design for punching shear strength with aci 318-95

ACI Structural Journal/July-August 1999 539

ACI Structural Journal, V. 96, No. 4, July-August 1999.Received October 13, 1997, and reviewed under Institute publication policies. Copy-

right © 1999, American Concrete Institute. All rights reserved, including the making of copiesunless permission is obtained from the copyright proprietors. Pertinent discussion includingauthor’s closure, if any, will be published in the May-June 2000 ACI Structural Journal ifthe discussion is received by January 1, 2000.

ACI STRUCTURAL JOURNAL TECHNICAL PAPER

Brittle punching failure of flat plates can occur due to the transferof shearing forces and unbalanced moments between slabs andcolumns. Design of connections of columns to flat plates to insuresafety against punching failure is presented. This paper covers thedesign procedure in most practical situations, including interior,edge, and corner columns; prestressed and nonprestressed slabs;slabs with openings; and slabs with shear reinforcement. The ACI318-95 Building Code requirements are adhered to where appli-cable. Numerical examples are presented to demonstrate thedesign procedure. Seismic design considerations are not discussedin this paper.

Keywords: columns (supports); connections; flat concrete plates; pre-stressed concrete; punching shear; raft foundations; reinforced concrete;shear strength; slabs; structural design.

INTRODUCTIONThe punching shear resistance of concrete flat plates

frequently needs to be increased by the provision of droppanels or by shear reinforcement. The latter solution is moreacceptable architecturally, and is often more economical.This paper gives the details of punching shear design of flatplates without drop panels, with or without shear reinforce-ment. Requirements of the ACI 318-951 Building Code fordesign of slabs against punching are reviewed. The designsteps are presented, adhering to the code requirements whenthey apply. Most conditions that occur in practice are consid-ered for slabs with or without prestressing, including slabs withopenings in the column vicinity. Interior, edge, and cornercolumn-slab connections subjected to shear and momenttransfer are considered. The design steps are demonstrated bycomputed examples. This paper presents a complete designprocedure for punching shear. Reference is made to an availablecomputer program that can be used for the design. When droppanels are used, the design procedure for flat plates applies withan additional provision that is also discussed.

The ACI 318-951 Building Code allows the use of shearheads, in the form of steel I- or channel-shaped sections, asshear reinforcement in slabs. Because at present this type israrely used, it will not be discussed here. The two mostcommon types of shear reinforcement are shown in Fig. 1.To save space in this paper, the arrangements of the rein-forcement with the two types are shown in a single top viewin Fig. 1(a). Fig. 1(b) and (c) are a pictorial view and a crosssection showing, respectively, details of conventional stir-rups and stud shear reinforcement (SSR). The vertical legs ofthe stirrups or the stems of the studs intersect the shear cracksand prevent their widening (Fig. 2). Because the intersectioncan occur at any section of the stirrup leg or the stud stem,the leg or the stem should be as long as possible and must beanchored as closely as possible to the top and bottomsurfaces of the slab (observing the cover requirements forcorrosion and fire protection).

Effective anchorage is essential to develop the yieldstrength of the shear reinforcement of both types. With stirrups

[Fig. 1(b)], the anchorage is provided by hooks, bends, andthe longitudinal flexural reinforcing bar lodged at thecorners. Before the force in a stirrup leg reaches its yieldstrength, the concrete inside the hooks or bends crushes or

Title no. 96-S60

Design for Punching Shear Strength with ACI 318-95by Amin Ghali and Sami Megally

Fig. 1—Types of shear reinforcement considered: (a) shearreinforcements (top view); (b) stirrups; and (c) stud shearreinforcement alternate details (Section A-A).

ACI Structural Journal/July-August 1999540

splits, causing slip, thus preventing development of the fullstrength of the stirrup, particularly in thin slabs. For thisreason, ACI 318R-951 emphasizes that stirrups can be used,provided they are well-anchored, and requires that the stirrupsbe closed and enclose a longitudinal bar at each corner[Fig. 1(b)]. The Canadian Standard CSA-A23.3-942 doesnot permit use of stirrups as shear reinforcement in slabsthinner than 300 mm (12 in.).

The SSR relies on mechanical anchorage by heads at bothends of the stem or by a head at one end and a steel strip weldedto several studs. The steel strip holds the studs in a vertical posi-tion and insures the appropriate spacing between them until theconcrete is cast. The size of the anchor heads must be large

enough to insure that the full yield strength of the stud can bedeveloped with negligible slip of the anchorage. Experimentsshow that this can be achieved with anchor heads of area nine to10 times the cross-sectional area of the stud.

RESEARCH SIGNIFICANCEThis paper outlines the steps of design for punching shear

strength in accordance with ACI 318-95. However, the codedoes not cover all situations encountered in practice. Forthese situations, the design is based on research.

