# Soil Mechanics Questions and Answers – Stress Distribution – Boussinesq Equations

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

1. The problems due to stress distribution in soils due to a concentrated load was studied by _____________
a) G.B Airy
b) Terzaghi
c) Darcy
d) Boussinesq

Explanation: Boussinesq in 1885 studied and solved the problems of stress distribution in soils due to a concentrated loads acting at the ground surface. Darcy gave the law of flow of water through soils. The stress function was introduced by G.B Airy in 1862.

2. The assumption made by Boussinesq in the solutions is by the ____________
a) theory of plasticity
b) theory of elasticity
c) yield point
d) failure point

Explanation: Boussinesq in 1885 studied and solved the problems of stress distribution in soils due to a concentrated loads acting at the ground surface, by assuming a suitable stress function. The assumptions made are based on theory of elasticity.

3. The assumption of Boussinesq equation is that the soil is ______________
a) elastic
b) semi-elastic
c) plastic
d) semi-plastic

Explanation: Boussinesq solved the problems of stress distribution in soils due to a concentrated loads acting at the ground surface. The assumptions made are based on theory of elasticity. Therefore, the soil is considered to be elastic.

4. The assumption of Boussinesq equation is that the soil is ______________
a) non-homogeneous
b) homogeneous
c) plastic
d) semi-plastic

Explanation: Boussinesq solved the problems of stress distribution in soils due to a concentrated loads acting at the ground surface. The assumptions made are based on theory of elasticity. Therefore, the soil is considered to be elastic and of homogeneous nature.

5. The assumption of Boussinesq equation is that the soil is ______________
a) semi-infinite
b) infinite
c) finite
d) semi- finite

Explanation: Semi-infinite condition is when one of the dimension extends to infinity. If XY pane is considered to be ground surface and the z-axis as depth, then this condition is known as semi-infinite.
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6. The Boussinesq equation representing the polar radial stress is ___________
a) $$σ_R=\frac{3Q}{2} \frac{cos⁡β}{R^2}$$
b) $$σ_R=\frac{3Q}{2π} \frac{cos⁡β}{R^2}$$
c) $$σ_R=\frac{3Q}{2π} \frac{cos⁡β}{R}$$
d) $$σ_R=\frac{3Q}{2π} \frac{cos⁡β}{R^3}$$

Explanation: Boussinesq showed that the polar radial stress is given by,
$$σ_R=\frac{3Q}{2π} \frac{cos⁡β}{R^2}$$ where σR is the polar radial stress
$$cos⁡β=\frac{z}{R}$$
R is the polar radial coordinate=√(r2+z2).

7. The Boussinesq equation representing the vertical stress is ___________
a) $$σ_z=\frac{3}{2π} \left[\frac{1}{1+(\frac{r}{z})^2} \right]^{5/2}$$
b) $$σ_z=\frac{3Q}{2π} \left[\frac{1}{1+(\frac{r}{z})^2} \right]^5$$
c) $$σ_z=\frac{3Q}{2πz^2}\left[\frac{1}{1+(\frac{r}{z})^2} \right]^{\frac{5}{2}}$$
d) $$σ_z=\frac{3Q}{2π} \left[\frac{1}{1+(\frac{r}{z})^2} \right]^2$$

Explanation: Boussinesq showed that the polar radial stress is given by,
$$σ_R=\frac{3Q}{2π} \frac{cos⁡β}{R^2}$$
Boussinesq’s vertical stress σz is given by,
σzRcos2 β
∴ $$σ_z=\frac{3Q}{2πz^2}\left[\frac{1}{1+(\frac{r}{z})^2} \right]^{\frac{5}{2}}$$ where, σz is the vertical stress
Q is the point load acting at the ground surface
r is the radial horizontal distance
z is the vertical distance.

8. The Boussinesq equation representing the tangential stress is ___________
a) $$τ_{rz}=\frac{3}{2πz^3} \left[\frac{1}{1+(\frac{r}{z})^2} \right]^{5/2}$$
b) $$τ_{rz}=\frac{3Qr}{2π} \left[\frac{1}{1+(\frac{r}{z})^2} \right]^5$$
c) $$τ_{rz}=\frac{3Qr}{2πz^3} \left[\frac{1}{1+(\frac{r}{z})^2} \right]^{\frac{5}{2}}$$
d) $$τ_{rz}=\frac{3Q}{2πz^3} \left[\frac{1}{1+(\frac{r}{z})^2} \right]^2$$

Explanation: Boussinesq showed that the polar radial stress is given by,
$$σ_R=\frac{3Q}{2π} \frac{cos⁡β}{R^2}$$
Boussinesq’s tangential stress σz is given by,
$$τ_{rz}=\frac{1}{2} σ_R sin2β$$
∴ $$τ_{rz}=\frac{3Qr}{2πz^3} \left[\frac{1}{1+(\frac{r}{z})^2} \right]^{5/2}$$ where, τrz is the tangential stress
Q is the point load acting at the ground surface
r is the radial horizontal distance
z is the vertical distance.

9. The Boussinesq influence factor is given by ____________
a) $$K_B=\frac{3Q}{2πz} \left[\frac{1}{1+(\frac{r}{z})^2} \right]^{5/2}$$
b) $$K_B=\frac{3Q}{2π} \left[\frac{1}{1+(\frac{r}{z})^2} \right]^{\frac{5}{2}}$$
c) $$K_B=\frac{3}{2π} \left[\frac{1}{1+(\frac{r}{z})^2} \right]^{\frac{5}{2}}$$
d) $$K_B=\frac{3}{2πz} \left[\frac{1}{1+(\frac{r}{z})^2} \right]^{\frac{5}{2}}$$

Explanation: The Boussinesq influence factor is given by,
$$K_B=\frac{3}{2π} \left[\frac{1}{1+(\frac{r}{z})^2} \right]^{\frac{5}{2}}$$ where the KB= Boussinesq influence factor which is a function of r/z ratio which is a dimensionless factor.

10. The intensities of pressure below a point load where r=0 on axis of loading is ____________
a) $$σ_z=\frac{0.4775Q}{z^2}$$
b) $$σ_z=\frac{0.7Q}{z^2}$$
c) $$σ_z=\frac{0.4775Q}{z^3}$$
d) $$σ_z=\frac{0.8Q}{z}$$

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 r=0,
We get,
$$σ_z=\frac{0.4775Q}{z^2}.$$

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