PC_CE_BSD_7._us_mp.mcdx Copyright 011 Knovel Corp. Building Structural Design homas P. Magner, P.E. 011 Parametric echnology Corp. Chapter 7: Reinforced Concrete Column and Wall Footings 7. Pile Footings Disclaimer While Knovel and PC have made every effort to ensure that the calculations, engineering solutions, diagrams and other information (collectively Solution ) presented in this Mathcad worksheet are sound from the engineering standpoint and accurately represent the content of the book on which the Solution is based, Knovel and PC do not give any warranties or representations, express or implied, including with respect to fitness, intended purpose, use or merchantability and/or correctness or accuracy of this Solution. Array origin: ORIGIN 0 Description Pile supported footings or pile caps are used under columns and walls to distribute the load to the piles. he plan dimensions of a pile footing are determined by the number and the arrangement of the piles and the required minimum spacing between the piles. his application determines minimum required pile cap thickness and the maximum size and minimum number of reinforcing bars for pile caps with from to 0 piles in a group. Pile diameter and spacing, and the size of the supported rectangular pier or wall may be specified by the user. he application uses the Strength Design Method of ACI 318-89. he required input includes the pile capacity at service load (unfactored loads) and the plan dimensions of the column or wall. he user may also enter material properties and variables designated as "project constants" in this application. he material properties include strength of the concrete and the reinforcement, unit weight of concrete, unit weight of reinforced concrete, and the preferred reinforcement ratio. he project constants include pile diameter, minimum pile spacing, edge distance, and multiples used for rounding the pile coordinates, the pile cap depth and the plan dimensions of the pile caps. Copyright 011 Knovel Corp. Page 1 of 41
PC_CE_BSD_7._us_mp.mcdx Reference: ACI 318-89 "Building Code Requirements for Reinforced Concrete." (Revised 199) Input Input Variables Enter service load pile capacity, number of piles in group, and pier or wall dimensions. Pile capacity at service loads: Number of piles in group: Column width: Column depth: P cap 80 kip N 8 C x 4 in C y 4 in If the footing supports a wall, enter any wall length greater than or equal to the dimension of the footing parallel to the wall. Copyright 011 Knovel Corp. Page of 41
PC_CE_BSD_7._us_mp.mcdx Computed Variables he following variables are calculated in this document: Ps F Pu qu h X Y x' y' total service load capacity of the piles combined load factor for dead and live load total factored load capacity of the piles and the pile cap factored load capacity of one pile total pile footing thickness the longer dimension of the pile cap the shorter dimension of the pile cap pile coordinates in the X direction from the centroid of the pile group pile coordinates in the Y direction from the centroid of the pile group Input Constants he following variables are normally constant for a given project and may be defined by the user. he variables s, SzP, SzD and E are used by this application to calculate the pile coordinates and plan dimensions of the pile caps. Minimum pile spacing: Multiple for rounding pile spacing: s 3 ft SzP 0.5 in Copyright 011 Knovel Corp. Page 3 of 41
