Formula Sheets: Compressibility & Consolidation of Soils | Soil Mechanics - Civil Engineering (CE) PDF Download

 Page 1


GA TE CE 2026 F orm ula Sheet: Soil Mec hanics - P ermeabilit y
and Seepage
1. P e rmeabilit y - One Dimensional Flo w
• Darcy’s La w:
q =k·i·A
where:
– q : Disc harge rate (m³/s)
– k : Co e?icien t of p ermeabilit y (m/s)
– i : Hydraulic gradien t, i=
h
L
– h : Head loss (m)
– L : Length of flo w path (m)
– A : Cross-sectional area (m²)
• V elo cit y of flo w:
v =k·i
where:
– v : Darcy v elo c it y (m/s)
• Seepage v elo cit y:
v
s
=
v
n
=
k·i
n
where:
– v
s
: Seepage v elo cit y (m/s)
– n : P orosit y
• Co e?icien t of p ermeabilit y (empirical, Hazen’s form ula):
k =C·D
2
10
where:
– C : Constan t (t ypically 100–150 for sands, cm/s)
– D
10
: Effectiv e grain size (mm)
2. Se epage through Soils - T w o-Dimensional Flo w
• Laplace equation for t w o-dimensional flo w:
?
2
h
?x
2
+
?
2
h
?z
2
=0
where:
– h : Hydraulic head (m)
– x , z : Co ordinates in horizon tal and v ertical directions
• T otal disc harge through a flo w net:
Q=k·H·
N
f
N
d
where:
– Q : T otal disc harge (m³/s)
– H : T otal head loss (m)
– N
f
: Num b er of flo w c hannels
– N
d
: Num b er of equip oten tial drops
1
Page 2


GA TE CE 2026 F orm ula Sheet: Soil Mec hanics - P ermeabilit y
and Seepage
1. P e rmeabilit y - One Dimensional Flo w
• Darcy’s La w:
q =k·i·A
where:
– q : Disc harge rate (m³/s)
– k : Co e?icien t of p ermeabilit y (m/s)
– i : Hydraulic gradien t, i=
h
L
– h : Head loss (m)
– L : Length of flo w path (m)
– A : Cross-sectional area (m²)
• V elo cit y of flo w:
v =k·i
where:
– v : Darcy v elo c it y (m/s)
• Seepage v elo cit y:
v
s
=
v
n
=
k·i
n
where:
– v
s
: Seepage v elo cit y (m/s)
– n : P orosit y
• Co e?icien t of p ermeabilit y (empirical, Hazen’s form ula):
k =C·D
2
10
where:
– C : Constan t (t ypically 100–150 for sands, cm/s)
– D
10
: Effectiv e grain size (mm)
2. Se epage through Soils - T w o-Dimensional Flo w
• Laplace equation for t w o-dimensional flo w:
?
2
h
?x
2
+
?
2
h
?z
2
=0
where:
– h : Hydraulic head (m)
– x , z : Co ordinates in horizon tal and v ertical directions
• T otal disc harge through a flo w net:
Q=k·H·
N
f
N
d
where:
– Q : T otal disc harge (m³/s)
– H : T otal head loss (m)
– N
f
: Num b er of flo w c hannels
– N
d
: Num b er of equip oten tial drops
1
3. Flo w Nets
• Hydraulic gradien t at a p oin t:
i=
?h
L
f
where:
– ?h : Head drop across one equip oten tial (m)
– L
f
: Length of flo w path b et w een equip oten tials (m)
• Shap e factor of flo w net:
S =
N
f
N
d
where:
– S : Shap e factor (dimensionless)
4. U plift Pressure
• Uplift pressure at a p oin t:
u=?
w
·h
w
where:
– u : Uplift pressure (kN/m²)
– ?
w
: Unit w eigh t of w ater (9.81 kN/m³)
– h
w
: Piezometric head at the p oin t (m)
• T otal uplift force:
F
u
=u·A
where:
– F
u
: T otal uplift force (kN)
– A : Area sub jected to uplift pressure (m²)
5. Piping
• Critical h ydraulic gradien t (piping condition):
i
c
=
G-1
1+e
where:
– i
c
: Critical h ydraulic gradien t
– G : Sp ecific gra vit y of soil solids
– e : V oid ratio
• F actor of safet y against piping:
FS
piping
=
i
c
i
exit
where:
– i
exit
: Exit h ydraulic gradien t at do wnstream end
2
Page 3


