Design for Punching Shear Strength With ACI 318-95

ACI STRUCTURAL JOURNAL

TECHNICAL PAPER

Title no. 96-S60

Design for Punching Shear Strength with ACI 318-95 by Amin Ghali and Sami Megally Brittle punching failure of flat plates can occur due to the transfer of shearing forces and unbalanced moments between slabs and columns. Design of connections of columns to flat plates to insure safety against punching failure is presented. This paper covers the design procedure in most practical situations, including interior, edge, and corner columns; prestressed and nonprestressed slabs; slabs with openings; and slabs with shear reinforcement. The ACI 318-95 Building Code requirements are adhered to where applicable. Numerical examples are presented to demonstrate the design procedure. Seismic design considerations are not discussed in this paper. Keywords: columns (supports); connections; flat concrete plates; prestressed concrete; punching shear; raft foundations; reinforced concrete; shear strength; slabs; structural design.

INTRODUCTION The punching shear resistance of concrete flat plates frequently needs to be increased by the provision of drop panels or by shear reinforcement. The latter solution is more acceptable architecturally, and is often more economical. This paper gives the details of punching shear design of flat plates without drop panels, with or without shear reinforcement. Requirements of the ACI 318-951 Building Code for design of slabs against punching are reviewed. The design steps are presented, adhering to the code requirements when they apply. Most conditions that occur in practice are considered for slabs with or without prestressing, including slabs with openings in the column vicinity. Interior, edge, and corner column-slab connections subjected to shear and moment transfer are considered. The design steps are demonstrated by computed examples. This paper presents a complete design procedure for punching shear. Reference is made to an available computer program that can be used for the design. When drop panels are used, the design procedure for flat plates applies with an additional provision that is also discussed. The ACI 318-951 Building Code allows the use of shear heads, in the form of steel I- or channel-shaped sections, as shear reinforcement in slabs. Because at present this type is rarely used, it will not be discussed here. The two most common types of shear reinforcement are shown in Fig. 1. To save space in this paper, the arrangements of the reinforcement with the two types are shown in a single top view in Fig. 1(a). Fig. 1(b) and (c) are a pictorial view and a cross section showing, respectively, details of conventional stirrups and stud shear reinforcement (SSR). The vertical legs of the stirrups or the stems of the studs intersect the shear cracks and prevent their widening (Fig. 2). Because the intersection can 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 be anchored as closely as possible to the top and bottom surfaces of the slab (observing the cover requirements for corrosion and fire protection). Effective anchorage is essential to develop the yield strength of the shear reinforcement of both types. With stirrups ACI Structural Journal/July-August 1999

Fig. 1—Types of shear reinforcement considered: (a) shear reinforcements (top view); (b) stirrups; and (c) stud shear reinforcement alternate details (Section A-A). [Fig. 1(b)], the anchorage is provided by hooks, bends, and the longitudinal flexural reinforcing bar lodged at the corners. Before the force in a stirrup leg reaches its yield strength, the concrete inside the hooks or bends crushes or ACI Structural Journal, V. 96, No. 4, July-August 1999. Received October 13, 1997, and reviewed under Institute publication policies. Copyright © 1999, American Concrete Institute. All rights reserved, including the making of copies unless permission is obtained from the copyright proprietors. Pertinent discussion including author’s closure, if any, will be published in the May-June 2000 ACI Structural Journal if the discussion is received by January 1, 2000.

539

ACI member Amin Ghali is a professor of civil engineering at the University of Calgary, Alberta, Canada. He is a member of ACI Committees 373, Circular Concrete Structures Prestressed with Circumferential Tendons; and 435, Deflection of Concrete Building 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 Engineering at the University of Calgary. He received his PhD from the University of Calgary in 1998 and his BSc from Ain-Shams University, Egypt, in 1988. His research interests include structural analysis, the finite element method, and seismic design of reinforced concrete structures.

enough to insure that the full yield strength of the stud can be developed with negligible slip of the anchorage. Experiments show that this can be achieved with anchor heads of area nine to 10 times the cross-sectional area of the stud. RESEARCH SIGNIFICANCE This paper outlines the steps of design for punching shear strength in accordance with ACI 318-95. However, the code does not cover all situations encountered in practice. For these situations, the design is based on research. ACI 318-95 Code requirements ACI 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-reduction factor (φ = 0.85); vu is the maximum shear stress caused by the transfer of a factored shearing force Vu and bending moments Mux and Muy between the slab and column and acting at critical section centroid Fig. 2—Interception of cracks by vertical shear reinforcement.

