guide to seismic design of punching shear reinforcement in flat … · 2019-10-03 · chapter...

ACI 421.2R-10

Reported by Joint ACI-ASCE Committee 421

Guide to Seismic Design ofPunching Shear Reinforcement

in Flat Plates

Guide to Seismic Design of Punching Shear Reinforcementin Flat Plates

First PrintingApril 2010

ISBN 978-0-87031-374-5

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Guide to Seismic Design of Punching Shear Reinforcement in Flat Plates

Reported by Joint ACI-ASCE Committee 421

ACI 421.2R-10

Simon J. Brown Amin Ghali* James S. Lai Edward G. Nawy

Pinaki R. Chakrabarti Hershell Gill Mark D. Marvin Eugenio M. Santiago

William L. Gamble* Neil L. Hammill Sami Hanna Megally* Thomas C. Schaeffer

Ramez Botros Gayed* Theodor Krauthammer Michael C. Mota Stanley C. Woodson

*Member of the subcommittee that prepared this guide.The committee would like to thank Frieder Seible for his contribution to this guide.

Mahmoud E. KamaraChair

ACI Committee Reports, Guides, Manuals, and Commentariesare intended for guidance in planning, designing, executing,and inspecting construction. This document is intended for theuse of individuals who are competent to evaluate thesignificance and limitations of its content and recommendationsand who will accept responsibility for the application of thematerial it contains. The American Concrete Institute disclaimsany and all responsibility for the stated principles. The Instituteshall not be liable for any loss or damage arising therefrom.

Reference to this document shall not be made in contractdocuments. If items found in this document are desired by theArchitect/Engineer to be a part of the contract documents, theyshall be restated in mandatory language for incorporation bythe Architect/Engineer.

During an earthquake, the unbalanced moments transferred at flat plate-column connections can produce significant shear stresses that increasethe vulnerability of these connections to brittle punching shear failure. Thisguide provides recommendations for designing flat plate-column connectionswith sufficient ductility to withstand lateral drift without punching shearfailure or loss of moment transfer capacity. This guide treats reinforcedconcrete flat plates with or without post-tensioning.

Keywords: ductility; flat plate; post-tensioning; punching shear; seismicdesign; shear reinforcement; stud shear reinforcement.

421

ACI 421.2R-10 supersedes ACI 421.2R-07 and was adopted and published April 2010.Copyright © 2010, American Concrete Institute.All rights reserved including rights of reproduction and use in any form or by any

means, including the making of copies by any photo process, or by electronic ormechanical device, printed, written, or oral, or recording for sound or visual reproduc-tion or for use in any knowledge or retrieval system or device, unless permission inwriting is obtained from the copyright proprietors.

CONTENTSChapter 1—Introduction, p. 421.2R-2

1.1—General1.2—Scope1.3—Objective1.4—Remarks

Chapter 2—Notation and definitions, p. 421.2R-32.1—Notation2.2—Definitions

Chapter 3—Lateral story drift, p. 421.2R-53.1—Lateral-force-resisting systems3.2—Limits on story drift ratio3.3—Effects of gravity loads on story drift capacity3.4—Design recommendations for flat plates with and

without shear reinforcement

Chapter 4—Minimum shear and integrity reinforcements in flat plates, p. 421.2R-7

.2R-1

421.2R-2 ACI COMMITTEE REPORT

Chapter 5—Assessment of ductility, p. 421.2R-8

Chapter 6—Unbalanced design moment,p. 421.2R-9

6.1—Frame analysis6.2—Simplified elastic analysis6.3—Upper limit for Mu

Chapter 7—Design of shear reinforcement,p. 421.2R-11

7.1—Strength design7.2—Summary of design steps7.3—ACI 318 provisions

Chapter 8—Post-tensioned flat plates,p. 421.2R-13

8.1—General8.2—Sign convention8.3—Post-tensioning effects8.4—Effective compressive stress fpc8.5—Extension of punching shear design procedure to

post-tensioned flat plates8.6—Research on post-tensioned flat plates

Chapter 9—References, p. 421.2R-179.1—Referenced standards and reports9.2—Cited references

Appendix A—Verification of proposed minimum amount of shear reinforcement for earthquake-resistant flat plate-column connections,p. 421.2R-18

Appendix B—Verification of upper limit to unbalanced moment to be used in punching shear design, p. 421.2R-18

Appendix C—Notes on properties of shear-critical section, p. 421.2R-20

C.1—Second moments of areaC.2—Equations for γv

Appendix D—Design examples, p. 421.2R-21D.1—GeneralD.2—Example 1: Interior flat plate-column connectionD.3—Example 2: Edge flat plate-column connectionD.4—Example 3: Corner flat plate-column connectionD.5—Example 4: Use of stirrups—Interior flat plate-

column connectionD.6—Example 5: Interior flat plate-column connection of

Example 1, repeated using SI unitsD.7—Post-tensioned flat plate structureD.8—Example 6: Post-tensioned flat plate connection

with interior columnD.9—Example 7: Post-tensioned flat plate connection

with edge column

Appendix E—Conversion factors, p. 421.2R-30

CHAPTER 1—INTRODUCTION1.1—General

Brittle punching failure can occur due to the transfer ofshear forces combined with unbalanced moments betweenslabs and columns. During an earthquake, significanthorizontal displacement of a flat plate-column connectionmay occur, resulting in unbalanced moments that induceadditional slab shear stresses. As a result, some flat platestructures have collapsed by punching shear in past earth-quakes (Berg and Stratta 1964; Yanev et al. 1991; Mitchellet al. 1990, 1995). During the 1985 Mexico earthquake(Yanev et al. 1991), 91 waffle-slab and solid-slab buildingscollapsed, and another 44 buildings suffered severe damage.Hueste and Wight (1999) studied a building with a post-tensioned flat plate that experienced punching shear failuresduring the 1994 Northridge, CA, earthquake. Their studyprovided a relationship between the level of gravity load andthe maximum story drift ratio that a flat plate-columnconnection can undergo without punching shear failure. Thedisplacement-induced unbalanced moments and resultingshear forces at flat plate-column connections, althoughunintended, should be designed to prevent brittle punchingshear failure. Even when an independent lateral-force-resisting system is provided, flat plate-column connectionsshould be designed to accommodate the moments and shearforces associated with the displacements during earthquakes.

1.2—ScopeIn seismic design, the displacement-induced unbalanced

moment and the accompanying shear forces at flat plate-column connections should be accounted for. This demandmay be effectively addressed by changes in dimensions ofcertain members, or their material strengths (for example,shear walls and column sizes), or provision of shear reinforce-ment or a combination thereof. This guide does not addresschanges in dimensions and materials of such members, butfocuses solely on the punching shear design of flat plateswith or without shear reinforcement.

This guide, supplemental to ACI 421.1R, focuses on thedesign of flat plate-column connections with or withoutshear reinforcement that are subject to earthquake-induceddisplacement; reinforced concrete flat plates with or withoutpost-tensioning are treated in the guide. Slab shear reinforcementcan be structural steel sections, known as shearheads, orvertical rods. Although permitted in ACI 318, shearheads arenot commonly used in flat plates. Stirrups and shear studreinforcement (SSR), satisfying ASTM A1044/A1044M, arethe most common types of shear reinforcement for flatplates. Shear stud reinforcement is composed of vertical rodsanchored mechanically near the bottom and top surfaces ofthe slab. Forged heads or welded plates can be used as theanchorage of SSR; the area of the head or the plate is sufficientto develop the yield strength of the stud, with negligible slipat the anchorage. The design procedure recommended in thisguide was developed based on numerical studies (finiteelement method) and experimental research on reinforcedconcrete slabs subjected to cyclic drift reversals that simulateseismic effects. The finite element analyses, supplemental to

SEISMIC DESIGN OF PUNCHING SHEAR REINFORCEMENT IN FLAT PLATES 421.2R-3

the experimental research, used software, constitutive relations,and models that were subject to extensive verifications bycomparing the results with the behavior observed in tests(Megally and Ghali 2000b).

Structural integrity reinforcement near the bottom of theslab extending through the columns should be provided asrequired by ACI 318. This document supplements ACI352.1R and ACI 421.1R, which, respectively, includerecommendations such as extending a minimum amount ofbottom integrity reinforcement through the column core andprovide details of design for shear reinforcement in flatplates. ACI 352.1R also provides recommendations for thedesign of flat plate-column connections without slab shearreinforcement subjected to moment transfer in the inelastic-response range. The equations of this guide predict punchingshear strength and drift capacity, assuming that adequateflexural reinforcement is provided at the flat plate-columnconnections; the present guide does not address the requiredflexural reinforcement.

1.3—ObjectiveThe objective is to provide a design recommendation for

flat plate-column connections with sufficient ductility toaccommodate the displacement of the selected lateral-force-resisting system without punching shear failure or loss ofmoment transfer capacity. The objective covers reinforcedconcrete slabs with or without post-tensioning.

1.4—RemarksThis guide gives recommendations for the design of shear

reinforcement, considering ductility, that supplement theprovisions of ACI 318 for punching shear design. The term“ductility,” used throughout this guide, is the ratio ofdisplacement at ultimate strength to the displacement atwhich yielding of the flexural reinforcement occurs. For flatplate-column connections, there is no unique definition forthese two displacements. Pan and Moehle (1989) define theultimate and yield displacements by a graphical bilinearidealization of the experimental load-displacement response,considering the displacement at a specified load levelbeyond the peak load.

ACI 318 allows the analysis of flat plate-column frames asequivalent plane frames. When the frame is not designated aspart of the lateral-force-resisting system and is subjected tohorizontal displacements, the width of slab strip to beincluded in the frame model and how to account for cracking(ACI 318, Section R13.5.1.2) are modeling parameters thatsignificantly affect the resulting computed values of themoments transferred between slabs and columns. This guidecontains a procedure that determines an upper limit momentthat can be transferred between the slab and column whenthe connection is subjected to an earthquake.

Chapter 3 defines the story drift that should be considered
in design. Chapter 4 recommends a minimum amount of shear reinforcement for certain cases. Chapter 5 describes means of increasing the shear strength of a flat plate-column
connection and compares the associated ductilities. Chapter 6
presents a method to calculate the unbalanced moment
required for design. Chapter 7 and Appendix D discuss relevantprovisions of ACI 318 and provide the design procedure andexamples for interior, edge, and corner flat plate-columnconnections. Chapter 8, expanded in this edition, provides

recommendations relevant to post-tensioned flat plates.

CHAPTER 2—NOTATION AND DEFINITIONS2.1—NotationAs = area of flexural reinforcing bars, in.2 (mm2)Av = cross-sectional area of shear reinforcement

on one peripheral line, in.2 (mm2)bo = length of perimeter of shear-critical section, in.

(mm)Cd = displacement amplification factor (ASCE/SEI 7)c1 to c6 = dimensions used in Fig. 8.3, defining a post-

tensioning tendon profilecx , cy = column dimensions in the x- and y-directions,

respectively, in. (mm)DRu = ultimate story drift ratio at peak strength (in

experiments), or design story drift ratio of aflat plate-column connection

d = average of distances from extreme compressionfiber to the centroid of the tension reinforce-ment positioned in two orthogonal directions,in. (mm)

Ec = elastic modulus of concrete, psi (MPa)e = tendon eccentricity, measured from the

midsurface of the slab to the centroid of thepost-tensioned tendon; e is positive for atendon situated below midsurface, in. (mm)

fc′ = specified concrete strength, psi (MPa)fpc = average in-plane compressive stress

produced by effective post-tension forces intwo orthogonal directions, psi (MPa)

fps = stress in post-tensioned reinforcement atnominal flexural strength of slab, psi (MPa)

fpy = specified yield strength of post-tensionedreinforcement, psi (MPa)

fse = effective stress in a post-tensioned tendonafter accounting for all post-tension losses, psi(MPa)

fy = specified yield strength of flexural reinforce-ment, psi (MPa)

fyt = specified yield strength of shear reinforce-ment, psi (MPa)

h = slab thickness, in. (mm)Ic = moment of inertia of gross section of column,

in.4 (mm4)IE = occupancy importance factor (ASCE/SEI 7)Iec = equivalent moment of inertia of columns

accounting for torsional members in accor-dance with ACI 318, Section 13.7.5 of (referto plane frame idealization [Fig. 6.1])

Is = second moment of area of slab accounting forcracking (refer to plane frame idealization[Fig. 6.1])


Jc = property of assumed critical section analogousto polar moment of inertia, defined by Eq. (C-1),taken from ACI 318

Jx , Jy = property of assumed critical section of anyshape, equal to d multiplied by the secondmoment of perimeter about principal x- or y-axis, respectively

Jxy = d times product of inertia of assumed shear-critical section about nonprincipal axes x and y

Kc = end rotational stiffness of column, momentper unit rotation

Kec = end rotational stiffness of equivalent column,moment per unit rotation

l = span length, in. (mm)lc = story height, in. (mm)lx , ly = projections of shear-critical section on its

principal axes x and y, respectively, in. (mm)M = unbalanced moment transferred between the

slab and the column, in.-lb (N·mm)MOx, MOy = unbalanced moment about an axis parallel to

the principal x- or y-axis and passing throughO, the column’s centroid; positive MOx orMOy is in the same direction as positive Muxor Muy (Fig. 3.2), in.-lb (N·m)

Mp′ = primary bending moment due to post-tensioning, in.-lb (N·mm)

Mp′′ = secondary (indeterminate) post-tensionedbending moment, in.-lb (N·mm)

Mpr = probable flexural strength, in.-lb (N·mm)(refer to Section 6.3)

Mprestress = bending moment produced by post-tensioning forces, in.-lb (N·mm)

Mu = ultimate unbalanced moment, at peakstrength, transferred between the slab and the column at shear-critical section centroid, in.-lb (N·mm); this definition applies wherecapacity is considered. Where demand isconsidered, Mu is factored unbalancedmoment in design

Mux, Muy = components of the unbalanced moment Mutransferred between the slab and the column;positive directions of Mux and Muy are definedin Fig. 3.2

Pe = absolute value of effective post-tensioningforce per unit slab width, in.-lb (N·mm)

s = spacing between peripheral lines of shearreinforcement, in. (mm)

so = spacing between the first peripheral line ofshear reinforcement and column face, in. (mm)

V = shear force transferred between the columnand the slab, lb (N)

Vc = nominal punching shear capacity of a flatplate-column connection with no shearreinforcement, lb (N)

Vp = shear force produced by post-tensioningforces or vertical component of effectivepost-tensioning force crossing the shear-criticalsection at d/2 from column face, lb (N)

Vp′ = primary shear force due to post-tensioning,lb (N)

Vp′′ = secondary (indeterminate) post-tensioningshear force, lb (N)

Vu = ultimate shear force transferred between theslab and the column, lb (N)

(Vu/φVc) = in presence of shear reinforcement (Eq. (7-1)),

vc is nominal shear strength (expressed instress units) provided by concrete, psi (MPa).In the absence of shear reinforcement, vc isnominal shear capacity expressed in stress units= Vc/(bod) (Vc is given by Eq. (3-1) to (3-3))

vn = nominal shear strength (expressed in stressunits), psi (MPa)

vs = nominal shear strength (expressed in stressunits) provided by shear reinforcement, psi(MPa)

vu = maximum shear stress at critical section, psi(MPa)

wc = weight of concrete per unit volumewD = service dead load per unit areawL = service live load per unit areawsd = superimposed dead load per unit areax, y = coordinates of point of maximum shear stress

on the critical section with respect to centroidalprincipal axes x and y, respectively. Also assubscripts, x and y refer to the same principalaxes

x, y = axes parallel to slab edges at the centroid ofthe shear-critical section at a corner column

αm = factor used in calculation of the designmoment for earthquake-resistant flat plate-column connections

αs = factor that adjusts vc for support typeβ = ratio of long side to short side of concentrated

load or reaction areaβr = aspect ratio of the shear-critical section at d/2

from column faceγv = fraction of unbalanced moment transferred

by vertical shear stresses at flat plate-columnconnections

δe = story drift used in elastic frame analysis, in.(mm)

δu = design story drift, including inelastic defor-mations, in. (mm)

θ = clockwise rotation angle of x-axis to x-axis,or y-axis to y-axis

θp = angle between tangent to the tendon and thecentroidal axis

λ = modification factor reflecting the reducedmechanical properties of lightweightconcrete, relative to normalweight concreteof the same compressive strength

φ = strength reduction factor according to ACI 318κ = fraction of service dead load to be balanced

by effective post-tensioningρ = slab reinforcement ratioρp = ratio of post-tensioning reinforcement


CHAPTER 3—LATERAL STORY DRIFT
3.1—Lateral-force-resisting systems
Flat plate-column frames without beams and without alateral-force-resisting system consisting of more rigidelements that limit the lateral displacements are notpermitted by ACI 318 in Seismic Design Categories (SDC)D, E, and F. In SDC D, E, and F, flat plate buildings need torely on a lateral-force-resisting system that limits lateraldisplacement. Flat plates will experience the same lateraldisplacements as the lateral-force-resisting system, andshould be designed to do so without losing their capability tosupport gravity loads during or after an earthquake.

