Equiv Frame to Conc Shearwalls

8
Equiualent Frame Method Applied to Gonqete Sheatwalls uildings that incorporate concreteshearwalls as struc- tural elements to resist both vertical and lateral loads are commonplace. Shearwall and cou- pled shearwall structures have been found to be economical up to the 30 to 40 story range, and shear- wall,/frame structures have shown their effectiveness up to 50 stories.l The calculation of stresses and deflection in a simple shearwall re- quires only rudimentary bending theory. Often however, one or more columns of door and window open- ings create two or more shearwalls coupled together at each floor. Un- Iike the simple shearwall,the analy- sis of coupled shearwalls is by no means trivial. Finally, the shear- wall,/frame system adds yet another degreeof complexity. There are four main methods for analyzingcoupled shearwalls: Scale model testing - Typically used in research facilities to vali- date or confirm proposed theories, it is not normally employed by de- sign offices due to resource, time, and cost restrictions. Lamina Method - Also referred to as the continuous medium method, it replaces the individual coupling beams between shearwalls with a continuous, uniform, homo- geneousmedium, referred to as a lamina. It assumes that the point of counterflexure occurs in the mid- span of the coupling beams, that the walls deflect equally when sub- jected to horizontal loads, and that the walls resistthe loads in propor- tion to their stiffness. The method takes into considerationthe contri- bution made to the shearwalls by the bending and shearin connecting beams. It is a hand method that in its pure form is tedious, but graph- ical methods proposed by research- ers assist in removing the drudgery from the method.2'3 However, it is limited to relatively high shear- walls, with constant floor heights and uniform openings. Finite Element Method (FEM) - This method partitions a complex elementinto smaller componentsof a finite size and number. The ge- ometry of these finite elements are simpler than the boundaries of the overall element. Usually the analy- sis is based on assumed displace,- ment functions. Because of the number of calculations required, even for simple elements, this method is limited to computer ap- plications. Even so, with large com- plex elements,idealized into small, numerous finite elements, compu- tation time can be significant. It is gaining wider use, and may be the most appropriate method of analy- sis for some complex problems. Equivalent Frame Method (EFM) Also referred to as the wide column analogy, it replaces the coupled shearwall components with an ide- alized frame structure that behaves identically to the shearwall. This idealized structure is resolved using matrix analysis techniques. A first order linear elastic analysis is per- formed. Although possibleto carry out a matrix analysis by hand, it quickly becomes time consuming and complex as the size of the structure increases. Like the FEM, analyzing a structure in matrix form is ideally suited to the digital com- puter. The matrix method of analy- sis is less computation intensive than the FEM, and consequently is of great interest to practicing engi- neers. Its adaptability and flexibil- ity have made it popular in engi- neering offices. The equivalent frame method provides a good balance of effect- iveness, efficiency, and easeof use. The application of the EFM to con- crete shearwalls is by no means a new topic. But this paper attempts to assemble information that has ei- ther been scatteredin a number of sources or typically has been cov- ered as an aside. Considering the number of structures that rely on shearwalls to resist part or all of the lateral loads imposed on them, a more unified presentation of the in- formation is warranted. Background The two main procedures of matrix analysis are the flexibility and the stiffness methods. The flexibility (or force) method is a generalization of the Maxwell-Mohr method, devel- oped by J.C Maxwell in 1864, and O. C. Mohr a decade later. By writ- ing compatibility equations in terms of flexibility coefficients and se- lected redundants, statically inde- terminate structures are analyzed. The problem with this method is that the choice of redundantsis not unique, and an inappropriate one November1991 65

Transcript of Equiv Frame to Conc Shearwalls

Page 1: Equiv Frame to Conc Shearwalls

Equiualent FrameMethod Applied toGonqete Sheatwalls

ui ld ings that incorporateconcrete shearwalls as struc-tural elements to resist bothvertical and lateral loads are

commonplace. Shearwall and cou-pled shearwall structures have beenfound to be economical up to the 30to 40 story range, and shear-wall,/frame structures have showntheir effectiveness up to 50 stories.l

The calculat ion of stresses anddeflection in a simple shearwall re-quires only rudimentary bendingtheory. Often however, one or morecolumns of door and window open-ings create two or more shearwallscoupled together at each floor. Un-Iike the simple shearwall, the analy-sis of coupled shearwalls is by nomeans tr iv ia l . Final ly, the shear-wall,/frame system adds yet anotherdegree of complexity.

