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Title no.85-553

Comparison of Analysis Procedures for Two-way slabs

@ffi

by Mary Theresa Cano and Richard E. Klingner

Two-way reinforced concrete slabs act with columns and walls to Jorm structural systems Jor resisting graviry and loteral loads. Current analysis approaches for such systems usually involve finite elements or equivalent frames (effective beam widths or equivalent Jrame properties). Each approach has advantages and disadvantages. As currently used, neither is completely suitoble for anolyzing twoway slgQ systems under combined gravity and lateral loads. Thi*development, advantdges, and limitatiorc of each approach arg.* discussed, with emphosis on the equivalent frame approach. AkY equivalenl frame analysis method is proposed that iyvolves explicit modeling of attached lronsverse torsional membersQateral deflections calculated by various slab onalysis methods aie compared with publishedlgcerimental results Jor a multistory slab system under loteral loads.'Slab moments calculated by variow slab analysis methods ore compared with each.o\her for idealized flat-plate ond two-way slab-on-beam structurei;lihe explicit transyerse torsionol member method is found to give good results for drifts and slab-biam actibrls and is recommended for analysis and design of tio-way slab systems under combined gravity and lateral loads.

Keywords: concrete slabs; lateral pressure; reinforced concrete; structural analysis; structural design; two-way slabs.

A common structural engineering problem is the design of two-way, reinforced concrete slab systems-flat plates, flat slabs, and two-way slabs on beams. Starting

with a trial slab thickness based on deflection or punching shear considerations, the designer must provide for satisfactory strength and stiffness under combinations of gravity and lateral loads. This rbquires that actions within the slab system be computed. Designers differ as to the best procedures for this.l Available approaches include those involving finite ele-

ments and those involving equivalent frames. These approaches can produce widely differing results, and each has advantages and disadvantages. Thus, it is sometimes

difficult for a designer to

select an appro-

priate analysis method and interpret the results for de-

OBJECTIVES AND SCOPE The objectives of this paper are: l. To review available analysis approaches for twoway slabs. ACI Structural Journal

/

November-December 1988

2. To discuss co_mpulel:aided m_ejhoels based on each approach. 3. To compare _numerr_qa! les,qllq from each method. 4. To recommend analysis methods for two-way slab systems.

This study concerns two-way slab systems of reinforced concrete under gravity and lateral loads. It is limited to analysis methods that are readily adaptable to computer-aided solutions. Methods based on the equivalent frame concept are emphasized; yield-line methods2 and strip design methods3 are not covered. Discussion is confined to obtaining design moments; steel placement and minimum reinforcement requirements are not addressed. ANALYSIS APPROACHES FOR TWO.WAY SLAB SYSTEMS Behavior of two-way slab systems under gravity and lateral loads is complex. Unlike planar frames, in which beam moments are transferred directly to columns, slab moments are transferred indirectly, due to the.t_ots,ion"g! fleailihly-,of the slab. Also, slab moments from gravity Ioads can "leak" from loaded to unloaded spans; this must be accounted foi in inatviii. fhJ need to model torsional flexibility and moment leakag6. has given rise to two main analysis approaches for two-way slabs:

those involving finite elements, and those involving equivalent frames (using effective slab widths or equivalent frame properties).

Finite element approach Slab behavioral modes can be modeled directly using finite element methods, typically involving plate bending elements.a,s Because many elements are usually required to achieve.good results, finite element approaches are expensive for large structures. Also, the _ Received-Sept. 10, 1987, and reviewed under Institute publication policies. Copyright @ 1988, American Concrete Institute. All righti reserved, iricluding the making of co-pies unless.permission is obtained from the copyright proprietors. Pertinent discussion will be published in the September-Octbbei ISES ACt Structural Journal if re'ceived by May l, 1989.

597

ACI member Mary Theresa Cano receiyed a BS in Architecturul Engineering and an MS in Civil Engineering from the IJniversity of Texas at Austin. Ms. Cano has worked as a design engineer, and is currently an engineer wilh the Bridge Division of the Texos State Department of Highways and public Trans_

portation, Austin.

ACI member Richard E. Klingner

is an associate professor o! civil engineering, The University of Texas at Austin. He is a member of ACI Committees 531, Concrete Masonry Struclures; 349, Concrete Nuclear Structwes; and joint ACIASCE Committee 442, Response of Concrete Buildings to Loterol Forces.

applicability of linear elastic analysis is questionable when calculated slab stresses exceed cracking values. For these reasons, direct use of finite element approaches is not discussed further here. Equivalent frame approach To reduce the complexity and cost often associated with finite element analyses, the equivalent frame approach can be used indirectly to compute equivalent beam widths or equivalent frame properties. In this approach, a three-dimensional slab structure is idealized as two independent sets of parallel planar frames, crossing each other (usually at right angles). This general classification should not be confused with the specific analysis method known as the ACI equivalent frame method, to be discussed later in this paper. Two familiar examples of the equivalent frame approach are the effective beam width and the transverse torqional member procedures.. -

..-_

.ljfegqjve b-eam-width procedurA.;-The effective beam-width procedure was developed for analyzing two-way slab systems subjected tojglgt4l,lggdq,and has been used primarily for flat slabs and fla-t plates. This method incorporates the effects of slab torsional flexibility, but,aet _BgirrgnifQkece. an errectiv; ;id;h- factor cy is obtained such that a slab of effective width culr, subjected to uniform support rotation 0, would have a total support moment equal to that of the original slab (width lr, varying 0). Once the effective beam width is

determined, a conventional planar frame analysis is carried out.

