Geotechnical Engineering Formulas for Civil Engineering Exam

 Page 1


 
 
 
 
Short Notes for Soil Mechanics & Foundation Engineering 
Properties of Soils 
Water content 
• 100
W
S
W
w
W
=? 
 W W = Weight of power 
  W S = Weight of solids 
 
Void ratio 
• 
v
s
V
e
V
= 
            V v = Volume of voids 
            V =  Total volume of soil 
 
Degree of Saturation 
• 100
w
v
V
S
V
=? 
V w = Volume of water 
 V v = Volume of voids 
0 = S= 100 
for perfectly dry soil : S = O 
for Fully saturated soil : S = 100% 
 
 
Air Content 
• 1
a
c
v
V
as
V
= = -    V a = Volume of air 
S r + a c = 1 
% Air Void 
• 
Volume of air
% 100 100
Total volume
a
a
V
n
V
= ? = ?
 
 
Unit Weight 
• Bulk unit weight 
sw
s w a
WW W
V V V V
?
+
==
++
 
 
• Dry Unit Weight 
s
d
W
V
? =
 
o Dry unit weight is used as a measure of denseness of soil 
• Saturated unit weight: It is the ratio of total weight of fully saturated soil sample to its total 
volume. 
sat
sat
W
V
? = 
• Submerged unit weight or Buoyant unit weight 
Page 2


 
 
 
 
Short Notes for Soil Mechanics & Foundation Engineering 
Properties of Soils 
Water content 
• 100
W
S
W
w
W
=? 
 W W = Weight of power 
  W S = Weight of solids 
 
Void ratio 
• 
v
s
V
e
V
= 
            V v = Volume of voids 
            V =  Total volume of soil 
 
Degree of Saturation 
• 100
w
v
V
S
V
=? 
V w = Volume of water 
 V v = Volume of voids 
0 = S= 100 
for perfectly dry soil : S = O 
for Fully saturated soil : S = 100% 
 
 
Air Content 
• 1
a
c
v
V
as
V
= = -    V a = Volume of air 
S r + a c = 1 
% Air Void 
• 
Volume of air
% 100 100
Total volume
a
a
V
n
V
= ? = ?
 
 
Unit Weight 
• Bulk unit weight 
sw
s w a
WW W
V V V V
?
+
==
++
 
 
• Dry Unit Weight 
s
d
W
V
? =
 
o Dry unit weight is used as a measure of denseness of soil 
• Saturated unit weight: It is the ratio of total weight of fully saturated soil sample to its total 
volume. 
sat
sat
W
V
? = 
• Submerged unit weight or Buoyant unit weight 
 
 
 
 
'
sat w
? ? ? =-
 
sat
? = unit wt. of saturated soil 
? = unit wt. of water 
• Unit wt. of solids:  
s
s
s
W
V
? = 
Specific Gravity 
True/Absolute Special Gravity, G 
• Specific gravity of soil solids (G) is the ratio of the weight of a given volume of solids to the 
weight of an equivalent volume of water at 4 ?. 
.
ss
s w w
W
G
V
?
??
== 
 
• Apparent or mass specific gravity (G m):  
 or  or 
.
d sat
m
ww
W
G
V
? ? ?
??
==
 
where, ? is bulk unit wt. of soil 
? = ? sat for saturated soil mass 
? = ? d for dry soil mass 
G m < G 
 
Relative density (I D) 
• To compare degree of denseness of two soils. 
1
 
D
Shear strength
Compressi t
I
bili y
? ? 
max
max min
% 100
D
ee
I
ee
-
=?
-
 
min
min max
11
    -    
% 100
11
    -    
dd
D
dd
I
??
??
=? 
 
