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**Compressibility And Consolidation**

**COEFFICIENT OF COMPRESSIBILITY (a _{v})**

e

e

ΔV = Change in volume in m^{3}, or cm^{3}.

V_{0} = Initial volume in m^{3} or cm^{3}.

ΔH = Change in depth in ‘m’ or ‘cm’.

H_{0} = original depth in ‘m’ or ‘cm’.

Remember: Consolidation settlement is a function of effective stress and not the function of total stress.

**COEFFICIENT OF COMPRESSION (C _{c}) **

(a)

(**b)** **C _{C} = 0.009 (W_{L}- 10)** For undisturbed clays of low to medium senstivity W

**(c) C _{C} = 0.007 ( W_{L}- 7)** For remoulded soil of low sensitivity

**(d) C _{C} = 0.40 (e_{0}- 0.25)** For undisturbed soil of medium sensitivity e

(e) For remoulded soil of low sensitivity.** C _{C} =1.15(e_{0}- 0.35) **

**(f) C =0.115 w** where, w = Water content

**OVER CONSOLIDATION RATIO**

**O.C.R >1**

For over consolidated soil.**O.C.R = 1**

For normally consolidated soil.**O.C.R < 1**

For under consolidated clay

**DIFFERENTIAL EQUATION OF 1-D CONSOLIDATION**

where,

u = Excess pore pressure,

= Rate of change of pore pressure

C_{V} = Coefficient of consolidation

= Rate of change of pore pressure with depth.

**COEFFICIENT OF VOLUME COMPRESSIBILITY:**

where, e_{0} = Initial void ratio

m_{v} = Coefficient of volume compressibility

**COMPRESSION MODULUS:**

where, E_{c} = Compression modulus.

**DEGREE OF CONSOLIDATION (i)**

where, % U = % degree of consolidation.

U = Excess pore pressure at any stage.

U_{i} = = Initial excess pore pressureat

At t = 0, U = u_{1} ⇒ %u= 0%

at t = ∞ , u = 0 ⇒ %u =100%**(ii)**

where, e_{f} = Void ratio at 100% consolidation

i.e., of t =∞

e = Void ratio at time ‘t’

e_{0}= Initial void ratio i.e, at t = 0

**(iii)**

where, ΔH = Final total settlement at the end of completion of primary consolidation i.e., at t =∞

Δh = Settlement occurred at any time ‘t’.

**TIME FACTOR**

where, T_{v} = Time factor

C_{V} = Coeff. of consolidation in cm^{2}/sec.

d = Length of drainage path

t = Time in ‘sec’**d= H _{o}/2** For 2-way drainage

where, H

**(i)**

, if u ≤ 60% T_{50} = 0.196

**(ii)** , if u > 60%

**METHOD TO FIND 'C _{v}' **

where,T

t

d = Length of drainage path.

**(ii) Logarithm of Time Fitting method**

where,T_{50 }= Time factor of 50% consolidation

t_{50} = Time of 50% consolidation**Remember: **Square root of time fitting method is better for soil having higher secondary consolidation.

**COMPRESSION RATIO:****(i) Initial compression Ratio**

where, R_{i} = Initial reading of dial gauge.

R_{0} = Reading of dial gauge at 0% consolidation

R_{f} = Final reading of dial gauge after secondary consolidation.

**(ii) Primary Consolidation Ratio**

where, R100 = Reading of dial gauge at 100% primary consolidation.

**(iii) Secondary Consolidation Ratio**

**TOTAL SETTLEMENT**

S = S_{i} + S_{P}+S_{s}

where S_{i} = Initial settlement, S_{p}=Primary Settlement S_{s} = Secondary settlement**(i) Initial Settlement**

For cohesionless soil.

where,

where, C_{r} = Static one resistance in kN/m^{2}

H_{0} = Depth of soil sample

for cohesive soil.

where, I_{t} = Shape factor or influence factor, A = Area.**Remember: **For square footing , A = B^{2} and I_{t} = 1

-for strip footing**(ii) Primary Settlement**

s_{c1} = Settlement for over consolidated stage

s_{c1} = Settlement normally consolidated stage

**(iii) Secondary Settlement**

where, H_{0} ~ H_{100}

H_{100} = Thickness of soil after 100% primary consolidation.

e_{100} = Void ratio after 100% primary consolidation.

t_{2} = Average time after t_{1} in which secondary consolidation is calculated.

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