Riveted bracket under eccentric load: If a bracket is subjected to a moment M about the centroid of a rivet group, the tangential force induced in any rivet located at radius r (at right angles to its radius vector) is given by which relation?

Introduction / Context:
Eccentrically loaded riveted or bolted connections experience a primary direct shear from the resultant force and a secondary shear due to the moment about the group centroid. The latter creates a rotational effect, distributing tangential forces proportional to rivet radii.

Given Data / Assumptions:

• Moment M acts about the centroid of the rivet group.
• Rivet i is at radius r_i from the centroid; candidate rivet is at radius r.
• Linear elastic deformation and proportional load sharing in rotation are assumed.

Concept / Approach:
Equilibrium of moments due to rotational shear components requires Σ(F_ti * r_i) = M. With an assumption that F_ti ∝ r_i, we write F_ti = k * r_i. Substituting gives k * Σ(r_i^2) = M, hence k = M / Σ(r_i^2). Therefore, for a rivet at radius r, F_t = k * r = M * r / Σ(r_i^2).

Step-by-Step Solution:
1) Assume tangential force varies linearly with radius: F_ti = k * r_i.2) Enforce moment equilibrium: Σ(F_ti * r_i) = Σ(k * r_i^2) = M.3) Solve for k: k = M / Σ(r_i^2).4) Force in a specific rivet at radius r: F_t = k * r = M * r / Σ(r_i^2).

Verification / Alternative check:
This expression is widely used for bolted/riveted group design under eccentric loading and matches results from classical connection analysis texts.

Why Other Options Are Wrong:
Forms using Σ(r_i) misrepresent the energy or moment distribution; the square summation arises from proportionality to radius.M/r alone neglects the group effect and overestimates load in outer rivets.

Common Pitfalls:
Forgetting to combine primary direct shear with secondary shear vectorially; ignoring directionality when composing rivet resultants.

Final Answer:
F_t = M * r / Σ(r_i^2)