Effective beam widths so derived depend on the assumed stiffnesses of the columns and of the beam-column connection regions. Typical of the results of such methods are the effective widths obtained by Khan and Sbarounis.6 Though strictly applicable only to slabs with boundary conditions and cracking consistent with the assumptions of their original derivations, such results are qften used for a wide-r4!ge__glcases. -Tronsverse

tirs|onat

mem_uei

iiiiii,ii.*rn" t urt-

verse torsionai mem6er pro..dur. was developed following extensive testing of two-way slabs.T-e Those por-

tions of the slab attached to the columns and transto the direction of the span (plus the transverse beams, if any), are assumed to act as transverse torsional members, transferring moments from slabs to columns. These transverse members are assumed rigid except in torsion. Moment transfer is treated as occurring directly over the column width c, and along the torsional members. The rotational stiffness of the joint verse

598

is determined as a function of the torsional stiffnesses

of the transverse members on each sid-e of the joint and of the flexural stiffnesses of the columns above and below the joint. DESIGN METHODS BASED ON THE TRANSVERSE TORSIONAL MEMBER PROCEDURE The transverse torsional member procedure accounts both for slab torsional flexibility "rrd -o.n"n'i*ffi[" and hai bbtir indorpoi6lefi nto'seneiai i[eCi]iC d.esiin methods. Two of these are the ACI equivalent frame

method (ACI EFM),,o and the extended equivalent frame method.rr-r3 A new method, termed the explicit transverse torsional member method,ta is also presented. In all such methods, member actions are computed, distributed to column and middle strips, and then used for slab design. AC! equivatent frame method The ACI EFM'o first requires thar the building be idealized as a series of equivalent planar frames (Fig. 1). The actual three-dimensional frame is assumed to be composed o{_{gb:!ggm; (horizontal elements with flexural stiffness I(,) supported on an assemblage of columns (vertical elements with flexural stiffness K") ana qransyerse torsional members (horizontal elements with torsional stiffn6ss K).' ?iie'equivalent planar frame is composed of slab-beams (horizontal elements with flexural stiffness .&',) supported by equivalent columns (vertical elements with flexural stiffness _r(,", defined as follows)

(1/K*): (t/DK) + (t/K,)

(t)

This notation conforms to that of ACI3l8-83.'o The flexibility of the equivalent column is the sum of the flexibilities of the actual columns and attached transverse torsional members. Required member stiffnesses K,, K", and K, are defined as follows.to Torsional member stiffness K,-The transverse torsional member concept was first proposed by Corley.T Moment transferred frorn slab to column was originally assumed to be uniformly distributed across the width of the slab. Jirsa later proposed8 a triangular moment variation (maximum intensity over the column, zero at each edge of the equivalent frame). The corresponding torsional stiffness K, was then obtained by approximate procedurese

K,:

DgEC/lrU

- @r/l)13

A)

For slabs with beams, K, lBq.(2)l is increased by the factor I"u/1,, using the notation of ACI 318-83.t0 Eq. (2) for K, was intended to apply to cracked slabse and was calibrated using the results of gravity-load tests.e,15'r6 Column stiffness &-K" is independent of K, and is calculated conventionally, using the actual column mo-

ment of inertia between the slabs, and an infinite moment of inertia within the slabs. ACI Structural Journal

/

November-December 1g88

lzl

/z

lz//2

i

(a) Definition

-n_

of equivalent frame-plan

vrcw

(b) Members of three-dimensionol structure, Detait A

K"" (c) Members of equivalent frame, Detail A

Fig. |-Member con/igurations assumed in ACI equivalent frame method Slab stiffness K.-Slab stiffness is calculated conventionally, including the effects of column capitals and drop panels. The moment of inertia of that portion between the ienter of column and face of column, bracket, or capital is then incieased by ihe factor l/(l - cr/lr)2, both to match test results and to account for the increased flexural stiffness of the slab-column con-

nection region.

For flat pldtes under uniformly distributed gravity loads, slab moments calculated by the ACI EFM were found to differ from measured values by at most 15 percent at interior columns, but by much more than

at,qlleri,or columns.e For.two-way slabs on beams, i-t,hiq drscrepancies-of l0 to 20 percent were observed

at some locations.e Despite discrepancies in the distribution of

moments, the

ACI EFM provides sufficient flexural

strength to resist the total factored static moment for each equivalent frame span. .

_ _F.g_B,a dgsigner,s viewpoint, the ACI EFM has three -ilisadvantages: First, rtime-consuming computations are required for member stiffnesses K,, K,, K", and K"".

ACI Structural Journal

/

November_December 19gg

Second, ES. (2) (for the equivalent column stiffness K*) was developed for gravityJoad analyses only. The ACI EFM can be applied correctly to lateral-load cases only if the slab-beam stiffnesses K, are reduced for the effects of cracking.to

Third, the ACI EFM is strictly applicable only to single-story substructures. The stiffnesses K"" of the

equivalent columns above and below a joint depend on the stiffnesses of the attached torsional members framing into the joint and also on the stiffnesses of the columns above and below the joint. Based on such calculations, a single stiffness (K*), is assigned to the equivalent column at a given level. In applying the ACI EFM to multistory structures, however, analogous calculations are performed for each level; on the next higher level, the same equivalent column can have a different stiffness (rQr. This discrepancy can be avoided by applying the ACI EFM to single-story substructures only.

Extended equivalent frame methods

In

rqspgqse

to these disadvantages, the ACI EFM of

was extended by Vanderbilt and others, in the form

599

(a) Extended equivalent column

method (three-dimensional method)

(c) Extended equivalent staL

method (three-dimensional model)

DISTR IBUTE

DISTRIBUTE KI

*ffic

[t,*Ktr]

IG, I

-[^-+ftJ

l-__&__l

LEF; (b) Extended equivalent column method using nrneuz (two-di-

mensional method)

(d) Extended equivalent slab method using nrn.e,tta (two-dimensional model)

FLEXURAL MEMBER (e) Special member used in EFRAME

TORSIONAL

MEMBERS

Fig. 2-Extended equvalent frame methods (Vonderbilttr)

the extended equivalent column K"" and. extended equivalent slab K* methoals.il-r3 The extended equivalent column method tFig. 2(a)l uses conventional beam elements (no attached torsional members at ends). Column elements incorporate the flexural flexibilities of the columns, plus the torsional flexibilities of the attached torsional members [Fig. 2(e)1, distributed to column elements above and below the joint in proportion to the flexural stiffnesses of those columns[Fig. 2(b)]. Tfre equivaient slab method IFig. 2(c)l uses conven_ tional column elements (no attached torsional members at ends). Slab-beam elements incorporate the flexural flexibility of the beams and slabs, plus the torsional

flexibilities of the attached torsional members [Fig. to slab-beam elements on each siOe of the joint in proportion to the flexural stiffnesses of 2(e)1, distributed

those slab-beams [Fig. 2(d)].

