Thread Bending Stress⚠ unverified
Mechanical / Power Screws · Compute the bending stress in a screw thread modelled as a cantilever
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| F | F | N | 1.0 | Axial load on the screw |
| t | t | m | 1.0 | Axial thickness of the thread at its base |
| b | b | m | 1.0 | Width of the thread section |
| h | h | m | 1.0 | Radial height of the thread, in metres (m). The bending moment arm is taken as ``h / 2`` |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | σ | Pa | Bending stress in the thread, in pascals (Pa). Returns 0.0 if ``b * t**2`` is not positive |
The science & history
Understanding the Parameters
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Axial load $F$ — the load the thread carries, applied along its flank. It creates the bending moment that tries to peel the thread off at its root.
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Base thickness $t$ — the axial thickness of the thread where it joins the core (the "fixed end" of the cantilever). It enters as $t^2$ (the section modulus of the rectangular thread base is $bt^2/6$), so it is the dominant strength lever — a thicker thread root is far stronger in bending. This is a reason square and Acme threads (thick roots) are used for power transmission over thin V-threads.
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Width $b$ — the length of thread resisting bending, taken as the engaged circumference (≈ $\pi d$). Stress falls linearly with $b$: more engaged thread shares the bending load.
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Thread height $h$ — sets the moment arm $h/2$ (load assumed to act at mid-height). A taller thread has a longer lever arm and thus higher root bending stress.
Derivation (Approaching a Proof)
Treat one turn of the thread as a cantilever beam projecting radially from the screw core, built in at its root and loaded by the axial thread force $F$ distributed along its flank.
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Bending moment. Taking the resultant load $F$ to act at the mid-height of the thread, the moment at the root (the fixed end) is $$M = F\,\frac{h}{2}.$$
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Section modulus. The thread root is a rectangular section of width $b$ and thickness $t$, whose section modulus is $$S = \frac{b t^2}{6}$$ (from $I/c = (bt^3/12)/(t/2)$; see Section Modulus).
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Flexure formula. The bending stress is moment over section modulus (Beam Bending Stress): $$\sigma = \frac{M}{S} = \frac{F(h/2)}{b t^2/6} = \frac{6 M}{b t^2} = \frac{3 F h}{b t^2}.$$
The cantilever model is an idealisation — real thread stress is concentrated at the fillet where the thread meets the core (a stress-concentration factor applies), and the load is shared unevenly among engaged threads — but it captures the dominant $1/t^2$ dependence that makes thread root thickness the key to bending strength.
Dimensional check. $[\sigma] = \dfrac{\text{N}\cdot\text{m}}{\text{m}\cdot\text{m}^2} = \dfrac{\text{N}}{\text{m}^2} = \text{Pa}$. ✓
History and Development
The cantilever-thread bending model is the classical way to check thread (and gear-tooth) root strength, appearing in Shigley and echoing the Lewis bending equation for gear teeth (a gear tooth is analysed as exactly this kind of loaded cantilever). It sets the thread proportions of power screws and, together with the tensile and shear checks, bounds the load a screw and nut can transmit.
Related Concepts: Thread Stress Tensile, Thread Shear Stress, Beam Bending Stress, Section Modulus, Gear Bending Stress Lewis
Notes: Cantilever idealisation (moment arm $h/2$; $M$ computed internally from $F$ and $h$). Apply a root-fillet stress-concentration factor and account for uneven thread-load sharing for design. Thick-root threads (square/Acme) resist bending far better than V-threads.