# Soil Mechanics Questions and Answers – Elasticity – Boundary Condition Equations

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This set of Soil Mechanics Multiple Choice Questions & Answers (MCQs) focuses on “Elasticity – Boundary Condition Equations”.

1. The boundary condition equation for X, where X is the component of the surface force in x-direction per unit area is ___________
a) X = σy m+τyz n+τxy l
b) X = σz n+τzx l+τzy m
c) X = σx l+τyx m+τzx n
d) X = σy l+τyx m+τzx n

Explanation: Let the area of ABC=ds.

area OBC=ds cos(N,x)=ds.l
area OAB=ds cos(N,y)=ds.m
area OAC=ds cos(N,z)=ds.n
Resolving all the forces in x-direction and equating them to zero, we get,
X = σx l+τyx m+τzx n.

2. The boundary condition equation for Y, where Y is the component of the surface force in y-direction per unit area is ___________
a) Y = σy m+τyz n+τxy l
b) Y = σz n+τzx l+τzy m
c) Y = σx l+τyx m+τzx n
d) Y = σy l+τyx m+τzx n

Explanation: Let the area of ABC=ds.

area OBC=ds cos(N,x)=ds.l
area OAB=ds cos(N,y)=ds.m
area OAC=ds cos(N,z)=ds.n
Resolving all the forces in y-direction and equating them to zero, we get,
Y = σy m+τyz n+τxy l.

3. The boundary condition equation for Z, where Z is the component of the surface force in z-direction per unit area is ___________
a) Z = σy m+τyz n+τxy l
b) Z = σz n+τzx l+τzy m
c) Z = σx l+τyx m+τzx n
d) Z = σy l+τyx m+τzx n

Explanation: Let the area of ABC=ds.

area OBC=ds cos(N,x)=ds.l
area OAB=ds cos(N,y)=ds.m
area OAC=ds cos(N,z)=ds.n
Resolving all the forces in z-direction and equating them to zero, we get,
Z = σz n+τzx l+τzy m.

4. The matrix form of the boundary condition equations is _____________
a) $$\begin{bmatrix} \overline{X}\\ \overline{Y}\\ \overline{Z} \end{bmatrix} = \begin{bmatrix} σ_{xx} & τ_{xy} & τ_{xz} \\ τ_{yx} & σ_{yy} & τ_{yz} \\ τ_{zx} & τ_{zy} & σ_{zz} \end{bmatrix} \begin{bmatrix} l \\ m \\ n \end{bmatrix}$$
b) $$\begin{bmatrix} \overline{X}\\ \overline{Y}\\ \overline{Z} \end{bmatrix} = \begin{bmatrix} σ_{zz} & τ_{xy} & τ_{xz} \\ τ_{yx} & σ_{yy} & τ_{yz} \\ τ_{zx} & τ_{zy} & σ_{xx} \end{bmatrix} \begin{bmatrix} l \\ m \\ n \end{bmatrix}$$
c) $$\begin{bmatrix} \overline{X}\\ \overline{Y}\\ \overline{Z} \end{bmatrix} = \begin{bmatrix} σ_{xx} & τ_{zz} & τ_{xz} \\ τ_{yx} & σ_{yy} & τ_{yz} \\ τ_{zx} & τ_{zy} & σ_{zz} \end{bmatrix} \begin{bmatrix} l \\ m \\ n \end{bmatrix}$$
d) $$\begin{bmatrix} \overline{X}\\ \overline{Y}\\ \overline{Z} \end{bmatrix} = \begin{bmatrix} σ_{xx} & τ_{yy} & τ_{xz} \\ τ_{yx} & σ_{yy} & τ_{yz} \\ τ_{zx} & τ_{yy} & σ_{zz} \end{bmatrix} \begin{bmatrix} l \\ m \\ n \end{bmatrix}$$

Explanation: The boundary condition equations are given by,
X = σx l+τyx m+τzx n
Y = σy m+τyz n+τxy l
Z = σz n+τzx l+τzy m,
∴ The matrix form is given by,
$$\begin{bmatrix} \overline{X}\\ \overline{Y}\\ \overline{Z} \end{bmatrix} = \begin{bmatrix} σ_{xx} & τ_{xy} & τ_{xz} \\ τ_{yx} & σ_{yy} & τ_{yz} \\ τ_{zx} & τ_{zy} & σ_{zz} \end{bmatrix} \begin{bmatrix} l \\ m \\ n \end{bmatrix}$$

5. The partial differential of normal stress in y-direction in terms of effective stress is given by __________
a) $$\frac{∂σ_z{‘}}{∂z}$$
b) $$\frac{∂σ_z{‘}}{∂z}-γ_w\frac{∂h}{∂z}$$
c) $$\frac{∂σ_z{‘}}{∂z}+γ_w\frac{∂h}{∂z}$$
d) $$\frac{∂σ_z{‘}}{∂z}*γ_w\frac{∂h}{∂z}$$

Explanation: Since, the normal stress in y-direction in terms of effective stress is given by,
σz= σz’+γw(h-he)
differentiating partially with respect to z,
$$\frac{∂σ_z}{∂z} = \frac{∂σ_z{‘}}{∂z} +γ_w \frac{∂(h-h_e)}{∂z}$$
But since $$\frac{∂h_e}{∂z}=0,$$
∴ $$\frac{∂σ_z}{∂z} = \frac{∂σ_z{‘}}{∂z} +γ_w \frac{∂h}{∂z}.$$

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

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

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

Explanation: 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 $$\frac{∂σ_y}{∂y} = \frac{∂σ_y{‘}}{∂y}+γ_w \frac{∂h}{∂y}$$
∴ $$\frac{∂τ_{xy}}{∂x}+\frac{∂σ_y{‘}}{∂y}+\frac{∂τ_{zy}}{∂z}+γ_w\frac{∂h}{∂y}=0$$

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

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

Sanfoundry Global Education & Learning Series – Soil Mechanics.