Both methods reduce the slab system to a planar frame which is then analyzed conventionally. Both in_ clude the effects of slab torsional flexibility under lat_

eral and gravity loads. Because the equivalent slab 600

method cannot reproduce the effects of moment leak_ age under gravity loads, it should be used for cases in_ volving lateral loads only.tt While removing some of the disadvantages of the ACI EFM, these methods require a special computer

programrT to handle the equivalent beam and slab ele_ ments. Also, hand computation is required to distrib_

ute the torsional member flexibilities to the columns above and below a joint, or to slab-beams on either side of a joint.

Explicit transverse torsional member method To eliminate these drawbacks, a modification of the preceding models, termed the explicit transverse tor_ sional member method,ra is proposed. As shown in Fig.

3, conventional columns are connected indirectly by two conventional slab-beam elements, each with half the stiffness of the actual slab_beam. The indirect con_ ggUqll, made using explicit transverse Gi"fi;;"r_ bers, permits the mo-deling of moment feaEag_e qllgell qq_ql9b-loilio-riitnixuiliji.wnilJtneresurtinsf rameis ;onpi#;; thiils-n6t a ierious comptication. Speciat_ ACI Structural Journal / November-December 19gg

purpose programs such as ETAqs,18 available in micro_

computer as well as mainfiame versions, are widely used for analyzing three-dimensional structures. Because the transverse torsional members are present only in the analytical model, their lengths are arbitrary, provided that their torsional stiffnesses are consistently defined, as explained later. The explicit transverse torsional member method has several advantages. Structural modeling is simple and direct, requiring very few hand computations. Also, computed member actions in the slab-beams and transverse torsional members can be used directly for design of slabs and spandrels, respectively. Finally, this method can even be used for true three-dimensional analysis of slab systems under combined gravity and lateral loads. Two sets of equivalent frames, each running parallel to one of the building,s two principal plan orientations, can be combined to form a single threedimensional model. This single model can be used to calculate actions in all members (slabs, columns, and spandrels) under as many combinations of gravity and lateral loads as desired.

COMPUTER APPLICATION OF SLAB ANALYSIS METHODS Computer application of the preceding methods requires that each equivalent frame's geometry, material characteristics, and member properties be defined. For all except the explicit transverse torsional member method, a planar frame is used. The secant modulus of concrete is usually computed according to ACI 31883.t0 Procedures for calculating member properties are summarized in Tables I through 3 and are discussed in the following.

EXPLICIT TORSIONAL MEMBERS

Fig. 3-Three-dimensional model of equivolent frame using explicit transyerse torsional member methodta Effective width method For computer analysis, column stiffnesses are computed conventionally and beam stiffnesses are computed using the effective width /r. Cracking, if present, should generally be accounted for separately. Most effective width methods do not address this isiue. In this faper, additional stiffness reduction factors of 0.33 and 0.70 were used for flat plates and two-way slabs on beams, respectively.

ACI equivatent frame method For computer analysis, the equivalent stiffness K"" of each column is computed from the joints at each end of the column. The column inertia is set so that the column's rotational end stiffness coefficient 4EJ"/h" equals the average of the two Ku" values. Because r("" is strictly the stiffness of a joint rather than a column, this pro-

Table 1-Modeling idealizations used in computer.aided analysis methods for two.way stabsof member Slab analysis method

Effective width method (Khan and

Columns

Modeled with effective width

Sbarounis6)

factor

ACI equivalent

Modeled normally

frame method8

Attached torsional

SIab-beams

members

Modeled normally

Not modeled

Column stiffness is combined with attached torsional member stiffness to give equivalent column stiffness. If column stiffness differs as determined from each end, average

Attached torsional member stiffness is combined with column stiffness to

with ends incorporating torsional flexibilities of

Special column element

Torsional stiffness of attached torsional member is modeled using special

bers

torsionally flexible ends

cu

give equivalent column stiffness

properties are used Extended equivalent column method

Modeled normally

(Vanderbilt,,)

attached torsional mem-

Extended equivalent slab method

column elements with

Special beam elements with ends

(Vanderbilt")

incorporating torsional fl exibilities of attached torsional

Explicit transverse torsional member

Modeled normally

members

Modeled normally

method'4

ACI Structural Journal

November-December

1

988

Modeled normally

Table

2-computation of member properties for analyses of flat plates GravityJoad analvsis Method

Effective width (Khan

Uncracked

,

LateralJoad analysis Uncracked

Cracked

Should not

and Sbarounisu)

Should not be

be used*

used*

Extended equivalent column method

May be used

Should not be

(Vanderbilt")

Extended equivalent slab method (Vanderbiltrr)

Explicit transverse torsional member methodr4

Il=1"

Cracked

May be used

I!

--

ot

May be used Ij -- 0.33 a lul

r|

May be used

May be used

:

usedl

Il=Io

be used"

Should not be used"'t

May be used

Il=Io

Il, = 0.33 I'

May be used

May be used

May be used

May be used

Should not

Il=ro

K,' =

K,

r; -- 0.33 rb Ki = 0.33 K,

ri,

0.33 Io

May be used

Il=Io

I;

--

0.33 r"

K: -- 0.33 K,

K,' = K,

*Does not exhibit moment leakage under gravity loads. Ia = 0.44 for flat plate example of this paper. I Not recommended for gravityJoad analyses using [ < Iu.

