Design of Pile Cap

14. Design of Pile Cap 14.1 Determination of Pile Cap Dimension Minimum number of piles n= where

PD + PL Qe

P D , PL

= dead and live loads on pile cap

Qe

= effective bearing capacity of pile

Qe = Qa − 20

kN 3

2

⋅ ( 3⋅ D) ⋅ H

m

D

= pile dimension

H

= depth of foundation

20

kN 3

= average unit weight of soil and concrete

m Qa

= allowable bearing capacity of pile with FS = 2.5 .. 4

Distance between piles = 2D .. 4D Distance from pile to concrete face =

D ≤ 150mm .. 200mm 2

Reactions of piles

Page-237-

Ri =

P + n

Mx⋅ yi n

∑

( yk) 2

k=1

My⋅ xi

+

n

∑

≤ Qu

( xk) 2

k=1

P = 1.2⋅ PD + 1.6⋅ PL Mx , My

= bending moments about x- and y-axis

xi , yi

= location of pile

Qu = Qa⋅

1.2⋅ 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

= punching shear

Vu

ϕVc = punching shear strength ϕ = 0.75 is strength reduction factor for shear Vu =

∑ Routside = Qu⋅ noutside

⎡⎡ ⎤⎤ 4⋅ f'c⋅ b0⋅ d ⎢⎢ ⎥⎥ ⎢⎢ ⎛ ⎥⎥ 4⎞ ⎜ 2 + ⎟ ⋅ f'c⋅ b0⋅ d ⎥⎥ ⎢ ⎢ ϕVc = ϕ⋅ min ⎝ β⎠ ⎢⎢ ⎥⎥ ⎢⎢ ⎛ αs⋅ d ⎞ ⎥⎥ + 2 ⋅ f' ⋅ b ⋅ d ⎜ ⎟ ⎢⎢ b c 0 ⎥⎥ ⎣⎣ ⎝ 0 ⎠ ⎦⎦

(in psi)

⎡⎡ ⎤⎤ 0.33⋅ f'c⋅ b0⋅ d ⎢⎢ ⎥⎥ ⎢⎢ ⎛ 1 + 2 ⎞ ⋅ f' ⋅ b ⋅ d ⎥⎥ 0.17 ⋅ ⎜ ⎟ c 0 ⎥⎥ ϕVc = ϕ⋅ min ⎢⎢ ⎝ β⎠ ⎢⎢ ⎥⎥ α ⋅ d ⎛ s ⎞ ⎢⎢ ⎥⎥ ⎢⎢ 0.083⋅ ⎜ b + 2⎟ ⋅ f'c⋅ b0⋅ d ⎥⎥ ⎣⎣ ⎝ 0 ⎠ ⎦⎦

(in MPa)

αs =

40

for interior columns

30

for edge columns for conner columns

20 β

= ratio of long side to short side of column Page-239-

= effective depth of footing

d

(

) (

)

critical section parameter b0 = ⎡ bc + d + hc + d ⎤is ⎣ ⎦⋅2 critical section for two−way shear S

S

D

S

D

D

D

S

B

critical section for one−way shear

L

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

⎡⎡ ⎤⎤ 0.33⋅ f'c⋅ b0⋅ d ⎢⎢ ⎥⎥ ⎢⎢ ⎛ 1 + 2 ⎞ ⋅ f' ⋅ b ⋅ d ⎥⎥ 0.17 ⋅ ⎜ ⎟ c 0 ⎥⎥ ϕVconer = ϕ⋅ min ⎢⎢ ⎝ β⎠ ⎢⎢ ⎥⎥ α ⋅ d ⎛ s ⎞ ⎢⎢ ⎥⎥ ⎢⎢ 0.083⋅ ⎜ b + 2⎟ ⋅ f'c⋅ b0⋅ d ⎥⎥ ⎣⎣ ⎝ 0 ⎠ ⎦⎦ Page-240-

(in MPa)

