Soil Mechanics Questions and Answers – Stress Distribution – Vertical Pressure – 2

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This set of Soil Mechanics Multiple Choice Questions & Answers (MCQs) focuses on “Stress Distribution – Vertical Pressure – 2”.

1. For maximum vertical stress, the shear stress is _________ if the load is 30 kN and r=4m.
a) 0.4356 kN/m2
b) 0.1359 kN/m2
c) 0.1518 kN/m2
d) 0.3625 kN/m2

Explanation: Given,
r=4m
Q=30 kN
$$τ_{rz}=\frac{0.0725Q}{r^2}$$
$$τ_{rz}=\frac{0.0725*30}{4^2}$$
∴ τrz=0.1359kN/m2.

2. What will be the intensity of shear stress at a depth of 4m and at a radial distance of 1m from concentrated load of 20 kN?
a) 0.4356 kN/m2
b) 0.244 kN/m2
c) 0.652 kN/m2
d) 0.128 kN/m2

Explanation: Given,
Z=4m
Q=20 kN
r=1
The Boussinesq’s shear stress τrz is given by,
$$τ_{rz}=\frac{3Qr}{2πz^3}\left[\frac{1}{1+(\frac{r}{z})^2}\right]^{\frac{5}{2}}$$
∴ $$τ_{rz}=\frac{3*20*1}{2π4^3}\left[\frac{1}{1+(\frac{1}{4})^2}\right]^{\frac{5}{2}}$$
∴ τrz=0.128 kN/m2.

3. If r/z ratio is 2 and load of 20 kN is acting at a point, then the vertical pressure at a depth 6m is ____________
a) 0.4356 kN/m2
b) 0.244 kN/m2
c) 0.1518 kN/m2
d) 4.72*10-3 kN/m2

Explanation: Given,
r/z=2
Q=20 kN
Z=6m
$$σ_z=\frac{0.0085Q}{z^2}$$
$$σ_z=\frac{0.0085*20}{6^2}$$
σz=4.72*10-3 kN/m2.
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4. The Boussinesq influence factor for r/z ratio equal to 1 is given by ____________
a) 0.3840
b) 0.5465
c) 0.0844
d) 0.2312

Explanation: Given,
r/z=1
The Boussinesq influence factor is given by,
$$K_B=\frac{3}{2π} \left[\frac{1}{1+(\frac{r}{z})^2}\right]^{\frac{5}{2}}$$
$$K_B=\frac{3}{2π} \left[\frac{1}{1+1^2}\right]^{\frac{5}{2}}$$
KB=0.0844.

5. When the maximum vertical stress is 0.235 kN/m2 at a radial distance of 4m from the point load is __________ kN.
a) 42.34
b) 10.56
c) 20.76
d) 30.65

Explanation: Given,
z)max=0.235 kN/m2
r=4m
since the maximum vertical stress is
$$(σ_z)_{max}=\frac{0.0888Q}{r^2}$$
∴ $$Q=\frac{(σ_z)_{max}r^2}{0.0888}$$
∴ $$Q=\frac{0.235*4^2}{0.0888}$$
Q=42.34 kN.

6. The Boussinesq’s vertical pressure σz under a uniformly loaded circular area is given by ________
a) $$σ_z=q\left[1-\left[\frac{1}{1+(\frac{a}{z})^2}\right]^{\frac{3}{2}}\right]$$
b) $$σ_z=q\left[1+\left[\frac{1}{1+(\frac{a}{z})^2}\right]^{\frac{3}{2}}\right]$$
c) $$σ_z=q\left[\frac{1}{1+(\frac{a}{z})^2}\right]^{\frac{3}{2}}$$
d) $$σ_z=q\left[1-\left[\frac{1}{1+(\frac{a}{z})^2}\right]^{\frac{5}{2}}\right]$$

Explanation: The Boussinesq’s vertical pressure σz under a uniformly loaded circular area is given by,
$$σ_z=q\left[1-\left[\frac{1}{1+(\frac{a}{z})^2}\right]^{\frac{3}{2}}\right]$$

where, q=load intensity per unit area
z= depth of point.

