Compressibility and Consolidation 5
- Coefficient of Compressibility (a_{v})
a_{v} = e_{1 }- e_{2}/σ_{2 }- σ_{1}
e_{1} = Void ratio at effective stress
e_{2} =Void ratio at effective stress ΔV/V_{0} = ΔH/H_{0}
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'. - Coefficient of Compression (C_{c})
(i)
↓
(ii) C_{c} = 0.009(W_{L}-10)
For undisturbed soil of medium sensitivity.
W_{L} = % liquid limit.
(iii) C_{c }= 0.009(W_{L}-7)
For remolded soil of low sensitivity
(iv) C_{c} = 0.40(e_{0}-0.25)
For undisturbed soil of medium sensitivity e_{0 = Initial void} ratio(v) For remoulded soil of low sensitivity.
C_{c} = 1.15(e_{0 }- 0.35)
(vi) C_{c} = 0.115w where, w = Water content - Over consolidation ratio
O.C.R = Maximum effective stress applied in the past/Existing effective stress
O.C.R > 1 For over consolidated soil.
O.C.R = 1 For normally consolidated soil.
O.C.R < 1 For under consolidated soil.
Differential Equation of 1-D Consolidation
where, u = Excess pore pressure.
∂u / ∂t = Rate of change of pore pressure
C_{v} = Coefficient of consolidation
∂u / ∂z = Rate of change of pore pressure with depth.
- Coefficient of volume compressibility m_{v} = a_{v}/1+e_{0} where, e_{0} = Initial void ratio
m_{v} = Coefficient of volume compressibility
Compression modulus
E_{c} = 1/m_{v }where, E_{c} =Compression modulus. - Degree of consolidation
(i) %U = (1-(U/U_{1})x100) where,
%U = % degree of consolidation.
U = Excess pore pressure at any stage.
U_{1} = = Initial excess pore pressure
at t = 0, u = u_{1} ⇒ %u = 0%
at t = ∞, u = 0 ⇒ %u = 100%
(ii) %u = (e_{0}-e/e_{0}-e_{f})x100 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) %u = (Δh / ΔH) x 100 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
T_{v} = C_{v}.(t/d_{2}) 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_{0}/2 For 2-way drainage
d = H_{0} For one-way drainage.
where, H_{0} = Depth of soil sample.
(i) T_{v} = (π/4)(u)^{2} ... if u ≤ 60% T_{50} = 0.196
(ii) T_{v} = -0.9332log_{10}(1-u)-0.0851...
if u > 60%
Method to find 'Cv'
- Square Root of Time Fitting Method
C_{v} = (T_{90}.d^{2})/t_{90} where,
T_{90} = Time factor at 90% consolidation
t_{90} = Time at 90% consolidation
d = Length of drainage path. - Logarithm of Time Fitting Method
C_{v} = T_{50}.d^{2}/t_{50}
where, T_{50} = Time factor at 50% consolidation
t_{50} = Time of 50% consolidation.
Compression Ratio
- 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. - Primary Consolidation Ratio
where, R_{100} = Reading of dial gauge at 100% primary consolidation. - Secondary Consolidation Ratio
r_{i}+r_{p}+r_{s} = 1
Total Settlement
S = S_{i }+ S_{p} + S_{s} where, S_{i} = Initial settlement
S_{p} = Primary settlement
S_{s }= Secondary settlement
- Initial Settlement
For cohesionless soil.
where, C_{s} = 1.5(C_{r}/σ_{0})
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. - Primary Settlement
(i)
(ii)
(iii)
(iv)
= Settlement for over consolidated stage
= Settlement for normally consolidation stage
- 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
Permeability
- Permeability of Soil
The permeability of a soil is a property which describes quantitatively, the ease with which water flows through that soil. - Darcy's Law
Darcy established that the flow occurring per unit time is directly proportional to the head causing flow and the area of cross-section of the soil sample but is inversely proportional to the length of the sample.
(i) Rate of flow (q)
qα(Δh/L)A → q = KiAWhere, q = rate of flow in m^{3}/sec.
K = Coefficient of permeability in m/s
I = Hydraulic gradient
A = Area of cross-section of sample
i = H_{L}/L where, H_{L} = Head loss = (H_{1} – H_{2})
i = tanθ(dy/dx)
(ii) Seepage velocity
Vs = V/n where, V_{s} = Seepage velocity (m/sec)
n = Porosity & V = discharge velocity (m/s)
(iii) Coefficiency of percolation
K_{P} = K/n where, K_{P} = coefficiency of percolation and n = Porosity. - Constant Head Permeability Test
K = QL/tH_{L}A where, Q = Volume of water collected in time t in m^{3}.
Constant Head Permeability test is useful for coarse grain soil and it is a laboratory method. - Falling Head Permeability Test or Variable Head Permeability Test
K = 2.303aL/At(log_{10})(h_{1 }/ h_{2})
a = Area of tube in m^{2}
A = Area of sample in m^{2}
t = time in 'sec'
L = length in 'm'
h_{1} = level of upstream edge at t = 0
h_{2 }= level of upstream edge after 't'. - Konzey-Karman Equation
Where, C = Shape coefficient, ∼5mm for spherical particle
S = Specific surface area = Area/Volume - For spherical particle.