PC_CE_BSD_7._us_mp.mcdx Multiple for rounding pile cap plan dimensions: Minimum edge distance from center of pile: SzD 1 in E 1 ft + 3 in Ratio of live load to dead load: Pile diameter at top of pile: Pile embedment into pile cap: Clearance between reinforcement and top of pile: Multiple for rounding footing depths: R 1 d p 8 in e 4 in cl 3 in SzF 1 in Material Properties Enter the values of f'c, fy, wc, wrc, kv and kw. Specified compressive strength of concrete: Specified yield strength of reinforcement: Unit weight of concrete: Unit weight of reinforced concrete: Shear strength reduction factor for lightweight concrete kv = 1 for normal weight, 0.75 for alllightweight and 0.85 for sand-lightweight concrete (ACI 318, 11..1..): f' c 4 ksi f y 60 ksi w c 145 pcf w rc 150 pcf k v 1 Copyright 011 Knovel Corp. Page 4 of 41
PC_CE_BSD_7._us_mp.mcdx Weight factor for increasing development and splice lengths kw = 1 for normal weight and 1.3 for lightweight aggregate concrete (ACI 318, 1..4.): k w 1 Factors for use of lightweight concrete are included, but it would be unusual to use lightweight concrete for a pile cap. Modulus of elasticity of reinforcement (ACI 318, 8.5.): Strain in concrete at compression failure (ACI 318, 10.3.): Strength reduction factor for flexure (ACI 318, 9.3..1): Strength reduction factor for shear (ACI 318, 9.3..3): E s 9000 ksi ε c 0.003 ϕ f 0.9 ϕ v 0.85 Reinforcing bar number designations, diameters, and areas: No [ 0 1 3 4 5 6 7 8 9 10 11 1 13 14 15 16 17 18] d b A b [ 0 0 0 0.375 0.5 0.65 0.75 0.875 1.00 1.18 1.7 1.41 0 0 1.693 0 0 0.57] in [ 0 0 0 0.11 0.0 0.31 0.44 0.60 0.79 1.00 1.7 1.56 0 0.5 0 0 0 4.00] in Bar numbers, diameters and areas are in the vector rows (or columns in the transposed vectors shown) corresponding to the bar numbers. Individual bar numbers, diameters and areas of a specific bar can be referred to by using the vector subscripts as shown in the example below. Copyright 011 Knovel Corp. Page 5 of 41
PC_CE_BSD_7._us_mp.mcdx Pile Coordinates and Plan Dimensions of Pile Caps Dimensions used to compute pile coordinates: s0 0 ft s1 SzP ceil s sin ( 45 deg) s1 = 4.5 ft SzP s SzP ceil s sin ( 60 deg) SzP s =.65 ft s3 s + s s3 = 4.15 ft s4 s + s s4 = 5.65 ft Angle between the centerline of the two piles in the base row and a centerline drawn between either of the two piles in the base row to the pile at the apex of the 3-pile group: α atan s α = 60.55 deg s Pile coordinates expressed in terms of pile spacing. he number within the column operator indicates the number of piles in the group: x 1 [ s0 s0 ] y 1 [ s0 s0 ] x s s y [ s0 s0 ] Copyright 011 Knovel Corp. Page 6 of 41
PC_CE_BSD_7._us_mp.mcdx x 3 s0 s s x 4 s s s s x 5 s1 s1 s0 s1 s1 y 3 s s 3 3 s 3 y 4 s s s s y 5 s1 s1 s0 s1 s1 x 6 [ s s0 s s s0 s] y 6 s s s s s s x 7 s s s s0 s s s y 7 [ s s s0 s0 s0 s s ] x 8 s s0 s s s s s0 s y 8 [ s s s s0 s0 s s s ] x 9 [ s s0 s s s0 s s s0 s] y 9 [ s s s s0 s0 s0 s s s] x 10 s s0 s 3 s s s s s s0 s y 10 [ s s s s0 s0 s0 s0 s s s ] x 11 3 s s s s s s0 s 3 s s s s y 11 [ s s s s s0 s0 s0 s s s s ] Copyright 011 Knovel Corp. Page 7 of 41
PC_CE_BSD_7._us_mp.mcdx x 1 3 s s s s 3 s s s s 3 s s s s y 1 x 13 [ s s s s s0 s0 s0 s0 s s s s] [ s s0 s s s s s0 s s s s s0 s] y 13 s s s s s s0 s0 s0 s s s s s x 14 s s0 s 3 s s s s 3 s s s s s s0 s y 14 s3 s3 s3 s s s s s s s s s3 s3 s3 x 15 y 15 [ s s s0 s s s s s0 s s s s s0 s s] [ s s s s s s0 s0 s0 s0 s0 s s s s s] x 16 3 s s s s 3 s s s s 3 s s s s 3 s s s s y 16 s s s s s s s s s s s s s s s s Copyright 011 Knovel Corp. Page 8 of 41
PC_CE_BSD_7._us_mp.mcdx x 17 [ s s s s0 s s s s s0 s s s s s0 s s s ] y 17 s s s s s s s s0 s0 s0 s s s s s 3 s 3 s x 18 [ s s0 s s s s s0 s s s s s0 s s s s s0 s] y 18 s s s s s s s s s0 s0 s s s s s 3 s 3 s 3 s x 19 [ s4 s s s4 s0 s4 s s s4 s0 s4 s s s4 s0 s4 s s s4 ] y 19 s s s s s s s s s s0 s s s s s 3 s 3 s 3 s 3 s x 0 [ s s s0 s s s s s0 s s s s s0 s s s s s0 s s] y 0 s s s s s s s s s s s s s s s 3 s 3 s 3 s 3 s 3 s Pile coordinates in the X and Y directions from the center of the pile group being designed: i 0 N 1 x' x i i, N Copyright 011 Knovel Corp. Page 9 of 41
