Industry: Engineering Product: ModelRisk Application: Material Fatigue Analysis
A materials group qualifying a 7075-T6 aluminium for a fatigue-critical part ran a 36-coupon test programme across six stress levels and fitted a Basquin S-N curve through the data. The least-squares fit gave a clean mean curve, and read off at 1e7 cycles it returned an endurance limit comfortably above the 150 MPa service design stress. On the mean fit the material looked fine. But a single best-fit line through scattered coupon data is one draw, not a guarantee — and the question that actually governs the design is the probability that the true allowable falls below the design stress.
Rebuilt in ModelRisk, the characterization was treated as an uncertain fit rather than a fixed curve. Bootstrapping the coupons and propagating the fit uncertainty over 40,000 refits, the fitted endurance limit had a median of 163 MPa but a 5th percentile of 150.6 MPa — landing the B-basis allowable barely 1 MPa above the design stress, with a 4.2% probability that the endurance limit is below it. The mean curve's reassurance was an artifact of ignoring the scatter in its own coefficients.
A fitted S-N curve has three sources of uncertainty that a single mean line discards: the slope and intercept are estimated from a finite sample and have standard errors; the coupons scatter about the line by a residual sigma on log-life; and different material heats carry systematic biases. Designing to the mean curve quietly assumes all three are zero. The aerospace and pressure-equipment codes correct for this with a mean-minus-k-sigma knock-down, but the right value of that knock-down — and the residual probability of being unsafe — can only come from propagating the fit uncertainty. ModelRisk does exactly that.
The model produces the allowable, it does not consume a service spectrum — that integration is a separate exercise. Its inputs are:
The lead chart is the characterization itself: the 36 coupon points, the mean fitted S-N curve (log10 N = 17.4 - 4.7 log10 S), and the design-allowable curve drawn a mean-minus-3-sigma knock-down below it. At the 200 MPa finite-life reference stress that knock-down corresponds to a 5x reduction in allowable life — from a mean-curve median of 3.8 million cycles to an allowable median of 0.77 million cycles. The 150 MPa service design stress line shows how close the allowable band runs to the operating point at high cycles.
The headline distribution is the endurance limit — the stress at 1e7 cycles — across the 40,000 refits. It has a mean of 163 MPa and a median of 163 MPa, with a P5 of 150.6 MPa and a P95 of 174 MPa. The B-basis allowable (the P5 of the fit) therefore sits 12 MPa below the mean, and the shaded region left of the 150 MPa design stress captures the 4.2% probability that the true endurance limit is below the stress the part will actually see. A 4% chance of an under-strength material is precisely the kind of number a mean-curve check cannot surface.
Designers must choose how many sigma of margin to carry, and this chart prices that choice. At the 200 MPa reference, the median allowable life falls from 3.8 million cycles at k = 0 (the mean curve) to 2.2 million at k = 1, 1.3 million at k = 2 and 0.77 million at k = 3 — a 5x knock-down at the standard B-basis level, with each additional sigma of reliability margin costing roughly half a decade of life. This makes the safety-versus-life trade-off explicit instead of hidden inside a code factor.
A tornado on the P5 endurance-limit allowable, baseline 145 MPa, ranks where the uncertainty comes from. S-N slope uncertainty dominates by a wide margin (an 84.7 MPa spread), confirming that the few coupons at the low-stress, long-life end of the curve carry enormous leverage over the extrapolated allowable. Per-heat material bias is the next contributor (13.9 MPa spread), then coupon life scatter (9.0 MPa) and coupon count (5.2 MPa). The implication for the test programme is sharp: adding coupons at the low-stress end, where the slope is anchored, buys far more allowable confidence than simply running more coupons anywhere.
Designing to the mean S-N curve would have qualified the material outright. The probabilistic characterization showed the B-basis allowable lands within 1 MPa of the design stress with a 4.2% chance of being below it — a margin too thin to accept. The group used the result two ways: it set the design allowable on the P5 fit rather than the mean, and it redirected the next test batch to additional low-stress coupons, the inputs the tornado identified as controlling the slope and therefore the allowable.