Residual Strength and R-Curve

Residual Strength Method

Residual Strength: Toughness, Net Section and the Feddersen Diagram, the R-Curve, Stiffened Panels and Leak Before Break

What the tool does

The residual strength of a cracked structure is the gross stress it can still carry. A damage-tolerant structure is sized so that, with the largest crack that may be present between inspections, the residual strength stays above the limit load. The tool draws the residual strength against crack size for a through crack in sheet or plate, finds the critical crack size at the required stress, and adds three analyses that the single-number toughness does not cover: the R-curve of thin sheet, the stiffened panel with crack arrest, and leak before break of a pressurized shell.

Conventions

• σ is the remote stress on the gross (uncracked) section, tension positive. a is the half-length of a center crack (tip to tip 2a), the length of an edge crack, and the length measured from the hole edge of each of two cracks at a hole. W is the full width, t the thickness.
• K = β σ √(π a). The geometry factors β are those of the Crack Growth tool: 1 for the infinite plate, √sec(π a / W) for the center crack of finite width (the form MIL-HDBK-5J uses to reduce its panel tests), the Brown–Srawley polynomial for the single edge crack, the Tada form for the double edge crack, Newman's tabulated values for two cracks at a hole, or an entered polynomial in a / W.
• Stress intensities are plain numbers in ksi√in or MPa√m, chosen by a select that does not follow the unit switch (1 ksi√in = 1.0988 MPa√m). Stresses and lengths follow the unit system.
• Load level: the damaged structure is required to carry limit load (14 CFR 25.571(b) lists the limit conditions and says they are to be considered as static ultimate loads, that is, with no further factor of safety). Every residual-strength margin is therefore written as failure stress ÷ limit stress − 1 and its box says LIM. An entered ultimate stress is divided by FS to recover the limit stress. The uncracked section is checked against F_tu at ultimate load (box ULT) and the net section against F_ty at limit load × FS_yld (box LIM).

Fracture toughness and thickness

Fracture is taken to occur when K reaches a critical value. In thick sections under plane strain this is the material property K_Ic. In thinner sections the plastic zone at the crack tip is no longer small against the thickness, the constraint is lost and the critical value K_c rises to a plane-stress maximum that can be two or three times K_Ic. The NASGRO material files describe the transition with

K_c = K_Ic [1 + B_k exp(−(A_k t / t₀)²)],   t₀ = 2.5 (K_Ic / F_y)²,

where t₀ is the thickness that satisfies the plane-strain size requirement and F_y the typical yield strength of the fit. The tool reproduces to the printed digit the fourteen K_c values that DOT/FAA/AR-05/15 prints with its K_Ic, yield strength, A_k, B_k and thickness for aluminum alloys; the default is its 2024-T3 sheet set (30 ksi√in, 53 ksi, 1.0, 1.5), which gives 74.3 ksi√in at 0.10 in. A K_c measured at the thickness can be entered instead. MIL-HDBK-5J reports such values as the apparent toughness K_app = σ_c √(π a₀ sec(π a₀ / W)), computed from the initial crack length and the maximum stress of center-cracked panels, and keeps only tests whose net-section stress at failure was below 0.8 F_ty.

Irwin's estimate of the plastic zone is r_y = (1 / (x π)) (K / F_ty)² with x = 2 in plane stress and x = 6 in plane strain. A section is in plane strain when thickness, crack size and ligament all exceed 2.5 (K_Ic / F_ty)², about fifty times the plane-strain plastic zone radius. Optionally the K_c curve is evaluated at the effective crack size a + r_y; at fracture K = K_c, so r_y is a constant and σ = K_c / (β(a + r_y) √(π (a + r_y))). The correction belongs with toughness values that were reduced from tests in the same way. Apparent values are computed with the physical crack size, so the default is no correction.

The residual strength diagram

Three limits bound the strength of a cracked panel.

• Fracture: σ_K = K_c / (β √(π a)). It grows without bound as a → 0.
• Net section: the ligament cannot carry more than a stress F on its own area, so σ ≤ F × (net area / gross area). F is the yield strength (the conservative, usual choice) or a flow stress between yield and ultimate.
• Uncracked strength F_tu at a = 0, checked separately at ultimate load.

Without further refinement the residual strength is the lower of the first two. For the center-cracked panel Feddersen proposed a construction that fits panel tests of many widths: draw the idealized curve σ = K_c / √(π a), and join it to the net-section strength F at a = 0 and to zero strength at 2a = W by straight lines tangent to the curve. Because the tangent to K_c / √(π a) at a point (a_t, σ_t) has the slope −σ_t / (2 a_t), it meets the stress axis at 1.5 σ_t and the crack-length axis at 3 a_t. The two tangent points follow in closed form:

left: σ₁ = 2F / 3, a₁ = (9 / (4π)) (K_c / F)²;   right: 2a₂ = W / 3, σ₂ = K_c / √(π W / 6).