The lateral-force-resisting system should have sufficientstiffness to control the lateral displacement of the structureand limit the maximum lateral story drift to the valuesprescribed by the general building code, and as discussed inSection 3.2. The story drift is defined as the lateral displacement

3.2—Limits on story drift ratioIt has been frequently recommended that flat plate structures

should have the capability to withstand a design story driftratio of at least 0.015, including inelastic deformations(Sozen 1980; ACI 352.1R; Pan and Moehle 1989).

In the force-based design approach, a static elastic analysisof the lateral-force-resisting system is performed to determinethe elastic story drift δe (due to forces specified in the Interna-tional Building Code (IBC 2006), or ASCE/SEI 7). Thisvalue is then multiplied by factors, given in IBC 2006 orASCE/SEI 7, to obtain the design story drift δu.

IBC 2006 requires that the calculated δu (depending on thelateral-force-resisting system and the seismic design category),including inelastic deformation, not exceed 0.007 to 0.025 ofthe story height. The code also requires that the flat plate-column connection be adequate to carry the vertical load andthe induced moments and shears resulting from the calculateddesign story drift.

The design story drift ratio may reach the upper limit ofIBC 2006 when the flat plate-column connections areprovided with slab shear reinforcement or when the gravityload produces low punching shear stress as discussed inChapter 4.

of one level, or floor, relative to the level above or below.The story drift ratio is defined as the story drift divided bythe story height. The story height is defined as the distancebetween the midsurfaces of the consecutive flat plates at topand bottom of the story of interest.

Unbalanced moments due to design displacements aretransferred between slabs and columns and are a direct resultof the lateral displacement experienced by the entirebuilding. The transfer of shear forces and unbalancedmoments increases the risk of a punching shear failureduring an earthquake.

ACI 352.1R provides connection design recommendationsfor flat plate-column frames and considers that such framesmay be adequate as the lateral-force-resisting system inregions of low and moderate seismic risk. This guideexcludes the selection or the design of the lateral-force-resisting systems; it includes the punching shear design offlat plates, with and without shear reinforcement, subjected

to gravity loads combined with unbalanced moments inducedby the lateral drift of the flat plate-column connections.

Fig. 3.1—Effect of gravity loads on lateral drift capacity ofinterior flat plate-column connections (Megally and Ghali1994, 2000d; Hueste and Wight 1999).

2.2—DefinitionsACI provides a comprehensive list of definitions through

an online resource, “ACI Concrete Terminology,” http://terminology.concrete.org. Definitions provided hereincomplement that resource.

design displacement—total lateral displacementexpected for the design-basis earthquake as required by thegoverning code for earthquake-resistant design.

design story drift—design displacement of one level, orfloor, relative to the level above or below.

design story drift ratio—design story drift divided by thestory height.

ductility—ratio of displacement at ultimate strength to thedisplacement at which yield of the flexural reinforcementoccurs.

flat plate—flat slab without column capitals or drop panels.lateral-force-resisting system—portion of the structure

composed of members designed to resist forces related toearthquake effects.

shear stud reinforcement (SSR)—reinforcementcomposed of vertical rods anchored mechanically near thebottom and top surfaces of the slab (ASTM A1044/A1044M).

3.3—Effects of gravity loads on story drift capacityFigure 3.1 (Megally and Ghali 1994, 2000d; Hueste and

Wight 1999) shows the variation of the ultimate story driftratio DRu for interior flat plate-column connections transferringgravity shear forces Vu with the ratio (Vu/φVc) whensubjected to reversals of cyclic drift. The experimentalvalues of DRu are compared with the design story drift δ atpeak strength, divided by the story height. Vu is the ultimateshear force transferred between the slab and the column atfailure;Vc is the nominal punching shear strength of the flatplate-column connection without shear reinforcement in theabsence of moment transfer. Each data point represents thecombination of (Vu/φVc) and the corresponding ultimate


Vc = (3-1)

Vc = (3-2)

Vc = (3-3)

2 4β---+⎝ ⎠

⎛ ⎞ bodλ fc′

αsd

bo

--------- 2+⎝ ⎠⎛ ⎞ bodλ fc′

4bodλ fc′

Fig. 3.2—Critical sections for punching shear at d/2 fromcolumn face. The arrows for Vu and Mu represent the forcesexerted by the column on the slab in their positive directions.x and y are centroidal principal axes of the critical section.

story drift ratio DRu at which a test specimen failed inpunching shear.

Curves 1, 2, and 3 shown in Fig. 3.1 fit the results ofexperiments reported in the literature on interior flat plate-column connections without shear reinforcement, withconventional stirrups, and with SSR, respectively (Pan andMoehle 1989, 1992; Robertson and Durrani 1992; Wey andDurrani 1992; Hawkins et al. 1975; Islam and Park 1976;Dilger and Brown 1995; Brown 2003; Dilger and Cao 1991;Cao 1993). The strength reduction factor φ is taken equal to1.0 in this figure because the strength of materials is knownfor the experiments. In preparing Fig. 3.1, Vc is used as thesmallest of Eq. (3-1) through (3-3)

Inch-pound units
SI units
Vc = (3-1)

Vc = (3-2)

1 2β---+⎝ ⎠

⎛ ⎞ bodλ fc′6

----------------------

αsd

bo

--------- 2+⎝ ⎠⎛ ⎞ bodλ fc′

12----------------------

Vc = (3-3)

where bo is the perimeter of the critical section for shear at(d/2) from the column face (Fig. 3.2); d is the average of thedistances from extreme compression fiber to the centroid oftension reinforcement in two orthogonal directions; fc′ is thespecified concrete compressive strength; β is the ratio of thelong side to the short side of the column; and αs = 40, 30, and20 for interior, edge, and corner flat plate-column connections,respectively. The arrows for Vu and Mu shown in Fig. 3.2(a)through (c) represent the forces exerted by the column on theslab in their positive directions.

Figure 3.1 indicates that the flat plate-column connectioncapability to experience story drift without failure decreaseswith increasing magnitude of the applied gravity load. Thesolid horizontal line shown in Fig. 3.1 represents the designstory drift ratio of 0.015, which is frequently adopted as aminimum drift capacity. This horizontal line intersectsCurve 1 at (Vu/φVc) ≅ 0.40, indicating that slabs withoutshear reinforcement can satisfy the required 0.015 designstory drift ratio only if Vu does not exceed 0.40φVc. The sameconclusion can be seen from a similar graph containing moredata (Kang and Wallace 2005). The horizontal dashed lineplotted in Fig. 3.1, corresponding to the 0.025 design storydrift ratio limit specified by IBC 2006 (for structures satis-fying specified conditions), intersects Curve 1 at approximately(Vu/φVc) ≅ 0.25. This suggests that for DRu = 0.025, the slabshould be designed with shear reinforcement when Vuexceeds approximately 0.25φVc. Section 3.4 conservativelyrecommends provision of shear reinforcement when DRu =0.025 and (Vu /φVc) > 0.20. This complies with ACI 318,Section 21.13.6.

Figure 3.3 is similar to Fig. 3.1, but the two curves shownin Fig. 3.3 represent the results of experiments of edge flatplate-column connections (Megally and Ghali 2000a). Thegraphs indicate that for DRu = 0.015, the flat plate should bedesigned with shear reinforcement when Vu > 0.40φVc. Acomparison of Fig. 3.1 and 3.3 indicates that a somewhat

13---bodλ fc′

Fig. 3.3—Effect of value of Vu on lateral drift capacity ofedge flat plate-column connections (Megally and Ghali2000a).


CHAPTER 4—MINIMUM SHEAR AND INTEGRITYREINFORCEMENTS IN FLAT PLATES

Seismic design of non-prestressed flat plates can bewithout shear reinforcement only when (Vu /φVc) is limited,as discussed in Section 3.4. In addition, the maximum shear

3.4—Design recommendations for flat plates with and without shear reinforcement

As discussed in Section 3.1, the lateral-force-resisting

structural system should have sufficient stiffness to controlthe story drift. Slab shear reinforcement is required when themaximum shear stress at d/2 from the column face exceedsφvc, where vc is the value given by Eq. (3-1) to (3-3) divided

by bod. In addition, flat plate-column connections shouldhave shear reinforcement equal to or exceeding theminimum amount given in Chapter 4, except when the valueof Vu is less than 0.20φVc, and φ = 0.75 (ACI 318, Section9.3.2.3). This requirement ensures that the connections cansustain the design story drift ratio DRu = 0.025. If it can beshown by analysis that when the maximum story drift ratioDRu, including the inelastic deformations, is between 0.015and 0.025, shear reinforcement is required when Vu exceedsφVc(0.70 –20DRu). This is the same as required by ACI 318,Section 21.13.6(b), that shear reinforcement be providedwhen DRu exceeds [0.035 – 0.05Vu/(φVc)].

Curve 1 in Fig. 3.1 shows that for slabs without shearreinforcement, DRu can reach 0.015 only when Vu ≤ 0.40φVc.The same curve also shows conservatively that DRu canreach 0.025 without shear reinforcement when Vu ≤ 0.20φVc.The same two conclusions can be reached if Curve 1 isreplaced by the bilinear graph of Hueste and Wight (1999),plotted in the same figure. The bilinear graph represents anapproximation of the same test data used in Fig. 3.1 for slabswithout shear reinforcement combined with data from sevenmore references.
stress due to Vu, combined with that due to the unbalanced
moment Mu, computed according to the linear stress distri-bution assumption of ACI 318, should not exceed φvc ,expressed in stress units (the smallest of Eq. (3-1) through(3-3) divided by bod), at the critical section at d/2 from the

column face. For ductility, connections that do not satisfyboth conditions should have slab shear reinforcement thatsatisfies Eq. (4-1).

Inch-pound units

(4-1)

SI units

(4-1)

where Av is the area of shear reinforcement in each periph-eral line parallel to the column faces (typical peripheral linesare shown in Fig. 4.1(a) to (c)); s is the spacing between

vsAv fyt

bos----------- 3 fc′≥=

vsAv fyt

bos----------- 1

4--- fc′≥=

peripheral lines of shear reinforcement(s ≤ 0.5d for stirrups;s ≤ 0.75d for SSR); bo is the perimeter of the critical sectionat d/2 from the column face (Fig. 3.2); and fyt is the specified

yield strength of the shear reinforcement. The distancebetween the column face and the outermost peripheral line ofshear reinforcement (Fig. 4.1) should not be less than 3.5d.The case that requires the minimum reinforcement forductility is further clarified in Section 7.2.

ACI 318 and ACI 421.1R provide equations to calculatethe maximum shear stress at the shear-critical section d/2from the column face; when the maximum shear stressexceeds φVc /(bod), shear reinforcement is required forstrength. Also for strength, the extent of the shear-reinforcedzone is determined by limiting the maximum shear stress at thecritical section at d/2 from the outermost peripheral line of shearreinforcement 2φ psi (0.17φ MPa) (ACI 318, Section11.11.7.2). When shear reinforcement is required forstrength, its extent should exceed the minimum 3.5d.

The minimum shear reinforcement recommended previouslywas based on a review of experiments by Islam and Park(1976); Hawkins et al. (1975); Dilger and Brown (1995);Dilger and Cao (1991); Megally and Ghali (2000a); andMegally (1998). In these experiments, flat plate-columnconnections with different amounts of stirrups or SSRsustained drift ratios higher than 0.025 without a punchingshear failure. Appendix A summarizes the results of these

fc′ fc′

experiments.The minimum shear reinforcement recommended

increases the shear strength and ensures that the flat plate-column connections can support factored gravity loads afterexperiencing inelastic deformations due to cyclic driftduring an earthquake. Experiments (Megally 1998) haveshown that this can be achieved by the provision of shearreinforcement, as recommended in the present guide. Thedesign of shear reinforcement, other than the minimumamount, is discussed in Chapter 7.

The structural integrity reinforcement defined in Section 5.3of ACI 352.1R should also be provided to increase theresistance of the structural system to progressive collapse.This reinforcement can be more than that required by thedetailing rules in Chapter 13 of ACI 318.

higher limit on (Vu/φVc) can be allowed for edge connectionsthan for interior connections for the same design story driftratio. For simplicity, a single limit on Vu is recommended inSection 3.4. In addition to the limitation on Vu, the absenceof shear reinforcement should be allowed only when themaximum shear stress is less than that given in Chapter 4.

Figures 3.1 and 3.3 show that the curves representingexperiments of flat plate-column connections with shearreinforcement fall well above the horizontal lines. Curves 2and 3 of Fig. 3.1 are not perfect fits because of the scatter ofexperimental data. The uncertainty of the curves, however,does not change the conclusion that no limit on (Vu/φVc) isneeded for slabs with shear reinforcement exceeding theminimum amount given in Chapter 4 to sustain the 0.025design story drift ratio required by IBC 2006. Figure 3.1 alsoshows that the drift capacity for slabs with SSR is higher thanslabs with stirrups.


Fig. 4.1—Critical sections for punching shear outside theshear-reinforced zone.

CHAPTER 5—ASSESSMENT OF DUCTILITYThe methods for increasing punching shear strength of flat

plate-column connections include increasing the flat platethickness, improving the material properties, providing shearreinforcement, or a combination of these methods. Theeffects of these different methods for enhancing shearstrength on ductility are substantially different.

The most common types of shear reinforcement in flatplates are the vertical legs of stirrups and SSR. Stirrups maynot be as effective as SSR in flat plates because of inefficientanchorage of the stirrups, unless they are closed and detailedin accordance with ACI 318, Section R11.11.3, which states:“Anchorage of stirrups according to the requirements of12.13 is difficult in slabs thinner than 10 in. (250 mm).” WithSSR, the anchorage is provided mechanically by forgedheads or by a forged head at one end and a steel strip (calleda rail) welded at the other end.

Figure 5.1 (Megally and Ghali 2000a) shows load-deflection

responses for five slabs with a thickness of 6 in. (150 mm).The tested specimens, with plan dimensions of 75 x 75 in.2

(1.9 x 1.9 m2), represent full-size connections of slabs withinterior columns of size 10 x 10 in.2 (250 x 250 mm2) andsquare panels of spans equal to 16 ft (4.8 m). Axial load ismonotonically applied on the column, transferring a shearforce, but no unbalanced moment, to the slab, which issimply supported on four edges. The slabs have the sameflexural reinforcement layout and properties; the flexuralreinforcement ratio is equal to 0.014 in each of the twoorthogonal directions. The concrete compressive strengthsfc′ ranged from 4200 to 5800 psi (29 to 40 MPa). The slabsdiffer only in the method used to increase punching shearstrength. Control Slab AB1 (Mokhtar et al. 1985) has nomeans for increasing punching shear strength. Figure 5.1,taken from Megally and Ghali (2000a), includes Slabs I andII having drop caps. Slabs I and II (Megally 1998) havethickness increases from 6 to 9 in. (150 to 225 mm) overareas, 17 x 17 in.2 (430 x 430 mm2) and 37 x 37 in.2 (950 x950 mm2), respectively, with the column at their centers.Slab B (Ghali and Hammill 1992) has closed stirrups,detailed according to ACI 318, Fig. R11.11.3(c). (It is notedthat since 2002, ACI 318 has permitted stirrups only in slabswith d ≥ 6 in. [150 mm]). Slab AB5 (Mokhtar et al. 1985) hasSSR. The nominal shear strengths, expressed in stress units,provided by the shear reinforcement (Eq. (4-1)) in Specimens Band AB5, are: 440 psi (3.0 MPa) for Specimen B, based onmeasured fyt , and 270 psi (1.9 MPa) for Specimen AB5,based on nominal fyt . The corresponding nominal shearstrengths provided by shear reinforcement, Vs = vsbod =Avfytd/s are 110 and 70 kips (480 and 320 kN). Punchingshear failure occurred at a section within the shear-reinforcedzone in Specimen B and at a section outside the shear-reinforcedzone in Specimen AB5. The extent of the shear reinforcementzones from column faces was 3.8d and 5.7d in Specimens Band AB5, respectively. The initial portions of the experimentalcurves shown in Fig. 5.1 for Slabs AB1 and AB5 areunavailable. The conclusions drawn from these curves,however, are unaffected.


