a practical method to estimate diaphragm deformation of double tee diaphragm zhengoliva pca 419

8/12/2019 A Practical Method to Estimate Diaphragm Deformation of Double Tee Diaphragm Zhengoliva Pca 419

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Wei Zheng, Ph.D., P.E.Senior Structural EngineerFlad & AssociatesMadison, Wisconsin

Michael G. Oliva, Ph.D.ProfessorDepartment of Civil andEnvironmental EngineeringUniversity of WisconsinMadisonMadison, Wisconsin

The in-plane flexibility of untopped precast concrete double-teediaphragms is often ignored in current design practice. The performanceof parking structures during earthquakes and subsequent researchstudies have shown, however, that in-plane flexibility of diaphragmsmay contribute to the development of large displacements, a concernfor stability. The objective of this paper is to provide designers witha practical approach to judge the linear elastic in-plane flexibility ofdiscretely connected untopped double-tee diaphragms. Based on theresults of detailed finite element model analyses of commonly used

untopped diaphragms, a simplified rational approximation is used toestablish equivalent beam models. Methods of defining linear elasticstiffness parameters of the equivalent beam are derived and can bedirectly used in manual calculations or computer analysis. Comparedto a complex finite element analysis, the proposed equivalent beammodel can predict the linear elastic in-plane diaphragm deformationunder wind or seismic load with reasonable accuracy for design.A deflection calculation example is provided in Appendix B to illustratethe proposed approach.

A Practical Method to EstimateElastic Deformation of PrecastPretopped Double-Tee Diaphragms

2 PCI JOURNAL

Double tees are commonly

used to form floor and roof

diaphragms for parking struc-tures, commercial and industrial build-

ings, and other structures. Two types

of diaphragms are used. In some parts

of the United States, designers prefer

to use discretely-spaced mechanical

flange-to-flange connectors to join ad-

jacent pretopped double tees and cre-

ate untopped double-tee diaphragms.In other locations and particularly in

high seismic zones, however, 2 to 4 in.

(51 to 102 mm) thick reinforced cast-

in-place concrete slabs (topping) maybe overlaid on double tees and across

joints, or they may be combined with

mechanical connectors to create con-

nections between double tees and form

topped double-tee diaphragms.

Both double-tee floor and roof sys-

tems were developed primarily for car-

rying vertical gravity loads, but theyare also key elements in transferring


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lateral wind or earthquake forces tolateral load-resisting systems, such asshear walls. In parking structures, thedouble-tee diaphragms are divided byramps into one-bay diaphragm seg-ments. Even though the span-to-widthratio of individual double-tee dia-phragm segments often range from 3 to5, the in-plane flexibility of double-tee

diaphragms is often neglected in de-sign practice.

As a result of structural failures thatoccurred during the 1994 Northridgeearthquake, the response of parkingstructures to seismic events has beenthe subject of investigation. Observa-tions after the earthquake indicated thatthe damage to some parking structuresmay have been caused by gravity load-resisting systems failing due to largelateral displacements (drifts) of the floor

at locations away from shear walls.1-3

Shear walls were expected to resist

lateral loads. Separate column and in-verted tee framing, intended to onlyresist the gravity loads, made up thegravity load system. Large lateraldisplacement of the columns, unre-strained by stiff diaphragms, may haveled to instability under vertical load.

Subsequent analytical studies4,5 ontopped double-tee diaphragms have in-dicated that the flexural deformation of

individual diaphragm segments causedlateral displacements several timeslarger than the supporting shear wallstory drifts. The structural system usedto carry vertical (gravity) loads to theground is often not designed to be ableto sustain such large drifts.

In a recent analytical investiga-tion,6 the influence of elastic in-planeflexibility of untopped (or pretopped)double-tee diaphragms on the seismicbehavior of parking structures was also

addressed. That study revealed that:1. The dynamic response of

parking structures with anuntopped diaphragm may besubstantially different from thedynamic response based on arigid diaphragm assumption,due to more participation ofdiaphragm-driven vibrationmodes;

2. The diaphragm shear forcesdeveloped at stories varied with

the in-plane flexibility and mightbe larger than specified values

estimated from building code

methods; and3. Predictions of elastic diaphragm

deformation show lateral

displacement demands on thegravity loadbearing system

much larger than would be

expected considering only storydrift at supporting shear walls.

Two studies5,6have suggested usinga design procedure that recognizes thata parking structure floor is composed

of flexible diaphragm segments.

Some precast concrete designers

have argued that double-tee diaphragmsshould be designed to remain elastic

during seismic events. This argument

is very convincing when the conceptof inelasticity is considered. If yielding

is allowed to develop in a precast dia-

phragm, it is likely to be concentrated

within one or a few joints adjacent toshear walls since all joints are normal-

ly provided with the same connectionsand the same strength.

When the deformability of existing

mechanical connectors is examined,6

it becomes obvious that the demanddeveloped in the yielding joints would

exceed the limited connector capaci-

ties. Thus, maintaining elastic behav-ior in mechanically connected precast

diaphragms seems imperative. For

this design approach, the diaphragmshears obtained from model building

code estimation procedures need to be

increased by an overstrength or magni-fication factor since they are the design

yield values of the lateral load resisting

system, not the expected peak values.

This approach for an elastic designload is presented in the Fifth Edition of

the PCI Design Handbook.7 Still, the

elastic in-plane flexibility of double-tee diaphragms should remain a major

concern in design because of excessivestory drifts that may be imposed on

gravity load-resisting systems causing

instability.

The International Building Code

(IBC 2003)8 and its predecessor, the

Uniform Building Code (UBC 1997),9

clearly stipulate that diaphragms shall

not be considered rigid for the purpose

of distributing story shear and torsional

moment when the maximum lateral de-

formation of the diaphragm is more thantwo times the average story drift of the

associated story. The average story drift

is based on displacement of the load-

resisting system, such as shear walls.

Both codes also stipulate that the in-

plane deflection of the diaphragm shall

not exceed the permissible deflection of

the attached elements, such as columns.

The P-effects on such elements shall

be considered. Since there is no infor-

mation or analytical method suggested

in the codes for determining the in-plane flexibility of untopped double-tee

diaphragms, these code provisions are

often not satisfactorily followed.

