Beam-Column with Crippling: Method

Beam-Column with Crippling: Exact Beam-Column Moments, Crippling, Johnson–Euler Column and the Interaction Check

What the tool does

A stringer or stiffener between frames carries its share of the panel compression and often a lateral load as well: cabin pressure on the skin it is riveted to, a fitting or equipment load, or end moments from the frames. The axial load magnifies the bending, the bending adds to the compression on one flange, and a thin-walled section fails locally, by crippling, before its material yields. The tool treats one stringer with its effective sheet as a pin-ended beam-column of constant EI, computes the exact moments and deflections of Timoshenko and Gere's Chapter 1 for a uniform load, a concentrated load and two end couples superposed at the applied axial load, finds the extreme-fibre stresses on the free flange and the sheet face, and checks the section three ways: the compressed fibre against the crippling stress, the stringer plus sheet as a Johnson–Euler column, and the aerospace stringer interaction R_a + R_b = 1 with the amplified bending stress, which combines the two and gives the margin. A fibre in net tension is checked against F_tu.

Conventions

The sheet lies at the bottom of the section and the stringer stands on it by its attachment flange; the free fibre is the point of the stringer farthest from the sheet (the outstanding flange of a Z or channel, the top of a hat, the tip of an angle leg or tee stem) and the sheet fibre is the outer face of the sheet (the attachment flange when no sheet is entered). The lateral loads w and Q are positive when they act from the stringer toward the sheet, the direction in which cabin pressure pushes the skin; the end couples M₁ (at end A, x = 0) and M₂ (at end B) are positive when they compress the free flange; the deflection y is positive toward the sheet; P is compression. With this set a strut under positive w, positive Q or two equal positive couples has a positive moment everywhere, its free flange in compression and its sheet in tension, and the Timoshenko forms apply with their printed signs. Two couples of opposite sign bend the strut in reverse curvature.

Section, crippling allowable and effective sheet

The Z, channel, hat, angle and tee sections are built from their centre-line dimensions (flange areas b t, web area b_w t, bend radii ignored); a section drawn in the Section Properties tool or typed as A, I, centroid height and overall height may be used instead. The crippling stress is Gerard's correlation of the NACA Handbook of Structural Stability: for formed Z and channel sections F_cc/F_cy = β [(t²/A)(E/F_cy)^1/3]^0.75 with β = 4.00 for 2017-T3 and 2024-T3 and 3.42 for alclad 7075-T6 (TN 3784 eq. 61, Table 14), for extruded Z and channel sections β = 2.14 α (b_w/t)^0.25 with α = 0.80 (eq. 63), for angles, hats and T-type extrusions the generalized form F_cc/F_cy = β_g [(g t²/A)(E/F_cy)^1/2]^m with g the number of flanges plus cuts (angle 2, tee 3, hat 11) and the constants of TN 3785 Table 4 (β_g = 0.665 formed angles, 0.630 for 7075-type alloys, 0.558 extruded angles and hats, 0.55 formed multicorner sections, m = 0.85; tees β_g = 0.670, m = 0.40). Every value is capped at F_cy and the handbook's validity limit (3/4 F_cy, 0.92 F_cy for formed angles) is flagged when exceeded. A crippling stress from a test or from the Crippling Strength tool can be entered instead and replaces Gerard's value in every check; NACA Report 1255 did exactly this for the Z of the default case, taking 27.5 ksi from the tests of its reference 25 where Gerard's formula gives 30.4 ksi.

After the sheet buckles the strip next to the stringer still works at the stringer stress over the von Kármán effective width 2w = 1.9 t_s √(E/σ), half on each side; the tool evaluates it at F_cc, the stress at which the stringer fails, as Bruhn's panel method does, and caps it at the pitch. The stringer plus this strip is the column section: its area, centroid, moment of inertia, radius of gyration and the two fibre distances c_free and c_sheet are reported and drawn.

Beam-column moments and deflections

For a pin-ended strut of constant EI under compression P the deflection obeys EI y″ + P y = −M₀(x), where M₀ is the first-order moment of the lateral loads. The equation is linear in those loads at a given P, so the exact solutions of the three load types are superposed. With k = √(P/EI), u = kL/2 and the Euler load P_e = π²EI/L², Timoshenko and Gere give for a uniform load w (eq. 1-7 and 1-8)

M(x) = (wL²/4u²)[cos(u(1 − 2x/L))/cos u − 1], M_mid = (wL²/8) · 2(sec u − 1)/u², δ_mid = (5wL⁴/384EI) · 12(2 sec u − 2 − u²)/(5u⁴);

for a concentrated load Q at a distance a from end A, with c = L − a (eq. 1-3 to 1-6),

M(x) = Q sin(kc) sin(kx)/(k sin kL) for x ≤ a and the mirror image beyond, so that a load at midspan gives M_mid = (QL/4) · tan u/u and δ_mid = (QL³/48EI) · 3(tan u − u)/u³;

and for end couples M₁ and M₂ (eq. 1-19)

M(x) = [M₂ sin kx + M₁ sin k(L − x)]/sin kL, which for equal couples gives M_mid = M₀ sec u.