γ vy M uy γ vx M ux V v u = -------u- + ---------------y + ---------------x bo d Jx Jy

(2)

where bo is length of perimeter of shear critical section; d is the distance from extreme compression fiber to centroid of longitudinal tension reinforcement; the subscripts x and y refer to centroidal axes in directions of both spans; (x, y) are coordinates of the point at which vu is maximum and J is a property of critical section “analogous to polar moment of inertia.” Figure 3 indicates the positive directions of x and y axes, the force Vu , and moments Mux and Muy ; in this figure and others in this paper, the arrows represent the directions of force and moments exerted by the column on the slab. In Fig. 3, x and y are replaced by x and y if they are principal axes. γvx and γvy are fractions of the moments transferred by eccentricity of shear about the x and y axes, respectively 1 γ v = 1 – ------------------------------2 1 + --- b 1 ⁄ b 2 3 Fig. 3—Critical sections for two-way shear in slabs at d/2 from column face: (a) interior column; (b) edge column; and (c) corner column. splits, causing slip, thus preventing development of the full strength of the stirrup, particularly in thin slabs. For this reason, ACI 318R-951 emphasizes that stirrups can be used, provided they are well-anchored, and requires that the stirrups be closed and enclose a longitudinal bar at each corner [Fig. 1(b)]. The Canadian Standard CSA-A23.3-942 does not permit use of stirrups as shear reinforcement in slabs thinner than 300 mm (12 in.). The SSR relies on mechanical anchorage by heads at both ends of the stem or by a head at one end and a steel strip welded to several studs. The steel strip holds the studs in a vertical position and insures the appropriate spacing between them until the concrete is cast. The size of the anchor heads must be large 540

(3)

ACI 318-951 defines b1 and b2, respectively, as widths of shear critical section measured in direction of the span for which moment is determined and perpendicular to it. Thus, when calculating γvy for the rectangular critical section shown 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 for critical sections having shapes other than a closed rectangle. In absence of shear reinforcement, the code requires that the nominal shear stress of nonprestressed slabs be the smallest of (using lb and in. units) 4-⎞ f ′ v n = v c = ⎛ 2 + ---⎝ β c⎠ c

(4)

αs d v n = v c = ⎛ -------- + 2⎞ f c′ ⎝ bo ⎠

(5)

ACI Structural Journal/July-August 1999

v n = v c = 4 f c′

(6)

where vc is the nominal shear stress provided by concrete; βc is ratio of long side to short side of column; fc′ is specified concrete 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 shear reinforcement provided. When shear reinforcement is used, ACI 318-951 expresses the nominal shear stress as v n = v c + v s ≤ 6 f c′

(7)

v c = 2 f c′

(8)

A v f yv v s = ------------bo s

(9)

where vs is nominal shear stress provided by shear reinforcement; Av is area of shear reinforcement within a distance s; fyv is specified yield strength of shear reinforcement; and s is spacing of shear reinforcement. The upper limit for s is 0.5d. Shear reinforcement must be extended for a sufficient distance until the critical section outside the shear-reinforced zone (Fig. 4) satisfies Eq. (1) with vn = vc = 2 f c′ . Other provisions for prestressed slabs and slabs with openings in the column vicinity will be discussed in the following sections. Prestressed slabs For prestressed slabs with no shear reinforcement, ACI 318-951 replaces Eq. (4) to (6) by v n = v c = β p f c′ + 0.3f pc + V p ⁄ b o d