This paper provides a simple method

that designers can use to check defor-

mations in untopped precast concrete

double-tee diaphragms. Existing infor-

mation on diaphragm components and

analysis is reviewed, and then the de-

velopment of a detailed finite element

model (FEM) for predicting diaphragm

behavior is briefly explained. The de-

velopment of a simple method, based

on the FEM, and using beam modeling

for the diaphragm, is shown with ex-

ample calculations of deformation in a

prototype structure.

OBJECTIVES AND SCOPE

This paper presents a simple method

for predicting elastic in-plane displace-

ments of untopped double-tee dia-

phragms, considering both flexural andshear deformation at joints between

Table 1. Comparison of predicted diaphragm deformations.

Accurate FEM

analysis*

FEM with

uniform connector

stiffness

Uniform

connector stiffness

and planar

behavior in joints

Diaphragm in-plane

deflection (in.)0.224 0.296 0.239

* Correct FEM analysis with different axial stiffness under tension and compression.

Approximate FEM analysis with uniform connector stiffness (tension value) in connectors. Model with uniform connector stiffness (tension value) and rigid flange edges.


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pretopped precast members as well as

within members.

First, behavioral simplifications

are proposed based on analytical re-

sults from detailed planar finite ele-

ment analysis of typical untopped

diaphragms. Then, with those assump-

tions, the elastic flexural deformation

and shear deformation in the pretopped

double tees and their flange-to-flangeconnections are smeared into deforma-

tions of an equivalent beam.

The elastic stiffness parameters of

the equivalent beam are derived and

can be easily used in manual calcu-

lations or computer analysis. Shear

deformation at the connection of the

diaphragm to the remaining lateral

load system is also considered and

smeared into the shear deformation of

the equivalent beam.

The proposed equivalent beam

model is intended to provide an eas-

ily calculated and reasonable estimate

of the elastic in-plane deformation of

an untopped diaphragm. The method

is suitable for the design of untopped

double-tee diaphragms to resist wind

and seismic loads.

BACKGROUND

The impact of in-plane flexibility infield-topped double-tee diaphragms on

seismic performance of parking struc-

tures has been analytically investigated

with complex finite element model-

ing at Lehigh University.5The model,

however, was based on flexural defor-

mation of topped diaphragms only and

did not include shear deformations of

double tees or shear sliding in the joints

between adjacent mechanically con-nected double tees.

In follow-up studies,10,11static analy-ses of detailed finite element modelswere conducted for both topped andpretopped mechanically connecteddouble-tee diaphragms. The analyticalmodel of the diaphragm, including wireand chord steel, was similar to that ad-

opted in the previous research, but me-chanical connectors were added usingnonlinear springs possessing both ten-sion (axial) and shear resistance. Fortopped diaphragms, the shear-frictioneffect due to the steel reinforcementcrossing the cracked concrete toppingsection at the joints was considered anddetermined by using empirical data.

In the pretopped diaphragm model,contact friction occurring at joints inregions of compression was included

by using friction-capable gap elements.The investigation proposed a practicalmethod to calculate elastic in-plane de-flection based on an equivalent elasticmodulus, determined by calibrating thedata with finite element analysis. Theequivalent elastic modulus was ob-tained for three types of connectors.

For other connectors with differentproperties, the equivalent elastic mod-ulus needs to be determined either byfurther calibrating with a finite element

model or by an interpolating method.For typical dry mechanical connec-tion systems preferred in pretoppeddiaphragms, assuming contact frictionat joints in the region of compressionmay unconservatively estimate a highdiaphragm stiffness.

Nakaki12 proposed another rationalmethod for calculating the deformationof diaphragms. The diaphragm elastic

flexural stiffness was derived basedon the cracked section property of amonolithic diaphragm, which is similarto a cracked beam model. For toppedand untopped diaphragms, with dis-crete cracks along precast panel joints,the elastic tension strength of the re-inforcement in cast-in-place toppingor the discrete connectors at joints is

converted into an equivalent web rein-forcement in a monolithic diaphragm.

The elastic modulus of the equiva-lent web reinforcement is proportionedso that the tension deformation of top-ping reinforcement, or discrete connec-tors at joints, can be simulated. A simi-lar treatment, however, is not includedfor chord steel reinforcing. Therefore,the tension deformation of chord steelnear the joints in a discrete cracked dia-phragm is not explicitly included in the

Nakaki model.For shear deformation, the Nakaki

method does not make a distinction be-tween discretely connected diaphragmsand monolithic diaphragms. It simplyassumes that the concrete shear mod-ulus, G, is equal to 0.4Eand uses thecracked monolithic diaphragm proper-ties to determine the diaphragm sheardeformation, regardless of the shearstiffness of discrete connectors acrossthe joints in an untopped diaphragm.

When the method is applied to anuntopped diaphragm, the stiffness con-tribution of the double-tee flange isomitted in the deflection calculation.Thus, for untopped diaphragms, thismethod does not recognize any benefi-cial effect on reducing the diaphragmdeflection by using thicker double-teeflanges or mechanical connectors withhigher shear stiffness. The Nakakimethod may not be appropriate for es-timating the deformation of untopped

diaphragms, especially for commondiaphragms with a span-to-width ratioof 3, in which the shear deformationcan be a significant source of the totaldiaphragm deformation.

In a recent analytical investigationon the impact of elastic behavior inuntopped double-tee diaphragms,6 adetailed two-dimensional finite ele-ment diaphragm model with discretemechanical connections betweendouble tees was developed. Planar fi-

nite elements were used to model theindividual double-tee members. The

Fig. 1. Segmentextracted from

a double-tee

diaphragm(plan view).


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March-April 2005 5

discrete mechanical connectors joiningthe double-tee flanges were represent-ed by truss elements (or springs) foraxial and shear behavior. The connec-tor properties were determined fromexperimental data.

The finite element model was usedto predict elastic in-plane deformationof a set of untopped double-tee dia-

phragms. It included both shear defor-mation in the double-tee members andshear sliding at joints between adjacentmembers.