GA TE CE 2026 F orm ula Sheet: Soil Mec hanics - P ermeabilit y
and Seepage
1. P e rmeabilit y - One Dimensional Flo w
• Darcy’s La w:
q =k·i·A
where:
– q : Disc harge rate (m³/s)
– k : Co e?icien t of p ermeabilit y (m/s)
– i : Hydraulic gradien t, i=
h
L
– h : Head loss (m)
– L : Length of flo w path (m)
– A : Cross-sectional area (m²)
• V elo cit y of flo w:
v =k·i
where:
– v : Darcy v elo c it y (m/s)
• Seepage v elo cit y:
v
s
=
v
n
=
k·i
n
where:
– v
s
: Seepage v elo cit y (m/s)
– n : P orosit y
• Co e?icien t of p ermeabilit y (empirical, Hazen’s form ula):
k =C·D
2
10
where:
– C : Constan t (t ypically 100–150 for sands, cm/s)
– D
10
: Effectiv e grain size (mm)
2. Se epage through Soils - T w o-Dimensional Flo w
• Laplace equation for t w o-dimensional flo w:
?
2
h
?x
2
+
?
2
h
?z
2
=0
where:
– h : Hydraulic head (m)
– x , z : Co ordinates in horizon tal and v ertical directions
• T otal disc harge through a flo w net:
Q=k·H·
N
f
N
d
where:
– Q : T otal disc harge (m³/s)
– H : T otal head loss (m)
– N
f
: Num b er of flo w c hannels
– N
d
: Num b er of equip oten tial drops
1
3. Flo w Nets
• Hydraulic gradien t at a p oin t:
i=
?h
L
f
where:
– ?h : Head drop across one equip oten tial (m)
– L
f
: Length of flo w path b et w een equip oten tials (m)
• Shap e factor of flo w net:
S =
N
f
N
d
where:
– S : Shap e factor (dimensionless)
4. U plift Pressure
• Uplift pressure at a p oin t:
u=?
w
·h
w
where:
– u : Uplift pressure (kN/m²)
– ?
w
: Unit w eigh t of w ater (9.81 kN/m³)
– h
w
: Piezometric head at the p oin t (m)
• T otal uplift force:
F
u
=u·A
where:
– F
u
: T otal uplift force (kN)
– A : Area sub jected to uplift pressure (m²)
5. Piping
• Critical h ydraulic gradien t (piping condition):
i
c
=
G-1
1+e
where:
– i
c
: Critical h ydraulic gradien t
– G : Sp ecific gra vit y of soil solids
– e : V oid ratio
• F actor of safet y against piping:
FS
piping
=
i
c
i
exit
where:
– i
exit
: Exit h ydraulic gradien t at do wnstream end
2
6. Capillarit y
• Capillary rise in soil:
h
c
=
4T cosa
?
w
d
where:
– h
c
: Heigh t of c apillary rise (m)
– T : Surface tension of w ater (N/m, t ypically 0.073 N/m)
– a : Con tact angle (degrees)
– d : Diameter of capillary tub e or p ore s ize (m)
– ?
w
: Unit w eigh t of w ater (9.81 kN/m³)
• Capillary pressure:
p
c
=?
w
·h
c
where:
– p
c
: Capillary pressure (kN/m²)
7. Seepage F orce
• Seepage force p er unit v olume:
j =i·?
w
where:
– j : Seepage force p er unit v olume (kN/m³)
– i : Hydraulic gradien t
• T otal seepage force:
F
s
=j·V
where:
– F
s
: T otal seepage force (kN)
– V : V olume of soil affecte d (m³)
8. Principle of Effectiv e Stress
• Effectiv e stress:
s
'
=s-u
where:
– s
'
: Effectiv e stress (kN/m²)
– s : T otal stress (kN/m²)
– u : P ore w ater pressure (kN/m²)
• P ore w ater pressure:
u=?
w
·h
w
where:
– h
w
: Heigh t of w ater column (m)
3
Page 4