Between the tangent points the failure is controlled by K_c; outside them it is a net-section failure and the strength lies on the tangents, below the K_c curve. The two points coincide when W = (27 / (2π)) (K_c / F)². A narrower panel fails by net-section yielding at every crack length, the envelope degenerates to the straight line F (1 − 2a / W), and no fracture test on such a panel measures K_c. This is why tough thin sheet needs wide test panels: the default 2024-T3 sheet needs more than 10.7 in. The rule of sizing the initial crack of a center-cracked test panel to one third of the width comes from the same construction.

The tool takes the residual strength as the lowest of the K_c stress with the finite-width factor, the net-section stress and the Feddersen envelope, tabulates the three against crack size, and names the governing regime. The critical crack size at limit load is the root of residual strength = limit stress.

A note on the sources. DOT/FAA/CT-93/69 describes the construction in words that agree with the closed form above (tangent from the flow stress touching the curve at two thirds of it; tangent through the end of the width), but its Figure 2-37 labels the right tangent point at a half crack length of W/3 and the left one at 28 ksi for a flow stress of 48 ksi, where 2/3 × 48 = 32. The tool follows the closed form, which is also checked against a numerical tangent construction.

The R-curve

In thin sheet a crack does not run when K first reaches a critical value. It tears slowly under rising load, and the stress intensity needed to keep it moving rises with the amount of tearing. The crack-growth-resistance curve K_R(Δa), measured on wide panels (ASTM E561), is to a good approximation a property of the material and thickness and independent of the initial crack length. During stable tearing the applied and the resisting stress intensity are equal,

β(a₀ + Δa) σ √(π (a₀ + Δa)) = K_R(Δa),

and the crack becomes unstable when, in addition, the applied K rises with crack length faster than the resistance: ∂K/∂a ≥ dK_R/dΔa. Graphically the K curve for the instability stress is tangent to the R-curve drawn from a₀. The tool solves the equivalent statement: the equilibrium stress σ(Δa) = K_R(Δa) / (β √(π (a₀ + Δa))) has its maximum at the tangent point. It is found by a scan and a golden-section refinement, which also handles R-curves given as tables with kinks or a plateau. For smooth power-law curves the result agrees with a two-equation root solve to eight figures.

The R-curve is a pasted table, interpolated by a monotone cubic and held constant beyond its last point, or the power law K_R = K₀ + c Δa^p fitted to the table by least squares. Three results follow: the stress at which tearing starts, K_R(0) / (β √(π a₀)); the instability stress σ_c with the stable extension Δa_c; and the two toughness numbers that a test of this panel would report, K_c = K_R(Δa_c) at the tangent point and the apparent K_app = β(a₀) σ_c √(π a₀). Neither is a constant. The table of K_c against initial crack length and width shows how much they move, and it marks the panels in which the net section yields before the tangent point is reached, where the R-curve stress cannot be developed. An infinite plate with a power-law exponent above one half has no tangent point at all; the page says so when the stress is still rising at the end of the data.

The default points are chart readings of the estimated R-curve for 2024-T3 printed in DOT/FAA/CT-93/69 Volume I (Figure 2-28). With them the tool reproduces that handbook's graphical example for a 20 in panel: K_c 124 against the printed 126 ksi√in and σ_c 24.5 against 25 ksi for 2a₀ = 8 in; K_c 106 against 105 at the 15.2 in crack that is critical at 10 ksi.

Stiffened panels

Stringers riveted across the path of a skin crack carry part of the load that the cracked skin sheds. Two factors describe the effect, both functions of the half crack length: the reduction of the skin stress intensity, β = K / (σ √(π a)), which dips sharply as the crack tip passes a stringer, and the stringer load-concentration factor L, the ratio of the largest load in the most highly loaded stringer to its remote load. The panel strength diagram plots, against a, the skin fracture stress K_c / (β √(π a)) and the stringer failure stress F_st / L.

• At a given stress the first crossing of the skin curve is the critical crack length. A crack that starts to run there is arrested where the skin curve rises again through that stress, provided the stringer curve has stayed above it.
• The arrest capability from an initial crack a₀ is the largest stress for which that holds: the maximum over a ≥ a₀ of the lower of the skin curve and the running minimum of the stringer curve. It is skin-critical when the peak of the skin curve past the stringer sets it and stringer-critical when the stringer fails first.

β and L are either entered as a table against a / b (b the stringer spacing) or computed.

Entered table. Two sets of chart readings can be loaded and edited, both for a stringer stiffness ratio μ = 0.30 and a rivet pitch of b / 6. The first is read from Poe's design graphs in NASA TR R-358, Figures 14(c) and 15(c), for a crack midway between two riveted stringers (about ±0.02 in β, ±0.03 in L). The second is read from his Figures 2(b) and 3(b) in NASA TM X-71947 for a crack centered on a broken stringer, L being that of the two stringers next to the broken one (about ±0.03 in β and ±0.1 in L; ±0.1 in β where the crack is shorter than 0.2 b or its tip is within 0.1 b of the first intact stringer, where the printed curves merge and fall steeply). Both reports present their results only as graphs. For bonded stringers, flexible fasteners or another configuration, paste the factors of an analysis of that panel.