Computation of member properties for analyses of two.way slabs on- beams Table 3

Method Effective width

(ACI T-beam width'o) Extended equivalent column method

(Vanderbilt")

Extended equivalent slab method (Vanderbiltrr)

Explicit transverse torsional member method'a

GravityJoad analysis

Lateral-load analysis

Uncracked

Cracked

Should not be

Should not be used*

Il=Io

Il

May be used

May be used

May be used

used*

May be used

Il -- Io Ii =L

May be used

Ii=1"

May be used

Il=Io Ki __ K,

r!=L

Uncracked

Il = I*,' I! = 0.75 I,

Should not be

II :

l--L

May be used

Il I!

IO

May be used

used"'5

Il=Io

May be used

May be used

Il : I"r,t Ki = o.33 K, | = 0.'1s L

Cracked

May be used

= I,nf

=

Iq|

=

0.75

I"

Should not be useds

May be used

Il -- Iu Ki=K, I::1"

Il = I"tt.t Kl = 0.33 K, I! = 0.75 I,

*Does not exhibit moment leakage under gravity loads. I I"nt : 0,42 Ib for two-way slab on beami example of this paper, I I*, : 0.70 Io for two-way slab on beams example of this paper. r Not recommended for graviryJoad analyses using .{j < 1r.-

cedure cannot be applied consistently to multistory structures. Other required member properties are the beam area

A,

length

Z, shear area Au, and moment of

inertia 1u. Beam properties are based on the gross section, using the full slab width /r. Effects of cracking need not be considered explicitly for gravity loads, because the formula for K"" was calibrated using cracked test specimens.T,e However, this may not be sufficient for a gravity-loaded slab previously cracked by lateral loads. For lateral load cases, slab-beam cracking should be considered by reducing the slab-beam's moment of inertia by a reduction factor (usually 0.25 to 0.33 for

flat slabs or flat plates'o).

the slab-beam stiffness should be multiplied by a reduction factor of 0.33 for flat slabs or flat plates.r0'rr

Extended equivalent slab methodir As noted, this model should not be used for gravity loads.tr Gross member properties are used as in the preceding. Transverse torsional member stiffnesses are distributed to special torsional end members on each side of a joint in proportion to relative slab-beam stiffnesses. When lateral loads are involved, the slab-beam stiffness should be multiplied by a reduction factor of 0.33 for flat slabs or flat plates.to,tt

Explicit transverse torsional member method Gross member properties are used for slab_beams and columns. Area, moment of inertia, and shear area are calculated conventionally. For computer input, the torsional stiffness ,I of the transverse torsional mem_

Extended equivalent column methodll For gravity loads, gross member properties are used as in the preceding. Transverse torsional member stiffnesses are distributed to special torsional end members above and below each joint in proportion to relative column stiffnesses. This process can be carried out automatically by the computer program EFRAME.T'7 Reduction of slab-beam stiffnesses for'gravity-load cases is not recommendedlt because it results in erroneous moments at exterior columns. For lateral load cases,

. Using an arbitrary length

602

ACI Structural Journal

bers is calculated by the following procedure

K, = D9Ec/lrll-(cr/tr)13 [Eq. (2),

/

Z for the torsional

repeated] members,

November_December lggg

c

2'- 9"

l'-9" Tv p. 1

Meosured

v I

*

Iv

v

Comouled EFE'WIDTH (o: O.5l) EXPI-ICIT.

Kec, ETABS Kes, ETABS

FLOOR 7' x 6'-0" x t7a" (

Fig.

4-NRC

model used

for drift

typicol

)

o.oo5

comporisons

:

J;G/L

(3)

L/G

(4)

K,;

Effects of cracking need not be considered explicitly for gravity loads, because the ACI expression for K, is con_ sistent with some cracking.T,e However, this may not be sufficient for a gravity-loaded slab previously cracked by lateral loads. When Iateral loads are present, sla6_ beam cracking should be considerea Uy muttiptying the slab-beam's moment of inertia by. a reduction factor of 0.33 for flat slabs or flat plates.'o

Comparison of different methods: Lateral drift calculation To compare the accuracy and ease of use of the preceding methods, each was used to compute the de_ flections of a small-scale, multistory flat plate test specimen,re,2o cited also by Vanderbilt.i This specimen, tested under the auspices of the Canadian National Re_ yargh Council (NRC), is referred to here as the NRC

Model, and is shown in Fig. 4. Measured lateral deflec_ tions of a transverse frame of the NRC model were compared with those computed by the following four

methods: a. Effective width method (Khan and Sbarounis)6

b. Explicit transverse torsional member methodra

c..

Extended equivalent column method (Vanderbilt)lr -

d, Extended equivalent slab method (Vanderbill;rr Because cracking was not observed in the NRC Model,2o all member properties were calculated neglect_ ing cracking. Since the computer program EFRAMET,T?

ACI Structural Journal

/

o

,ors

Fig. S-Loteral drift of transverse frome, NRC model

Therefore

J.

I

DRTFT (in)

the torsional stiffness ,I, of each is then calculated that

K,i:

0.o

November-December lggg

was not available, the extended equivalent column and extended equivalent slab methods were implemented using the ETABS program. Calculated results,o using ETABS were within 5 percent of published results, ob_ tained by Vanderbilt using EFRAME,,T for the'same structure. Shearing deformations were neglected throughout. Computation of member propertie-s is de_ scribed in detail in Appendix B of Reference 14. The results, shown in Fig. 5, indicate that for this example the e ff ectivg-.yi"* 'ffililtTfiiriee[r. -g-re1,! o d gi ves very .accur ate lalelal dri_fls, other rnlit giving very slmllar answers, all overestimate drift by"ar, as much as 20 percent. In evaluating these results, it should be noted that only the effective width method was developed as_

suming an uncracked slab. A typical slab structuie would be more likely to have some cracking, and its lateral drifts would be closer to the values ;ffiffi; ;; either of the equivalent frame methods. ihe effective widih method and the explicit transverse torsional member method were judged much easier to implement than either of the extended equivalent frame methods. While this observation probably would have changed slightly had the EFRAME program been available, it is advantageous for a method to require only standard

analysis programs.

