determine the initial radius of curvature a

Question

Determine the initial radius of curvature:

 A carriage spring, defiantly loaded and simply supported at its ends, has 12 steel plates each 60 mm wide by 6 mm thick. If the longest plate is 700 mm, determine the initial radius of curvature if the maximum stress is 150 N/mm² and the plates are ultimately straight.

By neglecting loss of energy at impact, find out the height from which 200 N may be dropped centrally on the spring.   E = 200 GPa.

Solution

Simply supported carriage spring :

n = 12

b = 60 mm

t = 6 mm

l = 700 mm

σ_b = 150 N/mm²

E = 200 GPa = 200 × 103 N/mm²

σ  =  3Wl  / 2 nb t²

∴ W = 3703 N

Δ=  3Wl  / 8 nbt ² E

=3 × 3703 × 700³/8 × 12 × 60 × 6³ × 200 × 10³ = 15.3 mm

R₀  = l ²/ 8 Δ =700²/ 8 × 15.3 = 4003 mm = 4 mm

U = Change in PE

(½) × 3703 × 15.3 = 200 × (h + 15.3)

∴ h = 126 mm