Close-coiled helical spring under axial load: for a spring with n coils, mean radius R, wire diameter d, and axial load W, what is the total compression (deflection) in terms of the shear modulus G?

Introduction / Context:
Close-coiled helical springs are common in mechanical and civil engineering devices (balances, suspensions, bearings). When loaded axially, the spring wire primarily undergoes torsion. This question asks for the standard deflection formula, highlighting how geometry and material properties control compliance.

Given Data / Assumptions:

• Spring is close-coiled (helix angle small; axial load causes mainly torsion in the wire).
• n = number of active coils, R = mean coil radius, d = wire diameter.
• W = axial load; G = shear modulus of the spring material.
• Linear elastic behavior; no coil clash or large-deformation effects.

Concept / Approach:
Under axial load, each coil twists. The total angle of twist over the wire length converts to axial deflection. Using torsion of circular shafts and spring geometry, the standard result for a close-coiled spring is obtained, showing strong sensitivity to wire diameter (d^4) and mean radius (R^3).

Step-by-Step Solution:
Torsion in wire: angle of twist θ = (T * L) / (J * G), with T ≈ W * R.Total wire length L = 2 * π * R * n for close-coiled spring.Polar moment for circular section: J = (π * d^4) / 32.Axial deflection δ relates to total twist by δ = θ * R.Combining and simplifying gives δ = (8 * W * R^3 * n) / (G * d^4).

Verification / Alternative check:
Stiffness k = W / δ = (G * d^4) / (8 * R^3 * n) is a widely cited result; inverting yields the same δ expression.

Why Other Options Are Wrong:

• Options with E (Young's modulus) are incorrect for close-coiled springs dominated by torsion; G must appear.
• Different constants (16, 64) or powers of R and d do not match the derived torsion-based relationship.

Common Pitfalls:

• Using E instead of G.
• Forgetting n in total length, or misusing wire diameter in J.

Final Answer:
δ = (8 * W * R^3 * n) / (G * d^4)