There are four main methods foranalyzing coupled shearwalls:

Scale model testing - Typicallyused in research faci l i t ies to val i -date or confirm proposed theories,it is not normally employed by de-sign offices due to resource, t ime,and cost restrictions.

Lamina Method - Also referredto as the cont inuous mediummethod, it replaces the individualcoupling beams between shearwallswith a continuous, uniform, homo-geneous medium, referred to as alamina. It assumes that the point ofcounterf lexure occurs in the mid-span of the coupl ing beams, thatthe walls deflect equally when sub-jected to horizontal loads, and thatthe walls resist the loads in propor-

tion to their stiffness. The methodtakes into consideration the contri-but ion made to the shearwal ls bythe bending and shear in connectingbeams. It is a hand method that inits pure form is tedious, but graph-ical methods proposed by research-ers assist in removing the drudgeryfrom the method.2'3 However, i t isl imi ted to relat ively high shear-walls, with constant f loor heightsand uniform openings.

Fini te Element Method (FEM) -This method partit ions a complexelement into smaller components ofa f in i te s ize and number. The ge-ometry of these finite elements aresimpler than the boundaries of theoverall element. Usually the analy-s is is based on assumed displace,-ment funct ions. Because of thenumber of calculat ions required,even for s imple elements, th ismethod is l imited to computer ap-plications. Even so, with large com-plex elements, idealized into small,numerous finite elements, compu-tation time can be significant. It isgaining wider use, and may be themost appropriate method of analy-sis for some complex problems.

Equivalent Frame Method (EFM)Also referred to as the wide columnanalogy, i t replaces the coupledshearwall components with an ide-alized frame structure that behavesident ical ly to the shearwal l . Thisidealized structure is resolved usingmatrix analysis techniques. A firstorder l inear elastic analysis is per-formed. Although possible to carryout a matr ix analysis by hand, i t

quickly becomes t ime consumingand complex as the s ize of thestructure increases. Like the FEM,analyzing a structure in matrix formis ideally suited to the digital com-puter. The matrix method of analy-s is is less computat ion intensivethan the FEM, and consequently isof great interest to practicing engi-neers. Its adaptability and flexibil-i ty have made i t popular in engi-neering offices.

The equivalent f rame methodprovides a good balance of effect-iveness, efficiency, and ease of use.The application of the EFM to con-crete shearwal ls is by no means anew topic. But this paper attemptsto assemble information that has ei-ther been scattered in a number ofsources or typically has been cov-ered as an aside. Consider ing thenumber of structures that re ly onshearwalls to resist part or all of thelateral loads imposed on them, amore unified presentation of the in-formation is warranted.

BackgroundThe two main procedures of matrixanalysis are the flexibil i ty and thestiffness methods. The flexibility (orforce) method is a generalization ofthe Maxwell-Mohr method, devel-oped by J.C Maxwell in 1864, andO. C. Mohr a decade later. By writ-ing compatibility equations in termsof f lexibi l i ty coeff ic ients and se-lected redundants, statically inde-terminate structures are analyzed.The problem with th is method isthat the choice of redundants is notunique, and an inappropriate one

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EquivalentFrame

Outline of coupledshear wall (deflectedshape undsr lateral load)

i *'+4 P' -Lr T b 1 Lr l

Fig. 1 - Coupled shearwal l subject to hor izontal loadingwith equivalent f rame super imposed.u

Fig. 2 - Coupled shearwal l and concrete f rame withequivalent f rame super imposed

can lead to significantly increasedcalculation requirements.

In the stiffness (or displacement)method, no input beyond definingthe structural model is required tocarry out the analysis. The un-known quantit ies are the joint dis-placements, and the number of un-knowns is equal to the degree ofk inemat ic indeterminacy of thestructure. The directness of this ap-proach makes i t super ior to theflexibility method for applicabilityto computer analysis.

The principle of superposition isfundamental to the application ofthe stiffness and flexibility methodsof analysis. For this principle to ap-ply, the structure must be l inearlyelast ic, which means that i t mustsatisfy the following three require-ments:o The mater ia l of the structuremust be elastic, having a l inear re-lationship between stress and strain(Hooke's law).r The displacements of the struc-ture are small. so that calculationsinvolving the overall dimensions ofthe structure can be based on i tsoriginal dimensions.,o No interaction exists between ax-ial and flexural effects in the mem-bers ( the ef fect that axial forces

have on members, when combinedwith even small deflections is com-monly referred to as the P-A effect,which is nonlinear. This topic wil lbe covered later in the article, alongwith other considerations).