The factors 2(sec u − 1)/u², tan u/u and sec u are the exact amplifications of the three cases; they approach 1 as P → 0 and grow without bound as P → P_e. Their series in α = P/P_e begin 1 + 1.028α, 1 + 0.822α and 1 + 1.234α (the deflection factors 1 + 1.003α, 1 + 1.028α and 1 + 1.028α), so the usual magnification factor 1/(1 − α) = 1 + α + α² + … is close to exact for the deflection of a uniformly loaded strut and about 3 % low for the moment of a strut under equal couples at moderate α. The tool prints the exact peak moment and its station, the first-order peak, the exact amplification, and beside them the approximate C_m M₀/(1 − P/P_e) with C_m = 1 for lateral loads and C_m = 0.6 + 0.4 M₁/M₂ for couples alone (the ratio positive for single curvature, the smaller couple on top); the exact value is the one carried into the checks. For unequal couples the station of the stationary moment is tan kx* = (M₁/M₂ − cos kL)/sin kL, and when it falls outside the span the larger end couple is the peak. The charts draw y(x) and M(x) with and without the axial load.

The moment diagram is always the pin-ended one. The end-fixity coefficient c of the frame attachment enters only the column check through L′ = L/√c; a strut with restrained ends also has end moments from the restraint, which must be entered as M₁ and M₂ if they are to be included (the Beam Column Strength tool handles struts whose axial load, section or loads change along the length).

Fibre stresses and the three checks

With compression positive, f_free = P/A + M c_free/I and f_sheet = P/A − M c_sheet/I along the strut; the tool reports the largest compression, the fibre and station where it occurs, and the largest tension if the bending overcomes the axial compression on one fibre. Crippling check: the compressed flange must stay below F_cc, M.S. = F_cc/f_c − 1 (the extreme-fibre check of Bruhn A5 with P/A included). Column check: the Johnson parabola F_c = F_cc − F_cc²(L′/ρ)²/(4π²E) from the crippling stress at zero length, tangent to the Euler curve at L′/ρ = π√(2E/F_cc) where the stress is F_cc/2, Euler beyond (Bruhn C7); P_c = F_c A and R_a = P/P_c. Interaction: R_a + R_b = 1 with R_b = f_b/F_cc, the bending stress on the compressed flange carrying the exact beam-column amplification, M.S. = 1/(R_a + R_b) − 1. Since F_c ≤ F_cc the interaction is never less conservative than the plain stress sum, so it is the lower bound and the margin of the page; the stress check shows how much of the difference is the column effect. Tension: a fibre in net tension is checked against F_tu.

Margins of the form allowable/applied − 1 scale the allowable against the loads at the entered P. Because the amplification itself grows with P, the factor λ by which all loads could grow together before R_a + R_b reaches 1 is a little smaller; the tool finds it by bisection on the exact solution and prints λ − 1 beside the ratio margin. Limit loads are multiplied by the factor of safety before any comparison; the allowables (F_cc, F_c, F_tu) are ultimate values.

Default example

The default inputs are the Z-stringer and sheet of the numerical example of NACA Report 1255 (t_w = 0.064 in, b_w = 2.56 in, b_A = 0.512 in, b_F = 1.024 in, t_s = 0.102 in, b_s = 3.06 in, 2024-T3 with E = 10,600 ksi, F_cy = 43.6 ksi, F_tu = 64 ksi from MIL-HDBK-5J, crippling stress 27.5 ksi as the report took it from tests) as a 19.6 in strut between frames (L/ρ = 20, the slenderness of the report's tests) under 9,000 lb of compression and 20 lb/in of pressure load toward the sheet (6.5 psi over the pitch), at ultimate. The sheet is fully effective (2w = 3.80 in > 3.06 in), the column section has A = 0.5743 in², I = 0.5515 in⁴, ρ = 0.980 in; P_e = 150,200 lb, α = 0.060, and the midspan moment of 960.4 lb-in grows to 1,023.3 lb-in (factor 1.0655 against 1.0637 from 1/(1 − α)). The free flange carries 15.67 + 3.59 = 19.26 ksi against F_cc = 27.5 ksi (M.S. +0.43); the Johnson–Euler allowable is 26.78 ksi (P_c = 15,377 lb), R_a = 0.585, R_b = 0.130, and the interaction gives M.S. +0.40 with a proportional load factor of 1.39. The closed forms were checked against an independent scipy solution of the differential equation and against the site's Beam Column Strength tool (Gittleman's tabular method), whose node moments for five uniform struts agree with them within 0.002 %.

Limits

Pin-ended struts of constant section and constant axial load (restrained ends only through c in the column check and through entered end couples); compression P below the Euler load; one plane of bending about the axis parallel to the sheet; the effective width held at F_cc along the whole span; no shear, torsion, inter-rivet buckling or wrinkling of the sheet (see the Stiffened Panel Compression tool), no lateral-torsional instability of an unsupported free flange, no plastic bending. The centre-line section model ignores bend radii; load a drawn section for exact properties.