find out the strain energy storeda close coiled

Question

Find out the strain energy stored:

A close coiled spring is subjected into an axial moment of 16 N-m. If the spring contains 15 coils with wire diameter of 10 mm and mean coil radius of 10 cm. Find out the following:

(a) the strain energy stored,

(b) maximum bending stress,

(c) axial twist.

 Take E = 200 GN/m².

Solution

M  = 16 N-m = 16 × 10³  N-mm , n = 15, d = 10 mm, R = 100 mm,

l = 2 π R n = 2π × 100 × 15 = 9434.8 mm , I = (π/64) d ⁴  = (π/64) (10)⁴  = 490.9 mm⁴

(a)   U = M 2 l / 2 EI = ((16 × 10³)² × 9424.8)/ (2 × (200 × 10³) (490.9)) = 12287.4 N-m = 12.3 N-m

(b)   σ= 32 M / π d ³  =32 × 16 × 10³ / π × 10³ = 163 N/mm²

(c)    U =( 1 /2 )M φ

⇒ 12.3 = (1 /2)× 16 × φ

∴          φ = 1.54 rad