I t is not necessary to know thespecifics of the theory behind ma-tr ix methods of f rame analysis toapply i t , but a pract ic ing engineermust be aware, at least conceptu-ally, of the underlying principles. Itis only in th is way that the engineercan have a better understanding ofthe types of structures to whichsuch analyses are applicable, as wellas the method's l imitations.

Equivalent Frame MethodEquivalent FrameIn applying matrix analysis to con-crete shearwalls, the coupled shear-walls are replaced with a centerlineframe that displays the same behav-ior as the elements being modelled.This model is referred to as anequivalent frame. Fig. I shows theessence of the approach, where thecenter lines of walls and connectingbeams form the members of theframe.

Section Propert iesThe sectional properties of the col-umns in the equivalent frame are

generally those for the correspond-ing wall sections, since the structureis assumed to behave in a l inearelastic fashion. Note however thatfor squat wal ls ( length is greaterthan 2 t imes height) shear def lec-tion predominates; for slender walls(height is greater Ihan 2 t imeslength) bending def lect ion governs;and for walls in between, a combi-nat ion of shear and bending con-trols the deflection.a The combineddeflection for a cantilever fixed atthe support, subject to a uniformlydistributed load is:

wLa 0.6wL2A-aror-

gg1 T

GA

(L^o,n",,) * (L,n"o,)

wherelr : uniformly distributed loadZ : lengthE : Young's modulus of elasticityG : shear modulus1 : moment of inertiaA: area

The combincd def lect ion for acantilever fixed at the support, sub-ject to a concentrated load at i tsfree end is:o' '

PL3 I.2PL1 -- I-tot 3EI cA

(L,o^",,) * (L,*",)

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Pecknold, Allenand Darvall

Carpenter,Mehrain and Aalami

Mehrain and AalamiAllen and Darvall

Khan and Sbarounissqualeround

Brotchie, square

Brotchie, round

I . . .I no|ot -

- I flexiblet

Square PanelSquare ColumnPoisson's Ration = 0For use wih program usingzero size joinb

- Exrapolated Curve

Fin i - Ff fect ivc s lah width.s|Y. v

0 0.05 0.10 0.15 0.20 0.25c/g

Fig. 4 - Comparison of several invest igat ions intoeffect ive s lab width.n

where P : concentrated loadThe foregoing is provided to al-

low the engineer to consider sheardeformation for walls with smallheight to depth ratios, where a re-duced moment of inertia may be inorder.

Shear def lect ion must also beconsidered to proper ly model thebehavior of the beams connectingthe shearwalls. As can be seen fromFig. 1, the connect ing beams canundergo relatively large deforma-tions (especially those in the upperportion of the frame). The equiva-lent f rame beams have the samearea as the connecting beams, butthe moments of inertia must incor-porate a shear def lect ion correc-tion. The following formulas havebeen proposed:6't

6 ' h3^/ l2f _ -___L_'o- l+2.8(h. /b) 'z

where6 : thickness of connecting beam/ro : depth of connecting beamD : width of opening1, : reduced moment of inertia ofconnecting beam.And

Ibrl

| + 2.4(d/b)3 (l + u)

where4 : moment of inertia of connect-ing beamd : depth of connecting beamu : Poisson's ratio

The addi t ional hor izontal sec-t ions between the frame columnsand the connecting beams are stiffended elements that rotate but donot bend. Theoretically, they shouldhave infinite areas and moments ofinertia. Programs exist that do al-low for end sections of beams to beinfinitely rigid, but for many ma-tr ix analysis programs, extremelylarge section properties can createerrors or large inaccuracies in theresults. If however, small inaccura-cies are acceptable, then perfect ri-gidity is not required. The follow-ing can be used to determine theproperties of the stiff ended beamelement (Fig. 1): '

A"/Ar : 100 (e/f)

I"/\: 100(e/f)3 + 300(e/f)'1+ 300(e/f)

wheree : length of stiff ended section

f : half length of connecting beamA" : atea, of stiff ended section1" : moment of inert ia of st i f fended section

Ar : ajea of connecting beam1r : moment of inertia of connect-ing beam

Loading

To model the effect of horizontalloads on the equivalent frame, theexterior loading on the shearwall isreplaced by an equivalent concen-trated load applied at the joints (in-tersection of columns and beams) ofthe frame.

