Question

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

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

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

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

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

The question was posed to me in my homework.

This question is from Elasticity in portion Elements of Elasticity of Geotechnical Engineering I

Answer 1

The correct choice is (d) \(\frac{∂σ_x{‘}}{∂x}+\frac{∂τ_{yx}}{∂y}+\frac{∂τ_{zx}}{∂z}+γ_w  \frac{∂h}{∂x}=0\)

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

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

Since σx= σx’+γw(h-he) and \(\frac{∂σ_x}{∂x} = \frac{∂σ_x{‘}}{∂x}+γ_w \frac{∂h}{∂x}\)

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