16 design of pile cap
TRANSCRIPT
14. Design of Pile Cap
14.1 Determination of Pile Cap Dimension
Minimum number of piles
nPD PL+
Qe=
where PD PL, = dead and live loads on pile cap
Qe = effective bearing capacity of pile
Qe Qa 20kN
m33 D⋅( )2⋅ H⋅−=
D = pile dimension
H = depth of foundation
20kN
m3= average unit weight of soil and concrete
Qa = allowable bearing capacity of pile with FS 2.5= 4..
Distance between piles = 2D 4D..
Distance from pile to concrete face = D2
150mm≤ 200mm..
Reactions of piles
Page-237-
RiPn
Mx yi⋅
1
n
k
yk( )2∑=
+My xi⋅
1
n
k
xk( )2∑=
+ Qu≤=
P 1.2 PD⋅ 1.6 PL⋅+=
Mx My, = bending moments about x- and y-axis
xi yi, = location of pile
Qu Qa1.2 PD⋅ 1.6 PL⋅+
PD PL+⋅=
Page-238-
Suggested minimum cener-to-center pile spacing by several building codes are as follows:
1.2 Determination of Depth of Pile Cap
A. Case of Two-way Shear
Vu ϕVc≤
where Vu = punching shear
ϕVc = punching shear strength
ϕ 0.75= is strength reduction factor for shear
Vu Routside∑= Qu noutside⋅=
ϕVc ϕ min
4 f'c⋅ b0⋅ d⋅
24β
+⎛⎜⎝
⎞⎟⎠
f'c⋅ b0⋅ d⋅
αs d⋅
b02+
⎛⎜⎝
⎞⎟⎠
f'c⋅ b0⋅ d⋅
⎡⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎦
⎡⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎦
⋅= (in psi)
ϕVc ϕ min
0.33 f'c⋅ b0⋅ d⋅
0.17 12β
+⎛⎜⎝
⎞⎟⎠
⋅ f'c⋅ b0⋅ d⋅
0.083αs d⋅
b02+
⎛⎜⎝
⎞⎟⎠
⋅ f'c⋅ b0⋅ d⋅
⎡⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎦
⎡⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎦
⋅= (in MPa)
αs 40
30
20
= for interior columns
for edge columnsfor conner columns
β = ratio of long side to short side of column
Page-239-
d = effective depth of footing
b0 bc d+( ) hc d+( )+⎡⎣ ⎤⎦ 2⋅= is critical section parameter
critical sectionfor two−way shear
critical sectionfor one−way shear B
L
S S DD
SS
DD
B. Case of Pucnhing at coner pile
Ru ϕVcorner≤
where Ru = reaction of pile
ϕVcorner = punching shear strength at the coner pile
D
d/2
b0
ϕVconer ϕ min
0.33 f'c⋅ b0⋅ d⋅
0.17 12β
+⎛⎜⎝
⎞⎟⎠
⋅ f'c⋅ b0⋅ d⋅
0.083αs d⋅
b02+
⎛⎜⎝
⎞⎟⎠
⋅ f'c⋅ b0⋅ d⋅
⎡⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎦
⎡⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎦
⋅= (in MPa)
Page-240-
b0 π D d+( )⋅= for circular pile
b0 D d+( ) 4⋅= for square pile
C. Case of Beam Shear or One-way Shear
Vu1 ϕVc1≤ Vu2 ϕVc2≤
where Vu1 Vu2, = beam shears
ϕVc1 ϕVc2, = beam shear strength
Vu1 Rleft∑= Qu nleft⋅= Vu2 Rtop∑= Qu ntop⋅=
ϕVc1 ϕ 2⋅ f'c⋅ B⋅ d⋅= (in psi)
ϕVc1 ϕ 0.17⋅ f'c⋅ B⋅ d⋅= (in MPa)
ϕVc2 ϕ 2⋅ f'c⋅ L⋅ d⋅= (in psi)
ϕVc2 ϕ 0.17⋅ f'c⋅ L⋅ d⋅= (in MPa)
14.3 Determination of Steel Area
Section: Rectangular singly reinforced.
