footing ptc ce bsd 7.2 us mp

7/30/2019 Footing PTC CE BSD 7.2 Us Mp

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PTC_CE_BSD_7.2_us_mp.mcdx

Mathcad Enabled Content Copyright 2011 Knovel Corp.

Building Structural Design Thomas P. Magner, P.E. 2011 Parametric Technology Corp.Chapter 7: Reinforced Concrete Column and Wall Footings

7.2 Pile Footings

Disclaimer

While Knovel and PTC have made every effort to ensure that the calculations, engineering solutions,diagrams and other information (collectively Solution) presented in this Mathcad worksheet aresound from the engineering standpoint and accurately represent the content of the book on which the

Solution is based, Knovel and PTC do not give any warranties or representations, express or implied,including with respect to fitness, intended purpose, use or merchantability and/or correctness oraccuracy of this Solution.

Array or ig in:

RIGIN 0

Description

Pile supported footings or pile caps are used under columns and walls to distribute the load to thepiles. The plan dimensions of a pile footing are determined by the number and the arrangement of thepiles and the required minimum spacing between the piles.

This application determines minimum required pile cap thickness and the maximum size and minimumnumber of reinforcing bars for pile caps with from 2 to 20 piles in a group. Pile diameter and spacing,and the size of the supported rectangular pier or wall may be specified by the user. The applicationuses the Strength Design Method of ACI 318-89.

The required input includes the pile capacity at service load (unfactored loads) and the plandimensions of the column or wall. The user may also enter material properties and variablesdesignated as "project constants" in this application.

The material properties include strength of the concrete and the reinforcement, unit weight ofconcrete, unit weight of reinforced concrete, and the preferred reinforcement ratio.

The project constants include pile diameter, minimum pile spacing, edge distance, and multiples usedfor rounding the pile coordinates, the pile cap depth and the plan dimensions of the pile caps.

Mathcad Enabled ContentCopyright 2011 Knovel Corp.All rights reserved.

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Reference: ACI 318-89 "Building Code Requirements for Reinforced Concrete." (Revised 1992)

Input

Input Variables

Enter service load pile capacity, number of piles in group, and pier or wall dimensions.

Pile capacity at service loads: Pcap 80 kipNumber of piles in group: N 8

Column width: Cx 24 inColumn depth: Cy 24 inIf the footing supports a wall, enter any wall length greater than or equal to the dimension of thefooting parallel to the wall.


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Computed Variables

The following variables are calculated in this document:

Ps total service load capacity of the piles

F combined load factor for dead and live load

Pu total factored load capacity of the piles and the pile cap

qu factored load capacity of one pile

h total pile footing thickness

X the longer dimension of the pile cap

Y the shorter dimension of the pile cap

x' pile coordinates in the X direction from thecentroid of the pile group

y' pile coordinates in the Y direction from thecentroid of the pile group

Input Constants

The following variables are normally constant for a given project and may be defined by the user. Thevariables s, SzP, SzD and E are used by this application to calculate the pile coordinates and plandimensions of the pile caps.

Minimum pile spacing: s 3 ftMultiple for rounding pile spacing: SzP 0.5 in


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Multiple for rounding pile

cap plan dimensions: SzD 1 inMinimum edge distancefrom center of pile: E +1 ft 3 inRatio of live load to dead load: R 1

Pile diameter at top of pile: dp 8 inPile embedment into pile cap: e 4 inClearance betweenreinforcement and top of pile: cl 3 inMultiple for roundingfooting depths: SzF 1 inMaterial Properties

Enter the values of f'c, fy, wc, wrc, kv and kw.

Specified compressive strength of concrete: f'c 4 ksiSpecified yield strength of reinforcement: fy 60 ksiUnit weight of concrete: wc 145 pcfUnit weight of reinforced concrete: wrc 150 pcfShear strength reduction factor for lightweightconcrete kv=1 for normal weight, 0.75 for all-lightweight and 0.85 for sand-lightweight concrete(ACI 318, 11.2.1.2.): kv 1


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Weight factor for increasing development and

splice lengths kw=1 for normal weight and 1.3for lightweight aggregate concrete(ACI 318, 12.2.4.2): kw 1

Factors for use of lightweight concrete are included, but it would beunusual to use lightweight concrete for a pile cap.

