Eccentric Weld Primary Shear⚠ unverified
Mechanical / Welds · Compute the primary shear stress from a direct load
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| F | F | N | 1.0 | Applied load on the weld group |
| A | A | m^2 | 1.0 | Throat area of the weld group |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | τprime | Pa | Primary (direct) shear stress, in pascals (Pa) |
The science & history
Understanding the Parameters
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Applied load $F$ — the full external load. In the primary term its point of application is ignored; only its magnitude and the total area matter. The eccentricity re-enters through the secondary term.
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Throat area $A$ — the sum of the throat areas of every weld segment in the group, $A = \sum 0.707\,h_i L_i$. Using the throat (not the leg) keeps the stress consistent with fillet-weld design.
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Primary shear $\tau'$ — a uniform, direction-following stress: it points parallel to the applied load everywhere on the weld. This is what makes the later vector addition with the (position-dependent) secondary shear meaningful.
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Why "primary/secondary" — the split is the elastic (vector) method of weld-group analysis. It is exact for the linear-elastic idealisation and conservative in practice, and it underlies the classic bracket-weld design tables.
Derivation (Approaching a Proof)
Replace the eccentric load $F$ acting at eccentricity $e$ by a statically equivalent system at the weld-group centroid: the same force $F$ through the centroid, plus a couple $M = F e$. Superpose their effects.
The force through the centroid is resisted by the weld throat in direct shear. Assuming the throat area carries it uniformly (the elastic-method assumption), the stress is force over area:
$$\tau' = \frac{F}{A}.$$
This is the primary term. The couple $M = Fe$ produces the secondary (torsional) shear $\tau'' = M r/J$ handled separately; the design stress is $\lvert \boldsymbol{\tau'} + \boldsymbol{\tau''}\rvert$ at the worst-located weld point.
Dimensional check. $\dfrac{F}{A} = \dfrac{\text{N}}{\text{m}^2} = \text{Pa}$ — a stress, as required.
History and Development
The primary/secondary decomposition is the elastic vector method for eccentrically loaded fastener and weld groups, taught in Shigley and the AISC manual for a century. It mirrors the identical treatment of eccentrically loaded bolt groups: reduce the load to a centroidal force plus a couple, compute a uniform "direct" component and a radially varying "torsional" component, then add them as vectors. A later, less-conservative alternative — the instantaneous-centre-of-rotation method — captures weld ductility for design tables, but the elastic method remains the transparent hand-calculation baseline.
Related Concepts: Eccentric Weld Secondary Shear, Weld Torsional Shear, Eccentric Load Angle, Eccentric Weld Load, Weld Group Polar Moment, Shear Stress
Notes: Primary = uniform direct shear $F/A$ (load direction). Add the secondary torsional shear $\tau'' = Fer/J$ vectorially for the peak stress. $A$ is the total throat area, $\sum 0.707\,h_iL_i$.