Composite Hole and Joint Strength

Composite Hole and Joint Method

Notched Laminates and Bolted Composite Joints: Stress Concentration, Characteristic Distances, Bearing and Bypass

Why a laminate needs its own notch analysis

A ductile metal yields at a hole, the peak stress stops rising, and at failure the net section is close to uniformly stressed. A fibre laminate has no yield point. If it were perfectly brittle it would fail when the stress at the edge of the hole reached the unnotched strength, that is at 1/K_T of that strength. Tests fall between the two limits: matrix cracks, splits along the fibres and small delaminations form in the highly stressed zone and blunt the notch, and they do so over a zone whose size belongs to the material, not to the hole. A small hole is therefore much less damaging than a large one with the same stress concentration factor (MIL-HDBK-17-3F sections 5.4.4 and 6.3.2.2). The methods on this page describe that with one fitted length (Whitney and Nuismer) or one fitted factor (Hart-Smith), and they are only as good as the tests the parameter was fitted to.

Axes and the laminate

The load acts along x, the 0° direction of the stacking code; the net section through the notch is the y axis. The laminate is treated as a homogeneous orthotropic plate with the in-plane constants E_x, E_y, G_xy and ν_xy. With a stacking code they come from classical laminate theory, E_x = 1/(h a₁₁), E_y = 1/(h a₂₂), G_xy = 1/(h a₆₆), ν_xy = −a₁₂/a₁₁, with a the in-plane block of the inverse of the laminate stiffness matrix. That treatment needs a balanced, symmetric laminate; the page says so when the code is not. Whitney and Nuismer drew the load along y and the net section along x, so their σ_y(x, 0) is this page's σ_x(0, y).

The unnotched strength σ₀ should be a measured laminate strength. When it is left blank the page uses the first-ply-failure stress of the stacking code by the maximum-strain criterion: the remote stress at which the first ply reaches one of its allowable strains under N_x alone. In tension that is usually transverse cracking of the 90° plies, well below fibre failure, so the default is conservative.

Stress concentration of a round hole

Lekhnitskii solved the hole in an anisotropic plate with two complex potentials Φ₁(z₁), Φ₂(z₂), z_k = x + μ_ky, where μ₁, μ₂ are the roots with positive imaginary part of μ⁴/E_x + (1/G_xy − 2ν_xy/E_x) μ² + 1/E_y = 0. The stresses are σ_x = σ + 2 Re[μ₁² Φ₁′ + μ₂² Φ₂′], σ_y = 2 Re[Φ₁′ + Φ₂′], τ_xy = −2 Re[μ₁ Φ₁′ + μ₂ Φ₂′]. Mapping each z_k plane so that the hole edge becomes the unit circle, z_k = R[(1 − iμ_k)ζ_k + (1 + iμ_k)/ζ_k]/2, the traction-free edge under a remote stress σ along x is satisfied by Φ_k = A_k/ζ_k with A₁ = −iσR/(2(μ₁ − μ₂)) and A₂ = −A₁. At the edge of the hole on the net section this gives

K_T = 1 + √(2(√(E_x/E_y) − ν_xy) + E_x/G_xy).

K_T is 3 for an isotropic or quasi-isotropic plate, above 3 when the 0° plies dominate (about 5 for a carbon/epoxy cross-ply, above 6 for a unidirectional laminate) and below 3 for laminates rich in ±45° plies. The page evaluates the exact stress across the net section from the potentials and plots it beside the approximation it uses in the criteria, the polynomial of Konish and Whitney:

σ_x(0, y)/σ = 1 + ½ρ² + (3/2)ρ⁴ − ½(K_T − 3)(5ρ⁶ − 7ρ⁸),   ρ = R/y.

The first three terms are Kirsch's isotropic solution; the last term brings the edge value to K_T without changing the load carried. For twenty random laminates of 0°, ±45° and 90° plies with K_T between 2.7 and 4.0 the polynomial stays within 4 % of the exact stress, and the strength ratios of the two criteria computed from it stay within 2 % of those computed from the exact field. The page prints the difference for the laminate entered and the criteria on the exact field in the results table.