Several experimental tests havebeen conducted on typical mechani-cal connectors which join double-teepanels.13-16The main focus of most ofthese experimental studies was on theconnector strength, although a few in-vestigators also examined stiffness.17One such study that addressed stiff-

ness in detail was a test program re-cently conducted at the University ofWisconsin-Madison.6,16

DIAPHRAGM FINITEELEMENT MODEL

The proposed simple beam modelis based on well-defined mechanicalconnector behavior measured in ex-perimental testing and on experiencederived from complex finite element

(FEM) analyses of a series of untoppeddiaphragms with discrete connections.The detailed FEM analysis was car-ried out using planar finite elements tomodel double-tee members and specialjoint elements with the measured dis-crete connector properties.

Tests have shown that the compres-sion stiffness of a mechanical connec-tor may be ten times the tension stiff-ness.6This high compression stiffnessand initial joint gap prevent the con-

crete flanges from ever coming directlyinto contact with one another duringlinear elastic response. This means thatonly the connector compression andshear stiffnesses needed to be modeledat the joints rather than adding contactelements having a friction-shear com-ponent as included in the Farrow andFleischman model.10,11

Using discrete elements to modelthe mechanical joint connectors poseda particular challenge. In a real flange,

the forces transferred through connec-tors are distributed into the flange con-

crete gradually through the connectorsanchor bars and create a complex three-dimensional stress field. Since the con-nector test results actually includedlocal deformations that occurred in thedouble-tee flange, the connector ele-ments in the model were joined to aseries of nodes in the region around theconnector location to avoid the concen-

trated load and local deformation prob-lem that might occur in the FEM.

When modeling different axial com-pression and axial tension stiffnessesof chords and mechanical connectors,analytical studies using the detailedfinite element model showed that theneutral axis of the diaphragm, with thediaphragm in flexure, was eccentricfrom the diaphragm center by about20 percent of the diaphragm depth. Anassumption that the compression stiff-

ness of connectors and chords at a jointis the same as the tension stiffness isadopted here to simplify calculations.It places the neutral axis at mid-depthand can overestimate rotation in thejoints between double tees, resulting inan overestimate of deflection.

If this incorrect assumption is used,calculated in-plane diaphragm deflec-tions are 20 to 30 percent larger thanthey should be. This error is compensat-ed for, however, when the model withequal tension and compression connec-tor stiffness is also assumed to have arigid flange edge (i.e., plane sectionassumption along joints). The error in

predicted diaphragm deformation wasreduced to 7 to 15 percent. The flexureresults from these three model varia-tions for one diaphragm configurationare listed in Table 1.

Results from the finite element mod-eling indicated that a series of model-ing assumptions might be acceptablefor a simplified beam model:

1. The axial compression stiffnessof the connectors and chord steelat joints could be assumed equalto their tension stiffness;

2. The axial force distribution toconnectors at a joint, due toflexure of the diaphragm, couldbe assumed to vary linearly(plane section assumption);

3. The shear force transferredacross the joint could beassumed to be uniformly

distributed to the connectorsat the joint because of the highflange stiffness; and

4. The total deflection of thediaphragm could be assumedas the sum of deflections ofa monolithic diaphragm withsection equal to the double-tee flanges plus additionaldeformation caused by shear androtation in the joints.

The assumptions reached on thebasis of the finite element results ap-pear to relate well to the physical be-havior except for the first assumption.Considering the high transverse stiff-

Fig. 2 Comparison ofin-plane deflectionbetween equivalentbeam and direct

calculation. Note: 1in. = 25.4 mm.

Fig. 3. Assumed

stress in embeddedsteel reinforcing bar.


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6 PCI JOURNAL

ness that a precast concrete double tee

possesses and the low stiffness of the

mechanical connectors, the second, orrigid edge assumption appears logical.

A similar argument can be made for

the third assumption; namely, the tee

is nearly rigid compared to the shear

stiffness of mechanical connectors.

The fourth assumption uses superposi-

tion as commonly accepted for linearelastic behavior.

The one assumption that may bequestionable is the first. The neutral

axis of the joint will not be located

at mid-depth under in-plane flexure,

as was observed in the accurate FEM

model, since the connectors have a

higher compression axial stiffness than

tension. When considering forces in-

side the uncracked double-tee flange,however, the neutral axis would be as-

sumed to be at mid-depth because of ac-tual uniform stiffness across the depth

(during linear elastic condition). There

is a discontinuity between double-tee

members and the joints. This may be

considered a disturbed region mak-

ing selection of a simple approximateanalysis method difficult.

The use of a detailed FEM analy-

sis, with appropriate connector axial

stiffnesses, provided a set of initial as-

sumptions for a simplified beam model.

A comparison with the detailed FEManalysis is used again later to judge the

error and acceptability of the proposedbeam model.

EQUIVALENT BEAM MODEL

Consider that a segment consisting of

a double-tee member and one joint con-

nection at the right edge of the member

is extracted from a complete double-

tee diaphragm system (see Fig. 1). The

connectors that would be on the leftside of the member would be included

as part of an adjacent segment. Based

on elementary beam theory,18with an

in-plane bending moment M assumed

constant over the segment, the flexural

rotation, 1, occurring in the double-tee

panel is:

1=Mb

EcI (1)

where

b = width of double-tee flange(see Fig. 1)

Ec = elastic modulus for concreteof double tee

I = in-plane transverse momentof inertia of double-tee flangesection (webs ignored)

The flexural rotation occurring atthe connection joint can be determinedusing a plane section assumption at theflange edge. Suppose a rotation 2oc-

curs at the joint between two doubletees. Then, from equilibrium of forcesin the axial direction of the connectorsacross the joint:

i

Kai (y ri)2=0 (2)

whereKai = axial stiffness of ith

connector or chord elementri = distance from ith connector or

chord to top end of double teey = distance from neutral axis to

top end of double teeSince 2 0, Eq. (2) can be rear-

ranged as:

i

Kai (y ri)=0 (3)

Assuming that the compression stiff-ness of connector and chord, respec-tively, is the same as their tension stiff-ness, the neutral axis will be located atthe center of the section; thus, y = d/2,where d is the depth of the diaphragm

(see Fig. 1).From equilibrium in flexure:

i

Kai (y ri)22=M (4)

Rearranging Eq. (4) results in thefollowing equation:

2=M

i

Kai (y ri)2

(5)

Thus, the in-plane flexural stiffnessof a joint between double tees, denotedas , can be expressed as:

= i

Kai (y ri)2 (6)

The parameter given in this formcan be used in general for any dia-phragm. Precasters often place connec-tors closer together near the mid-regionof a double tee. Close spacing near themidspan will not be effective in resist-ing flexure as witnessed by the squaredterm in the stiffness factor.

