Bolt Pattern Analysis: The Elastic Method for Bolt Groups

What the elastic method is for

When a bracket is bolted down and loaded off-centre, the bolts don't share the load equally — the ones farthest from the load work hardest. The elastic method is the classic way to find how much force each bolt actually sees. It's the same treatment in Shigley's Mechanical Engineering Design, Blodgett's Design of Weldments (applied to weld groups), and the AISC Steel Construction Manual — and it's what lets you check the worst-loaded bolt against its strength rather than guessing.

The core idea: bolts as point areas

Every bolt is assumed identical and treated as a unit "point area" at its location. That's valid because a single bolt's own cross-section is tiny compared with how far the patternspreads the bolts apart — so the group's geometry, not the individual bolt size, governs the force distribution. From the bolt coordinates you compute the pattern's centroid and its moments of inertia, and everything follows from those.

In-plane shear: direct + torsional

An in-plane load on a bolt group splits into two parts:

• Direct shear — the total force shared equally, F/n on every bolt.
• Torsional shear — if the force acts off the centroid, it also applies a twisting moment Mz. That's distributed by the polar moment of inertia J = Ixx + Iyy: each bolt gets Mz·r/J, perpendicular to its radius from the centroid, so bolts farther out carry more.

The two components add as vectors at each bolt; the bolt with the largest resultant is the one to check. An off-centre load is first reduced to an equivalent force-plus-moment at the centroid (standard moment transfer) before distributing.

Out-of-plane moments: tension

A moment that tries to pry the bracket off the surface puts the bolts into tension, distributed by the general unsymmetric-bending formula. The rigorous version keeps the product of inertia Ixy, so it handles a genuinely asymmetric custom bolt layout correctly — many simplified treatments assume Ixy = 0 (principal axes aligned with the pattern), which only holds when the pattern is symmetric about an axis. The neutral axis is taken at the centroid, valid while the joint faces stay in contact (this is not a cracked-section concrete-anchor prying analysis).