Tension Clips and Fittings: Method

Tension Clips and Fittings: Leg Bending, Prying, the End Pad as a Plate and the Fitting Factor

Why a clip is weak in tension

An angle clip joins two parts at right angles: its attached leg is riveted or bolted to a web and works in shear, its outstanding leg is bolted to the mating part. When the joint is pulled apart, the bolt and the attached leg are not in line. The load has to travel across the bend, and it does so by bending both legs. A thin sheet in bending is far weaker than the same sheet in tension, so the tension strength of a clip is a small fraction of the strength of its bolt, and it depends on one dimension above all: the distance from the bolt to the bend. The design rule that follows is to keep the bolt as close to the heel as the bend radius and the washer allow, to use the thickest clip that can be formed, and to carry real tension loads through a machined fitting with walls in line with the load.

Statics of the outstanding leg

Cut the outstanding leg at the tangent point of the bend (the heel section) and treat it as a strip of beam. The attached leg pulls with T at the heel, the bolt holds with B at a distance e_h = g_b − t_c − r_i, and the toe of the leg, a_t beyond the bolt, may bear on the mating part with a prying force Q. Equilibrium of forces and moments gives

B = T + Q,    T e_h = M_h + Q a_t,

where M_h is the bending moment at the heel and Q a_t the moment at the bolt line. There are three unknowns (B, Q, M_h) and two equations: the split of T e_h between the heel and the bolt line depends on how the joint restrains the leg. The tool offers four answers and prints all of them.

Cantilever. The mating part gives no rotational restraint at the bolt: M_h = T e_h, Q = 0, B = T. This is the largest heel moment.

Fixed-guided. The bolted joint keeps the leg flat at the bolt line, as a bolt head and washer clamped against a stiff part do: the leg is a beam with both ends held against rotation and one end displaced, so M_h = Q a_t = T e_h / 2. The bolt-line moment is written as a toe couple, Q = T e_h / (2 a_t), which is charged to the bolt.

Elastic tee-stub. The heel does not rotate (two clips back to back, or one half of a tee), the toe is a contact support and the bolt a spring of stiffness k_b. With the curvature w″ = M / (EI), the slope of the leg at the bolt line from the heel side is (Q a_t e_h − T e_h² / 2) / (EI), and from the toe side −δ / a_t − Q a_t² / (3 EI), where δ = (T + Q) / k_b is the bolt stretch. Equating the two,

Q = T (e_h² / 2 − EI / (a_t k_b)) / (a_t (e_h + a_t / 3) + EI / (a_t k_b)),   not less than zero.

With a rigid bolt Q = 3 T e_h² / (2 a_t (3 e_h + a_t)); with a_t = e_h that is 3T/8. When the leg is stiff and the bolt soft the numerator turns negative: the toe lifts off and there is no prying. This is the rigid-flange limit, B = T.

Heel-toe lever. The heel takes no moment (a thin attached leg or web that rotates freely), and the leg is a lever about its toe: Q = T e_h / a_t, B = T (1 + e_h / a_t). This is the upper bound on the bolt load.

The two limits were measured on the latch fittings of a launch-vehicle nose fairing (NASA TM X-71862). Where the shell behind the fittings was stiff, the bolt load was 1.00 to 1.07 times the load in the skin, against 2.07 from the lever; where the shell was flexible, 1.9 to 2.4 against 2.48. Prying is real, and how much of it there is depends on the stiffness of what the fitting is attached to.

Effective width and the bending sections

The bolt load enters the leg under the washer and spreads on its way to the heel. The strip that works at the heel is taken as w_e = d_w + 2 e_h tan θ, not more than the clip width that belongs to one bolt, with the half angle θ entered (30° is a common conservative value, 45° a common upper one). At the bolt line the same strip loses the hole, w_n = w_e − D_h. The surface stresses are f = 6 M / (w t_c²).

The attached leg carries the load T in tension. Just below the bend, where it leaves the support of the web, it also carries the moment M_a = M_h + T (r_i + t_c / 2), because the line of T in the attached leg is r_i + t_c / 2 further from the bolt than the heel section. This is usually the critical section of a clip. The bend itself is treated as a rigid corner between the two tangent sections; the stress concentration at the inside of a tight radius is not computed.

Plastic bending strength

A rectangular section does not fail when its surface reaches F_tu: the stress redistributes through the thickness. Cozzone's method, as given in NASA TM X-73305 Section B4.5, replaces the true stress distribution at failure by a trapezoid with the stress f_max at the surface and f₀ at the neutral axis, which gives the bending modulus of rupture

F_b = f_max + (k − 1) f₀,    k = 2 Q c / I,

with k = 1.5 for a solid rectangle. The stress f₀ comes from the stress-strain curve of the material (the Cozzone tool of this site computes it); when none is entered the tool uses F_ty, which is a lower bound for a ductile alloy because at rupture all but a thin core of the section is strained beyond yield. Rupture is checked as M.S. = F_b / f − 1 at ultimate load. In the attached leg the tension and bending ratios are added linearly, R = f_t / F_tu + f_b / F_b, which is conservative against the fully plastic interaction of a rectangle, R_t² + R_b = 1.