If connector spacing is uniform and

the same type of connector is used atall locations, the calculation of in

Eq. (6) can be further simplified byassuming the total axial stiffness ofthe connectors to be spread uniformlyalong the connection joint. The Kaiterms in Eq. (6) are replaced with Ktand Kc for the flange connectors andchord connectors, respectively. An ad-ditional axial stiffness is present whena chord exists, in excess of the normal

flange connector stiffness Kt. This isKc Ktwhen no connector is placed atthe chord location.

Eq. (6) becomes:

= (Kc Kt)d2

2+ nKt

d2

12 (7)

where = 1 when a mechanical

connector is not placed atthe diaphragm edge (normalcase), or 0 when a connectoris placed at the chord location

Kc = tension stiffness of chordconnection

Kt = tension stiffness of eachflange connector

n = number of flange connections(including chord connections)

Therefore, the total flexural rotationover the segment, including a doubletee and a connection joint, is:

=1+ 2=Mb

EcI+

M

(8)

The first term, 1, represents thedeformation in the double-tee flangesdue to flexure and might be neglectedwhen an axially soft connector is usedand the flange is relatively rigid. Then,the flange would be assumed as a rigidbody and all of the flexural deforma-tion of the diaphragm would arise fromjoint deformation. The entire equationwill be continued here.

Under a shear force, V, over the seg-ment as shown in Fig. 1(b), the shear

distribution among the connectors isassumed to be uniform. The in-planeshear deflection occurring at the jointcan be expressed as:

1=V

(9)

whereKvi= shear stiffness of ith

connector = total shear stiffness of

connectors at a joint, i

Kvi

If all the connectors are identical,then v = (n 2)(Kv) when no me-


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March-April 2005 7

chanical connectors are placed at thediaphragm edges, or v = nKv whenconnectors are placed at the chord lo-cations. Closer spacing of connectorsnear the midspan of double tees willsimply add to the shear stiffness.

Note that the shear stiffness of chordconnectors is ignored here (conserva-tive) since little information is avail-

able to accurately model their stiffnesseven though the strength contributioncan be estimated from shear-friction.A shear term for the chords, similar tothat in Eq. (7), could be added if theshear stiffness of chords was known.

The shear deformation occurring inthe double-tee panel can be expressedas:

2=1.2Vb

AGc (10)

where

A = cross-sectional area ofa double-tee flange inlongitudinal section

Gc = shear modulus of concrete =Ec/[2(1 + )]

= Poissons ratio for concrete(usually taken as 0.17 to 0.3)

Thus, the total shear deformationin the segment including a double-teepanel and a connection joint is:

=1+2=

1.2Vb

AGc +

V

(11)

where1 = shear deformation of flange2 = shear deformation in joint = total shear deformation of

flange and jointConsider that a segment with a length

b was extracted from an equivalentbeam model of the diaphragm. Basedon elementary beam theory, the flex-ural rotation over the segment of theequivalent beam model is:

=Mb

EI (12)

whereE = elastic modulus of equivalent

beamI = moment of inertia of

equivalent beam, in whichthe moment is consideredconstant over the length ofthe segment

The shear deformation over the

same segment of the equivalent beammodel is:

=1.2Vb

AG (13)

whereA = cross-sectional area of

equivalent beamG = E/[2(1 + )] = shear

modulus of equivalent beam = Poissons ratio of material in

equivalent beam, in which

the shear is consideredconstant over the length ofthe segment

To reach deformation equivalencebetween the equivalent beam modeland the diaphragm, the segment of theequivalent beam model and the seg-ment of the double-tee diaphragm musthave the same deformation, i.e., = and=. Thus:

Mb

EI=

Mb

Ec

I+

M

(14)

1.2Vb

GA=

1.2Vb

GcA+

V

(15)

Rearranging Eqs. (14) and (15) andtaking the thickness and Poissons ratioof the equivalent beam model as thoseof the actual double-tee flange yieldsEqs. (16) and (17):

d= d

1+GcA

1.2b

1+

EcI

b

(16)

where dis the depth of the equivalentbeam, and:

E=Ec

d

d

1+G

cA

1.2b

(17)

These parameters define the equiva-lent beam model of the double-tee dia-phragm.

The shear deflection from the equiv-

alent beam is smaller than that froman accurate calculation as shown inFig. 2. This is attributed to the missingdeformation in the left joint where thediaphragm is attached to a support (atlocation 0 in Fig. 2). Since the equiv-alent beam properties were calculatedfor the segment in Fig. 1, the deforma-tion on the left end of the diaphragmis missing. To reach the correct deflec-tion at midspan, the shear sliding at theend support should be smeared into the

shear deformation occurring within theequivalent beam model.

A modification of the shear modu-lus of the equivalent beam model ismade and a new equivalent depth dand modulusEare given in Eqs. (18)and (19):

d= d

1+ kGcA

1.2b

1+EcI

b

(18)

E=E

d

d

1+ kG

cA

1.2b

(19)

wherek = a joint modification factork = (m+ 2)/mwith an even

number of double teesk = (m+ 2)2/m2with an odd

number of double teesm = number of double teesin diaphragm span beingconsidered

The in-plane deflection of untoppeddiaphragms can be estimated using theequivalent beam either by manual cal-culation or structural analysis software.For instance, the midspan deflection ofa one-span simply supported untoppeddiaphragm can be calculated using thedeflection formula from elementary

beam theory.

18

The properties of the section, such asA and I, are calculated assuming thatthe beam section is equal to the lon-gitudinal cross section of the double-tee flange with the equivalent depth drather than the actual depth. For morecomplicated support conditions, struc-tural analysis software can be usedwith theEand d.

