2. Load Limiter Torsion Bar Yield Calibration for a 50th Percentile Male Occupant

Calibration Target and Physical Model

The load limiter in a retractor with a torsion bar works by plastic torsion of a steel bar as webbing is extracted. Once the bar yields, the belt force plateaus at a level set by the bar's torsional yield torque. For a 50th percentile male (Hybrid III 50M) in a frontal sled test, a common calibration target for the shoulder belt force plateau is approximately 3.5 kN ± 0.3 kN during the load-limiting phase, after the pretensioner has completed its stroke. The task here is to select a torsion bar diameter d and a material yield strength S_y that together produce this plateau, then check that manufacturing tolerances do not push the plateau outside the acceptance band.

The governing relation comes from torsion of a solid circular shaft. The torsional yield torque is:

T_y = (π · d³ · τ_y) / 16

where τ_y is the shear yield strength. Using the von Mises yield criterion, τ_y = S_y / √3. The belt force plateau is related to the bar torque through the retractor spool radius r_s and the webbing wrap geometry. For a first-order calibration we use the effective moment arm r_eff (spool radius plus half webbing thickness):

F_plateau = T_y / r_eff = (π · d³ · S_y) / (16 · √3 · r_eff)

This is the design equation. It shows that plateau force scales with the cube of diameter and linearly with yield strength, so diameter is the dominant lever. A 5% diameter change produces roughly a 15.8% force change, while a 5% yield strength change produces only 5%. This asymmetry drives the tolerance strategy: control diameter tightly, and treat material strength as a coarser adjustment.

Worked Calibration Example

Assume a retractor with effective moment arm r_eff = 21.5 mm and a target plateau of F_target = 3.50 kN. Rearranging the design equation for the required product d³ · S_y:

d³ · S_y = (16 · √3 · r_eff · F_target) / π
         = (16 · 1.7321 · 0.0215 m · 3500 N) / 3.1416
         ≈ 663.6 N·m

Now evaluate candidate combinations. A common load-limiter steel is a low-carbon drawing-quality grade with specified yield in the 350–550 MPa range. Two candidate pairs:

Candidate E (d = 11.0 mm, S_y = 500 MPa) lands almost exactly on target. Candidate D is slightly low; a small diameter increase to 10.7 mm at 520 MPa would also work. The choice between them depends on available bar stock and the tolerance stack-up, which we evaluate next.

Tolerance Stack-Up on Plateau Force

Plateau force is a product of three toleranced quantities: diameter, yield strength, and effective moment arm. Using the logarithmic derivative of the design equation, the relative variance is:

(σ_F / F)² = (3 · σ_d / d)² + (σ_Sy / S_y)² + (σ_reff / r_eff)²

For candidate E, assume these 3σ manufacturing tolerances: diameter ±0.05 mm (σ_d = 0.0167 mm on d = 11.0 mm), yield strength ±30 MPa (σ_Sy = 10 MPa on 500 MPa), and moment arm ±0.3 mm (σ_reff = 0.1 mm on 21.5 mm). Computing each relative term:

• Diameter: 3 × 0.0167 / 11.0 = 0.00455
• Yield strength: 10 / 500 = 0.0200
• Moment arm: 0.1 / 21.5 = 0.00465

Combining in quadrature: σ_F/F = √(0.00455² + 0.0200² + 0.00465²) ≈ 0.0210, so σ_F ≈ 0.0210 × 3.51 kN ≈ 0.074 kN. The 3σ band is ±0.22 kN, giving a worst-case plateau range of roughly 3.29–3.73 kN. That fits inside the ±0.3 kN acceptance band, but with limited margin. Yield strength dominates the stack-up (its term is more than four times the diameter term), so tightening the material specification — for example, ±20 MPa instead of ±30 MPa — is the most effective way to buy margin.

Verification and Failure Symptoms

After selecting the bar, verify the plateau in a quasi-static webbing pull or a sled test. Pull webbing at a controlled rate (typically 100–500 mm/min for quasi-static) and record force versus extraction stroke. The plateau should be flat within the acceptance band over the usable stroke. If the measured plateau is high, check whether the bar diameter is at the upper tolerance limit or the material yield is above nominal. If the plateau drifts upward with stroke, suspect work-hardening of the bar or spool friction buildup. If the plateau is low and noisy, look for bar buckling, spool misalignment, or a yield strength below specification. Because diameter dominates, a single diameter measurement on a few bars is not sufficient — measure across the lot and confirm the distribution, not just the mean.

Calibration Record

This record documents the selected combination, the dominant tolerance contributor, and the resulting plateau band. It is the basis for the acceptance decision and for any later change control on bar stock or material specification.