(10)

where Vp is the vertical component of all effective prestress forces crossing the critical section; fpc is average value of fpc in two vertical slab sections in perpendicular directions, with fpc being the compressive stress at section centroid after allowance for all prestress losses; and βp is the smaller of 3.5 and [(αs d/bo) + 1.5]. Eq. (10) can replace Eq. (4) to (6) only if the following conditions are satisfied: (a) no portion of the cross section of the column shall be closer than four times the slab thickness to a discontinuous edge; (b) fc′ shall not be taken greater than 5000 psi; and (c) fpc in each direction shall not be less than 125 psi nor be taken greater than 500 psi. In thin slabs, it is difficult to control the slope of tendon profile at the point it crosses a critical section. Thus, for practical considerations, the last term in Eq. (10) may be neglected or Vp reduced to account for the inaccuracy that can occur in the execution of the tendon profile. Within the shear-reinforced zone, vn is to be calculated using the same equations as for nonprestressed slabs. Section 11.5.4.1 of ACI 318-95 allows for prestressed members, spacing of shear reinforcement, s to reach 0.75h but not to exceed 24 in., where h is overall thickness of member. It is considered here that this limit is excessive in slabs, and it is recommended that the spacing should not exceed 0.75d. This is because the difference between d and h is more important in slabs than in beams and cracks could bypass the shear reinforcement, as shown in Fig. 2. ACI Structural Journal/July-August 1999

Fig. 4—Critical sections for two-way shear in slabs at d/2 from outermost peripheral line of shear reinforcement: (a) interior column; (b) edge column; and (c) corner column. Slabs with openings ACI 318-951 requires that effect of openings on punching shear resistance of a slab-column connection must be considered when openings are located at a distance less than 10 times the slab thickness from the column or when openings are located within the column strip. The effect of openings is taken into account by considering part of shear critical section to be ineffective. The ineffective part is that part of the critical section perimeter that is enclosed by straight lines projecting from the column centroid and tangent to the boundaries of the openings (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 given by Eq. (3), and increasing γf by the same amount of reduction. The symbol γf is the fraction of unbalanced moment transferred by flexure. For a corner column [Fig. 3(c)] or for an edge column [Fig. 3(b), in absence of Mux], the coefficient γv can be reduced to zero provided that Vu ≤ 0.5φVc or 0.75φVc for a corner or edge column, respectively; where Vc = vc bo d, with vc given by Eq. (4) to (6). For an interior column or an edge column [Fig. 3(b), in absence of Muy], γv can be reduced to (1.25γv by Eq. (3) - 0.25), provided that Vu ≤ 0.4φVc . For all slab-column connections, the optional reduction of γv below the value given by Eq. (3) is allowed only when ρ ≤ 0.375ρb ; where ρ is the ratio of nonpre541

where so is the distance between first peripheral line of studs and column face. ACI 421.1R-9214 considers a vertical branch of a stirrup to be less effective than a stud in controlling shear cracks because the stud stem is straight over its full length while the ends of the stirrup branch are curved, and the mechanical anchors at the stud ends insure that the yield strength is available at all sections of the stem; this is not the case with a vertical branch of a stirrup. For the same reasons, the Canadian Standard CSA-A23.3942 allows, in presence of shear studs, a value of vc 1-1/2 times the allowable value when stirrups are employed. The same approach is adopted in the remainder of the paper. Thus, when SSR is used, Eq. (7) and (8) will be replaced by v n = v c + v s ≤ 8 f c′

(12)

v c = 3 f c′

(13)

with

Fig. 5—Stud shear reinforcement arrangement: (a) rectangular columns; (b) orthogonal arrangement at circular columns; and (c) radial arrangement at circular columns. stressed tension reinforcement in the slab; and ρb is the value of ρ producing balanced strain conditions. The authors consider Section 13.5.3.3 unsafe. The justifications are given in the discussion of the code.3 Additional experimental data4-7 for interior columns giving further justification 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, justified by tests,8-13 ACI 421.1R-9214 suggests the following deviations from ACI 318 when SSR is used: 1) The nominal shear stress vn resisted by concrete and shear reinforcement [Eq. (7)] can be as high as 8 f c′ , instead of 6 f c′ . This enables use of thinner slabs; 2) The upper limits for so and s can be based on the value of vu at the critical section at d/2 from column face so ≤ 0.5d and s ≤ 0.75d when v u ⁄ φ ≤ 6 f c′ so ≤ 0.35d and s ≤ 0.5d when v u ⁄ φ ≤ 6 f c′

Parameter J The Code Commentary ACI 318R-951 gives an equation for the parameter J when the shear critical section has the rectangular shape shown in Fig. 3(a). The code commentary equation may be written in the form Jy = Iy + d 3(c1 + d)/6

(14)

(11.1) (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 reproduction plus handling at time of request.