METHOD TO ESTIMATE

TENSION STIFFNESS OFCHORD STEEL

Diaphragm chord members can actu-ally be created by casting a raised sec-tion of concrete, a pour strip or curb, ontop of the double-tee diaphragm alongtwo longitudinal edges. Chord steel re-inforcement is embedded in this pourstrip. The deformation behavior ofchord steel members at joints has notbeen specifically tested in full-scale

diaphragm segments. The behavior,however, might be reliably predicted


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8 PCI JOURNAL

from the engineering properties of thechord steel.

The following proposed method

can be used to estimate a conservativechord steel linear elastic tension stiff-ness in the absence of test data.

The force in the chord steel at thejoints will be developed through bondwith the flange concrete adjacent to

joints. Thus, the tensile stress in thechord steel will vary from a lowervalue within the double-tee flanges

to a higher value or its yield stress atthe joints. The development length forthe embedded reinforcement to de-velop its yield strength is defined in

ACI 318-02.19 The stress distributionin embedded reinforcement, at highstrains, has been determined by pull-

out tests.20,21 Based on these pull-outtest results, the stress distribution in

the chord steel on both sides of a jointis simplified as a parabolic curve (seeFig. 3).

When the steel is at a stress substan-

tially lower than the yield level, andthe concrete surrounding the bar is ata strain below the cracking strain, the

stress variation in the reinforcing barcould be significantly different. It islikely that the variation would be theinverse of the parabola shown in Fig.

3, high where it enters the concrete but

rapidly dropping off. An average be-tween those two parabolic distributions

might be a linear variation, as indicatedby the dotted line in Fig. 3. A parabolicassumption will provide a low (conser-

vative) chord stiffness estimate.The linear elastic deformation of the

chord steel model between double-tee

flanges is controlled by the assumedstress distribution along the chord steelas it becomes anchored in a flange. Thedeformation can be determined by in-

tegrating the chord steel strain over thedevelopment length on both sides of

the joint and taking the chord deforma-tion as:

s= 2ld

0

dx

= 2ld

0

f

Es

dx

=21

Es

2

3fmaxld

=4fmaxld3Es

(20)

wherefmax= T/AsAs = cross-sectional area of chord

steelEs = elastic modulus of chord steelld = development length of chord

steel from ACIT = tension in chord steel

Then, the tension stiffness of chordsteel at the joint between double tees,Ks, can be taken as:

Ks=T

s=

AsEs4

3ld

=3

4AsEs

ld (21)

or, if a linear stress is assumed alongthe anchorage length:

Ks=AsE

ld (22)

where Ksis the chord stiffness.

EXAMPLES ANDDISCUSSION

To illustrate and evaluate the equiva-lent beam method, an example is given

for a one-span untopped diaphragmwith plan dimension of 120 60 ft (36.6

18.3 m). The diaphragm is attachedto one-story shear walls at both ends,

as shown in Fig. 4. In this example, thedouble tees are connected to adjacentpanels by nine discrete mechanical

connectors spaced at 6 ft (1.8 m).Four types of mechanical connec-

torsare considered, ranging from light-

duty type connectors such as a hairpinor bent wing connector, to heavy-dutytype connectors such as a structural-

tee connector. The characteristic shearbehavior of the bent wing connectoris shown in Fig. 5. This connector is

similar to a widely marketed commer-cial connector that has been rigorouslytested with test results published.22Thebehavior of the other connectors is de-

scribed elsewhere.6,16

Chords are cast in pour-strips alongboth longitudinal edges of the dia-

phragm with three No. 6 (19 mm) em-bedded steel reinforcing bars. The

thickness of the double-tee flange is4 in. (102 mm) if it is a floor and 2 in.(51 mm) if it is a roof. Concrete strengthis taken as 5000 psi (34.5 MPa). Five

diaphragm configurations, four asfloors, and one as a roof (see Table 2),are examined.

Drag beams are assumed to transfershear forces to the walls at either end ofthe diaphragm and deformations in thedrag beams should be included in nor-

mal calculations, but are excluded here

to focus on the diaphragm alone. Re-gardless of the diaphragm type, the total

horizontal lateral (in-plane) load on thediaphragm is assumed to be 123.5 kips(549 kN) from an IBC 20038 calcula-

tion (0.156 times the seismic weight ofthe diaphragm) uniformly distributedover the diaphragm area. The concrete

elastic modulus Ec= 4070 ksi (28063MPa) and Poissons ratio is taken as= 0.3. The steel modulus is 29,000 ksi(200000 MPa).

Table 2. Parameters for different diaphragm configurations in example.

Analysis

case

Type of

connector

Thickness

of panel, t

(in.)

Chord

steel

Stiffness of

connector

in shear, Kv

(kips/in.)*

Stiffness of

connector in

tension, Kt

(kips/in.)*

Tension yield

strength of

connector

(kips)

Stiffness of

connectors in

compression

(kips/in.)

Case 1 Bent wing 4 3 No. 6 300 16 3.5 3000

Case 2 Structural tee 4 3 No. 6 600 160 11.9 3000

Case 3 Bent wing 2 3 No. 6 300 16 3.5 1500

Case 4 Hairpin 4 3 No. 6 500 70 7.8 3000

Case 5 Stud-to-plate 4 3 No. 6 350 150 7.4 3000

*Connector tension stiffness and yield is based on test results presented in Reference 6.

Connector compression stiffness is based on test results presented in Reference 6.Note: 1 in. = 25.4 mm; 1 kip = 4.45 kN; No. 6 bar = 19 mm diameter.


8/12

March-April 2005 9

Calculation of the midspan dia-phragm deflection for one of the casesof this example is illustrated in a step-by-step procedure in Appendix B usingthe proposed equivalent beam model.The stiffness parameters of the equiva-lent beams are shown in Table 3 andare based on measured connector prop-erties.6In Table 4, the calculated mid-

span deflections are tabulated and com-pared with deflections from the Nakakimethod12 and more detailed planar fi-nite element model analysis.

