Question

Three bars of equal length and having, cross-sectional areas in ratio 1:2:4, are all subjected to equal load. Compare their strain energy.

Answer 1

Let A, 2A, 4A be the areas.

Loads are equal = P

Each have equal length = L

Now σ₁ = P/A; σ₂ = P/2A; σ₃ = P/4A;

Since strain energy = U = σ₂.Vol/2E

Strain energy in first bar = U₁ = σ₁².Vol/2E = (1/2E)(P/A)².AL = P^2.L/2AE

Strain energy in second bar = U₂ = σ₂^2.Vol/2E = (1/2E)(P/2A)².2AL = P².L/4.AE

Strain energy in third bar = U₃ = σ₃².Vol/2E = (1/2E)(P/4A)².4AL = P².L/8AE 

Now;

U₁ : U₂ : U₃ = 1 : 1/2 : 1/4