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# Terzaghi’s Theory of One Dimensional Consolidation – 3

If v is velocity at the entry of the soil element of size dx, dz and width dy as shown in the figure. The quantity of water entering the element is ___________

a) v

b) vdxdy

c) vdzdy

d) vdzdx

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Answer: b

Explanation: Since the quantity of water q is equal to velocity of flow time the area of cross-section through which it flows,

∴ q=v*A. The area of element is dxdy.

∴ The quantity of water entering the element is vdxdy.

If v is velocity at the entry of the soil element of size dx, dz and width dy as shown in the figure. The quantity of water leaving the element is ___________

a) v

b) vdxdy

c) (v+∂v∂zdz)dxdy

d) (v+∂v∂zdz)dxdz

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Answer: c

Explanation: Since the quantity of water q is equal to velocity of flow time the area of cross-section through which it flows,

∴ q=v*A. The area of element is dxdy. The velocity leaving the element is (v+∂v∂zdz). Therefore, the quantity of water leaving the element is (v+∂v∂zdz)dxdy.

If v is velocity at the entry of the soil element of size dx, dz and width dy as shown in the figure. The net quantity of water squeezed out of the element per unit time is ___________

a) ∆q=v

b) ∆q=vdxdy

c) Δq=∂v∂zvdzdy

d) Δq=∂v∂zdxdydz

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Answer: d

Explanation: The net quantity of water squeezed out of the element per unit time is given by,

Δq=qout−qin=(v+∂v∂zdz)dxdy−vdxdy

∴ Δq=∂v∂zdxdydz.

Decrease in the volume of the soil is equal to _________

a) volume of water squeeze out

b) volume of water entering the soil element

c) volume of water leaving the soil element

d) volume of soil in wet state

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Answer: a

Explanation: Since the water occupies the voids between the soil particles and when the force is applied, the water is squeezed out and there is decrease in volume of soil. Therefore, decrease in the volume of the soil is equal to the volume of water squeeze out.

The decrease in volume is given by the equation ___________

a) ∆v=-V_{0} ∆σ’

b) ∆v=-a_{v} V_{0} ∆σ’

c) ∆v=-m_{v} ∆σ’

d) ∆v=-m_{v} V_{0} ∆σ’

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Answer: d

Explanation: The decrease in the volume of the soil is equal to the volume of water squeeze out. Thus, the decrease in volume is given by the equation,

∆v=-m_{v} V_{0} ∆σ’ where the negative sign indicates the decrease in volume.

The change of volume per unit time is __________

a) ∂ΔV∂z=−V0Δσ′

b) ∂ΔV∂z=−avV0Δσ′

c) ∂ΔV∂z=−mv∂Δσ′∂t

d) ∂ΔV∂z=−mvdxdydz∂Δσ′∂t

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Explanation: Since the decrease in volume is given by the equation,

∆v=-m

_{v}V

_{0}∆σ’ where V

_{0}=dxdydz,

differentiating the equation, we get,

∂ΔV∂z=−mvdxdydz∂Δσ′∂t where, m

_{v}is the coefficient of volume change.

The partial differentiation of velocity of water in soil element with respect to depth z in terms of coefficient of volume change is given by ____________

a) ∂v∂z=mv∂2u¯¯¯∂t2

b) ∂v∂z=mvu¯¯¯∂t2

c) ∂v∂z=∂2u¯¯¯∂t2

d) ∂v∂z=mv∂u¯¯¯∂t

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Answer: d

Explanation: From the volume of water squeezed out, ∆q=∂v∂z dxdydz and change of volume per unit time, ∂ΔV∂z=−mvdxdydz∂Δσ′∂t

we get, ∂v∂z=−mv∂Δσ′∂t —————–(1)

since, ∆σ=∆σ’+u, partally differentiating it with respect to time t,

∂Δσ′∂t=−∂u¯¯¯∂t ————————(2)

∴ from (1) and (2),

∂v∂z=mv∂u¯¯¯∂t.

The term ∂u¯¯¯∂t in terms of coefficient of volume change and coefficient of permeability is given by ____________

a) ∂u¯¯¯∂t=kmvγw∂u¯¯¯∂z

b) ∂u¯¯¯∂t=kmvγw∂u¯¯¯∂z

c) ∂u¯¯¯∂t=kmvγw∂2u¯¯¯∂z2

d) ∂u¯¯¯∂t=kmvγw∂u¯¯¯∂z

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Answer: c

Explanation: The decrease in volume is given by,

∆v=-m_{v} V_{0} ∆σ’—————–(1)

also ∂v∂z=mv∂u¯¯¯∂t ——————————-(2)

∴ from (1) and (2), we get,

∂u¯¯¯∂t=kmvγw∂2u¯¯¯∂z2.

The coefficient of consolidation is given by ___________

a) Cv=qmvγw

b) Cv=1mvγw

c) Cv=kmv

d) Cv=kmvγw

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Answer: d

Explanation: The coefficient of consolidation is given by,

Cv=kmvγw

where, k=coefficient of permeability

m_{v}=coefficient of volume change

γ_{w}=unit weight of water.

The coefficient of consolidation in terms of voids ratio e and coefficient of compressibility is ____________

a) Cv=k(1+e0)avγw

b) Cv=k(1−e0)avγw

c) Cv=k(e0)avγw

d) Cv=(1+e0)avγw

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Answer: a

Explanation: Since we know the coefficient of volume change mv=av1+e0

and the coefficient of consolidation is given by,

Cv=kmvγw. Therefore, combining the two equations, we get,

Cv=k(1+e0)avγw.

The term coefficient of consolidation C_{v} used in equation is adopted to indicate the effect of permeability only.

a) True

b) False

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Answer: b

Explanation: The term coefficient of consolidation C_{v} used in equation is adopted to indicate the combined effects of the permeability and the compressibility of the soil on the rate of volume change.

The units of C_{v} is ___________

a) cm/s

b) cm

c) s

d) cm^{2}/s

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Answer: d

Explanation: The coefficient of consolidation C_{v} is given by,

Cv=kmvγw

where, k is in cm/s

m_{v} in cm^{2}/kN

and γ_{w} in kN/cm^{3}.

combining the units of these in the formula,

The units of C_{v} is cm^{2}/s.