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Arrangement of shear reinforcement Figure 5(a) shows the typical arrangement of stud shear reinforcement at rectangular columns. Each group of studs on a line perpendicular to the column face are welded to a steel strip or spaced in a steel trough [Fig. 1(c)]. ACI 421.1R9214 recommends that, in a direction parallel to the column face, the maximum distance g between the steel strips, or troughs, be less than 2d. This limitation is to insure that the studs confine the concrete and prevent widening of shear cracks 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 of circular columns. The distance g between stud rails in the vicinity of circular columns should not exceed 2d as shown in Fig. 5(b) and (c). The authors recommend the orthogonal rather than the radial arrangement of stud rails. This is because with the radial arrangement of stud rails, shear studs placed in the forms in their appropriate design locations are more likely to interfere with the bars of the flexural reinforcement mesh. When stirrups are used, they should be placed in rows parallel to the column [Fig. 1(a)]. In the direction parallel to the column faces, the distance g between stirrup legs [Fig. 1(b)] should satisfy the requirement g ≤ 2d, or because stirrups are less effective than shear studs, a more restrictive limit should apply.

where Iy is the second moment of area of the critical section about the y axis. It can be verified that with the column sizes and slab thicknesses used in practice, the difference (Jy – Iy), which is equal to the second term in Eq. (14), does not exceed 3 percent of Iy . ACI 318-95 and its commentary define J as an “analogous to polar moment of inertia” and do ACI Structural Journal/July-August 1999

Fig. 6—Equations for γv applicable for critical sections at d/2 from column face and outside shear-reinforced zone. not give equations for J when the critical section has shapes other than rectangular. The vertical shear stress vu calculated by Eq. (2) has a vertical resultant component equal to Vu , but has moment components slightly smaller than γvx Mux and γvy Muy . In other words, the component Vu combined with γvx Mux and γvy Muy are not in equilibrium with the shear stress in the critical section. Replacing Jx and Jy in Eq. (2) by the critical section area’s second moments Ix and Iy about the centroidal principal axes x and y, respectively, gives linearly varying stress vu , whose resultants exactly satisfy equilibrium. With this replacement, the equation for the shear stress vu at any point of the critical section becomes γ vy M uy γ vx M ux V v u = -------u- + ---------------y + ---------------x bo d Ix Iy

1 γ vx = 1 – ----------------------------2 1 + --- l y ⁄ l x 3 1 γ vy = 1 – ----------------------------2 1 + --- l x ⁄ l y 3

(16)

(17)

At edge columns (15)

This equation applies when the critical section has any shape. Use of this equation avoids the ambiguity in calculating the parameter J, which has no known meaning in mechanics. Coefficient γv Numerous experiments have shown that the empirical Eq. (3) adopted by ACI 318-95 is satisfactory for interior columns where the critical section, at d/2 from column faces, has the shape of the perimeter of a closed rectangle. At the same location, the critical section for edge and corner columns has three or two sides, respectively [Fig. 3(b) and (c)]. Outside the shear-reinforced zone, the critical section follows the perimeter of a closed or open polygon, whose sides are not all parallel to a column face (Fig. 4). Problems arise15 when the empirical Eq. (3), allowed by ACI 318-95 for critical sections having the shape of a closed rectangle, is employed for corner columns. Similar design problems may arise when employing Eq. (3) for edge columns. Elgabry and Ghali16 showed by numerous finite element analyses that Eq. (3) does not apply for all cases and for all critical sections. They gave the following equations for γv to ACI Structural Journal/July-August 1999

cover all cases and all shapes of the critical section encountered in design (Fig. 6). At interior columns

γvx = same as Eq. (16)

(18)

l 1 γ vy = 1 – ----------------------------------------------- when ----x < 0.2, γ vy = 0 (19) 2 ly 1 + --- ( l x ⁄ l y ) – 0.2 3 At corner columns γvx = 0.4

(20)

γvy = same as Eq. (19)

