Shear-Resistant Web: Method

Shear-Resistant Web: Buckling of a Stiffened Web and the Stiffness Its Stiffeners Need

What the tool does

A built-up beam, spar or bulkhead with a thin sheet web and transverse stiffeners can be designed in two ways. If the web is allowed to buckle, it carries the load as a field of diagonal tension and the stiffeners become compression posts: that is the diagonal-tension web, analysed by the companion tool from NACA Technical Note 2661. If the web is thick enough, or the stiffeners close enough, that it does not buckle up to the design load, the web works in true shear, the stiffeners carry no load from it, the stresses follow the plate-girder formulas and the beam is called shear-resistant (TN 2661, Section 1). This tool checks a web against the second design philosophy: the nominal web shear stress, the buckling stress of the panel between flanges and stiffeners with the restraint the flanges and stiffeners provide, the margin against buckling at limit load, the shear strength of the sheet at ultimate, the stiffness the stiffeners need to act as nodal lines of the buckled web, the stiffener as a column, the rivets, and the weight per unit length. When the ultimate load lies above the buckling load, as it often does in a web designed to be shear-resistant only at limit, the tool also gives the diagonal-tension factor k at ultimate and the loads it puts into the stiffeners and rivets, and points to the Diagonal Tension Web tool for the complete post-buckling analysis.

Nominal web shear stress

The shear flow in a built-up beam is q = V Q/I; its variation over the depth of a thin web is small, and the plate-girder formula τ = V/(h t), with h the depth between the flange centroids, is the usual design value (TN 2661, equations (2) and (4)). The tool evaluates both for the rivet load and uses the plate-girder value for the web. The shear V is the shear carried by the web in the bay analysed, the external shear less the vertical component of the flange forces in a tapered beam. The limit value is the design shear the web must carry unbuckled; the ultimate value, the limit times the ultimate factor (1.5 for aircraft), is used for the strength checks.

Buckling stress of the web

The panel between two flanges and two stiffeners is a rectangular plate in shear whose edges are neither simply supported nor clamped: a thick flange angle restrains the web nearly fully, a stiffener as thin as the web hardly at all. Kuhn, Peterson and Levin (TN 2661, Section 4.2, equation (32)) wrote the critical stress as the simply-supported plate value corrected by two empirical restraint coefficients:

τ_cr,e = k_ss E (t/d_c)² [R_h + ½ (R_d − R_h)(d_c/h_c)³] for d_c ≤ h_c,

and the same with d_c and h_c and R_d and R_h exchanged when the stiffener spacing exceeds the depth. Here d_c and h_c are the clear dimensions of the panel between the attached legs, k_ss is the theoretical coefficient of a simply supported plate for the panel's aspect ratio (Figure 12(a) of the report, which folds π²/(12(1 − ν²)) into the coefficient: 8.3 for a square panel, 4.9 for a very long one), and R_h and R_d are the restraint coefficients along the stiffeners and along the flanges, read from Figure 12(b) against the ratio of the member thickness to the web thickness. On the upper curve, for a web sandwiched between two members (double stiffeners, two flange angles), a thickness ratio of 1 gives R = 1.0, simple support, and the curve flattens towards 1.62, the clamped value, at a ratio of 3; the lower curve, for one member on one side of the web, gives 0.70 at a ratio of 1, because a member no thicker than the web bends with it and provides less than simple support. The bracket weights the two restraints by the panel proportions: a long narrow panel between stiffeners feels mainly the stiffener restraint. If the formula gives less than the buckling stress of the whole web with the stiffeners disregarded (a long plate of width h_c with the flange restraint), the stiffeners are too weak to matter and the latter value is used (Note 2 of Section 4.2). The coefficients were fitted to NACA beam tests and are the same ones the diagonal-tension analysis uses, so the two tools agree on the buckling stress of a given web.

For comparison the tool prints the values of NACA Technical Note 3781 (Gerard and Becker, Figure 22) for the same panel with all four edges simply supported and all four clamped, τ = k_s π² E/(12(1 − ν²)) (t/b)² with b the short side and the parabolic fits k_s = 5.34 + 4 (b/a)² and 8.98 + 5.6 (b/a)², which follow the exact Southwell–Skan and Budiansky–Connor curves within about 3 percent; the TN 2661 value normally falls between them, and below the simply-supported value when a single stiffener no thicker than the web gives R_h < 1.

Above the proportional limit the critical stress is reduced by the plasticity factor η = E_sec/E (Stowell, TN 1681; Gerard, TN 3781 equation (A5)), the secant modulus being taken on the Ramberg–Osgood compression curve ε = σ/E + 0.002 (σ/F_cy)ⁿ at the equivalent compressive stress √3 τ; because η depends on the stress, τ_cr = η(τ_cr) τ_cr,e is solved as a fixed point. A blank exponent leaves the stress elastic.