AnalysisWith the conf igurat ion of theequivalent f rame establ ished, thesection properties determined, andthe loading prepared, the equiva-lent frame is ready for analysis us-ing one of the various matrix anal-ysis programs available.

Other considerationsThe foregoing establishes a methodfor analyzing coupled shearwalls.But, i t is unl ikely that a bui ld ingwill have just one coupled shearwallact ing in isolat ion. More real ist i -cally, buildings will possess numer-ous shearwalls of differing configu-rations, shearwalls acting togetherwith elevator, /stair cores, shear-wal ls, / f rames, or a combinat ionthereof. These give r ise to otherfactors that require consideration,

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including the connecting elements,interact ion wi th f rames, supportconditions, symmetry, torsion, andvarying opening widths.

Connect ing elementsWhere the connecting elements incoupled shearwalls are well definedbeams, the section properties of thebeams are as discussed previously.But in many residential buildingsand hotels, the flat concrete floorslabs are the connecting elements.The interaction that occurs betweencoupled shearwalls depends on thebending stiffness of the connectingelernents. In the case of flat slabs.the depth of the connect ing ele-ments is merely the floor slab thick-ness, but the width is not quite asobvious.

An assessment made bySchwaighofer on a system com-prised of two 21.3 ft (6.50 m) wideby 194.4 tt (59.25 m) high concreteshearwal ls spaced 5.3 f t (1.63 m)apart, and connected by 8 in. (203mm) thick concrete f loor s labsshowed little difference in shearwallinteraction when using a slab widthof 2r.3 f t (6.50 m) and 10.7 f t (3.25m) respectively.6 For larger shear-wall spacing, the same may not betrue. Except for very high and lowslab stiffness values, the stiffness ofthe system is greatly dependent onthe slab stiffness.T Thus, when flatslabs are the connecting elementsbetween coupled shearwalls, it is upto the engineer to consider the sys-tem on a case by case basis, reviewthe literature, and decide on an ap-propriate effective slab width.

Interaction with framesMany contemporary buildings em-ploy a combination concrete frameand shearwal l (Fig. 2) . Since thestiffness of concrete frames is sen-si t ive to the s lab st i f fness, theshearwall/frame combination wil la lso display a sensi t iv i ty to s labstiffness.

Many studies have been carriedout to determine the most appropri-ate values for slab stiffnesses, anddi f ferent models have been pro-posed. One of the more well knownwas developed by Khan and Sba-rounis, who in 1964 proposed effec-tive widths of slabs varying from 20to 60 percent of the slab width, de-pending on the geometry of theframe (Fig. 3)8.

However, in 1983 Vanderbilt andCorley compared the results of sev-

eral investigations with respect toslab widths. This comparisonshowed ef fect ive widths varyingfrom 20 percent of the slab width togreater than the full slab width (Fig.4) 'g. In 1988 Cano and Kl ingnercompared different analysis proce-dure for two way slabs, which in-cluded the effective width method(Khan and Sbarounis) , the ACIequivalent frame method, and theextended equivalent column andslab methods (Vanderbilt).r0 The re-sults from the various analyses werecompared to test resul ts f rom asmall scale multistory flat plate testspecimen prepared by the NationalResearch Council of Canada.

For lateral dr i f t . the ef fect ivewidth method provided the mostaccurate resul ts. wi th the othermethods giving similar results, butover estimating the drift. Thus, likethe effective slab width in coupledshearwalls. the determination of aneffective slab width for frames islef t to the discret ion of the enei-neer.