PC_CE_BSD_7._us_mp.mcdx x' = [ 3 0 3 1.5 1.5 3 0 3] ft y' y i i, N y' = [.65.65.65 0.000 0.000.65.65.65 ] ft Absolute values of pile coordinates in the X and Y directions from the center of the pile group being designed: x_abs x' x_abs = [ 3 0 3 1.5 1.5 3 0 3] ft y_abs y' y_abs = [.65.65.65 0 0.65.65.65 ] ft Absolute values of pile coordinates in the X and Y directions from the center of the pile group being designed minus 1/ the pier width, or the maximum absolute pile coordinate if the pile center is less than half the pier width from the center of the group: x_face if x_abs C x, x_abs C x, max i i i (x_abs) x_face = [.000 3.000.000 0.500 0.500.000 3.000.000 ] ft y_face if y_abs C y, y_abs C y, max i i i (y_abs) Copyright 011 Knovel Corp. Page 10 of 41
PC_CE_BSD_7._us_mp.mcdx y_face = [ 1.65 1.65 1.65.65.65 1.65 1.65 1.65 ] ft Distances ax and ay from face of pier in the X and Y directions to the closest pile center, or 0 ft if all pile centers are less than or equal to 1/ the pier width or depth from the pile group center: a x if max (x_abs) C x, 0 ft, min (x_face) a y if max (y_abs) C y, 0 ft, min (y_face) a x = 6 in a y = 19.5 in Plan dimensions pile caps as function of the number of piles: FX (N) max x N min N x + E FY (N) max y N min N y + 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 ft Dimensions A and B for the 3-pile group rounded up to a multiple of SzD: A E 1 1 SzD ceil sin (α) tan (α) SzD A= 18 in Copyright 011 Knovel Corp. Page 11 of 41
PC_CE_BSD_7._us_mp.mcdx B E 1+ tan (α) 1 1 SzD ceil sin (α) SzD B= 19 in Plan area of pile cap, CapArea: CapArea if FX (3) N 3, X Y, FX (3) FY (3) A ( FY (3) B) CapArea = 65.875 ft Example: No = 5 d b5 = 0.65 in A b5 = 0.31 in 5 he following values are computed from the entered material properties. Modulus of elasticity of concrete (for values of wc between 90 pcf and 155 pcf (ACI 318, 8.5.1): E c w 1.5 c 33 f' c psi E c = 3644 ksi pcf psi Strain in reinforcement at yield stress: ε y f y ε y = 0.0007 E s Factor used to calculate depth of equivalent rectangular stress block (ACI 318, 10..7.3): β 1 if f' f' c 4 ksi c 4 ksi f' c 8 ksi, 0.85 0.05, if f' c 4 ksi, 0.85, 0.65 ksi Copyright 011 Knovel Corp. Page 1 of 41
PC_CE_BSD_7._us_mp.mcdx β 1 = 0.85 Reinforcement ratio producing balanced strain conditions (ACI 318, 10.3.): ρ b β 1 0.85 f' c E s ε c ρ b =.851% E s ε c + f y f y Maximum reinforcement ratio (ACI 318, 10.3.3): ρ max 3 4 ρ b ρ max =.138% Minimum reinforcement ratio for beams (ACI 318, 10.5.1, Eq. (10-3)): ρ min 00 lbf ρ = in min 0.333% f y Shrinkage and temperature reinforcement ratio (ACI 318, 7.1..1): f y ρ temp if f y 50 ksi,.00, if f y 60 ksi,.00.000, if.0018 60 ksi.0018 60 ksi.0014,,.0014 60 ksi f y f y Preferred reinforcement ratio: ρ 3 8 ρ b Any reinforcement ratio from min (the minimum reinforcement ratio for flexure) to 3/4 b (the maximum reinforcement ratio) may be specified. Numerical values may also be entered directly, such as 1.0 % or 0.01. Flexural coefficient K, for rectangular beams or slabs, as a function of (ACI 318, 10.): (Moment capacity Mn = K( F, where F = bd) Copyright 011 Knovel Corp. Page 13 of 41
PC_CE_BSD_7._us_mp.mcdx K (ρ) ϕ f K (ρ) = 0.53 ksi ρ ρ f y 1 f y 0.85 f' c Limit the value of f'c for computing shear and development lengths to 10 ksi by substituting f'c_max for f'c in formulas for computing shear and development lengths (ACI 318, 11.1., 1.1.): f' c_max if f' c > 10 ksi, 10 ksi, f' c Nominal "one way" shear strength per unit area in concrete (ACI 318, 11.3.1.1, Eq. (11-3), 11.5.4.3): v c f' c_max k v psi v c = 16 psi psi Basic tension development length ldbt (ACI 318, 1.. and 1..3.6): No. 3 through No. 11 bars: n 3 11 X1 n f y 0.04 A bn lbf f' c_max X n f y 0.03 d bn f' c_max psi psi l dbtn if X1 > X, X1, X n n n n l dbt = [ 0 0 0 10.7 14. 17.8 1.3 4.9 30 37.9 48. 59.] in Copyright 011 Knovel Corp. Page 14 of 41