Modulus of elasticity ofreinforcement (ACI 318, 8.5.2):

Es 29000 ksiStrain in concrete at compressionfailure (ACI 318, 10.3.2):

c 0.003

Strength reduction factor forflexure (ACI 318, 9.3.2.1):

f 0.9

Strength reduction factor forshear (ACI 318, 9.3.2.3):

v 0.85

Reinforcing bar number designations, diameters, and areas:

NoT

0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8

db T

0 0 0 0.375 0.5 0.625 0.75 0.875 1.00 1.128 1.27 1.41 0 0 1.693 0 0 0 2.257 in

Ab T0 0 0 0.11 0.20 0.31 0.44 0.60 0.79 1.00 1.27 1.56 0 0 2.25 0 0 0 4.00 in 2Bar numbers, diameters and areas are in the vector rows (or columns in thetransposed vectors shown) corresponding to the bar numbers. Individualbar numbers, diameters and areas of a specific bar can be referred to byusing the vector subscripts as shown in the example below.


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Pile Coordinates and Plan Dimensions of Pile Caps

Dimensions used to compute pile coordinates: s0 0 ft

s1 2 SzP ceil

s sin 45 degSzP

=s1 4.25 ft

s2 SzP ceil

s sin 60 degSzP

=s2 2.625 ft

s3 +s

2s2 =s3 4.125 ft

s4 +s2 s =s4 5.625 ftAngle between the centerline of the two piles in the base row and acenterline drawn between either of the two piles in the base row to the pile atthe apex of the 3-pile group:

atan

2 s2

s

= 60.255 deg

Pile coordinates expressed in terms of pile spacing. The number within the

column operator indicates the number of piles in the group:

x 1T

s0 s0 y 1T

s0 s0

x 2T

s

2

s

2

y 2

Ts0 s0


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x

3

T

s0

s

2

s

2

y3

T2 s2

3

s2

3

s2

3

x 4T

s

2

s

2

s

2

s

2

y 4

T

s

2

s

2

s

2

s

2

x 5T

s1

2

s1

2

s0 s1

2

s1

2

y 5T

s1

2

s1

2

s0 s1

2

s1

2

x 6T

s s0 s s s0 s y 6T

s

2

s

2

s

2

s

2

s

2

s

2

x 7T

s

2

s

2s s0 s

s

2

s

2

y 7

Ts2 s2 s0 s0 s0 s2 s2

x 8T

s s0 s s

2

s

2s s0 s

y 8

Ts2 s2 s2 s0 s0 s2 s2 s2

x 9T

s s0 s s s0 s s s0 s y 9T

s s s s0 s0 s0 s s s

x 10T

s s0 s 3 s

2

s

2

s

2

3 s

2s s0 s

y 10

Ts2 s2 s2 s0 s0 s0 s0 s2 s2 s2

x 11T

3 s

2

s

2

s

2

3 s

2s s0 s

3 s

2

s

2

s

2

3 s

2

y 11

Ts2 s2 s2 s2 s0 s0 s0 s2 s2 s2 s2


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x

12

T3 s

2

s

2

s

2

3 s

2

3 s

2

s

2

s

2

3 s

2

3 s

2

s

2

s

2

3 s

2

y 12T

s s s s s0 s0 s0 s0 s s s s

x 13T

2 s2 s0 2 s2 s2 s2 2 s2 s0 2 s2 s2 s2 2 s2 s0 2 s2

y 13T

s s s s2

s2

s0 s0 s0 s2

s2

s s s

x 14T

s s0 s 3 s

2

s

2

s

2

3 s

2

3 s

2

s

2

s

2

3 s

2s s0 s

y 14T

s3 s3 s3s

2

s

2

s

2

s

2

s

2

s

2

s

2

s

2s3 s3 s3

x 15T

2 s s s0 s 2 s 2 s s s0 s 2 s 2 s s s0 s 2 s

y 15T

s s s s s s0 s0 s0 s0 s0 s s s s s

x 16T

3 s

2

s

2

s

2

3 s

2

3 s

2

s

2

s

2

3 s

2

3 s

2

s

2

s

2

3 s

2

3 s

2

s

2

s

2

3 s

2

y 16T

3 s

2

3 s

2

3 s

2

3 s

2

s

2

s

2

s

2

s

2

s

2

s

2

s

2

s

2

3 s

2

3 s

2

3 s

2

3 s

2


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x 17T

s2 s2 2 s2 s0 2 s2 s2 s2 2 s2 s0 2 s2 s2 s2 2 s2 s0 2 s2 s2 s2

y 17T

3 s

2

3 s

2s s s

s

2

s

2s0 s0 s0

s

2

s

2s s s

3 s

2

3 s

2

x 18T

2 s2 s0 2 s2 s2 s2 2 s2 s0 2 s2 s2 s2 2 s2 s0 2 s2 s2 s2 2 s2 s0 2 s2

y 18T

3 s2

3 s2

3 s2

s s s2

s2

s2

s0 s0 s2

s2

s2

s s 3 s2

3 s2

3 s2

x 19T

s4 s2 s2 s4 s0 s4 s2 s2 s4 s0 s4 s2 s2 s4 s0 s4 s2 s2 s4

y

19

T

3 s

2

3 s

2

3 s

2

3 s

2 s

s

2

s

2

s

2

s

2 s0

s

2

s

2

s

2

s

2 s

3 s

2

3 s

2

3 s

2

3 s

2

x 20T

2 s s s0 s 2 s 2 s s s0 s 2 s 2 s s s0 s 2 s 2 s s s0 s 2 s

y 20T

3 s

2

3 s

2

3 s

2

3 s

2

3 s

2

s

2

s

2

s

2

s

2

s

2

s

2

s

2

s

2

s

2

s

2

3 s

2

3 s

2

3 s

2

3 s

2

3 s

2

Pile coordinates in the X and Y directions from thecenter of the pile group being designed:

i 0 N 1

x'i

x,i N


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=

T

x' 3 0 3 1.5 1.5 3 0 3 fty'

iy

,i N

=Ty' 2.625 2.625 2.625 0.000 0.000 2.625 2.625 2.625 ftAbsolute values of pile coordinates in the X and Y directionsfrom the center of the pile group being designed:

x_absx'

=Tx_abs 3 0 3 1.5 1.5 3 0 3 ft

y_absy'

=Ty_abs 2.625 2.625 2.625 0 0 2.625 2.625 2.625 ftAbsolute values of pile coordinates in the X and Y directions from the centerof the pile group being designed minus 1/2 the pier width, or the maximumabsolute pile coordinate if the pile center is less than half the pier width fromthe center of the group:

x_facei

if

,,x_abs

iCx

2x_abs

iCx

2max x_abs

=Tx_face 2.000 3.000 2.000 0.500 0.500 2.000 3.000 2.000 ft

y_facei

if

,,y_absi

Cy

2y_abs

iCy

2max y_abs


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=

T

y_face 1.625 1.625 1.625 2.625 2.625 1.625 1.625 1.625 ftDistances ax and ayfrom face of pier in the X and Y directions to the closestpile center, or 0 ft if all pile centers are less than or equal to 1/2 the pier widthor depth from the pile group center:

ax if

,,max x_abs

Cx

20 ft min x_face

=ax 6 in

ay if

,,max y_abs

Cy

20 ft min y_face

=ay 19.5 in

Plan dimensions pile caps as function of the number of piles:

FX N +max x N min x N 2 E

FY N +max y N min y N 2 E

Plan dimensions X and Y of the pile cap being designed:

X FX N Y FY N

=X 8.5 ft =Y 7.75 ftDimensions A and B for the 3-pile group rounded up to a multiple of SzD:

A SzD ceil

2 E

1

sin

1

tan

SzD

=A 18 in


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B SzD ceil

E

+1 tan

1

sin 1

SzD

=B 19 inPlan area of pile cap, CapArea:

CapArea if

,,N 3 X Y FX 3 FY 3 FX 3 A

2FY 3 B

=CapArea 65.875 ft 2Example: =No

55 =db5

0.625 in =Ab5 0.31 in 2

The following values are computed from the entered material properties.

Modulus of elasticity of concrete (for values of wc between 90 pcf and155 pcf (ACI 318, 8.5.1):

Ec

wc

pcf

1.5

33f'c

psi psi =Ec 3644 ksiStrain in reinforcement at yield stress:

y fyEs

=y 0.00207

Factor used to calculate depth of equivalent rectangular stressblock (ACI 318, 10.2.7.3):

1 if

,,f'c 4 ksi f'c 8 ksi 0.85 0.05 f'c 4 ksiksi if ,,f'c 4 ksi 0.85 0.65


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=1 0.85

Reinforcement ratio producing balanced strain conditions (ACI 318, 10.3.2):

b 1 0.85 f'c

fy

Es c

+Es c fy=b %2.851

Maximum reinforcement ratio (ACI 318, 10.3.3):

max 34

b =max %2.138

Minimum reinforcement ratio for beams (ACI 318, 10.5.1, Eq. (10-3)):

min 200

fybfin 2 =min %0.333

Shrinkage and temperature reinforcement ratio (ACI 318, 7.12.2.1):

temp if

,,fy 50 ksi .002 if

,,fy 60 ksi .002 fy

60 ksi .0002 if

,,

.0018 60 ksify

.0014 .0018 60 ksi

fy.0014

Preferred reinforcement ratio:

3

8 b

Any reinforcement ratio frommin(the minimum reinforcement ratio for flexure) to 3/4b (themaximum reinforcement ratio) may be specified. Numerical values may also be entered directly, suchas 1.0 % or 0.01.