ACI 318-95 Code requirementsACI 318-951 requires that at a critical section at d/2 from

column face (Fig. 3)

vu ≤ φvn (1)

where vn is the nominal shear stress; φ is the strength-reduc-tion factor (φ = 0.85); vu is the maximum shear stress causedby the transfer of a factored shearing force Vu and bendingmoments Mux and Muy between the slab and column andacting at critical section centroid

(2)

where bo is length of perimeter of shear critical section; d isthe distance from extreme compression fiber to centroid oflongitudinal tension reinforcement; the subscripts x and yrefer to centroidal axes in directions of both spans; (x, y) arecoordinates of the point at which vu is maximum and J is aproperty of critical section “analogous to polar moment ofinertia.” Figure 3 indicates the positive directions of x and yaxes, the force Vu, and moments Mux and Muy ; in this figureand others in this paper, the arrows represent the directionsof force and moments exerted by the column on the slab. InFig. 3, x and y are replaced by x and y if they are principalaxes. γvx and γvy are fractions of the moments transferred byeccentricity of shear about the x and y axes, respectively

(3)

ACI 318-951 defines b1 and b2, respectively, as widths ofshear critical section measured in direction of the span forwhich moment is determined and perpendicular to it. Thus,when calculating γvy for the rectangular critical sectionshown in Fig. 3(a), b1 and b2 are respectively equal to (c1 +d) and (c2 + d). The code does not give an equation for γv forcritical sections having shapes other than a closed rectangle.

In absence of shear reinforcement, the code requires thatthe nominal shear stress of nonprestressed slabs be thesmallest of (using lb and in. units)

(4)

(5)

vuVu

bod--------

γvxMux

Jx

----------------yγvyMuy

Jy

----------------x+ +=

γv 1 1

1 23--- b1 b2⁄+

-------------------------------–=

vn vc 2 4βc

-----+⎝ ⎠⎛ ⎞ fc′= =

vn vcαsd

bo

--------- 2+⎝ ⎠⎛ ⎞ fc′= =

ACI member Amin Ghali is a professor of civil engineering at the University of Cal-gary, Alberta, Canada. He is a member of ACI Committees 373, Circular ConcreteStructures Prestressed with Circumferential Tendons; and 435, Deflection of ConcreteBuilding Structures; and Joint ACI-ASCE Committees 343, Concrete Bridge Design;and 421, Design of Reinforced Concrete Slabs.

ACI member Sami Megally is a postdoctoral associate in the Department of Civil Engi-neering at the University of Calgary. He received his PhD from the University of Cal-gary in 1998 and his BSc from Ain-Shams University, Egypt, in 1988. His researchinterests include structural analysis, the finite element method, and seismic design ofreinforced concrete structures.

Fig. 2—Interception of cracks by vertical shear reinforcement.

Fig. 3—Critical sections for two-way shear in slabs at d/2from column face: (a) interior column; (b) edge column;and (c) corner column.


(6)

where vc is the nominal shear stress provided by concrete; βcis ratio of long side to short side of column; fc′ is specifiedconcrete compressive strength; αs = 40 for interior columns;αs = 30 for edge columns; and αs = 20 for corner columns.

When vu > φvn, slab thickness must be increased or shearreinforcement provided. When shear reinforcement is used,ACI 318-951 expresses the nominal shear stress as

(7)

(8)

(9)

where vs is nominal shear stress provided by shear reinforce-ment; Av is area of shear reinforcement within a distance s;fyv is specified yield strength of shear reinforcement; and s isspacing of shear reinforcement. The upper limit for s is 0.5d.Shear reinforcement must be extended for a sufficientdistance until the critical section outside the shear-reinforcedzone (Fig. 4) satisfies Eq. (1) with vn = vc = 2 .

Other provisions for prestressed slabs and slabs with openingsin the column vicinity will be discussed in the followingsections.

Prestressed slabsFor prestressed slabs with no shear reinforcement, ACI

318-951 replaces Eq. (4) to (6) by

(10)

where Vp is the vertical component of all effective prestressforces crossing the critical section; fpc is average value of fpcin two vertical slab sections in perpendicular directions, withfpc being the compressive stress at section centroid afterallowance for all prestress losses; and βp is the smaller of 3.5and [(αs d/bo) + 1.5]. Eq. (10) can replace Eq. (4) to (6) onlyif the following conditions are satisfied: (a) no portion of thecross section of the column shall be closer than four times theslab thickness to a discontinuous edge; (b) fc′ shall not betaken greater than 5000 psi; and (c) fpc in each direction shallnot be less than 125 psi nor be taken greater than 500 psi.

In thin slabs, it is difficult to control the slope of tendonprofile at the point it crosses a critical section. Thus, for practicalconsiderations, the last term in Eq. (10) may be neglected orVp reduced to account for the inaccuracy that can occur in theexecution of the tendon profile.

Within the shear-reinforced zone, vn is to be calculatedusing the same equations as for nonprestressed slabs.

Section 11.5.4.1 of ACI 318-95 allows for prestressedmembers, spacing of shear reinforcement, s to reach 0.75hbut not to exceed 24 in., where h is overall thickness ofmember. It is considered here that this limit is excessive inslabs, and it is recommended that the spacing should notexceed 0.75d. This is because the difference between d andh is more important in slabs than in beams and cracks couldbypass the shear reinforcement, as shown in Fig. 2.

vn vc 4 fc′= =

vn vc vs 6 fc′≤+=

vc 2 fc′=

vsAv fyv

bos--------------=

fc′

vn vc βp fc′ 0.3fpc Vp bod⁄+ += =Slabs with openings

ACI 318-951 requires that effect of openings on punchingshear resistance of a slab-column connection must be consid-ered when openings are located at a distance less than 10 timesthe slab thickness from the column or when openings arelocated within the column strip. The effect of openings is takeninto account by considering part of shear critical section to beineffective. The ineffective part is that part of the criticalsection perimeter that is enclosed by straight lines projectingfrom the column centroid and tangent to the boundaries of theopenings (Example 3).