Yield is checked at limit load times the yield factor against k_y F_ty on the elastic surface stress. With k_y = 1 this is first yield of the surface, which in a section with a shape factor of 1.5 comes long before any permanent set that could be measured; a larger k_y, up to the shape factor, accepts yielding through part of the thickness.

What the beam model leaves out

As a thin clip is pulled, its legs straighten and begin to carry the load as a cable, in tension, and the bend opens. Tests of clips therefore give loads well above the bending strength computed here, and design manuals carry test-based allowable curves for standard clips. Those curves belong to their publishers and are not reproduced; the tool computes from mechanics, states its assumptions, and lets a test or handbook allowable be entered so that its margin is printed beside the computed ones. The computed values are the safe side: they are what the clip can carry without relying on large deformation.

Pull-through, bolt and fasteners

The bolt load B, prying included, shears the leg around the washer: f = B / (π d_w t_c) against F_su. The bolt itself is checked as B against its ultimate tensile strength.

The attached-leg fasteners carry T in shear. In a single clip the moment M_a at the top of the attached leg is reacted by the heel pressing on the web and a pull on the fastener row at h₁ from the heel, the kick load K_f = M_a / (h₁ n_f). Fasteners under tension and shear are checked with R_t² + R_s³ = 1 (NASA TM-106943, eq. 56); the margin is λ − 1, where λ is the factor by which both loads can grow together until (λ R_t)² + (λ R_s)³ = 1. In a back-to-back pair the two heel moments balance through the web, there is no kick load and the fasteners work in double shear.

Limit loads of the tee

If the leg is allowed to form plastic hinges, three mechanisms are possible, and the least of their loads is the limit load of the leg and bolt: hinges at the heel and at the bolt line, T = (M_p + M_pb) / e_h; a hinge at the heel with the bolt failing under prying, T = (M_p + P_tb a_t) / (a_t + e_h), from moments about the toe; and the bolt failing with no prying, T = P_tb. Here M_p = F_b w_e t_c² / 6 and M_pb = F_b w_n t_c² / 6. These are the tee-stub modes of steel connection design (Douty and McGuire; Kulak, Fisher and Struik; the AISC prying action procedure), written as statics. The tool prints them as a comparison; they do not include the attached leg.

Bathtub, channel and angle fittings

A tension fitting puts walls in line with the load. The bolt bears on an end pad; the pad is joined to side walls and to a base, which run along the load and are fastened to the skin or web in shear. The pad is a short, thick plate loaded by the washer and held along the edges where it joins the walls and the base; its fourth edge is free. The tool solves it as a Kirchhoff plate: energy D/2 ∫∫ [w_xx² + w_yy² + 2 ν w_xx w_yy + 2 (1 − ν) w_xy²] dx dy with conforming bicubic (Bogner-Fox-Schmit) elements, the bolt load uniform over the washer circle, the mesh refined around it and graded toward every corner where a built-in edge meets a free one. Moments per unit width, M_x = −D (w_xx + ν w_yy), are proportional to the bolt load and independent of the pad thickness, so each result is a coefficient: σ = 6 M / t_p² = β B / t_p².

Two limits of edge restraint are solved. Built in: walls and a base about as thick as the pad hold its edges against rotation; the largest moments are then at the edges, and the walls and base must carry them as local bending, 6 M / t_w². Simply supported: thin walls rotate with the pad; no moment enters them and the stress under the bolt is at its largest. The real pad lies between the two.

The moment along a built-in edge is not read from the curvature of the elements, which converges slowly there, but from the reactions of the held edge rotations. For a virtual rotation of the edge, v = N(x) φ(y) with N the slope function of the edge nodes, virtual work gives a(w, v) − (p, v) = ± ∫ M_n φ ds, so the edge moment, written in the same cubic functions along the edge, follows from a one-dimensional system. The reactions carry the accuracy of the energy: for the clamped square plate under uniform load they give the mid-side moment 0.05133 q a² to five figures with six elements per side. Where a built-in edge runs into a free edge the moment rises to a peak a few per cent of the edge length before the corner and then reverses steeply (the corner is a singular point of plate theory); the last 5 % of such an edge is left out of the maximum, and the graded mesh resolves the peak. In a mesh study of six pads (three, two and one built-in edge pairs, centred and offset bolts, aspect ratios to 3) the edge maxima are within 0.2 % of a mesh about five times finer, and the moment under the bolt within 0.3 %. The solver is also checked against the series solutions of the Plate Bending tool under the same rectangular patch load, against an independent polynomial Ritz solution, and against the statics of the pad as a whole.