Flexural coefficient K, for rectangular beams or slabs, as a function of(ACI 318, 10.2):(Moment capacity Mn =K(F, where F =bd2)


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K f

1

fy

2 0.85 f'c

f

y

=K 0.523 ksiLimit the value of f'c for computing shear and development lengths to10 ksi by substituting f'c_max for f'c in formulas for computing shearand development lengths (ACI 318, 11.1.2, 12.1.2):

f'c_max if ,,>f'c 10 ksi 10 ksi f'cNominal "one way" shear strength per unit area inconcrete (ACI 318, 11.3.1.1, Eq. (11-3), 11.5.4.3):

vc kv 2 f'c_max

psi psi =vc 126 psi

Basic tension development length ldbt(ACI 318, 12.2.2 and 12.2.3.6):

No. 3 through No. 11 bars: n 3 11

X1n

0.04 Abn fy

f'c_max lbfX2

n0.03 dbn

fy

f'c_max

psipsi

ldbtnif ,,>X1

nX2

nX1

nX2

n

=Tldbt 0 0 0 10.7 14.2 17.8 21.3 24.9 30 37.9 48.2 59.2 in


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No. 14 bars: ldbt14

0.085

fy in 2f'c_max lbf

=ldbt14

80.6 in

No. 18 bars ldbt180.125

fy in 2f'c_max lbf

=ldbt18118.6 in

Tension development length (ACI 318, 12.2.1):

No. 3 through No. 11 bars:

ldtnif

,,kw ldbtn12 in kw ldbtn if ,,>kw ldbtn 0 in 12 in 0 in

=Tldt 0 0 0 12 14.2 17.8 21.3 24.9 30 37.9 48.2 59.2 inNo. 14 bars: ldt14

kw ldbt14=ldt14

80.6 inNo. 18 bars ldt18

kw ldbt18=ldt18

118.6 inSolution

Service load:

Ps Pcap N =Ps 640 kipCombined load factor for dead +live load:

F +1.4 1.7 R

+1 R=F 1.55


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Factored load Pu:

Pu F N Pcap =Pu 992 kipFactored load per pile:

quPu

N=qu 124 kip

The longer dimension of the pile cap:

=X 8.5 ftThe shorter dimension of the pile cap:

=Y 7.75 ftRange variable i from 0 to the number of piles N, minus 1:

i 0 N 1

Pile coordinates in the X direction from the centerline of the pile group, starting from top left to bottomright:

=Tx' 3 0 3 1.5 1.5 3 0 3 ftPile coordinates in the Y direction from the centerline of the pile group, starting from left to right andtop to bottom:

=Ty' 2.625 2.625 2.625 0 0 2.625 2.625 2.625 ft


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Solution to determine the minimum required footing depth for flexure, df

Bending moment about the Y axis as function of distance x2 from Y axis:

My x2 i

qu if ,,x'i x2 0 ft x'i x2

Bending moment about the Y axis at face of pier:

=My

Cx

2

558 ip ft

Bending moment about the X axis as function of distance y2 from X axis:

Mx y2 i

qu if ,,y'i y2 0 ft y'i y2

Bending moment about the X axis at face of pier:

=Mx

Cy

2

604.5 ip ft

Effective footing width for bending about the X axis, at face of pier:

Xf if

,,N 3 X +A

+E

2

3s2

Cy

2

X A

Y B

=Xf 8.5 ft


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Effective footing widths for flexure and shear vary with distance from the pile group centroid for the 3-

pile group. The remaining pile caps are rectangular in shape and the full width is effective for flexureor shear at any section.

