Question

The partial differential of normal stress in x-direction in terms of effective stress is given by __________

(a) \(\frac{∂σ_x{‘}}{∂x}\)

(b) \(\frac{∂σ_x{‘}}{∂x}-γ_w  \frac{∂h}{∂x}\)

(c) \(\frac{∂σ_x{‘}}{∂x}+γ_w  \frac{∂h}{∂x}\)

(d) \(\frac{∂σ_x{‘}}{∂x}*γ_w  \frac{∂h}{∂x}\)

The question was posed to me in an interview for internship.

The query is from Elasticity topic in section Elements of Elasticity of Geotechnical Engineering I

Answer 1

The correct answer is (c) \(\frac{∂σ_x{‘}}{∂x}+γ_w  \frac{∂h}{∂x}\)

Explanation: Since, the normal stress in x-direction in terms of effective stress is given by,

σx= σx’+γw(h-he)

differentiating partially with respect to x,

\(\frac{∂σ_x}{∂x} =  \frac{∂σ_x{‘}}{∂x} + γ_w  \frac{∂(h-h_e)}{∂x}\)

But since \(\frac{∂h_e}{∂x}=0,\)

∴ \(\frac{∂σ_x}{∂x} = \frac{∂σ_x{‘}}{∂x+γ_w} \frac{∂h}{∂x}.\)