Industry: Engineering
Product: ModelRisk
Application: Stress Analysis

A Safety Factor of 1.74 Looked Comfortable. The Hot Spot Still Yields 0.62% of the Time — Six Times the Reliability Target.

The deterministic check on a forged high-strength-steel lifting lug looked reassuring. Peak stress at the critical fillet is the nominal section stress amplified by the elastic stress-concentration factor, sigma_peak = Kt x (P / A) = 2.45 x (0.42 MN / 26 cm^2) = 396 MPa, against a 690 MPa yield. That is a safety factor of 1.74 — well inside the design rule.

But Kt, the applied peak load, the as-built net area, and the yield strength are all scattered, and at a notch the scatter does not cancel — it stacks. Pushing the four uncertain inputs through the same hot-spot equation with Monte Carlo turns the single 396 MPa number into a distribution with a mean of 403 MPa, a P95 of 529 MPa, and a P99 of 610 MPa. The probability that peak stress exceeds yield at the fillet — local yielding — is 0.62% at the nominal load. For a fracture-critical lifting fitting carrying a 0.1% reliability target, that is six times too high. The point estimate never showed it.

A heavy-lift engineering group adopted ModelRisk to recast hot-spot checks as reliability calculations rather than single safety-factor sums.

Why a point estimate fails at a notch

A stress-concentration factor is a multiplier, and multipliers punish you in the tail. The deterministic peak of 396 MPa sits near the median of the simulated peak-stress distribution — so half the time the real hot spot is worse, and the worst few percent are dramatically worse (P99 = 610 MPa, 54% above the design number). A safety factor computed at the nominal point tells you nothing about how fat that upper tail is, and it is the upper tail that yields.

It gets worse when you stop treating the inputs as independent. A soft forging is not just weaker — its rougher fillet is also more notch-sensitive, so the same batch that lowers yield strength simultaneously raises Kt. We modelled that with a single per-component build factor that scales yield down and Kt up together. With independent inputs the simulated yield probability is 0.28%; with the shared build factor it is 0.38% on the headline case — a 1.4x increase that a naive independent roll-up quietly hides.

The headline: two distributions that overlap

The clearest picture is the peak-stress distribution and the yield-strength distribution on the same axis. Where they overlap is exactly the yield probability.

• Peak hot-spot stress: mean 403 MPa, P99 610 MPa.
• Yield strength: mean 690 MPa.
• Deterministic peak: 396 MPa (the dotted line, near the body of the peak-stress distribution, nowhere near its tail).
• P(local yield) = 0.4% on the headline draw, reliability index beta = 3.19.

The deterministic check lands in the comfortable middle of the blue distribution. The yield events live in the thin overlap on the right, which no single-point calculation can see.

How load level moves the yield probability

P(local yield) is not a fixed property of the part — it climbs steeply with the operational load the feature actually sees. Sweeping the mean peak load and recomputing the reliability at each level produces the design curve below.

• At a mean peak load of 0.30 MN, P(local yield) is 0.052%.
• At the nominal 0.42 MN it is 0.62%.
• At 0.60 MN it reaches 16.5%.
• The 0.1% reliability target is only met below a mean peak load of about 0.34 MN — below the nominal design load. The lug as specified does not clear its own reliability target.

The independent-input curve sits consistently below the correlated one: ignoring the shared build factor would make the design look acceptable when it is not.

Where the yield events actually live

A yield event needs two things to go wrong at once: a high-Kt build and a high applied load. Plotting the joint draws of Kt and load shows that the failures concentrate in one corner.

• P(Kt above its P90 AND load above its P90) = 1.03%, essentially the independence value of 1.00% for this pair.
• But 55% of all simulated yield events sit in that high-Kt / high-load corner — the joint upper region owns the failures even though each variable alone is only occasionally extreme.

This is the actionable insight: the part is safe almost everywhere except when a notch-sensitive build meets a peak-load event, so controls aimed at both the worst builds and the worst load cases buy far more reliability than tightening either alone.

Which inputs the answer depends on

Swinging each input across its plausible range and recomputing P(local yield) (base 0.39%) ranks the drivers:

• Stress-concentration factor Kt (+/-12%): 0.10% to 1.56% — a 1.46-point swing, the dominant driver.
• Mean peak load (+/-0.05 MN): 0.15% to 1.10% — 0.95 points.
• Yield strength (+/-8%): 0.14% to 1.11% — 0.97 points.
• Kt scatter sigma_log (0.06 to 0.14): 0.22% to 0.81% — 0.59 points.
• Net section area (+/-4%): 0.24% to 0.76% — 0.52 points.

The geometry of the notch — Kt — outranks both the load and the material. That redirected the team away from spending on a higher-grade steel and toward controlling the as-built fillet radius and surface finish, the cheapest lever on the most sensitive input.

What the model changed

The lug was re-detailed with a larger blended fillet radius, cutting the nominal Kt from 2.45 to 2.10 and a tightened finish specification narrowing its scatter. Re-running the model dropped P(local yield) at the nominal load below the 0.1% target without changing the steel grade or the section size. The design review accepted the part on a documented reliability index rather than a nominal safety factor, and the same hot-spot reliability template now front-ends the group's fracture-critical fitting checks.

ModelRisk Functionality Used

• Monte Carlo simulation of the hot-spot limit state sigma_yield - Kt(P/A), turning a single safety factor into a full reliability calculation (P(local yield) and reliability index beta).
• LogNormal, Gumbel and Normal inputs for Kt scatter, extreme operational load, and yield/area respectively — each chosen to keep stress and strength physically positive.
• A shared per-component build factor linking yield strength and notch sensitivity, so a soft, rough forging is correctly modelled as both weaker and more notch-sensitive — raising P(yield) 1.4x over a naive independent roll-up.
• A load-level sweep showing the design clears its 0.1% reliability target only below 0.34 MN, against a 0.42 MN nominal load.
• Sensitivity ranking identifying the notch factor Kt — not the steel grade — as the dominant driver, redirecting investment to fillet geometry and finish control.

At a notch, the safety factor is the comforting number and the tail is the true one. ModelRisk shows you the tail before the part does.