Question

The equilibrium equation in Z-direction in terms of effected stress for a saturated soil body is given by __________

(a) \(\frac{∂σ_x{‘}}{∂x}+\frac{∂τ_{yx}}{∂y}+\frac{∂τ_{zx}}{∂z}++γ_w \frac{∂h}{∂z}=0\)

(b) \(\frac{∂τ_{xy}}{∂x}+\frac{∂σ_y{‘}}{∂y}+\frac{∂τ_{zy}}{∂z}+γ_w  \frac{∂h}{∂z}=0\)

(c) \(\frac{∂τ_{xz}}{∂x}+\frac{∂τ_{yz}}{∂y}+\frac{∂σ_z{‘}}{∂z}+γ’+γ_w  \frac{∂h}{∂z}=0\)

(d) \(\frac{∂σ_x{‘}}{∂x}+\frac{∂τ_{yx}}{∂y}+\frac{∂τ_{zx}}{∂z}=0\)

This question was addressed to me in examination.

I'd like to ask this question from Elasticity topic in portion Elements of Elasticity of Geotechnical Engineering I

Answer 1

Correct choice is (c) \(\frac{∂τ_{xz}}{∂x}+\frac{∂τ_{yz}}{∂y}+\frac{∂σ_z{‘}}{∂z}+γ’+γ_w  \frac{∂h}{∂z}=0\)

Best explanation: The equilibrium equations in terms of total stresses formed by summing all forces on z-direction will include seepage force. Therefore, the equilibrium equations in terms of total stresses formed by summing all forces on z-direction is,

\(\frac{∂τ_{xz}}{∂x}+ \frac{∂τ_{yz}}{∂y}+\frac{∂σ_z}{∂z}+γ=0.\)

Since σz= σz’+γw(h-he) and \(\frac{∂σ_z}{∂z} = \frac{∂σ_z{‘}}{∂z}+γ_w  \frac{∂h}{∂z}\)

∴  \(\frac{∂τ_{xz}}{∂x}+\frac{∂τ_{yz}}{∂y}+\frac{∂σ_z{‘}}{∂z}+γ’+γ_w  \frac{∂h}{∂z}=0\)