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-