Required strength:
Mu1 Ri∑ xihc2
−⎛⎜⎝
⎞⎟⎠
⋅=
Mu2 Ri∑ yibc2
−⎛⎜⎝
⎞⎟⎠
⋅=
Example 14.1
Required strength PD 361.5kN:= PL 91.27kN:=
Foundation depth H 1.5m:=
bc 250mm:= hc 250mm:=Column dimensions
Material f'c 25MPa:= fy 400MPa:=
Page-241-
Allowable bearing capacity of pile Qa 65kN:=
Dimension of pile D 200mm:=
Solution
Effective bearing capacity of soil
Qe Qa 20kN
m33 D⋅( )2⋅ H⋅− 54.2 kN⋅=:=
Number of piles
nPD PL+
Qe8.354=:= Use n 9:=
Pile spacing S 3 D⋅ 600 mm⋅=:=
Dimension of pile cap
B 2 S⋅ 2 D⋅+ 1.6m=:= L B 1.6m=:=
1600
1600
600 600 200200
600
600
200
200
Determination of depth of pile cap
Depth of pile cap h 350mm:=
d h 50mm 14mm+14mm
2+⎛⎜
⎝⎞⎟⎠
− 279 mm⋅=:=
Page-242-
Design bearing capacity of pile
Qu Qa1.2 PD⋅ 1.6 PL⋅+
PD PL+⋅ 83.241 kN⋅=:=
Two-way shear from the face of column distane d2
139.5 mm⋅=
Vu Qu 8⋅ 665.929 kN⋅=:=
Two-way shear strength
b0 bc d+ hc+ d+( ) 2⋅ 2.116m=:=
ϕ 0.75:=
βhcbc
1=:= αs 40:= for interoir column
ϕVc ϕ min
0.33MPaf'c
MPa⋅ b0⋅ d⋅
0.17 12β
+⎛⎜⎝
⎞⎟⎠
MPa⋅f'c
MPa⋅ b0⋅ d⋅
0.083αs d⋅
b02+
⎛⎜⎝
⎞⎟⎠
MPa⋅f'c
MPa⋅ b0⋅ d⋅
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦
⋅:=
ϕVc 730.575 kN⋅=
Two_way_shear "is not critical" ϕVc Vu≥if
"is critical" otherwise
:=
Two_way_shear "is not critical"=
Beam shears or one way shear
Vu1 Qu 3⋅ 249.723 kN⋅=:=
Vu2 Qu 3⋅ 249.723 kN⋅=:=
Beam shear strengths
ϕVc1 ϕ 0.17⋅ MPaf'c
MPa⋅ B⋅ d⋅ 284.58 kN⋅=:=
Page-243-
ϕVc2 ϕ 0.17⋅ MPaf'c
MPa⋅ L⋅ d⋅ 284.58 kN⋅=:=
Beam_shear "is not critical" ϕVc1 Vu1≥ ϕVc2 Vu2≥∧if
"is critical" otherwise
:=
Beam_shear "is not critical"=
Steel reinforcements in direction L 1.6m=
b B 1.6m=:= d 279 mm⋅=
Mu Qu Shc2
−⎛⎜⎝
⎞⎟⎠
⋅ 3⋅ 118.619 kN m⋅⋅=:=
RMu
0.9 b⋅ d2⋅1.058 MPa⋅=:=
ρ 0.85f'cfy
⋅ 1 1 2R
0.85 f'c⋅⋅−−
⎛⎜⎝
⎞⎟⎠
⋅ 0.00271=:=
ρmin max0.25MPa
f'cMPa
⋅
fy
1.4MPafy
,
⎛⎜⎜⎜⎝
⎞⎟⎟⎟⎠
0.0035=:=
As max ρ ρmin, ( ) b⋅ d⋅ 15.624 cm2⋅=:=
As0π 14mm( )2⋅
41.539 cm2⋅=:=
sL FloorAs0Asb
10mm, ⎛⎜⎜⎝
⎞⎟⎟⎠
150 mm⋅=:= nLB 50mm 2⋅−
sL1+ 11=:=
As_L nL As0⋅ 16.933 cm2⋅=:=
Shrinkage steel reinforcement
As.t 0.0018 b⋅ h⋅ 10.08 cm2⋅=:= < As_L 16.933 cm2⋅=
Page-244-
Steel reinforcements in direction B 1.6m=