PC_CE_BSD_7._us_mp.mcdx No. 14 bars: l dbt14 f y in 0.085 lbf f' c_max l dbt14 = 80.6 in No. 18 bars l dbt18 f y in 0.15 lbf f' c_max l dbt18 = 118.6 in ension development length (ACI 318, 1..1): No. 3 through No. 11 bars: l dtn if k w l dbtn 1 in, k w l dbtn, if k w l dbtn > 0 in, 1 in, 0 in l dt = [ 0 0 0 1 14. 17.8 1.3 4.9 30 37.9 48. 59. ] in No. 14 bars: l dt14 k w l dbt14 = 80.6 in l dt14 No. 18 bars l dt18 k w l dbt18 = 118.6 in l dt18 Solution Service load: P s P cap N = 640 kip P s Combined load factor for dead + live load: F 1.4 + 1.7 R 1+ R F= 1.55 Copyright 011 Knovel Corp. Page 15 of 41
PC_CE_BSD_7._us_mp.mcdx Factored load Pu: P u F N P cap = 99 kip Factored load per pile: P u q u P u N q u = 14 kip he longer dimension of the pile cap: X= 8.5 ft he shorter dimension of the pile cap: Y= 7.75 ft Range 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 bottom right: x' = [ 3 0 3 1.5 1.5 3 0 3] ft Pile coordinates in the Y direction from the centerline of the pile group, starting from left to right and top to bottom: y' = [.65.65.65 0 0.65.65.65] ft Copyright 011 Knovel Corp. Page 16 of 41
PC_CE_BSD_7._us_mp.mcdx Solution to determine the minimum required footing depth for flexure, df Bending moment about the Y axis as function of distance x from Y axis: M y (x) i q u if x' x, 0 ft, x' x i i Bending moment about the Y axis at face of pier: M y C x = 558 kip ft Bending moment about the X axis as function of distance y from X axis: M x (y) i q u if y' y, 0 ft, y' y i i Bending moment about the X axis at face of pier: M x C y = 604.5 kip ft Effective footing width for bending about the X axis, at face of pier: X f if N 3, X, A+ E+ 3 s C y X A Y B X f = 8.5 ft Copyright 011 Knovel Corp. Page 17 of 41
PC_CE_BSD_7._us_mp.mcdx Effective footing widths for flexure and shear vary with distance from the pile group centroid for the 3- pile group. he remaining pile caps are rectangular in shape and the full width is effective for flexure or shear at any section. Effective footing width at face of pier, for bending about the Y axis: Y f Y if ( N 3) + C x A, 0 ft, C x A X A ( Y B) Y f = 7.75 ft Maximum bending moment per unit width: M u M x C y M y C x M x C y M y C x if >,, X f Y f X f Y f = M u kip 7 ft ft Minimum required effective footing depth for flexure at specified reinforcement ratio (ACI 318, 10.): d f M u K (ρ) d f = 11.735 in Solution to determine the minimum required footing depth for one-way beam shear, dbm Shear Vux(x) in the X direction as a function of any specified positive distance x from the Y axis (ACI 318, 15.5.3): Copyright 011 Knovel Corp. Page 18 of 41
PC_CE_BSD_7._us_mp.mcdx V ux (x) i if x' C x + d p, 0, if x' x + i i d p,, 1 if x' x i d p x' + d p x i,, 0 d p q u Shear Vuy(y) in the Y direction as a function of any specified positive distance y from the X axis (ACI 318, 15.5.3): V uy (y) i if y' C y + i d p,, 0 if y' y + i d p, 1, if y' y i d p y' + d p y i, 0, d p q u Since the 3-pile group is not rectangular the effective width in the X and Y directions varies with distance from the pile group centroid. he shear must be checked at both the inside edge of the piles and at distance d from the face of the pier. he following functions are for calculating the effective widths and required depths at any positive distance from the pile group centroid, and the required depths at distance d from the face of