Support condit ionsAll two dimensional matrix analysisprograms allow at least three typesof supports. A roller support is re-strained against movement in justone pr incipal axis. A pinned sup-port is restrained against movementin both principal axes. A fixed sup-port is restrained against movementin both principal axes, and also re-sists rotation. Many frame analysisprograms check for stabi l i ty onlysuperficially, so it is worth notingthat for a two dimensional analysis,a minimum of three restraint com-ponents are required for static equi-l ibr ium. This can be made up ofone fixed support, one pinned sup-port and one rol ler support , orthree roller supports not located onthe same parallel l ines or arc.r' Forthree dimensional f rames. an in-spection of the model may prove tobe the best way to determine stabil-ity, and can be accomplished by as-sessing whether the f rame cantranslate along a plane, or rotateabout an axis in an unrestrainedmanner. l l

Usual ly, non yielding supportsare assumed at the founding level ofvertical elements of the equivalentframe, but conditions arise wherevert ical and rotat ional f lexibi l i tymay be required. ' Computer pro-grams exist that provide the ability

to model inclined roller supports,spr ing supports, and supports re-strained partially against rotation.In the event that an engineer doesnot have access to such a program,these support conditions can be rea-sonably approximated.r l An in-clined roller can be modeled with ashort, stiff, pin-ended member in-clined at the required angle (Fig. 5).

Spring supports (used to simulatespecific soil conditions for instance)can be modeled with pin endedmembers whose properties approxi-mate the spring constant required(Fig. 6). Partial rotational restraints(somewhere between a fixed and apinned condi t ion) can be repre-sented with an element perpendicu-Iar to the actual member to be par-tially restrained, with properties ap-proximating the degree of restraint(Fie. 7).

SymmetryThe engineer can take advantage ofbuildings that possess a symmetricstructural layout by apportioningthe lateral forces to all shearwalland frame bents in a single opera-tion (Fig. 8). The frames are ideal-ized using EFM and linked at eachf loor wi th beams hinged at theirends. The link beams should theo-retically be made infinitely stiff. Butas discussed previously, this pres-ents a problem with some matr ixanalysis programs, so the beamsshould be made suff ic ient ly st i f fsuch that their axial deformationsare negligible. This is based on theassumption that the floor slabs actas rigid diaphragms.

For long narrow buildings, or forbuildings whose lateral load resist-ing elements are almost as stiff asthe floor slab diaphragm, then thedistribution of horizontal forces isnot the same as that derived assum-ing a r ig id diaphragm.T The engi-neer should be aware of conditionswhere the validity of the rigid dia-phragm assumption should be ques-tioned in order to not be misled byinvalid results.

Even when the structural layout isnot symmetr ic, i t may be that agiven frame within the structuredisplays symmetry. Engineers cantake advantage of favorable geom-etry to significantly reduce the sizeof the frame to be analyzed.

In two dimensional matrix analy-sis, if the l ine of symmetry is coin-cident wi th a column l ine (even

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Structure beingsupported

Short, stiff (large area)pin ended member slqpedat angle to simulateinclinod roller

Clockwise from above left:

Fig. 5 - Model for incl ined rol ler . l l

F ig. 6 - Model for spr ing supports."

Fig. 8 - Lateral force distr ibut ion between ele-ments ( for symmetr ic plan layouts, d istr ibut ion oflateral loads can be achieved in a s ingle opera-t ion by l inking elements.)

Fig.7 - Model for spr ing supports."

Structure beingsupported

Horizor$al mernberto simulate oartial restraint.Spring constant is 3El/1. Adjust| & L to model degree of resEaint.

Structure beingsupported

Member with sectionpropenies to simulate

being modeled.

Vertical pin ended memb€rto simulde elastic soilsupport conditions. Sff ingconstent is AE/L. AdjustA&Ltoapprdximatesoil behavior.

FRAMING PLAN

W2E

Shd, stifi (lilge a@)pin ended mmbsslir*ing fril6 ate@h floor.

number of bays), then only one halfof the frame needs to be analyzed,with one half of the lateral forcesappl ied (Fig. 9) . The sect ionalpropert ies of the column on thesymmetry line are likewise reducedby one half. Finally, the column onthe l ine of symmetry is restrainedfrom axial deformation with rollersupports introduced at the columnjoints at each floor. This is to sim-ulate the cantilever bending actionof the frame where. for horizontalloads. the neutral axis is assumed topass through the line of symmetry.

I f the l ine of symmetry passesthrough the center of a bay (oddnumber of bays), again one half ofthe frame is analyzed, with one halfof the hor izontal forces appl ied(Fig. 10). To simulate the neutralaxis of the frame subjected to hori-zontal loads, roller supports are in-troduced where the beams intersect

the line of symmetry at each floor.In three dimensional matrix anal-

ysis, p lanes of symmetry can beused to al low the analysis of onehalf (for one plane of symmetry), orone quarter (for two planes of sym-metry) of the three dimensionalframe. The procedure is similar tothe two dimensional case. but addi-t ional support e lements must beplaced so that the hal f or quarterframe cannot rotate when subjectedto horizontal loads (Fig. 1l).

