Eccentric Weld Secondary Shear⚠ unverified
Mechanical / Welds · Compute the secondary shear stress from torsion
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| F | F | N | 1.0 | Applied load on the weld group |
| e | e | m | 1.0 | Eccentricity of the load from the weld group centroid |
| J | J | m^4 | 1.0 | Polar moment of inertia of the weld group |
| r | r | m | 1.0 | Radial distance from the weld group centroid to the point of interest |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | τdoubleprime | Pa | Secondary (torsional) shear stress, in pascals (Pa) |
The science & history
Understanding the Parameters
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Twisting couple $M = Fe$ — the eccentricity $e$ converts the applied load into a torque about the centroid. Larger eccentricity means larger secondary shear, which is why brackets are detailed to keep the load line close to the weld centroid.
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Radial distance $r$ — the secondary shear grows linearly from zero at the centroid to a maximum at the weld point farthest from it. The governing point is the one with the largest $r$ and the least favourable direction relative to the primary shear.
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Polar moment $J$ — the weld group's resistance to twist, $J = I_x + I_y$ (see Weld Group Polar Moment). A group spread far from its centroid has a large $J$ and therefore low torsional stress.
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Direction — $\tau''$ acts perpendicular to the radius $r$ (like the shear in a shaft under torsion), which is essential for the vector sum with the load-parallel primary shear.
Derivation (Approaching a Proof)
Reduce the eccentric load to a centroidal force plus a couple $M = Fe$. The couple twists the (thin) weld group in its own plane. Model the throat area as a thin planar region; under in-plane torsion the elastic shear at a point varies linearly with its distance from the centroid, exactly as in the torsion of a shaft:
$$\tau'' = \frac{M\,r}{J} = \frac{F\,e\,r}{J}.$$
Here $J = \int r^2\,dA$ is the polar second moment of the weld's throat area about the centroid. The linear $r$-dependence and the $J$ in the denominator both come straight from the torsion formula $\tau = T\rho/J$; the weld group borrows it wholesale by treating the throat area as the twisted section.
Dimensional check. $\dfrac{F\,e\,r}{J} = \dfrac{\text{N}\cdot\text{m}\cdot\text{m}}{\text{m}^4} = \dfrac{\text{N}}{\text{m}^2} = \text{Pa}$ — a stress, as required.
History and Development
The torsional term completes the elastic vector method for eccentric weld and bolt groups (Shigley, AISC). Historically the weld group was first reduced to a "line weld", its polar moment computed per unit throat, and the peak point found by inspection; the vector sum of direct and torsional shear then sized the weld. The method is a direct transplant of Coulomb's torsion theory onto a planar weld pattern — a neat example of one classical result (shaft torsion) solving an apparently different problem (a twisted line of weld metal).
Related Concepts: Weld Torsional Shear, Eccentric Weld Primary Shear, Eccentric Load Angle, Eccentric Weld Load, Weld Group Polar Moment, Torsional Shear Stress
Notes: Duplicate of Weld Torsional Shear ($\tau = Fer/J$). Acts perpendicular to $r$; add vectorially to the primary shear $F/A$. Maximum at the largest-$r$ weld point. $J = I_x + I_y$ from Weld Group Polar Moment.