Formula Sheets: Earth Pressure & Retaining Walls | Soil Mechanics - Civil Engineering (CE) PDF Download

 Page 1


GA TE CE 2026 F orm ula Sheet: F oundation Engineering
1. Sub-surface In v estigations
• Standard P enetration T est (SPT) N-v alue correction:
N
60
=N ·
E
r
60
·
C
N
·C
E
·C
B
·C
R
0.77
where:
– N
60
: Corrected SPT N-v alue
– N : Field-measured SPT N-v alue
– E
r
: Hammer e?iciency (%)
– C
N
: Ov erburden correction factor, C
N
=
v
100
s
'
0
– s
'
0
: Effectiv e o v erburden pressure (kP a)
– C
E
, C
B
, C
R
: Corrections for energy , b orehole diameter, and ro d length
• Cone P enetration T est (CPT) cone resistance:
q
c
=
P
A
c
where:
– q
c
: Cone resistance (kP a)
– P : F orce on cone (kN)
– A
c
: Cross-s ectional area of cone (m²)
• Plate Load T est - Bearing capacit y:
q
u
=
P
u
A
p
where:
– q
u
: Ultimate b earing capacit y (kP a)
– P
u
: Ultimate load from plate load test (kN)
– A
p
: Area of plate (m²)
• Mo dulus of subgrade reaction:
k
s
=
q
d
where:
– k
s
: Mo dulus of subgrade reaction (kN/m³)
– q : Applied pressure (kP a)
– d : Settlemen t of plate (m)
2. Earth Pressure Theories - Rankine
• A ctiv e earth pressure co e?icien t:
K
a
=
1-sin?
1+sin?
= tan
2
(
45
?
-
?
2
)
where:
– K
a
: A ctiv e earth pressure co e?icien t
– ? : Angle of in ternal friction (degrees)
1
Page 2


GA TE CE 2026 F orm ula Sheet: F oundation Engineering
1. Sub-surface In v estigations
• Standard P enetration T est (SPT) N-v alue correction:
N
60
=N ·
E
r
60
·
C
N
·C
E
·C
B
·C
R
0.77
where:
– N
60
: Corrected SPT N-v alue
– N : Field-measured SPT N-v alue
– E
r
: Hammer e?iciency (%)
– C
N
: Ov erburden correction factor, C
N
=
v
100
s
'
0
– s
'
0
: Effectiv e o v erburden pressure (kP a)
– C
E
, C
B
, C
R
: Corrections for energy , b orehole diameter, and ro d length
• Cone P enetration T est (CPT) cone resistance:
q
c
=
P
A
c
where:
– q
c
: Cone resistance (kP a)
– P : F orce on cone (kN)
– A
c
: Cross-s ectional area of cone (m²)
• Plate Load T est - Bearing capacit y:
q
u
=
P
u
A
p
where:
– q
u
: Ultimate b earing capacit y (kP a)
– P
u
: Ultimate load from plate load test (kN)
– A
p
: Area of plate (m²)
• Mo dulus of subgrade reaction:
k
s
=
q
d
where:
– k
s
: Mo dulus of subgrade reaction (kN/m³)
– q : Applied pressure (kP a)
– d : Settlemen t of plate (m)
2. Earth Pressure Theories - Rankine
• A ctiv e earth pressure co e?icien t:
K
a
=
1-sin?
1+sin?
= tan
2
(
45
?
-
?
2
)
where:
– K
a
: A ctiv e earth pressure co e?icien t
– ? : Angle of in ternal friction (degrees)
1
• P assiv e earth pressure co e?icien t:
K
p
=
1+sin?
1-sin?
= tan
2
(
45
?
+
?
2
)
where:
– K
p
: P assiv e earth pressure co e?i cien t
• A ctiv e earth pressure (cohesionless soil):
s
a
=K
a
·? ·z
where:
– s
a
: A ctiv e earth pressure (kP a)
– ? : Unit w eigh t of soil (kN/m³)
– z : Depth b elo w surface (m)
• A ctiv e earth pressure (cohesiv e soil):
s
a
=K
a
·? ·z -2c
v
K
a
where:
– c : Cohesion (kP a)
• T otal activ e force:
P
a
=
1
2
K
a
?H
2
-2c
v
K
a
H
where:
– P
a
: T otal activ e force p er unit length (kN/m)
– H : Heigh t of retaining w all (m)
3. Earth Pressure Theories - Coulom b
• A ctiv e earth pressure co e?icien t:
K
a
=
cos
2
(?-a)
cos
2
a·cos(d +a)
[
1+
v
sin(?+d)sin(?-ß)
cos(d+a)cos(a-ß)
]
2
where:
– a : Angle of w all bac k with v ertical (degrees)
– d : Angle of w all friction (degrees)
– ß : Angle of bac kfill slop e (degrees)
• P assiv e earth pressure co e?icien t:
K
p
=
cos
2
(?+a)
cos
2
a·cos(d-a)
[
1-
v
sin(?+d)sin(?+ß)
cos(d-a)cos(a-ß)
]
2
• T otal activ e force:
P
a
=
1
2
K
a
?H
2
• T otal passiv e force:
P
p
=
1
2
K
p
?H
2
2
Page 3


