Casagrande:
k=1.4e2k0.85
Dr (%) Description
0 – 20 Very Loos e
20 – 40 Loose
40 – 70 Medium Dense
70 – 85 Dense
85 – 100 Very Dense
LI State
LI < 0 Semisolid
0 < LI < 1 Pla stic
LI > 1 Liquid
Volume
e=Vv
Vs
n=Vv
V
S=Vw
Vv
Weight
ω=Ww
Ws
0 < e < ∞
e= n
1−n
0 < n < 1
n= e
1+e
Se=Gsω
ɤ=W
V
ɤd=WS
V
ɤd=ɤ
1+ω
Unit Weight:
ɤ=(Gs+Gsω)ɤw
1+e
ɤ=(Gs+Se)ɤw
1+e
When S=0:
ɤd=Gsɤw
1+e
When S=100%:
ɤsat =(Gs+e)ɤw
1+e
ɤsub =ɤsat −ɤw
ɤsub =(Gs−1)ɤw
1+e
ɤzav =Gsɤw
1+Gsω
Specific Gravity of Solid:
Gs=ɤs
ɤw
Bulk Specific Gravity:
g=Gs(1−n)
Relative Compaction:
R= ɤd
ɤd𝑚𝑎𝑥
Relative Density/
Density Index:
Dr=e𝑚𝑎𝑥 −e
e𝑚𝑎𝑥 −e𝑚𝑖𝑛
Dr=1
ɤd𝑚𝑖𝑛−1
ɤd
1
ɤd𝑚𝑖𝑛−1
ɤd𝑚𝑎𝑥
Atterberg Limits
PI=LL−PL
LI=ω−PL
LL−PL
SI=PL−SL
CI=LL−ω
LL−PI
SL=m1−m2
m2−V1−V2
m2ɤw
SL=e
Gs ; SR=m2
V2ɤw
G𝑠=1
1
SR−SL
GI=(F−35)[0.2+0.005(LL−40)]
+0.01(F−15)(PI−10)
Ac=PI
μ; St=quund
qurem
μ = % passing 0.002mm
PI Description
0 Non-plastic
1-5 Sl ightly plastic
5-10 Low plasticity
10-20 Medium plasticity
20-40 High plasticity
>40 Very High plastic
Ac Class
AC < 0.7 Inactive
0.7 < AC < 1.2 Normal
AC > 1.2 Active
Sieve Analysis
Uniformity
Coefficient:
Cu=D60
D10
Coeff. of Gradation
or Curvature:
Cc=(D30)2
D60∙D10
Sorting
Coefficient:
So=√D75
D25
Suitability Number:
Sn=1.7√ 3
(D50)2+1
(D20)2+1
(D10)2
Permeability
v=ki ; i =∆h
L ; v𝑠=v
n
Q=vA=kiA
Constant Head Test:
k= QL
Aht
Falling/Variable Head Test:
k=aL
At𝑙𝑛h1
h2
Stratified Soil
for Parallel flow:
keq =h1k1+h2k2+...+hnkn
H
for Perpendicular flow:
keq =H
h1
k1+h2
k2+...+hn
kn
Pumping Test:
Unconfined:
k= Q 𝑙𝑛r1
r2
π(h12−h22)
Confined:
k= Q 𝑙𝑛r1
r2
2πt(h1−h2)
Hazen Formula
k=c∙D102
Samarasinhe:
k= C3∙en
1+e
Kozeny-Carman:
k= C1∙e2
1+e
Stresses in Soil
Effective Stress/
Intergranular Stress:
pE=pT−pw
Pore Water Pressure/
Neutral Stress:
pw=ɤwhw
Total Stress:
pT=ɤ1h1+ɤ2h2+...+ɤnhn
Flow Net / Seepage
Isotropic soil:
q=kHNf
Nd
Non-Isotropic soil:
q=√kxkzHNf
Nd
Compressibility of Soil
Compression Index, CC:
Cc=0.009(LL−10%)
Cc=e−e′
𝑙𝑜𝑔∆P+Po
Po
For normally consolidated clay:
S=e−e′
1+eH
S= CcH
1+e𝑙𝑜𝑔∆P+Po
Po
With Pre-consolidation pressure, Pc:
when (△P+Po) < Pc:
S= CsH
1+eo𝑙𝑜𝑔∆P+Po
Po
when (△P+Po) > pc:
S= CsH
1+e𝑙𝑜𝑔Pc
Po+CcH
1+e𝑙𝑜𝑔∆P+Po
Pc
Over Consolidation Ratio (OCR):
OCR=pc
po; OCR = 1
Coefficient of Compressibility:
av=∆e
∆P
Coefficient of Volume Compressibility:
mv=∆e
∆P
1+eave
Coefficient of Consolidation:
Cv=Hdr2Tv
t
Coefficient of Permeability:
k=mvCvɤw
Equipotential line ----
Flow line ----
Lateral Earth Pressure
ACTIVE PRESSURE:
pa=1
2kaɤH2−2cH√ka
For Inclined:
ka=cosβcosβ−√cos2β−cos2Ø
cosβ+√cos2β−cos2Ø
For Horizontal:
ka=1−sinØ
1+sinØ
If there is angle of friction α bet. wall and soil:
ka=cos2Ø
cos α [1 + √sin (Ø + α) sin Ø
cosα ]2
PASSIVE PRESSURE:
pP=1
2kPɤH2+2cH√kP
For Inclined:
kP=cosβcosβ+√cos2β−cos2Ø
cosβ−√cos2β−cos2Ø
For Horizontal:
kP=1+sinØ
1−sinØ
If there is angle of friction α bet. wall and soil:
kP=cos2Ø
cos α [1 − √sin (Ø − α) sin Ø
cosα ]2
AT REST:
ko=1−sinØ
(for one layer only)
Swell Index, CS:
Cs=1
5Cc
(for normally consolidated soil)
△e → change in void ratio
△P → change in pressure
Hdr → height of drainage path
→ thickness of layer if drained 1 side
→ half of thickness if drained both sides
Tv → factor from table
t → time consolidation
Shear Strength of Soil
Ө → angle of failure in shear
Ø → angle of internal friction/shearing resistance
C → cohesion of soil
θ=45°+Ø
2
TRI-AXIAL TEST:
σ1 → maximum principal stress
→ axial stress
△σ → additional pressure
→ deviator stress
→ plunger pressure
σ3 → minimum principal stress
→ confining pressure
→ lateral pressure
→ radial stress
→ cell pressure
→ chamber pressure
Normally consolidated:
sinØ= r
σ3+r
Cohesive soil:
sinØ= r
x+σ3+r
tanØ=c
x
Unconsolidated-
undrained test:
c=r
Unconfined
compression test:
σ3=0
DIRECT SHEAR TEST:
σn → normal stress
σs → shear stress
Normally consolidated soil:
tanØ=σS
σN
Cohesive soil:
tanØ= σS
x+σN=c
x
σS=c+σNtan∅
Nf → no. of flow channels [e.g. 4]
Nd → no. of potential drops [e.g. 10]
1
2
3
4
1
2
3
4
5
6
7
8
9
10
NOTE:
Quick
condition:
pE=0
Capillary Rise:
hcr =C
eD10