Effective footing width at face of pier, for bending about the Y axis:

Yf Y if

,,+N 3 Cx A 0 ft

Cx A

X AY B

=Yf 7.75 ftMaximum bending moment per unit width:

Mu if

,,>

Mx

Cy

2

Xf

My

Cx

2

Yf

Mx

Cy

2

Xf

My

Cx

2

Yf

=Mu 72 ip ft

ftMinimum required effective footing depth for flexure at specified reinforcement ratio (ACI 318, 10.2):

df Mu

K =df 11.735 in

Solution to determine the minimum required footing depth for one-way beam shear, dbm

Shear Vux(x2) in the X direction as a function of any specified positive distance x2 from the Y axis(ACI 318, 15.5.3):


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Vux x2

i

if

,,x'i

+Cx2

dp2

0 if

,,x'i

+x2 dp2

1 if

,,x'i

x2 dp2

0

+x'i

dp

2

x2

dp

qu

Shear Vuy(y2) in the Y direction as a function of any specified positive distance y2 from the X axis(ACI 318, 15.5.3):

Vuy y2

i

if

,,y'i

+Cy2

dp2

0 if

,,y'i

+y2 dp2

1 if

,,y'i

y2 dp2

0

+y'i

dp

2y2

dp

qu

Since the 3-pile group is not rectangular the effective width in the X and Y directions varies withdistance from the pile group centroid. The shear must be checked at both the inside edge of the pilesand at distance d from the face of the pier.

The following functions are for calculating the effective widths and required depths at any positivedistance from the pile group centroid, and the required depths at distance d from the face of the pier.

The largest required depth at distance d from the face of pier or at the inside edge of the pilesdetermines the minimum required depth for the 3-pile group.

Effective footing width for one-way shear in the Y direction for the3-pile group, as a function of any positive distance y2 from the X axis:

X3 y2 +A +E max y' y2

X A

Y B

Minimum required depth for one-way shear in the Y direction for the 3-pile group as a function of anypositive distance y2 from the X axis:

dy3 y2 if

,,y2

Cy

20 ft if

,,


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Effective footing width for one-way shear in the X direction for the

3-pile group, as a function of any positive distance x2 from the Y axis:

Y3 x2 Y if

,,+

Cx

2x2

A

20 ft +Cx 2 x2 A

Y B

X A

Minimum required depth for one-way shear in the X direction for the 3-pile group as a function of anypositive distance x2 from the Y axis:

dx3 x2 if

,,x2

Cx

20 ft if

,,


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Minimum required depth for one-way shear in the Y direction for the 3-pile group at distance d from

face of pier:

GuessValues

Constrain

ts

Solver

d 1 ft

d +max y'dp

2

Vuy

+

Cy

2

d

v vc X3

+

Cy

2d

d

dy3_d d Find d


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Functions for rectangular pile caps

The following Mathcad solve blocks contain the functions for determining the required depth for one-

way shear for the rectangular pile caps.

Minimum required depth for one-way shear in the X direction at distance d from face of pier:

GuessValues

Constraints

Solver

d 1 ft

d +max x'dp

2

Vux

+

Cx

2d

v vc Y

d

dx d Find d

Minimum required depth in the Y direction at distance d from face of pier:

GuessValues

Constraint

s

Solver

d 1 ft

d +max y'dp

2

Vuy

+

Cy

2d

v vc X

d

dy d Find d


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d 1 ftDepth required for one-way shear in the Y direction:

dy if

,,N 3 dy d max

dy3

max y'

dp

2

dy3_d d

=dy 19.015 inDepth required for one-way shear in the X direction:

dx if

,,N 3 dx d max

dx3

max x'

dp

2

dx3_d d

=dx 21.171 inMinimum depth required for beam shear:

dbm max

dxdy

=dbm 21.171 in


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Calculations to determine the minimum required footing depth for peripheral shear, dper(ACI 318, 11.12.2.1)

Critical Sections for Peripheral Shear(d is the effective depth and dp is the pile diameter)

Perimeter of critical section for slabs and footings expressed as a function of d (ACI 318, 11.12.1.2):

bo

d 2 ++Cx

Cy

2 d

Ratio of the longer to the shorter pier dimension (ACI 318 11.12.2.1):

c if

,,Cx CyCx

Cy

Cy

Cx

=c 1


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Nominal "two way" concrete shear strength per unit area in slabs and footings, expressed as afunction of effective depth d. Since piers or columns are at the center of the pile cap they areconsidered "interior" columns for calculating two way shear, and s is equal to 40. (ACI 318 11.12.2.1,Eqs. (11-36), (11-37) and (11-38)):

s 40

vcp d min

+24

cs d

bo d

4

kv f'c_max

psi psi

Function V1(d) computes a vector with elements equal to 0 if the pile produces shear on the criticalsection and 1 if it does not:

V1 d

x'

+Cx d dp

2

y'

+Cy d dp

2

Function V2(d) computes a vector with elements equal to 1 if the full reaction of the pile producesshear on the critical section and 0 if it does not:

V2 d

>+

x'

++Cx d dp

2

y'

++Cy d dp

2

0

Function V3(d) computes a vector with elements equal to 1 if a portion of the pile reaction producesshear on the critical section and 0 if it does not:

V3 d 1 V1 d V2 d


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Function V4(d) computes a vector with elements equal to the the larger difference between the x' or y'pile coordinate and the corresponding distance to the shear section minus half the pile diameter or 0 ft.