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 where

Vu1 , Vu2

Vu2 ≤ ϕVc2 = 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:

⎛

hc ⎞

⎝

2⎠

⎛

bc ⎞

⎝

2⎠

Mu1 =

∑

Ri⋅ ⎜ xi −

Mu2 =

∑

Ri⋅ ⎜ yi −

⎟

⎟

Example 14.1 Required strength

PD := 361.5kN

Foundation depth

H := 1.5m

Column dimensions

bc := 250mm

hc := 250mm

Material

f'c := 25MPa

fy := 400MPa

Page-241-

PL := 91.27kN

Allowable bearing capacity of pile

Qa := 65kN

Dimension of pile

D := 200mm

Solution Effective bearing capacity of soil Qe := Qa − 20

kN 3

2

⋅ ( 3⋅ D) ⋅ H = 54.2⋅ kN

m

Number of piles n :=

PD + PL

Pile spacing

Qe

= 8.354

n := 9

Use

S := 3⋅ D = 600⋅ mm

Dimension of pile cap B := 2⋅ S + 2⋅ D = 1.6 m 600

600

200

200

600

1600

600

200

200

L := B = 1.6 m

1600

Determination of depth of pile cap Depth of pile cap

h := 350mm d := h − ⎛⎜ 50mm + 14mm +

⎝

Page-242-

14mm ⎞ ⎟ = 279⋅ mm 2 ⎠

Design bearing capacity of pile Qu := Qa⋅

1.2⋅ PD + 1.6⋅ PL PD + PL

= 83.241⋅ kN

Two-way shear from the face of column distane

d = 139.5⋅ mm 2

Vu := Qu⋅ 8 = 665.929⋅ kN Two-way shear strength

(

)

b0 := bc + d + hc + d ⋅ 2 = 2.116 m ϕ := 0.75 β :=

hc bc

=1

αs := 40

for interoir column

⎡⎡ ⎤⎤ f'c ⎢⎢ ⎥⎥ 0.33MPa⋅ ⋅ b0⋅ d MPa ⎢⎢ ⎥⎥ ⎢⎢ ⎥⎥ f'c 2⎞ ⎢ ⎢ ⎛ ϕVc := ϕ⋅ min 0.17⋅ ⎜ 1 + ⎟ MPa⋅ ⋅ b0⋅ d ⎥⎥ ⎢⎢ ⎥⎥ MPa ⎝ β⎠ ⎢⎢ ⎥⎥ f' α ⋅ d ⎛ ⎞ ⎢⎢ ⎥⎥ c s ⎢⎢ 0.083⋅ ⎜ b + 2⎟ MPa⋅ MPa ⋅ b0⋅ d ⎥⎥ ⎣⎣ ⎝ 0 ⎠ ⎦⎦ ϕVc = 730.575⋅ kN Two_way_shear :=

"is not critical" if ϕVc ≥ Vu "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⋅ 0.17MPa⋅

f'c ⋅ B⋅ d = 284.58⋅ kN MPa

Page-243-

f'c ϕVc2 := ϕ⋅ 0.17MPa⋅ ⋅ L⋅ d = 284.58⋅ kN MPa Beam_shear :=

"is not critical" if ϕVc1 ≥ Vu1 ∧ ϕVc2 ≥ Vu2 "is critical" otherwise

Beam_shear = "is not critical" Steel reinforcements in direction L = 1.6 m b := B = 1.6 m

d = 279⋅ mm

hc ⎞ ⎛ Mu := Qu⋅ ⎜ S − ⎟ ⋅ 3 = 118.619⋅ kN⋅ m 2⎠ ⎝ Mu

R :=

2

= 1.058⋅ MPa

0.9⋅ b⋅ d

f'c ⎛ ρ := 0.85⋅ ⋅ ⎜ 1 − fy

1 − 2⋅

⎝

R ⎞ = 0.00271 0.85⋅ f'c ⎟

⎠

⎞ ⎛ f'c ⎜ 0.25MPa⋅ ⎟ MPa 1.4MPa ⎟ ⎜ ρmin := max , = 0.0035 ⎜ fy fy ⎟ ⎝ ⎠

(

)

