Diagonal Tension Web

Diagonal Tension Web

Diagonal Tension Web: Kuhn's Engineering Theory of Incomplete Diagonal Tension

What the tool does

The tool analyses one bay of a plane, stiffened shear web, the web of a built-up spar or bulkhead with flanges and transverse uprights, that buckles in shear before it fails and then carries the load as a field of diagonal tension. It follows the procedure of Section 4 of NACA Technical Note 2661, A Summary of Diagonal Tension, Part I: Methods of Analysis (Kuhn, Peterson and Levin, 1952), the report that collected the NACA work of 1940 to 1952 into one method and that Bruhn, Niu and the aircraft companies adopted. For the entered geometry, material and web shear it gives the buckling stress of the web, the loading ratio, the diagonal-tension factor k, the angle of the diagonal tension, the average and maximum stresses in the uprights, the peak nominal web stress and its allowable, the upright allowables for forced crippling and column failure, the secondary stresses in the flanges, the rivet loads, a margin of safety for every check and the web shear that would fail each mode. The default inputs are the report's own numerical example 1, NACA test beam I-40-4Da, so the first page reproduces the printed values of Section 7.

Where the method comes from

Wagner showed in 1929 to 1931 (NACA Technical Memoranda 604 to 606) that a very thin web does not fail when it buckles: it folds into diagonal waves and works as a set of tension diagonals, the uprights becoming compression posts and the flanges chords, so that the web-stiffener system acts as a truss and can carry many times its buckling load. His theory of pure diagonal tension assumes the web buckles at an infinitesimal load and carries no diagonal compression at all. Real webs lie between that limit and the shear-resistant web that never buckles. Kuhn's engineering theory (NACA Report 697 of 1940, Technical Note 1364 of 1947 with Peterson, and the summary of 1952) interpolates between the two with a single empirical factor k, calibrated on the upright stresses measured in 32 NACA test beams, and supplies the failure criteria from about 200 beam tests and 140 web-shear tests, all collected in Part II (Technical Note 2662). The method is deliberately simple, because every airplane contains hundreds of such elements, and deliberately conservative: its allowables sit at the lower edge of the test scatter bands.

The procedure, step by step

Effective upright area (Section 4.1). No web sheet is counted with the upright. A double upright, one angle on each side of the web, has A_Ue = A_U. A single upright is loaded eccentrically by the web; the stress that matters is the one in the fibres at the web, and equation (22) expresses this as an effective area A_Ue = A_U / (1 + (e/ρ)²), where e is the distance from the web median plane to the centroid of the upright and ρ its radius of gyration about the axis parallel to the web. Typical angles and zees have A_Ue/A_U of 0.4 to 0.6.

Buckling stress (Section 4.2, equation (32)). τ_cr,elastic = k_ss E (t/d_c)² [R_h + ½(R_d − R_h)(d_c/h_c)³] for d_c < h_c (the roles of d_c and h_c and of R_d and R_h swap otherwise), with k_ss the theoretical coefficient of a simply supported plate (Figure 12(a)) and R_h, R_d empirical restraint coefficients that depend on how much thicker than the web the uprights and the flange legs are (Figure 12(b); 1.0 corresponds to simple support, 1.62 to clamped edges, and a very thin stiffener gives less than simple support because it bends with the web). If the formula gives less than the buckling stress of the web with the uprights disregarded, the latter is used. Above the proportional limit the plasticity correction of Figure 12(c) applies; the tool holds both curves, 24S-T3 and Alclad 75S-T6, digitized from the report.

Loading ratio and diagonal-tension factor (Sections 4.3 and 4.4). The nominal web shear stress is τ = S_w/(h_e t), with S_w the shear carried by the web (the external shear less the vertical component of the flange forces in a tapered beam) and h_e the depth between flange centroids. The empirical law of equation (27), k = tanh(0.5 log₁₀(τ/τ_cr)), gives k = 0 at and below buckling and k → 1 as the loading ratio grows without bound; a web loaded to 45 times its buckling stress, like the example beam, has k = 0.68 and is far from pure diagonal tension. The total stress is then split into a true shear part (1 − k)τ and a diagonal-tension part kτ.