V4 d

+V3 d

>x'

+Cx d dp

2

0 ft

x'

+Cx d dp

2

V3 d

>y'

+Cy d dp

2

0 ft

y'

+Cy d dp

2

Shear Vup(d) on the peripheral shear section at distance d/2 from the face of the pier, expressed as afunction of d (ACI 318, 15.5.3):

Vup d

+V2 d

V4 d

dp

qu

Guess value of d: d dbm

d'per d root

,

Vup d

bo d d v vcp d d

d'per if ,,++Cx dbm X +Cy dbm Y dbm d'per d

=d'per 19.816 inMinimum depth for peripheral shear dper:

dper if ,,++Cx d'per X +Cy d'per Y dbm d'per=dper 19.816 in


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If either dimension of the critical shear section is greater than thecorresponding pile cap dimension, peripheral shear is not critical and the

depth for peripheral shear is defined as equal to the depth required forone-way shear.

Calculation to determine the minimum required depth for deep beam shear dbm2(ACI 318, 11.8, using Eq. (11-30) for shear strength)

Shear spans in the X and Y directions are one-half the distances between the closest pile row and theface of the pier (See ACI 318, Chapter 11 Notation, and 11.8):

=0.5 ax 0.25 ft =0.5 ay 0.813 ftMoment and shear in the X and Y directions on the critical shear sections at distance 0.5ax and 0.5ayfrom the face of the pier (ACI 318, 15.5.3):

=My

+

Cx

20.5 ax

465 ip ft


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28/41


=Vux

+Cx

2

0.5 ax

356.5 kip

=Mx

+

Cy

20.5 ay

302.25 ip ft

=Vuy

+

Cy

2 0.5 ay

372 kipRatios of moment to shear Rx and Ry, at the critical sections for deep beam shear in the X and Ydirections:

Rx

My

+

Cx

20.5 ax

Vux

+

Cx

20.5 ax

=Rx 1.304 ft

Ry

Mx

+

Cy

20.5 ay

Vuy

+

Cy

2 0.5 ay

=Ry 0.813 ft

Multipliers for use in ACI 318, Eq. (11-30):

KX d if

,,3.5 2.5

Rx

d2.5 2.5 3.5 2.5

Rx

d


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KY d if

,,3.5 2.5Ry

d

2.5 2.5 3.5 2.5Ry

d

Nominal deep beam shear stress at factored load in the X and Y directions (ACI 318, 11.8.7, Eq.(11-30)) with w conservatively assumed equal to minimum temperature reinforcement, temp:

vcx d min

X d

+1.9

f'c

psi 2500 tempd

Rx

psi

6f'

cpsi psi

vcy d min

Y d

+1.9

f'c

psi 2500 temp d

Ry

psi

6f'c

psi psi

Effective depth required in the X direction for deep beam shear:

Guess value of d: d dx

dbm2_x root

,

Vux

+

Cx

20.5 ax

v vcx d if

,,N 3 Y Y3

+

Cx

20.5 ax

d d

=dbm2_x 21.381 in =vcx dbm2_x 210.9 psi


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Effective depth required in the Y direction for deep beam shear:

Guess value of d: d dy

dbm2_y root

,

Vuy

+

Cy

20.5 ay

v vcy d if

,,N 3 X X3

+

Cy

20.5 ay

d d

=dbm2_y 16.556 in =vcy dbm2_y 259.2 psiDepth required for deep beam shear is the larger value of dbm2:

dbm2 max

dbm2_xdbm2_y

=dbm2 21.381 in


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Solution to determine the minimum required depth for beam shear or peripheral shear for onecorner pile, dcor

Critical Section forBeam Shearfor 1 corner pile

Critical Section forPeripheral Shearfor 1 corner pile

Required depth for beam shear for one corner pil e

Guess value of d: d 1 ft

dcor_1 root

,qu

v vc

++dp 2 d 2 2 E

d d

=dcor_1 14.515 inRequired depth for peripheral shear for one corner pi le

Nominal "two way" concrete shear strength per unit area in slabs and footings,expressed as a function of effective depth d. For a corner pile s is equal to 20.Since c is 1 for piles in this application, ACI 318 Eq. (11-36) is not critical (ACI318, 11.12.2.1, Eqs. (11-37) and (11-38)):