(6-1)δu δeCd

IE

------⎝ ⎠⎛ ⎞=

Fig. 5.1—Load-deflection curves of slabs with differentmeans of increasing punching shear strength. Slabssubjected only to shear force V with no unbalancedmoment. The deflection is measured in the direction of V(Megally and Ghali 2000a).

Figure 5.1 indicates that the punching shear strength isincreased slightly with the presence of stirrups; however,increases in slab thickness and SSR provided a more significantincrease in punching shear strength. The specimens withincreased slab thickness, Slabs I and II, experienced a brittlefailure with small slab deflections compared with the slabwith SSR, for which the failure may be considered ductile.The stirrups provided an insignificant increase in ductilityfor such a thin slab. Stirrups are assumed more efficientwhen used in slabs thicker than 10 in. (250 mm), as suggestedin ACI 318, Section R11.11.3; however, no experimentaldata on slabs with this thickness could be found. Figure 5.1and results of experiments conducted on flat plate-columnconnections subjected to cyclic moment reversals (Dilgerand Brown 1995; Dilger and Cao 1991; Megally and Ghali2000a,b) indicate that SSR effectively increases punchingshear strength for earthquake-resistant flat plate-columnconnections even when the slab is relatively thin. Note thatFig. 5.1, taken from Megally and Ghali (2000a), includesSlabs I and II having drop caps; however, this guide islimited to flat plates.

CHAPTER 6—UNBALANCED DESIGN MOMENTMoments are transferred between flat plates and their

supporting columns as the flat plate-column connectionsexperience the lateral displacement induced on the lateral-force-resisting system. Thus, flat plate-column connectionsshould have a punching shear strength able to resist thefactored shear force Vu and factored unbalanced moment Mudue to gravity loads combined with the design story driftduring an earthquake.

The methods in Sections 6.1 and 6.2 are recommended for

6.1—Frame analysisA flat plate-column frame, including its lateral-force-

resisting system, can be analyzed for gravity and lateral

seismic loads using linear frame analysis techniques (Ghaliet al. 2003). Unless the layout of columns is highly irregular,the moments of inertia of the slab and its supporting columnsare determined according to the equivalent frame method ofACI 318. A static analysis can be performed using theseismic lateral forces specified by IBC 2006 or ASCE/SEI 7;alternatively, a dynamic analysis may be performedaccording to the two references.

To account for the effect of cracking in non-prestressedflat plates, Vanderbilt and Corley (1983) recommend consid-ering the moment of inertia of the slab as equal to one-third thevalue of the uncracked slab strip—from panel centerline topanel centerline—to obtain a conservative estimate of the storydrift. ACI 318, Section R13.5.1.2, recommends the use of 25to 50% of the uncracked moment of inertia of the slab tocompute the elastic story drift δe. According to ASCE/SEI 7, δeis related to the maximum design story drift δu , includinginelastic deformations by

where Cd (= 1.25 to 6.5) and IE (= 1.0 to 1.5) are dimensionlessfactors specified by ASCE/SEI 7, depending on the inherentinelastic deformability of the lateral-force-resisting systemand the occupancy importance of the structure, respectively.To include the P-Δ effect, the factor inside the brackets inEq. (6-1) should be multiplied by an appropriate coefficientspecified in ASCE/SEI 7.

Flat plate-column frames not designated as part of thelateral-force-resisting system experience the same lateraldisplacements as those of the lateral-force-resisting system.Thus, flat plate-column connections should be designed totransfer shears and moments associated with the δu valuesobtained from Eq. (6-1). Accurate determination of shearsand moments associated with δu is not possible, but fordesign purposes, the simplified approach presented inSection 6.2 is deemed adequate. The design moments and

shears for flat plate-column connections should be deter-mined from the aforementioned elastic frame analysisaccording to the IBC 2006 or ASCE/SEI 7 static lateral forceanalysis method. The unbalanced moment caused by thefactored vertical forces that exist during an earthquakeshould be added to the moments obtained by the analysisdiscussed in this section. The value of Mu is the smaller ofthe upper limit given in Section 6.3 and the total factoredunbalanced moment determined as described previously.

To avoid underestimation of the unbalanced momentstransferred between the slab and the columns for a givenvalue of δe , the unbalanced moments should be determinedby an elastic analysis, but with the moment of inertia of theslab equal to 50% of the value of the uncracked slab andusing the values of the moments of inertia of the uncrackedcolumns (ACI 318, Section R13.5.1.2). There is no consensuson the 50% value in this recommendation; a one-third valuehas been also recommended.

the calculation of the value of Mu. An upper limit for Mu isgiven in Section 6.3 and Appendix B.


6.3—Upper limit for MuThe value of the unbalanced moment corresponding to the

displacement δe can be higher than the value that producesductile flexural failure. Provision of shear reinforcement inthis case would not increase the value of the unbalancedmoment strength. For slabs having low flexural reinforcementratios, Eq. (6-2) may limit the value of M . Based on finite

(6-2)MuMpr

αm

---------≤

uelement analyses (Megally and Ghali 2000b) and experiments(Megally and Ghali 2000a) on flat plate-column connectionstransferring shear combined with moment reversals, anupper limit can be set for the value of δe when computing theshear strength of earthquake-resistant flat plate-columnconnections

where Mpr is the sum of the absolute values of the probableflexural strengths of opposite critical section sides of width(cx + d) or (cy + d) when the transferred moment is about thex- or y-axis, respectively. The variables cx and cy are thecolumn dimensions in the x- and y-directions, respectively(Fig. 3.2). The probable flexural strength should be based on

the assumption that the force in the flexural tensile reinforcementis 1.25(As fy) (ACI 318, Chapter 21), where fy is the specifiedyield strength of the reinforcement, and As is the cross-sectionalarea of the reinforcing bars normal to the two opposite sidesof the shear-critical section (Fig. 3.2). The top bars on oneside plus the bottom bars on the opposite side within theaforementioned width should be included, with carefulconsideration of the anchorages of bars. The right-hand sideof Eq. (6-2) represents the magnitude of the unbalanced

Fig. 6.1—Plane frame idealization of flat plate-columnconnections.
6.2—Simplified elastic analysis
Instead of the analysis suggested in Section 6.1, theunbalanced moments transferred between the slab and thecolumns under the elastic story drift δe can be determined bya linear analysis of simplified equivalent frames as shown inFig. 6.1(a) and (b) for interior and exterior columns (edge orcorner), respectively. The slab is assumed to be simplysupported at locations of contraflexure lines assumed atmidspan. The column is assumed to have hinged supports atcontraflexure points assumed approximately at midheight ofthe story. For the frame models shown in Fig. 6.1, it isassumed that the lengths of spans adjacent to the column areequal l and the story heights above and below the consideredlevel are equal lc. The moments of inertia of the column andthe slab are calculated according to ACI 318, Sections13.5.1.2 and 13.7.5 (considering torsional membersconnecting the column to the beam). Figure 6.1(c) indicatesthe second moment of areas of the frame members.

The horizontal displacement δe is introduced at the upperends of the columns as shown in the figure. The value of δe ,representing the elastic story drift, can be estimated by anyrational analysis; for example, it can be calculated using anelastic analysis of the lateral-force-resisting systemsubjected to the lateral forces as specified by IBC 2006 orASCE/SEI 7. The unbalanced moment due to the horizontalseismic forces transferred between the column and theconnected slab is equal to the sum of the end moments at thecolumn ends above and below the flat plate-column connection.Additional unbalanced moments caused by factored verticalforces that can exist during an earthquake should also beconsidered. The smaller of the total unbalanced momentcalculated using this procedure and the upper limit given inSection 6.3 should be used.


moment that will develop the yield strength of the flexuralreinforcement. When this occurs without a punching shearfailure, the flat plate-column connection will experiencesubstantial drift and not lose the ability to transfer gravityloads, thus avoiding collapse. Based on the finite elementresults, the empirical coefficient αm (Eq. (6-2)) is expressedby Eq. (6-3) or (6-4)

αm = 0.85 – γv – (6-3)

αm = 0.55 – γv – + 10ρ (6-4)

where γv is the fraction of moment transferred by verticalshear stresses in the slab. Equations for γv , depending on theshape of the critical section, are given in Appendix C.

βr

20------⎝ ⎠

⎛ ⎞

βr

40------⎝ ⎠

⎛ ⎞
Equation (6-3) applies for interior connections transferring
Mux or Muy (Fig. 3.2(a)) and also for edge and cornerconnections transferring Mux (with Muy = 0 [Fig. 3.2(b) and(c)]). Equation (6-4) applies for exterior connections trans-ferring Muy (with Mux = 0). In Eq. (6-3) and (6-4), βr isequal to (ly /lx) or (lx /ly) when the transferred moment isabout the x- or y-axis, respectively, where lx and ly areprojections of the critical section at d/2 from the column faceon its principal axes x and y, respectively (Fig. 3.2). In Eq. (6-4),ρ is the reinforcement ratio of tensile flexural reinforcingbars passing through the projection of the shear-criticalsection (Fig. 3.2) in the direction in which moments aretransferred. In Fig. 3.2(b), for example, ρ is the reinforce-ment ratio of the tensile flexural reinforcing bars passingthrough Side BC of the critical section for the edge flat plate-column connection (when Mux = 0). For the corner flat plate-column connection shown in Fig. 3.2(c), αm is calculated byEq. (6-4) with ρ given by

ρ = ρxcos2θ + ρysin2θ (6-5)

where ρx is the reinforcement ratio for x-direction barscrossing critical section Side BC; ρy is the reinforcementratio for y-direction bars crossing critical section Side AB;and θ is the angle between the nonprincipal and principalaxes. The value of Mu, given by Eq. (6-2), can be substantiallygreater than Mpr , meaning that the transfer of unbalancedmoment can mobilize the flexural strength of the slab over awidth considerably greater than (cx + d) or (cy + d).

When Eq. (6-2) is used to determine the upper limit forMuy at an interior connection (with Mux = 0 [Fig. 3.2(a)]),Mpr is the sum of the absolute values of the positive andnegative probable flexural strength of the two oppositeSections AD and BC, respectively (Fig. 3.2(a)). For an edgeflat plate-column connection (with Mux = 0 [Fig. 3.2(b)]),Mpr should be calculated for the negative or the positiveprobable flexural strength of the critical section side parallelto the free edge (Side BC). The connection should bedesigned to resist Vu combined with each of the two moments.

For an edge flat plate-column connection transferring Mux(with Muy = 0 [Fig. 3.2(b)]), Mpr is the sum of the absolutevalues of the positive and negative probable flexural strengthof the two opposite Sections CD and AB, respectively.

For a corner flat plate-column connection, Mpr is calculatedfor the negative or positive probable flexural strength of slabstrips with widths equal to projections of the critical sectionon its principal axes. The connection should be designed forpunching shear considering the positive or negative probableflexural strength when Mpr is transferred about the principaly-axis (Fig. 3.2(c)). The connection should be checkedconsidering the sum of the absolute values of the positive andnegative probable flexural strength when Mpr is transferredabout the principal x-axis (Fig. 3.2(c)). When the unbalancedmoment transfer has each of its components (Mux or Muy ≠ 0),the upper limit for each can be conservatively consideredseparately. This means that for the calculation of the upperlimit of Mux, Muy can be ignored. Similarly the calculation ofthe upper limit of Muy can be done assuming Mux = 0 (usingEq. (6-2) and (6-4)).

In a flat plate-column connection (Fig. 3.2(a) or (b)) trans-ferring constant Vu combined with Muy of increasing magnitude,yielding is reached in the flexural reinforcing bars passingthrough critical section Sides AD and BC in Fig. 3.2(a) orSide BC in Fig. 3.2(b). The sum of the absolute values of thepositive probable flexural strength of Side AD and negativeprobable flexural strength of Side BC (Fig. 3.2(a)) representsthe product αmMuy. Similarly, αmMuy represents the negativeprobable flexural strength of Side BC in Fig. 3.2(b). Thequantity αmMuy can be determined from the results ofnonlinear finite element analyses. The results of such analyses,which have helped in developing the empirical Eq. (6-3) and(6-4), are presented in Appendix B. The validity of the finite

element software (ANATECH Consulting Engineers1995) and the finite element models were verified by physicalexperiments (Megally and Ghali 2000b).

CHAPTER 7—DESIGN OF SHEAR REINFORCEMENT

vn = vc + vs (7-1)

7.1—Strength designThe computation of punching shear strength should follow

the recommendations found in ACI 421.1R. When shearreinforcement is provided, the nominal shear strength(expressed in stress units) is given by

where vc and vs are the nominal shear strengths (expressed instress units) provided by the concrete and shear reinforce-ment, respectively. ACI 421.1R limitsvn to 8 or 6psi (0.67 or 0.5 MPa), respectively, when the shearreinforcement is SSR or stirrups. These two limits should beincreased by 25% in seismic design when the shear stressdue to Vu/φ alone does not exceed 4 psi (0.33MPa). This is because the maximum shear stress is causedmainly by Mu , rather than by Vu, occurring at only a point ora side of the critical section. The limit vn = 10 psi(0.83 MPa) has been exceeded in tests (Ritchie and Ghali

fc′ fc′fc′ fc′

fc′ fc′

fc′fc′


7.2—Summary of design stepsThe design story drift, including inelastic deformations,

is controlled by the lateral-force-resisting structuralsystem, as described previously. The steps explained previ-ously for computing the punching shear strength of flatplate-column connections are illustrated by the flowchartshown in Fig. 7.1. In the penultimate step in the figure, Mu

Fig. 7.1—Steps for punching shear design of earthquake-resistant flat plate-columnconnections.
should be calculated taking into account the upper limitspecified in Section 6.3. Then the equations found in ACI
421.1R should be used to calculate the maximum shearstress and the amount of shear reinforcement, whenrequired, and to verify that the minimum amount of reinforce-ment is provided. Design examples of slab shear reinforcementrequired in flat plate-column connections are presented inAppendix D.

2005). The value of vc in Eq. (7-1) is limited to 1.5λ psi(0.125λ MPa) in seismic design (ACI 421.1R).

With shear reinforcement and absence of unbalancedmoment, the shear reinforcement can increase Vu/φ from Vcto the upper limit 2Vc or 1.5Vc with SSR or stirrups,respectively. Equation (7-1) recommends increasing theallowable shear stress due to the combination of Vu and Muonly when the shear stress due to Vu /φ alone is relativelysmall compared with its upper limit.

fc′fc′

7.3—ACI 318 provisionsACI 318-08, Section 21.13.6, is relevant to shear reinforce-

ment at flat plate-column connections not designated as partof the lateral-force-resisting system. ACI 421.1R recommenda-tions satisfy the requirements of ACI 318, including Section21.13.6. Shear reinforcement greater than the requirementsof ACI 318 is recommended when Vu/(φVc) > 0.4, accordingto Sections 21.13.6 and R21.13.6.

Section 21.13.6 requires that the shear-reinforced zoneextend at least four times the slab thickness from the faceof the column, and that vs shall be not less than 3.5 psi([7/24] MPa). ACI 318 waives these requirements whenthe design of shear reinforcement satisfies Section 11.12.6.2.

Section R21.13.6 comments that calculation of theunbalanced moment due to the design displacement is notneeded when, in Fig. 7.2, the point {Vu/(φVc), DRu} falls

fc′fc′

below the bilinear boundary (A-B-C). When the point fallswithin Zone 1 (Fig. 7.2), the minimum shear reinforcementaccording to Chapter 4 of the present guide is recommended.When the point falls in Zone 2, ACI 318-08 requires shearreinforcement.