Upright stresses and the angle of the folds (Sections 4.5 and 4.6). In pure diagonal tension the upright carries the vertical component of all the diagonals in its bay, σ_U = −τ d t tan α / A_Ue. In incomplete diagonal tension only the fraction k of the stress is diagonal tension, and a width 0.5(1 − k) d of sheet, which has not yet lost its ability to carry compression, works with the upright, so equation (30a) reads σ_U = −k τ tan α / (A_Ue/(d t) + 0.5(1 − k)). The angle α follows from the strain compatibility of the truss, tan²α = (ε − ε_F)/(ε − ε_U) (equation (30c)): the diagonal strain of the web, ε = (τ/E)[2k/sin 2α + (1 − k)(1 + μ) sin 2α], must agree with the shortening of the uprights and of the flanges. Because σ_U depends on α and α on σ_U, the report solves the pair by successive approximation and offers Figure 14 (σ_U/τ against k for each A_Ue/dt) as its result; the figure was drawn with sin 2α taken as 1 and the flange strain neglected, and the tool reproduces exactly that, tabulating the cycles, and prints the exact solution beside it. The stress is largest at mid-height of the upright, the "gusset effect"; Figure 15 gives σ_Umax/σ_U from d/h_U and k, the ratio being 1.8 for a very short bay just after buckling and 1 in pure diagonal tension. The report calls Figure 15 tentative. For the allowable web stress a second angle is needed, the angle the folds would take in pure diagonal tension, α_PDT, from Wagner's equation (15), tan⁴α = (1 + h t/(2A_F)) / (1 + d t/A_Ue).

Web strength (Sections 4.7 and 4.8). The peak nominal web shear stress is τ_max = τ(1 + k C₂), where C₂ is the stress-concentration factor of Figure 18 for flanges flexible enough to sag between the uprights and so unload the diagonals attached near mid-bay; it depends on Wagner's flange-flexibility factor ωd = 0.7 d ⁴√(t/((I_C + I_T) h_e)) and is negligible below ωd = 1.2. The alternative form τ′_max = τ(1 + k² C₁)(1 + k C₂) also carries the angle factor C₁ = 1/sin 2α − 1. The allowable is empirical: Levin's 97 tests of long 24S-T3 and Alclad 75S-T6 webs in nearly pure shear (Technical Note 1756) gave the "basic allowable" τ*_all of Figure 19, the line 10 percent below the mean, against k; it falls from the shear strength of the unbuckled sheet at k = 0 (27.4 ksi for 24S-T3, 34.8 ksi for Alclad 75S-T6) to about half the diagonal tensile strength at k = 1, because in pure diagonal tension the sheet stress is 2τ. The curve for the angle α_PDT is built from the top curve: at k = 1 the tensile stress scales as 1/sin 2α, at k = 0 the angle is immaterial, and the difference varies linearly in k. The allowable is multiplied by 1.1 for tight rivets, 0.9 for bolt heads bearing directly on the sheet, and in proportion to the web's ultimate strength when a value other than the specification minimum is justified. τ_max compared with the α_PDT curve is the report's recommended procedure and gives the tool's web margin; τ′_max compared with the 45° curve is shown beside it. The dashed line of Figure 19(a) is the allowable yield stress of a 24S-T3 web, half the tensile yield at k = 1; the tool checks it at limit load.

Upright strength (Sections 4.9 to 4.11). Nearly every upright failure in the tests was forced crippling: the shear buckle of the web forces the free edge of the attached leg into a wave, the strip along that edge is a beam-column under the upright compression and the lateral push of the buckle, and large deformations grow long before any elastic local-buckling load. The empirical allowable for the maximum upright stress is σ₀ = C k^2/3 (t_U/t)^1/3 ksi, equations (36) and (37), with C = 21 and 26 for double and single uprights of 24S-T3 and 26 and 32.5 for 75S-T6, the thickness ratio measuring the "relative sturdiness" of the upright; the constants are the lower edge of a ±20 percent scatter band, and σ₀ is multiplied by E_sec/E when it exceeds the proportional limit (the tool uses a Ramberg-Osgood curve when an exponent is entered). Column failure is checked with the effective length of equation (35), L_e = h_U / √(1 + k²(3 − 2d/h_U)) for d < 1.5h, an empirical fit to the restraint of the web against bowing (Wagner's theoretical curves of Figure 6 are too optimistic), the allowable being the column curve for a solid section; the tool uses Euler's curve above and the Johnson parabola below the transition π√(2E/F_cy). A riveted double upright develops that strength only if its upright-to-web rivets carry the longitudinal shear R_R = 100 Q h_U/(b L_e) kips (equation (40), from the double-angle column tests of Kuhn and Moggio); a shortfall reduces the column allowable by Figure 21. A single upright must keep σ_U below the column yield stress and its average stress σ_U A_Ue/A_U below the column allowable at a slenderness h_U/(2ρ), a rule that limits the two-half-wave deformation seen on slender single uprights.

