Formula Sheets: Pile Foundation

 Page 1


GA TE CE 2026 F orm ula Sheet: F oundation Engineering - Piles
and Pile Groups
1. Axial Load Capacit y of Piles in Sands and Cla ys
• Ultimate axial load capacit y of a pile:
Q
u
=Q
p
+Q
f
where:
– Q
u
: Ultimate axial load c apacit y (kN)
– Q
p
: End-b earing capacit y (kN)
– Q
f
: Skin friction capacit y (kN)
• End-b earing capacit y:
Q
p
=q
p
·A
p
where:
– q
p
: Unit end-b earin g capacit y (kP a)
– A
p
: Cross-sectional area of pile tip (m²)
• End-b earing in sands:
q
p
=s
'
v
·N
q
where:
– s
'
v
: Effectiv e o v erburden stress at pile tip (kP a)
– N
q
: Bearing capacit y fac tor (function of ?
'
)
• End-b earing in cla ys (undrained):
q
p
=c
u
·N
c
where:
– c
u
: Undrained shear strength (kP a)
– N
c
: Bearing capacit y factor (t ypically 9 for deep piles)
• Skin friction capacit y:
Q
f
=
?
(f
s
·A
s
)
where:
– f
s
: Unit skin friction (kP a)
– A
s
: Surface area of pile shaft in soil la y er (m²)
• Skin friction in sands:
f
s
=K ·s
'
v
·tand
where:
– K : Earth pressure co e?icien t (t ypically 0.5–1.0)
– d : F riction angle b et w een pile and soil (degrees)
• Skin friction in cla ys (undrained):
f
s
=a ·c
u
where:
– a : A dhesion factor (0.4–1.0, dep ends on c
u
)
• Safe axial load capacit y:
Q
safe
=
Q
u
FS
where:
– FS: F actor of safet y (t ypically 2.5–3)
1
Page 2


GA TE CE 2026 F orm ula Sheet: F oundation Engineering - Piles
and Pile Groups
1. Axial Load Capacit y of Piles in Sands and Cla ys
• Ultimate axial load capacit y of a pile:
Q
u
=Q
p
+Q
f
where:
– Q
u
: Ultimate axial load c apacit y (kN)
– Q
p
: End-b earing capacit y (kN)
– Q
f
: Skin friction capacit y (kN)
• End-b earing capacit y:
Q
p
=q
p
·A
p
where:
– q
p
: Unit end-b earin g capacit y (kP a)
– A
p
: Cross-sectional area of pile tip (m²)
• End-b earing in sands:
q
p
=s
'
v
·N
q
where:
– s
'
v
: Effectiv e o v erburden stress at pile tip (kP a)
– N
q
: Bearing capacit y fac tor (function of ?
'
)
• End-b earing in cla ys (undrained):
q
p
=c
u
·N
c
where:
– c
u
: Undrained shear strength (kP a)
– N
c
: Bearing capacit y factor (t ypically 9 for deep piles)
• Skin friction capacit y:
Q
f
=
?
(f
s
·A
s
)
where:
– f
s
: Unit skin friction (kP a)
– A
s
: Surface area of pile shaft in soil la y er (m²)
• Skin friction in sands:
f
s
=K ·s
'
v
·tand
where:
– K : Earth pressure co e?icien t (t ypically 0.5–1.0)
– d : F riction angle b et w een pile and soil (degrees)
• Skin friction in cla ys (undrained):
f
s
=a ·c
u
where:
– a : A dhesion factor (0.4–1.0, dep ends on c
u
)
• Safe axial load capacit y:
Q
safe
=
Q
u
FS
where:
– FS: F actor of safet y (t ypically 2.5–3)
1
2. Pile Load T est
• Ultimate load from pile load test (Da visson’s criterion):
Q
u
= Load at settlemen t =
4B
120
+
Q
elastic
·L
A
e
E
p
where:
– Q
u
: Ultimate load from load-settlemen t curv e (kN)
– B : Pile diameter or width (mm)
– Q
elastic
: Load in elastic range (kN)
– L : Pile length (m)
– A
e
: Cross-sectional area of pile (m²)
– E
p
: Mo dulus of elasticit y of pile m aterial (kP a)
• Safe load from pile load test:
Q
safe
=
Q
u
FS
where:
– FS: F actor of safet y (t ypically 2–2.5 for load test)
• Settlemen t at w orking load:
S
w
=
Q
w
·L
A
e
·E
p
+S
soil
where:
– S
w
: Settlemen t at w orking load (m)
– Q
w
: W orking load (kN)
– S
soil
: Soil settlem en t comp onen t (m)
3. Pil e under Lateral Loading
• Lateral load capacit y (Broms’ metho d, short pile in sand):
H
u
=
3?Bz
2
K
p
2
where:
– H
u
: Ultimate lateral load capacit y (kN)
– ? : Unit w eigh t of soil (kN/m³)
– B : Pile diameter or width (m)
– z : Depth of fixit y (m)
– K
p
: P assiv e earth pressure co e?i cien t, K
p
=
1+sin?
1-sin?
– ? : Angle of in ternal friction (degrees)
• Lateral load capacit y (short pile in cla y):
H
u
= 9c
u
Bz
where:
– c
u
: Undrained shear strength (kP a)
• Deflection under lateral load (elastic metho d):
y =
H
k
h
z
where:
– y : Lateral deflection (m)
– H : Applied lateral load (kN)
– k
h
: Co e?icien t of horizon tal subgrade reaction (kN/m³)
2
Page 3


