250+ TOP MCQs on Stress Distribution – Vertical and Horizontal Pressure and Answers

Soil Mechanics Multiple Choice Questions on “Stress Distribution – Vertical and Horizontal Pressure”.

1. In simple radial distribution, the three stress components σ_r, σ_θ and τ_rθ are given by ___________

a) (σ_r=K frac{Q cos⁡θ}{r}, σ_θ=0 ,and, τ_{rθ}=0 )
b) σ_r=KQ, σ_θ=0 and τ_rθ=0
c) (σ_r=frac{Q cos⁡θ}{r}, σ_θ=0 ,and, τ_{rθ}=0)
d) σ_r=0, σ_θ=0 and τ_rθ= 0
Answer: a
Clarification: At any radial distance r and polar angle θ, Mitchell found that the three stress components σ_r, σ_θ and τ_rθ are given by,
(σ_r=K frac{Q cos⁡θ}{r}, σ_θ=0 ,and, τ_{rθ}=0. )
Where K is a constant to be found by boundary conditions. The above solution is valid only if it satisfies the equilibrium equations and the compatibility equation.

2. The equilibrium equation in polar coordinates is given by _____________
a) (frac{1}{r} frac{∂τ_{rθ}}{∂θ}+frac{σ_r-σ_θ}{r}=0)
b) (frac{∂σ_r}{∂r}+frac{∂τ_{rθ}}{∂θ}+frac{σ_r-σ_θ}{r}=0)
c) (frac{∂σ_r}{∂r}+frac{1}{r} frac{∂τ_{rθ}}{∂θ}+frac{σ_r-σ_θ}{r}=0)
d) (frac{∂σ_r}{∂r}+frac{1}{r} frac{∂τ_{rθ}}{∂θ}=0)
Answer: c
Clarification: The equilibrium equations in polar coordinates are given by,
1. (frac{∂σ_r}{∂r}+frac{1}{r} frac{∂τ_{rθ}}{∂θ}+frac{σ_r-σ_θ}{r}=0 )
2. (frac{1}{r}frac{∂σ_θ}{∂θ}+frac{∂τ_{rθ}}{∂r}+frac{2τ_{rθ}}{r}=0.)

3. The equilibrium equation in polar coordinates is given by _____________
a) (frac{1}{r} frac{∂τ_{rθ}}{∂θ}+frac{σ_r-σ_θ}{r}=0)
b) (frac{∂σ_r}{∂r}+frac{∂τ_{rθ}}{∂θ}+frac{σ_r-σ_θ}{r}=0)
c) (frac{∂σ_r}{∂r}+frac{1}{r} frac{∂τ_{rθ}}{∂θ}+frac{σ_r-σ_θ}{r}=0)
d) (frac{∂σ_r}{∂r}+frac{1}{r} frac{∂τ_{rθ}}{∂θ}=0)
Answer: c
Clarification: The equilibrium equations in polar coordinates are given by,
1. (frac{∂σ_r}{∂r}+frac{1}{r} frac{∂τ_{rθ}}{∂θ}+frac{σ_r-σ_θ}{r}=0 )
2. (frac{1}{r}frac{∂σ_θ}{∂θ}+frac{∂τ_{rθ}}{∂r}+frac{2τ_{rθ}}{r}=0.)

4. The compatibility equation in terms of stress components in polar coordinates are given by ____________
a) ((frac{∂^2}{∂r^2} +frac{1}{r} frac{∂}{∂r}+frac{1}{r^2} frac{∂^2}{∂θ^2} )(σ_r+σ_θ )=0)
b) ((frac{∂^2}{∂r^2} +frac{1}{r} frac{∂}{∂r}+frac{1}{r^2} frac{∂^2}{∂θ^2} )(σ_θ )=0)
c) ((frac{∂^2}{∂r^2} +frac{1}{r} frac{∂}{∂r}+frac{1}{r^2} frac{∂^2}{∂θ^2} )(σ_r )=0)
d) ((frac{∂^2}{∂r^2} +frac{1}{r} frac{∂}{∂r}+frac{1}{r^2} frac{∂^2}{∂θ^2} )(σ_r+σ_θ )=1)
Answer: a
Clarification: The compatibility equation is the additional equation to solve the stress problem. The compatibility equation in terms of stress components in polar coordinates are given by,
((frac{∂^2}{∂r^2} +frac{1}{r} frac{∂}{∂r}+frac{1}{r^2} frac{∂^2}{∂θ^2} )(σ_r+σ_θ )=0.)

5. In simple radial distribution, if (σ_r=K frac{Q cos⁡θ}{r},) then the value of K is ________

a) K=(frac{2}{2α+sin2α})
b) K=2α+sinα
c) K=2α-sinα
d) K=sinα
Answer: a
Clarification: Considering the equilibrium of the wedge aob,
We have (KQ(α+frac{1}{2} sin2α)=Q)
∴ K=(frac{2}{2α+sin2α}.)

6. When the ground is horizontal, (α=frac{π}{2}) in constant K. What will be the radial stress σ_r due to vertical line load?

a) (σ_r=frac{Q cos⁡θ}{r})
b) (σ_r=frac{2Q cos⁡θ}{πr})
c) (σ_r=frac{Q sin⁡θ}{r})
d) (σ_r=frac{2Q sin⁡θ}{r})
Answer: b
Clarification: At any radial distance r and polar angle θ, Mitchell found that the radial stress component σ_r is given by,
(σ_r=Kfrac{Q cos⁡θ}{r} ,where, K=frac{2}{2α+sin2α})
When the ground is horizontal =(frac{π}{2},)
∴ (σ_r=frac{2Q cos⁡θ}{πr}.)

7. The relation between the stress component in x-direction on a horizontal plane in Cartesian coordinates and polar coordinates for vertical line load is ___________
a) σ_x=σ_r tan^2⁡θ
b) σ_x=σ_r cosec^2⁡θ
c) σ_x=σ_r cos⁡θ
d) σ_x=σ_r sin^2⁡θ
Answer: d
Clarification: From the figure,

On a horizontal plane, the relation between the stress component in x-direction in Cartesian coordinates and polar coordinates is,
σ_x=σ_r sin^2⁡θ.