The shear-resistant requirement and the strength of the web

The web is shear-resistant when the limit shear stress is below the critical stress; the tool's first margin is τ_cr/τ_L − 1 at limit load. At ultimate load a web that has not buckled fails by shearing the sheet, so the nominal stress is compared with the ultimate shear strength F_su of the material; a web that buckles between limit and ultimate carries the excess by diagonal tension, and the NACA engineering theory measures how far that has developed by k = tanh(0.5 log₁₀(τ/τ_cr)) (TN 2661, equation (27)): k = 0 at the buckling load, 0.06 at 1.3 times it, 0.46 at ten times. The tool prints k at ultimate and uses it for the stiffener compression and the rivet loads below; for the web itself the allowable of a buckled web is the empirical curve of the diagonal-tension tool, not F_su, which is why a web expected to buckle well below ultimate should be analysed there.

Stiffness required of the stiffeners

A transverse stiffener is useful only if it stays straight while the web buckles, forming a nodal line so that the panel size is the spacing d and not the whole bay. How stiff it must be is a question with a long history and several answers, and the tool prints four of them side by side, each evaluated for the moment of inertia I_u of the stiffener about the median plane of the web, the axis about which a stiffener attached to a buckling web bends (for a single angle, its centroidal inertia plus A_u e_u²).

Bruhn, equation C10.8. Wagner's 1930 treatment of the stiffened web as an orthotropic plate, with Seydel's shear-buckling coefficient for a plate much stiffer across than along (8.125, so that 4 × 8.125 = 32.5, rounded to 33), gives the general buckling load of the whole web, S = (33 E/h) (t³/12 · (I_u/d)³)^1/4. Setting it equal to the applied shear and solving for the inertia gives I_u = 2.29 (d/t) (V h/(33 E))^4/3, with 2.29 = 12^1/3. The tool evaluates it at the limit shear, the load up to which the web must not buckle. It is a general-buckling condition, not a nodal-line condition, and is usually the least demanding of the four.

Bruhn, Figure C10.9. The later chart, attributed to Chance Vought, makes the general buckling stress of the stiffened web equal to the local buckling stress of the simply supported panel between stiffeners, so that the stiffeners are just stiff enough for the panel buckling to be the critical mode; for a panel whose short side is d this is I_u/(d t³) = 0.0217 (h √K_s/d)^8/3 with K_s = k_s π²/(12(1 − ν²)), and the constant is again 12^1/3/33^4/3. It does not depend on the load.

Moore, NACA Technical Note 862 (1942). R. L. Moore of the Aluminum Research Laboratories tested two 17S-T plate girders with eight sizes of angle stiffener at five spacings, 60 tests in all, measuring the lateral deflections of web and stiffeners. No definite buckling load could be observed, so he defined an adequate stiffener as one whose deflection at the design load stayed under 0.020 in and under a quarter of the web deflection, and found that the required ratio of the stiffener's flexural rigidity to that of the web panel between stiffeners, λ = 12 E I (1 − μ²)/(E d t³) with μ = 1/3, followed λ = 14/(d/h_c)³ (his b/d), h_c being the clear depth. Stiffeners of that size developed 1 to 1.5 times the simply-supported buckling load of their panels. He judged the sizes larger than the theoretical requirements of Timoshenko and of the Moisseiff–Lienhard bridge formula and "hardly likely to be classed as too large"; the formula is the most demanding of the four.

Stein and Fralich, NACA Technical Note 1851 (1949). The exact theory: an infinitely long simply supported plate with identical, equally spaced transverse stiffeners of zero torsional stiffness, solved by the Lagrangian multiplier method for the whole range of stiffener stiffness E I/(D d), D = E t³/(12(1 − ν²)), at panel aspect ratios h_c/d of 1, 2 and 5 (Table I of the report). For weak stiffeners the plate behaves as an orthotropic plate; as the stiffness grows the critical stress rises along two further branches (buckling antisymmetric over two bays, and periodic over four bays), and with rigid stiffeners it exceeds the simply-supported panel value because the plate is continuous across the stiffeners. The stiffness at which the critical stress first reaches the simply-supported value of the panel between stiffeners, the report's "cut-off" and the value older theory had taken from the orthotropic curve, is the nodal-line requirement; the report shows that the older cut-off was unconservative. The tool takes the lowest of the three branches at each stiffness, finds the cut-off stiffness for each tabulated ratio and interpolates log-log in h_c/d, flagging ratios outside 1 to 5. Bleich fitted the same results with I = 2.5 h t³ (h/d − 0.7 d/h), the form quoted in Galambos's stability guide; it lies within 2 to 25 percent of the Table I cut-offs and is printed for comparison.

The margin box takes the largest of the four requirements. For the default web Moore's test formula asks for about 70 percent more than the Stein–Fralich theory, which asks for three times Bruhn's C10.8 value; all four are well below the inertia of the 0.75 × 0.75 × 0.063 in angle. The choice between them is a design-policy question: the Collier Aerospace note on the Bruhn equations records finite-element studies in which stiffeners sized to the Bruhn criteria raised the buckling load but bowed with the web, while the Bleich/Stein–Fralich size kept them nearly straight.