b L 1.6m=:= d 279 mm⋅=
Mu Qu Sbc2
−⎛⎜⎝
⎞⎟⎠
⋅ 3⋅ 118.619 kN m⋅⋅=:=
RMu
0.9 b⋅ d2⋅1.058 MPa⋅=:=
ρ 0.85f'cfy
⋅ 1 1 2R
0.85 f'c⋅⋅−−
⎛⎜⎝
⎞⎟⎠
⋅ 0.00271=:=
ρmin max0.25MPa
f'cMPa
⋅
fy
1.4MPafy
,
⎛⎜⎜⎜⎝
⎞⎟⎟⎟⎠
0.0035=:=
As max ρ ρmin, ( ) b⋅ d⋅ 15.624 cm2⋅=:=
sB FloorAs0Asb
10mm, ⎛⎜⎜⎝
⎞⎟⎟⎠
150 mm⋅=:= nBB 50mm 2⋅−
sB1+ 11=:=
As_B nB As0⋅ 16.933 cm2⋅=:=
Shrinkage steel reinforcement
<As.t 0.0018 b⋅ h⋅ 10.08 cm2⋅=:= As_B 16.933 cm2⋅=
1600
1600
11DB14@150
Page-245-
Example 14.2
Required strength
PD 1794.572kN:= PL 427.5kN:=
MD 25.55kN m⋅:= ML 12.65kN m⋅:=
Pile cap depth H 1.5m:=
bc 400mm:= hc 400mm:=Column stud
Material f'c 25MPa:= fy 400MPa:=
Dimension of pile D 300mm:=
Allowable bearing capacity of pile Qa 367.8kN:=
Solution
Effective bearing capacity of soil
Qe Qa 20kN
m33 D⋅( )2⋅ H⋅− 343.5 kN⋅=:=
Number of piles
nPD PL+
Qe6.469=:= Use n 7:=
Location of pile
X
1000−
500−
500−
0
500
500
1000
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠
mm:= Y
0
750
750−
0
750
750−
0
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠
mm:=
Dimension of pile cap
B max Y( ) D+( ) 2⋅:= L max X( ) D+( ) 2⋅:=B
L⎛⎜⎝
⎞⎟⎠
2100
2600⎛⎜⎝
⎞⎟⎠
mm⋅=
Page-246-
X012
L−
L
L
L−
L−
⎛⎜⎜⎜⎜⎜⎝
⎞⎟⎟⎟⎟⎟⎠
⋅:= Y012
B−
B−
B
B
B−
⎛⎜⎜⎜⎜⎜⎝
⎞⎟⎟⎟⎟⎟⎠
⋅:=
2− 1− 0 1 2
2−
1−
1
2Pile Locations
Reaction of pile
Pu 1.2PD 1.6PL+:= Mu 1.2MD 1.6ML+:=
ORIGIN 1:= n rows X( ):= n 7=
i 1 n..:= RiPun
Mu Xi⋅
1
n
k
Xk( )2∑=
+:=
Page-247-
R
388.389
396.872
396.872
405.355
413.839
413.839
422.322
⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝
⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠
kN⋅=
Ultimate bearing capacity of pile
Qu Qa1.2 PD⋅ 1.6PL+
PD PL+⋅ 469.664 kN⋅=:=
Pile "is OK." max R( ) Qu≤if
"is not good." otherwise
:= Pile "is OK."=
Depth of pile cap h 750mm:=
d h 50mm 20mm+20mm
2+⎛⎜
⎝⎞⎟⎠
− 670 mm⋅=:=
Punching shear from the face of column distane d2
335 mm⋅=
Graphs
2− 1− 0 1 2
2−
1−
1
2
critical section of two-way shear
Vu Qu 6⋅ 2817.985 kN⋅=:=
Page-248-
Punching shear strength
b0 bc d+ hc+ d+( ) 2⋅ 4.28m=:=
d 670 mm⋅= ϕ 0.75:=
βhcbc
1=:= αs 40:= for interior
ϕVc ϕ min
0.33MPaf'c
MPa⋅ b0⋅ d⋅
0.17 12β
+⎛⎜⎝
⎞⎟⎠