(21)

where lx and ly are projections of the critical section on principal axes x and y, respectively. The safety of design using the above equations has been verified using published experimental results.16 Inclined axes The shear critical sections for corner columns, and for all columns when the slab has nonsymmetric openings, have principal axes x and y inclined to the column faces. In these cases, it may be more convenient to calculate the shear stress at 543

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

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

points with coordinates (x, y) referring to centroidal but nonprincipal axes using the following equation to replace Eq. (2) ⎛ M x I y – M y I xy⎞ ⎛ M y I x – M x I xy⎞ V v u = -------u- + ⎜ -----------------------------y + -⎟ x ⎟ ⎜ -----------------------------bo d ⎝ I I – I 2 ⎠ ⎝ I I – I2 ⎠ x y

xy

x y

(22)

xy

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

Ix =

Mx = γvx Mux cosθ + γvx Muysinθ

(23)

My = –γvx Mux sinθ + γvy Muycosθ

(24)

∫y

(25)

–2

da ; I y =

∫x

–2

∫

da ; I xy = xy da

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 Iy of the critical section may be determined by summation of the contributions of straight segments m

I xy =

∑

i=1

m

I xyi ; I x =

∑

i=1

m

I xi ; I y =

∑ Iyi

(26)

i=1

where m is the total number of segments, and i refers to the i-th segment. A typical straight segment AB is shown in Fig. 8; its contributions to Ix y , Ix , and Iy may be calculated by 2

2 1⁄2

( l ) AB = [ ( x B – x A ) + ( y B – y A ) ] 544

(27)

d ( l ) AB ( I xy ) AB = --------------( 2x A y A + 2x B y B + x A y B + x B y A ) 6

(28)

d ( l ) AB 2 2 ( I x ) AB = --------------( yA + yB + yA yB ) 3

(29)

d ( l ) AB 2 2 ( I y ) AB = --------------( xA + xB + xA xB ) 3

(30)

where d is effective depth; (xA , yA) and (xB , yB) are the coordinates of the segment ends A and B. The angle θ between the principal x axis and the x axis is given 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 steps The data required for design of slab-column connections are: d, c1, c2, Vu , MuxO, MuyO, and fc′ [Fig. 3(a) and (b)]. It is required to determine whether d is sufficient for safety against punching without the use of shear reinforcement and if not, design the necessary shear reinforcement. The symbols MuxO and MuyO are the unbalanced moments at the column centroid. When working with nonprincipal axes x, y [Fig. 3(c)], the given moments will be MuxO and MuyO and Steps 1 and 2 of the design given below will be changed. The first critical section to be considered is at d/2 from the column face. The steps of design when x and y are principal axes are: Step 1—Replace Vu , MuxO, and MuyO by their statical equivalents Vu , Mux , and Muy at the centroid of the critical section considered [Fig. 3(a) and (b) or 4(a) and (b)] M ux = M uxO + V u y O ; M uy = M uyO + V u x O

(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. ACI Structural Journal/July-August 1999

Step 2—Using the applicable equation for γv selected from Fig. 6, determine γvx and γvy . Calculate vu by Eq. (15). Step 3—If vu ≤ φvn [given by Eq. (4) to (6)], no shear reinforcement is required. If (vu /φ) > vn limit , d must be increased; where vn limit = 6 f c ′ or 8 f c ′ when stirrups or studs are used as shear reinforcement, respectively. When vn < 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 determined using Eq. (7) to (9). When stud shear reinforcement is used, use Eq. (12), (13), and (9). Step 5—Extend the shear reinforcement zone by increasing the number of peripheral lines of studs. Repeat Steps 1 and 2 for a critical section at d/2 outside the outermost peripheral line of shear reinforcement (Fig. 4). If vu ≤ 2φ f c′ , extension of shear reinforcement is sufficient; if not, extend the shear reinforcement farther away from column and repeat Steps 1 and 2 until this requirement is satisfied. Revision of Steps 1 and 2 when nonprincipal axes are used Step 1 revised—Replace Vu, MuxO, and MuyO by their statical equivalents Vu, Mux and Muy at the centroid of the critical section considered [Fig. 3(c) or 4(c)] M ux = M uxO + V u y O ; M uy = M uyO + V u x O