A point in Zones 1 and 2 represents the case when Vu >0.4φVc. With such a high value of Vu combined withunbalanced moment reversals, ductility can be ensured onlywith shear reinforcement. For interior column-flat plateconnections, refer to Cao (1993) and for edge column-flat


CHAPTER 8—POST-TENSIONED FLAT PLATES

Fig. 7.2—Requirement for shear reinforcement criterion(ACI 318).

plate connections, refer to Megally and Ghali (2000a,c). Thisguide is consistent with ACI 318-08 in recommending shearreinforcement in Zone 3 (zone of relatively high DRu). Inaddition, for ductility purposes, the minimum shear reinforce-ment, according to Chapter 4, should be provided.
8.1—General
The design recommendations of Chapter 7 are applied inthis chapter to post-tensioned flat plate-column connections.Section 8.4 discusses additional internal forces—induced by
post-tensioning tendons in statically indeterminate structures—
that need to be considered in the design for punching shear.The compressive stress, produced by effective post-tensioned forces, is required in the design equations. Asimplified equation that calculates the effective compressivestress is presented in Section 8.4. Design equations applicablefor post-tensioned connections are summarized in Section 8.5.
Findings of research conducted on behavior of post-
tensioned flat plate-column connections subjected toearthquake-simulated effects are discussed in Section 8.6.
Design examples of the connection of a post-tensioned flat
plate with an interior and an edge column are presented inSections D.7 through D.9.

Fig. 8.1—Banded-distributed tendons.
Figure 8.1 shows a scheme for placing the post-tensioning
tendons in two-way flat plates that is recommended by ACI423.3R. In this scheme, tendons are closely spaced in narrowbands over support lines in one direction—commonly thelonger one—and evenly distributed in the perpendiculardirection. In addition to the constructibility advantage of thisscheme, both directions can be designed for the maximumpossible tendon drape. The banded and distributed tendonsdo not touch each other, except over the supports.

8.2—Sign conventionThe positive sign convention for bending moment Mprestress

and shear force Vp produced by post-tensioning is indicatedin Fig. 8.2(a) and (b). The variables Vp and Mprestress dependmainly on the effective post-tensioning force and the profileof the tendons, expressed by eccentricity e, where e ismeasured from the midsurface of the slab to the centroid ofthe post-tensioning tendon; e is positive for a tendon situatedbelow midsurface (Fig. 8.2(c)).

8.3—Post-tensioning effectsIn a statically determinate structure, post-tensioning produces

zero reactions. In statically indeterminate structures, post-tensioning produces reactions and internal forces due to therestraints imposed at the supports. The reactions and thecorresponding internal forces are called “secondary.” Forchecking the design strength, the secondary shear force Vp′′ andthe secondary bending moment Mp′′ are required by ACI 318.

Fig. 8.2—Sign convention of: (a) Mprestress; (b) Vp; and (c)tendon eccentricity and slope.


Fig. 8.3—Cross section in a unit width of a post-tensioned flat plate: (a) tendon profile; and (b) post-tensioned equivalent forces.

Fig. 8.4—Nonprismatic modeling of flat plate-column framesfor effect of gravity loads and post-tensioning effects.

Tendon profiles are commonly composed of parabolicsegments. Figure 8.3(a) depicts a typical profile of post-tensioning tendons in adjacent spans of a slab. Figure 8.3(b)represents the forces exerted by tendons on the concrete.Before using this system of forces in the analysis, it shouldbe verified that it is self-equilibrating; that is, it produces noreactions if applied on a statically determinate structure. Thesymbol Pe in Fig. 8.3(b) is the absolute value of the effectivepost-tensioning force per unit slab width, assumed constantover the length of the tendon. θp = (de/dx) is the anglebetween the tangent to the tendon and the centroidal axis(Fig. 8.2(c)), whereas e is the eccentricity of the tendon. The

effective prestressing force is equal to the jacking force lessthe losses due to anchor setting, friction, shrinkage and creepof concrete, and relaxation of post-tensioning steel.

In defining the geometry of the tendon profile (Fig. 8.3(a)),the α-values should be selected so that the tendon has thesame slope on both sides of the inflection points: 2, 4, 6, and 8.The equations in Fig. 8.3(b) are based on the assumption thatthe tendon has zero slope at the end (θp1 = 0) and each of thesegments 2-3-4 and 6-7-8 is a continuous parabola. Thelength of the inverted segments over internal supports istypically between 20 to 30% of the span of the panel, l (forexample, α45 + α56 ≈ 0.20 to 0.30). Section 8.6 gives anexample of a tendon profile. For ease of construction and formaximum load balancing by post-tensioning, tendons shouldbe placed in the banded direction so that their highest andlowest points are at the same level as the innermost bars ofthe top and bottom flexural reinforcement mesh. The distributedtendons at their highest points over the supports should bebelow and touching the banded tendons. At anchorages, thetendons are commonly at the midsurface of the slab.

The effect of post-tensioning is used in the analysis of theequivalent frame in Fig. 8.4. Where justified by constructionsequence, Vp′′ and Mp′′ can be calculated by applying thepost-tensioning forces (Fig. 8.3(b)) on the equivalent frameof Fig. 8.4. Reactions will form a set of forces in equilibrium,representing the secondary (statically indeterminate) reactions.The shear force Vp and the bending moment Mprestressresulting from the analysis are the internal forces includingthe secondary forces. Thus, the secondary shear force andbending moment at any section are

Vp′′ = Vp – Vp′ (8-1)


8.4—Effective compressive stress fpcThe compressive stress fpc (= Pe/h) is the average in-plane

stress produced by effective post-tension forces in one oftwo orthogonal directions. Consider the tendon profile in atypical interior panel of a flat plate (Fig. 8.5(a)). The force Pe

(a)

(b)

Fig. 8.5—(a) Tendon profile in typical interior span; and(b) variation of fpc with span length l, and the geometricparameter α.
that balances a fraction κ of the service dead load per unit
area of the slab’s surface, wD , is

(8-5)

The tendon is assumed to have a common tangent at thepoint of inflection in Fig. 8.5(a). The geometric symbols α,h, l, and hc are defined in Fig. 8.5(a); wc is the weight ofconcrete per unit volume; wsd is the superimposed dead loadper unit area of the floor. Consistent units for force andlength should be used. For example, the basic units lb and ft(or N and m) can be used for all the symbols in Eq. (8-5).Figure 8.5(b) shows the variation of fpc with l and α, forwhich the cover of post-tensioning tendon at top or bottom istaken as (h – hc)/2 = 1.57 in. (40.0 mm). Figure 8.5(b) showsthat for the range of l = 20 to 39 ft (6 to 12 m), fpc variesbetween 175 and 230 psi (1.21 and 1.59 MPa). The sameordinates of the graph in Fig. 8.5(a) can be used for any valueof wsd by noting that fpc is proportional to (wch + wsd).

Pe κwD1 2α–( )2

l2

8 1 2α–( )hc

-----------------------------=

8.5—Extension of punching shear design procedure to post-tensioned flat plates

The shear strength of concrete (in stress units) at a criticalsection at d/2 from the column face where shear reinforcementis not provided, is given by ACI 318 as

Inch-pound units

(8-6)

SI units

(8-6)

where βp is the smaller of 3.5 and [(αsd/bo) + 1.5]; λ is amodification factor related to unit weight of concrete; fpc isthe average value of the compressive stresses at centroid ofcross section in two directions (after allowance for all post-tension losses); Vp is the vertical component of the effectivepost-tension forces crossing the shear-critical section.Equation (8-6) is applicable only if:

a. No portion of the column cross section is closer to adiscontinuous edge than 4h;

vc βpλ fc′ 0.3fpcVp

bod--------+ +=

vc βpλfc′

12--------- 0.3fpc

Vp

bod--------+ +=

Mp′′ = Mprestress – Mp′ (8-2)

where Vp′ and Mp′ are the primary shear force andbending moment due to post-tensioning. Their values atany section are

Vp′ = –Peθp = –Pe(de/dx) (8-3)

Mp′ = –Pee (8-4)

The secondary shear force and bending moment diagramsare composed of straight lines whose ordinates can becalculated by considering only the secondary reactions onthe structure as a free body.

Commonly the value of Pe is chosen to produce upwardload (that is, w24 and w68 [Fig. 8.3(b)]) that balances 75 to85% of the dead load. Section 8.5 gives an equation for thepost-tensioning level necessary to balance a fraction κ of thedead load in an interior span.

To maintain integrity, ACI 318 requires, as minimum,either: 1) two tendons pass through the column cage in eachdirection, or 2) two bottom non-prestressed bars pass in eachdirection within the column core and be anchored at exteriorsupports. ACI 318 also specifies arrangement of the non-prestressed reinforcing bars required to supplement post-tensioned tendons to meet the flexural strength demand anda minimum amount of non-prestressed reinforcement tocontrol cracking in the vicinity of the column.


b. in Eq. (8-6) is not taken greater than 70 psi (5.8MPa); and

c. fpc in each direction is not less than 125 psi (0.86 MPa),nor taken greater than 500 psi (3.45 MPa).

If any of the above conditions is not satisfied, the shearstrength is taken the same as for non-post-tensioned flatplates (Eq. (3-1), (3-2), and (3-3)). When shear reinforcement isrequired for post-tensioned flat plate-column connections,its design is the same as for non-post-tensioned flat plates(Eq. (4-1)).

Special care should be exercised in computing Vp in Eq. (8-6), due to the sensitivity of its value to the in-place tendonprofile. When it is uncertain that the actual construction willmatch design assumptions, a reduced or zero value for Vpshould be used in Eq. (8-6). It is noted that post-tensioningtendons in the banded direction commonly produce muchhigher confining stress in the vicinity of the column than theaverage stress, fpc used by the ACI 318 Code in Eq. (8-6).

In calculating the upper limit of Mu for post-tensioned flatplates (Section 6.3, Eq. (6-2)), the probable flexural strengthMpr should be determined using a tensile stress 1.25fy in thenon-post-tensioned steel and fpc in the post-tensionedtendons; with fps being the stress in unbonded tendons at thenominal strength (ACI 318), calculated by Eq. (8-7) or (8-8)depending on the span-depth ratio of the slab.

Inch-pound unitsFor l/h ≤ 35

(8-7)

For l/h > 35

(8-8)

SI unitsFor l/h ≤ 35

(8-7)

For l/h > 35

(8-8)

In Eq. (8-7) and (8-8), ρp is the ratio of the post-tensioningreinforcement; fse is the effective stress in tendons afteraccounting for all post-tension losses; and fpy is the yieldstrength of the post-tensioning steel.

fc′

fps fse 10,000fc′

100ρp

-------------- the least of: fpy

fse 60,000 psi+⎩⎨⎧

≤+ +=

fps fse 10,000fc′

300ρp


fse 30,000 psi+⎩⎨⎧

≤+ +=

fps fse 70fc′

100ρp


fse 414 MPa+⎩⎨⎧

≤+ +=

fps fse 70fc′

300ρp


fse 207 MPa+⎩⎨⎧

≤+ +=

8.6—Research on post-tensioned flat platesHawkins (1981) tested a number of unbonded post-tensioned

flat plate-column connections subjected to a combination of

shear force and unidirectional moment reversals of increasingmagnitudes until failure. A non-prestressed reinforcedconcrete specimen with stirrup shear reinforcement wasincluded in this test series. The purpose of this control specimenwas to compare its behavior, in terms of ductility and energyabsorption, with the post-tensioned specimens. A comparisonof the response of the post-tensioned flat plate-columnconnections with the control specimen showed that, aftersignificant reversed cyclic loading, the post-tensioned speci-mens exhibited more ductility than the non-prestressedspecimen. Performance of post-tensioned connections inresidual shear capacity tests was better than that of thecontrol specimen.

Martinez-Cruzado et al. (1994) tested a number of post-tensioned flat plate-column connections under simulatedbiaxial earthquake loading. The test specimens weresubjected to different levels of gravity load to investigate theinfluence of gravity load on the response to biaxial lateralload of increasing magnitude. The tendons were banded inone direction and distributed in the perpendicular direction,and non-prestressed reinforcement was provided in thecolumn vicinity to satisfy the ACI 318 requirements. Noreinforcement was provided near the bottom slab surface.The test results showed that the level of gravity load appliedon the slab was an important factor in the performance of theconnection. Similar to non-prestressed concrete slabs (Fig. 3.1and 3.3), the strength and the deformation of post-tensionedflat plate-column connections decreased as the gravity loadincreased. In addition, the results of the tests implicitlyshowed that providing post-tensioned tendons in accordancewith ACI 318, Section 18.12.4 was effective in preventingcomplete collapse of the flat plate.

Experimental and analytical (finite-element) research(Kang and Wallace 2005; Ritchie and Ghali 2005; Gayedand Ghali 2006; and Gayed 2005) considered the seismicdesign procedure for punching shear reinforcement (studshear reinforcement) in post-tensioned flat plates. Two testseries, representing full-scale edge and interior column-slabconnections, were conducted by Ritchie and Ghali (2005)and Gayed and Ghali (2006), respectively. The connectionsin both series were reinforced with post-tensioned tendonsand non-prestressed steel bars. The amount of the tworeinforcement types was varied in each series, withoutchanging the sum of their contribution to flexural strength ofthe slab. The effective compressive stress fpc varied between0 and 160 psi (0.0 and 1.1 MPa). All connections in bothseries were provided with headed SSR, designed as outlinedin Chapters 6 and 7. The test specimens, subjected to shearingforce combined with drift reversals to simulate earthquakeeffects, exhibited ductile behavior (DRu > 0.025) withoutmuch difference due to the variation of fpc. The trend of theresults as well as nonlinear finite-element analyses (Gayed2005) indicated that increasing fpc up to 200 psi (1.4 MPa) hadno adverse effect on either the strength or the ductility that couldbe achieved without punching failure. When Eq. (8-6) orEq. (3-1), (3-2), and (3-3), combined with the condition vu< φvc is not satisfied, shear reinforcement is required and itsdesign, as recommended in Chapters 6 and 7, applies for post-


tensioned flat plates with fpc = 60 to 200 psi (0.4 to 1.4 MPa). Itis noted that Gayed (2005) limited fpc to 200 psi (1.4 MPa),although higher values may occur in practice.

Kang and Wallace (2005) conducted two shake-table testsof one-third scale, two-story, two-bay flat plate-columnframes. The slabs in the two test specimens were post-tensioned in one and non-post-tensioned in the other. The twoframes were subjected to gravity loads combined withuniaxial base accelerations of increasing intensity. Both testswere provided with headed shear stud reinforcement, designedin accordance with ACI 421.1R. Drift ratio DRu = 0.04 and0.03 were reached, without loss of strength, in the post-tensioned and the non-post-tensioned connections, respectively.
CHAPTER 9—REFERENCES
9.1—Referenced standards and reportsThe standards and reports listed below were the latest

edition at the time this document was prepared. Becausethese documents are revised frequently, the reader is advisedto contact the proper sponsoring group if it is desired to referto the latest version.

American Concrete Institute318 Building Code Requirements for Structural Concrete352.1R Recommendations for Design of Slab-Column

Connections in Monolithic Reinforced ConcreteStructures

421.1R Guide to Shear Reinforcement for Slabs423.3R Recommendations for Concrete Members

Prestressed with Unbonded Tendons

American Society of Civil EngineersASCE/SEI 7 Minimum Design Loads for Buildings and

Other Structures

ASTM InternationalA1044/1044M Standard Specification for Steel Stud

Assemblies for Shear Reinforcement ofConcrete

International Code CouncilIBC 2006 International Building Code

The above publications may be obtained from thefollowing organizations:

American Concrete Institute38800 Country Club DriveFarmington Hills, MI 48331www.concrete.org

American Society of Civil Engineers1801 Alexander Bell DriveReston, VA 20191www.asce.org

ASTM International100 Barr Harbor DriveWest Conshohocken, PA 19428www.astm.org

International Code Council500 New Jersey Avenue, NW6th FloorWashington, DC 20001www.iccsafe.org

9.2—Cited referencesANATECH Consulting Engineers, 1995, “ANATECH

Concrete Analysis Program ANACAP,” Version 2.1, User’sGuide, San Diego, CA.

Berg, G. V., and Stratta, J. L., 1964, “Anchorage and theAlaska Earthquake of March 27, 1964,” American Iron andSteel Institute, New York, 63 pp.

Brown, S. J., 2003, “Seismic Response of Slab-ColumnConnections,” PhD thesis, Department of Civil Engineering,University of Calgary, Calgary, AB, Canada.

Cao, H., 1993, “Seismic Design of Slab-Column Connec-tions,” MSc thesis, Department of Civil Engineering,University of Calgary, Calgary, AB, Canada.

Dilger, W. H., and Brown, S. J., 1995, “Earthquake Resis-tance of Slab-Column Connections,” Institut für Baustatikund Konstruktion, ETH Zürich, Sept., pp. 22-27.

Dilger, W. H., and Cao, H., 1991, “Behaviour of Slab-Column Connections Under Reversed Cyclic Loading,”Proceedings of the 2nd International Conference of High-Rise Buildings, China.