Although strictly speaking not anitem covered by symmetry, lumpingtechniques represent anothermethod by which the engineer canreduce the size of the model to beanalyzed.T With any computer anal-ysis, the intent is to model the be-havior of the structure being ana-lyzed. It may be possible to achievethis, within reasonable l imits, witha lumped model as well as the full

(unlumped) model. In this process,the area and location of the verticalelements in the lumped model arethe same as in the unlumped model.

If n (two or more) typical floorsin the frame are lumped into onefloor, the moment of inertia, andthe area of the horizontal memberin the lumped model should be ntimes that of the unlumped model.The height of the column in thelumped model would be made ntimes that of the unlumped model.Final ly, the moment of inert ia ofthe column in the lumped modelwould be n ' t imes the moment ofinert ia of the unlumped column.The technique could be quite usefulwhen carry ing out a prel iminarymatrix analysis of a frame in a tallbui ld ing. When using the tech-nique, the engineer should use bothcare and caution to ensure that thecorrect modell ins of the structural

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Page 6: Equiv Frame to Conc Shearwalls

SymlLil

(?nerry

3) 4;

i

Abl,2 Abz,i Ab3,4 = Ab2,3lbE;4= lbz;3 iF l r l |be,3

N

Nioi<t si l i -

* i r !

<:3!i1$

Roller suppons restrainingvertical movement are addedat each floor.

t5 :

I

Ab4,F = Abr,zih4,5 = ht,z i

I L+,s = :./ - ----- ' """ ' /Lt,z I

Ac, Ab= area olcolumn &beamlc, lb = moment oi inertia ot column & beam

L = tengttr

Structures wi th Torsion" andcludes a worked out example ofapplication.

Varying opening widthsI t is possible to use the EFM tomodel coupled shearwalls with non-uniform openings simply by shift-ing the center l ine of the wal l e le-ments (Fig. l2) . However, th is ig-nores the rotation of the walls at theoffsets, resulting in deflections thatare s l ight ly l iberal , but accurateenough to predict the overal l be-havior of the bui ld ing. ' Since thebehavior in the zone of the wal lof fset may not be correct ly pre-dicted using the EFM, the finite ele-ment method should be consideredfor a detai led assessment of thestress distributions in the zone.

P-A EffectAs stated earlier, this is the term forthe effect that axial forces have onmembers that undergo even smalldeflections. To accurately take thiseffect into consideration, a secondorder non-linear analysis includingthe effect of sway is required. How-ever, the ACI reinforced concretecode gives provis ions for the ap-proximate evaluation of this effect,through the use of a moment mag-nification factor (ACI 318-89, sec-t ion 10.11). '3 Nevertheless, thecommentary of the ACI does en-dorse the use of second order frameanalyses to direct ly include thesway, or P-A, effect.

If such a program is not availableto the designer, then it is possible totake the P-A effect into considera-t ion by modify ing the f i rst order(EFM) analysis.T The technique re-quires the following procedure:o The horizontal and vertical load-ing is applied to the structure, andusing the EFM analysis, the f i rstorder displacements (Ai) at eachstory are obtained.o Knowing the story displacementand the accumulated gravity load ateach f loor the addi t ional storyshears caused by the P-A momentcan be calculated. The net story

in-its

o

<5i l -

<3

I Lr,z i Lz,s I Le,+ =f f i " " 'l l lzg

l lI Only half of frame I

is analyzed

Flg. 9 - Symmetry l ine coincident wi th column. '

wt2E'

EI

tl

E

. )aDYY

lI Only half of frame l

is analyzed

Fig. 10 - Symmetry l ine through center of bay. '

Roller suppons restrainingvertical movement are add6dat each floor.

4

Abi,4 =,4bl,z:lbir,4;lhi,t :

< i Ef l iT

<:-9L}!\a

Ls,+ = t,Lt,z I

Ac, Ab = area ot column & beamlc, lb = moment ot inertia of column & beamL = lengrh

behavior is not compromised forthe sake of reduced computations.

