Deflection:

Under the axial load the free end of the spring shall move vertically down. The amount by which the free end moves is called as the deflection of the spring, mentioned by Δ. Clearly the free end moves due to twist in the wire of the spring which may be regarded as a shaft of length l,

l = 2π R n

where n = Number of coils.

The amount by which the open section of wire shall move down is R × twist.

The angle of twist of length l from T/J  = G θ/l is θ= Tl /GJ and putting for l from

Eq. (13.6)

J =  (π /32 )d⁴      and T = WR

θ= 2WR² n π× 32/ G π d ⁴ =64W R² n/ G d ⁴

∴          Δ= R θ

or

Δ= 64WR³ n / Gd ⁴

Strain energy stored in spring

U = (1 /2 )W Δ

or         U =32W 2 R³ n / Gd ⁴

 It can be good to check that strain energy per unit volume

 (Divide Eq. by volume = π d ²/4 × l ).

Stiffness is described as load per unit deflection i.e.

K = W / Δ

or         K = Gd ⁴ / 64 R³ n