Stiffener as a column

Up to the buckling load the stiffeners of a shear-resistant web carry no load from the web. If the web buckles before ultimate, the diagonal tension compresses them by σ_u = k τ tan α/(A_ue/(d t) + ½(1 − k)) (TN 2661 equation (30a), with the angle α from the Figure 14 iteration and, for a single stiffener, the effective area A_ue = A_u/(1 + (e_u/ρ_u)²) of equation (22)); an applied load P_u, say a rib reaction introduced at the stiffener, adds P_u/A_u. The allowable is the column curve, Euler above the transition π√(2E/F_cy) and the Johnson parabola below, at the effective length L_e = h_u/√(1 + k²(3 − 2 d/h_u)) of equation (35), which is h_u itself when k = 0. The forced-crippling failure of the attached leg, the usual failure of uprights in a diagonal-tension web, is not checked here: it belongs to the companion tool.

Rivets and weight

The web-to-flange rivets carry the shear flow at the rivet line, V Q/I with Q the static moment of the flange about the neutral axis, or the plate-girder value V/h; the tool uses the larger, computes I and Q from the flange area and the web when they are not entered, and multiplies the flow by (1 + 0.414 k) when the web buckles before ultimate (equation (34) of TN 2661, which interpolates between V/h for the shear-resistant web and √2 V/h for pure diagonal tension). The load per rivet is the flow times the pitch divided by the number of rows, checked against the rivet shear allowable and, through P/(D t), against the bearing strength of the web. The stiffener-to-web rivets carry no theoretical load in an unbuckled web; once the web buckles, the folds try to lift the sheet off a single stiffener or split a double one, and equations (41) and (42) ask for a tensile strength of 0.22 t F_tu or 0.15 t F_tu per unit run, a criterion the report labels tentative; the tool prints it, with a margin when a rivet tensile allowable is entered. The weight per unit length of beam, web plus stiffeners plus flanges, is given so that a thicker web with wider spacing can be compared with a thinner web with more stiffeners: for a given depth and load the lightest shear-resistant web is found by trading t against d along the buckling curve of Figure 1.

Charts

Figure 1 draws the critical shear stress of the web against the spacing ratio d/h with everything else held (the clear-spacing gap and the restraint coefficients kept): the TN 2661 curve with and without plasticity, the TN 3781 simply-supported and clamped bounds, the limit and ultimate shear stresses as horizontal lines and the design point; where the curve crosses the limit line is the widest spacing that keeps the web shear-resistant. Figure 2 draws the four stiffener requirements (and Bleich's fit) against d/h on a logarithmic scale with the entered stiffener, which shows at a glance how much margin each criterion leaves and how fast the requirement grows as the stiffeners are moved closer together, as (h/d)³ in Moore's formula and about (h/d)² in the theory.

Units and inputs

Lengths, areas, inertias, stresses, forces and the density go through the site's unit dictionary (inches or millimetres, ksi or MPa, pounds or newtons); the analysis itself is unit-consistent and its only unit-bound constant is Moore's Poisson's ratio of 1/3, so the SI page gives the same ratios and margins as the USCS page. Optional inputs left blank take their stated defaults or skip their check: the clear depth and spacing default to the full ones, the stiffener column length to the clear depth, the beam inertia and flange static moment to the lumped-flange values, a blank Ramberg–Osgood exponent leaves the buckling stress elastic, a blank density skips the weight, blank rivet data skip the rivet margins. The loads may be entered at limit or at ultimate with the ultimate factor; the shear-resistant check is always at limit and the strength checks at ultimate.

Limits

The restraint coefficients of TN 2661 come from NACA beam tests on 24S-T and 75S-T aluminium webs with angle stiffeners and flanges, and the report calls them empirical; a stiffener with a wide attached flange, a closed section or a stiffener of a much stiffer material than the web is outside their range. Moore's formula was fitted to two girders of one alloy at d/h_c from 0.25 to 1 and one or two stiffener thicknesses; Stein and Fralich treat an infinitely long plate with simply supported flanges and torsionally weak stiffeners at aspect ratios 1 to 5. The buckling stress is that of a flat, unloaded-in-plane panel: a web with in-plane bending stress, cut-outs, joggles or a concentrated load introduced in the bay needs the combined-load interaction of TN 3781 or a local analysis. The plasticity factor E_sec/E is Gerard's recommendation for aluminium plates in shear, with the equivalent stress √3 τ; the alloy-specific curve of TN 2661 Figure 12(c) agrees with it within about 10 percent for 24S-T3. Beyond the buckling load the web's own strength is governed by the diagonal-tension allowables, which this tool does not evaluate.

Reading the results

The governing check is the smallest margin in the summary table. A positive buckling margin at limit load with a negative one at ultimate (k > 0) is the normal state of a web designed to be shear-resistant at limit only: it is acceptable if the diagonal-tension analysis of the companion tool clears the web, stiffeners and rivets at ultimate. A negative buckling margin at limit means the web is not shear-resistant; thicken it, close the stiffeners or design it as a diagonal-tension web. A stiffener below the Stein–Fralich requirement does not form a nodal line and the panel size for the buckling stress is in doubt; below Bruhn's C10.8 the whole web can buckle as one orthotropic plate at the limit shear.