the pier. he largest required depth at distance d from the face of pier or at the inside edge of the piles determines the minimum required depth for the 3-pile group. Effective footing width for one-way shear in the Y direction for the 3-pile group, as a function of any positive distance y from the X axis: X 3 (y) A+ ( E + max (y') y) X A Y B Minimum required depth for one-way shear in the Y direction for the 3-pile group as a function of any positive distance y from the X axis: d y3 (y) if y C y, 0 ft, if V uy (y) < y C y,, ϕ v v c X 3 (y) V uy (y) y C y ϕ v v c X 3 (y) Copyright 011 Knovel Corp. Page 19 of 41
PC_CE_BSD_7._us_mp.mcdx Effective footing width for one-way shear in the X direction for the 3-pile group, as a function of any positive distance x from the Y axis: Y 3 (x) Y if C x + x A, 0 ft, C x + x 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 any positive distance x from the Y axis: d x3 (x) if x C x, 0 ft, if V ux (x) < x C x,, ϕ v v c Y 3 (x) V ux (x) x C x ϕ v v c Y 3 (x) Minimum required depth for one-way shear in the X direction for the 3-pile group at distance d from face of pier: Guess Values d 1 ft d max (x') + d p Constraints Solver V ux C x + d = ϕ v v c Y 3 C x + d d d x3_d (d) Find (d) Copyright 011 Knovel Corp. Page 0 of 41
PC_CE_BSD_7._us_mp.mcdx Minimum required depth for one-way shear in the Y direction for the 3-pile group at distance d from face of pier: Guess Values d 1 ft d max (y') + d p Constraints V uy C y + d = ϕ v v c X 3 C y + d d Solver d y3_d (d) Find (d) Copyright 011 Knovel Corp. Page 1 of 41
PC_CE_BSD_7._us_mp.mcdx Functions for rectangular pile caps he following Mathcad solve blocks contain the functions for determining the required depth for oneway shear for the rectangular pile caps. Minimum required depth for one-way shear in the X direction at distance d from face of pier: Guess Values d 1 ft d max (x') + d p Constraints V ux C x + d =d ϕ v v c Y Solver d x (d) Find (d) Minimum required depth in the Y direction at distance d from face of pier: Guess Values Constraints Solver d 1 ft d max (y') + d p V uy C y + d =d ϕ v v c X d y (d) Find (d) Copyright 011 Knovel Corp. Page of 41
PC_CE_BSD_7._us_mp.mcdx d 1 ft Depth required for one-way shear in the Y direction: d y d if N 3, d y (d), max y3 max (y') d y3_d (d) d y = 19.015 in Depth required for one-way shear in the X direction: d p d x d if N 3, d x (d), max x3 max (x') d x3_d (d) d x = 1.171 in d p Minimum depth required for beam shear: d bm max d x d d bm = 1.171 in y Copyright 011 Knovel Corp. Page 3 of 41
PC_CE_BSD_7._us_mp.mcdx Calculations to determine the minimum required footing depth for peripheral shear, dper (ACI 318, 11.1..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, 11.1.1.): b o (d) C x + C y + d Ratio of the longer to the shorter pier dimension (ACI 318 11.1..1): C x β c if C x C y,, β c = 1 C x C y C y Copyright 011 Knovel Corp. Page 4 of 41
PC_CE_BSD_7._us_mp.mcdx Nominal "two way" concrete shear strength per unit area in slabs and footings, expressed as a function of effective depth d. Since piers or columns are at the center of the pile cap they are considered "interior" columns for calculating two way shear, and s is equal to 40. (ACI 318 11.1..1, Eqs. (11-36), (11-37) and (11-38)): α s 40 v cp (d) + 4 β c min α s d b o (d) 4 k v f' c_max psi psi Function V1(d) computes a vector with elements equal to 0 if the pile produces shear on the critical section and 1 if it does not: V1 (d) x' C x + d d p y' C y + d d p Function V(d) computes a vector with elements equal to 1 if the full reaction of the pile produces shear on the critical section and 0 if it does not: V (d) x' C x + d + d p + y' C y + d + d p > 0 Function V3(d) computes a vector with elements equal to 1 if a portion of the pile reaction produces shear on the critical section and 0 if it does not: V3 (d) 1 V1(d) V (d) Copyright 011 Knovel Corp. Page 5 of 41