2

As := max ρ , ρmin ⋅ b⋅ d = 15.624⋅ cm As0 :=

π⋅ ( 14mm) 4

2

2

= 1.539⋅ cm

⎞ ⎛ As0 sL := Floor ⎜ , 10mm⎟ = 150⋅ mm A ⎜ s ⎟ ⎝ b ⎠

nL :=

B − 50mm⋅ 2 + 1 = 11 sL

2

As_L := nL⋅ As0 = 16.933⋅ cm Shrinkage steel reinforcement

2

2

< As_L = 16.933⋅ cm As.t := 0.0018⋅ b⋅ h = 10.08⋅ cm

Page-244-

Steel reinforcements in direction B = 1.6 m b := L = 1.6 m

d = 279⋅ mm

bc ⎞ ⎛ Mu := Qu⋅ ⎜ S − ⎟ ⋅ 3 = 118.619⋅ kN⋅ m 2⎠ ⎝ R :=

Mu

= 1.058⋅ MPa

2

0.9⋅ b⋅ d

ρ := 0.85⋅

f'c ⎛ ⋅ 1− fy ⎜

1 − 2⋅

⎝

R ⎞ = 0.00271 0.85⋅ f'c ⎟

⎠

⎞ ⎛ f'c ⎜ 0.25MPa⋅ ⎟ MPa 1.4MPa ⎟ ⎜ ρmin := max , = 0.0035 ⎜ fy fy ⎟ ⎝ ⎠

(

)

2

As := max ρ , ρmin ⋅ b⋅ d = 15.624⋅ cm

⎞ ⎛ As0 sB := Floor ⎜ , 10mm⎟ = 150⋅ mm As ⎜ ⎝

⎟ ⎠

b

nB :=

B − 50mm⋅ 2 + 1 = 11 sB

2

As_B := nB⋅ As0 = 16.933⋅ cm Shrinkage steel reinforcement

2

As.t := 0.0018⋅ b⋅ h = 10.08⋅ cm

<

1600

11DB14@150

1600

Page-245-

2

As_B = 16.933⋅ cm

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

Column stud

bc := 400mm

hc := 400mm

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 − 20

kN 3

2

⋅ ( 3⋅ D) ⋅ H = 343.5⋅ kN

m

Number of piles n :=

PD + PL Qe

= 6.469

Use

n := 7

Location of pile

⎛ −1000 ⎞ ⎜ −500 ⎟ ⎜ ⎟ ⎜ −500 ⎟ X := ⎜ 0 ⎟ mm ⎜ ⎟ 500 ⎜ ⎟ ⎜ 500 ⎟ ⎜ ⎟ ⎝ 1000 ⎠

⎛ 0 ⎞ ⎜ 750 ⎟ ⎜ ⎟ ⎜ −750 ⎟ Y := ⎜ 0 ⎟ mm ⎜ ⎟ 750 ⎜ ⎟ ⎜ −750 ⎟ ⎜ ⎟ ⎝ 0 ⎠

Dimension of pile cap B := ( max( Y) + D) ⋅ 2

L := ( max( X) + D) ⋅ 2

Page-246-

⎛ B ⎞ ⎛ 2100 ⎞ ⎜ ⎟=⎜ ⎟ ⋅ mm ⎝ L ⎠ ⎝ 2600 ⎠

⎛ −L ⎞ ⎜L ⎟ 1 ⎜ ⎟ X0 := ⋅ ⎜ L ⎟ 2 ⎜ ⎟ −L ⎜ ⎟ ⎝ −L ⎠

⎛ −B ⎞ ⎜ −B ⎟ ⎟ 1 ⎜ Y0 := ⋅ ⎜ B ⎟ 2 ⎜ B ⎟ ⎜ ⎟ ⎝ −B ⎠ Pile Locations 2

1

−2

−1

0

1

−1

−2

Reaction of pile Pu := 1.2PD + 1.6PLMu := 1.2MD + 1.6ML ORIGIN := 1

n := rows ( X)

