Crippling Strength: Method

Crippling Strength of Thin-Walled Compression Members

What crippling is

A thin-walled compression member made of flat elements (an angle, channel, zee, hat or extrusion) buckles locally long before the material yields: each flat element bows between the corners at its own critical stress. Local buckling is not failure. After buckling the load shifts to the corners, which stay straight and carry a rising stress while the buckled plate middles carry little more than their buckling stress. The section fails when the corner stress reaches a large fraction of the compressive yield strength and the corner itself distorts; the average stress on the section at that load is the crippling stress F_cs and F_cs·A is the crippling load P_cc. Crippling is the strength of a short column, the plateau of the column curve: a longer member fails at the Euler or Johnson stress, with F_cs as the cutoff of the column curve.

Gerard (NACA TN 3784, 1957) showed from a large body of tests that, for a given material, the crippling stress depends on two geometric quantities only: the developed length-to-thickness ratio, written as A/t², and the number of corners (or the number of flanges and cuts, g). The reason is physical: a corner is what supports the unloaded edges of the plates meeting there, so the more corners a given developed length has, the higher the average stress at which the corners reach yield.

Conventions

• Every width b is a centre-line width measured from a corner (the intersection of the element centre lines) to the free edge or to the next corner; A = Σ b t is the section area and Σ b the developed length. Bend radii are not subtracted (the reports used the actual area of the formed specimens, which differs by a few percent for R/t = 3).
• An element has one edge free when it ends at a free edge (a flange, a leg, a lip), no edge free when both edges are corners (a web, a tube side).
• Thicknesses may differ in an extrusion (stem and flange of a tee). Gerard's abscissa then uses the area-weighted mean thickness t̄ = A/Σb (TN 3785 eq. 7); Needham's and the element method use each element's own thickness.
• Stresses are compressive and positive; F_cy is the compressive yield strength of the sheet or extrusion and E its compression modulus.

Gerard's generalized method (NACA TN 3784 and TN 3785)

Gerard combined two correlation schemes that had both fitted the test data: the Gallaher–Needham form σ̄_f = C (t²/A)^m for one material, and Schuette's form σ̄_f/σ_cr = α (σ_cy/σ_cr)ⁿ, which carries E and σ_cy. Eliminating the buckling stress between them gives a non-dimensional formula with the range of validity of the second and the simplicity of the first. The forms used by the tool are:

• Angle-type sections (two elements meet at every junction: angles, plates between supports, tubes, hats, lipped channels and zees, formed multicorner sections), TN 3785 eq. (2):
  F_cs/F_cy = β_g [ (g t²/A) (η̄E/F_cy)^1/2 ]^0.85,
  with g the number of flanges plus the number of cuts needed to divide the section into angle elements (TN 3785 figure 6: angle 2, plate 3, tee 3, cruciform 4, channel or zee 5, I or H 7, hat and lipped channel 11, rectangular tube 12). β_g = 0.558 for extrusions (tubes, angles and V-groove plates fall on one line, TN 3785 Table 4). For formed sections the coefficient depends on how much forming raised the yield strength of the corners: formed angles take β_f = 0.59, 0.62, 0.65 and 0.66 for corner yield ratios of 1.0, 1.1, 1.2 and above 1.25 (TN 3784 Table 13; Needham's 2024-T3 angles formed in the heat-treated condition gave 0.665, alclad 7075-T6 gave 0.630), and multicorner sections take β_c = 1.1, 1.2 or 1.3 in the corner form of TN 3784 eq. (75), which the tool converts with β_g = β_c (c/g)^0.85 (TN 3785 eq. 3; g = 3c − 1 for an open formed section). The formula holds up to F_cs/F_cy = 3/4 (0.92 for formed angles with a corner ratio above 1.25, 0.9 for formed multicorner sections with β_c = 1.3); beyond the cutoff the report sets F_cs equal to the plastic buckling stress, which the tool does not compute, so the cutoff value is used as a conservative estimate.
• T-type sections (three or more elements meet at a junction: tee, cruciform, I and H extrusions), TN 3785 eq. (5):
  F_cs/F_cy = 0.670 [ (g t²/A) (η̄E/F_cy)^1/2 ]^0.40, valid to 3/4.
  The opposed flanges of a T junction keep the junction straight after buckling, so these sections cripple at a higher stress than angle-type sections of the same A/t² and the exponent is much gentler. Crockett's and Ramberg and Levy's tee tests and Heimerl's H-extrusions (five alloys) lie within 5 percent of the line.
• Two-corner sections (Z, channel and J, formed or extruded), TN 3784 eq. (61):
  F_cs/F_cy = β [ (t²/A) (η̄E/F_cy)^1/3 ]^0.75,
  with β = 4.00 for 2017-T3, 2024-T3 and alclad 2024-T3 formed in the heat-treated condition (R/t about 3), 3.42 for alclad 7075-T6, 3.78 for FS-1h magnesium, 3.6 for extrusions and an estimated 3.3 for sections formed without corner strengthening (Tables 8 and 14). The one-third power of E/F_cy comes from the way the buckling coefficient and A/t² of a Z or channel both vary with the flange-to-web ratio. The range of validity is F_cs/F_cy ≤ [ β/2.14 · (t/b_w)^0.25 ]^4/3 (eq. A7, about 2 (t/b_w)^1/3 for β = 3.6), drawn as the b_w/t = 20, 30 and 40 cutoff lines of the report's figure 16; beyond it F_cs = σ_cr again.

