determine strain energy stored in coiled helical

Question

Determine Strain energy stored in coiled helical spring:

For a close coiled helical spring subjected to an axial load of 200 N with 10 coils of wire of diameter 18 mm and built with coil diameter of 200 cm, determine

1.      strain energy stored,

2.      axial deflection, and

3.      maximum shear stress in the wire.

Take G = 80 GN/m².

Solution

 W = 200 N, n = 10, d = 8 mm, D = 20 cm = 200 mm, R = D/2 = 100 mm

Neglecting effect of direct shear since (d /2 D)  < < 1.

Axial deflection,  Δ=  64 W R³ n / Gd ⁴

=64 (200) (100)³ (10) /(80 × 10³ ) (18)⁴

= 15.24 mm

Strain energy, =( ½) W Δ

= (½) × 200 × (15.24 /1000) = 1.524 N-m

Maximum shear stress,  τ_max  = 16 W R  /π d³

= 16 (200) (100) / π (18)³

= 17.5 N/mm²