Compressibility & Consolidation | Civil Engineering SSC JE (Technical) - Civil Engineering (CE) PDF Download

Compressibility And Consolidation

COEFFICIENT OF COMPRESSIBILITY (a_v)

e₁ = Void Ratio at effective stress
e₂ = Void ratio at effective stress  

 

ΔV = Change in volume in m³, or cm³.
V₀ = Initial volume in m³ or cm³.
ΔH = Change in depth in ‘m’ or ‘cm’.
H₀ = 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 (e₀- 0.25) For undisturbed soil of medium sensitivity e₀ = Initial void ratio

(e) For remoulded soil of low sensitivity.
     C_C =1.15(e₀- 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, e₀ = 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 = u₁ ⇒ %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’
e₀= 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 cm²/sec.
d  = Length of drainage path
t  = Time in ‘sec’
d= H_o/2 For 2-way drainage d =H₀ For one-way drainage.
where, H₀ = Depth of soil sample.

(i)
 , if u ≤ 60% T₅₀ = 0.196 

(ii)  , if u > 60%

METHOD TO FIND 'C_v' 
(i) Square Root of Time Fitting method
 
where,T₉₀ = Time factor at 90% consolidation
t₉₀ = Time at 90% consolidation
d   = Length of drainage path.

(ii) Logarithm of Time Fitting method
 
where,T₅₀ = Time factor of 50% consolidation
t₅₀ = 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.
R₀ = 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/m²
H₀ = Depth of soil sample
 
for cohesive soil.
where, I_t = Shape factor or influence factor, A = Area.
Remember: For square footing , A = B² 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, H₀ ~ H₁₀₀
H₁₀₀ = Thickness of soil after 100% primary consolidation.
e₁₀₀ = Void ratio after 100% primary consolidation.
t₂  = Average time after t₁ in which secondary consolidation is calculated.