The equivalent beam deflection pre-dictions are very close to the accuratefinite element model predictions. Ifthe double tees were considered rigid[the first term in Eq. (8) being zero]the displacements would be predictedas 7 to 10 percent smaller. The maxi-mum error between the beam model

and FEM, nearly 19 percent, occurs inCase 2 where the axially stiff structuraltee type connector is used.

Case 5, with the axially stiff stud-to-plate connector, has a similar error(17 percent). Fortunately, these connec-tors are not common for practical usebecause the high axial stiffness causesresistance to volume change in thestructure and is likely to result in long-term flange deterioration problems.

The error ranges from 7 to 14 percent

on the conservative (overestimate) sidein most of the other cases. The only un-derestimate of deflection occurs in Case3 due to the plane section assumptionat the joint. The thinner 2 in. (51 mm)flange of Case 3 develops more localflange deformation in the FEM analy-sis due to the discrete connector forceson the thin concrete.

The plane section assumption ignoresthis deformation and overestimates thediaphragm stiffness. The model devel-

oped by Farrow and Fleischman10,11

pre-dicted a lower deformation, 0.115 in.(3 mm), for the Case 1 diaphragm.23The lower prediction occurs becausethe Farrow and Fleischman model in-cluded a shear-friction component inthe joint compression region.

For dry mechanical connectionsin untopped diaphragms, it may beunconservative to assume shear-fric-tion can develop between the flangesbecause measured connector com-

pression stiffness is very high and theflanges are unlikely to come in con-

tact. Since untopped parking structuresuse pretopped [4 in. (102 mm)] double

tees, the equivalent beam model ap-pears to be acceptable, and conserva-

tive, for parking structure displacementestimates.

In the Nakaki method,12 the shear

stiffness of connectors at a joint is notincluded in the deflection calculation.

The method does not reflect the benefi-cial effect on reducing the diaphragm

deformation from using connectorswith a high shear stiffness. The flange

thickness of the double tees is actuallyomitted in the deflection calculation.

Thus, the calculated midspan deflec-

tion of a diaphragm with 4 in. (102 mm)thick flanges is the same as that of one

with 2 in. (51 mm) thick flanges. Over-all, the deflection obtained from the

Nakaki method is larger than that from

a more detailed finite element analysis

or the equivalent beam model.

SUMMARY ANDCONCLUSIONS

A simplified deflection analysis was

performed to provide precast designers

with a practical approach for calculat-

ing linear elastic in-plane flexibility andlateral deformation of untopped precast

double-tee floor diaphragms that use

discrete mechanical flange connectors.Complex finite element modeling

and mechanical connector properties

derived from tests served as the basis to

develop the simplified analysis meth-od. Simplifying assumptions, justified

through the accurate FEM model, led to

the development of an equivalent beam

model for practical applications. The

Fig. 5. Shear test results and schematic of the bent wing connector. Face plate at edgeof tee flange is white. Note: 1 in. = 25.4 mm; 1 kip = 1.45 kN.

Fig. 4. Example of one-span diaphragm layout. Note: 1 ft = 0.305 m.


9/12

10 PCI JOURNAL

equivalent model was shown, throughexample calculations and compari-son with the FEM analysis, to provide

deflection prediction accuracy within14 percent of that of a much more com-plicated and accurate analysis.

Using the proposed equivalent beammodel, the flexibility of linear elastic

pretopped double-tee diaphragms can be

modeled with reasonable accuracy, anddeflections can be calculated from con-

ventional beam equations. To achieve asatisfactory analytical model and reflectthe flexibility of untopped double-tee

diaphragms, both the shear and tensionstiffnesses of mechanical connectors are

required. The model is appropriate forpretopped diaphragms with:

1. Closely spaced flange

connectors [less than 10 ft(3.1 m)].

2. Connectors that have axialtension stiffness between 16and 160 kips/in. (2.8 and

28 kN/mm).3. Similar connectors used

throughout the diaphragm.

4. Chords formed from specialmechanical connectors or a

reinforced pour strip.The equivalent beam model can be

directly established by considering

the diaphragm to be a horizontal beam

with a width equal to the double-teeflange thickness and span equal to the

diaphragm span, but setting the depth,

d, and the material elastic modulus,E,

to equivalent values as given in Eqs.

(18) and (19).

The proposed model provides a sim-

plified but reasonably accurate and

practical method for predicting the

linear elastic deflection of double-tee

diaphragms resisting lateral in-plane

loading. There is certainly the oppor-

tunity to further improve and verifythe proposed method through tests on

chords, full diaphragm testing, and

building field studies. At present, pre-

cast concrete diaphragm design should

be based on maintaining elastic behav-

ior because of the low deformability of

mechanical flange connectors.

Further research is desirable, through

testing and analysis, to understand the

behavior of precast diaphragms when

the linear elastic capacity threshold is

exceeded. The likelihood of concentra-tion of inelasticity in a few joints and

the accompanying deformation demand

during an extreme event level earth-

quake motion should be identified.

ACKNOWLEDGMENTS

The research reported in this paper

was partially funded by the National

Science Foundation (NSF) under Grant

No. CMS-9412906. The following PCI

Producer Members also participated infunding portions of the study through

the PCI Research and Development

Program: Atlanta Structural Concrete,Blakeslee Prestress, Concrete Technol-ogy, Ferreri Concrete Structures, Me-tromont Prestress Corporation, Rocky

Mountain Prestress, Spancrete Indus-tries, Spancrete Midwest Inc., TindallCorporation, and Wells Concrete.

The opinions, findings and conclu-sions expressed in this paper are those

of the authors and do not necessarilyreflect the views of the sponsoring or-ganizations.

The technical input provided by anindustry advisory panel is also grate-fully acknowledged. Special recogni-

tion is due the PCI JOURNAL review-ers who provided invaluable commentsand recommendations to the authors.

REFERENCES

1. EERI, Northridge Earthquake ofJanuary 17, 1994 Reconnaissance

Report, Volume 2, William T. Holmes

and Peter Sommers (Technical Editors),Earthquake Spectra, EarthquakeEngineering Research Institute,

Supplement C to Volume 11, January

1996.

