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2024-T3 centre-cracked sheet 12 in wide: K_c residual strength, Feddersen tangents, R-curve instability, panel arrest and leak before break

Tool	Residual Strength and R-Curve
Source	U.S. Department of Transportation (1993). Damage Tolerance Assessment Handbook, Volume I: Introduction, Fracture Mechanics, Fatigue Crack Propagation. DOT/FAA/CT-93/69.I, Volpe National Transportation Systems Center, Cambridge, MA. Sections 2.4.1 and 2.4.2 (R-curve example, Figure 2-28; Feddersen diagram); Forman, R. G., et al. (2005). Fatigue Crack Growth Database for Damage Tolerance Analysis. DOT/FAA/AR-05/15 (K_c constants of 2024-T3 sheet); Poe, C. C., Jr. (1971). NASA Technical Report R-358 (Figures 14(c), 15(c)).
Level	Professional

Problem

A 2024-T3 sheet 12 in wide and 0.10 in thick (F_ty 47 ksi, F_tu 64 ksi) carries a limit gross stress of 15 ksi and holds a centre crack of 2a = 3 in. Find the fracture toughness at this thickness, the residual strength with the net-section and Feddersen limits, the critical crack size at limit load, the instability stress from the R-curve, the arrest capability of the same skin with riveted stringers 6 in apart, and whether a 20 in radius shell of this sheet at 150 psi leaks before it breaks.

Inputs to Enter

Centre crack of finite width: a_0 1.5 in, W 12 in, t 0.10 in; gross stress 15 ksi entered as a limit stress, FS 1.5, FS_yld 1.15
K_c from K_Ic 30 ksi√in, fit yield 53 ksi, A_k 1.0, B_k 1.5 (DOT/FAA/AR-05/15, 2024-T3 sheet L-T); F_ty 47 ksi, F_tu 64 ksi (MIL-HDBK-5J Table 3.2.3.0(b), A basis); net-section strength F_ty; Feddersen tangents on; no plastic-zone correction
R-curve points (Δa in, K_R ksi√in): 0 41, 0.25 74, 0.5 92, 1.0 113, 1.5 126, 2.0 131, 2.5 132 (readings of DOT/FAA/CT-93/69 Figure 2-28), interpolated
Stiffened panel: stringer spacing 6 in, stringer strength 78 ksi, 24 rows of a/b, β, L read from NASA TR R-358 Figures 14(c) and 15(c) (μ = 0.30, rivet pitch b/6)
Leak before break: R 20 in, p 150 psi, pressure factor 1.0, crack length 2 t, Folias bulging factor

Open the Residual Strength and R-Curve tool (loaded with this example)

Answers

t_0 = 2.5 (30 / 53)² = 0.8010 in; K_c = 30 (1 + 1.5 exp(−(0.10 / 0.801)²)) = 74.30 ksi√in (printed 74.3 in DOT/FAA/AR-05/15); plane-stress plastic zone 0.398 in
β = √sec(π a / W) = 1.040; K_c stress 74.30 / (1.040 √(π · 1.5)) = 32.90 ksi; net-section yield 35.25 ksi; Feddersen envelope 33.87 ksi; residual strength 32.90 ksi
Feddersen tangent points: σ_1 = 2 F_ty / 3 = 31.33 ksi at a_1 = 1.790 in, σ_2 = 29.64 ksi at a_2 = W/6 = 2 in; least width for a K_c-controlled failure 10.74 in
Critical crack size at 15 ksi: a = 3.966 in (2a = 7.93 in), on the right tangent side of the K_c range
R-curve: tearing starts at 18.15 ksi; instability at 35.93 ksi after 0.950 in of stable extension, K_c = 111.4 and K_app = 81.1 ksi√in; the net section (60.7 ksi) yields first, so the diagram value governs
Stiffened panel: skin 35.32 ksi at the entered crack, arrest capability 40.69 ksi just past the first stringer (skin-critical), stringer 72.06 ksi
Leak before break: hoop stress 30 ksi, M = 1.004, K = 16.88 ksi√in against 74.30; critical through crack 2a = 2.08 in (20.8 t)
M.S.: residual strength +1.19, R-curve +1.40, net-section yield +1.04, panel arrest +1.71, stringer +3.80, leak before break +3.40 (all LIM); uncracked section +1.84 (ULT)

Notes

The tool's default inputs are this panel; press Submit. The case is illustrative, not a printed example. Checked by test_residual_strength.php (740 checks): the fourteen K_c fits of DOT/FAA/AR-05/15 to the printed digit, the R-curve example of DOT/FAA/CT-93/69 Volume I (K_c 126 and 25 ksi at 2a = 8 in; K_c 105 at the 15.2 in crack critical at 10 ksi, both graphical, reproduced within 3 percent from readings of its Figure 2-28), Example 4.4.1 and Example 4.5.1 of the USAF Damage Tolerant Design Handbook, the closed-form Feddersen tangents against a numerical construction, and scipy root solves for the tangency, the critical crack sizes, the panel and the bulging factors. The stringer factors are chart readings (about ±0.02), not tabulated values.