COMPARISON OF DIFFERENT ANALYSIS METHODS: IDEALIZED FLAT.PLATE To compare the accuracy and convenience of differ-

ent slab analysis methods, the same four methods were

used to compute slab moments in an idealized two_story

flat-plate frame: a. Effective wldth method (Khan and Sbarounis)6

603

tll if

As noted previously, the ef_fectjve

INTERIOR EOUIVALENT

_!en{g{ equivalent glab

Slob thickness 8"

All col. 24"x24"

Fig. 6-Three-dimensional view of ideolized two-story flot-plate example

wid_1h and 9a m91!q{s_. cannot model moment

leakage, and hence should not be used for gravity-load analyses. In analyzing slab systems for combined gravity and lateral loads, the preceding two methods can only be used if the analysis is split into two parts: either of the preceding two methods is used for the lateralload portion of the analysis, and other methods (such as the extended_equjvalent qolumn or the explicit transverse torsional member methods) are used for the gravityJoad portion. The results are then combined manually. This process is referred to here as a "two-model" analysis. The other two methods (extended equivalent column and explicit transverse torsional member methods) can be used for gravityJoad as well as lateral-load .analyses. Using

a single model, results for different load cases can be computed and combined automati-

cally.

b. Explicit transverse torsional member methodra c. Extended equivalent column method (Vanderbilt)" d. Extended equivalent slab method (Vanderbilt)" As shown in Fig. 6, the idealized example frame has 20-ftbay widths, a uniform l2-ft. story height, and24-

in. square columns. The slab thickness of 8 in. was seIected based on the shear and moment transfer and deflection provisions of ACI 318-83,'0 assuming a dead load of self-weight plus 15 lb/ft2, and a live load of 50 lb,/ft'?. The frame was analyzed for gravity loads and also for lateral loads of 20lb/ft'z. Member properties were calculated as shown in Table 2. Joints were considered ri gid, sh e aring 4eloflgalLb B!*y-e,f 9 t9g!99q94, and member actions were computed at member faces. Table 4

-

Slab moment results for gravity loading Results for gravity loading are shown in Table 4. The extended equivalent column method, which is.almost identical to the ACI EFM,rt is used as the standard of comparison for gravity loading. For gravity loading of an uncracked structure, the effective width method and the extended equivalent slab method give expectedly poor results. The extended equivalent column method and the explicit transverse torsional member method give good results.

To u_sq q $Letj flq!:p_!el_e-q94_elfor ggavity g! ygll.as laleral loading, the slabs shquld be qraq[ed. However, when the slabs alone are cracked, both extended equiv-

Comparison of slab moments, idealized flat plate example Slab moments at Level 1, kip-in.

Exterior Interior column column

Load case and analysis method Gravity pattern loads, uncracked (1.4D Effective width methodu Extended equivalent column method'l

Interior column

Exterior column

+ l.7L)

Extended equivalent slab method" Explicit transverse torsional mbmber method,o

Cravity pattern loads, cracked slabs Effective width method" Extended equivalent column method" -Extended equivalent slab method'r

Explicit transverse torsional member method'o

1629

r699

1675

850

1980

t'176

lt97

665

762

854

1995

1790

661 292

683

6?5

866

700

t46

257

299.

881

7t3

882

2041

1787

3'7

-37 _40

553

Gravity pattern loads, crack slabs, and torsional members

Explicit transverse torsional member method,ra

Ki :

0.33 K,

Lateral loads, cracked slabs

Effective width methodu Extended equivalent column methodl Extended equivalent slab method'l Explicit transverse torsional member method,la Ki = K,

Lateral loads, cracked slabs

+

- 3'1

-50 -49 -49

45 43

37

40 40

44

-40 -39

31

-23

23

39

-37

-45

37

-43

50 49 49

-31

-16

-44

torsional

members

Explicit transverse torsional member method,'o

K: :

0.33 K,

Combined loads 0.75 (1.4D + l.7L + l.7W) Two-model method (Extended column uncracked + effective width cracked) Extended equivalent column method, cracked Explicit transverse torsional member method,'o K,' : 0.33 K,

604

-38 52'1

I

550

t268

1396

882 571

t446 t534

123'7

1316 1365

1318

ACI Structural Journal

1420 t35't 14'71

/

6s9

982 647

November-December 1988

alent frame methods and the explicit transverse tor_ sional member method give erroneous results fo1 grav; ity-load cas-es. As shown in Table 4, decreasinl the slab-beamstif f nes5_i4qgagqstherl"b-";;*;;;:

p o it s, lelh er hAqtqigAfqll eA _4,rt, ;r a u e ex p6ctEd-ihe-rea;n-iilahE is ifiaTiri" .;iffi* t.

IO x 4

l2 x 2O girders

_

lnterior equivolent '

"*

conliiictrid to the torsional members rather than to the slab-beams. When the latter are made more flexible, the increased relative stiffness of the torsiorrt-;;;;;;$ii causes support moments to increase.il,ra The solution to this problem is to include the effects of cracking in the transverse torsional members as well as the slabs. This modification is easy to carry out with the explicit transverse torsional *.-L., method. Table 4 shows the results for Ki : 0.33 K,. -Bggu-lts are close

to those of the extended equivalent cotrrr--metnoO :' ' Cuncraakeil

case)

Slab moment results for laterat loading Lateral loading results are also shown in Table 4. When slabs alone are cracked, all four meth.ods give similar results. When transverse torsional members are also crackedln the case of the explicit transverse tor_

sional member method, moments are decreased slightly.

Slab moment resutts for combined loading (gravity plus laterat) Combined loading results are also shown in Table 4. Using the "two-model', procedure, gravity_load mo_

ments calculated using the extended equivalent column method (uncracked members) are combined with lat_ eral-load moments calculated using the extended equiv8 alen!_ qla!_me1ho,{ (crackbd slabs). Using tfre t,single_

model"'procedure, gravity- and iateral_ltad moments are calculated using the extended equivalent column method, as well as the explicit transverse torsional member method. Cracking is taken into account for the

slab-beams in each case.(Ii = 0.33 1r). Because the ex_ tended equivalent column rirettrOa does not permit

easy

incorporation of cracking of the torsional members,

ii

2O spondrels

frome

l.2ao" +

2+o"

l.

