factor of safety FS (default 1.5); where a requirement forbids buckling below limit load, enter the limit stresses as ultimate with no factor. The chart of k against a/b shows where the element sits on its garland curve. With the section option on, the element table, the lower bound, the exact section stress, the governing element, its equivalent ε and the section stress against half-wave length follow.