Gayed, R. B., 2005, “Design for Punching Shear at InteriorColumns in Prestressed Concrete Slabs: Resistance to Earth-quakes,” PhD dissertation, University of Calgary, Calgary,AB, Canada, 318 pp.

Gayed, R. B., and Ghali, A., 2006, “Seismic-ResistantJoints of Interior Columns with Prestressed Slabs,” ACIStructural Journal, V. 103, No. 5, Sept.-Oct., pp. 710-719.

Ghali, A., and Hammill, N., 1992, “Effectiveness of ShearReinforcement in Slabs,” Concrete International, V. 14, No. 1,Jan., pp. 60-65.

Ghali, A.; Neville, A. M.; and Brown, T. G., 2003, Struc-tural Analysis: A Unified Classical and Matrix Approach,fifth edition, Spon Press, 844 pp.

Hawkins, N. M., 1981, “Lateral Load Resistance ofUnbonded Post-Tensioned Flat Plate Construction,” PCIJournal, V. 26, No. 1, Jan., pp. 94-117.

Hawkins, N. M.; Mitchell, D.; and Hanna, S. N., 1975,“Effects of Shear Reinforcement on the Reversed CyclicLoading Behaviour of Flat Plate Structures,” CanadianJournal of Civil Engineering, V. 2, pp. 572-582.

Hueste, M. B. D., and Wight, J. K., 1999, “NonlinearPunching Shear Failure Model for Interior Slab-ColumnConnections,” Journal of Structural Engineering, ASCE,V. 125, No. 9, Sept., pp. 997-1008.

Islam, S., and Park, R., 1976, “Tests on Slab-ColumnConnections with Shear and Unbalanced Flexure,” Journal ofStructural Division, ASCE, V. 102, No. ST3, Mar., pp. 549-568.

Kang, T. H. K., and Wallace, J. W., 2005, “DynamicResponses of Flat Plate Systems with Shear Reinforcement,”ACI Structural Journal, V. 102, No. 5, Sept.-Oct., pp. 763-773.

Martinez-Cruzado, J. A.; Qaisrani, A. N.; and Moehle, J. P.,1994, “Post-Tensioned Flat Plate Slab-Column Connections


APPENDIX B—VERIFICATION OF UPPER LIMITTO UNBALANCED MOMENT TO BE USED IN

PUNCHING SHEAR DESIGNAs mentioned in Section 6.3, the quantity αmMu can be
determined from the results of a nonlinear finite element
analysis (Megally and Ghali 2000b). Results of such ananalysis, which have helped in selecting the empirical Eq. (6-3)and (6-4), are presented in Tables B.1 and B.2 for interior and
edge flat plate-column connections, respectively (Fig. B.1(a)
and (b)). The slab is simply supported on four or three edges
(Fig. B.1(a) or (b), respectively). The magnitude of the
constant force Vu, applied through the axis of the column, isone of the variables in the tables. The bottom reinforcementratio in all slabs is 0.65%, whereas the top reinforcementratio ρ is variable. The yield strength of the flexural reinforcingbars is 60 ksi (414 MPa), except as noted otherwise.

APPENDIX A—VERIFICATION OF PROPOSED MINIMUM AMOUNT OF SHEAR REINFORCEMENT

FOR EARTHQUAKE-RESISTANT FLAT PLATE-COLUMN CONNECTIONS

As discussed in Chapter 4, the recommended minimumamount of shear reinforcement is based on experiments(Islam and Park 1976; Hawkins et al. 1975; Dilger andBrown 1995; Dilger and Cao 1991; Megally and Ghali2000a; Megally 1998). In these experiments, flat plate-column connections with different amounts of stirrups orSSR experienced substantial drifts without punching shearfailure. The slab thickness in the tests was 6 in. (150 mm)except in Islam and Park’s tests, where the thickness was 3.5 in.(89 mm). Table A.1 and Fig. A.1 summarize results of these

Fig. A.1—Shear strength provided by slab shear reinforcement.

experiments; the figure is a plot of the drift capacity versus(vs/ ). The figure shows that all of the tested flat plate-column connections have a ratio (vs/ ) exceeding 3. Thetable indicates that even when the SSR is spaced at s = 0.75d,flat plate-column connections can experience story drift ratioDRu much higher than 0.025 without punching shear failureor significant loss of moment transfer capacity. This holdstrue for cases even when (Vu/φVc) is relatively high. For thisreason, the minimum amount of shear reinforcement specifiedby Eq. (4-1) is somewhat smaller than what has been used inthe experiments.

fc′fc′

Subject to Earthquake Loading,” 5th U.S. National Conferenceon Earthquake Engineering, V. 2, Chicago, IL, pp. 139-148.

Megally, S., 1998, “Punching Shear Resistance ofConcrete Slabs to Gravity and Earthquake Forces,” PhDdissertation, Department of Civil Engineering, University ofCalgary, Calgary, AB, Canada, 468 pp.

Megally, S., and Ghali, A., 1994, “Design Considerationsfor Slab-Column Connections in Seismic Zones,” ACI Struc-tural Journal, V. 91, No. 3, May-June, pp. 303-314.

Megally, S., and Ghali, A., 2000a, “Seismic Behavior ofSlab-Column Connections,” Canadian Journal of CivilEngineering, V. 27, No. 1, Feb., pp. 84-100.

Megally, S., and Ghali, A., 2000b, “Punching of ConcreteSlabs Due to Column’s Moment Transfer,” Journal of Struc-tural Engineering, ASCE, V. 126, No. 2, Feb., pp. 180-189.

Megally, S., and Ghali, A., 2000c, “Seismic Behavior of EdgeColumn-Slab Connections with Stud Shear Reinforcement,”ACI Structural Journal, V. 97, No. 1, Jan.-Feb., pp. 53-60.

Megally, S., and Ghali, A., 2000d, “Punching ShearDesign of Seismic-Resistant Slab-Column Connections,”ACI Structural Journal, V. 97, No. 5, Sept.-Oct., pp. 720-730.

Mitchell, D.; DeVall, R. H.; Saatcioglu, M.; Simpson, R.;Tinawi, R.; and Tremblay, R., 1995, “Damage to ConcreteStructures Due to the 1994 Northridge Earthquake,” CanadianJournal Civil Engineering, V. 22, No. 4, Apr., pp. 361-377.

Mitchell, D.; Tinawi, R.; and Redwood, R. G., 1990,“Damage to Buildings Due to the 1989 Loma Prieta Earth-quake—A Canadian Code Perspective,” Canadian Journalof Civil Engineering, V. 17, No. 10, Oct., pp. 813-834.

Mokhtar, A. S.; Ghali, A.; and Dilger, W. H., 1985, “StudShear Reinforcement for Flat Concrete Plates,” ACI JOURNAL,Proceedings V. 82, No. 5, Sept.-Oct., pp. 676-683.

Pan, A. D., and Moehle, J. P., 1989, “Lateral DisplacementDuctility of Reinforced Concrete Flat Plates,” ACI Struc-tural Journal, V. 86, No. 3, May-June, pp. 250-258.

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

Ritchie, M., and Ghali, A., 2005, “Seismic-ResistantConnections of Edge Columns with Prestressed Slabs,” ACIStructural Journal, V. 102, No. 2, Mar.-Apr., pp. 314-323

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

Sozen, M. A., 1980, “Review of Earthquake Response of Rein-forced Concrete Buildings With a View to Drift Control, State-of-the-Art in Earthquake Engineering,” 7th World Conferenceon Earthquake Engineering, Istanbul, Turkey, pp. 119-174.

Vanderbilt, M. D., and Corely, W. G., 1983, “Frame Analysisof Concrete Buildings,” Concrete International, V. 5, No. 12,Dec., pp. 33-43.

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

Yanev, P. I.; Gillengerten, J. D.; and Hamburger, R. O., 1991,“The Performance of Steel Buildings in Past Earthquakes,”American Iron and Steel Institute and EQE Engineering, Inc.,97 pp.


Table B.1—Nonlinear finite element results of fraction of unbalanced moment transferred by flexure in interior flat plate-column connections

(1) (2) (3) (4) (5) (6) (7)

Slab lx/ly ρ, % Vu, kip (kN) αm-FE αm αm-FE/αm

I1 1.00 0.61 0 (0) 0.40 0.40 1.00

I2 1.00 0.61 16.9 (75.2) 0.39 0.40 0.98

I3 1.00 0.61 33.7 (149.9) 0.37 0.40 0.93

I4 1.00 0.61 67.4 (299.8) 0.43 0.40 1.08

I5 1.00 1.22 0 (0) 0.39 0.40 0.98

I6 1.00 1.22 16.9 (75.2) 0.39 0.40 0.98

I7 1.00 1.22 33.7 (149.9) 0.37 0.40 0.93

I8 1.00 1.22 67.4 (299.8) 0.44 0.40 1.10

I9* 1.00 1.22 33.7 (149.9) 0.37 0.40 0.93

I10† 1.00 1.22 33.7 (149.9) 0.39 0.40 0.98

I11 1.00 2.44 0 (0) 0.45 0.40 1.13

I12 1.00 2.44 16.9 (75.2) 0.42 0.40 1.05

I13 1.00 2.44 33.7 (149.9) 0.44 0.40 1.10

I14 1.00 2.44 67.4 (299.8) 0.52 0.40 1.30

I15 1.00 2.44 101.2 (450.2) 0.57 0.40 1.43

I16‡ 1.00 2.44 0 (0) 0.44 0.40 1.10

I17§ 1.00 2.44 33.7 (149.9) 0.47 0.40 1.18

I18|| 1.00 2.44 33.7 (149.9) 0.44 0.40 1.10

II9‡ 1.00 2.44 33.7 (149.9) 0.43 0.40 1.08

I20# 1.00 2.44 33.7 (149.9) 0.45 0.40 1.13

I21 1.00 2.44 101.2 (450.2) 0.55 0.40 1.38

I22 0.39 1.22 0 (0) 0.64 0.54 1.19

I23 0.39 1.22 22.5 (100.1) 0.64 0.54 1.19

I24 0.39 1.22 33.7 (149.9) 0.62 0.54 1.15

I25 0.39 1.22 45.0 (200.2) 0.60 0.54 1.11

I26 0.39 1.22 67.4 (299.8) 0.64 0.54 1.19

I27 0.66 1.22 0 (0) 0.53 0.47 1.15

I28 0.66 1.22 33.7 (149.9) 0.47 0.47 1.00

I29 0.66 1.22 67.4 (299.8) 0.55 0.47 1.17

I30 1.52 1.22 0 (0) 0.34 0.32 1.06

I31 1.52 1.22 33.7 (149.9) 0.30 0.32 0.94

I32 1.52 1.22 67.4 (299.8) 0.37 0.32 1.16

I33 2.57 2.44 0 (0) 0.19 0.20 0.95

I34 2.57 2.44 16.9 (75.2) 0.19 0.20 0.95

I35 2.57 2.44 33.7 (149.9) 0.20 0.20 1.00

Mean 1.09

Standard deviation 0.12

Coefficient of variation 0.11

*fy = 44 ksi (303 MPa).†fy = 73 ksi (503 MPa).‡Slab with shear reinforcement of 3/8 in. (9.5 mm) diameter (12 shear reinforcingbars in each peripheral line).§Slab with shear reinforcement of 1/8 in. (3.2 mm) diameter (12 shear reinforcingbars in each peripheral line).||Slab with shear reinforcement of 1/4 in. (6.4 mm) diameter (12 shear reinforcing barsin each peripheral line).#Slab with shear reinforcement of 1/2 in. (12.7 mm) diameter (12 shear reinforcingbars in each peripheral line).

Table A.1—Ultimate story drift ratios of flatplate-column connections transferring cyclic moment reversals

(1) (2) (3) (4) (5) (6) (7) (8) (9)

Slab Reference

Type of shear

reinforce-ment

fc′ , psi (MPa) Vu/φVc s/d

fyt , ksi

(MPa) vs/√fc′ 100DRu

6CS

Islam and Park

(1976)Stirrups

4090 (28.2) 0.24 0.55 42.1

(290) 5.37 4.0

7CS 4310(29.7) 0.24 0.55 44.1

(304) 4.50 3.7

8CS 3210(22.1) 0.27 0.55 42.5

(293) 3.11 5.0

SS1

Hawkinset al.

(1975)Stirrups

4000(27.6) 0.49 0.33 66.6

(459) 9.54 3.5

SS3 3750(25.9) 0.48 0.33 66.0

(455) 9.85 4.1

SS4 4000(27.6) 0.47 0.33 66.0

(455) 9.54 5.5

SS5 4670 (32.2) 0.42 0.33 67.1

(463) 3.80 4.9

SJB-1

Dilger and Brown (1995)†

SSR

4669(32.2) 0.48 0.57 66.7

(460) 5.88 5.5

SJB-2 4974 (34.3) 0.47 0.57 66.7

(460) 5.70 5.7

SJB-3 4698(32.4) 0.48 0.57 66.7

(460) 5.87 5.0

SJB-4 5757(39.7) 0.43 0.57 66.7

(460) 5.30 6.4

SJB-5 4843(33.4) 0.47 0.57 66.7

(460) 5.78 7.6

SJB-8* 5075(35.0) 0.46 0.57 66.7

(460) 5.64 5.7

SJB-9* 4524(31.2) 0.49 0.57 66.7

(460) 5.98 7.1

CD3

Dilger and Cao

(1991)SSR

5162(35.6) 0.92 0.79 53.8

(371) 4.43 3.5

CD4 4974(34.3) 0.62 0.79 53.8

(371) 4.51 4.8

CD6 4553(31.4) 0.65 0.39 53.8

(371) 12.3 5.4

CD7 4147(28.6) 0.51 0.79 53.8

(371) 4.94 5.6

MG-3

Megallyand Ghali(2000a)

SSR

4872(33.6) 0.56 0.75 54.7

(377) 4.03 5.4

MG-4 4684(32.3) 0.86 0.75 54.7

(377) 4.11 4.6

MG-5 4104(28.3) 0.31 0.75 54.7

(377) 4.39 6.5

MG-6 4365(30.1) 0.59 0.44 54.7

(377) 7.23 6.0

MG-10 Megally(1998) SSR 4307

(29.7) 0.60 0.75 54.7(377) 4.28 5.2

*Flat plate-column connections transferring biaxial moments Mux = Muy (Fig. 3.2(a)).†Test variable in this series was concentration of slab flexural reinforcement in columnvicinity.


APPENDIX C—NOTES ON PROPERTIES OF SHEAR-CRITICAL SECTION

The equations in Section C.1 and C.2 are taken from ACI421.1R. They are given as follows because they are used inthe present document.

C.1—Second moments of areaIn this report, the shear stresses caused by unbalanced

moments are computed using parameters Jx and Jy ratherthan the Jc property that has been used for several decades.The value of Jy is smaller than Jc; in most cases, the differ-ence is only a few percent. The use of Jx or Jy instead of Jcfor a critical section of general shape is recommended inACI 421.1R; the relevant material is summarized in the presentsubsection.