Behind the pad the load is carried by the walls and the base as a section in tension. The bolt line is not at the centroid of that section, so the section also bends: σ = T / A + T e y / I for a channel, and the unsymmetrical-bending form with both second moments and the product of inertia for an angle fitting. The section is taken as free to bend; the skin it is fastened to restrains it, so this is the conservative end. If the structure behind the fitting is flexible, the fitting rocks on its toe and pries on the bolt as the clip does; the heel-toe factor multiplies the bolt load for that.

Bending of the fitting bolt

The pad does not stay square to the bolt: unless the bolt sits on a line of symmetry in both directions, the plate surface under it rotates, θ = |∇w| at the bolt line, taken from the plate solution at ultimate load. The model is an imposed rotation. The head (or nut) is seated flat on the pad and turns with it; the other end bears on the mating part, which either stays square or, for two like fittings bolted pad to pad, rotates the opposite way (relative rotation 2 θ). The bolt is free to slide in its clearance hole, so no shear goes with the rotation and the grip length L_g is in pure bending,

M = E_b I_b θ / L_g,    I_b = π D_b⁴ / 64.

The head cannot pass more moment than the bolt load acting at the rim of the washer, B d_w / 2, at which it begins to tip on the pad; the smaller value is used. This is the case NASA TM-106943 describes for a flange much stiffer than the bolt, which "tends to rotate as a rigid body, forcing the head of the bolt to rotate"; it is an upper bound, because a washer that seats by local yielding, or a bolt that is not snug, passes less. The memorandum combines tension, bending and shear in a bolt as (R_t + R_b)² + R_s³ = 1 (eq. 60); the fitting bolt carries no shear, so the margin is 1 / (R_t + R_b) − 1 with R_t = B / P_tb, the heel-toe factor included in B. The bending allowable is put on the same section as the entered tensile strength, M_ab = k_bb P_tb D_b / 8, which is P_tb as a stress on the shank area times the section modulus π D_b³ / 32; k_bb = 1 is the elastic value, and a modulus of rupture of the bolt material allows up to the shape factor 1.7 of a round bar. Preload is not modelled.

Tension in the attachment rows of a fitting

The bolt line lies e_f above the faying surface (bolt height plus base thickness unless entered), and the attachment reacts the load in the faying surface, so the fitting puts the moment T e_f onto the structure. It presses the heel of the fitting, the edge of its base under the end pad, down and lifts the tail. The rows are entered as stations x_r from the heel with n_r fasteners each, and two elastic patterns are printed. About the heel (used unless the other is chosen): the fitting turns as a rigid body about its heel, the base carries the compression there in bearing and each fastener stretches in proportion to its station, F = T e_f x_r / Σ n_r x_r². It is the default because fasteners in clearance holes do not carry compression, the base does. About the pattern centroid: the fasteners alone carry the moment, F = T e_f (x_r − x̄) / Σ n_r (x_r − x̄)², the choice for a base that cannot bear at its heel (a shim gap, a joggle, a soft skin); it gives the larger tension and puts the rows near the pad into compression. The most loaded fastener is checked with its shear T / Σ n_r by R_t² + R_s³ = 1. The fitting is taken as rigid and the rows as equally stiff; prying of a thin base between the rows is not included.

Material values

The default material values are the MIL-HDBK-5J design values of 2024-T3 bare sheet 0.010 to 0.128 in thick, A basis, longitudinal grain direction (Table 3.2.3.0(b)), read from the handbook database of this site. A material saved from the MIL-HDBK-5J Allowables tool fills E, F_tu, F_ty, F_su and ν from the Material dropdown, at its saved temperature, grain direction and basis.

Factors

The entered load is an ultimate or a limit load. Strength checks (bending rupture, pull-through, bolt, fasteners) are written at ultimate load: a limit load is multiplied by the factor of safety, an ultimate load is used as entered. Yield checks are written at limit load times the yield factor. The fitting factor multiplies both. Under 14 CFR 25.619 the factor of safety of 25.303 (1.5) is multiplied by the highest pertinent special factor, and 25.625 sets a fitting factor of at least 1.15 for each fitting whose strength is not proven by limit and ultimate load tests, applied to the fitting, its means of attachment and the bearing on the joined members; casting factors (25.621) and bearing factors (25.623) take its place when they are larger.

Not covered

Bearing and tear-out at the holes, bolt preload and the bending of the bolt of a clip, the curved-beam stress at the inside of the bend, fatigue (a clip that is repeatedly bent at its heel has a short life), the strength of the web and of the mating part, prying of the base of a fitting between its attachment rows, fillets and tapered walls of machined fittings, and the membrane strength of a clip at large deflection.