GA TE CE 2026 F orm ula Sheet: F oundation Engineering
1. Sub-surface In v estigations
• Standard P enetration T est (SPT) N-v alue correction:
N
60
=N ·
E
r
60
·
C
N
·C
E
·C
B
·C
R
0.77
where:
– N
60
: Corrected SPT N-v alue
– N : Field-measured SPT N-v alue
– E
r
: Hammer e?iciency (%)
– C
N
: Ov erburden correction factor, C
N
=
v
100
s
'
0
– s
'
0
: Effectiv e o v erburden pressure (kP a)
– C
E
, C
B
, C
R
: Corrections for energy , b orehole diameter, and ro d length
• Cone P enetration T est (CPT) cone resistance:
q
c
=
P
A
c
where:
– q
c
: Cone resistance (kP a)
– P : F orce on cone (kN)
– A
c
: Cross-s ectional area of cone (m²)
• Plate Load T est - Bearing capacit y:
q
u
=
P
u
A
p
where:
– q
u
: Ultimate b earing capacit y (kP a)
– P
u
: Ultimate load from plate load test (kN)
– A
p
: Area of plate (m²)
• Mo dulus of subgrade reaction:
k
s
=
q
d
where:
– k
s
: Mo dulus of subgrade reaction (kN/m³)
– q : Applied pressure (kP a)
– d : Settlemen t of plate (m)
2. Earth Pressure Theories - Rankine
• A ctiv e earth pressure co e?icien t:
K
a
=
1-sin?
1+sin?
= tan
2
(
45
?
-
?
2
)
where:
– K
a
: A ctiv e earth pressure co e?icien t
– ? : Angle of in ternal friction (degrees)
1
• P assiv e earth pressure co e?icien t:
K
p
=
1+sin?
1-sin?
= tan
2
(
45
?
+
?
2
)
where:
– K
p
: P assiv e earth pressure co e?i cien t
• A ctiv e earth pressure (cohesionless soil):
s
a
=K
a
·? ·z
where:
– s
a
: A ctiv e earth pressure (kP a)
– ? : Unit w eigh t of soil (kN/m³)
– z : Depth b elo w surface (m)
• A ctiv e earth pressure (cohesiv e soil):
s
a
=K
a
·? ·z -2c
v
K
a
where:
– c : Cohesion (kP a)
• T otal activ e force:
P
a
=
1
2
K
a
?H
2
-2c
v
K
a
H
where:
– P
a
: T otal activ e force p er unit length (kN/m)
– H : Heigh t of retaining w all (m)
3. Earth Pressure Theories - Coulom b
• A ctiv e earth pressure co e?icien t:
K
a
=
cos
2
(?-a)
cos
2
a·cos(d +a)
[
1+
v
sin(?+d)sin(?-ß)
cos(d+a)cos(a-ß)
]
2
where:
– a : Angle of w all bac k with v ertical (degrees)
– d : Angle of w all friction (degrees)
– ß : Angle of bac kfill slop e (degrees)
• P assiv e earth pressure co e?icien t:
K
p
=
cos
2
(?+a)
cos
2
a·cos(d-a)
[
1-
v
sin(?+d)sin(?+ß)
cos(d-a)cos(a-ß)
]
2
• T otal activ e force:
P
a
=
1
2
K
a
?H
2
• T otal passiv e force:
P
p
=
1
2
K
p
?H
2
2
4. Stabilit y of Slop es - Infinite Slop es
• F actor of safet y (cohesionless soil, infinite slop e):
FS =
tan?
tanß
where:
– FS: F actor of safet y
– ß : Slop e angle (degrees)
• F actor of safet y (cohesiv e soil, infinite slop e):
FS =
c
?zsinßcosß
+
tan?
tanß
where:
– z : Depth to failure plane (m)
• Critical depth (cohesiv e soil):
z
c
=
c
?(sinßcosß -cos
2
ßtan?)
5. Stabilit y of Slop es - Finite Slop es (Bishop’s Simplified Metho d)
• F actor of safet y (Bishop’s metho d):
FS =
?
[
c
'
bi+(Wi-uibi)tan?
'
cos?i+
sin?
i
tan?
'
FS
]
?
W
i
sin?
i
where:
– c
'
: Effectiv e cohesion (kP a)
– ?
'
: Effectiv e angle of in ternal friction (degrees)
– b
i
: Width of slice i (m)
– W
i
: W eigh t of slice i (kN)
– u
i
: P ore w ater pressure at base of slice i (kP a)
– ?
i
: Angle of slice base with horizon tal (degrees)
• W eigh t of slice:
W
i
=?h
i
b
i
where:
– h
i
: Heigh t of slice i (m)
– ? : Unit w eigh t of soil (kN/m³)
3