TorsionThe ef fects of tors ion on lateralload resist ing elements can resul tfrom either an asymmetrical struc-tural layout, unbalanced wind load-ing, or eccentrically applied earth-quake loads. Where shearwal l orshearwall/frame structures are sub-iect to eccentric loads the distribu-

tion of lateral forces to the variouselements is non-trivial. If a three-dimensional, matrix-analysis com-puter program is avai lable, theanalysis can be greatly facil i tated.In the event that the engineer doesnot have access to a three dimen-sional frame analysis program, itcan be done either by hand or withthe help of a spreadsheet. Mac-Leod'2 discusses in detai l the"Component Stiffness Method for

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SymmetryLrne

\

shear at a given level is the alge-braic sum of the story shears fromthe column above and below thefloor.r The f inal second order def lec-tions are calculated from the firstorder deflections as follows:

I\ -1 - DPLi/Hh

whereEP : cumulative vertical loadAi : first order story swayh : story heightH : horizonral shearo The addi t ional story shears areadded to the applied loads, and thestructure is re-analyzed (first orderanalysis) , using the new lateralloads.

Thus, a first order (EFM) analy-sis can be used to give second orderdef lect ions, moments, and forces.These can then be used in the de-sign of members, without resortingto the moment magnifier method.

The designer should be aware,however, that ideal ly the f lexuralstiffness (E1) needs to reflect theamount of re inforcing, extent ofcracking, creep, reduced stiffnessdue to axial loads, as well as the in-elastic behavior of concrete and thevariation of EI along the length ofa member f rom cracked to un-cracked regions. In practice, simpli-fying assumptions are used to com-pute EI, since it would be impracti-cal to consider the flexural stiffnessof each member of a hish r isebuilding.?

SummaryOne of the strengths of the equiva-lent frame method of analysis is itsappl icabi l i ty to a var iety of con-crete shearwall configurations andshapes. It is unencumbered by theneed for ei ther constant f loorheights or uniform openings (as inthe Lamina Method). Virtually anytype of horizontal and vertical loads( including temperature induced)may be appl ied to the ideal ized

Movernent iny directionresrained.

Fig. 11 - Symmetry for 3D f rame. '

SymmetryLine

'/

Movement iny directionrestrain€d.

Movement inz directionrstrarned.Y

structure. It can also be used in theanalysis of shearwal ls interact ingwith frames.

With any method of analysis, it isimportant to have a general sense ofhow the structure wil l behave andthe order of magnitude of resultantreactions. This is particularly truewhen using computers to assist inthe analysis of structures. I f theoutput varies greatly from the ex-pected solut ion Lhe engineer mustdetermine whether the results con-tain an error (and if so, why) or ifthe structure is behaving in an uh-anticipated fashion.

All computer analysis techniquesand programs have basic underly-ing assumptions bui l t into them.Further, their output is directly andstrictly related to the input, which inturn is based on further assump-tions made by the user. It is up tothe engineer to understand the limi-tat ions of the assumptions (pro-gram based and user made) andverify the results. The EFM can as-sist in this verif ication process byallowing the engineer to quickly re-model the structure and test theoutcome of differing assumptions.In many cases the validity of com-puter analyses can st i l l be crosschecked with s impl i f ied manualanalysis techniques. The engineermust also be sufficiently aware ofthe limitations of the EFM to know

Fig. 12 - Coupled shearwal l wi thnon-uni form openings. '

when a more detai led analyt icaltechnique may be appropriate.

References1. "Modern Mult i -Story Concrete Bui ld-

ings," Canadian Portland Cement Associa-t ion, Canada, 1989, p. 3.

2. Rosman, Riko, "Approximate Analysisof Shearwalls Subjected to Lateral Loads,"ACI JounNal, Proceedings, V. 61, No. 6,June 1964, pp.711-133.

3. Schwaighofer, J. , "Tables for theAnalysis of Shearwal ls wi th Two Vert icalRows of Openings," Publicotion 71-27, Uni-versity of Toronto Department of Civil En-gineer ing, Nov. 1971.

November 1991

Page 8: Equiv Frame to Conc Shearwalls

4. Glanvi l le, John I . , and Hatzinikolas,Michael A., "Engineered Masonry Design,"Winston House Enterprises, Winnipeg, Can-ada, 1989, p. 160.

