An eccentric load on a bolt group is one of the most common connection checks in steelwork - a bracket, a cleat, a beam-to-column fin plate where the load does not pass through the centre of the bolts. It is also a check with two very different methods, and choosing between them is a deliberate trade of effort against economy.
Where the eccentricity comes from
Whenever the line of action of a force misses the centroid of the bolt group, that offset multiplies into a moment, . The group must now carry both the direct force and the moment , and the bolts furthest from the centroid pick up the most. The whole question is how that combined demand is shared out among the bolts - and that is where the two methods diverge.
The elastic method
The elastic (or vector) method assumes the bolt group rotates rigidly about its centroid. Each bolt carries a force from the direct load shared equally, plus a force from the moment that grows in proportion to its distance from the centroid. The two are added as vectors, and the most heavily loaded bolt - usually a corner one - governs the design.
It is simple, fast, and always conservative, because it assumes a linear distribution right up to the point the worst bolt reaches capacity, ignoring any redistribution.
The plastic (instantaneous-centre) method can show 20-40% more capacity for the same bolts, because it lets the group redistribute once the corner bolts yield.
The plastic (instantaneous-centre) method
The instantaneous-centre method recognises that bolts are ductile: once the most-loaded bolt reaches its capacity, it does not fail, it deforms and lets the others take more. The group rotates about an instantaneous centre (not the centroid), and each bolt's force depends on its deformation through a non-linear load-slip relationship. The result is a higher, more realistic capacity for the same arrangement - at the cost of an iterative calculation.
When to use which
The choice is genuinely situational:
- Quick sizing, or a connection that is not weight-critical - the elastic method wins on speed and transparency. If it passes comfortably, there is no reason to do more work.
- A tight connection where bolt count or steel weight matters - the instantaneous-centre method earns its extra effort by unlocking capacity the elastic method leaves on the table.
- When in doubt - start elastic. It is conservative, so if it passes you are safe; only reach for the plastic method when the elastic check is the thing forcing an extra bolt row.
Both methods are standard and accepted; neither is "more correct" in isolation. The elastic method is a safe lower bound, the plastic method a sharper estimate of the true strength. Pick the one that matches the stakes of the connection, and remember that the bolt resistances themselves - shear, bearing and the relevant partial factors - come from the same EC3 framework you would use for any other connection check.