Close-coiled helical spring under axial load: For a closely-coiled helical spring of mean coil diameter D, wire diameter d, n turns, modulus of rigidity C (G), and axial load W, the deflection δ is

Introduction / Context:
Close-coiled helical springs store energy primarily by twisting of the wire. The load–deflection relation links geometry (D, d, n) and material shear modulus to the axial compliance.

Given Data / Assumptions:

• Close-coiled (helix angle small), pure torsion of wire dominates.
• Axial load W causes torque in each coil.
• Modulus of rigidity C (often denoted G) is constant.

Concept / Approach:
Each coil behaves like a torsion bar of circular section. Summing angular twist over n coils and converting to axial deflection yields the well-known expression δ = 8 * W * D^3 * n / (C * d^4).

Step-by-Step Solution:

Verification / Alternative check:
Dimensional check: δ has units of length; numerator W * D^3 * n divided by C * d^4 (stress per shear-strain scale) yields length.

Why Other Options Are Wrong:

• Other powers of D and d do not arise from torsion theory; they would give incorrect stiffness scaling.

Common Pitfalls:
Using E instead of C (G); mixing up D (mean diameter) with radius; forgetting the power d^4 in the denominator.

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