Relative Compaction 
• Indicate: Degree of denseness of cohesive + cohesionless soil 
 
max
D
c
D
R
?
?
=  
Relative Density 
• Indicate: Degree of denseness of natural cohesionless soil 
Some Important Relationships 
• Relation between ,
d
??  
1
d
w
?
? =
+
 
(ii) 
1
s
V
V
e
=
+
   (iii) 
1
s
W
W
w
=
+
 
Page 3


 
 
 
 
Short Notes for Soil Mechanics & Foundation Engineering 
Properties of Soils 
Water content 
• 100
W
S
W
w
W
=? 
 W W = Weight of power 
  W S = Weight of solids 
 
Void ratio 
• 
v
s
V
e
V
= 
            V v = Volume of voids 
            V =  Total volume of soil 
 
Degree of Saturation 
• 100
w
v
V
S
V
=? 
V w = Volume of water 
 V v = Volume of voids 
0 = S= 100 
for perfectly dry soil : S = O 
for Fully saturated soil : S = 100% 
 
 
Air Content 
• 1
a
c
v
V
as
V
= = -    V a = Volume of air 
S r + a c = 1 
% Air Void 
• 
Volume of air
% 100 100
Total volume
a
a
V
n
V
= ? = ?
 
 
Unit Weight 
• Bulk unit weight 
sw
s w a
WW W
V V V V
?
+
==
++
 
 
• Dry Unit Weight 
s
d
W
V
? =
 
o Dry unit weight is used as a measure of denseness of soil 
• Saturated unit weight: It is the ratio of total weight of fully saturated soil sample to its total 
volume. 
sat
sat
W
V
? = 
• Submerged unit weight or Buoyant unit weight 
 
 
 
 
'
sat w
? ? ? =-
 
sat
? = unit wt. of saturated soil 
? = unit wt. of water 
• Unit wt. of solids:  
s
s
s
W
V
? = 
Specific Gravity 
True/Absolute Special Gravity, G 
• Specific gravity of soil solids (G) is the ratio of the weight of a given volume of solids to the 
weight of an equivalent volume of water at 4 ?. 
.
ss
s w w
W
G
V
?
??
== 
 
• Apparent or mass specific gravity (G m):  
 or  or 
.
d sat
m
ww
W
G
V
? ? ?
??
==
 
where, ? is bulk unit wt. of soil 
? = ? sat for saturated soil mass 
? = ? d for dry soil mass 
G m < G 
 
Relative density (I D) 
• To compare degree of denseness of two soils. 
1
 
D
Shear strength
Compressi t
I
bili y
? ? 
max
max min
% 100
D
ee
I
ee
-
=?
-
 
min
min max
11
    -    
% 100
11
    -    
dd
D
dd
I
??
??
=? 
 
Relative Compaction 
• Indicate: Degree of denseness of cohesive + cohesionless soil 
 
max
D
c
D
R
?
?
=  
Relative Density 
• Indicate: Degree of denseness of natural cohesionless soil 
Some Important Relationships 
• Relation between ,
d
??  
1
d
w
?
? =
+
 
(ii) 
1
s
V
V
e
=
+
   (iii) 
1
s
W
W
w
=
+
 
 
 
 
 
• Relation between e and n 
1
e
n
e
=
+
    or    
1
n
e
n
=
-
 
• Relation between e, w, G and S: 
Se = w. G 
• Bulk unit weight () ? in terms of G, e, w and 
w
? ? , G, e, S r, 
w
? 
()
1
rw
G eS
e
?
?
+
=
+
 
(1 )
(1 )
w
Gw
e
?
?
+
=
+
    {Srxe = wG} 
• Saturated unit weight ( .) sat ? in terms of G, e & 
w
?  
S r = 1 .
1
sat w
Ge
e
??
+ ??
=
??
+
??
 
• Dry unit weight ()
d
? in terms of G, e and 
w
? 
S r = 0 
(1 )
11
1
w w a w
d
G G G
wG
e wG
S
? ? ? ?
?
-
= = =
++
+
 
• Submerged unit weight ( ') ? in terms of G, e and 
w
? 
sat w
? ? ? = - = 
1
'.
1
w
G
e
??
- ??
=
??
+
??
 
• Relation between degree of saturation (s) w and G 
1
(1 )
w
W
S
W
G
?
?
=
+-
 
 
 
• Calibration of Hydrometer 
 
 
• Effective depth is calculated as 
1
1
2
H
e
j
V
H H h
A
??
= + -
??
??
??
 
where, H 1 = distance (cm) between any hydrometer reading and neck. 
h = length of hydrometer bulb 
V H = volume of hydrometer bulb 
Page 4


 
 
 
 
Short Notes for Soil Mechanics & Foundation Engineering 
Properties of Soils 
Water content 
• 100
W
S
W
w
W
=? 
 W W = Weight of power 
  W S = Weight of solids 
 
Void ratio 
• 
v
s
V
e
V
= 
            V v = Volume of voids 
            V =  Total volume of soil 
 
Degree of Saturation 
• 100
w
v
V
S
V
=? 
V w = Volume of water 
 V v = Volume of voids 
0 = S= 100 
for perfectly dry soil : S = O 
for Fully saturated soil : S = 100% 
 
 
Air Content 
• 1
a
c
v
V
as
V
= = -    V a = Volume of air 
S r + a c = 1 
% Air Void 
• 
Volume of air
% 100 100
Total volume
a
a
V
n
V
= ? = ?
 