Optional values for fraction γv for moment transfer by shear

ACI 318-951 introduced for the first time Section 13.5.3.3,which permits the option of reducing γv from the value givenby Eq. (3), and increasing γf by the same amount of reduc-tion. The symbol γf is the fraction of unbalanced momenttransferred by flexure. For a corner column [Fig. 3(c)] or foran edge column [Fig. 3(b), in absence of Mux], the coefficientγv can be reduced to zero provided that Vu ≤ 0.5φVc or0.75φVc for a corner or edge column, respectively; where Vc= vc bod , with vc given by Eq. (4) to (6). For an interiorcolumn or an edge column [Fig. 3(b), in absence of Muy], γvcan be reduced to (1.25γv by Eq. (3) - 0.25), provided that Vu≤ 0.4φVc. For all slab-column connections, the optionalreduction of γv below the value given by Eq. (3) is allowedonly when ρ ≤ 0.375ρb; where ρ is the ratio of nonpre-

Fig. 4—Critical sections for two-way shear in slabs at d/2from outermost peripheral line of shear reinforcement: (a)interior column; (b) edge column; and (c) corner column.

542 ACI Structural Journal/July-August 1999

stressed tension reinforcement in the slab; and ρb is the valueof ρ producing balanced strain conditions.

The authors consider Section 13.5.3.3 unsafe. The justifi-cations are given in the discussion of the code.3 Additionalexperimental data4-7 for interior columns giving further justi-fication of this opinion are given in Appendix A.*

Allowable values for nominal shear stress and spacing of stud shear reinforcement

Because of the superiority of anchorage of the SSR, justi-fied by tests,8-13 ACI 421.1R-9214 suggests the followingdeviations from ACI 318 when SSR is used:

1) The nominal shear stress vn resisted by concrete andshear reinforcement [Eq. (7)] can be as high as 8 ,instead of 6 . This enables use of thinner slabs;

2) The upper limits for so and s can be based on the valueof vu at the critical section at d/2 from column face

so ≤ 0.5d and s ≤ 0.75d when (11.1)

so ≤ 0.35d and s ≤ 0.5d when (11.2)

*The Appendix is available in xerographic or similar form from ACI headquarters,where it will be kept permanently on file, at a charge equal to the cost of reproductionplus handling at time of request.

fc′fc′

vu φ 6 fc′≤⁄

vu φ 6 fc′≤⁄

where so is the distance between first peripheral line of studsand column face.

ACI 421.1R-9214 considers a vertical branch of a stirrup tobe less effective than a stud in controlling shear cracksbecause the stud stem is straight over its full length while theends of the stirrup branch are curved, and the mechanicalanchors at the stud ends insure that the yield strength is avail-able at all sections of the stem; this is not the case with avertical branch of a stirrup.

For the same reasons, the Canadian Standard CSA-A23.3-942 allows, in presence of shear studs, a value of vc 1-1/2times the allowable value when stirrups are employed. Thesame approach is adopted in the remainder of the paper.Thus, when SSR is used, Eq. (7) and (8) will be replaced by

(12)

with

(13)

Arrangement of shear reinforcementFigure 5(a) shows the typical arrangement of stud shear

reinforcement at rectangular columns. Each group of studson a line perpendicular to the column face are welded to asteel strip or spaced in a steel trough [Fig. 1(c)]. ACI 421.1R-9214 recommends that, in a direction parallel to the columnface, the maximum distance g between the steel strips, ortroughs, be less than 2d. This limitation is to insure that thestuds confine the concrete and prevent widening of shearcracks over the perimeter of the critical section.

Stud rails can be arranged in two orthogonal directions[Fig. 5(b)] or radial directions [Fig. 5(c)] in the vicinity ofcircular columns. The distance g between stud rails in thevicinity of circular columns should not exceed 2d as shownin Fig. 5(b) and (c). The authors recommend the orthogonalrather than the radial arrangement of stud rails. This isbecause with the radial arrangement of stud rails, shear studsplaced in the forms in their appropriate design locations aremore likely to interfere with the bars of the flexural rein-forcement mesh.

When stirrups are used, they should be placed in rowsparallel to the column [Fig. 1(a)]. In the direction parallel tothe column faces, the distance g between stirrup legs [Fig.1(b)] should satisfy the requirement g ≤ 2d, or because stir-rups are less effective than shear studs, a more restrictivelimit should apply.

Parameter JThe Code Commentary ACI 318R-951 gives an equation

for the parameter J when the shear critical section has therectangular shape shown in Fig. 3(a). The code commentaryequation may be written in the form

Jy = Iy + d3(c1 + d)/6 (14)

where Iy is the second moment of area of the critical sectionabout the y axis. It can be verified that with the column sizesand slab thicknesses used in practice, the difference (Jy – Iy),which is equal to the second term in Eq. (14), does notexceed 3 percent of Iy . ACI 318-95 and its commentarydefine J as an “analogous to polar moment of inertia” and do

vn vc vs 8 fc′≤+=

vc 3 fc′=

Fig. 5—Stud shear reinforcement arrangement: (a) rectangularcolumns; (b) orthogonal arrangement at circular columns;and (c) radial arrangement at circular columns.


not give equations for J when the critical section has shapesother than rectangular.

