Weld Group Polar Moment⚠ unverified
Mechanical / Welds · Compute the polar moment of inertia of a weld group
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| b | b | m | 1.0 | Width of the weld group |
| h | h | m | 1.0 | Height of the weld group |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | J | m^4 | Polar moment of inertia of the weld group, in metres to the fourth power (m^4) |
The science & history
Understanding the Parameters
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Both dimensions matter — unlike bending (governed by $h^3$ alone), twist resistance draws on both $b$ and $h$: $J = bh(b^2+h^2)/12$. A tall, narrow group resists bending well but twists easily; a square group resists twist best for a given area.
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Polar = sum of two rectangular moments — $J = I_x + I_y$. This is the perpendicular-axis theorem, valid for a planar area: the polar moment about an axis normal to the plane equals the sum of the two in-plane second moments.
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Role of $J$ — it converts a twisting couple $M = Fe$ into a shear stress via $\tau = Mr/J$. Maximising $J$ (spreading the welds from the centroid) minimises the torsional shear at the critical corner.
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Approximation caveat — the solid-rectangle $J$ overstates a real thin-line weld group's polar moment, so it under-predicts torsional stress; the exact throat-line sum is conservative by comparison.
Derivation (Approaching a Proof)
For a planar area the polar second moment about an axis perpendicular to the plane through the centroid is, by definition, $J = \int r^2\,dA$ where $r^2 = x^2 + y^2$. Splitting the integral,
$$J = \int (x^2 + y^2)\,dA = \int y^2\,dA + \int x^2\,dA = I_x + I_y.$$
This is the perpendicular-axis theorem. For a solid rectangle, $I_x = bh^3/12$ (bending about the horizontal axis) and $I_y = hb^3/12$ (about the vertical axis), so
$$J = \frac{b\,h^3}{12} + \frac{h\,b^3}{12} = \frac{b\,h\,(b^2 + h^2)}{12}.$$
For a discrete weld pattern the same theorem applies to the summed line/throat contributions of each segment about the group centroid.
Dimensional check. Each term is $\text{m}\cdot\text{m}^3 = \text{m}^4$, so $J = \dfrac{b h^3}{12} + \dfrac{h b^3}{12}$ is in $\text{m}^4$ — correct for a polar second moment of area.
History and Development
The perpendicular-axis theorem and the rectangle's second moments are foundations of classical mechanics of materials (Euler, Navier). Their use for weld groups is the torsional half of Blodgett's line method (Design of Welded Structures, 1966): tabulate the unit polar moment $J_u$ of standard weld patterns, then scale by the throat to size an eccentrically loaded connection. The solid-rectangle form here is the coarse approximation those tabulated line properties refine.
Related Concepts: Weld Group Moment Of Inertia, Weld Torsional Shear, Eccentric Weld Secondary Shear, Polar Moment solid shaft, Polar Moment Hollow
Notes: $J = I_x + I_y$ (perpendicular-axis theorem). Solid-rectangle (bounding-box) approximation — real groups sum line/throat contributions, giving smaller $J$. Both $b$ and $h$ matter for twist resistance.