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# Stress Distribution – Boussinesq Equations

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

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Answer: d

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.

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

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Answer: b

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.

The assumption of Boussinesq equation is that the soil is ______________

a) elastic

b) semi-elastic

c) plastic

d) semi-plastic

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Answer: a

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.

The assumption of Boussinesq equation is that the soil is ______________

a) non-homogeneous

b) homogeneous

c) plastic

d) semi-plastic

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Answer: b

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.

The assumption of Boussinesq equation is that the soil is ______________

a) semi-infinite

b) infinite

c) finite

d) semi- finite

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Answer: a

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.

The Boussinesq equation representing the polar radial stress is ___________

a) σR=3Q2cosβR2

b) σR=3Q2πcosβR2

c) σR=3Q2πcosβR

d) σR=3Q2πcosβR3

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Answer: b

Explanation: Boussinesq showed that the polar radial stress is given by,

σR=3Q2πcosβR2 where σ_{R} is the polar radial stress

cosβ=zR

R is the polar radial coordinate=√(r^{2}+z^{2}).

The Boussinesq equation representing the vertical stress is ___________

a) σz=32π[11+(rz)2]5/2

b) σz=3Q2π[11+(rz)2]5

c) σz=3Q2πz2[11+(rz)2]52

d) σz=3Q2π[11+(rz)2]2

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Answer: c

Explanation: Boussinesq showed that the polar radial stress is given by,

σR=3Q2πcosβR2

Boussinesq’s vertical stress σ_{z} is given by,

σ_{z}=σ_{R}cos^{2} β

∴ σz=3Q2πz2[11+(rz)2]52 where, σ_{z} 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.

The Boussinesq equation representing the tangential stress is ___________

a) τrz=32πz3[11+(rz)2]5/2

b) τrz=3Qr2π[11+(rz)2]5

c) τrz=3Qr2πz3[11+(rz)2]52

d) τrz=3Q2πz3[11+(rz)2]2

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Answer: c

Explanation: Boussinesq showed that the polar radial stress is given by,

σR=3Q2πcosβR2

Boussinesq’s tangential stress σ_{z} is given by,

τrz=12σRsin2β

∴ τrz=3Qr2πz3[11+(rz)2]5/2 where, τ_{rz} 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.

The Boussinesq influence factor is given by ____________

a) KB=3Q2πz[11+(rz)2]5/2

b) KB=3Q2π[11+(rz)2]52

c) KB=32π[11+(rz)2]52

d) KB=32πz[11+(rz)2]52

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Answer: c

Explanation: The Boussinesq influence factor is given by,

KB=32π[11+(rz)2]52 where the K_{B}= Boussinesq influence factor which is a function of r/z ratio which is a dimensionless factor.

The intensities of pressure below a point load where r=0 on axis of loading is ____________

a) σz=0.4775Qz2

b) σz=0.7Qz2

c) σz=0.4775Qz3

d) σz=0.8Qz

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Answer: a

Explanation: Boussinesq’s vertical stress σ_{z} is given by,

σz=3Q2πz2[11+(rz)2]52

Substituting r=0,

We get,

σz=0.4775Qz2.