The vertical shear stress vu calculated by Eq. (2) has avertical resultant component equal to Vu, but has momentcomponents slightly smaller than γvx Mux and γvy Muy. In otherwords, the component Vu combined with γvx Mux and γvy Muyare not in equilibrium with the shear stress in the criticalsection. Replacing Jx and Jy in Eq. (2) by the critical sectionarea’s second moments Ix and Iy about the centroidal prin-cipal axes x and y, respectively, gives linearly varying stressvu , whose resultants exactly satisfy equilibrium. With thisreplacement, the equation for the shear stress vu at any pointof the critical section becomes

(15)

This equation applies when the critical section has any shape.Use of this equation avoids the ambiguity in calculating theparameter J, which has no known meaning in mechanics.

Coefficient γvNumerous experiments have shown that the empirical Eq. (3)

adopted by ACI 318-95 is satisfactory for interior columnswhere the critical section, at d/2 from column faces, has theshape of the perimeter of a closed rectangle. At the same loca-tion, the critical section for edge and corner columns has threeor two sides, respectively [Fig. 3(b) and (c)]. Outside theshear-reinforced zone, the critical section follows the perim-eter of a closed or open polygon, whose sides are not allparallel to a column face (Fig. 4). Problems arise15 when theempirical Eq. (3), allowed by ACI 318-95 for critical sectionshaving the shape of a closed rectangle, is employed for cornercolumns. Similar design problems may arise when employingEq. (3) for edge columns.

Elgabry and Ghali16 showed by numerous finite elementanalyses that Eq. (3) does not apply for all cases and for allcritical sections. They gave the following equations for γv to

cover all cases and all shapes of the critical section encoun-tered in design (Fig. 6).

At interior columns

(16)

(17)

At edge columns

γvx = same as Eq. (16) (18)

when (19)

At corner columns

γvx = 0.4 (20)

γvy = same as Eq. (19) (21)

where lx and ly are projections of the critical section on prin-cipal axes x and y, respectively.

The safety of design using the above equations has beenverified using published experimental results.16

Inclined axesThe shear critical sections for corner columns, and for all

columns when the slab has nonsymmetric openings, haveprincipal axes x and y inclined to the column faces. In thesecases, it may be more convenient to calculate the shear stress at

vuVu

bod--------

γvxMux

Ix

----------------yγvyMuy

Iy

----------------x+ +=

γvx 1 1

1 23--- ly lx⁄+

-----------------------------–=

γvy 1 1

1 23--- lx ly⁄+

-----------------------------–=

γvy 1 1

1 23--- lx ly⁄( ) 0.2–+

-----------------------------------------------–=lx

ly

---- 0.2 γvy 0=,<

Fig. 6—Equations for γv applicable for critical sections at d/2 from column face andoutside shear-reinforced zone.


points with coordinates (x, y) referring to centroidal but nonprin-cipal axes using the following equation to replace Eq. (2)

(22)

where Mx and My are statical equivalents of γvx Mux and γvyMuy given by [Fig. 7(a)]

Mx = γvx Muxcosθ + γvxMuysinθ (23)

My = –γvx Muxsinθ + γvyMuycosθ (24)

(25)

where da is elemental area of the critical section.In general, the periphery of shear critical section is

composed of straight segments. The values of Ix y , Ix , and Iyof the critical section may be determined by summation ofthe contributions of straight segments

; ; (26)

where m is the total number of segments, and i refers to thei-th segment. A typical straight segment AB is shown in Fig. 8;its contributions to Ix y , Ix , and Iy may be calculated by

(27)

vuVu

bod--------

MxIy MyIxy–

IxIy Ixy2–

-------------------------------⎝ ⎠⎜ ⎟⎛ ⎞

yMyIx MxIxy–

IxIy Ixy2–

-------------------------------⎝ ⎠⎜ ⎟⎛ ⎞

x+ +=

Ix y2–

∫ da Iy x2–∫ da Ixy xy ad∫=;=;=

Ixy Ixyi

i 1=

m

∑= Ix Ixi

i 1=

m

∑= Iy Iyi

i 1=

m

∑=

l( )AB xB xA–( )2 yB yA–( )2+[ ]1 2⁄

=

(28)

(29)

(30)

where d is effective depth; (xA, yA) and (xB, yB) are the coor-dinates of the segment ends A and B.

The angle θ between the principal x axis and the x axis isgiven by

tan2θ = –2Ixy/(Ix – Iy) (31)

The positive sign convention for θ is indicated in Fig. 7.The equations presented in this section apply when the x

and y axes are principal or not. But, when they are principal,x ≡ x; y ≡ y; Ix y = 0; θ = 0, and Eq. (22) reduces to Eq. (15).

Design stepsThe data required for design of slab-column connections

are: d, c1, c2, Vu, MuxO, MuyO, and fc′ [Fig. 3(a) and (b)]. It isrequired to determine whether d is sufficient for safetyagainst punching without the use of shear reinforcementand if not, design the necessary shear reinforcement. Thesymbols MuxO and MuyO are the unbalanced moments at thecolumn centroid. When working with nonprincipal axes x,y [Fig. 3(c)], the given moments will be MuxO and MuyO andSteps 1 and 2 of the design given below will be changed.