(33)

where xO and yO are coordinates of the column centroid. Step 2 revised—Transform Mux and Muy to their statical equivalents Mux and Muy in directions of principal axes [Fig. 7(b)] M ux = M ux cos θ – M uy sin θ

(34)

M uy = M ux sin θ + M uy cos θ

(35)

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

Fig. 9—Arrangement of shear studs in vicinity of interior column in Example 1. Step 1—Vu = 110 kips; Mux = 400 kip-in.; and Muy = 250 kip-in. Step 2—Properties of the critical section at d/2 from column 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)] 3

3

110 × 10 - 0.445 ( 400 × 10 ) ( 12.9 ) v u = ---------------------+ ---------------------------------------------------------- + 3 87 ( 5.75 ) 50.20 × 10 3

Computer program STDESIGN An available computer program, STDESIGN,17 which follows the above mentioned procedure, can be employed for punching shear design to reduce the time consumed by designers. The program designs stud shear reinforcement when shear reinforcement is required. It is usable on IBM compatible microcomputers. DESIGN EXAMPLES This section of the paper demonstrates the design procedure mentioned earlier by means of numerical examples of connection of a flat plate with interior and edge rectangular columns. The following data are valid for all the columns considered 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 with diameter 3/8 in.; flexural reinforcement bar diameter = 1/2 in.; d = 7 - 0.75 - 0.5 = 5.75 in.

0.356 ( 250 × 10 ) ( 8.9 ) ------------------------------------------------------ = 293 psi 3 28.68 × 10 Step 3—vn = 253 psi [Eq. (6)]; vu > φvn (= 215 psi); shear reinforcement is required. Step 4—Select 3/8-in. diameter studs with the arrangement shown in Fig. 9. vu/φ = 345 psi < 6 f c ′ (= 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 f c ′ (= 506 psi) [Eq. (12)]. vu < φvn (= 297 psi); shear reinforcement is adequate. Step 5—Properties of critical section at d/2 from the outermost 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 or Eq. (16) and (17)]. The maximum shear stress is at (7.2, 35.1) in. [Eq. (15)] 3

Example 1: Interior column (Fig. 9) Given: Vu = 110 kips; MuxO = 400 kip-in.; MuyO = 250 kip-in. ACI Structural Journal/July-August 1999

3

0.415 ( 400 × 10 ) ( 35.1 ) 110 × 10 v u = ----------------------------- + ---------------------------------------------------------+ 3 208.9 ( 5.75 ) 669.5 × 10 545

Fig. 10—Arrangement of shear studs in vicinity of edge column in Example 2. 3

0.386 ( 250 × 10 ) ( 7.2 ) ------------------------------------------------------ = 101 psi < 2φ f c′ ( = 108 psi ) 3 575.1 × 10 This indicates that the extension of the shear-reinforced zone 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 equivalent forces at critical section centroid are: Vu = 60 kips; Mux = 0; Muy = 527 kip-in. Step 2—Properties of the critical section at d/2 from column 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)] 3

3

0.291 ( 527 × 10 ) ( 4.0 ) 60 × 10 v u = -------------------------- + ------------------------------------------------------ = 269 psi 3 55.5 ( 5.75 ) 7.544 × 10 Step 3—vn = 253 psi [Eq. (6)]; vu > φvn (= 215 psi); shear reinforcement is required. Step 4—Select 3/8-in.-diameter studs with the arrangement shown in Fig. 10. vu/φ = 316 psi < 6 f c ′ (= 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 f c ′ (= 506 psi) [Eq. (12)]. vu < φvn (= 309 psi); shear reinforcement is adequate. Step 5—Properties of critical section at d/2 from the outermost peripheral line of studs: bo = 105.1 in.; Iy = 64.83 × 103 in.4; γvy = 0.278 [Fig. 6 or Eq. (19)]. 546

Fig. 11—Interior column with opening in its vicinity in Example 3: (a) effective critical section at d/2 from column face; and (b) arrangement of shear studs and effective critical section outside shear-reinforced zone. 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 shear stress is at (–21.1, 31.1) in. [Eq. (15)] 3

3 60 × 10 - 0.278 ( – 87 × 10 ) ( – 21.1 ) v u = ---------------------------+ ------------------------------------------------------------3 105.1 ( 5.75 ) ( 64.83 × 10 )

