1. The plastic equilibrium stress equation in terms of flow valve is _____
a) σ1=2c∛Nφ+σ3 Nφ
b) σ1=2c√Nφ+σ3 Nφ
c) σ1=2cNφ-σ3 Nφ
d) σ1=Nφ+σ3 Nφ
Explanation: The stress condition during plastic equilibrium is given by,
\(σ_1=2c \,tan(45°+\frac{φ}{2})+σ_3 tan^2 (45°+\frac{φ}{2}),\)
since, \(N_φ= tan^2 (45°+\frac{φ}{2}),\)
∴ σ1=2c√Nφ+σ3 Nφ.
2. The Mohr Coulomb equation in terms of stress components in x-z plane is ______________
a) \(\sqrt{((\frac{σ_z-σ_x}{2})^2+τ_{xz}^2))} -\frac{σ_z+σ_x}{2} sinφ=c cosφ\)
b) \(\sqrt{((\frac{σ_z-σ_x}{2})^2+τ_{xz}^2))} \frac{σ_z+σ_x}{2} sinφ=c cosφ\)
c) \(\sqrt{((\frac{σ_z-σ_x}{2})^2+τ_{xz}^2))} = c cosφ\)
d) \(\sqrt{((\frac{σ_z-σ_x}{2})^2+τ_{xz}^2))} -\frac{σ_z+σ_x}{2} sinφ=cosφ\)
Explanation: The Mohr Coulomb equation is given by,
\(\frac{σ_1-σ_3}{2}-\frac{σ_1+σ_3}{2} sinφ=c cosφ,\)
Where, σ_1=major principal stress and σ3=minor principal stress.
∴ in terms of stress components in x-z plane it is,
\(\sqrt{((\frac{σ_z-σ_x}{2})^2+τ_{xz}^2))} -\frac{σ_z+σ_x}{2} sinφ=c cosφ,\)
where, σz=stress in z-direction
σx= stress in x-direction.
3. Combing the three equations, \(\sqrt{((\frac{σ_z-σ_x}{2})^2+τ_{xz}^2))} -\frac{σ_z+σ_x}{2} sinφ=c cosφ\), \(\frac{∂σ_x}{∂x}+\frac{∂τ_{xz}}{∂z}=0\), and \(\frac{∂τ_{xz}}{∂x}+\frac{∂σ_x}{∂z}+γ=0\), lead to the equation called ___________
a) Kolter’s equation
b) Terzaghi’s equation
c) Darcy’s equation
d) Skempton’s equation
Explanation: The material just in the verge of flowing plastically is still in static equilibrium and will satisfy the equations of, \(\frac{∂σ_x}{∂x}+\frac{∂τ_{xz}}{∂z}=0, \,and\, \frac{∂τ_{xz}}{∂x}+\frac{∂σ_x}{∂z}+γ=0.\) These equations with the Mohr Coulomb equation give the Kolter’s equation.
4. The solution of Kolter’s equation gives ___________
a) permeability of soil
b) orientation of slip lines
c) specific gravity of grains
d) seepage pressure
Explanation: The solution of Kolter’s equation for a given boundary condition gives the orientation of slip planes together with the stress at each point at the failure zone.
5. In active state, the major principal stress σ1 is ____________
a) horizontal direction
b) vertical direction
c) can be both vertical and horizontal direction
d) in no direction
Explanation: In active state, the major principal stress denoted by σ1 is in vertical direction. While in the passive state, the major principal stress σ1 is in horizontal direction.
6. The coefficient of active earth pressure is_______ than the coefficient of passive pressure.
a) less than
b) greater than
c) equal to
d) insufficient data
Explanation: The coefficient of active earth pressure is less than that of the coefficient of passive earth pressure. The coefficient of active earth pressure is also less than that of the coefficient of earth pressure at rest.
7. The coefficient earth pressure at rest is _______________
a) less than coefficient of active pressure
b) greater than coefficient of active pressure
c) equal to coefficient of active pressure
d) one
Explanation: The coefficient earth pressure at rest is greater than that of the coefficient of active earth pressure. But, he coefficient earth pressure at rest is less than the coefficient of passive earth pressure.
8. For earth pressure at rest, there will be no ______
a) vertical stress
b) shear stress
c) horizontal stress
d) both vertical and horizontal stress
Explanation: For earth pressure at rest, there is horizontal stress as well as the vertical stress. For earth pressure at rest, there will be no shear stresses.
9. The lateral strain in the horizontal direction is _______
a) \(∈_h=\frac{1}{E} [σ_h-μ(σ_v)] \)
b) \(∈_h=\frac{1}{E} [σ_h-μ(σ_h-σ_v)]\)
c) \(∈_h=\frac{1}{E} [σ_h-μ(σ_h+σ_v)]\)
d) \(∈_h=\frac{1}{E} [σ_h-(σ_h-σ_v)]\)
Explanation: The lateral strain ∈h in the horizontal direction is given by,
\(∈_h=\frac{1}{E} [σ_h-μ(σ_h-σ_v )],\)
where, σh =horizontal stress
σv =vertical stress
E=elastic modulus
μ= Poisson’s ratio.
10. The lateral earth pressure at rest with respect to Poisson’s ratio is _______
a) σh=2μ(σh-σv)
b) σh=μ(σh-σv)
c) σh=μ(σh+σv)
d) σh=-μ(σh-σv)
Explanation: The earth pressure at rest has the condition corresponding to ∈h=0.
Since, \(∈_h=\frac{1}{E} [σ_h-μ(σ_h-σ_v)],\)
∴ σh=μ(σh+σv).