The first critical section to be considered is at d/2 fromthe column face. The steps of design when x and y are prin-cipal axes are:

Step 1—Replace Vu, MuxO, and MuyO by their staticalequivalents Vu , Mux, and Muy at the centroid of the criticalsection considered [Fig. 3(a) and (b) or 4(a) and (b)]

; (32)

where xO and yO are coordinates of the column centroid.Appropriate signs for the force and moments must be used;the positive sign convention is indicated in Fig. 3.

Ixy( )AB

d l( )AB

6--------------- 2xAyA 2xByB xAyB xByA+ + +( )=

Ix( )AB

d l( )AB

3--------------- yA

2 yB2 yAyB+ +( )=

Iy( )AB

d l( )AB

3--------------- xA

2 xB2 xAxB+ +( )=

Mux MuxO VuyO+= Muy MuyO VuxO+=

Fig. 7—Transformation of moments: (a) use of Eq. (23) and(24); and (b) use of Eq. (34) and (35).

Fig. 8—i-th segment of shear critical section.


Step 2—Using the applicable equation for γv selectedfrom Fig. 6, determine γvx and γvy. Calculate vu by Eq. (15).

Step 3—If vu ≤ φvn [given by Eq. (4) to (6)], no shearreinforcement is required. If (vu/φ) > vn limit , d must beincreased; where vn limit = 6 or 8 when stirrups orstuds are used as shear reinforcement, respectively. Whenvn < vu/φ ≤ vn limit , go to Step 4.

Step 4—Select Av and s such that Eq. (1) is satisfied.When conventional stirrups are used, vn is determinedusing Eq. (7) to (9). When stud shear reinforcement is used,use Eq. (12), (13), and (9).

Step 5—Extend the shear reinforcement zone byincreasing the number of peripheral lines of studs. RepeatSteps 1 and 2 for a critical section at d/2 outside the outer-most peripheral line of shear reinforcement (Fig. 4). If vu ≤2φ , extension of shear reinforcement is sufficient; ifnot, extend the shear reinforcement farther away fromcolumn and repeat Steps 1 and 2 until this requirement issatisfied.

Revision of Steps 1 and 2 when nonprincipal axes are used

Step 1 revised—Replace Vu, MuxO, and MuyO by their stat-ical equivalents Vu, Mux and Muy at the centroid of the criticalsection considered [Fig. 3(c) or 4(c)]

; (33)

where xO and yO are coordinates of the column centroid.Step 2 revised—Transform Mux and Muy to their statical equiv-

alents Mux and Muy in directions of principal axes [Fig. 7(b)]

(34)

(35)

Using the appropriate equation for γv selected from Fig. 6,determine γvx and γvy. Apply Eq. (23), (24), and (22) tocalculate vu.

Computer program STDESIGNAn available computer program, STDESIGN,17 which follows

the above mentioned procedure, can be employed forpunching shear design to reduce the time consumed bydesigners. The program designs stud shear reinforcementwhen shear reinforcement is required. It is usable on IBMcompatible microcomputers.

DESIGN EXAMPLESThis section of the paper demonstrates the design proce-

dure mentioned earlier by means of numerical examples ofconnection of a flat plate with interior and edge rectangularcolumns. The following data are valid for all the columnsconsidered here: c1 = 12 in.; c2 = 20 in.; slab thickness = 7 in.;concrete cover = 0.75 in.; normal weight concrete is used; f ′c= 4000 psi; fyv = 50 ksi; stud shear reinforcement is used withdiameter 3/8 in.; flexural reinforcement bar diameter = 1/2 in.;d = 7 - 0.75 - 0.5 = 5.75 in.

Example 1: Interior column (Fig. 9)Given: Vu = 110 kips; MuxO = 400 kip-in.; MuyO = 250 kip-in.

fc′ fc′

fc′

Mux MuxO VuyO+= Muy MuyO VuxO+=

Mux Mux θcos Muy θsin–=

Muy Mux θsin Muy θcos+=

Step 1—Vu = 110 kips; Mux = 400 kip-in.; and Muy = 250kip-in.

Step 2—Properties of the critical section at d/2 fromcolumn face: bo = 87 in.; Ix = 50.20 × 103 in.4; Iy = 28.68 ×103 in.4; γvx = 0.445; γvy = 0.356 [Fig. 6 or Eq. (16) and (17)].

The maximum shear stress is at the point (8.9, 12.9) [Eq. (15)]

Step 3—vn = 253 psi [Eq. (6)]; vu > φvn (= 215 psi); shearreinforcement is required.

Step 4—Select 3/8-in. diameter studs with the arrange-ment shown in Fig. 9.

vu/φ = 345 psi < 6 (= 379 psi); so ≤ 0.5d; s ≤ 0.75d.Select so = 2.25 in.; s = 4 in.; Av = 1.104 in.2; vs = 159 psi[Eq. (9)]; vc = 190 psi [Eq. (13)].

vn = 190 + 159 = 349 psi < 8 (= 506 psi) [Eq. (12)].vu < φvn (= 297 psi); shear reinforcement is adequate.Step 5—Properties of critical section at d/2 from the outer-

most peripheral line of studs: bo = 208.9 in.; Ix = 669.5 × 103

in.4; Iy = 575.1 × 103 in.4; γvx = 0.415; γvy = 0.386 [Fig. 6 orEq. (16) and (17)].