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s 20 bo d

++dp d4

2 E

vcp d min

+s d

bo d2

4

kv f'c_max

psi psi

dcor_2 root

,

qu

v vcp d bo d d d

=dcor_2 12.507 inDepth required for one corner pile is the larger value of dcor:

dcor max

dcor_1dcor_2

=dcor 14.515 inDepth required for a single interior p ile, done

Nominal "two way" concrete shear strength per unit area for one interior pile,expressed as a function of effective depth d. For an interior pile s is equal to 40Since c is 1 for piles in this application ACI 318 Eq. (11-36) is not critical. (ACI

318 11.12.2.1, Eqs. (11-37) and (11-38)):

s 40 bo d +dp d

vcp d min

+s d

bo d2

4

kvf'c_max

psi psi


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done root ,bo d d v vcp d qu d

=done 10.126 in

Depth required for two interior pil es, dtwo

Nominal "two way" concrete shear strength per unit area for two interior piles, expressed as a functionof effective depth d. For interior piles s is equal to 40 (ACI 318, 11.12.2.1, Eqs. (11-37) and (11-38)):

s 40

c d ++s d dp

+dp d

bo d + +dp d +2 s d

vcp d min

+24

c d

+s d

bo d2

4

kv f'c_max

psi psi

Depth, dtwo, required for two adjacent piles is equal to done unless overlapping shear perimetersrequire a greater depth when spacing s is less than done +dp:

dtwo if ,,+dp done s done root ,bo d d v vcp d 2 qu d

=dtwo 10.126 in


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_ _ _ _ _ p

Minimum thickness for a footing on piles (ACI 318, 15.7):

dmin 12 inMinimum required effective depth, de

Largest effective depth determined by flexure, one-way beam shear, peripheral shear, deep beamshear, shear on a single corner pile, a single interior pile, two adjacent piles, or the minimum thicknessrequired for a footing on piles de:

=

dfdbmdperdbm2dcordoned

twodmin

11.73521.17119.81621.38114.51510.12610.126

12.000

in de max

dfdbmdperdbm2dcordoned

twodmin

=de 21.381 in

Solution to determine the largest permissib le reinforcing bar sizes

Maximum available lengths in the X direction for development of reinforcement:

Ldx X Cx

2

3 in =Ldx 3 ft


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Maximum available lengths in the Y direction for development of reinforcement. The availabledevelopment length for the 3-pile group is calculated from face of pier to the pile at the apex of the

group:

Ldy f

,,N 3 Y Cy

2

+2

3s2 E

Cy

2

sin

3 in =Ldy 2.625 ftIndex numbers of maximum bar sizes determined by available development lengths:

index0 0 index 0 n if ,,ldtn Ldx n index0bx index

0=bx 8

index0

0 index0 n

if

,,ldtnLdy n index0

by index0

=by 8

Sizes of the largest permissible reinforcing bars for the X and Y directions. Since the 3-pile group usesthree equal bands of reinforcement the maximum bar size is defined as the smaller bar size for eitherthe X or Y direction:

BarSize_X if

,,N 3 bx min

bxby

=BarSize_X 8

bx BarSize_X =dbbx1 in

BarSize_Y if

,,N 3 by min

bxby

=BarSize_Y 8


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by BarSize_Y =dbby1 in

Total pile cap thickness rounded up to the nearest multiple of SzF:

h SzF ceil

++++de dbbx

0.5 dbby

cl e

SzF

=h 30 in

Pile cap weight:

CapWt CapArea h wrc =CapWt 24.703 kipCalculation to determine the required number of reinforcing bars in the X and Y directions

Effective depths in the X and Y directions:

dex h e cl 0.5 dbbx

=dex 22.5 in

dey h e cl dbbx0.5 dbby

=dey 21.5 inTotal required flexural reinforcement area in the X direction:

Asx_flex

Yf dex

1

1

2 My

Cx

2

f Yf dex

2 0.85 f'c

0.85 f'cfy

=Asx_flex 5.646 in 2


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Minimum required reinforcement area for shrinkage and temperature in the X direction (ACI 318, 7.12):

Asx_temp temp h Y =Asx_temp 5.022 in 2Larger required reinforcement in the X direction:

Asx max

Asx_flex

sx_temp

=Asx 5.646 in 2

Total required reinforcement area for flexure in the Y direction:

