4. Belt element formulation in PAM-CRASH

Seatbelt restraint modeling in PAM-CRASH is built on a small family of dedicated belt elements that behave very differently from ordinary shell or solid elements. A belt is a long, thin, highly anisotropic structure that can carry membrane tension, slide through hardware, fold onto itself, and go slack under compression. Reproducing that physics with generic elements is impractical, so PAM-CRASH provides specialized belt formulations whose behavior is governed by element type, slip ring definition, and a tension-only material law. Understanding these three ingredients is the prerequisite for any occupant protection model that produces trustworthy belt forces and chest deflections.

Belt element types and their roles

PAM-CRASH belt modeling typically combines three element categories, each responsible for a distinct part of the restraint path:

• Belt elements (membrane/seatbelt elements) — one-dimensional or membrane-like elements that carry the axial tension of the webbing. They are defined along the belt centerline and are the only elements that actually transmit restraint force to the occupant.
• Slip ring elements — massless, frictionless (or friction-controlled) nodes that let the belt slide through a D-ring, buckle, or retractor anchor. They do not carry tension themselves; they redirect and redistribute it.
• Retractor and pretensioner elements — special belt elements that impose a pay-out/retraction law, and in pretensioned systems apply an initial pull-in stroke before the crash pulse develops.

The distinction matters because a common modeling error is to treat the D-ring as a rigid constraint. If the belt is pinned at the D-ring instead of passing through a slip ring, the shoulder-belt length is fixed, the lap and shoulder segments cannot share webbing, and the model will over-predict belt force and under-predict occupant excursion. The slip ring exists precisely to allow the webbing to migrate between segments.

Slip ring definition and the sliding constraint

A slip ring is defined at a node or a small set of nodes that the belt passes through. In the PAM-CRASH input, the belt element connectivity is arranged so that the belt is continuous through the slip ring node, and the slip ring is declared as a sliding condition rather than a fixed attachment. The essential parameters are:

• Slip ring node / position — the geometric location of the D-ring, buckle, or anchor through which the belt slides.
• Friction coefficient — optional Coulomb friction between webbing and hardware. Zero friction gives free sliding; a small value (commonly 0.1–0.3 for webbing over a steel D-ring) is more realistic and prevents unrealistically free webbing migration.
• Slip direction / routing — the belt must be routed so that the incoming and outgoing segments are geometrically consistent with the hardware; a mis-routed slip ring produces spurious belt length changes.

Mechanically, the slip ring enforces equal tension on both sides of the ring (minus friction). If the shoulder segment tension is T_s and the lap segment tension is T_l, then with friction coefficient μ and wrap angle θ, the capstan relation gives T_s = T_l · e^μθ. With μ = 0 this reduces to T_s = T_l, which is the ideal slip ring. This is why a slip ring is not a load path in itself — it is a tension equalizer and a routing device.

Tension-only behavior and why it is essential

Webbing cannot push. Under a compressive or slack condition the belt simply folds and carries no load. PAM-CRASH belt elements therefore use a tension-only formulation: the element stiffness is active only when the element is in tension (positive axial strain), and the element offers essentially zero resistance when slack. This is implemented through a tension-only flag on the belt material or element definition, combined with a stress–strain curve that is defined only for positive strain.

The practical consequences are significant:

• No compressive stiffness — a slack belt does not push the occupant away from the seat, which is physically correct and prevents non-physical forces during the early phase of a crash pulse.
• Load path activation — the belt only begins to restrain once it is taut. The time at which the belt becomes taut (the "belt slack take-up" time) directly controls when occupant loading starts.
• Numerical stability — tension-only elements can snap between slack and taut states, so a small amount of numerical damping or a smooth transition in the material curve is often used to avoid convergence issues.

A typical belt material definition uses a piecewise linear or tabulated force–strain curve. For a 50 mm wide polyester webbing, a representative engineering curve might be:

STRAIN    FORCE_PER_WIDTH (N/mm)
0.000     0.0
0.005     8.0
0.010     18.0
0.020     32.0
0.040     48.0
0.080     62.0
0.120     70.0

These values are illustrative only and must be replaced with the actual webbing certification data for the restraint system being modeled. The key point is that the curve is defined for positive strain only; negative strain produces zero force.

Putting the three ingredients together

The interaction between element type, slip ring, and tension-only behavior is best understood as a signal chain: the occupant moves forward, the belt geometry changes, the slip ring redistributes webbing between lap and shoulder segments, and the tension-only belt elements carry load only where the webbing is taut. The SVG below shows this chain for a three-point belt with a D-ring slip ring and a retractor.

Practical modeling checklist

When building or reviewing a PAM-CRASH belt model, the following checks catch the majority of formulation errors:

The belt formulation is not a detail to be finalized after the occupant model is built. It determines when and how restraint force enters the system, and therefore shapes every downstream result — chest deflection, head excursion, and injury criteria. Getting the element type, slip ring, and tension-only behavior right at the start is far cheaper than diagnosing a physically inconsistent belt response after a full crash simulation has run.