The maximum shear stress is at (7.2, 35.1) in. [Eq. (15)]

vu110 103×87 5.75( )----------------------- 0.445 400 103×( ) 12.9( )

50.20 103×----------------------------------------------------------+ +=

0.356 250 103×( ) 8.9( )

28.68 103×------------------------------------------------------- 293 psi=

fc′

fc′

vu110 103×

208.9 5.75( )----------------------------- 0.415 400 103×( ) 35.1( )

669.5 103×----------------------------------------------------------+ +=

Fig. 9—Arrangement of shear studs in vicinity of interiorcolumn in Example 1.


This indicates that the extension of the shear-reinforcedzone is adequate (Fig. 9).

Example 2: Edge column (Fig. 10)Given: Vu = 60 kips; MuxO = 0; MuyO = 820 kip-in.Step 1—The above forces act at column centroid O

whose coordinates are (-4.9, 0.0) in. Statical equivalentforces at critical section centroid are: Vu = 60 kips; Mux = 0;Muy = 527 kip-in.

Step 2—Properties of the critical section at d/2 fromcolumn face: bo = 55.5 in.; Iy = 7.544 × 103 in.4; γvy = 0.291[Fig. 6 or Eq. (19)].

The maximum shear stress is at (4.0, 12.9) in. [Eq. (15)]


Step 4—Select 3/8-in.-diameter studs with the arrange-ment shown in Fig. 10.

vu/φ = 316 psi < 6 (= 379 psi); so ≤ 0.5d; s ≤ 0.75d.Select so = 2.25 in.; s = 4 in.; Av = 0.773 in.2; vs = 174 psi[Eq. (9)]; vc = 190 psi [Eq. (13)].

vn = 190 + 174 = 364 psi < 8 (= 506 psi) [Eq. (12)].vu < φvn (= 309 psi); shear reinforcement is adequate.Step 5—Properties of critical section at d/2 from the outer-

most peripheral line of studs: bo = 105.1 in.; Iy = 64.83 × 103

in.4; γvy = 0.278 [Fig. 6 or Eq. (19)].

0.386 250 103×( ) 7.2( )

575.1 103×------------------------------------------------------- 101 psi 2φ fc′ 108 psi=( )<=

vu60 103×

55.5 5.75( )-------------------------- 0.291 527 103×( ) 4.0( )

7.544 103×------------------------------------------------------- 269 psi=+=

fc′

fc′

The coordinates of column centroid O are (–15.1, 0.0) in.Statical equivalent forces at critical section centroid are: Vu= 60 kips; Mux = 0; Muy = -87 kip-in. The maximum shearstress is at (–21.1, 31.1) in. [Eq. (15)]

This indicates that the extension of the shear-reinforcedzone is adequate (Fig. 10).

Example 3: Interior column near slab opening (Fig. 11)

Given Vu = 110 kips; MuxO = 400 kip-in.; MuyO = 250 kip-in.

vu60 103×

105.1 5.75( )----------------------------- 0.278 87– 103×( ) 21.1–( )

64.83 103×( )-------------------------------------------------------------+=

107= psi 2φ fc′ 108 psi=( )<

Fig. 10—Arrangement of shear studs in vicinity of edgecolumn in Example 2.

Fig. 11—Interior column with opening in its vicinity inExample 3: (a) effective critical section at d/2 from columnface; and (b) arrangement of shear studs and effective crit-ical section outside shear-reinforced zone.


Step 1—The above forces act at column centroid O whosecoordinates are (–1.2, –0.9) in. Statical equivalent forces atcritical section centroid are: Vu = 110 kips; Mux = 301 kip-in.;Muy = 118 kip-in. [Eq. (33)].

Step 2—Properties of the critical section at d/2 fromcolumn face: bo = 76.6 in.; Ix = 46.67 × 103 in.4; Iy = 23.36× 103 in.4; Ix y = –3.992 × 103 in.4

The projections of critical section on principal axes x andy are 21.7 in. and 28.3 in., respectively. Eq. (16) and (17)give: γvx = 0.432; γvy = 0.369. Transform Mux and Muy to prin-cipal directions [Eq. (34) and (35)]: Mux = 278 kip-in.; Muy =166 kip-in. The parts of these moments transferred by eccen-tricity of shear: γvx Mux = 120 kip-in. and γvy Muy = 61.2 kip-in. Transform these moments to the x and y directions [Eq. (23)and (24)]: Mx = 128 kip-in., and My = 40.7 kip-in.

The maximum shear stress is at the point (7.7, 12 in.)[Eq. (22)]


Step 4—Select 3/8-in.-diameter studs with the arrange-ment shown in Fig. 11(b).

vu/φ = 355 psi < 6 (= 379 psi); so ≤ 0.5d; s ≤ 0.75d.Select so = 2.25 in.; s = 4 in.; Av = 1.104 in. 2; vs = 180 psi[Eq. (9)].

vc = 190 psi [Eq. (13)].vn = 190 + 180 = 370 psi < 8 (= 506 psi) [Eq. (12)].vu < φvn (= 315 psi); shear reinforcement is adequate.Step 5—Properties of the critical section at d/2 from the

outermost peripheral line of shear studs:bo = 204.5 in.; Ix = 843.6 × 103 in.4; Iy = 635.0 × 103 in.4;