R = Radius of spherical particle.
S = 6/√ab
When particles are not spherical and of variable size. If these particles passes through sieve of size 'a' and retain on sieve of size 'n'.
e = void ratio
μ = dynamic viscosity, in (N - s/m^{2})
γ_{w} = unit weight of water in kN/m^{3}
- Allen Hazen Equation
K=C.D^{2}_{10} Where, D_{10 }= Effective size in cm. k is in cm/s C = 100 to 150 - Lioudens Equation
log_{10}KS^{2}= a + b.n
Where, S = Specific surface area
n = Porosity.
a and b are constant.
Consolidation equation K = C_{v}m_{v}γ_{w}
Where, C_{v }= Coefficient of consolidation in cm^{2}/sec
m_{v} = Coefficient of volume Compressibility in cm^{2}/N - Capillary Permeability Test i = h_{0 }+ h_{c}/x where, S = Degree of saturation
K = Coefficient of permeability of partially saturated soil.
where h_{c} = remains constant (but not known as depends upon soil)
= head under first set of observation,
n = porosity, h_{c} = capillary height
Another set of data gives,
= head under second set of observation
For S = 100%, K = maximum. Also, k_{u} ∝ S. - Permeability of a stratified soil
(i) Average permeability of the soil in which flow is parallel to bedding plane,
k_{eq}∼k_{x}(ii) Average permeability of soil in which flow is perpendicular to bedding plane.
k_{eq}∼k_{z}(iii) For 2-D flow in x and z direction
(iv) For 3-D flow in x, y and z direction k_{eq} = (k_{x}.k_{y}.k_{z})^{1/3}
Coefficient of absolute permeability (k_{0})
Effective Stress, Capilarity, Seepage
- Seepage Pressure and Seepage Force
Seepage pressure is exerted by the water on the soil due to friction drag. This drag force/seepage force always acts in the direction of flow.
The seepage pressure is given by
P_{S} = hγ_{ω} where, P_{s} = Seepage pressure
γ_{ω} = 9.81 kN/m^{3}
Here, h = head loss and z = length
(i) F_{S} = hAγ_{ω} where, F_{s} = Seepage force
(ii) f_{s} = iγ_{ω} where, f_{s} = Seepage force per unit volume.
i = h/z where, I = Hydraulic gradient. - Quick Sand Condition
It is condition but not the type of sand in which the net effective vertical stress becomes zero, when seepage occurs vertically up through the sands/cohesionless soils.
Net effective vertical stress = 0
i_{c} = (G - 1)/(1 + e) where, i_{c} = Critical hydraulic gradient.
2.65 ≤ G ≤ 2.70 0.65 ≤ e ≤ 0.70
To Avoid Floating Condition
i < i and F.O.S = i_{c}/i > 1
Laplace Equation of Two Dimensional Flow and Flow Net: Graphical Solution of Laplace Equation
(i)
where, ∅ = Potential function = kH
H = Total head and k = Coefficient of permeability
(ii) … 2D Laplace equation for Homogeneous soil.
where, ∅ = k_{X} H and ∅ = k_{y} H for Isotropic soil, k_{x}= k_{y}
Seepage discharge (q)
q = kh.(N_{f}/N_{d}) where, h = hydraulic head or head difference between upstream and downstream level or head loss through the soil.
- Shape factor = N_{f}/N_{d}
- N_{f} = N_{ψ} - 1
where, N_{f }= Total number of flow channels
N_{ψ }= Total number of flow lines. - N_{d} = N_{∅ }- 1
where, N_{d} = Total number equipotential drops.
N_{∅ }= Total number equipotential lines. - Hydrostatic pressure = U = h_{w}γ_{w}
where, U = Pore pressure h_{w} = Pressure head
h_{w} = Hydrostatic head – Potential head - Seepage Pressure
P_{s} = h'γ_{w} where, h' = h - (2h/N_{d})
- Exit gradient,
where, size of exit flow field is b x b.
and ΔH = h/N_{d} is equipotential drop.
Phreatic Line
It is top flow line which follows the path of base parabola. It is a stream line. The pressure on this line is atmospheric (zero) and below this line pressure is hydrostatic.
- Phreatic Line with Filter
Phreatic line (Top flow line).
↓
Follows the path of base parabola
CF = Radius of circular arc =
C = Entry point of base parabola
F = Junction of permeable and impermeable surface
S = Distance between focus and directrix
= Focal length.
FH = S
(i) q = ks where, q = Discharge through unit length of dam.
(ii)
(iii) - Phreatic Line without Filter (i) For ∝ < 30°
q = k a sin^{2} ∝
(ii) For ∝ > 30°
q = k a sin ∝ tan ∝ and