η̄ is the cladding reduction factor of TN 3784 eq. (66), η̄ = (1 + 3 (σ_cl/σ_cr) f)/(1 + 3 f), with f the cladding fraction of the thickness (0.10 for alclad 2024-T3, 0.08 for alclad 7075-T6) and σ_cl/σ_cr the ratio of the cladding yield stress to the buckling stress of the section; the report used overall values of 1/2 and 1/3, giving η̄ = 0.885 and 0.871. It is 1.0 for bare sheet and extrusions and enters the abscissa only, so its effect is small.

In the tool the abscissa x is the bracket of these formulas; the reports plot its inverse, A/(g t²) (F_cy/η̄E)^1/2, and both are printed. The test data of TN 3784 scatter within about ±10 percent of the lines for angles and two-corner sections and ±5 percent for multicorner and T-type sections.

Needham's method (angle elements)

Needham (1954) tested formed angles, channels and square tubes of 2024-T3 and alclad 2024-T3 and 7075-T6 and proposed dividing a formed section into angle elements, one per corner, by cutting every web at mid-width. Each angle element with legs a and b, mean leg b′ = (a + b)/2 and thickness t fails at

F_e = C_e √(F_cy E) / (b′/t)^0.75, C_e = 0.316 (both edges free, a free angle), 0.342 (one edge free, an angle cut from a channel), 0.366 (no edge free, an angle cut from a tube),

as the method is reproduced in Bruhn's Analysis and Design of Flight Vehicle Structures, C7.7. The section crippling load is the sum of the element loads F_e A_e, and F_cs the load divided by the area, each element stress capped at F_cy. Needham argued that the angle, not the plate, is the right unit because the corner governs failure and the cut falls in the least sensitive part of the plate; the method also carries the raised yield strength of formed corners implicitly, because its constants were fitted to formed sections. Gerard's Table 4 of TN 3784 lists Needham's own fits for the free angle as C̄ (t/b)^0.75 with C̄ = 225.0, 200.1 and 259.0 ksi for the three materials; divided by √(F_cy E) they give 0.327, 0.309 and 0.318, a mean of 0.318 against the 0.316 of the design table. The method is not defined for a T junction (three elements at a corner), so the tool reports it as not applicable for tees, I sections, cruciforms and typed element lists.

Element method

The element methods of the aircraft companies (Bruhn C7.4, after Crockett) sum the failing loads of the flat plate elements, each read from a curve of failing stress against b/t for one edge free and for no edge free. The tool uses Needham's functional form for those curves so that the three methods share one basis: F_i = C √(F_cy E) (t/b)^0.75 with C = 0.316 for an element with one edge free and C = 0.366 × 2^0.75 = 0.616 for an element supported at both edges. The first constant is Needham's free-angle value: each flange of an equal angle is a plate with one edge free and width equal to the leg, so the angle result is the flange result. The second is Needham's tube value rewritten for the full plate width: his square-tube angle element has legs b/2, and (2t/b)^0.75 = 2^0.75 (t/b)^0.75. With these constants a square tube's four sides each give the simply supported plate value, Needham's method and the element method coincide for a channel whose flanges are half the web (within 0.3 percent), and the no-edge-free constant reproduces Gerard's extruded-tube and V-groove-plate line within 3 percent over 0.2 ≤ (t/b)√(E/F_cy) ≤ 0.3 and within 8 percent at 0.5. The section value is the area-weighted mean, element stresses capped at F_cy.

Which method to trust

Gerard's method rests on the widest body of tests (several hundred specimens of five aluminum alloys and two magnesium alloys, formed and extruded) and is the only one of the three with material-specific coefficients for extrusions, for alclad 7075-T6 and for sections formed without corner strengthening; it is the method the later design handbooks reproduce. Needham's method and the element method carry constants fitted to 2024-T3 sections formed in the heat-treated condition, so for an extrusion their free-flange values run 10 to 20 percent above Gerard's extruded-angle line at the same b/t, and for a material with a flatter stress-strain curve (7075-T6) they are on the high side as well. The tool takes the minimum of the applicable methods as the governing value unless a method is selected, and marks the governing row.

Reading the results

• Figure 1 draws the section with the free edges marked and the Gerard cuts dashed; the count g printed under it is flanges plus cuts, and the class (angle-type, T-type, two-corner) fixes the formula.
• Table 1 lists every flat element with b/t, its edge condition and its element-method stress; Table 2 the Needham angle elements with b′; Table 3 the three section values and P_cc = F_cs A.
• The margin of safety compares the governing crippling load with the applied load at the ultimate level (a limit load is multiplied by the factor of safety). Crippling is an ultimate failure, so no yield margin is written.
• Figure 2 draws the Gerard line of the section's class against the abscissa on a log scale with the cutoff plateau and the section's point; the Needham and element values are placed at the same abscissa for comparison.

Limits

• The member must be short enough not to buckle as a column (L′/ρ below about 20); otherwise use F_cs as the cutoff of a Johnson–Euler column curve (Column Stability tool).
• Lips and bulbs count as corners only when they are large enough to make the flange act as a web (TN 3784, "Lips and Bulbs"); a small lip should be lumped into its flange and the section treated as having one corner fewer.
• Bend angles other than 90° do not change the crippling stress (Roy and Schuette's tests, TN 3784 figure 21) provided the radius of gyration is not reduced enough to cause primary instability.
• Curved elements, sections with holes, elevated temperature (enter the properties at temperature; Heimerl and Roberts found the room-temperature relations to hold), and the sheet acting with a stiffener (Stiffened Panel Compression tool) are outside this tool.
• The coefficients are those of the 1957 reports for aluminum and magnesium alloys; for steel and titanium sections the Gerard forms are commonly used with the same coefficients, but the reports contain no such tests.