Belt Stress⚠ unverified
Mechanical / Flexible Elements · Compute the tensile stress in a belt cross-section
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| F | F | N | 1.0 | Belt tension force |
| w | w | m | 1.0 | Belt width |
| t | t | m | 1.0 | Belt thickness |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | σ | Pa | Belt tensile stress, in pascals (Pa) |
The science & history
Understanding the Parameters
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Tension $F$ — use the tight-side tension $F_1$ (plus centrifugal tension $T_c$ at high speed), since that governs. The stress varies around the belt as the tension changes between the tight and slack spans.
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Cross-section $w\,t$ — the load-bearing area. Wider or thicker belts carry more load at the same stress, which is how belts are sized: pick $w, t$ so $\sigma$ stays under the allowable.
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Two independent limits — a belt drive is limited by the smaller of two ceilings: this strength limit ($\sigma \le \sigma_{allow}$) and the friction limit (slip, Belt Tension Ratio). Slow drives are often strength-limited; the design must satisfy both.
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Bending stress too — this is the tensile (axial) stress; a real belt also sees bending stress as it wraps the pulley (larger on small pulleys), which is why minimum pulley diameters are specified. The combined tensile + bending stress drives belt fatigue.
Derivation (Approaching a Proof)
A belt in tension carries a uniform axial force $F$ across its rectangular cross-section of width $w$ and thickness $t$. Assuming the stress is uniformly distributed (a thin belt, well away from the pulley), the axial stress is force over area:
$$A = w\,t, \qquad \sigma = \frac{F}{A} = \frac{F}{w\,t}.$$
This is elementary $\sigma = F/A$ (Shear Stress is the shear analog); the significance is in which $F$ to use (tight side, plus centrifugal) and in remembering the superimposed bending stress from wrapping the pulley.
Dimensional check. $\sigma = \dfrac{F}{w\,t} = \dfrac{\text{N}}{\text{m}\cdot\text{m}} = \dfrac{\text{N}}{\text{m}^2} = \text{Pa}$ — a stress, as required.
History and Development
Belt tensile stress is the strength side of belt-drive design, complementing the friction (capstan) side. Flat-belt design historically balanced leather/fabric belt strength against slip; modern belts (reinforced elastomer, with tension cords) are rated by allowable tension per unit width, and the drive is checked against both the strength limit here and the friction limit (Belt Tension Ratio). Combined with wrapping bending stress, it governs belt fatigue life and minimum pulley diameters.
Related Concepts: Belt Power, Maximum Belt Power, Belt Tension Ratio, Centrifugal Tension, Shear Stress, V-Belt Design
Notes: Use the tight-side tension ($F_1 + T_c$). Sets the strength limit — a drive is bounded by the smaller of strength ($\sigma \le \sigma_{allow}$) and friction (slip, Belt Tension Ratio). Add the pulley-wrap bending stress for fatigue (drives minimum pulley diameter).