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Compressibility And Consolidation
COEFFICIENT OF COMPRESSIBILITY (a_{v})
e_{1} = Void Ratio at effective stress
e_{2} = Void ratio at effective stress
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_{L} = liquid limit expressed in percent.
(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_{0} = Initial void ratio
(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 1D 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 2way drainage d =H_{0} For oneway drainage.
where, H_{0} = Depth of soil sample.
(i)
, if u ≤ 60% T_{50} = 0.196
(ii) , if u > 60%
METHOD TO FIND 'C_{v}'
(i) Square Root of Time Fitting method
where,T_{90} = Time factor at 90% consolidation
t_{90} = Time at 90% consolidation
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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