= 107 psi < 2φ f c′ ( = 108 psi ) This indicates that the extension of the shear-reinforced zone 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. ACI Structural Journal/July-August 1999

Fig. 12—Drop panels and shear capitals. Step 1—The above forces act at column centroid O whose coordinates are (–1.2, –0.9) in. Statical equivalent forces at critical 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 from column 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 and y 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 principal directions [Eq. (34) and (35)]: Mux = 278 kip-in.; Muy = 166 kip-in. The parts of these moments transferred by eccentricity of shear: γvx Mux = 120 kip-in. and γvy Muy = 61.2 kipin. 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)] 3

110 × 10 - 128 ( 23.36 ) – 40.7 ( – 3.992 ) v u = ------------------------+ ------------------------------------------------------------------ ( 12 ) 76.6 ( 5.75 ) 46.67 ( 23.36 ) – ( – 3.992 ) 2 40.7 ( 46.67 ) – 128 ( – 3.992 ) + ------------------------------------------------------------------ ( 7.7 ) = 302 psi 2 46.67 ( 23.36 ) – ( – 3.992 ) Step 3—vn = 253 psi [Eq. (6)]; vu > φvn (= 215 psi); shear reinforcement is required. Step 4—Select 3/8-in.-diameter studs with the arrangement shown in Fig. 11(b). vu/φ = 355 psi < 6 f c′ (= 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 f c′ (= 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 column centroid are (–3.8, –2.2) in. Statical equivalent forces at critical section centroid are: Vu = 110 kip; Mux = 158 kip-in.; and Muy = –168 kip-in. Following the same procedure as for the critical section at d/2 from column face, the maximum shear stress vu = 98 psi < 2φ f c′ (= 108 psi). This indicates that the extension of the shear-reinforced zone is adequate [Fig. 11(b)]. ACI Structural Journal/July-August 1999

Fig. 13—Arrangement of shear studs in raft foundations and walls. Circular columns The punching shear design steps described earlier in this paper are applicable for connections of slabs with circular columns. The circular column cross section is replaced by a square section so that the critical section at d/2 from the square column face will have the same perimeter length as for the critical section for the circular column. Slabs with drop panels and shear capitals A common solution used in practice to augment the punching shear strength of slab-column connections is to increase the slab thickness around the columns; this can be achieved by use of drop panels [Fig. 12(a)]. When drop panels are used, two critical sections must be investigated for punching shear strength, at d1/2 from column face and at d2/2 outside the drop panel, where d1 and d2 are effective depths of the slab inside and outside the drop panel, respectively. The two critical sections are checked following the design steps mentioned earlier. Plan dimensions are selected so that Eq. (1) is satisfied at the critical section outside the drop panel with vu determined by Eq. (15) and vn = vc = 2 f c ′ . Figure 12(b) shows what is known in practice as shear capital. It differs from drop panel in the plan dimensions. The shear capital is commonly small in size and is provided with no reinforcement other than the vertical bars of the column. The punching design is based on a critical section at d/2 outside the shear capital with the nominal shear stress vn given by Eq. (4) to (6). Recent experiments18 show that the punching failure with this type of capitals can be extremely brittle; therefore, this practice is not recommended by the authors. Other applications of stud shear reinforcement Stud 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 547

shear studs in the vicinity of a column in a raft foundation; the studs are mechanically anchored by heads at the top and by a steel strip at the bottom similar to Fig. 1(c). Figure 13(b) shows arrangement of shear studs with respect to other reinforcement in a wall. The figure can represent a vertical or a horizontal section. It is to be noted that the studs have double heads situated in the same plane as the outermost flexural reinforcement. Thus, the overall length of the studs, including the heads, should ideally be equal to the wall thickness minus the sum of the specified cover at the two wall faces. CONCLUSIONS A complete design procedure for slab-column connections against punching shear is presented. This design procedure satisfies the requirements of the ACI 318-95 Building Code. Equations based on research are used in the design procedure of practical design situations not covered by the ACI 318-95 Code. Design examples are presented. The design can be simplified by use of an available computer program.

