# Soil Mechanics Questions and Answers – Stress Distribution – Pressure Distribution Diagrams

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

1. ____________ is not the vertical pressure distribution diagram, which can be prepared by Boussinesq’s theory.
a) stress isobars
b) vertical pressure distribution on a horizontal plane
c) horizontal pressure distribution on a horizontal plane
d) vertical pressure distribution on a vertical plane

Explanation: By means of Boussinesq’s stress distribution theory, the following vertical pressure distribution diagrams can be prepared,

• stress isobars
• vertical pressure distribution on a horizontal plane
• vertical pressure distribution on a vertical plane.

2. An isobar is a curve connecting all points of _______ below the ground.
a) equal vertical pressure
b) unequal vertical pressure
c) equal horizontal pressure
d) unequal horizontal pressure

Explanation: An isobar is a curve or contour connecting all points below the ground surface or ground level of equal vertical pressure. In an isobar, at each and every point the vertical pressure is same.

3. An isobar is a curved surface of the shape of _________
a) circular
b) rectangle
c) bulb
d) hexagon

Explanation: An isobar is a spatial, curved surface of the shape of bulb. This is because the vertical pressure on a given horizontal plane is the same in all directions at all points located at equal radial distances around the axis of loading.
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4. The zone of soil in isobar is called __________
a) stress diagram
b) contour
c) pressure bulb
d) isotherm

Explanation: The zone in a loaded soil mass bounded by an isobar of given vertical pressure intensity is called pressure bulb. In any point inside the pressure bulb, the vertical pressure in a horizontal plane is same.

5. An isobar diagram consists of __________
a) family of isobars of various intensities
b) single isobar only
c) two isobars only
d) isobars of same intensities

Explanation: An isobar is a spatial, curved surface of the shape of bulb. This is because the vertical pressure on a given horizontal plane is the same in all directions at all points located at equal radial distances around the axis of loading. Considering parallel horizontal planes, there will be different vertical pressures. Therefore, An isobar diagram consists of family of isobars of various intensities.
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6. The vertical pressure distribution on any horizontal plane at a depth z below the ground due to concentrated load is ___________
a) $$σ_z=K_B \frac{Q}{z}$$
b) $$σ_z=K_B \frac{Q}{z^2}$$
c) $$σ_z=K_B \frac{z}{Q}$$
d) $$σ_z=K_B \frac{Q}{z^3}$$

Explanation: The Boussinesq’s vertical stress σz is given by,
$$σ_z=\frac{3Q}{2πz^2} \left[\frac{1}{1+(\frac{r}{z})^2}\right]^{\frac{5}{2}}$$
Representing $$K_B \,as\, \frac{3}{2π} \left[\frac{1}{1+(\frac{r}{z})^2}\right]^{\frac{5}{2}}$$
∴ we get,
$$σ_z=K_B \frac{Q}{z^2}.$$

7. If the r/z ratio is 0.5, then the vertical pressure on a horizontal plane is given by _________
a) $$σ_z=\frac{0.56Q}{z^3}$$
b) $$σ_z=\frac{0.4775Q}{z^3}$$
c) $$σ_z=\frac{0.2733Q}{z^2}$$
d) $$σ_z=\frac{0.4775Q}{z^2}$$

Explanation: Boussinesq’s vertical stress σz is given by,
$$σ_z=\frac{3Q}{2πz^2} \frac{3}{2π} \left[\frac{1}{1+(\frac{r}{z})^2}\right]^{\frac{5}{2}}$$
Substituting r/z=0.5,
We get,
$$σ_z=\frac{0.2733Q}{z^2}.$$

8. If the r/z ratio is unity, then the vertical pressure on a horizontal plane is given by _________
a) $$σ_z=\frac{0.56Q}{z^3}$$
b) $$σ_z=\frac{0.0844Q}{z^2}$$
c) $$σ_z=\frac{0.2733Q}{z^2}$$
d) $$σ_z=\frac{0.4775Q}{z^2}$$

Explanation: Boussinesq’s vertical stress σz is given by,
$$σ_z=\frac{3Q}{2πz^2} \left[\frac{1}{1+(\frac{r}{z})^2}\right]^{\frac{5}{2}}$$
Substituting $$\frac{r}{z}=1,$$
We get,
$$σ_z=\frac{0.0844Q}{z^2}.$$

9. If the r/z ratio is 2, then the vertical pressure on a horizontal plane is given by _________
a) $$σ_z=\frac{0.0085Q}{z^2}$$
b) $$σ_z=\frac{0.0844Q}{z^2}$$
c) $$σ_z=\frac{0.2733Q}{z^2}$$
d) $$σ_z=\frac{0.4775Q}{z^2}$$

Explanation: Boussinesq’s vertical stress σz is given by,
$$σ_z=\frac{3Q}{2πz^2} \left[\frac{1}{1+(\frac{r}{z})^2}\right]^{\frac{5}{2}}$$
Substituting $$\frac{r}{z}=2,$$
We get,
$$σ_z=\frac{0.0085Q}{z^2}.$$

10. The vertical stress distribution diagram on a horizontal plane due to a concentrated load is known as the influence diagram if the load is ______
a) zero
b) unity
c) two units
d) three units

Explanation: The influence diagram is a representation of vertical stress distribution diagram on a horizontal plane due to a concentrated load, at a depth z when it is plotted for a unit load Q=1.

11. The vertical stress is proportional to __________
a) $$\frac{z^2}{K_B}$$
b) $$\frac{K_B}{z^2}$$
c) $$\frac{K_B}{z^3}$$
d) $$\frac{z^3}{K_B}$$

Explanation: The Boussinesq’s vertical stress σz is given by,
$$σ_z=\frac{3Q}{2πz^2} \left[\frac{1}{1+(\frac{r}{z})^2}\right]^{\frac{5}{2}}$$
Representing $$K_B \,as\, \frac{3}{2π} \left[\frac{1}{1+(\frac{r}{z})^2}\right]^{\frac{5}{2}}$$
∴ we get,
$$σ_z=K_B \frac{Q}{z^2},$$
∴ $$σ_z∝\frac{K_B}{z^2}$$.

12. The maximum value of σz on vertical line is obtained at the point of intersection of vertical plane with radial line at the angle of ____________
a) 39°30’
b) 39°45’
c) 35°15’
d) 39°15’

Explanation: The vertical stress first increases, attains a maximum value and then decreases. It can be shown that the maximum value of σz on vertical line is obtained at the point of intersection of vertical plane with radial line at the angle of39°15’.

13. The maximum value of σz on vertical line is ____________
a) $$(σ_z)_{max}=\frac{0.0888Q}{r^2}$$
b) $$(σ_z)_{max}=\frac{0.0844Q}{z^4}$$
c) $$(σ_z)_{max}=\frac{0.2733Q}{z^3}$$
d) $$(σ_z)_{max}=\frac{0.4775Q}{z^3}$$

Explanation: The maximum value of σz on vertical line is obtained at the point of intersection of vertical plane with radial line at the angle of39°15’. The corresponding value of $$\frac{r}{z}$$ is 0.817.
∴ $$z=\frac{r}{0.817}=1.225r$$
KB=0.1332
$$σ_z=K_B \frac{Q}{z^2}$$
$$σ_z=0.1332 \frac{Q}{1.225r^2}$$
∴ $$(σ_z)_{max}=\frac{0.0888Q}{r^2}$$

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