General expressions for Jx and Jy are given, but first theexpressions for Jc and Jy for the case of interior rectangularcolumn are compared (Fig. 3.2(a)). ACI 318 defines Jc as aproperty of the assumed critical section analogous to polarmoment of inertia and gives Eq. (C-1), which is compared withEq. (C-2) for Jy for the same critical section:

(C-1)

(C-2)

Jcd cx d+( )3

6------------------------

d cy d+( ) cx d+( )2

2--------------------------------------------

d3 cx d+( )2

------------------------+ +=

Jyd cx d+( )3

6------------------------

d cy d+( ) cx d+( )2

2--------------------------------------------+=

Table B.2—Nonlinear finite element results of fraction of unbalanced moment transferred by flexure in edge flat plate-column connections

(1) (2) (3) (4) (5) (6) (7)

Slab lx/ly ρ, % Vu, kip (kN) αm-FE αm αm-FE/αm

E1 0.84 0.61 0 0.24 0.24 1.00

E2 0.84 0.61 16.9 (75.2) 0.22 0.24 0.92

E3 0.84 0.61 22.5 (100.1) 0.21 0.24 0.88

E4 0.84 0.61 34.1 (151.7) 0.25 0.24 1.04

E5 0.84 1.22 0 (0) 0.34 0.30 1.13

E6 0.84 1.22 16.9 (75.2) 0.29 0.30 0.97

E7 0.84 1.22 22.5 (100.1) 0.31 0.30 1.03

E8 0.84 1.22 28.1 (125.0) 0.32 0.30 1.07

E9 0.84 1.22 33.7 (149.9) 0.32 0.30 1.07

E10 0.84 1.22 38.1 (169.5) 0.37 0.30 1.23

E11 0.84 1.22 16.9 (75.2) 0.36 0.30 1.20

E12 0.30 1.22 16.9 (75.2) 0.70 0.49 1.43

E13 0.50 1.22 16.9 (75.2) 0.49 0.39 1.26

E14 1.28 1.22 16.9 (75.2) 0.25 0.23 1.09

E15 1.91 1.22 16.9 (75.2) 0.14 0.16 0.88

E16 2.33 1.22 16.9 (75.2) 0.13 0.12 1.08

E17 0.92 1.22 16.9 (75.2) 0.30 0.29 1.03

I18* 0.84 1.22 16.9 (75.2) 0.26 0.30 0.87

E19† 0.84 1.22 16.9 (75.2) 0.36 0.30 1.20

E20‡ 0.84 1.22 46.1 (205.1) 0.38 0.30 1.27

E21 0.84 1.83 16.9 (75.2) 0.34 0.36 0.94

E22 0.84 2.44 0 (0) 0.51 0.43 1.19

E23 0.84 2.44 16.9 (75.2) 0.45 0.43 1.05

E24 0.84 2.44 22.5 (100.1) 0.45 0.43 1.05

E25 0.84 2.44 33.7 (149.9) 0.46 0.43 1.07

E26 0.84 2.44 45.0 (200.2) 0.48 0.43 1.12

E27 0.84 2.44 56.9 (253.1) 0.52 0.43 1.21

E28‡ 0.84 2.44 0 (0) 0.40 0.43 0.93

E29‡ 0.84 2.44 16.9 (75.2) 0.40 0.43 0.93

E30‡ 0.84 2.44 33.7 (149.9) 0.43 0.43 1.00

E31‡ 0.84 2.44 67.7 (301.1) 0.54 0.43 1.26

Mean 1.08

Standard deviation 0.14

Coefficient of variation 0.13

*fy = 43.5 ksi (300 MPa).†Slab with twice the number of reinforcing bars but with ρ = 1.22%.‡Slab with shear reinforcement of 3/8 in. (9.5 mm) diameter (seven shear reinforcingbars in each peripheral line).

Fig. B.1—Plan views of flat plate-column connectionsanalyzed by the finite element method.

The ratio αm-FE (finite element) to αm (Eq. (6-3) or (6-4))in Column 7 of Tables B.1 or B.2 has an average value of1.09 or 1.08, respectively, with the coefficient of variationequal to 0.11 or 0.13, respectively. These results substantiatethe proposed Eq. (6-2) through (6-4).


The expression for Jy is the same as the expression for Jc, withthe last term dropped. The third term on the right-hand side ofEq. (C-1) is normally much smaller than the other two terms.

General expressions for bo, Jx , and Jy of a shear-criticalsection composed of straight segments are

(C-3)

(C-4)

(C-5)

where bo is the length of the perimeter of the shear-criticalsection; Jx and Jy = d multiplied by the second moments ofthe perimeter of the shear-critical section about its centroidalprincipal axes x and y, respectively; xi and yi are coordi-nates of i; xj and yj are coordinates of j, with i and j beingthe extremities of a typical straight segment of the perimeter;and lij is the length of the segment ij.

ACI 318 requires that the shear stress, vu due to Vu,(γvx Mux) and (γvyMuy) vary linearly over the shear-criticalsection. The shear stress distribution that satisfies thisrequirement is expressed as

(C-6)

where x and y are coordinates, with respect to the centroidalprincipal axes, of any point on the perimeter of the shear-criticalsection. The centroidal principal axes, the only orthogonalaxes about which the product of inertia is nil, must be usedto obtain a stress vu whose resultants are: Vu, (γvxMux) and(γvyMuy). Figure 3.2(c) shows an example of a shear-criticalsection whose centroidal principal axes x and y do not coincidewith the centroidal axes x and y, that are parallel to the sidesof the column. The principal axis x or y makes an angle θ(measured in clockwise direction) with the axis x or y, given by

(C-7)

where Jx and Jy = d multiplied by the second moments ofthe perimeter of the shear-critical section about the axes x andy, respectively (Eq. (C-4) and (C-5), substituting x and y for xand y). For a shear-critical section whose perimeter iscomposed of straight segments, the product of inertia isgiven by

(C-8)

where and are coordinates of i, and and arecoordinates of j, with i and j being the extremities of a typical

bo lij∑=

Jxd3--- lij yi

2 yiyj yj2+ +( )[ ]∑=

Jyd3--- lij xi

2 xixj xj2+ +( )[ ]∑=

vuVu

bod-------- γvx

Mux

Jx

---------y γvyMuy

Jy

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

2θtan2J

xy–

Jx Jy–---------------=

Jxy

d6--- lij 2xiyi xiyj xjyi 2xjyj+ + +( )[ ]∑=

xi yi xj yj

straight segment, whose length is lij. The coordinates (x,y) ofany point are related to the coordinates (x,y) by

x = xcosθ + ysinθ (C-9)

y = –xsinθ + ycosθ (C-10)

C.2—Equations for γvFor interior columns, the fraction γv of the unbalanced

moment that it transmitted by shear is defined in ACI 318-05and in ACI 421.1R by Eq. (C-11) and (C-12). The latter

(C-11)

(C-12)

γvx 1 1

1 23---

ly

lx

----+

---------------------–=

γvy 1 1

1 23---

lx

ly

----+

---------------------–=

source, however, also gives γv for edge and corner columns.Considering unbalanced moments Mux or Muy (Fig. 3.2), theequations for γvx and γvy are

Interior columns

Edge columns

(C-13)

(γvy = 0 when lx/ly ≤ 0.2) (C-14)

Corner columns

γvx = 0.4 (C-15)

γvy = same as for edge column (C-16)

γvx 1 1

1 23---

ly

lx

----+

---------------------–=

γvy 1 1

1 23---

lx

ly

---- 0.2–+

----------------------------------–=

APPENDIX D—DESIGN EXAMPLESD.1—General

The examples are worked out only in inch-pound units tomake it easier for the reader; conversion factors are given inAppendix E; Example 1 is repeated in Example 5 using SI

units. The examples consider a flat plate for a building, withequal span lengths l = 20 ft (6.1 m) in two orthogonal direc-tions, story height lc = 12 ft (3.66 m), and the slab thicknessh = 8 in. (203 mm). The lateral-force-resisting structuralsystem is composed of shear walls that limit the story driftratio, including inelastic deformations, to DRu = 0.02. Thefollowing are design examples of the shear reinforcementrequired for interior, edge, and corner flat plate-columnconnections, for Vu combined with earthquake-factored


D.2—Example 1: Interior flat plate-column connection

The value of (Vu)D+L+E, given as follows, corresponds tothe load combination that governs the design. The loadfactors used to calculate (Vu)D+L+E are in accordance withACI 318. The values of (My)D and (My)L due to nonfactoreddead and live loads, respectively, are given. The given valuesare (Vu)D+L+E = 78 kips, (My)D = 0.0, and (My)L = 0.0. Thecross-sectional dimensions of the column are cx = cy = 16 in.(Fig. D.1); the reinforcement ratios of the top and bottomflexural bars are 1.0 and 0.5%, respectively.

The critical section shown in Fig. 3.2(a) has bo = 90.5 in.;Vc is governed by Eq. (3-3): Vc = 4(90.5)(6.625) /1000= 151.7 kips; and (Vu/φVc) = 78/(0.75 × 151.7) = 0.69.

The unbalanced moment caused by seismic forces is obtainedfrom an elastic analysis of the frame shown in Fig. 6.1(a). Theprescribed elastic story drift δe is given by Eq. (6-1)

4000

= 0.75 in.

where (Cd/IE) = 3.85 according to the ASCE/SEI 7(assuming that the lateral-force-resisting system consists of

δeDRulc

Cd/IE( )------------------ 0.02 12 12××

3.85-----------------------------------= =

shear walls). For the frame shown in Fig. 6.1(a), the momentof inertia of the slab Is = 5120 in.4 This value is equal to one-half the gross moment of inertia of an 8 in. slab that is 20 ftwide. The reduction is made to reflect some of the effects ofcracking on the force distribution. The equivalent moment ofinertia of the column Iec = 1320 in.4 This value is obtainedaccording to the equivalent frame method of ACI 318 byassuming that the flexibility of the equivalent column equalsthe sum of the flexibility of the actual column and thetorsional strip of the slab. In the determination of the equivalentmoment of inertia, the column lengths were taken as 72 in.(Fig. 6.1(a)) with 4 in. long rigid parts at the joints and withthe other ends simply supported, as shown in Fig. 6.1(c). Theequivalent moment of inertia is computed as Iec = Ic(Kec/ΣKc),where Kec and Kc are end rotational stiffnesses (moment perunit rotation) of equivalent column and the actual column,respectively.

The unbalanced moment due to δe , obtained from elasticanalysis of the frame in Fig. 6.1(a), is (My)E = 1670 in.-kipAccording to ACI 318-08, Section 9.2, the factored unbalancedmoment is

Muy = 1.2(My)D + 1.0(My)L + 1.0(My)E (D-1)

Muy = 1.2(0) + 1.0(0) + 1.0(1670) = 1670 in.-kip

The flexural strengths provided by the top and bottomflexural reinforcement, based on a stress of 1.25fy, of criticalsection Sides BC and AD (Fig. 3.2(a)), are 660 and 350 in.-kip,respectively. Thus, Mpr = 660 + 350 = 1010 in.-kip. For thisinterior connection, γvy = 0.4; βr = 1.0; and αm = 0.4 (Eq. (6-3)).It follows that the upper limit of Muy is (Eq. (6-2))

(Muy)upper limit = = 2530 in.-kip

Because the upper limit is higher than Muy = 1670 in.-kip, theflat plate-column connection should be designed to resist Vu =78 kips and Muy = 1670 in.-kip. In this structure, the drift due toseismic forces does not change Vu for this interior column.

The maximum factored shear stress due to Vu and Muy(computed according to the eccentric shear stress model ofACI 318) at the critical section at d/2 from the column face(Fig. 3.2(a)) is vu = 278 psi and (vu /φ) = 5.9 psi. Thisvalue is less than 6 psi; therefore, stirrups or SSR can beused. The SSR shown in Fig. D.1 is designed according to

660 350+0.4

------------------------

fc′fc′

Fig. D.1—Arrangement of SSR in interior flat plate-columnconnection of Examples 1 and 5.

ACI 421.1R. For the chosen shear reinforcement, Av = 2.36in.2 (12 studs of diameter 1/2 in.); s = 4.625 in., the nominalshear strength (expressed in stress units) provided by shearreinforcement (Eq. (4-1)) is vs = [2.36(50,000)]/[90.5(4.625)] =282 psi, which is greater than the minimum: 3 = 190 psi.The distance between the column face and the outermostperipheral line of SSR is 25.75 in., which is greater than(3.5d = 23.2 in.). The details of the stud arrangement shownin Fig. D.1 satisfy the requirement for the minimum amountof shear reinforcement. ACI 421.1R recommends that so ≤0.4d; the value used in this example and Examples 2, 3, and

fc′

moment Muy (Fig. 3.2). Other data are concrete cover = 0.75 in.(19 mm); fc′ = 4000 psi (27.6 MPa); Ec = 3600 ksi (24.8 GPa);fy = 60 ksi (414 MPa); flexural reinforcement nominal diameter= 0.625 in. (15.9 mm); diameter of shear reinforcement =0.50 in. (12.7 mm); fyt = 50 or 60 ksi (345 or 414 MPa) forSSR or stirrups, respectively; and d = 8 – 0.75 – (0.625) =6.625 in. (168 mm). The slab has been designed to supportgravity loads using the Direct Design Method of ACI 318,and the flexural reinforcement ratios have been determined.The strength reduction factor φ according to ACI 318-08,Section 9.3.2.3, is 0.75. The value of the dimensionlesscoefficient (Cd /IE) is 3.85. The dead load D is 120 lb/ft2

(5.75 kN/m2) and the live load L is 50 lb/ft2 (2.39 kN/m2).Load factors as given in ACI 318-08, Section 9.2.1 are used.


D.3—Example 2: Edge flat plate-column connectionThe edge flat plate-column connection in Fig. 3.2(b) is

designed for DRu = 0.02 in the x-direction (same drift ratioas in Example 1). The cross-sectional dimensions of thecolumn are cx = cy = 16 in. (Fig. D.2). Ratios of top and
(Muy)Case II = 0.90(296) + 1.0(–1212) = –946 in.-kip
Fig. D.2—Arrangement of SSR in the edge flat plate-columnconnection of Example 2.

bottom flexural reinforcements, running perpendicular to theslab edge, are 0.6 and 0.4%, respectively.

Consider Cases I and II, which will produce extremestresses at Side BC and at the outer extremities (Points A andD) of the shear-critical section (Fig. 3.2(b)). For thisexample, the outer column face is at the slab edge. Theunbalanced moments My and MOy about axes through thecentroid of the shear-critical section at d/2 from the columnface and at column’s centroid are related by

My = MOy + VxO (D-2)

where xO = –5.22 in. = the x-coordinate of the column’scentroid, O (Fig. 3.2(b)).

The values of VD , VL , (My)D , and (My)L without loadfactors are: VD = 22.6 kips; VL = 9.4 kips; (My)D = 296 in.-kip;and (My)L = 123 in.-kip These moments cause tension on topof the slab.

The unbalanced moment caused by seismic forces isobtained from an elastic analysis of the frame shown inFig. 6.1(b) subjected to lateral displacement in each of twodirections. The prescribed elastic story drift δe is ±0.75 in. asfor the interior column in Example 1. For the frame shown inFig. 6.1(b), Is = 5120 in.4 and Iec = 1320 in.4 The unbalancedmoment and shearing force obtained from the frame analysis,considering their statical equivalents at the centroid of theshear-critical section (Eq. (D-2)), are

(My)E = ±1212 in.-kip; (V)E = ±10.6 kips

Using ACI 318 load factors, the factored unbalancedmoment for Case I (1.2D +1.0L + 1.0E) is

(Muy)Case I = 1.2(296) + 1.0(123) + 1.0(1212) = 1691 in.-kip

The factored unbalanced moment for Case II (0.9D + 1.0E) is

The probable flexural strengths provided by the top andbottom flexural reinforcement of critical section Side BC(Fig. 3.2(b)) are 417 and 285 in.-kip, respectively. For thisedge connection, γvy = 0.35 (Eq. (C-14)); βr = 0.85; and αm= 0.24 (Eq. (6-4)) for negative moment of resistance, and αm= 0.21 (Eq. (6-4)) for positive moment of resistance. There-fore, the upper limits of |Muy| are given by (Eq. (6-2))

Upper limit |Muy| = 417/0.24 = 1740 in.-kip > 1691 in.-kip

Upper limit |Muy| = 285/0.21 = 1360 in.-kip > 946 in.-kip

Thus, the connection should be designed to resist:(a) Case I: Vu = 1.2(22.6) + 1.0(9.4) + 1.0(10.6) = 47.1 kips,

combined with Muy = 1691 in.-kip; and(b) Case II: Vu = 0.9(22.6) + 1.0(–10.6) = 9.7 kips,

combined with Muy = –946 in.-kip.The perimeter of the critical section bo = 61.25 in. The

parameter Jy of the critical section about the centroidal prin-cipal y-axis is Jy = 16,770 in.4 The fraction of the unbal-anced moment transferred by vertical shear stresses is (ACI421.1R)

(D-3)

The projections of the critical section on the principal axesare lx = 19.3 in. and ly = 22.6 in., and Eq. (D-3) gives γvy =0.35. The shear stress at Point A or B (Fig. 3.2(b)) isdetermined by

(vu)A or B = (xA or xB) (D-4)

γvy 1 1

1 23---

lx

ly

---- 0.2–+

----------------------------------–=

Vu

bod-------- γvy

Muy

Jy

---------+

4 is s = 2.625 in. The nominal shear strength v multiplied

o nby φ should exceed vu = 278 psi. In this example

vn = vs + vc = 282 + 1.5 = 377 psi

Thus, φvn = 0.75(377) = 283 psi, which is greater than278 psi. The maximum shear stress at the critical section atd/2 from the outermost peripheral line of studs is

vu = 80 psi < 2φ = 95 psi

Thus, the stud shear reinforcement and its extension(Fig. D.1) are adequate.