GA TE CE 2026 F orm ula Sheet: Soil Mec hanics - P ermeabilit y
and Seepage
1. P e rmeabilit y - One Dimensional Flo w
• Darcy’s La w:
q =k·i·A
where:
– q : Disc harge rate (m³/s)
– k : Co e?icien t of p ermeabilit y (m/s)
– i : Hydraulic gradien t, i=
h
L
– h : Head loss (m)
– L : Length of flo w path (m)
– A : Cross-sectional area (m²)
• V elo cit y of flo w:
v =k·i
where:
– v : Darcy v elo c it y (m/s)
• Seepage v elo cit y:
v
s
=
v
n
=
k·i
n
where:
– v
s
: Seepage v elo cit y (m/s)
– n : P orosit y
• Co e?icien t of p ermeabilit y (empirical, Hazen’s form ula):
k =C·D
2
10
where:
– C : Constan t (t ypically 100–150 for sands, cm/s)
– D
10
: Effectiv e grain size (mm)
2. Se epage through Soils - T w o-Dimensional Flo w
• Laplace equation for t w o-dimensional flo w:
?
2
h
?x
2
+
?
2
h
?z
2
=0
where:
– h : Hydraulic head (m)
– x , z : Co ordinates in horizon tal and v ertical directions
• T otal disc harge through a flo w net:
Q=k·H·
N
f
N
d
where:
– Q : T otal disc harge (m³/s)
– H : T otal head loss (m)
– N
f
: Num b er of flo w c hannels
– N
d
: Num b er of equip oten tial drops
1
3. Flo w Nets
• Hydraulic gradien t at a p oin t:
i=
?h
L
f
where:
– ?h : Head drop across one equip oten tial (m)
– L
f
: Length of flo w path b et w een equip oten tials (m)
• Shap e factor of flo w net:
S =
N
f
N
d
where:
– S : Shap e factor (dimensionless)
4. U plift Pressure
• Uplift pressure at a p oin t:
u=?
w
·h
w
where:
– u : Uplift pressure (kN/m²)
– ?
w
: Unit w eigh t of w ater (9.81 kN/m³)
– h
w
: Piezometric head at the p oin t (m)
• T otal uplift force:
F
u
=u·A
where:
– F
u
: T otal uplift force (kN)
– A : Area sub jected to uplift pressure (m²)
5. Piping
• Critical h ydraulic gradien t (piping condition):
i
c
=
G-1
1+e
where:
– i
c
: Critical h ydraulic gradien t
– G : Sp ecific gra vit y of soil solids
– e : V oid ratio
• F actor of safet y against piping:
FS
piping
=
i
c
i
exit
where:
– i
exit
: Exit h ydraulic gradien t at do wnstream end
2
6. Capillarit y
• Capillary rise in soil:
h
c
=
4T cosa
?
w
d
where:
– h
c
: Heigh t of c apillary rise (m)
– T : Surface tension of w ater (N/m, t ypically 0.073 N/m)
– a : Con tact angle (degrees)
– d : Diameter of capillary tub e or p ore s ize (m)
– ?
w
: Unit w eigh t of w ater (9.81 kN/m³)
• Capillary pressure:
p
c
=?
w
·h
c
where:
– p
c
: Capillary pressure (kN/m²)
7. Seepage F orce
• Seepage force p er unit v olume:
j =i·?
w
where:
– j : Seepage force p er unit v olume (kN/m³)
– i : Hydraulic gradien t
• T otal seepage force:
F
s
=j·V
where:
– F
s
: T otal seepage force (kN)
– V : V olume of soil affecte d (m³)
8. Principle of Effectiv e Stress
• Effectiv e stress:
s
'
=s-u
where:
– s
'
: Effectiv e stress (kN/m²)
– s : T otal stress (kN/m²)
– u : P ore w ater pressure (kN/m²)
• P ore w ater pressure:
u=?
w
·h
w
where:
– h
w
: Heigh t of w ater column (m)
3
9. Quic ksand Condition
• Critical h ydraulic gradien t for quic ksand:
i
c
=
?
'
?
w
=
G-1
1+e
where:
– ?
'
: Submerged unit w eigh t (kN/m ³)
• Condition for quic ksand:
i=i
c
• Effectiv e stress at quic ksand:
s
'
=0
4