PC_CE_BSD_7._us_mp.mcdx 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' > + C x + d d p 0 ft x' C x + d d p V3 (d) y' > C y + d d p 0 ft y' C y + d d p Shear Vup(d) on the peripheral shear section at distance d/ from the face of the pier, expressed as a function of d (ACI 318, 15.5.3): V up (d) V4 (d) V (d) + q u d p Guess value of d: d d bm d' per (d) root V up (d) ϕ v v cp (d), d b o (d) d d' per if C x + d bm X + C y + d bm Y, d bm, d' per (d) d' per = 19.816 in Minimum depth for peripheral shear dper: d per if C x + d' per X + C y + d' per Y, d bm, d' per d per = 19.816 in Copyright 011 Knovel Corp. Page 6 of 41
PC_CE_BSD_7._us_mp.mcdx If either dimension of the critical shear section is greater than the corresponding pile cap dimension, peripheral shear is not critical and the depth for peripheral shear is defined as equal to the depth required for one-way shear. Calculation to determine the minimum required depth for deep beam shear dbm (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 the face of the pier (See ACI 318, Chapter 11 Notation, and 11.8): 0.5 a x = 0.5 ft 0.5 a y = 0.813 ft Moment and shear in the X and Y directions on the critical shear sections at distance 0.5ax and 0.5ay from the face of the pier (ACI 318, 15.5.3): C x M y + 0.5 a x = 465 kip ft Copyright 011 Knovel Corp. Page 7 of 41
PC_CE_BSD_7._us_mp.mcdx C x V ux + 0.5 a x = 356.5 kip M x C y + 0.5 a y = 30.5 kip ft V uy C y + 0.5 a y = 37 kip Ratios of moment to shear Rx and Ry, at the critical sections for deep beam shear in the X and Y directions: C x M y + 0.5 a x R x V ux C R x = 1.304 ft x + 0.5 a x M x C y + 0.5 a y R y V uy C R y = 0.813 ft y + 0.5 a y Multipliers for use in ACI 318, Eq. (11-30): KX (d) if 3.5.5 R x < 1, 1, if 3.5.5 R x >.5,.5, 3.5.5 R x d d d Copyright 011 Knovel Corp. Page 8 of 41
PC_CE_BSD_7._us_mp.mcdx KY (d) if 3.5.5 R y < 1, 1, if 3.5.5 R y >.5,.5, 3.5.5 R y d d 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: v cx (d) v cy (d) KX (d) 1.9 f' c + 500 ρ temp d psi min psi R x 6 f' c psi psi KY (d) 1.9 f' c + 500 ρ temp d psi min psi R y 6 f' c psi psi Effective depth required in the X direction for deep beam shear: Guess value of d: d d x V ux C x + 0.5 a x d bm_x root, ϕ v v cx (d) if N 3, Y, Y 3 C d d x + 0.5 a x d bm_x = 1.381 in v cx d bm_x = 10.9 psi Copyright 011 Knovel Corp. Page 9 of 41
PC_CE_BSD_7._us_mp.mcdx Effective depth required in the Y direction for deep beam shear: Guess value of d: d d y V uy C y + 0.5 a y d bm_y root, ϕ v v cy (d) if N 3, X, X 3 C d d y + 0.5 a y d bm_y = 16.556 in v cy d bm_y = 59. psi Depth required for deep beam shear is the larger value of dbm: d bm max d bm_x d bm_y d bm = 1.381 in Copyright 011 Knovel Corp. Page 30 of 41
PC_CE_BSD_7._us_mp.mcdx Solution to determine the minimum required depth for beam shear or peripheral shear for one corner pile, dcor Critical Section for Beam Shear for 1 corner pile Critical Section for Peripheral Shear for 1 corner pile Required depth for beam shear for one corner pile Guess value of d: d 1 ft d cor_1 q u root, ϕ v v c d p + d+ E d d d cor_1 = 14.515 in Required depth for peripheral shear for one corner pile 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 0. Since c is 1 for piles in this application, ACI 318 Eq. (11-36) is not critical (ACI 318, 11.1..1, Eqs. (11-37) and (11-38)): Copyright 011 Knovel Corp. Page 31 of 41