vn vs vu Vc

= = = =

Vp

=

Vu x, y

= =

x, y

=

αs βc βp γv

= = = =

θ

=

ρ ρb φ

= = =

shear reinforcement nominal shear stress of critical section nominal shear stress provided by shear reinforcement maximum shear stress at critical section due to applied forces pure shear capacity of slab-column connection with no shear reinforcement vertical component of effective prestress forces crossing critical section applied shearing force at failure coordinates of point of maximum shear stress in critical section with respect to centroidal principal axes x and y coordinates of point of maximum shear stress in critical section with respect to centroidal nonprincipal axes x, y factor which adjusts vc for support type ratio of long side to short side of concentrated load or reaction area constant used to compute vc in prestressed slabs fraction of unbalanced moment transferred by eccentricity of shear at slab-column connections angle of inclination of principal axes x and y with respect to centroidal axes x, y, respectively ratio of nonprestressed tension reinforcement reinforcement ratio producing balanced strain conditions strength reduction factor = 0.85

ACKNOWLEDGMENTS

REFERENCES

This study was funded by a grant from the Natural Sciences and Engineering Research Council of Canada that is gratefully acknowledged.

1. 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 to Building 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 SlabColumn 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 SlabColumn 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 Seismic Behavior 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 with Shear 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 Concrete Slabs,” Proceedings, ASCE, V. 107, ST12, Dec. 1981, pp. 2403-2420. 9. Andrä, H. P., “Strength of Flat Slabs Reinforced with Stud Rails in the Vicinity of the Supports (Zum Tragverhalten von Flachdecken mit Dubelliesten—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 Reinforcement 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-Column Connections with Stud Shear Reinforcement Subjected to Shear-Moment Transfer,” 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 Prestressed Concrete Edge Slab-Column Connections with and without Stud Shear Reinforcement,” Canadian Journal of Civil Engineering, V. 16, No. 6, 1989, pp. 807-819. 14. ACI Committee 421, “Shear Reinforcement for Slabs (ACI 421.1R92),” American Concrete Institute, Farmington Hills, Mich., 1993, 11 pp. 15. Elgabry, A. A., and Ghali, A., “Transfer of Moments between Columns 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 SlabColumn Connections,” ACI Structural Journal, V. 93, No. 2, Mar.-Apr. 1996, pp. 187-196. 17. Ghali, A., (revised by N. Hammill, 1995), Computer Program STDESIGN, Decon, Brampton, Ontario, Canada. 18. Megally, S., “Punching Shear Resistance of Concrete Slabs to Gravity and Earthquake Forces,” PhD dissertation, Department of Civil Engineering, University of Calgary, Alberta, Canada, June 1998, 468 pp.

CONVERSION FACTORS 1 in. 1 ft 1 kip 1 ft-kip 1 psi f c ′ , psi

= = = = = =

25.4 mm 0.3048 m 4.448 kN 1.356 kN-m 6.89 × 10-3 MPa 0.083 f c ′ , MPa

NOTATION Av

= cross-sectional area of shear reinforcement on line parallel to perimeter of column = width of critical section for shear, at d/2 from column face, b1 measured in direction of span for which moments are determined = width of critical section for shear, at d/2 from column face, b2 measured in direction perpendicular to b1 bo = length of perimeter of critical section c1, c2 = dimensions of column measured in two span directions d = effective depth of slab = effective depths of slab inside and outside drop panel, d1, d2 respectively fc′ = specified compressive strength of concrete fpc = compressive stress in concrete (after allowance for all prestress losses) at centroid of cross section resisting externally applied loads = specified yield strength of shear reinforcement fyv g = spacing between stirrup vertical branches or shear studs in direction parallel to column face h = slab thickness = second moments of area of critical section about principal I x , Iy axes x and y, respectively I x . Iy = second moments of area of critical section about axes x and y, respectively Ix y = product of inertia of area of critical section about axes x and y J = 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 and column about nonprincipal axes x and y, respectively, at critical section centroid MuxO , MuyO,= factored unbalanced moments transferred between slab and MuxO , MuyO column about axes x, y, x and y, respectively, at column centroid s = spacing between peripheral lines of shear reinforcement = spacing between first peripheral line of shear reinforcement so and column face = nominal shear stress provided by concrete in presence of vc

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ACI Structural Journal/July-August 1999