Finite width

The stress field above is for an infinite plate. For a hole in a strip of width W the infinite-plate strength is divided by Y = K_T(W)/K_T(∞) = (2 + (1 − D/W)³)/(3(1 − D/W)), the isotropic ratio (Whitney, NASA CP-2278 p. 248). It is 1.03 for the usual coupon with W/D = 6 and grows quickly below W/D = 3. For a centre crack the factor is √(sec(πa/W)) (MIL-HDBK-17-3F eq. 7.8.4.1.1(b)); the polynomial Whitney prints for the same correction, 1 + 0.128(2a/W) − 0.288(2a/W)² + 1.52(2a/W)³, differs from it by about 1 % or less up to 2a/W = 0.7. Both are isotropic corrections applied to an orthotropic plate, the practice of both sources.

The Whitney-Nuismer criteria

Point stress: failure when the stress at a fixed distance d₀ ahead of the notch equals σ₀. Average stress: failure when the mean stress over a fixed distance a₀ ahead of the notch equals σ₀. With the polynomial, for a hole of radius R (ξ_p = R/(R + d₀), ξ_a = R/(R + a₀)):

σ_N/σ₀ = 2/(2 + ξ_p² + 3ξ_p⁴ − (K_T − 3)(5ξ_p⁶ − 7ξ_p⁸)),   σ_N/σ₀ = 2(1 − ξ_a)/(2 − ξ_a² − ξ_a⁴ + (K_T − 3)(ξ_a⁶ − ξ_a⁸)).

Both tend to 1 as the hole vanishes and to 1/K_T as it grows: the hole-size effect. For a centre crack of half length a the stress on the crack line is σ x/√(x² − a²) and the criteria give √(1 − ξ_p²) and √((1 − ξ_a)/(1 + ξ_a)) with ξ = a/(a + d₀) and a/(a + a₀). For long cracks they become fracture mechanics with an apparent toughness K_Q = σ₀√(2πd₀) or σ₀√(πa₀/2), equal when a₀ = 4d₀.

The distances are found by fitting notched tests. Whitney (NASA CP-2278) shows d₀ = 0.0437 in and a₀ = 0.163 in for (0/90)_4s T300/5208 with holes, and d₀ = 0.0358 in, a₀ = 0.143 in for a (0/±45/90)_2s glass/epoxy with cracks; the defaults 0.04 in and 0.15 in are the round values usually quoted from Whitney and Nuismer's work. They change with the lay-up, the material and the sign of the load: compression needs its own values (MIL-HDBK-17-3F section 6.3.2.2). The same data scatter that the fit averages over is why the handbook asks for B-basis values of the notched strength or of d₀ itself.

Mar-Lin

For large cracks and slits Mar and Lin wrote the infinite-plate strength as σ_N = H_c(2a)⁻ⁿ, with H_c a composite fracture toughness and n the order of the singularity. n = 0.5 is classical fracture mechanics; the handbook quotes 0.25 to 0.35 from the singularity of a matrix crack ending at a fibre and treats H_c and n in practice as two fitted numbers (section 7.8.4.1.1). A lower n makes large notches less damaging; the handbook warns that choosing it from small-notch data alone can over-predict the strength of a large notch. The relation has no upper limit for short cracks, so the page caps it at σ₀. H_c is entered as a plain number in ksi·inⁿ or MPa·mmⁿ.

Filled holes and compression

A hole filled with a close-fit but unloaded fastener behaved as an open hole in tension in Hart-Smith's tests. In compression the fastener can carry load across the hole, so the filled hole is stronger than the open one; Crews and Naik measured that as a higher compressive damage-onset stress and explained it by the contact of the bolt with both sides of the hole. The page analyses the hole as open, which is the conservative reading in compression, and uses the compressive unnotched strength when the remote load is negative.

The fastener hole: Hart-Smith's method

Hart-Smith (NASA CR-144899) works with stress concentration factors on the net-section stress P/(t(w − d)) of a strip of width w (the fastener pitch) with a hole of diameter d at a distance e from the end. For an elastic isotropic material he fitted theory and photoelastic data with

loaded hole: k_te = 2 + (w/d − 1) − 1.5 Θ (w/d − 1)/(w/d + 1), Θ = 1.5 − 0.5/(e/w) for e/w ≤ 1, Θ = 1 for e/w ≥ 1 (eqs 1 and 2);

open hole: k_te = 2 + (1 − d/w)³ (eq. 8).