Ix y = -80.99 × 103 in.4.The projections of critical section on principal axes x and

y are 73.7 in. and 78.7 in., respectively. Eq. (16) and (17)give: γvx = 0.408; γvy = 0.392. The coordinates of columncentroid are (–3.8, –2.2) in. Statical equivalent forces at crit-ical section centroid are: Vu = 110 kip; Mux = 158 kip-in.; andMuy = –168 kip-in. Following the same procedure as for thecritical section at d/2 from column face, the maximum shearstress vu = 98 psi < 2φ (= 108 psi). This indicates that theextension of the shear-reinforced zone is adequate [Fig. 11(b)].

vu110 103×76.6 5.75( )-------------------------- 128 23.36( ) 40.7 3.992–( )–

46.67 23.36( ) 3.992–( )2–------------------------------------------------------------------ 12( )+=

40.7 46.67( ) 128 3.992–( )–

46.67 23.36( ) 3.992–( )2–------------------------------------------------------------------+ 7.7( ) 302 psi=

fc′

fc′

fc′

Circular columnsThe punching shear design steps described earlier in this

paper are applicable for connections of slabs with circularcolumns. The circular column cross section is replaced by asquare section so that the critical section at d/2 from thesquare column face will have the same perimeter length asfor the critical section for the circular column.

Slabs with drop panels and shear capitalsA common solution used in practice to augment the

punching shear strength of slab-column connections is toincrease the slab thickness around the columns; this can beachieved by use of drop panels [Fig. 12(a)]. When droppanels are used, two critical sections must be investigated forpunching shear strength, at d1/2 from column face and at d2/2outside the drop panel, where d1 and d2 are effective depthsof the slab inside and outside the drop panel, respectively.The two critical sections are checked following the designsteps mentioned earlier. Plan dimensions are selected so thatEq. (1) is satisfied at the critical section outside the droppanel with vu determined by Eq. (15) and vn = vc = 2 .

Figure 12(b) shows what is known in practice as shearcapital. It differs from drop panel in the plan dimensions. Theshear capital is commonly small in size and is provided with noreinforcement other than the vertical bars of the column. Thepunching design is based on a critical section at d/2 outside theshear capital with the nominal shear stress vn given by Eq. (4) to(6). Recent experiments18 show that the punching failure withthis type of capitals can be extremely brittle; therefore, this prac-tice is not recommended by the authors.

Other applications of stud shear reinforcementStud shear reinforcement can be used and designed using the

above equations to resist punching in raft foundations, footings,and in walls subjected to concentrated horizontal forces (e.g.,offshore structures). Fig. 13(a) represents the arrangement of

fc′

Fig. 12—Drop panels and shear capitals.

Fig. 13—Arrangement of shear studs in raft foundations andwalls.

ACI Structural Journal/July-August 1999548

shear studs in the vicinity of a column in a raft foundation; thestuds are mechanically anchored by heads at the top and by asteel strip at the bottom similar to Fig. 1(c).

Figure 13(b) shows arrangement of shear studs with respectto other reinforcement in a wall. The figure can represent avertical or a horizontal section. It is to be noted that the studshave double heads situated in the same plane as the outermostflexural reinforcement. Thus, the overall length of the studs,including the heads, should ideally be equal to the wall thick-ness minus the sum of the specified cover at the two wall faces.

CONCLUSIONSA complete design procedure for slab-column connections

against punching shear is presented. This design proceduresatisfies the requirements of the ACI 318-95 Building Code.Equations based on research are used in the design procedureof practical design situations not covered by the ACI 318-95Code. Design examples are presented. The design can besimplified by use of an available computer program.

ACKNOWLEDGMENTSThis study was funded by a grant from the Natural Sciences and Engineering

Research Council of Canada that is gratefully acknowledged.

CONVERSION FACTORS1 in. = 25.4 mm1 ft = 0.3048 m

1 kip = 4.448 kN1 ft-kip = 1.356 kN-m

1 psi = 6.89 × 10-3 MPa, psi = 0.083 , MPa

NOTATIONAv = cross-sectional area of shear reinforcement on line parallel

to perimeter of columnb1 = width of critical section for shear, at d/2 from column face,

measured in direction of span for which moments aredetermined

b2 = width of critical section for shear, at d/2 from column face,measured in direction perpendicular to b1

bo = length of perimeter of critical sectionc1, c2 = dimensions of column measured in two span directionsd = effective depth of slabd1, d2 = effective depths of slab inside and outside drop panel,

respectivelyfc′ = specified compressive strength of concretefpc = compressive stress in concrete (after allowance for all pre-

stress losses) at centroid of cross section resisting externallyapplied loads

fyv = specified yield strength of shear reinforcementg = spacing between stirrup vertical branches or shear studs in

direction parallel to column faceh = slab thicknessIx , Iy = second moments of area of critical section about principal

axes x and y, respectivelyIx . Iy = second moments of area of critical section about axes x and

y, respectivelyIx y = product of inertia of area of critical section about axes x and yJ = property of shear critical section defined by ACI 318-95

Code as “analogous to the polar moment of inertia”lx , ly = projections of critical section on principal axes x and y,

respectively.Mux, Muy = factored unbalanced moments transferred between slab and

column about principal axes x and y, respectively, at critical section centroid

Mux, Muy = factored unbalanced moments transferred between slab andcolumn about nonprincipal axes x and y, respectively, at criticalsection centroid