7. The Boussinesq influence factor for uniformly distributed circular area is given by ____________
a) $$K_B= \left[1-\left[\frac{1}{1+(\frac{a}{z})^2}\right]^{\frac{3}{2}}\right]$$
b) $$K_B= \left[1+\left[\frac{1}{1+(\frac{a}{z})^2}\right]^{\frac{3}{2}}\right]$$
c) $$K_B= \left[\frac{1}{1+(\frac{a}{z})^2}\right]^{\frac{3}{2}}$$
d) $$K_B= q\left[1-\left[\frac{1}{1+(\frac{a}{z})^2}\right]^{\frac{5}{2}}\right]$$

Explanation: The Boussinesq influence factor for uniformly distributed circular area is given by,
$$K_B= \left[1-\left[\frac{1}{1+(\frac{a}{z})^2}\right]^{\frac{3}{2}}\right]$$
where the KB= Boussinesq influence factor which is a function of r/z ratio which is a dimensionless factor.

8. If θ is the apex angle which the line joining the apex makes with the outer edge of the loading of a circular area, then the Boussinesq’s vertical pressure σz under a uniformly loaded circular area is given by ______________
a) σz=q[1-sin3θ]
b) σz=q[1-cos3θ]
c) σz=q[1-tan3θ]
d) σz=q[1-cos2θ]

Explanation: The Boussinesq’s vertical pressure σ_z under a uniformly loaded circular area is given by,
$$σ_z=q\left[1-\left[\frac{1}{1+(\frac{a}{z})^2}\right]^{\frac{3}{2}}\right].$$ If θ is the apex angle which the line joining the apex makes with the outer edge of the loading of a circular area, then the term,
$$\left[\frac{1}{1+(\frac{a}{z})^2}\right]^{\frac{3}{2}} = cos^3 θ$$
∴ σz=q[1-cos3θ].

9. The Boussinesq’s vertical pressure σz due to line load is given by ________
a) $$σ_z=\frac{5q’}{πz}\frac{1}{[1+\frac{x}{z}^2 ]^2}$$
b) $$σ_z=\frac{3q’}{πz}\frac{1}{[1+(\frac{x}{z})^2 ]^2}$$
c) $$σ_z=\frac{2q’}{πz}\frac{1}{[1+(\frac{x}{z})^2 ]^2}$$
d) $$σ_z=\frac{2q’}{z}\frac{1}{[1+⌊\frac{x}{z}⌋^2 ]^2}$$

Explanation: The Boussinesq’s vertical pressure σz due to line load is given by,
$$σ_z=\frac{2q’}{πz}\frac{1}{[1+(\frac{x}{z})^2 ]^2}$$
Where q’=line load intensity per unit length
X=horizontal distance from line load
Z= depth of point.

10. The Boussinesq’s vertical pressure σz due to line load at a point situated vertically below the line load is given by ________
a) $$σ_z=\frac{2q’}{πz}$$
b) $$σ_z=\frac{3q’}{πz}$$
c) $$σ_z=\frac{2q’}{πz}\frac{1}{[1+(z)^2 ]^2}$$
d) $$σ_z=\frac{2q’}{z}$$

Explanation: The Boussinesq’s vertical pressure σz due to line load is given by,
$$σ_z=\frac{2q’}{πz}\frac{1}{[1+(\frac{x}{z})^2 ]^2}$$ at a point situated vertically below the line load implies x=0
∴ $$σ_z=\frac{2q’}{πz}\frac{1}{[1+(\frac{0}{z})^2 ]^2}$$
∴ $$σ_z=\frac{2q’}{πz}.$$

11. If θ is the angle subtended by the edges of the strip load, then the Boussinesq’s vertical pressure σz due to strip load is given by ________
a) $$σ_z=\frac{q}{π}(θ+sinθ)$$
b) $$σ_z=\frac{q}{π}(θ-sinθ)$$
c) $$σ_z=\frac{q}{π}(sinθ)$$
d) $$σ_z=\frac{q}{π} θ$$

Explanation:

The vertical pressure due to elementary line load is given by,
$$∆σ_z=\frac{2q’}{πz}\frac{1}{[1+(\frac{0}{z})^2 ]^2}$$
When θ is the angle subtended by the edges of the strip load, the Boussinesq’s vertical pressure σz due to strip load is given by $$σ_z=\frac{q}{π}(θ+sinθ).$$

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