4000

fc′


Substitution of the values Vu combined with Muy in Cases Iand II, with d = 6.625 in., and xA = –13.22 in. or xB = 6.09 in.,gives the maximum shear stresses

Case I: (vu)B = 0.331 ksi = 331 psi; (vu)A = 0.351 ksi = 351 psi

Case II: (vu)A = 0.285 ksi = 285 psi

The maximum factored shear stress at the critical section atd/2 from the column face is vu = 351 psi at A or D (Fig. 3.2(b),Case I); (vu/φ) = 7.4 psi (0.62 MPa). The ratio vu/φis higher than 6 psi (0.5 MPa); thus, stirrups cannotbe used, but SSR can be used. The SSR shown in Fig. D.2 isdesigned in accordance with ACI 421.1R. For the chosenshear reinforcement, Av = 1.77 in.2 (nine studs); fyt = 50 ksi;and s = 3.25 in. The nominal shear strength (expressed instress units) provided by shear reinforcement (Eq. (4-1)) is vs= (177 × 50,000)/(61.25 × 3.25) = 444 psi. This value of vsis greater than the minimum of 3 = 190 psi. The distancebetween the column face and the outermost peripheral line ofSSR = 25.375 in., which is greater than 3.5d = 23.2 in. Thus,details of stud arrangement shown in Fig. D.2 satisfy therequirement of minimum amount of shear reinforcement.

The nominal shear strength vn multiplied by φ must exceedvu = 351 psi. In this example

vn = vs + vc = 444 + 1.5 = 539 psi

Thus, φvn = 0.75(539) = 404 psi, which is greater than351 psi. The maximum shear stress at the critical section atd/2 from the outermost peripheral line of studs (Fig. D.2)occurs at its Side AB in Case I

vu = 70 psi < 2φ = 95 psi


D.4—Example 3: Corner flat plate-column connection

The corner column-flat plate connection in Fig. 3.2(c) isdesigned assuming DRu = 0.02 in the x-direction (same DRuvalue as in Examples 1 and 2). The outer faces of the columnare at the slab edges. The cross-sectional dimensions of thecolumn are cx = cy = 20 in. (Fig. D.3). The ratios of top and

fc′ fc′fc′ fc′

fc′

4000

fc′

Fig. D.3—Arrangement of SSR in the corner flat plate-column connection of Example 3.

bottom flexural reinforcements, in the directions of the edgesof the slab, are 0.6 and 0.4%, respectively. It is assumed thatthe geometry and the loading are the same in the x- and y-directions; the design is for DRu in the x-direction only.Consider Cases I and II producing extreme shear stresses atPoints B and A, respectively (Fig. 3.2(c)).

The perimeter of the critical section is bo = 46.625 in. Theprincipal axes x and y, at the critical section’s centroid, are atan angle θ = 45 degrees (Fig. 3.2(c)). The coordinates of thecolumn’s centroid are xO = –10.59 in. and yO = 0. Theparameters Jx and Jy of the critical section at d/2 fromcolumn faces, determined by Eq. (C-4) and (C-5), are Jx =

27,960 in.4 and Jy = 6990 in.4 The moments MOy and Myabout axes through O and through the centroid of the criticalsection are related by

My = MOy + VuxO

where xO = –7.48 in. = x coordinate of O, the centroid of thecolumn.

The values of VD, VL, (My)D, and (My)L are VD = 12.7 kips;VL = 5.3 kips; (My)D = (Mx)D = 193 in.-kip; and (My)L =(Mx)L = 80 in.-kip

The unbalanced moment caused by seismic forces isobtained from an elastic analysis of the frame shown inFig. 6.1(b) subjected to lateral displacement δe. Theprescribed elastic story drift δe is ±0.75 in. as for the interiorcolumn in Example 1. For the frame shown in Fig. 6.1(b), Is= 2770 in.4 and Ic = 1150 in.4 The moment of inertia of theslab is one-half that of a gross slab strip with a width = 10 ft +c/2 = 130 in. The moment of inertia of the column is calcu-lated according to the equivalent frame method of ACI 318by considering flexibility of an equivalent column combinedwith a torsional strip.

Elastic analysis for the frame in Fig. 6.1(b) for δe = ±0.75 in.gives

(My)E = ±783 in.-kip; (Mx)E = +52 in.-kip; (Vu)E = ±7.0 kips

Using the load factors of ACI 318, consider:

Case I: 1.2D + 1.0L + 1.0E;

Case II: 0.9D + 1.0E;

Muy( )Case I 1.2 193( ) 1.0 80( ) 1.0 783( )+ + 1095 in.-kip= =

Mux( )Case I 1.2 193( ) 1.0 80( ) 1.0 52–( )+ + 260 in.-kip= =⎩⎨⎧

Muy( )Case II 0.9 193( ) 1.0 783–( )+ 609– in.-kip= =

Mux( )Case II 0.9 193( ) 1.0 52( )+ 226 in.-kip = =⎩⎨⎧


In the directions of the principal axes (Fig. 3.2(c)), the staticalequivalents to Mx combined with My are given by

Mx = Mxcosθ – Mysinθ; My = Mxsinθ + Mycosθ

With θ = 45 degrees:

Case I: Mux = –590 in.-kip; Muy = 958 in.-kip

Case II: Mux = 590 in.-kip; Muy = –271 in.-kip

The probable flexural strengths provided by the top andbottom flexural reinforcements crossing the projection of theshear-critical section on the principal y-axis (Fig. 3.2(c)) are608 and 415 in.-kip, respectively. For this corner connection,γvy = 0.27; βr = 0.5; and αm = 0.31 and 0.30 for negative andpositive moment resistances, respectively (Eq. (6-4)). Itfollows that the upper limits of |Muy| are (Eq. (6-2))

Upper limit |Muy|Case I = = 1960 in.-kip > 958 in.-kip

Upper limit |Muy|Case II = = 1380 in.-kip > 271 in.-kip

Similar calculation for upper limits for Mux indicates thatthey do not govern the design. Therefore, the connectionshould be designed to resist:

(a) Case I: Vu = 1.2(12.7) + 1.0(5.3) + 1.0(7.0) = 27.5 kips,combined with Muy = 958 in.-kip and Mux = –590 in.-kip;and

(b) Case II: Vu = 0.9(12.7) + 1.0(–7.0) = 4.4 kips,combined with Muy = –271 in.-kip and Mux = 590 in.-kip

The shear stress at any point (x,y) on the shear-criticalsection is

vu = (D-5)

The projections of the critical section on the x- and y-axes arelx = 16.48 in. and ly = 32.97 in. The fractions of unbalancedmoments transferred by shear are (Eq. (C-15) and (C-16))

= 0.267; γvx = 0.4

The maximum shear stress at Point A (–8.24 in., 16.48 in.[Fig. 3.2(c)]) occurs in Case I. Its value is

(vu)A =

= –0.352 ksi

6080.31----------

4150.30----------

Vu

bod-------- γvx

Mux

Jx

---------y γvyMuy

Jy

---------x+ +

γvy 1 1

1 23---

lx

ly

---- 0.2–+

----------------------------------–=

27.546.625 6.625( )----------------------------------- 0.4 590–( )

27,960------------------------- 16.48( ) 0.267 958( )

6990--------------------------- 8.24–( )+ +

The maximum shear stress at Point B (8.24 in., 0 in.)occurs also in Case I. Its value is

(vu)B = = 0.391 ksi

The maximum factored shear stress at the critical section atd/2 from the column face is vu = 391 psi (at Point B, Fig. 3.2(c),Case I); (vu /φ) = 8.2 psi (0.69 MPa). The SSR,shown in Fig. D.3, is designed in accordance with ACI421.1R. For the chosen shear reinforcement, Av = 1.18 in.2

(six studs); fyt = 50 ksi; and s = 2.875 in. The nominal shearstrength (expressed in stress units) provided by shear reinforce-ment (Eq. (4-1)) is vs = (1.18 × 50,000)/(46.625 × 2.875) =441 psi, which is greater than the minimum of 3 = 190 psi.The distance between the column face and the outermostperipheral line of SSR = 25.625 in., which is greater than3.5d = 23.2 in. Thus, details of stud arrangement, shown inFig. D.3, satisfy the requirement for the minimum amount ofshear reinforcement.

The nominal shear strength vn multiplied by φ must exceedvu = 391 psi. In this example

vn = vs + vc = 441 + 1.5 = 536 psi

Thus, φvn = 0.75(536) = 402 psi, which is greater than vu= 391 psi.

In each of Examples 1, 2, and 3, the shear reinforcement hascapacity equal to or exceeding the minimum requirement, theexcess is not large enough to allow the use of fewer studs.Using the properties of the shear-critical section at d/2 fromthe outermost studs (Fig. D.1 to D.3), it can be verified byEq. (D-5) that the maximum shear stress does not exceed2φ = 95 psi.

27.546.625 6.625( )----------------------------------- 0.4 590–( )

27,960------------------------- 0( ) 0.267 958( )

6990--------------------------- 8.24( )+ +

fc′ fc′

fc′

4000

fc′

D.5—Example 4: Use of stirrups—Interior flatplate-column connection

Repeat the design of the interior flat plate-column connectionof Example 1, with DRu = 0.005. Assume that cx = cy = 12 in.and Vu = 78 kips. The remaining data are as in Example 1.An elastic analysis of the frame in Fig. 6.1(a) for δe =0.005[lc /(Cd /IE)] = 0.19 in. gives (My)E = 243 in.-kip, usingIs = 5120 in.4 and Iec = 650 in.4 (equivalent second momentof area calculated by combining the column with a torsionalstrip, in accordance with ACI 318-08, Section 13.7.5).

Assuming that the diameter of the stirrups is 3/8 in. andmaintaining the 3/4 in. cover above the stirrups, the effectivedepth of the flexural reinforcement (having a diameter of5/8 in.) is: d = 8 – 3/4 – 5/8 – 3/8 = 6.25 in. The properties ofthe shear-critical section at d/2 from column face are: bo= 73 in.; Jy = 25,330 in.4; and γvy = 0.4.

φVc = φ(4λ bod)

φVc = 0.75[4(1.0) (73)(6.25)]/1000 = 87 kips

fc′

4000


Because Vu is greater than 0.4φVc = 87 × 0.4 = 35 kips, shearreinforcement is required. The maximum shear stress is

= 0.206 ksi = 206 psi

4 < (vu/φ) = 4.3 psi < 6

Shear reinforcement is required and stirrups or SSR can beused; choose stirrups.

With the choice and the arrangement of stirrups shown inFig. D.4, Av = 0.88 in.2. The nominal shear strength providedby the stirrups is

= 231 psi

vn = vs + 1.5λ = 326 psi

The distance between the column face and the outer-most peripheral line of stirrups is 24.5 in. > 3.5d =3.5(6.25) = 21.9 in.

vuVu

bod--------

γvyMuyx

Jy

-------------------+=

vu78

73 6.25×---------------------- 0.4 243 18.25 2⁄( )××

25,330------------------------------------------------------+=

fc′ fc′ fc′

vsAv fyt

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

vs0.88( ) 60,000( )

73 3.125( )-------------------------------------=

fc′

Fig. D.4—Arrangement of stirrups in interior flat plate-column connection of Example 4.
D.6—Example 5: Interior flat plate-column
connection of Example 1, repeated using SI unitsThe value of (Vu)D+L+E , given as follows, corresponds to

the load combination that governs the design. The load

factors used to calculate (Vu)D+L+E are in accordance withACI 318. Also given are the values of (My)D and (My)L dueto nonfactored dead and live loads. The given values are(Vu)D+L+E = 347 kN, (My)D = 0.0, and (My)L = 0.0. Thecross-sectional dimensions of the column are (Fig. D.1) cx =cy = 406 mm; the reinforcement ratios of the top and bottomflexural bars are 1.0 and 0.5%, respectively.

The critical section shown in Fig. 3.2(a) has bo = 2.299 mand Vc (Eq. (3-3)) = 0.33(2.299)(0.168) (103) = 675kN; and Vu/φVc = 347/(0.75 × 675) = 0.69.

The unbalanced moment caused by seismic forces isobtained from an elastic analysis of the frame, shown inFig. 6.1(a). The prescribed elastic story drift δe is given byEq. (6-1)

= 19 mm

where (Cd/IE) = 3.85 according to the ASCE/SEI 7(assuming that the lateral-force-resisting system consists ofshear walls). For the frame shown in Fig. 6.1(a), the momentof inertia of the slab Is = 2.13 × 10–3 m4. This value is equalto one-half the gross moment of inertia of a 203 mm slab that is6.1 m wide. The reduction is made to reflect some of theeffects of cracking on the force distribution. The equivalentmoment of inertia of the column is Iec = 0.549 × 10–3 m4.This value is obtained according to the equivalent framemethod of ACI 318 by assuming that the flexibility of theequivalent column equals the sum of the flexibility of theactual column and the torsional strip of the slab. In the deter-mination of the equivalent moment of inertia, the columnlengths were taken as 1.83 m (Fig. 6.1(a)) with 102 mm longrigid parts at the joints and with the other ends simplysupported, as shown in Fig. 6.1(c). The equivalent momentof inertia is computed as Iec = Ic(Kec/ΣKc), where Kec and Kc areend rotational stiffnesses (moment per unit rotation) of equiv-alent column and the actual column, respectively.

The unbalanced moment due to δe , obtained from elasticanalysis of the frame in Fig. 6.1(a), is (My)E = 189 kN·m.According to ACI 318-08, Section 9.2, the factored unbalancedmoment is

Muy = 1.2(My)D + 1.0(My)L + 1.0(My)E (D-1)

Muy = 1.2(0) + 1.0(0) + 1.0(189) = 189 kN·m

The flexural strengths provided by the top and bottomflexural reinforcement, based on a stress of 1.25 fy, of criticalsection Sides BC and AD (Fig. 3.2(a)) are 74.6 and 39.5 kN·m,respectively. Thus, Mpr = 74.6 + 39.5 = 114.1 kN·m. For thisinterior connection, γvy = 0.4; βr = 1.0; and αm = 0.4 (Eq. (6-3)).It follows that the upper limit of Muy is (Eq. (6-2))

(Muy)upper limit = = 285.3 kN·m

27.6

δeDRulc

Cd/IE( )------------------ 0.02 3.66×

3.85---------------------------= =

74.6 39.5+0.4

---------------------------


Because the upper limit is higher than Muy = 189 kN·m, theflat plate-column connection should be designed to resist Vu= 347 kN and Muy = 189 kN·m. In this structure, the drift dueto seismic forces does not change Vu for this interior column.