i := 1 .. n

Ri :=

Pu n

+

n=7 Mu⋅ Xi n

∑

( Xk) 2

k=1

Page-247-

2

⎛ 388.389 ⎞ ⎜ 396.872 ⎟ ⎜ ⎟ 396.872 ⎜ ⎟ ⎜ R = 405.355 ⎟ ⋅ kN ⎜ ⎟ 413.839 ⎜ ⎟ ⎜ 413.839 ⎟ ⎜ ⎟ ⎝ 422.322 ⎠ Ultimate bearing capacity of pile

Qu := Qa⋅

Pile :=

1.2⋅ PD + 1.6PL PD + PL

= 469.664⋅ kN

"is OK." if max( R) ≤ Qu

Pile = "is OK."

"is not good." otherwise Depth of pile cap

h := 750mm d := h − ⎛⎜ 50mm + 20mm +

⎝

Punching shear from the face of column distane

20mm ⎞ ⎟ = 670⋅ mm 2 ⎠ d = 335⋅ mm 2

Graphs

2

1

−2

−1

0

1

−1

−2

critical section of two-way shear Vu := Qu⋅ 6 = 2817.985⋅ kN Page-248-

2

Punching shear strength

(

)

b0 := bc + d + hc + d ⋅ 2 = 4.28 m d = 670⋅ mm ϕ := 0.75 β :=

hc bc

=1

αs := 40

for interior

⎡⎡ ⎤⎤ f'c ⎢⎢ ⎥⎥ 0.33MPa⋅ ⋅ b0⋅ d MPa ⎢⎢ ⎥⎥ ⎢⎢ ⎥⎥ f'c 2 ϕVc := ϕ⋅ min ⎢⎢ 0.17⋅ ⎛⎜ 1 + ⎞⎟ MPa⋅ ⋅ b0⋅ d ⎥⎥ ⎢⎢ ⎥⎥ MPa ⎝ β⎠ ⎢⎢ ⎥⎥ α ⋅ d f' ⎛ ⎞ ⎢⎢ ⎥⎥ s c ⎢⎢ 0.083⋅ ⎜ b + 2⎟ MPa⋅ MPa ⋅ b0⋅ d ⎥⎥ ⎣⎣ ⎝ 0 ⎠ ⎦⎦ ϕVc = 3548.655⋅ kN The_Cap :=

"is not punching" if ϕVc ≥ Vu "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

−2

−1

0

1

−1

−2

critical section of beam shear Page-249-

2

Vu1 := Qu⋅ 1 = 469.664⋅ kN

Vu2 := Qu⋅ 2 = 939.328⋅ kN

Beam shear strengths ϕVc1 := 0.75⋅ 0.17MPa⋅

f'c ⋅ B⋅ d = 896.962⋅ kN MPa

ϕVc2 := 0.75⋅ 0.17MPa⋅

f'c ⋅ L⋅ d = 1110.525⋅ kN MPa

The_Cap :=

"is not beam shear" if ϕVc1 ≥ Vu1 ∧ ϕVc2 ≥ Vu2 "is beam shear" otherwise

The_Cap = "is not beam shear" - Steel reinforcements in direction L = 2.6 m b := B = 2.1 m

d = 670⋅ mm

Ru := max( R) Mx :=

for i ∈ 1 ..