2. NIST, 1994 Northridge Earthquake:

Performance of Structures, Lifelines,

and Fire Protection System, NIST

Special Publication, National Institute

of Standards and Technology,

Gaithersburg, MD, 1994.

3. Iverson, J. K., and Hawkins, N. M.,

Performance of Precast/Prestressed

Concrete Building Structures During

the Northridge Earthquake, PCI

JOURNAL, V. 39, No. 2, March-April1994, pp. 38-55.

4. Wood, S. L., and Stanton, J. F.,

Performance of Precast Parking

Garages in the Northridge Earthquake:

Lesson Learned, Proceedings of

Structures Congress XIV, Volume 2,

American Society of Civil Engineers,

New York, NY, 1996, pp. 1221-1227.

5. Fleischman, R. B., Sause, R., Rhodes,A. B., and Pessiki, S., Seismic

Behavior of Precast Parking Structure

Diaphragms, PCI JOURNAL, V. 43,

No. 1, January-February 1998, pp.

38-53.6. Zheng, W., Analytical Method for

Assessment of Shear Capacity Demand

for Untopped Precast Double-Tee

Diaphragms Joined by Mechanical

Connectors, Ph.D. Dissertation,

University of Wisconsin-Madison,

Madison, WI, 2001.

7. PCI Design Handbook: Precast andPrestressed Concrete, Fifth Edition,

Table 4. Maximum in-plane deflection at midspan of diaphragm (in.).

Analysis

model

Equivalent

beam model

Nakaki

method

Planar finite

element model

Case 1 0.239 0.471 0.224

Case 2 0.184 0.229 0.154

Case 3 0.254 0.471 0.273

Case 4 0.203 0.338 0.178

Case 5 0.216 0.238 0.184

Note: 1 in. = 25.4 mm.

Table 3. Parameters of equivalent beam model.

Analysis

model

Thickness

of section,

t(in.)

Depth

of section,

d(in.)

Elastic

modulus,

E(ksi)

Shear

modulus,

G(ksi)

Case 1 4 683.33 295.34 113.59

Case 2 4 524.84 719.50 276.73

Case 3 2 685.52 550.85 211.87

Case 4 4 547.17 587.74 226.05

Case 5 4 661.92 351.66 135.25

Note: 1 in. = 25.4 mm; 1 ksi = 6.89 MPa.


10/12

March-April 2005 11

Precast/Prestressed Concrete Institute,Chicago, IL, 1999.

8. ICC, International Building Code,

International Code Council, FallsChurch, VA, 2003.

9. ICBO, 1997 Uniform Building Code,

International Conference of BuildingOfficials, Whittier, CA, 1997.

10. Farrow, K. T., and Fleischman, R. B.,Effect of Dimension and Detail on the

Capacity of Precast Parking StructureDiaphragms, PCI JOURNAL, V.48, No. 5, September-October 2003,

pp. 46-61.11. Fleischman, R. B., and Farrow, K. T.,

Seismic Design Recommendationsfor Precast Concrete Diaphragms in

Long Floor Span Construction, PCIJOURNAL, V. 48, No. 6, November-December 2003, pp. 46-62.

12. Nakaki, S. D., Design Guidelines forPrecast and Cast-in-Place ConcreteDiaphragms, The 1998 NEHRPProfessional Fellowship Report,

Earthquake Engineering ResearchInstitute, Berkeley, CA, 1998.

13. Venuti, W. J., Diaphragm Shear

Connectors Between Flanges of

Prestressed Concrete T-Beam, PCI

JOURNAL, V. 15, No. 1, February

1970, pp. 67-79.

14. Venuti, W. J., and Nazarian, D.,

Diaphragm Shear Connectors Between

Flanges of Prestressed Concrete T-

Beam, Report No. ST-0007-68, San

Jose State College, San Jose, CA, 1968.

15. Kallros, M. K., and Spencer R. A.,

An Experimental Investigation of theBehavior of Connections in Thin Precast

Concrete Panels Under Earthquake

Loading, MS Thesis, University of

British Columbia, Vancouver, British

Columbia, Canada, April 1987.

16. Pincheira, J. A., Oliva, M. G., and

Kusumo-Rahardjo, F. I., Tests on

Double-Tee Connectors Subjected to

Monotonic and Cyclic Loading, PCI

JOURNAL, V. 43, No. 3, May-June

1998, pp. 50-67.

17. Aswad, A., Selected Precast

Connections: Low Cycle Behaviorand Strength, Second U.S. National

Conference in Earthquake Engineering,

Stanford, CA, August 1979.

18. Timoshenko, S. P., and Goodier, J. N.,Theory of Elasticity, Third Edition,McGraw-Hill, New York, NY, 1970.

19. ACI Committee 318, Building CodeRequirements for Structural Concrete(ACI 318-02) and Commentary(ACI 318R-02), American ConcreteInstitute, Farmington Hills, MI, 2002.

20. ACI Committee 408, Bond StressThe State of Art, ACI Journal,

Proceedings, V. 63, No. 11, November1966, pp. 1161-1190.21. Alsiwat, J., and Saatcioglu, M.

Reinforced Anchorage Slip UnderMonotonic Loading, Journal of theStructural Division, American Societyof Civil Engineers, V. 118, No. 9,September 1992.

22. Oliva, M. G., Testing of the JVIFlange Connector for Precast ConcreteDouble-Tee Systems, Structures andMaterials Test Lab Report, Universityof Wisconsin-Madison, Madison, WI,June 2000 [www.jvi-inc.com].

