12-Multi Story Frame Structures

8/13/2019 12-Multi Story Frame Structures

1/77

Richard Liew, J.Y.; Balendra, T. and Chen, W.F.Multistory Frame StructuresStructural EngineeringHandbookEd. Chen Wai-Fah

Boca Raton: CRC Press LLC, 1999


2/77

Mu lt is t o r y Fr a m e S t r u c t u r e s

J. Y. Richard Liew

and T. Bal endra

Depar tm ent of Civ il Engineer ing,

Nat ional U niver sit y of Singapor e,

Singapore, Singapore

W. F. C hen

School of C ivi l Engineer ing,

Pur due U niver sity,

West Lafayette, IN

12.1 Classifi cati on of Buildi ng FramesRigid Frames SimpleFrames(Pin-ConnectedFrames) Brac-ing Systems Braced Frames vs. Unbraced Frames SwayFrames vs. Non- Sway Frames Classification of Tall Build-ing Frames

12.2 Composite Floor SystemsFloor Structures in Multi story Buildings Composite Floor

Systems Composite Beams and Girders Long-Span Floor-ing Systems Comparison of Floor Spanning Systems FloorDiaphragms

12.3 Design Concepts and Str uctural SchemesIntroductionGravit y Frames Bracing Systems Moment-Resisting Frames Tall Building Framing Systems Steel-Concrete CompositeSystems

12.4 Wind Effects on BuildingsIntroduction Characteristicsof Wi nd Wind Induced Dy-namicForces ResponseDuetoAlongWind ResponseDuetoAcrossWind Torsional Response Responseby Wind TunnelTests

12.5 Defini ng TermsReferencesFurther Reading

12.1 ClassificationofBuildingFrames

For building framedesign, it is useful to define variousframesystemsin order to simplify models ofanalysis. For example, in the caseof abraced fr ame, it is not necessary to separateframeand bracingbehavior becauseboth can beanalyzed with asinglemodel. On theother hand, for morecomplicatedthree-dimensional structuresinvolvingtheinteraction of different str uctural systems, simplemodelsare useful for preliminary design and for checking computer results. Thesemodels should beable tocapture the behavior of individual subframesand their effectson the overall structures.

Theremainder of thissection attemptsto descri bewhat aframed system represents, define when a

fr amed system can beconsidered to bebr aced by another system, what is meant by abracing system,and the differencebetween sway and non-sway frames. Vari ousstr uctural schemesfor tall buildi ngconstr uction are also given.

12.1.1 RigidFrames

A rigid frame derives its lateralstiffness mainly from the bendingrigidity of framemembers inter-connected by rigid joints. The joints shall be designed i n such a manner that they have adequate

c1999by CRC PressLLC


3/77

strength and sti ffness and negligible deformati on. The deformati on must be small enough to haveany significant influence on thedistri bution of internal forces and moments in the structure or onthe overall framedeformation.

A rigid unbraced frameshould becapableof resisti nglateral loadswithout relyingon an additional

bracing system for stabil it y. Theframe, by itself, hasto resist all the design forces, including gravityas well as lateral forces. At the same ti me, i t should haveadequate lateral sti ffness against sideswaywhen it i ssubjected to horizontal wind or earthquakeloads. Even though thedetaili ng of the rigidconnecti ons results in a less economic str ucture, ri gid unbraced frame systems have the followingbenefits:

1. Rigid connectionsare moreductile and therefore the structure performs better i n loadreversal sit uati onsor in eart hquakes.

2. From the architectural and functional points of view, it can beadvantageousnot to haveany tr iangulated bracing systemsor solid wall systemsin the buildi ng.

12.1.2 SimpleFrames(Pin-ConnectedFrames)

A simpleframerefers to a structural system in which the beamsand columnsare pinned connectedand the system is incapable of resisting any lateral loads. The stabil it y of the entire str ucture mustbeprovided for by attaching the simple frameto someform of bracing system. Thelateral loadsareresisted by the bracing systemswhilethe gravity loadsareresisted by both the simple frameand thebracing system.

In most cases, thelateral load responseof thebracingsystem is sufficiently small such that second-order effects may be neglected for t he design of the frames. Thus, the simple frames that are at-tached to the bracing system may beclassifi ed as non-sway frames. Figure12.1 showstheprincipalcomponentssimply frame and bracing systemof such a str ucture.

There are several reasonsof adopting pinned connectionsin the design of steel multistory frames:

1. Pin-jointed framesareeasier to fabricate and erect. For steel str uctures, it ismore conve-nient to join the websof the members without connecting the flanges.

2. Bolted connectionsarepreferred over welded connecti ons, which normally require weldinspection, weather protection, and surfacepreparation.

3. It is easier to design and analyze a building str ucture that can be separated i nto systemresisti ngvert ical loadsand system resistinghorizontal loads. For example, if all thegirdersare simply supported between the columns, the sizing of the simply supported girdersand thecolumnsis a straightforward task.