2ao"

+

Slory height = l2' Slob lhickness = 6" All columns, 16"x

16"

Fig. 7-Plan view of idealized two-story, two_woy slab with beams example

of 6 in. was selected based on the deflection provisions of ACI 318-83,'0 assuming a dead load of self_weight plus 15 lb/ft2, and a live load of SO lb/ftr. The frame was analyzed for gravity loads, afid also for lateral Ioads of 20 lb/ftr. Joints were considered rigid, shear_

ing deformations were neglected, and member actions were computed at member faces. The frames were analyzed using both two_model and

single-model procedures. For the effective width

method, widths of slab-beam members weiil as ildfined bt tfie T-beam width provisions of ACI 3t8-g3 (Refer_ ence 10). Effects of slab-beam cracking were accounted

for using an average effective moment of inertia, equal in this case to abofi 0.42 times the gross inertia.,. For the qlh.qr-fngtg4s, slab-beams and transverse torsional members were as defined by the ACI equivalent frame method.10 Effects of slab-beam cracking were ac_ counted for using an average effective moment of in_ ertia, equal in this case to about 0.70 times the gross inertia.ta Column stiffnesses were calculated using the gross inertia, multiplied by a reduction factor equal to 0.75 (Reference l0). procedures for calculating mem_ ber properties are summarized, in Table 3.

gives erroneous results for the combined load case. The

.

explicit transverse torsional riremlier method, on the other hand, gives results close to those of the two_

model method.

COMPARISON OF DIFFERENT ANALYSIS METHODS: tDEAL|ZED TWO.WAY SLAB ON BEAMS To compare the accuracy and convenience of differ_ ent slab analysis methods, the four methods were used to compute slab-beam moments in an idealized two_ story frame made up of two-way slabs on beams: a. Effective width method (ACI effective width for

T-beams).Io

b. Explicit transverse torsional member method.ra Extended equivalent column method (Vander_

- .-c.

bilt).'r d. Extended equivalent slab method (Vanderbilt).il As shown in Fig. 7, the idealized frame has two_way on beams, 20-ft spans, a uniform l2_ft story .sl1b1 height, and l6-in. sqrar. columns. The slab thickness ACI Structural Journal / November_December lggg

Slab.beam momenl results for gravity loading These results are shown in Table 5. The extended equivalent column method (uncracked case), almost identical to the ACI EFM,I is used as the standard for comparison for gravity loading. For gravity loading of the uncracked structure, the extended equivalent slab method gives expectedly poor results, ,hil" th. ,*_ tended equivalent column method and the explicit transverse torsional member method give almost iden_ tical results. When the slab-beams alone were cracked, the ACI effective width method and the extended equivalent slab method were much less accurate than the other methods. Both the,extended equivalent column method and the explicit transverse torsional member method gave slightly high results, although not as far off as those of the preceding flat-plate example. As before, this problem was resolved by using thi explicit trans_ verse torsional member method with cracked torsional members as well as slab-beams. Results are very close 605

ol srab-bearn moments, ideatized two.way I1l9J-_-comparison Deams example Slab-beam moments at Level

Load case and analysis method

Gravity pattern loads, uncracked (1.4D + lr7L\ ACI effective T-beam width meihod,o.r4 ' Extended equivalent column methodri Extended equivalent slab method', Explicit transverse torsional member method,n

Gravity pattern loads, cracked slabs ACI effective T-beam width method,o',.

Extended equivalent column method,, Extended equivalent slab method,, Explicit transverse torsional member method14

Gravity pattern loads, cracked slabs members

+

Exterior Int.rioffi column column 1457 196'7 948 20s7 1083 1584 952 20',12

513 895

1591

1908

805

99s

2065 t'709 2080

605 889

1

103

1002

l,

srab on

kip-in.

column

column

814 881

8'74

torsional

Explicit transverse torsional member method,'o

K,' =

0.70 K,

Lateral loads, cracked slab-beams + columns ACI effective T-beam width methodr0.,a Extended equivalent column method,, Extended equivalent slab method',

Explicit transverse torsional member method,'a

Ki :

K,

962 - 105 -164 - 163

2089 100 133

129

r

896

-95 95 - l0l l0l - 104 104 - 10t 101

-164

133

163

131

-99

99

606

1643

l3r6

1506

583 559

r683

- 100 - 133 - 129 - 133

105

t64 163 164

Lateral loads, cracked slabs + columns + torsional members Explicit transverse torsional member method,'a

Ki = 0.7O K,

Combined loads 0.75 (1.4D + t.7L + l.7Wl Two-model method (extended column unoacked + effective width cracked) Extended equivalent column method,'ciacked Explicit transverse torsional member method,'o K,' 0,70 K,

:

to those given by the extended equivalent column method (uncracked case).

Slab-beam moment results for laterat loading These are also given in Table 5. All methods ixcept the ACI effective width method give satisfactory results. Unlike the preceding flat-plate example, torsional member cracking using the explicit transverse torsional member method does not significantly decrease slabbeam moments under lateral load.

Slab.beam moment results for combined loading (gravity plus lateral) These results are also shown in Table 5. Using the "two-model" procedure, gravity-load moments calculated using the extended equivalent column method (no cracking) are combined with lateral-load moments calc u I at ed u s i n g t h e ex.t e n e-{ e qqi_vp!_e_n! { - Clab _ir1g!Lq*4'j (cracked slab-beams and coiumns). Using tha ;;sin;i;-

model" procedure, gravity- and lateral-load moments are calculated using the extended equivalent column method, as well as the explicit transverse torsional member method. Slab-beam and column cracking are considered identicalty in both cases, and torsional member cracking is considered in the Iatter method. As shown in Table 5, all three methods give acceptable results. The single-model procedures are much more convenient.