 
Unit Weight 
• Bulk unit weight 
sw
s w a
WW W
V V V V
?
+
==
++
 
 
• Dry Unit Weight 
s
d
W
V
? =
 
o Dry unit weight is used as a measure of denseness of soil 
• Saturated unit weight: It is the ratio of total weight of fully saturated soil sample to its total 
volume. 
sat
sat
W
V
? = 
• Submerged unit weight or Buoyant unit weight 
 
 
 
 
'
sat w
? ? ? =-
 
sat
? = unit wt. of saturated soil 
? = unit wt. of water 
• Unit wt. of solids:  
s
s
s
W
V
? = 
Specific Gravity 
True/Absolute Special Gravity, G 
• Specific gravity of soil solids (G) is the ratio of the weight of a given volume of solids to the 
weight of an equivalent volume of water at 4 ?. 
.
ss
s w w
W
G
V
?
??
== 
 
• Apparent or mass specific gravity (G m):  
 or  or 
.
d sat
m
ww
W
G
V
? ? ?
??
==
 
where, ? is bulk unit wt. of soil 
? = ? sat for saturated soil mass 
? = ? d for dry soil mass 
G m < G 
 
Relative density (I D) 
• To compare degree of denseness of two soils. 
1
 
D
Shear strength
Compressi t
I
bili y
? ? 
max
max min
% 100
D
ee
I
ee
-
=?
-
 
min
min max
11
    -    
% 100
11
    -    
dd
D
dd
I
??
??
=? 
 
Relative Compaction 
• Indicate: Degree of denseness of cohesive + cohesionless soil 
 
max
D
c
D
R
?
?
=  
Relative Density 
• Indicate: Degree of denseness of natural cohesionless soil 
Some Important Relationships 
• Relation between ,
d
??  
1
d
w
?
? =
+
 
(ii) 
1
s
V
V
e
=
+
   (iii) 
1
s
W
W
w
=
+
 
 
 
 
 
• Relation between e and n 
1
e
n
e
=
+
    or    
1
n
e
n
=
-
 
• Relation between e, w, G and S: 
Se = w. G 
• Bulk unit weight () ? in terms of G, e, w and 
w
? ? , G, e, S r, 
w
? 
()
1
rw
G eS
e
?
?
+
=
+
 
(1 )
(1 )
w
Gw
e
?
?
+
=
+
    {Srxe = wG} 
• Saturated unit weight ( .) sat ? in terms of G, e & 
w
?  
S r = 1 .
1
sat w
Ge
e
??
+ ??
=
??
+
??
 
• Dry unit weight ()
d
? in terms of G, e and 
w
? 
S r = 0 
(1 )
11
1
w w a w
d
G G G
wG
e wG
S
? ? ? ?
?
-
= = =
++
+
 
• Submerged unit weight ( ') ? in terms of G, e and 
w
? 
sat w
? ? ? = - = 
1
'.
1
w
G
e
??
- ??
=
??
+
??
 
• Relation between degree of saturation (s) w and G 
1
(1 )
w
W
S
W
G
?
?
=
+-
 
 
 
• Calibration of Hydrometer 
 
 
• Effective depth is calculated as 
1
1
2
H
e
j
V
H H h
A
??
= + -
??
??
??
 
where, H 1 = distance (cm) between any hydrometer reading and neck. 
h = length of hydrometer bulb 
V H = volume of hydrometer bulb 
 
 
 
 
 
Plasticity Index (I p):  
• It is the range of moisture content over which a soil exhibits plasticity. 
I p = W L - W p 
W L = water content at LL 
W p = water content at PL 
 
I p (%) Soil Description 
0 
1 to 5 
5 to 10 
10 to 20 
20 to 40 
> 40 
Non plastic 
Slight plastic 
Low plastic 
Medium plastic 
Highly plastic 
Very highly plastic 
 
Relative Consistency or Consistency – index (I c):  
LN
C
p
WW
I
I
-
= 
 
 
   0 
     
 1
C
N L C
NP
For W W I
For W I W
=
=
? ? = ?
?
?=
?
 
 
Liquidity Index (I L) 
NP
L
P
WW
I
I
-
= 
For a soil in plastic state I L varies from 0 to 1. 
 