4. It is morecost effectiveto reducethe horizontal drif t by meansof bracing systemsaddedto thesimple framing than to useunbraced framesystemswith rigid connections.

Actual connectionsin structuresdo not alwaysfall within thecategoriesof pinned or rigid connec-ti ons. Practical connections are semi-rigid in nature and therefore the pinned and rigid conditi onsare only idealizations. Modern design codesallow the design of semi-rigid framesusing the conceptof wind moment design ( type 2 connections). In wind moment design, the connecti on isassumedto becapableof transmitt ing only part of thebendingmoments( thosedueto thewind only). Recentdevelopment in the analysis and design of semi-rigid framescan be obtained from Chen et al. [15].Design guidanceis given in Eurocode3 [22].



4/77

FIGURE 12.1: Simple braced fr ame.

12.1.3 BracingSystems

Bracing systemsrefer to structuresthat can providelateral stabil it y to the overall framework. It maybe in the form of triangulated fr ames, shear wall/cores, or rigid-jointed fr ames. It is common tofind bracing systemsrepresented asshown in Figure12.2.They arenormally located in buildingstoaccommodate lift shaftsand staircases.

In steel str uctures, it is common to represent a bracing systemby a tr iangulated truss because,unlike concrete structures where all the joints are naturally continuous, the most immediate wayof making connections between steel members is to hinge one member to the other. As a result,common steel building str ucturesare designed to havebracing systemsin order to providesideswayresistance. Therefore, bracing can only beobtained by use of tr iangulated tr usses(Figure12.2a) or,exceptionally, by a very sti ff str ucture such asshear wall or core wall (Figure12.2b). The efficiencyof a building to resist lateral forces depends on the location and the types of the bracing systemsemployed, and the presenceor absenceof shear wallsand coresaround li ft shaftsand stair wells.

12.1.4 BracedFramesvs. UnbracedFrames

The main function of a bracing system i s to resist lateral forces. Building frame systems can beseparated into vert ical load-resistance and hori zontal load-resistance systems. In some cases, theverti cal load-resistance system also has some capabil it y to resist hori zontal forces. It is necessary,

therefore, to identi fy the two sources of resistance and to compare their behavior with respect tothe horizontal actions. However, this identification is not that obvious sincethe bracing is integralwithin the structure. Someassumptionsneed to bemade in order to define the two str uctures forthepurposeof comparison.

Figures 12.3 and 12.4 represent the structuresthat areeasy to define within onesystem: two sub-assemblies identi fying the bracing system and the system to be braced. For the str ucture shown inFigure 12.3,there is a clear separation of functions in which the gravity loads are resisted by t hehinged subassembly (Frame B) and the horizontal load loads are resisted by the braced assembly



5/77

FIGURE12.2: Common bracing systems: (a) vert ical trusssystem and (b) shear wall.

FIGURE12.3: Pinned connected fr amesspli t into two subassemblies.

(FrameA). In contrast, for the structure in Figure12.4, sincet hesecond sub-assembly (FrameB) isable to resist horizontal actionsaswell asvert ical actions, it i snecessary to assumethat practi cally all

the horizontal acti onsarecarried by the first sub-assembly (FrameA) in order to define thissystemasbraced.

Eurocode3 [ 22]givesa clear guidance in defini ng braced and unbraced frames. A fr ame may beclassified asbraced if itssway resistanceissupplied byabracing system in whichitsresponseto lateralloadsi ssufficiently sti ff for i t to beacceptably accurateto assumeall horizontal loadsare resisted bythe bracing system. Theframecan beclassifi ed asbraced if the bracing system reducesitshorizontaldisplacement by at least 80%.



6/77

FIGURE 12.4: Mi xed framesspli t into two subassembli es.

For theframeshown in Figure12.3, thehinged frame(FrameB) hasno lateral sti ffness, and FrameA (t russ frame) resists all lateral load. In thiscase, FrameB is considered to bebraced by FrameA.For theframeshown in Figure12.4, FrameB may beconsidered to beabraced framei f thefollowingdeflection cri teri on is satisfied:

1 A

B

0.8 (12.1)

whereA = lateral deflection calculated from the tr ussframe(FrameA) aloneB = lateral deflection calculated from FrameB alone

Alternatively, the lateral sti ffnessof Frame A under the applied lateral load should beat least fiveti meslarger t han that of FrameB:

KA 5KB (12.2)whereKA = lateral sti ffnessof FrameAKB = lateral sti ffnessof FrameB

12.1.5 SwayFramesvs. Non-Sway Frames

Thei denti fication of sway framesand non-sway framesi n abuildingis useful for evaluating safety ofstructuresagainst instability. In thedesignof multistory buildingframe, it isconvenient to isolatethecolumnsfrom theframeand treat thestability of columnsand thestability of framesasindependentproblems. For acolumn in abraced frame, it is assumed that thecolumnsarerestr icted at their endsfrom horizontal displacements and therefore are only subjected to end moments and axial loadsastr ansferred from theframe. It is then assumed that theframe, possibly bymeansof abracingsystem,sati sfiesglobal stabil it y checksand that the global stabil it y of the frame does not affect the columnbehavior. This gives the commonly assumednon-sway frame. The design of columns in non-swayfr ames follows the conventional beam-column capacity check approach, and the column effectivelength may be evaluated based on the column end restr aint condit ions. Interaction equati ons forvari ous cross-section shapes have been developed through years of research spent in the field ofbeam-column design [12].