SUMMARY This report has focused on the following analysis methods for two-way reinforced concrete slabs: 606

1..

t700

130'7

1324

131

163

1443 817 t418 9lt 1437 885

1509

1523

Effective width methods, exemplified by the

method of Khan and Sbarounis.6 2. Transverse torsional member methods. a. ACI equivalent frame method.to b. Extended equivalent slab method (Vanderbiltlr).

c. Extended equivalent column

method

(Vanderbiltrr).

d. Explicit transverse torsional member metfod.ra The ACI equivalent frame method was developed and calibrated for single-story substructures under gravity loads and cannot be consistently applied to multistory structures., In the extended equivalent column and extended equivalent slab niethods, the torsional stiffnesses K, of the transverse torsional members are distributed between adjoining columns or slabbeams, respectively. Both methods are used with a specialized computer program.rT The explicit transverse torsional member method accounts directly for moment leakage and slab torsional flexibility. It requires only conventional computer programs and can be specialized to either of the extended methods.

Lateral deflections computed by all methods were checked against results obtained from an uncracked, scale-model, flat-plate structure.re,20 While results from the effective width method were most accurate, the slightly greater drifts predicted by the other methods might be more reasonable in a real structure with some cracking. All methods were considered to give satisfactory deflection calculations. The preceding methods were also compared with respect to accuracy and convenience in calculating mo-

ACI Structural Journal

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November-December 19gB

ments for two idealized two_story structures, one a flat plate, and the other a two_way slab on beams. Gravity_ load moments, lateral-load moments, and combined gravity- and lateral-load moments were examined. gravity_load moments were compared .Calculated with those from the extended equivalent column

Table 6

-

Recomm

ffi l,m,:is$*,s*;*?T*f

Type of twc-way slabTwo-way slab on bearns

Il

= I*r,* = 0.7s I, Ki = 0.70 K,

method, previously verified for gravity_load mo_

I, r I

I!

ments.rr For the flat_plate frame and the two_way slab on beams, the extended equivalent column method and the explicit transverse torsional member method gave good results.

of ti

irEpo

4. Additional examples involving three_dimensional

5. Correlation with additional experimental data on drifts and member actions. ci tations

C

=

c2

=

corresporl:?lt'JBl"

""".r

o

Cross-sectional constant to define torsional properties [see Eq. (13-7), Chapter l3J Size of rectangular or.equivalent rectangular column, cap_ ital, or bracket measured transverse to the direction of the span for which moments are being determined, in. (Chap_

ters 11 and 13)

E G

= Modulus of elasticity of member (concrete) = Shearing modulus of member (concrete) Io, Il = Moment of inertia of beam 1g.oss ani cracked, respec_ tively)

Advantages of the explicit transverse torsional member method

f: = Moment of inertia of column (gross and cracked, respectively) Iur, : Effective moment of inertia (based on ACI T_beam width) f",

Because it gives reasonable results, does not require special computer programs, and permits

used

easy consider_

November-December 19gB

.Irpt.

structures.

*.*u.r

/

U.urn,

1. Wall loads on beams of two_way slabs. 2. Relation between computed spandrel torsions and ACI compatibility torsion provisions. 3. Simpler ways to include the effects of cracking in two-way slabs on beams.

iaiiiridtory--di:i{eciio; actions ""a were obtained using a single model based on eittrer the explicit transverse torsional member method,ra or the extended equivalent column method.n Recommended procedures for calculating member properties for use in all methods are given in Tables I through 3.

ACI Structural Journal

o,

method:

Ged rli.it 'qgjgionA;;;ue;-@@ way stabs on

umn axial loads, a single computer model, including the effects of cracking, can be used to calcurate laterar drifts and all member actions (slabs, walls, and col_ umns). Calculated slab actions can be assigned to col_ umn and middle strips, and slab design cin easily be

for two-way ,tut

Needed research Further research is needed on the following topics related to the expricit transverse torsionar member

CONCLUSIONS of their accuracy and relative simplicity, two methods were preferable for analysis of siabs u"ae. combined gravity and lateral loads. For flat plates, sat_ istaelqrlllefl ecliaa!_e-sd_nqe*u_"r_aeirsr!_tid;ffi E_ tained y{ry=1ji$-ls=ry9d"l

ation of cracking in both slab-beams and transverse torsional members, the explicit transverse torsional member method is proposed as a powerful method for analyzing two-way slab systems under combinations of gravity and lateral loads. Member stiffnesses for this method are as recommended in Table 6. Using this method, an entire three_dimensional building can be analyzed at once, under combinations of gravity and laterar loads in difierent plan directions. If beam end shears for the equivalent fru*er;;;; each direction are corrected to avoid doubling the col-

0.70 Io

completed. No hand calculations are required for load combinations. The explicit transverse torsional member method is believed to present current ACI equivalent frame method concepts in a form that is both powerful and convenient for design purposes.

Because

bdami;

:

*1"1,

Combined gravity-load and lateral_load moments were compared with those computed by a two_model procedure. The extended equivalent column method was used to calculate gravity_load moments. The effec_ tive width and extended equivalent slab methods were used for lateral-load calculations for flat plates and two-way slabs on beams, respectively. For the flat_ plate case, the explicit transverse toisional member method (using a single model) gave results very close to those obtained with the two_model procedure and was much more convenient. For the two_way slab on beams, both the extended equivalent column and the explicit transverse torsional member methods gave re_ sults very close to those obtained with the two_model procedure and were much more convenient.

vgise

#'nj':",*'ii:.'"''

I*, !,,

for effective beam width in two_way slab on

example

= I,u

J,

K"

K*

beams

Effective cracked moment of inertia used for beams in twoivay slab on beams example Gross moment of inertia of slabs or slab-beams Torsional stiffness of fl torsional member

= : = Flexural stiffness of column; moment per unit rotation (Chapter t3) : Flexural stiffness of equivalent column (Chapter 13).