PC_CE_BSD_7._us_mp.mcdx α s 0 b o (d) d p + d π 4 + E v cp (d) min α s d b o (d) 4 + k v f' c_max psi psi d cor_ root q u d, d ϕ v v cp (d) b o (d) d cor_ = 1.507 in Depth required for one corner pile is the larger value of dcor: d cor max d cor_1 d cor_ d cor = 14.515 in Depth required for a single interior pile, 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 40 Since c is 1 for piles in this application ACI 318 Eq. (11-36) is not critical. (ACI 318 11.1..1, Eqs. (11-37) and (11-38)): α s 40 b o (d) π d p + d v cp (d) min α s d b o (d) 4 + k v f' c_max psi psi Copyright 011 Knovel Corp. Page 3 of 41
PC_CE_BSD_7._us_mp.mcdx d one root b o (d) d ϕ v v cp (d) q u, d d one = 10.16 in Depth required for two interior piles, dtwo Nominal "two way" concrete shear strength per unit area for two interior piles, expressed as a function of effective depth d. For interior piles s is equal to 40 (ACI 318, 11.1..1, Eqs. (11-37) and (11-38)): α s 40 β c (d) s+ d+ d p + d d p b o (d) π d p + d + ( s+ d) v cp (d) + 4 β c (d) min α s d + b o (d) 4 k v f' c_max psi psi Depth, dtwo, required for two adjacent piles is equal to done unless overlapping shear perimeters require a greater depth when spacing s is less than done + dp: d two if d p + d one s, d one, root b o (d) d ϕ v v cp (d) q u, d d two = 10.16 in Copyright 011 Knovel Corp. Page 33 of 41
PC_CE_BSD_7._us_mp.mcdx Minimum thickness for a footing on piles (ACI 318, 15.7): d min 1 in Minimum required effective depth, de Largest effective depth determined by flexure, one-way beam shear, peripheral shear, deep beam shear, shear on a single corner pile, a single interior pile, two adjacent piles, or the minimum thickness required for a footing on piles de: d f d bm d per d bm d cor d one d two d min 11.735 1.171 19.816 1.381 = in 14.515 10.16 10.16 1.000 d e max d f d bm d per d bm d cor d one d two d min d e = 1.381 in Solution to determine the largest permissible reinforcing bar sizes Maximum available lengths in the X direction for development of reinforcement: L dx X C x 3 in L dx = 3 ft Copyright 011 Knovel Corp. Page 34 of 41
PC_CE_BSD_7._us_mp.mcdx Maximum available lengths in the Y direction for development of reinforcement. he available development length for the 3-pile group is calculated from face of pier to the pile at the apex of the group: L dy + if Y C y 3 s E C y N 3,, 3 in L dy =.65 ft sin (α) Index numbers of maximum bar sizes determined by available development lengths: index 0 index if l dtn L,, 0 0 n dx n index 0 bx index bx = 8 0 index 0 index if l dtn L,, 0 0 n dy n index 0 by index by = 8 0 Sizes of the largest permissible reinforcing bars for the X and Y directions. Since the 3-pile group uses three equal bands of reinforcement the maximum bar size is defined as the smaller bar size for either the X or Y direction: BarSize_X if N 3, bx, min bx by BarSize_X = 8 bx BarSize_X = 1 in d bbx BarSize_Y if N 3, by, min bx by BarSize_Y = 8 Copyright 011 Knovel Corp. Page 35 of 41
PC_CE_BSD_7._us_mp.mcdx by BarSize_Y = 1 in d bby otal pile cap thickness rounded up to the nearest multiple of SzF: h d e + d bbx + 0.5 d bby + cl + e SzF ceil SzF h= 30 in Pile cap weight: CapWt CapArea h w rc CapWt = 4.703 kip Calculation to determine the required number of reinforcing bars in the X and Y directions Effective depths in the X and Y directions: d ex h e cl 0.5 d bbx =.5 in d ex d ey h e cl d bbx 0.5 d bby = 1.5 in d ey otal required flexural reinforcement area in the X direction: A sx_flex Y f d ex 1 M y C x 1 ϕ f Y f d 0.85 f' c ex 0.85 f' c f y A sx_flex = 5.646 in Copyright 011 Knovel Corp. Page 36 of 41
