Please demonstrate both methods, initial and bonus.

Leaf springs are commonly used in low$-$cost or high$-$

stiffness vehicle suspensions, e$.$g$.$ golf cars, light$-$truck

rear axle. Consider a cold$-$drawn AISI $1040$ flat leaf

spring subjected to a fluctuating loading condition of

maximum and minimum stresses of $420$ MPa and $140$

MPa respectively for a total of $50,000$ cycles. The

loading condition of the spring then changes to maximum

and minimum stresses of $350$ MPa and $-200$ MPa

respectively. Estimate the remaining life of the spring

under the new loading condition using Palmgren$-$Miner

cumulative damage theory. Assume that the fully$-$

corrected endurance limit is $200$ MPa and S$\_($n$)$ is $90\%$ of

S$\_($ut$).$

Bonus Question:

Calculate the remaining life of the leaf spring in Question

$5$ using Manson's method for linear damage.