Tests of graphite/epoxy joints then showed that the factor reached at failure, k_tc = F_tu(w − d)t/P, plots as a straight line through (1, 1) against k_te: k_tc = 1 + C(k_te − 1) (eqs 16 and 23). C = 0 would be a material with complete relief and C = 1 one with none. His fits: 0.27 for quasi-isotropic and 0.41 for orthotropic T300/5208, 0.25 and 0.46 for Morganite II/1004, about 0.5 to 0.6 for graphite-glass hybrids. C plays the part that d₀ plays for the open hole and must come from tests of the lay-up.

A single fastener then fails the laminate at the lowest of the net-tension load F_tu(w − d)t/k_tc (eq. 14), the bearing load F_brdt (eq. 19) and the shear-out load 2F_sot(e − d/2) (eq. 17). The bearing strength is a test value for the thickness, fastener and clamp-up; the shear-out allowable is a test value too and is well below the in-plane shear strength of the laminate. Wide pitches fail in bearing and close pitches in net tension, and the strongest single row lies near the transition, at an efficiency of only about 40 % of the unnotched laminate for quasi-isotropic graphite/epoxy.

Bearing-bypass interaction

In a multi-row joint a fastener reacts a bearing load P_br while the bypass load P_by runs past its hole. Both load the same net section, and Hart-Smith found the peaks to add (eqs 27, 28, 32, 33):

k_bσ_br + k_tσ_by ≤ F_tu,   σ_br ≤ F_br,   k_b = k_tc(loaded)/(w/d − 1),   k_t = k_tc(open),

with σ_br = P_br/(dt) and σ_by = P_by/(t(w − d)). On a chart of bearing stress against bypass stress this is a sloping net-tension line cut off by a level bearing line. Crews and Naik's tests on a quasi-isotropic T300/5208 laminate (NASA TM-87705) confirmed both for damage onset in tension. In compression the mechanism differs: a close-fit fastener carries the laminate compression across the hole as extra bearing, so σ_br + σ_c ≤ F_br with σ_c the gross laminate stress (eq. 36), and a fastener too loose to do so leaves an open hole in compression, k_t|σ_by| ≤ F_cu. The page applies both. Crews and Naik found the same trend, a bearing strength that falls as compressive bypass is added.

Each fastener of the table is a point on that chart. The load factor λ is the number by which its bearing and bypass loads can be multiplied together before the point reaches the envelope, and the bearing-bypass margin is λ − 1 for the fastener with the smallest λ. The bearing loads are entered as magnitudes; a positive bypass load means a joint in tension and a negative one a joint in compression. How the load divides between the rows depends on the fastener and member flexibilities and is an input here.

Countersinks and geometry limits

A countersink replaces part of the cylindrical bearing surface by a cone. The page counts a share K_cs of the countersunk depth as bearing thickness, t_b = t − (1 − K_cs)t_cs, an entered convention (0.5 by default, the value of the site's Fastened Joint Strength tool) that countersunk bearing tests should replace, and compares the countersink depth with an entered share of the thickness; a countersink through the whole thickness leaves a knife edge. The end distance and pitch are compared with entered minima: the handbook's repair guidance is 2.5 diameters to an edge and 4 diameters between fasteners, Hart-Smith needed e/d of 4 to develop the full bearing strength, and the handbook calls w/d of 6 usual.

Load level

All checks are strength checks against ultimate allowables. The remote load and the fastener loads are used as entered when they are ultimate loads and are multiplied by the factor of safety when they are limit loads; every margin is written at ultimate load. There is no yield check.

Limits

One hole or crack at mid-width under a load along one principal axis of a balanced symmetric laminate. Not covered: biaxial and shear loading, off-axis laminates, holes near an edge or near each other, stacking-sequence and thickness effects other than through the fitted parameters, delamination, fastener failure, pull-through, single-shear bending, load sharing between rows, fatigue and environment.