Also

refers to Vanderbilt,s Extended Equivalent fiame Metfrod (equivalent column)

(K*)' = Flexural stiffnesses of equivalent columns on Levels I and (K*)' 2, respectively K^ = Refers to Vanderbilt,s extended equivalent frame method (equivalenr slab)

K"

:

Flexural stiffness of slab; moment per unit rotation

K,, K,'

=

Flexural stiffness of torsional members, uncracked and cracked respectively (refer to Chapter 13). r(,, refers to i,i

L

=

Arbitrary length of transverse iorsional members

(Chapter t3)

torsional member

607

: : =

which moments are being determined, measured center-tocenter of supports (ChaPter 13)

Y 67' No' 1 I ' Analysis for Slab Desigp." ACI Jcnxrr' hoceedings ' E' S' Eberhardt' C' A' by Nov. 1970, pp. 875-8&l- AIso, Discussion ProClosure' Hoffman, Ti Huang, J. C. Jofrirr' Y' K' Hanson' and

Effective width factor

ceedings

Length of span transverse to direction of the span for

Uniform rotation at column, used in deriving effective width factor o

CONVERSION FACTORS

l

ii

I ft = 0.305 m I in. = 25.4 mm I lb/ft2 = 4.882k9/m' I kiP : 4'448 kN I ksi = 6.895 MPa I Psi = 0.006895 MPa

ir

ii ri

ii

REFERENCES

questionnaire distribl."Design of Reinforced Concrete Slabs"'

uted by ACI-ASCE Committee 421' 1986'

Robert, and Gamble, William L" Reinforced Concrete pp' 274-464' Slabs, John Wiley & Sons, New York, 1980' and Con3. Hillerborg, Arne, S/rrp Method of Design' Cement

i.iuri,

crete Association, Wexham Springs,

191

5' 256 pp'

4. Pecknold, David A.,"Slab Effective Width for Equivalent No' 4 Apr' Frame Analysis," ACI JouRNAL, Proceedings V' 72' 1sls, pp. tl\-tsl . Also, Discussion bv F' H' Allen' P' leP' Darvall' n. g. Ctover, and D. A. Pecknold, Proceedings V' 72, No' 10' Oct' 1975, pp. 583-586. ACI S. Btias, ZiadM.,"Lateral Stiffness of Flat Plate Structures"' JounNer, Proceedings V. 80, No. 1, Jan'-Feb' 1983, pp' 50-54' 6. Khan, Fazlur R', and Sbarounis, John A',"Interaction of Shear Walls and Frames," Proceedings, ASCE, V' 90' ST3' Part l, June 1964, pp. 285- 335.

7. Corley, W. Gene; Sozen, Mete A.; and Siess, Chester P',"The Equivaleni Frame Analysis for Reinforced Concrete Slabs," S/ructural Research Serres No. 218, Department of Civil Engineering, University of Illinois, Urbana, June 1961, 168 pp' 8. Jirsa, James O.; Sozen, Mete A.; and Siess, Chester P',"Pattern Loadings on Reinforced Concrete Floor Slabs," Proceedings, ASCE,

V.95, 3T6, June 1969, PP. l1l7-1137' 9. Corley, W. Gene, and Jirsa, James O.,"Equivalent Frame

V. 68, No. 5, May l97l' pp' 397-4OI' 10'ACICommittee3lE'..B'JilditrgCodeRequirementsforRein. Deforced Concrete (ACI 318-83)"' American Concretelnstitute' Code RequireBuilding on pp, and "Commeatary 111 troit, 1983, pp' ments for Reinforced Concrete (ACI 318{3)" 155 of Unbraced Analysis Frame D.,"Equivaleot M. Vanderbilt, 11. structural Loads," Lateral Reinforced concrete Buildingi for Static Colorado Departmenr' Engineering Ci;il No. 36, Reporl Research State University, Fort Collins, Colorado, July 1981' ,

-

iZ. Vuna.rUitt, M. Daniel,"pquivalent Frame Aoalysis for Lateral Loads," Proceedings, ASCE, V. 105, ST10, Oct' 1979' pp' 19811980' pp' 1998. Also, Discussion by ZiadM. Elias, V. 106, ST7, July p' 245' 1981, Jan. 107, STl, 167l-1672, and Closure, V. 13. Vanderbilt, M. Daniel, and Corley, W. Gene,"Frame Analysis of Concrete Buildings," Concrete Internotionol: Design & Construction,Y.5, No. 12, Dec. 1983, pp.33-43. 14. Cano, M. T.,"Comparison of Analysis Procedures for TwoWay Slabs," MS Thesis, Depaftment of Civil Engineering, University of Texas, Austin, Aug. 1984. l5."Reinforced Concrete Floor Slabs - Research and Design," Bulletin No. 20, Reinforced Concrete Research Council, American Society of Civil Engineers, New York, 1918,209 pp. 16. Guralnick, Sidney A., and LaFraugh, Robert W.,"Laboratory Study of a 45-foot Square Flat Plate Structure," ACI JounNel, Proceedings V.60, No.9, Sept' 1963, pp. 1107-1185. 17. "EFRAME," computer program' National Information Service in Earthquake Engineering, University of California, Berkeley' 18. Wilson, E. L., Hollings, J. P. and Dovey, H' H',"Erars: Three-Dimensional Analysis of Building Systems (Extended Version)," computer program' National Information Service in Earthquake Engineering, BerkeleY, 1977. 19. Zelman, Maier I.; Heidebrecht, Arthur C.; Tso, W' K'; and Johnston, William A',"Practical Problems and Costs of Fabricating Multi-story Models," Models for Concrete Struclures, SP-24, American Concrete Institute, Detroit, 1970' pp. 159-185'

20. Hartley, G.; Rainer, J. H; and Ward, H. S',"Static and Dy-

namic Properties of a Reinforced Concrete Building Model," Building Research No. 140, Division of Building Research, National Research Council of Canada, Ottawa' Apr. 1919,24pp'

l

i l

j :]

608

ACI Structural Journal

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November-December 19BB