Eccentrically loaded bracket – force on a rivet due to moment A bracket is loaded by a force P with eccentricity e. If a rivet lies at a radial distance r_n from the group centroid, the additional resisting force on that rivet due to the moment is:

Introduction / Context:
For riveted or bolted connections subjected to an eccentric load, the load can be decomposed into a resultant force at the centroid and a moment about the centroid. The moment is resisted by a system of forces in the fasteners proportional to their distance from the centroid.

Given Data / Assumptions:

• Eccentric load P acting at eccentricity e from the centroid of the rivet group.
• Rivet at distance r_n from the centroid.
• Linear elastic distribution; fastener forces from moment vary linearly with radius.

Concept / Approach:
The total resisting moment from the group equals the applied moment M = P * e. With a linear law F_n = C * r_n, the moment balance gives Σ(F_n * r_n) = C * Σ(r_n^2) = P * e. Solving for C yields C = (P * e) / Σ(r^2), and the force in the nth rivet is F_n = C * r_n.

Step-by-Step Solution:

Verification / Alternative check:
Check dimensions: numerator has force × length × length, denominator length^2 → force, consistent.

Why Other Options Are Wrong:
Forms lacking Σ(r^2) ignore group stiffness; inversions like (P * e) / r_n or multiplying by r_n^2 misrepresent the linear distribution assumption.

Common Pitfalls:
Combining direct shear from P/number of rivets and moment-induced forces incorrectly (they must be vectorially combined per rivet).

Final Answer:
F = (P * e * r_n) / Σ(r^2)