Consist. Description I C I L 
Liquid 
Plastic 
 
 
 
 
Semi-
solid 
 
Solid 
Liquid 
Very soft  
soft 
medium 
stiff  
stiff 
Very stiff 
OR Hard 
 
Hard OR 
very hard 
<0 
0-0.25 
0.25-0.5 
0.50-0.75 
0.75-1.00 
 
 
>1 
 
 
>1 
>1 
0.75-1.00 
0.50-0.75 
0.25-0.50 
0.0-0.25 
 
 
< 0 
 
 
< 0 
 
Flow Index (I f) 
12
21
log10( / )
f
WW
I
NN
-
= 
 
 
Toughness Index (I t) 
P
T
F
I
I
I
= 
• For most of the soils:  0 < I T < 3 
Page 5


 
 
 
 
Short Notes for Soil Mechanics & Foundation Engineering 
Properties of Soils 
Water content 
• 100
W
S
W
w
W
=? 
 W W = Weight of power 
  W S = Weight of solids 
 
Void ratio 
• 
v
s
V
e
V
= 
            V v = Volume of voids 
            V =  Total volume of soil 
 
Degree of Saturation 
• 100
w
v
V
S
V
=? 
V w = Volume of water 
 V v = Volume of voids 
0 = S= 100 
for perfectly dry soil : S = O 
for Fully saturated soil : S = 100% 
 
 
Air Content 
• 1
a
c
v
V
as
V
= = -    V a = Volume of air 
S r + a c = 1 
% Air Void 
• 
Volume of air
% 100 100
Total volume
a
a
V
n
V
= ? = ?
 
 
Unit Weight 
• Bulk unit weight 
sw
s w a
WW W
V V V V
?
+
==
++
 
 
• Dry Unit Weight 
s
d
W
V
? =
 
o Dry unit weight is used as a measure of denseness of soil 
• Saturated unit weight: It is the ratio of total weight of fully saturated soil sample to its total 
volume. 
sat
sat
W
V
? = 
• Submerged unit weight or Buoyant unit weight 
 
 
 
 
'
sat w
? ? ? =-
 
sat
? = unit wt. of saturated soil 
? = unit wt. of water 
• Unit wt. of solids:  
s
s
s
W
V
? = 
Specific Gravity 
True/Absolute Special Gravity, G 
• Specific gravity of soil solids (G) is the ratio of the weight of a given volume of solids to the 
weight of an equivalent volume of water at 4 ?. 
.
ss
s w w
W
G
V
?
??
== 
 
• Apparent or mass specific gravity (G m):  
 or  or 
.
d sat
m
ww
W
G
V
? ? ?
??
==
 
where, ? is bulk unit wt. of soil 
? = ? sat for saturated soil mass 
? = ? d for dry soil mass 
G m < G 
 
Relative density (I D) 
• To compare degree of denseness of two soils. 
1
 
D
Shear strength
Compressi t
I
bili y
? ? 
max
max min
% 100
D
ee
I
ee
-
=?
-
 
min
min max
11
    -    
% 100
11
    -    
dd
D
dd
I
??
??
=? 
 
Relative Compaction 
• Indicate: Degree of denseness of cohesive + cohesionless soil 
 
max
D
c
D
R
?
?
=  
Relative Density 
• Indicate: Degree of denseness of natural cohesionless soil 
Some Important Relationships 
• Relation between ,
d
??  
1
d
w
?
? =
+
 
(ii) 
1
s
V
V
e
=
+
   (iii) 
1
s
W
W
w
=
+
 
 
 
 
 
• Relation between e and n 
1
e
n
e
=
+
    or    
1
n
e
n
=
-
 
• Relation between e, w, G and S: 
Se = w. G 
• Bulk unit weight () ? in terms of G, e, w and 
w
? ? , G, e, S r, 
w
? 
()
1
rw
G eS
e
?
?
+
=
+
 
(1 )
(1 )
w
Gw
e
?
?
+
=
+
    {Srxe = wG} 
• Saturated unit weight ( .) sat ? in terms of G, e & 
w
?  
S r = 1 .
1
sat w
Ge
e
??
+ ??
=
??
+
??
 
