Weld Group Moment Of Inertia⚠ unverified
Mechanical / Welds · Compute the second moment of area of a weld group (approximate)
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 | I | m^4 | Second moment of area of the weld group, in metres to the fourth power (m^4) |
The science & history
Understanding the Parameters
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Height $h$ — the dimension in the plane of bending, measured perpendicular to the neutral axis. Because it enters as $h^3$, doubling the group height increases bending resistance eightfold: deep brackets are vastly stiffer than shallow ones.
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Width $b$ — enters linearly; it scales $I$ but cannot compete with height. For a real weld group $b$ is closer to the throat's contribution than a solid width, which is the source of the approximation.
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Centroidal axis — $bh^3/12$ is taken about the axis through the centroid. If the neutral axis is elsewhere (asymmetric groups), the parallel-axis theorem must shift it — another reason the exact line method is preferred for real patterns.
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Role of $I$ — it converts a bending moment into a stress via $\sigma = My/I$; a larger $I$ means lower stress for the same moment.
Derivation (Approaching a Proof)
For a solid rectangle of width $b$ and height $h$, take the centroidal axis parallel to $b$. A horizontal strip of thickness $dy$ at distance $y$ has area $dA = b\,dy$. The second moment of area is
$$I = \int y^2\,dA = \int_{-h/2}^{+h/2} y^2\,b\,dy = b\left[\frac{y^3}{3}\right]_{-h/2}^{+h/2} = b\left(\frac{h^3}{24} + \frac{h^3}{24}\right) = \frac{b\,h^3}{12}.$$
For an actual weld group — discrete line segments rather than a filled area — one instead sums each segment's own second moment plus its $A d^2$ parallel-axis term about the group centroid, using throat area for $A$. The solid-rectangle result is the limiting approximation when the pattern is idealised as a full $b\times h$ block.
Dimensional check. $\dfrac{b\,h^3}{12} = \text{m}\cdot\text{m}^3 = \text{m}^4$ — a fourth power of length, as a second moment of area must be.
History and Development
The rectangle's $bh^3/12$ is one of the oldest results in strength of materials, tracing to the flexure theory of Bernoulli, Euler, and Navier. In weld design, the line method (Blodgett, Design of Welded Structures, 1966) adapts it: treat the weld as a line, tabulate the unit second moment $I_u$ for common patterns (single line, box, channel), then multiply by the throat to get the section's $I$. This calculator's solid-rectangle form is the coarse first approximation those tabulated line properties refine.
Related Concepts: Weld Group Polar Moment, Weld Bending Stress, Section Modulus, Rectangular Moment of Inertia, Eccentric Load Angle
Notes: Solid-rectangle (bounding-box) approximation — real weld groups use summed line/throat properties (parallel-axis), usually giving smaller $I$. $h^3$ dependence: height dominates. Taken about the centroidal axis.