GA TE CE 2026 F orm ula Sheet: F oundation Engineering - Piles
and Pile Groups
1. Axial Load Capacit y of Piles in Sands and Cla ys
• Ultimate axial load capacit y of a pile:
Q
u
=Q
p
+Q
f
where:
– Q
u
: Ultimate axial load c apacit y (kN)
– Q
p
: End-b earing capacit y (kN)
– Q
f
: Skin friction capacit y (kN)
• End-b earing capacit y:
Q
p
=q
p
·A
p
where:
– q
p
: Unit end-b earin g capacit y (kP a)
– A
p
: Cross-sectional area of pile tip (m²)
• End-b earing in sands:
q
p
=s
'
v
·N
q
where:
– s
'
v
: Effectiv e o v erburden stress at pile tip (kP a)
– N
q
: Bearing capacit y fac tor (function of ?
'
)
• End-b earing in cla ys (undrained):
q
p
=c
u
·N
c
where:
– c
u
: Undrained shear strength (kP a)
– N
c
: Bearing capacit y factor (t ypically 9 for deep piles)
• Skin friction capacit y:
Q
f
=
?
(f
s
·A
s
)
where:
– f
s
: Unit skin friction (kP a)
– A
s
: Surface area of pile shaft in soil la y er (m²)
• Skin friction in sands:
f
s
=K ·s
'
v
·tand
where:
– K : Earth pressure co e?icien t (t ypically 0.5–1.0)
– d : F riction angle b et w een pile and soil (degrees)
• Skin friction in cla ys (undrained):
f
s
=a ·c
u
where:
– a : A dhesion factor (0.4–1.0, dep ends on c
u
)
• Safe axial load capacit y:
Q
safe
=
Q
u
FS
where:
– FS: F actor of safet y (t ypically 2.5–3)
1
2. Pile Load T est
• Ultimate load from pile load test (Da visson’s criterion):
Q
u
= Load at settlemen t =
4B
120
+
Q
elastic
·L
A
e
E
p
where:
– Q
u
: Ultimate load from load-settlemen t curv e (kN)
– B : Pile diameter or width (mm)
– Q
elastic
: Load in elastic range (kN)
– L : Pile length (m)
– A
e
: Cross-sectional area of pile (m²)
– E
p
: Mo dulus of elasticit y of pile m aterial (kP a)
• Safe load from pile load test:
Q
safe
=
Q
u
FS
where:
– FS: F actor of safet y (t ypically 2–2.5 for load test)
• Settlemen t at w orking load:
S
w
=
Q
w
·L
A
e
·E
p
+S
soil
where:
– S
w
: Settlemen t at w orking load (m)
– Q
w
: W orking load (kN)
– S
soil
: Soil settlem en t comp onen t (m)
3. Pil e under Lateral Loading
• Lateral load capacit y (Broms’ metho d, short pile in sand):
H
u
=
3?Bz
2
K
p
2
where:
– H
u
: Ultimate lateral load capacit y (kN)
– ? : Unit w eigh t of soil (kN/m³)
– B : Pile diameter or width (m)
– z : Depth of fixit y (m)
– K
p
: P assiv e earth pressure co e?i cien t, K
p
=
1+sin?
1-sin?
– ? : Angle of in ternal friction (degrees)
• Lateral load capacit y (short pile in cla y):
H
u
= 9c
u
Bz
where:
– c
u
: Undrained shear strength (kP a)
• Deflection under lateral load (elastic metho d):
y =
H
k
h
z
where:
– y : Lateral deflection (m)
– H : Applied lateral load (kN)
– k
h
: Co e?icien t of horizon tal subgrade reaction (kN/m³)
2
4. Pile Group E?iciency
• Pile group e?iciency:
? =
Q
u, group
n ·Q
u, single
where:
– ? : Group e?iciency (t ypically 0.7–1.0)
– Q
u, group
: Ultimate capacit y of pile group (kN)
– Q
u, single
: Ultimate capacit y of single pile (kN)
– n : Num b er of piles in group
• Ultimate capacit y of pile group (cohesionless soils):
Q
u, group
= min(n ·Q
u
,A
b
·q
b
+P ·f
s
)
where:
– A
b
: Base area of group blo c k (m²)
– q
b
: End-b earing capacit y of group blo c k (kP a)
– P : P erimeter of group blo c k (m)
– f
s
: Unit skin friction along group p erimeter (kP a)
• Ultimate capacit y of pile group (cohesiv e soils):
Q
u, group
= min(n ·Q
u
,9c
u
A
b
+ac
u
PL)
where:
– L : Em b edded length of piles (m)
– a : A dhesion factor
5. Ne gativ e Skin F riction
• Negativ e skin friction force:
Q
n
=f
n
·A
s
where:
– Q
n
: Negativ e skin friction force (kN)
– f
n
: Unit negativ e s kin friction (kP a)
– A
s
: Surface area of pile shaft in settling la y er (m²)
• Negativ e skin friction in cla ys:
f
n
=a ·c
u
where:
– a : A dhesion factor (0.3–0.6 for negativ e friction)
– c
u
: Undrained shear strength (kP a)
• Negativ e skin friction in sands:
f
n
=K ·s
'
v
·tand
where:
– K : Earth pressure co e?icien t (t ypically 0.3–0.5)
– s
'
v
: Effectiv e o v erburden stress (kP a)
– d : F riction angle b et w een pile and soil (degrees)
• T otal load on pile including negativ e skin friction:
Q
total
=Q
applied
+Q
n
where:
– Q
applied
: Applied axial load ( kN)
3