MPa⋅f'c
MPa⋅ b0⋅ d⋅
0.083αs d⋅
b02+
⎛⎜⎝
⎞⎟⎠
MPa⋅f'c
MPa⋅ b0⋅ d⋅
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦
⋅:=
ϕVc 3548.655 kN⋅=
The_Cap "is not punching" ϕVc Vu≥if
"is punching" otherwise
:=
The_Cap "is not punching"=
Beam shears or one way shear from the face of column distance d 670 mm⋅=
2− 1− 0 1 2
2−
1−
1
2
critical section of beam shear
Page-249-
Vu1 Qu 1⋅ 469.664 kN⋅=:= Vu2 Qu 2⋅ 939.328 kN⋅=:=
Beam shear strengths
ϕVc1 0.75 0.17⋅ MPaf'c
MPa⋅ B⋅ d⋅ 896.962 kN⋅=:=
ϕVc2 0.75 0.17⋅ MPaf'c
MPa⋅ L⋅ d⋅ 1110.525 kN⋅=:=
The_Cap "is not beam shear" ϕVc1 Vu1≥ ϕVc2 Vu2≥∧if
"is beam shear" otherwise
:=
The_Cap "is not beam shear"=
- Steel reinforcements in direction L 2.6m=
b B 2.1m=:= d 670 mm⋅=
Ru max R( ):=
Mx
continue Xi 0=if
Mi Ru Xihc2
−⎛⎜⎝
⎞⎟⎠
⋅←
i 1rows X( )
2..∈for
M
:=
Mx
337.857
126.697
126.697
⎛⎜⎜⎜⎝
⎞⎟⎟⎟⎠
kN m⋅⋅=
Mu1 Mx∑ 591.251 kN m⋅⋅=:=
R1Mu1
0.9 b⋅ d2⋅0.697 MPa⋅=:=
ρ 0.85f'cfy
⋅ 1 1 2R1
0.85 f'c⋅⋅−−
⎛⎜⎜⎝
⎞⎟⎟⎠
⋅ 0.00177=:=
Page-250-
ρmin max0.25MPa
f'cMPa
⋅
fy
1.4MPafy
,
⎛⎜⎜⎜⎝
⎞⎟⎟⎟⎠
0.0035=:=
As max ρ ρmin, ( ) b⋅ d⋅ 49.245 cm2⋅=:=
As0π 22mm( )2⋅
43.801 cm2⋅=:=
s1 FloorAs0Asb
10mm, ⎛⎜⎜⎝
⎞⎟⎟⎠
160 mm⋅=:=
n1 ceilB 50mm 2⋅−
s11+⎛
⎜⎝
⎞⎟⎠
14=:=
Asx n1 As0⋅ 53.219 cm2⋅=:=
Top bars (shrinkage reinforcement)
As.t 0.0018 b⋅ h⋅ 28.35 cm2⋅=:=
As1π 18mm( )2⋅
42.545 cm2⋅=:=
st FloorAs1As.t
b
10mm, ⎛⎜⎜⎝
⎞⎟⎟⎠
180 mm⋅=:=
nt floorB 50mm 2⋅−
st1+⎛
⎜⎝
⎞⎟⎠
12=:=
- Steel reinforcements in direction B 2.1m=
b L 2.6m=:= d 670 mm⋅=
Page-251-
My
continue Yi 0=if
Mi Ru Yihc2
−⎛⎜⎝
⎞⎟⎠
⋅←
i 1rows Y( )
2..∈for
M
:=
My
0
232.277
232.277
⎛⎜⎜⎜⎝
⎞⎟⎟⎟⎠
kN m⋅⋅=
Mu2 My∑ 464.554 kN m⋅⋅=:=
R2Mu2
0.9 b⋅ d2⋅0.442 MPa⋅=:=
ρ 0.85f'cfy
⋅ 1 1 2R2
0.85 f'c⋅⋅−−
⎛⎜⎜⎝
⎞⎟⎟⎠
⋅ 0.00112=:=
ρmin max0.25MPa
f'cMPa
⋅
fy
1.4MPafy
,
⎛⎜⎜⎜⎝
⎞⎟⎟⎟⎠
0.0035=:=
As max ρ ρmin, ( ) b⋅ d⋅ 60.97 cm2⋅=:=
s2 FloorAs0Asb
10mm, ⎛⎜⎜⎝
⎞⎟⎟⎠
160 mm⋅=:=
n2 ceilL 50mm 2⋅−
s21+⎛
⎜⎝
⎞⎟⎠
17=:=
Asy n2 As0⋅ 64.623 cm2⋅=:=
Top bars (shrinkage reinforcement)
As.t 0.0018 b⋅ h⋅ 35.1 cm2⋅=:=
st FloorAs1As.t
b
10mm, ⎛⎜⎜⎝
⎞⎟⎟⎠
180 mm⋅=:=
Page-252-
nt floorB 50mm 2⋅−
st1+⎛
⎜⎝
⎞⎟⎠
12=:=
2− 1− 0 1 2
2−
1−
1
2
Bottom Steel Bar
Page-253-