Flanges and rivets (Sections 4.12 to 4.16). The horizontal component of the diagonal tension compresses both flanges by k S_w cot α/(2A_F) and the vertical component bends them between uprights, with a primary maximum moment k C₃ S_w d² tan α/(12 h_e) over each upright and half that at mid-bay; the report calls these moments conservative, notes that practice often neglects them, and suggests the block compressive strength as the allowable. The web-to-flange rivets carry R″ = S_w(1 + 0.414k)/h_R per inch, an interpolation between S_w/h_R for a shear-resistant web and √2 S_w/h_R for pure diagonal tension, h_R being the depth between the rivet lines (using h_e was found unconservative). The upright-to-flange rivets carry the upright load σ_U A_U (double) or σ_U A_Ue (single). The upright-to-web rivets must resist the tension set up by the web folds trying to lift the sheet off a single upright or to split a double one: 0.15 t σ_ult and 0.22 t σ_ult per inch, criteria the report labels tentative.

Charts and the predicted failing loads

The tool reproduces the report's design curves as digitized tables: Figure 12(a) k_ss, 12(b) R_h and R_d, 12(c) the plasticity correction, 15 σ_Umax/σ_U, 18 C₂ and C₃, 19 the basic allowables and the yield line, and 21 the riveted-column reduction; Figures 13, 14, 16 and 17 are the equations themselves and are evaluated directly. The first chart of the results page draws k against τ/τ_cr with the web's point, the second the allowable curves at 45° and at α_PDT with the two peak stresses. The last table scans the web shear through the whole procedure, since k, α and every allowable change with the load, and reports the shear at which each margin reaches zero, the P_W, P_FC and P_C of the Part II test tables; the smallest is the predicted failing shear of the bay.

Units and inputs

Lengths, areas, stresses and forces go through the site's unit dictionary (inches or millimetres, ksi or MPa, kips or kN); internally the analysis runs in the report's units, because its empirical constants, equations (36), (37) and (40) and every chart, are in ksi and kips. Optional inputs left blank skip their check: the rivet-line depth defaults to h_e, a blank flange inertia or ωd takes the flanges as rigid (C₂ = 0, C₃ = 1), a blank flange area takes them as very heavy in the angle of pure diagonal tension, blank fastener allowables skip the rivet margins, a blank Q skips the riveted-column check. The Load Level radio says whether the entered web shear is ultimate or limit. The strength margins (web rupture, forced crippling, column, flanges, rivets) are written at ultimate load (ULT): a limit shear is multiplied by the factor of safety FS (default 1.5), an ultimate shear is used as entered. The 24S-T3 yield check is written at limit load times the yield factor FS_yld (default 1.15; LIM): a limit shear is multiplied by FS_yld, an ultimate shear is divided by FS and multiplied by FS_yld, so FS stays active in both positions; the web is analysed again at that load because k, C₂ and the yield line depend on the load level. The single-upright F_cy and the flange F_cy,F boxes are strength criteria of the report (column-curve ceiling, block compressive strength) and are written at ultimate.

Limits of the method

Plane webs only: curved webs (fuselage skins) follow Sections 8 to 13 of the report with a different angle equation and different allowables. The allowable curves and the crippling constants exist only for 24S-T3 and Alclad 75S-T6 with web and uprights of the same alloy; for other alloys the report has no data, and the tool's material factor is only the proportional scaling the report permits for a higher guaranteed ultimate. Flanges with ωd above 2.5 and bays into which a large load is introduced (end bays, tip loads, fuselage reactions) are excluded from the report's accuracy statement; Section 3.7 asks for a doubler in such bays. Upright spacings beyond 1.5h take the full upright length as the column length; closed-section uprights have no forced-crippling data. Figure 15 and the rivet tension criteria are called tentative by the authors. The report estimates that its predictions are unconservative by more than 2 percent in fewer than 5 percent of cases; web strengths average 10 percent above the allowable, upright strengths 1.25 times the forced-crippling allowable, and the portal-frame effect of the appendix can add about 10 percent that the report declines to credit. The test tables of Part II, which the test script reproduces, use allowables corrected to the actual material properties and 5 percent for tight bolts; the tool's defaults use the specification values.

Reading the results

The margin of safety of each check is allowable/applied − 1 at ultimate load, in the site's margin boxes; the governing mode is the smallest. For the example beam the report finds the uprights slightly weak in column action (allowable 15.5 ksi against 16.9 ksi at the test failing load, about 8 percent conservative) and 25 percent and 15 percent of reserve against forced crippling and web rupture, and the beam did fail by column bowing of the uprights; the tool shows the same picture. A web that has not buckled (τ ≤ τ_cr) has k = 0: the uprights carry nothing, the web margin is the shear-resistant one at the k = 0 end of Figure 19, and the diagonal-tension checks are moot.