rows ( X) 2

continue if Xi = 0 hc ⎞ ⎛ Mi ← Ru⋅ ⎜ Xi − ⎟ 2⎠ ⎝

⎛⎜ 337.857 ⎟⎞ Mx = ⎜ 126.697 ⎟ ⋅ kN⋅ m ⎜ 126.697 ⎟ ⎝ ⎠

M Mu1 :=

R1 :=

∑ Mx = 591.251⋅ kN⋅m Mu1 2

= 0.697⋅ MPa

0.9⋅ b⋅ d

ρ := 0.85⋅

f'c ⎛ ⋅ ⎜1 − fy ⎜

⎝

1 − 2⋅

⎞⎟ = 0.00177 0.85⋅ f'c ⎟ ⎠ R1

Page-250-

⎛ f'c ⎞ ⎜ 0.25MPa⋅ ⎟ MPa 1.4MPa ⎟ ⎜ ρmin := max , = 0.0035 ⎜ fy fy ⎟ ⎝ ⎠

(

)

2

As := max ρ , ρmin ⋅ b⋅ d = 49.245⋅ cm 2

π⋅ ( 22mm) 2 As0 := = 3.801⋅ cm 4

⎞ ⎛ As0 s1 := Floor ⎜ , 10mm⎟ = 160⋅ mm As ⎜ ⎝

n1 := ceil ⎛⎜

⎝

⎟ ⎠

b

B − 50mm⋅ 2 + 1⎞⎟ = 14 s1

⎠

2

Asx := n1⋅ As0 = 53.219⋅ cm

Top bars (shrinkage reinforcement) 2

As.t := 0.0018⋅ b⋅ h = 28.35⋅ cm As1 :=

π⋅ ( 18mm) 4

2

2

= 2.545⋅ cm

⎞ ⎛ As1 st := Floor ⎜ , 10mm⎟ = 180⋅ mm As.t ⎜ ⎝

nt := floor ⎛⎜

⎝

b

⎟ ⎠

B − 50mm⋅ 2 + 1⎞⎟ = 12 st

⎠

- Steel reinforcements in direction B = 2.1 m b := L = 2.6 m

d = 670⋅ mm

Page-251-

My :=

for i ∈ 1 ..

rows ( Y) 2

⎛⎜ 0 ⎟⎞ My = ⎜ 232.277 ⎟ ⋅ kN⋅ m ⎜ 232.277 ⎟ ⎝ ⎠

continue if Yi = 0 hc ⎞ ⎛ Mi ← Ru⋅ ⎜ Yi − ⎟ 2⎠ ⎝ M Mu2 :=

R2 :=

∑ My = 464.554⋅ kN⋅m Mu2 2

= 0.442⋅ MPa

0.9⋅ b⋅ d

f'c ⎛ ρ := 0.85⋅ ⋅ ⎜ 1 − fy ⎜

⎞⎟ 1 − 2⋅ = 0.00112 0.85⋅ f'c ⎟ ⎠ R2

⎝

⎞ ⎛ f'c ⎜ 0.25MPa⋅ ⎟ 1.4MPa MPa ⎜ ⎟ ρmin := max , = 0.0035 ⎜ fy fy ⎟ ⎝ ⎠

(

)

2

As := max ρ , ρmin ⋅ b⋅ d = 60.97⋅ cm

⎞ ⎛ As0 s2 := Floor ⎜ , 10mm⎟ = 160⋅ mm As ⎜ ⎝

n2 := ceil ⎛⎜

⎝

b

⎟ ⎠

L − 50mm⋅ 2 + 1⎟⎞ = 17 s2

⎠

2

Asy := n2⋅ As0 = 64.623⋅ cm

Top bars (shrinkage reinforcement) 2

As.t := 0.0018⋅ b⋅ h = 35.1⋅ cm

⎞ ⎛ As1 st := Floor ⎜ , 10mm⎟ = 180⋅ mm As.t ⎜ ⎝

b

⎟ ⎠

Page-252-

nt := floor ⎛⎜

⎝

B − 50mm⋅ 2 + 1⎞⎟ = 12 st

⎠

2

1

−2

−1

0

−1

−2

Bottom Steel Bar

Page-253-

1

2