MuxO, MuyO,= factored unbalanced moments transferred between slab andMuxO, MuyO column about axes x, y, x and y, respectively, at column centroids = spacing between peripheral lines of shear reinforcementso = spacing between first peripheral line of shear reinforcement

and column facevc = nominal shear stress provided by concrete in presence of

shear reinforcementvn = nominal shear stress of critical sectionvs = nominal shear stress provided by shear reinforcementvu = maximum shear stress at critical section due to applied forcesVc = pure shear capacity of slab-column connection with no

shear reinforcementVp = vertical component of effective prestress forces crossing

critical sectionVu = applied shearing force at failurex, y = coordinates of point of maximum shear stress in critical

section with respect to centroidal principal axes x and yx, y = coordinates of point of maximum shear stress in critical section

with respect to centroidal nonprincipal axes x, yαs = factor which adjusts vc for support typeβc = ratio of long side to short side of concentrated load or reaction areaβp = constant used to compute vc in prestressed slabsγv = fraction of unbalanced moment transferred by eccentricity

of shear at slab-column connectionsθ = angle of inclination of principal axes x and y with respect to

centroidal axes x, y, respectivelyρ = ratio of nonprestressed tension reinforcementρb = reinforcement ratio producing balanced strain conditionsφ = strength reduction factor = 0.85

REFERENCES1. ACI Committee 318, “Building Code Requirements for Structural

Concrete (ACI 318-95) and Commentary,” American Concrete Institute,Farmington Hills, Mich., 1995, 369 pp.

2. Canadian Standards Association, “Design of Concrete Structures(CSA-A23.3-94),” Dec. 1994, 199 pp.

3. Ghali, A., and Megally, S., “Discussion of Proposed Revisions toBuilding Code Requirements for Reinforced Concrete (ACI 318-89)(Revised 1992) and Commentary (ACI 318R-89) (Revised 1992),”Concrete International, V. 17, No. 7, July 1995, pp. 77-82.

4. Wey, E. H., and Durrani, A. J., “Seismic Response of Interior Slab-Column Connections with Shear Capitals,” ACI Structural Journal, V. 89,No. 6, Nov.-Dec. 1992, pp. 682-691.

5. Pan, A. D., and Moehle, J. P., “Experimental Study of Slab-Column Connections,” ACI Structural Journal, V. 89, No. 6, Nov.-Dec.1992, pp. 626-638.

6. Robertson, I. N., and Durrani, A. J., “Gravity Load Effect on SeismicBehavior of Interior Slab-Column Connections,” ACI Structural Journal,V. 89, No. 1, Jan.-Feb. 1992, pp. 37-45.

7. Islam, S., and Park, R., “Tests on Slab-Column Connections withShear and Unbalanced Flexure,” ASCE Journal of Structural Division, V.102, No. 3, Mar. 1976, pp. 549-568.

8. Dilger, W. H., and Ghali, A., “Shear Reinforcement for ConcreteSlabs,” Proceedings, ASCE, V. 107, ST12, Dec. 1981, pp. 2403-2420.

9. Andrä, H. P., “Strength of Flat Slabs Reinforced with Stud Rails in theVicinity of the Supports (Zum Tragverhalten von Flachdecken mit Dubelli-esten—Bewchruing im Auflogerbereich),” Beton-und Stahlbetonbau,Berlin, V. 76, No. 3, Mar. 1981, pp. 53-57, and V. 76, No. 4, Apr. 1981,pp. 100-104.

10. Mokhtar, A. S.; Ghali, A.; and Dilger, W. H., “Stud Shear Reinforce-ment for Flat Concrete Plates,” ACI Structural Journal, V. 82, No. 5, Sept.-Oct. 1985, pp. 676-683.

11. Elgabry, A. A., and Ghali, A., “Tests on Concrete Slab-ColumnConnections with Stud Shear Reinforcement Subjected to Shear-MomentTransfer,” ACI Structural Journal, V. 84, No. 5, Sept.-Oct. 1987, pp. 433-442.

12. Mortin, J., and Ghali, A., “Connection of Flat Plates to Edge Columns,”ACI Structural Journal, V. 88, No. 2, Mar.-Apr. 1991, pp. 191-198.

13. Dilger, W. H., and Shatila, M., “Shear Strength of PrestressedConcrete Edge Slab-Column Connections with and without Stud ShearReinforcement,” Canadian Journal of Civil Engineering, V. 16, No. 6,1989, pp. 807-819.

14. ACI Committee 421, “Shear Reinforcement for Slabs (ACI 421.1R-92),” American Concrete Institute, Farmington Hills, Mich., 1993, 11 pp.

15. Elgabry, A. A., and Ghali, A., “Transfer of Moments betweenColumns and Slabs: Proposed Code Revisions,” ACI Structural Journal,V. 93, No. 1, Jan.-Feb. 1996, pp. 56-61.

16. Elgabry, A. A., and Ghali, A., “Moment Transfer by Shear in Slab-Column Connections,” ACI Structural Journal, V. 93, No. 2, Mar.-Apr.1996, pp. 187-196.

17. Ghali, A., (revised by N. Hammill, 1995), Computer ProgramSTDESIGN, Decon, Brampton, Ontario, Canada.

18. Megally, S., “Punching Shear Resistance of Concrete Slabs toGravity and Earthquake Forces,” PhD dissertation, Department of CivilEngineering, University of Calgary, Alberta, Canada, June 1998, 468 pp.

fc ′ fc ′

design for punching shear strength with aci 318-95

Documents