23. Farrow, K. T., Private Communication,May 10, 2004.

APPENDIX A NOTATION = 1 when mechanical connector

is not placed at diaphragmedge (normal case)

= 0 when connector is placed atchord location

A = cross-sectional area ofa double-tee flange inlongitudinal section

A = cross-sectional area ofequivalent beam

As = cross-sectional area of chordsteel

b = width of double-tee flange (seeFig. 1)

d = depth of equivalent beamd = depth of diaphragm (see Fig. 1)E = elastic modulus of equivalent

beamEc = elastic modulus of concrete in

double teeEs = elastic modulus of chord steelfmax = chord steel stress at joint, T/AsG = shear modulus of equivalent

beamGc = shear modulus of concreteI = in-plane transverse moment

of inertia of double-tee flangesection (webs ignored)

I = moment of inertia of

equivalent beamk = joint modification factor

Kai = axial stiffness of ith connectoror chord steel

Kc = tension stiffness of chordconnection

Ks = tension stiffness of chord steel

at joint between double teesKt = tension stiffness of each flange

connectorKvi = shear stiffness of ith connectorld = development length of

embedded steelL = span of diaphragmM = in-plane moment between

adjacent double teesm = number of double tees in

diaphragm span beingconsidered

n = number of flange connections(including chord connections)

ri = distance from ith connector orchord to top end of double tee(see Fig. 1)

T = tension in chord steelt = thickness of equivalent beamt = thickness of double-tee flangeV = in-plane shear force between

two double teesw = uniform lateral load on

diaphragm per unit length

x = distance from support ofequivalent beam (see Fig. 2)

y = distance from neutral axis totop end of double tee (seeFig. 1)

= shear deformation ofequivalent beam section

= total shear deformation offlange and joint

1 = shear deformation of flange2 = shear deformation in joint = strain in chord steel at joint = Poissons ratio for material of

equivalent beam = Poissons ratio for material of

actual concrete = in-plane flexural stiffness of

joint between double tees = total shear stiffness of

mechanical connectors injoint =

i

Ki

= flexural rotation in equivalentbeam

1 = flexural in-plane rotation ofsingle double tee

2 = flexural in-plane rotationwithin joint between doubletees

= total flexural rotation of jointand double tee


11/12

12 PCI JOURNAL

The midspan deflection of the single-span untopped dia-

phragm shown in Fig. 5 is calculated below using the equiva-

lent beam model. The diaphragm configuration is chosen as

Case 1, with the bent wing connector in a 4 in. flange, from

Table 1.

Starting data:

Connector shear stiffness Kvi= 300 kips/in.

Connector tension stiffness Kt= 16 kips/in.

Chord steel area, 3 No. 6 barsAs= 3(0.44)

= 1.32 sq in.

Number of connections,including two chords

n = 11

Number of double tees in span m = 12

Panel flange thickness t = 4 in.

Panel widthb= 10(12)

= 120 in.

Panel lengthd= 60(12)

= 720 in.

Diaphragm spanL= 120(12)

= 1440 in.

Total load on diaphragm V= 123.5 kips

Concrete strength fc= 5000 psi

Reinforcement strength fy= 60,000 psi

Elastic modulus of concrete Ec= 4070 ksi

Elastic modulus of steel Es= 29,000 ksi

Poissons ratio of concrete = 0.3

Calculations:

1. Calculate the development length for a No. 6 reinforcing

bar in accordance with ACI 318-02:

No. 6 rebar diameter, de= 0.75 in.

A 1 in. thick cover over chord steel is assumed;

therefore, cover dimension, c= 1 + de / 2 = 1 + 0.75/2 =

1.375 in.

No transverse reinforcement is used; therefore,

transverse reinforcement index, Ktr= 0

(c+ Ktr) / de= (0.375 + 0) / 0.75 = 1.833 < 2.5

Reinforcement location factor, = 1

Coating factor,= 1

= 1 < 1.7

Concrete aggregate factor, = 1

= 1

No. 6 rebar development length:

Ld= de3

40

fy

fc

c + Ktr

de

= 0.753

4060,000

5000

(1)(1)(1)(1)

1.375 + 00.75

= 26.03 in.

Conservatively, takeLd= 27 in.

2. Calculate chord steel tension stiffness at joints in

accordance with Eq. (21):

Kc=3

4AsEs

Ld

=3

4

1.32(29,000)

27

= 1063.33 kips/in.

3. Calculate shear stiffness at joints:

V= (n 2)KV= (11 2)(300) = 2700 kips/in.

4. Calculate rotation stiffness at joints in accordance with

Eq. (7):

No connector at chord steel location; therefore, = 1.

= (Kc Kt)d2

2+ nKt

d2

12

= 1063.33 (1)(16)7202

2+ 11(16)

7202

12

= 2.791 108kips-in.

5. Calculate double-tee panel properties:

Cross-sectional area,A= td= 4(720) = 2880 sq in.

Cross-sectional moment of inertia:

I = td3/12 = 4(720)3/12 = 1.244 108in.4

6. Calculate concrete shear modulus:

Gc=Ec/2(1 + ) = 29,000/2(1 + 0.3) = 1565.39 ksi

APPENDIX B EQUIVALENT BEAM EXAMPLE


12/12

March April 2005 13

7. Calculate shear modulus modification factor:

k= (m+ 2)/m= (12 + 2)/12 = 1.167

Section depth [Eq. (18)]:

d= d

1+kGcA

1.2b

1+EcI

rb

= 720

1 +1.167(1565.39)(2880)

1.2(2700)(120)

1+4070(1.244 108)

(2.791 108)(120)

= 683.51 in.

Elastic modulus [Eq. (19)]:

E=Ec

d

d

1+kGcA

1.2b

=

720683.51

1+1.167(1563.39)(2880)

1.2(2700)(120)

= 295.11 ksi

Section thickness, t= t= 4 in.

Poissons ratio, = = 0.3

Equivalent cross-sectional area:

A= dt= 683.51(4) = 2734.03 sq in.

Equivalent section modulus:

I= td3/12 = 4(683.51)3/12 = 1.064 108in.4

Equivalent shear modulus:

G=E/[2(1 + )] = 295.11/[2(1 + 0.3)] = 113.50 ksi

8. Calculate diaphragm midspan deflection:

Diaphragm span,L= 12(10)(12) = 1440 in.

Load on diaphragm, w= 123.5/L= 0.086 kips/in.

Deflection due to shear:

s= 1.2wL2/8GA

= 1.2(0.086)(1440)2/8(113.50)(2734.03)

= 0.086 in.

Deflection due to bending:

b= 5wL4/384EI

= 5(0.086)(1440)4/384(295.11)(1.064 108)

= 0.153 in.

Total deflection:

=s+b

= 0.086 + 0.153

= 0.239 in.

a practical method to estimate diaphragm deformation of double tee diaphragm zhengoliva pca 419

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