Computed. The tool solves Poe's problem for the entered μ = A_st E_st / (A_st E_st + b t E), rivet pitch p and stringer spacing b: an infinite sheet with a crack along a rivet row, equal stringers b apart, sheet and stringers strained equally far from the crack. The unknowns are the rivet forces Q_j. At every rivet the displacement of the cracked sheet, under the remote stress and all the rivet forces, is set equal to the displacement of the stringer under the remote stress and its own rivet forces, which gives a set of linear equations. Then

• β = 1 − Σ Q_j K̄_j / (σ √(π a)), where K̄_j is the stress intensity of a unit set of four symmetric forces at the rivet (equation 5 of TR R-358);
• L = 1 + Σ Q_j / (remote stringer load), summed over the rivets of one side of the most highly loaded stringer;
• for long cracks L tends to 1 / μ, the whole bay load carried by the stringer.

Three crack positions are offered: centered between two stringers, centered on an intact stringer, and centered on a broken stringer. For the broken stringer (TM X-71947) the stringer force is zero at the break, which fixes the sum of its rivet forces, and the stringer is free to move relative to the sheet there. Near a broken stringer β is above 1, because the stringer unloads into the sheet beside the crack; once the tip is past the first intact stringer β falls to about the intact value, and L of the adjacent stringers tends to 1 / μ + 1 / 2.

As in the reports the sheet is in plane stress with ν = 0.3, each rivet force is spread over a diameter p / 4, and the stringer is a strip 1.25 p wide; the last two have little effect. The displacement of the cracked sheet under a rivet force, a numerical integral in the report, is evaluated here in closed form: the stress that the force puts on the crack line has poles at the rivet position, its expansion in Chebyshev polynomials sums to a logarithm of ζ = (z − √(z² − a²)) / a, and the stress-intensity term that follows is identical to the report's equation 5. Rivets are taken one by one to max(10 p, min(a, b)) from the crack and lumped in growing groups out to 12 max(a, b); stringers are included to the first beyond the crack tip (one more with a broken stringer). Against a model more than three times larger the factors change by less than 0.5 % (1.5 % with a broken stringer). The computed curves lie on the printed ones: within 0.03 in β and 0.04 in L of the readings of TR R-358 Figures 14(c) and 15(c), and within 0.04 in β (0.10 at a / b = 0.1, where the printed curve is nearly vertical) and 0.13 in L of the readings of TM X-71947 Figures 2(b) and 3(b), the computed L being the higher. The page also prints the largest rivet force.

The rivets are rigid in this analysis and the stringers carry axial load only. Flexible fasteners, yielding of rivets or stringers and stringer bending raise β and change L; the diagram uses a constant K_c for the skin and does not check net-section yield of the skin or fastener shear.

Leak before break

A surface flaw in the wall of a pressure shell grows through the thickness before it grows long. If the through crack it leaves is stable at the operating stress, the shell leaks and the damage is found; if not, it bursts. The check takes an axial through crack of length 2a = n t in a cylinder of radius R under the hoop stress σ_h = p R / t. n = 2 is the classical choice (a semicircular flaw at break-through); requirements for pressure vessels use longer cracks, so n is an input. Pressure on the unsupported crack edges bulges them outward, which raises the stress intensity above the flat-sheet value: K = M σ_h √(π a). Four bulging factors are printed.

• Folias: M = √(1 + 1.61 a² / (R t)). His shallow-shell solution gives (σ_shell / σ_plate)² ≈ 1 − 0.49 λ² with λ⁴ = 12 (1 − ν²) a⁴ / (R t)²; 0.49 √(12 (1 − ν²)) is 1.60 for ν = 1/3 and 1.62 for ν = 0.3, the commonly quoted 1.61. Valid for short cracks.
• Newman: M = √(1 + 0.52 x + 1.29 x² − 0.074 x³), x = a / √(R t) ≤ 10, a fit to the numerical shell solutions of Erdogan and Kibler.
• Peters and Kuhn: M = 1 + 4.6 (2a) / R = 1 + 9.2 a / R, from burst tests of 2024-T3 and 7075-T6 cylinders.
• Anderson and Sullivan: M = 1 + C a / R with a bulge coefficient C fitted to burst tests of the material at temperature (9.5 for their room-temperature aluminum tanks).

The page also gives the critical through-crack length at the hoop stress, where M σ_h √(π a) = K_c, and the margin K_c / K − 1. Linear shell theory overestimates the bulging of long cracks at high pressure, where membrane tension stiffens the bulge, so the factors are conservative there. For a stiffened fuselage the frames restrain the bulge and other factors apply.

What is not covered

• Part-through (surface and corner) cracks, cracks under bending, mixed-mode loading and compression.
• J-integral resistance curves and other methods for fracture after gross yielding. The R-curve analysis is linear-elastic and applies while the net section is below yield.
• Buckling of the sheet along the crack edges, which lowers the strength of thin unrestrained panels (MIL-HDBK-5J distinguishes tests with and without buckling restraint).
• Stiffened panels with flexible or yielding fasteners, bonded or integral stringers, more than one broken stringer, a crack off the two symmetric positions, a / b above 3; multiple-site damage and the link-up of adjacent cracks.
• Crack growth to the critical size: use the Crack Growth tool with the critical crack size found here.