PC_CE_BSD_7._us_mp.mcdx Minimum required reinforcement area for shrinkage and temperature in the X direction (ACI 318, 7.1): A sx_temp ρ temp h Y A sx_temp = 5.0 in Larger required reinforcement in the X direction: A sx max A sx_flex A sx_temp A sx = 5.646 in otal required reinforcement area for flexure in the Y direction: A sy_flex if N, X f d ey 1 M x C y 1, ϕ f X f d 0.85 f' c ey 0.85 f' c f 0 in y A sy_flex = 6.414 in Minimum required reinforcement area for shrinkage and temperature in the Y direction (ACI 318, 7.1): A sy_temp ρ temp h X A sy_temp = 5.508 in Larger required reinforcement in the Y direction: A sy max A sy_flex A A sy = 6.414 in sy_temp Copyright 011 Knovel Corp. Page 37 of 41
PC_CE_BSD_7._us_mp.mcdx Minimum number of bars to limit spacing of reinforcement to 18 inches (ACI 318, 7.6.5): MinNumb_X ceil Y 6 in + 0.5 MinNumb_X = 6 18 in MinNumb_Y ceil X 6 in + 0.5 MinNumb_Y = 6 18 in Maximum bar areas corresponding to minimum specified spacing: MinA_x A sx MinNumb_X MinA_x = 0.941 in MinA_y A sy MinNumb_Y MinA_y = 1.069 in Index numbers of bar sizes determined by minimum specified spacing: index 0 index if 0 0 n ax index ax = 8 0 A bn MinA_x, n, index 0 index 0 index if 0 0 n ay index ay = 9 0 A bn MinA_y, n, index 0 Copyright 011 Knovel Corp. Page 38 of 41
PC_CE_BSD_7._us_mp.mcdx 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 reinforcement or the specified minimum spacing). he bar size in the Y direction of the -pile group is set at 3: BarSixe_X if M y C x =0 kip ft + ( ax bx), ax, bx BarSixe_X = 8 BarSize_Y if N, if M x C y =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 required 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 pile centers. Because the bands are at an angle to the axes of bending the reinforcement areas must be adjusted accordingly. otal required reinforcement area in each of 3 bands for the 3-pile group: A sx if N 3, A sx, if A sx A sy >,, sin (α) 1 + cos (α) A sx sin (α) A sy 1 + cos (α) A sx = 5.646 in Copyright 011 Knovel Corp. Page 39 of 41
PC_CE_BSD_7._us_mp.mcdx A sy if N 3, A sy, A sx A sy = 6.414 in Number of bars required in the X and Y directions. he number of bars in Y direction of the -pile group is set at a multiple of 18 inches: Numb_X A sx ceil Numb_X = 8 A bcx Numb_Y if N, ceil A sy, ceil X A bcy 18 in Numb_Y = 9 Summary Computed Variables Combined load factor for dead and live load: F= 1.55 otal service load capacity of the piles and the pile cap: = 640 kip P s otal factored load capacity of the piles and the pile cap: P u = 99 kip Number of reinforcing bars in the X direction: Numb_X = 8 Copyright 011 Knovel Corp. Page 40 of 41
PC_CE_BSD_7._us_mp.mcdx Bar size in the X direction: BarSize_X = 8 Number of reinforcing bars in the Y direction: Numb_Y = 9 Bar size in the Y direction: BarSize_Y = 8 he number and size of reinforcing bars for the 3-pile group is shown as equal in the X and Y directions. he 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 shown for the X and Y directions. Longer dimension of the pile cap: X= 8.5 ft Shorter dimension of the pile cap: Y= 7.75 ft otal pile footing thickness: h= 30 in otal pile cap weight: CapWt = 4.703 kip User Notices Equations and numeric solutions presented in this Mathcad worksheet are applicable to the specific 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 and spans, materials and other input parameters beyond the intended range may make some equations no longer applicable. Modify the equations as appropriate if your parameters fall outside of the intended range. For this Mathcad worksheet, the global variable defining the beginning index identifier for vectors and arrays, ORIGIN, is set as specified in the beginning of the worksheet, to either 1 or 0. If ORIGIN is set to 1 and you copy any of the formulae from this worksheet into your own, you need to ensure that your worksheet is using the same ORIGIN. Copyright 011 Knovel Corp. Page 41 of 41