Question

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

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

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

(c) \(\frac{∂τ_{xz}}{∂x}+\frac{∂τ_{yz}}{∂y}+\frac{∂σ_z{‘}}{∂z}+Z=0\)

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

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

Asked question is from Elasticity in chapter Elements of Elasticity of Geotechnical Engineering I

Answer 1

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

For explanation I would say: The equilibrium equations in terms of total stresses formed by summing all forces on y-direction will include seepage force. Therefore, the equilibrium equations in terms of total stresses formed by summing all forces on y-direction is,

\(\frac{∂τ_{xy}}{∂x}+ \frac{∂σ_y}{∂y}+\frac{∂τ_{zy}}{∂z}=0.\)

Since σy= σy’+γw(h-he) and \(\frac{∂σ_y}{∂y} =  \frac{∂σ_y{‘}}{∂y}+γ_w \frac{∂h}{∂y}\)

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