5. Hal l . A.S.. An Introduct ion to the Me-chanics of Solids, John Wiley & Sons Aus-tralasia Pty Ltd, Sydney, Australia, 1969, pp.

t16-124.6. Schwaighofer, Joseph, "Shearwal l

Structures," Structural Concrete Sympo's i rm, Universi ty of Toronto Department ofCivil Engineering & Portland Cement Asso-ciat ion. , Toronto, Canada, MaY 13&14,1971, pp. 118-145.

7. Taranath, Bungale S., Structural Anal-ysis and Design of Tall Buildings, McGraw-Hi l l Book Company, New York, 1988, pp.

49r-535 , 67 5-686.8. Khan, Fazlur R., Sbarounis, John A.,

" Interact ion of Shearwal ls and Frames,"Journal of the Structural Division, ASCE, V'90, ST3, June 1964, pp.285-335.

9. Vanderbilt, M. Daniel, and Corley, W.

Gene, "Frame Analysis of Concrete Build-ings," Concrete Internationrtl: Design &Construct ion, V. 5, No. 12, Dec. 1983, pp'

33-43.10. Cano, Mary Theresa, and Kl ingner,

Richard 8., "Comparison of Analysis Pro-cedures for Two-Way Slabs," ACI Struc-tural Journql , V. 85, No. 6, Nov.-Dec. 1988,pp.597-608.

11. Lutz, Leroy A., "Computer-AidedAnalysis and Design," Building Structural

Desing Handbook, Richard N. White andCharles G. Salmon, edi tors, John Wiley &Sons, New York, 1987, pp.530-541.

12. Macleod. Ia in A.. "Shearwal l -FrameInteract ion: A Design A\d," Engineer ingBulletin, Portland Cement Association, Sko-kie, 1970, 17 pp.

13. ACI Commit tee 318, "Bui ld ing CodeRequirements for Reinforced Concrete (ACI

318-89) and Commentary - (ACI 318R-89),"American Concrete Institute, Detroit, 1989,pp. 117-129.

14. Coul l , Alexander, and Choudhury, J.R., "stresses and Def lect ions in CoupledShearwalls," ACI JounNa,l, Proceedings, Y.64, No. 2, Feb. 1967, pp.65-72.

15. Coul l , Alexander, and ChoudhurY,J.R., "Analysis of Coupled Shearwal ls,"ACI JounNa. l , Proceedings, V. 64, No. 9,Sept. 1967, pp. 587-593.

16. Coul l , A. , and El Hag, A. A., "Ef fec-tive Coupling of Shearwalls by Floor Slabs,"ACI JounN,lt-, Proceedings, V. 72, No. 8,Aug. 1975, pp.429-431.

17. Duchesne, D. P. J. , and Humar, J ' L ' ,

"Engineer ing Software - a Consul tant 'sPerspective," Canadian Journal of Civil En-gineer ing, V.18, Apr. 1991, pp. 303-311.

18. Falk, Howard, "Microcomputer Soft-ware for Concrete Structural Design," Con-crete Internationol: Design & Construction,V. 7, No. 6, June 1985, pp.49-56.

19. Khan, A.H., and Staf ford Smith, B. ,

"simplified Method of Analysis for Deflec-

t ions and Stresses in Wal l -Frame Struc-

Ives." Building ond Environment, Y. ll,

No. 1, 1976, pp.69-78.20. Kong, F.K., et a l , Edi tors, Handbook

of Structural Concrete, Pitman Books Lim-

i ted, London, 1983, pp. 3l -1 to 3 '7-44.21. Schwaighofer, Joseph, and Col l ins,

Michael P., "Experimental Study of the Be-havior of Reinforced Concrete Coupl ingSlabs," ACI JounN,ql- , Proceedings, V. 74,

No. 3, Mar. 1977, pp. 123'127.22. Weaver, Wi l l iam Jr. , and Gere, James

M., "Matr ix Analysis of Framed Struc-tures," 3rd Edi t ion, Van Nostrand Rein-hold, New York, 1990.

Received and reviewed under Institute publi-

cation policies.

ACI member An-gelo Mattac-chione is Presi-dent of Prosum En-ninoor inn l l r l 2

structural consul t -ing f i rm in NorthYork, Ontar io. Hehas been act ive inthe design of numerous structures int imber, s l ructural steel , and rein-forced and post{ensioned concrete.

72 Authorized reprint from: November l99l lssue of ACI Concrete Internattonal