-
ACI 421.2R-10
Reported by Joint ACI-ASCE Committee 421
Guide to Seismic Design ofPunching Shear Reinforcement
in Flat Plates
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Guide to Seismic Design of Punching Shear Reinforcementin Flat
Plates
First PrintingApril 2010
ISBN 978-0-87031-374-5
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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
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inwriting is obtained from the copyright proprietors.
421.2R-1
ACI Committee Reports, Guides, Manuals, and Commentariesare
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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.
Guide to Seismic Design of Punching Shear Reinforcement in Flat
Plates
Reported by Joint ACI-ASCE Committee 421
ACI 421.2R-10
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.
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
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
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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
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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 consideredin
design. Chapter 4 recommends a minimum amount ofshear reinforcement
for certain cases. Chapter 5 describesmeans of increasing the shear
strength of a flat plate-columnconnection and compares the
associated ductilities. Chapter 6presents 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,
providesrecommendations 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])
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421.2R-4 ACI COMMITTEE REPORT
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
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SEISMIC DESIGN OF PUNCHING SHEAR REINFORCEMENT IN FLAT PLATES
421.2R-5
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).
CHAPTER 3—LATERAL STORY DRIFT3.1—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
displacementof 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.
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.
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
Fig. 3.1—Effect of gravity loads on lateral drift capacity
ofinterior flat plate-column connections (Megally and Ghali1994,
2000d; Hueste and Wight 1999).
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421.2R-6 ACI COMMITTEE REPORT
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
Vc = (3-1)
Vc = (3-2)
Vc = (3-3)
SI units
Vc = (3-1)
Vc = (3-2)
2 4β---+⎝ ⎠
⎛ ⎞ bodλ fc′
αsdbo
--------- 2+⎝ ⎠⎛ ⎞ bodλ fc′
4bodλ fc′
1 2β---+⎝ ⎠
⎛ ⎞ bodλ fc′6
----------------------
αsdbo
--------- 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. 3.2—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.
Fig. 3.3—Effect of value of Vu on lateral drift capacity ofedge
flat plate-column connections (Megally and Ghali2000a).
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SEISMIC DESIGN OF PUNCHING SHEAR REINFORCEMENT IN FLAT PLATES
421.2R-7
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.
3.4—Design recommendations for flat plates with and without
shear reinforcement
As discussed in Section 3.1, the
lateral-force-resistingstructural 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)
dividedby 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.
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 shearstress due to Vu,
combined with that due to the unbalancedmoment 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 betweenperipheral
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 specifiedyield 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 theseexperiments.
The minimum shear reinforcement recommendedincreases 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.
vsAv fytbos
----------- 3 fc′≥=
vsAv fytbos
----------- 14--- fc′≥=
fc′ fc′
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421.2R-8 ACI COMMITTEE REPORT
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-deflectionresponses 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.
Fig. 4.1—Critical sections for punching shear outside
theshear-reinforced zone.
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SEISMIC DESIGN OF PUNCHING SHEAR REINFORCEMENT IN FLAT PLATES
421.2R-9
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 forthe
calculation of the value of Mu. An upper limit for Mu isgiven in
Section 6.3 and Appendix B.
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
(6-1)
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 andshears 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.
δu δeCdIE------⎝ ⎠
⎛ ⎞=
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).
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421.2R-10 ACI COMMITTEE REPORT
6.2—Simplified elastic analysisInstead of the analysis suggested
in Section 6.1, the
unbalanced 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.
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 Mu. Based on finiteelement 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
(6-2)
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
MuMprαm---------≤ the assumption that the force in the flexural
tensile reinforcement
is 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.
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SEISMIC DESIGN OF PUNCHING SHEAR REINFORCEMENT IN FLAT PLATES
421.2R-11
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.Equation (6-3)
applies for interior connections transferringMux 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 + ρysin
2θ (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.
βr20------⎝ ⎠
⎛ ⎞
βr40------⎝ ⎠
⎛ ⎞
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 finiteelement software
(ANATECH Consulting Engineers1995) and the finite element models
were verified by physicalexperiments (Megally and Ghali 2000b).
CHAPTER 7—DESIGN OF SHEAR REINFORCEMENT
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
vn = vc + vs (7-1)
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′
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421.2R-12 ACI COMMITTEE REPORT
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.
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, Mushould be calculated taking
into account the upper limitspecified in Section 6.3. Then the
equations found in ACI421.1R 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.
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} fallsbelow 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
fc′fc′
fc′fc′
Fig. 7.1—Steps for punching shear design of earthquake-resistant
flat plate-columnconnections.
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SEISMIC DESIGN OF PUNCHING SHEAR REINFORCEMENT IN FLAT PLATES
421.2R-13
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.
CHAPTER 8—POST-TENSIONED FLAT PLATES8.1—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 bypost-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 flatplate with an
interior and an edge column are presented inSections D.7 through
D.9.
Figure 8.1 shows a scheme for placing the post-tensioningtendons
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. 7.2—Requirement for shear reinforcement criterion(ACI
318).
Fig. 8.1—Banded-distributed tendons.
Fig. 8.2—Sign convention of: (a) Mprestress; (b) Vp; and
(c)tendon eccentricity and slope.
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421.2R-14 ACI COMMITTEE REPORT
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)
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.
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SEISMIC DESIGN OF PUNCHING SHEAR REINFORCEMENT IN FLAT PLATES
421.2R-15
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.
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 Pethat
balances a fraction κ of the service dead load per unitarea 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–( )2l2
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′ 0.3fpcVpbod--------+ +=
vc βpλfc′
12--------- 0.3fpc
Vpbod--------+ +=
(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.
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421.2R-16 ACI COMMITTEE REPORT
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.
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-
fc′
fps fse 10,000fc′
100ρp-------------- the least of:
fpy
fse 60,000 psi+⎩⎨⎧
≤+ +=
fps fse 10,000fc′
300ρp-------------- the least of:
fpy
fse 30,000 psi+⎩⎨⎧
≤+ +=
fps fse 70fc′
100ρp-------------- the least of:
fpy
fse 414 MPa+⎩⎨⎧
≤+ +=
fps fse 70fc′
300ρp-------------- the least of:
fpy
fse 207 MPa+⎩⎨⎧
≤+ +=
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SEISMIC DESIGN OF PUNCHING SHEAR REINFORCEMENT IN FLAT PLATES
421.2R-17
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—REFERENCES9.1—Referenced standards and reports
The standards and reports listed below were the latestedition 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
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APPENDIX A—VERIFICATION OF PROPOSED MINIMUM AMOUNT OF SHEAR
REINFORCEMENT
FOR EARTHQUAKE-RESISTANT FLAT PLATE-COL