1. For finding out the active earth pressure for a backfill with sloping surface, the Rankine’s theory makes as additional assumption of ________
a) vertical and lateral stresses are normal to surcharge
b) vertical and lateral stresses are tangential to surcharge
c) vertical and lateral stresses are conjugate
d) vertical and lateral stresses are negligible
Explanation: The additional assumption of vertical and lateral stresses are conjugate is made as, it can be shown that stresses on a given plane at a given point is parallel to another plane, the stresses on the latter plane at the same point must be parallel to the first plane.
2. The vertical and the lateral pressures have the same angle of obliquity β.
a) True
b) False
Explanation: In Rankine’s theory, the additional assumption of the vertical and lateral stresses are conjugate is made. Being conjugate, both the vertical and the lateral pressures have the same angle of obliquity β.
3. The Rankine’s lateral pressure ratio is given by ________
a) \(K=\frac{\sqrt{cos^2 β-cos^2 φ}}{cosβ+\sqrt{cos^2 β-cos^2 φ}} \)
b) \(K=\frac{cosβ+\sqrt{cos^2 β-cos^2 φ}}{cosβ-\sqrt{cos^2 β-cos^2 φ}} \)
c) \(K=\frac{cosβ-\sqrt{cos^2 β-cos^2 φ}}{cosβ+\sqrt{cos^2 β-cos^2 φ}} \)
d) \(K=cosβ\frac{cosβ-\sqrt{cos^2 β-cos^2 φ}}{cosβ+\sqrt{cos^2 β-cos^2 φ}} \)
Explanation: The ratio K is the conjugate ratio or the Rankine’s lateral pressure ratio, which is given by,
\(K=\frac{cosβ-\sqrt{cos^2 β-cos^2 φ}}{cosβ+\sqrt{cos^2 β-cos^2 φ}}, \)
where, β=surcharge angle
φ=angle of internal friction.
4. For backfill with sloping surface, the coefficient of active earth pressure is given by ______
a) \(K_a=\frac{\sqrt{cos^2 β-cos^2 φ}}{cosβ+\sqrt{cos^2 β-cos^2 φ}} \)
b) \(K_a=\frac{cosβ+\sqrt{cos^2 β-cos^2 φ}}{cosβ-\sqrt{cos^2 β-cos^2 φ}} \)
c) \(K_a=\frac{cosβ-\sqrt{cos^2 β-cos^2 φ}}{cosβ+\sqrt{cos^2 β-cos^2 φ}} \)
d) \(K_a=cosβ\frac{cosβ-\sqrt{cos^2 β-cos^2 φ}}{cosβ+\sqrt{cos^2 β-cos^2 φ}} \)
Explanation: The vertical pressure in case of backfill with surcharge,
σ= γzcosβ,
Therefore, from the assumption of stresses being conjugate,
\(K_a=cosβ\frac{cosβ-\sqrt{cos^2 β-cos^2 φ}}{cosβ+\sqrt{cos^2 β-cos^2 φ}}. \)
5. When the surcharge angle reduces to zero, the coefficient of active earth pressure is given by
a) Ka=1
b) \(K_a=\frac{1-sinφ}{1+sinφ}\)
c) \(K_a=\frac{1+sinφ}{1-sinφ}\)
d) Ka=0
Explanation: For backfill with sloping surface, the coefficient of active earth pressure is given by,
\(K_a=cosβ\frac{cosβ-\sqrt{cos^2 β-cos^2 φ}}{cosβ+\sqrt{cos^2 β-cos^2 φ}}, \)
When the surcharge angle reduces to zero, β=0,
substituting this in the equation, we get,
\(K_a=\frac{1-sinφ}{1+sinφ}.\)
6. The principal stress relationship on a failure plane is given by _______
a) σ1=σ3 tan2 α
b) σ1=2c tanα-σ3 tan2 α
c) σ1=2c tanα+σ3
d) σ1=2c tanα+σ3 tan2 α
Explanation: The principal stress relationship on a failure plane is given by,
σ1=2c tanα+σ3 tan2 α,
Where, σ1=major principal stress
σ3=minor principal stress
\(α=(45°+\frac{φ}{2}), \)
φ=angle of internal friction
c=cohesion.
7. The Belli equation of lateral pressure of cohesive soil is ____________
a) pa=γzcot2 α-2c cotα
b) pa=γzcot2 α+2c cotα
c) pa=-2c cotα
d) pa=γzcot2 α/2c cotα
Explanation: Since the principal stress relationship on a failure plane is given by,
σ1=2c tanα+σ3 tan2 α,
σ1=γz and σ3=pa,
∴ γz=2c tanα+pa tan2 α
∴ pa=γzcot2 α-2c cotα.
8. The Belli equation at the ground surface is given by _________
a) pa=γzcot2 α-2c cotα
b) pa=γzcot2 α+2c cotα
c) pa=-2c cotα
d) pa=γzcot2 α/2c cotα
Explanation: Since the Belli equation of lateral pressure of cohesive soil is given by,
pa=γzcot2 α-2c cotα,
at the ground surface, z=0,
∴ pa=-2c cotα.
9. The tension at the top level of retaining wall reduces to zero at a depth ___________
a) \(z_0=\frac{q-2c cotα}{γ} \)
b) \(z_0=\frac{2c tanα}{γ} \)
c) \(z_0=\frac{2 cotα}{γ} \)
d) \(z_0=\frac{cotα}{γ} \)
Explanation: Since the Belli equation of lateral pressure of cohesive soil is given by,
pa=γzcot2 α-2c cotα,
When pa=0,
∴ \(z_0=\frac{2c tanα}{γ} \)
10. The depth at which the tension is zero for cohesive soils with retaining wall in terms Ka is _____
a) \(z_0=\frac{2cK_a}{γ} \)
b) \(z_0=\frac{2c}{γ} \frac{1}{√K_a } \)
c) \(z_0=\frac{2cK_a}{γ} \)
d) \(z_0=\frac{2c}{γ}(\frac{1}{K_a}) \)
Explanation: The tension at the top level of retaining wall reduces to zero at a depth,
\(z_0=\frac{2c tanα}{γ}, \)
Since, \(K_a=\frac{1}{tan^2 (45°+\frac{φ}{2})}, \)
∴ \(z_0=\frac{2c}{γ} \frac{1}{√K_a}. \)