Asy_flex if

,,N 2

Xf dey

1

1

2 Mx

Cy

2

f Xf dey2 0.85 f'c

0.85 f'c

fy

0 in 2

=Asy_flex 6.414 in2

Minimum required reinforcement area for shrinkage and temperature in the Y direction (ACI 318, 7.12):

Asy_temp temp h X =Asy_temp 5.508 in2Larger required reinforcement in the Y direction:

Asy max

Asy_flex

sy_temp

=Asy 6.414 in 2


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Minimum number of bars to limit spacing of reinforcement to 18 inches (ACI 318, 7.6.5):

MinNumb_X ceil

+

Y 6 in18 in 0.5

=MinNumb_X 6

MinNumb_Y ceil

+

X 6 in18 in 0.5

=MinNumb_Y 6

Maximum bar areas corresponding to minimum specified spacing:

MinA_x Asx

MinNumb_X=MinA_x 0.941 in 2

MinA_y Asy

MinNumb_Y=MinA_y 1.069 in 2

Index numbers of bar sizes determined by minimum specified spacing:

index0

0 index0 n

if

,,AbnMinA_x n index

0

ax index0

=ax 8

index0

0 index0 n

if

,,AbnMinA_y n index

0

ay index0

=ay 9


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Actual bar sizes in the X and Y directions.

Bar sizes in the X and Y directions (the smaller bar area determined by the required total reinforcementor the specified minimum spacing). The bar size in the Y direction of the 2-pile group is set at 3:

BarSixe_X if

,,+

My

Cx

2

0 kip ft

ax bx ax bx

=BarSixe_X 8

BarSize_Y if

,,N 2 if

,,+

Mx

Cy

2

0 kip ft

ay by ay by

3

=BarSize_Y 8

Subscript variables cx and cy defined as the bar sizes in the X and Y directions, respectively:

cx BarSize_X cy BarSize_Y

Adjustments requi red in reinforcement areas for the 3-pile group

In the the 3-pile group the reinforcement is customarily placed in 3 bands of equal area, over the pilecenters. Because the bands are at an angle to the axes of bending the reinforcement areas must beadjusted accordingly.

Total required reinforcement area in each of 3 bands for the 3-pile group:

Asx if

,,N 3 Asx if

,,>

Asx

2 sin Asy

+1 cos

Asx

2 sin Asy

+1 cos

=Asx 5.646 in 2


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Asy if ,,N 3 Asy Asx

=Asy 6.414 in 2Number of bars required in the X and Y directions. The number of bars in Y direction of the 2-pilegroup is set at a multiple of 18 inches:

Numb_X ceil

Asx

Abcx

=Numb_X 8

Numb_Y if

,,N 2 ceil

Asy

Abcy

ceil

X

18 in

=Numb_Y 9

Summary

Computed Variables

Combined load factorfor dead and live load: =F 1.55

Total service loadcapacity of the pilesand the pile cap: =Ps 640 kip

Total factored loadcapacity of the pilesand the pile cap:

=Pu 992 kipNumber of reinforcingbars in the X direction: =Numb_X 8


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Bar size inthe X direction: =BarSize_X 8

Number of reinforcingbars in the Y direction: =Numb_Y 9

Bar size inthe Y direction: =BarSize_Y 8

The number and size of reinforcing bars for the 3-pile group is shown as equal in the X and Ydirections. The reinforcement for the 3-pile group consists of three equal bands over the three

centerlines of the piles, with the number and size of bars of the bars in each band equal to that shownfor the X and Y directions.

Longer dimensionof the pile cap: =X 8.5 ftShorter dimensionof the pile cap: =Y 7.75 ft

Total pile footingthickness: =h 30 in

Total pile cap weight: =CapWt 24.703 kipUser Notices

Equations and numeric solutions presented in this Mathcad worksheet are applicable to thespecific example, boundary condition or case presented in the book. Although a reasonable effort

was made to generalize these equations, changing variables such as loads, geometries andspans, materials and other input parameters beyond the intended range may make someequations no longer applicable. Modify the equations as appropriate if your parameters falloutside of the intended range.For this Mathcad worksheet, the global variable defining the beginning index identifier for vectorsand arrays, ORIGIN, is set as specified in the beginning of the worksheet, to either 1 or 0. IfORIGIN is set to 1 and you copy any of the formulae from this worksheet into your own, you needto ensure that your worksheet is using the same ORIGIN.


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footing ptc ce bsd 7.2 us mp

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