The maximum factored shear stress due to Vu and Muy(computed according to the eccentric shear stress model ofACI 318) at the critical section at d/2 from the column face(Fig. 3.2(a)) is vu = 1.92 MPa and (vu/φ) = 0.49 MPa.This value is less than 0.5 MPa; therefore, stirrups orSSR can be used. The SSR shown in Fig. D.1 is designedaccording to ACI 421.1R. For the chosen shear reinforce-ment, Av = 1520 mm2 (12 studs of diameter 12.7 mm); and s= 117 mm, the nominal shear strength (expressed in stressunits) provided by shear reinforcement (Eq. (4-1)) is vs =[1520(345)]/[2299(117)] = 1.94 MPa, which is greater thanthe minimum 0.25 = 1.31 MPa. The distance betweenthe column face and the outermost peripheral line of SSR is654 mm, which is greater than 3.5d = 589 mm. Therefore, thedetails of the stud arrangement shown in Fig. D.1 satisfy therequirement for the minimum amount of shear reinforce-ment. ACI 421.1R recommends that so ≤ 0.4d; the value usedin this example and Examples 2, 3, and 4 is so = 67 mm. Thenominal shear strength vn multiplied by φ should exceed vu= 1.92 MPa. In this example

vn = vs + vc = 1.94 + 0.125 = 2.60 MPa

Thus, φvn = 0.75(2.60) = 1.95 MPa, which is greater than1.92 MPa. The maximum shear stress at the critical sectionat d/2 from the outermost peripheral line of studs is

vu = 0.55 MPa < 0.167φ = 0.66 MPa


fc′fc′

fc′

27.6

fc′

D.7—Post-tensioned flat plate structureThe design of earthquake-resistant post-tensioned flat

plate-column connections for punching shear, according tothe recommendations of this guide, is illustrated by thefollowing numerical examples. Consider a post-tensionedfloor following the generic geometric arrangement in Fig. 8.1,
having: lx × ly = 26.25 × 26.25 ft2 (8.000 × 8.000 m2) slab
thickness; h = 7.87 in. (200 mm) interior columns’ size, cx =19.7 in. (500 mm), cy = 11.8 in. (300 mm); square edgecolumns with side dimension = 11.8 in. (300 mm); and storyheight lc = 11.48 ft (3.500 m). The earthquake-excitedmotion is in the x-direction, where the structure has fivebays, with many bays in the y-direction. The structure has alateral-force-resisting system that limits the interstory driftratio, including inelastic deformations, to DRu = 0.015;assume that the ratio is Cd /IE = DRu lc /δe = 4.0. The entirestructure is assumed to be of normal-density concrete, wc =153 lb/ft3 (24 kN/m3), with a specified compressive strengthof fc′ = 4350 psi (30 MPa) (Ec = 3580 ksi [24.7 GPa]); andconcrete cover = 0.790 in. (20.0 mm). Non-prestressed No. 5reinforcement is of nominal diameter = 0.625 in. (16 mm)

and yield strength fy = 60 ksi (414 MPa); the reinforcementratios of the top and bottom non-prestressed flexural bars are0.7 and 0.5%, respectively. Post-tensioned reinforcement is0.6 in., seven-wire strands having the following properties:area of tendon Ap = 0.217 in.2 (140 mm2); ultimate strengthfpu = 270 ksi (1861 MPa); yield strength fpy = 243 ksi(1675 MPa); effective post-tensioning force per tendon =35.1 kips (156 kN); post-tensioning is required to balance noless than 85% of the total dead load (that is, κ ≥ 0.85, Section 8.4).The floor is designed for a superimposed dead load, wsd =27 lb/ft2 (1.3 kPa) and a service live load, wL = 50 lb/ft2 (2.4 kPa).If shear reinforcement is required, use headed SSR satisfyingASTM A1044/A1044M-05 with nominal diameter = 1/2 in.and fyt = 50 ksi (345 MPa). The average of distances fromextreme compression fiber to the centroid of the tension non-prestressed reinforcement positioned in two orthogonaldirections is d = 7.87 – 0.79 – 0.625 = 6.46 in. (164 mm). Itis required to design for punching shear at the first interiorcolumn (B-2) and at the edge column (B-1) (Fig. 8.1;Sections D.8 and D.9, respectively). In calculating d,

consider only the non-prestressed reinforcement. At theshear-critical sections, the tendons can be far away from theslab surfaces.

Elastic analysis of the equivalent frame in Fig. 8.4

subjected to service dead and live loads gives shearing forcesand unbalanced moments

At the first interior column (B-2)

VD = 97 kips; VL = 38 kips; (My)D = –265 in.-kip;(My)L = –104 in.-kip.

At the edge column (B-1)

VD = 36 kips; VL = 14 kips; (MOy)D = 523 in.-kip;(MOy)L = 206 in.-kip.

where MOy is unbalanced moment about an axis through O,the column’s centroid, parallel to the y-axis in Fig. 3.2(b);positive MOy is in the same direction as positive Muy. Froman elastic analysis of the equivalent frames in Fig. 6.1(a) and(b), subjected to an elastic imposed lateral displacement δe =DRulc /(Cd/IE) = 0.015(11.48 × 12)/4 = 0.512 in. (13.0 mm),the shearing force and unbalanced moment are


VE = ±0.382 kips and (My)E = ±1117 in.-kip


VE = ±3.272 kips and (MOy)E = ±492 in.-kip

For the banded tendons, substitution of κ = 0.85; α = 0.1;wD = wch + wsd = 127 lb/ft2 (6.1 kPa); l = 26.25 ft (8.00 m);h = 7.87 in. (200 mm); and hc = 4.33 in. (110 mm) in Eq. (8-5)gives (fpc)required = (Pe)required/h = 219 psi (1.51 MPa). Use


D.8—Example 6: Post-tensioned flat plate connection with interior column

The design steps, outlined in Section 7.2 (Chapter 7), arefollowed here for the interior column-flat plate connectionB-2 (Fig. 8.1). The load factors used to calculate Vu and Muyare in accordance with Sections 9.2.1 and 18.10.3 of ACI318-08. Consider two cases varying the sign of VE and ME

Case I: (1.2D + 0.5L + 1.0E + 1.0 Secondary effect of post-tensioning)

Vu = 1.2VD + 0.5VL + VE + Vp′′ = 1.2(97) + 0.5(38) + (–0.382) + (–0.171) = 134 kips (598 kN)

Muy = 1.2(My)D + 0.5(My)L + (My)E + Mp′′ = 1.2(–265) + 0.5(–104) + (–1117) + 4.83 = –1482 in.-kip (–167 kN·m)

Case II: (0.9D + 1.0E + 1.0 Secondary effect of post-tensioning)

Vu = 0.9VD + 0.5VE + Vp′′ = 0.9(97) + 0.382 – 0.171 = 87 kips (388 kN)

Muy = 0.9(My)D + (My)E + Mp′′ = 0.9(–265) + 1117 + 4.83 = 833 in.-kip (100 kN·m)

The shear-critical section at d/2 from the column face hasthe following properties: bo = 88.8 in. (2260 mm); lx = 26.1 in.(664 mm); ly = 18.3 in. (464 mm); γvy = 0.444 (Eq. (C-12));Jy = 59.5 × 103 in.4 (24.8 × 109 mm4) (Eq. (C-5)). The threeconditions warranting the use of Eq. (8-6) are satisfied. Twotendons in each of the x-direction and the y-directionintercept the shear-critical section at d/2 from the column(Fig. D.5(a)); the sum of the vertical components of these

Table D.1—Parameters defining profile of the banded tendons and the corresponding set of self-equilibrating forces exerted on a unit width of floor (Fig. 8.3)

Units c1 c2 c4 c5 = c9 c6 = c8

in.(mm)

2.17(55.0)

1.43(36.3)

3.56(90.3)

4.33(110)

3.47(88.0)

Units α12 α23 α34 α45 = α56 α67 = α78

– 0.150 0.291 0.459 0.100 0.400

Units w12 w24 w45 w56 = w89 w68

lb/ft2

(kPa)–169.3

(–8.107)87.27

(4.179)–400.6

(–19.185)–447.9

(–21.450)112.0

(5.363)

tendons at the location of the shear-critical section Vp = 4.7kips (21 kN). Ignoring Vp (refer to Section 8.5), Eq. (8-6)gives: vc = 306 psi (2.11 MPa) and Vc = 175 kips (780 kN).The values of Vu/(φVc) for Cases I and II are: 1.02 and 0.66,respectively. Case II with (My)E = –1117 in.-kip would result inMuy = –1351 in.-kip, which is smaller than the value of Case I.

The upper limit for Mu (Section 8.5) is greater than Muy inCase I and II. Thus, the punching shear design will proceedwith the values of Vu and Muy determined above.

The maximum factored shear stress vu at d/2 from thecolumn face, corresponding to Case I and II, is (Eq. (C-6))

Case I: vu = 379 psi = 7.7φ (2.61 MPa = 0.64φ )

Case II: vu = 238 psi = 4.8φ (1.64 MPa = 0.40φ )

fc′ fc′

fc′ fc′

sixteen 0.6 in. seven-wire tendons per panel to provide effec-tive post-tensioning force = 16(35.1 × 103)/(26.25 × 12) = 1782lb/in. The average compressive stress provided = 1782/7.87= 227 psi (1.56 MPa). Similar calculations with hc = 3.62 in.(92.0 mm) will show that 19 distributed tendons, providinga confining stress of 269 psi (1.85 MPa), are sufficient tobalance approximately the same load. In the banded direction,hc = h minus the sum of the cover at top and bottom, one bardiameter of the outermost top and bottom flexural reinforcingbars and one diameter of the post-tensioning sheathing(assumed 0.71 in. [18 mm]), hc = 7.87 – 2 × 0.79 – 2 × 0.625– 0.71 = 4.33 in. (110 mm). The distance from the nearestslab surface to the tendons’ centroid at Points 3, 5, and 7(Fig. 8.3(a)) = 1.77 in. (45 mm). In the direction of thedistributed tendons, hc is smaller by one diameter of the post-tensioning sheathing (hc = 4.33 – 0.71 = 3.62 in. [92.0 mm]).The average in-plane compression produced by the effectivepost-tension forces in the two orthogonal directions, fpc = 247psi (1.71 MPa). The selected post-tensioning, (16 × 35.1 × 103)/(26.25 × 12) = 1782 lb/in. (312 kN/m), produces in the end spana smaller balancing load given by (Fig. 8.3(b))

= 85.6 lb/ft2 (4.1 kPa)

The post-tensioning tendons are banded in the x-directionand uniformly distributed in the y-direction (Fig. 8.1).Table D.1 gives values of the parameters defining thebanded tendons’ profile, and the forces exerted by the post-tensioning tendons on the floor (Fig. 8.3(a) and (b)).Application of the forces in Table D.1 and Fig. 8.3(b) to theequivalent frame in Fig. 8.4(a) results in a set of reactions, atthe supports, that is in self-equilibrium. The internal forcesof the equivalent frame (Fig. 8.4 subjected to the forces inTable D.1 and Fig. 8.3(b)) minus the primary internal forcesgive the secondary internal forces, Vp′′ and Mp′′ .


Vp′′ = –0.171 kips and Mp′′ = ±4.831 in.-kip


Vp′′ = 0.140 kips and MOp′′ = –217 in.-kip

In this example, it is seen that VE, Vp′′, and Mp′′ are ofinsignificant magnitudes.

w232Pec2

α23l( )2----------------- 2 1782( ) 1.43( )

0.291 26.52×( )2----------------------------------------= =


Fig. D.5—Arrangement of tendons and headed SSR at: (a) interior column-flat plate connection B-2 (Fig. 8.1; Example 6);and (b) edge column-flat plate connection B-1 (Fig. 8.1; Example 7).

(vu/φ) in Case I is greater than vc = 304 psi (2.10 MPa); thus,shear reinforcement is required; the same value is greaterthan 6 psi (0.5 MPa), thus, stirrups cannot be used.

The shear reinforcement in Fig. D.5(a) provides nominalshear strength (Eq. (7-1), Eq. (4-1), and Section 7.1)

This value exceeds vu = 379 psi (2.61 MPa), indicatingthat the choice of studs and their spacing is adequate. Outsidethe shear-reinforced zone, the shear-critical section has theproperties (Fig. D.5(a)): bo = 272.6 in. (6.923 m); Jy = 1.608× 106 in.4 (669.3 × 10–3 m4); γvy = 0.411; xmax = 46.34 in.(1177 mm); the maximum shear stresses are

Case I: vu = 94 psi = 1.90φ (0.648 MPa = 0.158φ )

Case II: vu = 60 psi = 1.21φ (0.414 MPa = 0.101φ )

The values of (vu/φ) are less than than 2 psi (0.17MPa), indicating that the extent of the shear-reinforced zoneis adequate. The studs are placed on three lines runningperpendicular to the long side of the column; with thespacing between the lines <2d, which is a requirement ofACI 318-08, Section 11.12.

fc′ fc′

φvn 0.75 1.5 4350 10 0.196( ) 50,000( )88.8 2.56( )

----------------------------------------------+ 399 psi (2.75 MPa)= =

fc′ fc′

fc′ fc′

fc′ fc′

D.9—Example 7: Post-tensioned flat plate connection with edge column

Two load combinations are considered

Case I: (1.2D + 0.5L + 1.0E + 1.0 Secondary effect of post-tensioning)

Vu = 1.2VD + 0.5VL + VE + Vp′′ = 1.2(36) + 0.5(14) + 3.272 + 0.140

= 54.0 kips (240 kN)

MuOy = 1.2(MOy)D + 0.5(MOy)L + (MOy)E + MOp′′

= 1.2(523) + 0.5(206) + 492 + (–217) = 1006 in.-kip (113.6 kN·m)

Case II: (0.9D + 1.0E + 1.0 Secondary effect of post-tensioning)

Vu = 0.9VD + VE + Vp′′ = 0.9(36) – 3.272 + 0.140 = 29.3 kips (131 kN)

MuOy = 0.9(MOy)D + (MOy)E + MOp′′ = 0.9(523) – 492 – 217

= –238.3 in.-kip (–26.97 kN·m)

The shear-critical section at d/2 from the column face hasthe following properties: bo = 48.3 in. (1230 mm); lx = 15.0 in.(382 mm); ly = 18.3 in. (464 mm); xO = –4.45 in. (–113 mm);xA = –10.4 in. (–263 mm); xB = 4.69 in. (119 mm); γvy =0.345 (Eq. (C-14)); Jy = 7.81 × 103 in.4 (3.25 × 109 mm4)(Eq. (C-5)). The first of the three conditions warranting theuse of Eq. (8-6) is not satisfied. Thus, the shear strength ofthe connection will be calculated as for reinforced concrete


APPENDIX E—CONVERSION FACTORS1 in. = 25.4 mm1 ft = 0.3048 m

1 kip = 4.448 kN1 in.-kip = 0.113 kN·m

1 psi = 6.89 kPa = 6.89 × 10–3 MPa, psi = /12, MPafc′ fc′

connections without prestressing; vc is governed by Eq. (3-3):vc = 4 = 265 psi (1.83 MPa). The unbalancedmoments at the centroid of the shear-critical section at d/2from the column, corresponding to the above two cases are(Eq. (D-2))

Case I: Muy = MuOy + VuxO = 1006 + 54(–4.45) = 765 in.-kip (86.4 kN·m)

Case II: Muy = MuOy + VuxO = –238.3 + 29.3(–4.45) = –370 in.-kip (–41.8 kN·m)

The absolute values of the probable flexural strengthsprovided by the top and bottom flexural reinforcement ofcritical section side BC (Fig. 3.2(b)), based on stresses 1.25fyand fps, respectively, for the non-post-tensioned and post-tensioned reinforcement, are: 681 and 613 in.-kip (77.0 and69.3 kN·m), respectively. For this edge connection, γvy =0.345, βr = 0.82, and αm = 0.254 for negative momentresistance and αm = 0.235 for positive moment resistance(Eq. (6-4)). The upper limits of |Muy| are (Eq. (6-2))

Upper limit |Muy | = 681/0.254 = 2681 in.-kip (303.0 kN·m)

(> 765 in.-kip [86.4 kN·m])

Upper limit |Muy | = 613/0.235 = 2609 in.-kip (294.8 kN·m)

(> 370 in.-kip [41.8 kN·m])

The upper limits are greater than Muy, calculated for Case Iand II. Thus, the punching shear design will proceed with Vuand Muy determined above for the two cases.

The maximum factored shear stress vu at d/2 from thecolumn face, corresponding to Case I and II, is (Eq. (C-6)and Fig. D.5(b))

Case I: (vu)BC = 331 psi (2.28 MPa)

Case II: (vu)A and D = 264 psi (1.82 MPa)

4350

The value of (vu/φ)BC in Case I is greater than vc = 265 psi(1.83 MPa); thus, shear reinforcement is required. The samevalue is greater than 6 psi (0.5 MPa); thus, stirrupscannot be used.

Using the arrangement of the headed SSR in Fig. D.5(b),it can be verified that the nominal shear strength vn = 576 psi(3.97 MPa); φvn = 432 psi (2.98 MPa), exceeds (vu)BC =331 psi (2.28 MPa), indicating that the choice of studs andtheir spacing are adequate.

Outside the shear-reinforced zone, the shear-criticalsection has the properties (Fig. D.5(b)): bo = 111.3 in.(2.826 m); Jy = 96.11 × 103 in.4 (40.00 × 10–3 m4); γvy =0.294; xE = –22.72 in. (–577 mm); xF = 15.35 in. (390 mm); xO= –16.81 in. (–427 mm). The unbalanced moments Muy (by Eq.(D-2)), combined with the corresponding Vu and themaximum shear stresses, are

Case I: Vu = 54 kips; Muy = 1006 + 54(–16.81)= 98.26 in.-kip

(vu)F = 80 psi = 1.61φ (0.55 MPa = 0.134φ )

Case II: Vu = 29.3 kips; Muy = –238.3 + 29.3(–16.81)= –730.8 in.-kip

(vu)E = 92 psi = 1.85φ (0.64 MPa = 0.153φ )

The values of (vu /φ) are less than 2 psi (0.17MPa), indicating that the extent of the shear-reinforced zoneis adequate.

fc′ fc′

fc′ fc′

fc′ fc′

fc′ fc′

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Guide to Seismic Design of Punching Shear Reinforcementin Flat Plates



guide to seismic design of punching shear reinforcement in flat … · 2019-10-03 · chapter...

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