• Dry unit weight ()
d
? in terms of G, e and 
w
? 
S r = 0 
(1 )
11
1
w w a w
d
G G G
wG
e wG
S
? ? ? ?
?
-
= = =
++
+
 
• Submerged unit weight ( ') ? in terms of G, e and 
w
? 
sat w
? ? ? = - = 
1
'.
1
w
G
e
??
- ??
=
??
+
??
 
• Relation between degree of saturation (s) w and G 
1
(1 )
w
W
S
W
G
?
?
=
+-
 
 
 
• Calibration of Hydrometer 
 
 
• Effective depth is calculated as 
1
1
2
H
e
j
V
H H h
A
??
= + -
??
??
??
 
where, H 1 = distance (cm) between any hydrometer reading and neck. 
h = length of hydrometer bulb 
V H = volume of hydrometer bulb 
 
 
 
 
 
Plasticity Index (I p):  
• It is the range of moisture content over which a soil exhibits plasticity. 
I p = W L - W p 
W L = water content at LL 
W p = water content at PL 
 
I p (%) Soil Description 
0 
1 to 5 
5 to 10 
10 to 20 
20 to 40 
> 40 
Non plastic 
Slight plastic 
Low plastic 
Medium plastic 
Highly plastic 
Very highly plastic 
 
Relative Consistency or Consistency – index (I c):  
LN
C
p
WW
I
I
-
= 
 
 
   0 
     
 1
C
N L C
NP
For W W I
For W I W
=
=
? ? = ?
?
?=
?
 
 
Liquidity Index (I L) 
NP
L
P
WW
I
I
-
= 
For a soil in plastic state I L varies from 0 to 1. 
 
Consist. Description I C I L 
Liquid 
Plastic 
 
 
 
 
Semi-
solid 
 
Solid 
Liquid 
Very soft  
soft 
medium 
stiff  
stiff 
Very stiff 
OR Hard 
 
Hard OR 
very hard 
<0 
0-0.25 
0.25-0.5 
0.50-0.75 
0.75-1.00 
 
 
>1 
 
 
>1 
>1 
0.75-1.00 
0.50-0.75 
0.25-0.50 
0.0-0.25 
 
 
< 0 
 
 
< 0 
 
Flow Index (I f) 
12
21
log10( / )
f
WW
I
NN
-
= 
 
 
Toughness Index (I t) 
P
T
F
I
I
I
= 
• For most of the soils:  0 < I T < 3 
 
 
 
 
• When I T < 1, the soil is friable (easily crushed) at the plastic limit. 
 
 
 
• Shrinkage Ratio (SR) 
12
12
100
d
VV
V
SR
ww
-
?
=
-
 
V 1 = Volume of soil mass at water content w 1%. 
V 2 = volume of soil mass at water content w 2%.  
V d = volume of dry soil mass 
? 
1
1
100
()
d
d
s
VV
V
SR
WW
?? -
?
??
??
=
-
 
If w 1 & w 2 are expressed as ratio, 
1 2 1 2
12
12
( ) / ( ) /
,
dw
s
V V V V V
SR But w w
W W W
? --
= - =
-
 
?
1
 .
sd
d w w
W
SR
V
?
??
== 
 
Properties Relations
hip 
Governing 
Parameters 
Plasticity      ? Plasticity Index 
Better 
Foundation 
Material upon 
Remoulding 
     ?  Consistency 
Index 
Compressibility      ? Liquid Limit 
Rate of loss in 
shear strength 
with increase in 
water content 
     ? Flow Index 
Strength of 
Plastic Limit 
     ? Toughness 
Index 
 
 
Compaction of Soil 
 
 
Optimum moisture content 
max
()
1
d imum
optimum
w
?
? =
+
 
max
()
d imum
? = Maximum dry density 
 ? = Density of soil 
optimum
w = Optimum moisture content