Another reason for defining sway and non-sway fr ames is the need to adopt conventionalanalysis in which all the internal forces are computed on the basis of the undeformed geometr yof the str ucture. This assumption is valid if second-order effects are negligible. When there isan i nteracti on between overall frame stabil it y and column stabil it y, i t is not possible to i solate thecolumn. Thecolumn and the framehaveto act interactively in a sway mode. The design of swayfr ameshasto consider theframesubassemblageor the str ucture asa whole. Moreover, the presenceof inelasticity in the columnswill render somedoubtson theuseof thefamiliar concept of elasticeffectivelength [ 45, 46].



7/77

On the basis of the aboveconsiderations, a definiti on can be established for sway and non-swayframesas:

A framecan beclassified asnon-swayif it sresponseto in-planehori zontal forcesissufficiently

sti ff for it to be acceptably accurate to neglect any addit ional internal forces or momentsari sing from hori zontal displacements of its nodes.

Briti sh Code: BS5950:Part 1 [11]providesa procedure to distinguish between sway and non-swayframesasfollows:

1. Applyaset of notional horizontal loadsto theframe. Thesenoti onal forcesareto betakenas0.5% of the factored dead plusvert ical imposed loadsand are applied in isolation, i.e.,without thesimultaneousappli cation of actual vert ical or hori zontal loading.

2. Carr y out a first-order linear elastic analysis and evaluate the individual relative swaydeflection for each stor y.

3. If theactual framei suncladed, the framemay beconsidered to benon-sway if theinter-story deflection of every story satisfiesthe following limit:

3.6 multistory officesitu shore buildings

concretePr e- cast 36 110200 Fast Fai r- good Fai r- good Al l cat egor ies

concrete with cranagerequirements

Pre- 69 110200 Medi um Fai r-good Fai r-good Multi stor ystressed buildingsandconcrete bridges

12.3 DesignConceptsandStructural Schemes

12.3.1 Introduction

Mult istory steel fr amesconsist of acolumn and a beam interconnected to form athree-dimensionalstr ucture. A building framecan bestabil ized eit her by someform of bracing system (braced frames)or can bestabil ized byitself (unbraced frames). All buildi ngframesmust bedesigned to resist lateralload to ensureoverall stability. A common approach is to provideagravity framing system with oneor morelateral bracingsystem attached to it. Thistypeof framing system, which isgenerally referredto assimple braced frames, is found to becost-effectivefor multistory buildingsof moderate height(up to 20 stories).

For gravity frames, the beamsand columnsare pinned connected and the framesare not capableof resistingany lateral loads. Thestability of theentirestructureis provided by attaching thegravityframesto someform of bracing system. Thelateral loadsareresisted mainly by the bracing systems,while thegravity loadsareresisted byboth thegravit y frameandthebracing system. For buildingsofmoderateheight, the bracing systemsresponseto lateral forcesis sufficiently sti ff such that second-order effectsmay beneglected for thedesign of such frames.

In moment resisting frames, the beams and columns are rigidly connected to provide momentresistance at joints, which may be used to resist lateral forces in the absenceof any bracing system.However, moment j oints are rather costly to fabricate. In additi on, it takes a longer t ime to erect amoment framethan a gravity frame.

A cost-effectiveframingsystem for multistorybuildi ngscan beachieved byminimi zing thenumberof moment j oints, replacing field welding by field bolting, and combining variousfr aming schemeswith appropriatebracing systemsto mini mizeframedri ft. A multistory str ucture ismost economicaland efficient when i t can transmit the applied loads to the foundation by the shortest and most



26/77

direct routes. For ease of constr uction, the structural schemes should be simple enough, whichimplies repeti tion of member and joints, adoption of standard structural details, straightforwardtemporary works, and mini mal requirements for inter-related erection procedures to achieve theintended behavior of the completed structure.

Sizing of structural members should be based on the longest spans and largest att ri buted roofand/or floor areas. The same sections should be used for simi lar but less onerous cases. Simplestructural schemes are quick to design and easy to erect. It also providesa good benchmark forfurther refinement. Many building str uctures have to accommodate extensive services within thefloor zone. It is important that the engineer chooses a str uctural scheme (see Section12.2)whichcan accommodate the servicerequirements within the restr icted floor zoneto mini mizeoverall cost.

Schemedrawingsfor multistory building designsshould includethe following:

1. General arrangement of thestructureincludingcolumn and beam layout, bracing frames,and floor systems.

2. Criti cal and typical member sizes.

3. Typical cladding and bracing details.

4. Typical and unusual connection details.5. Proposals for fireand corrosion protection.

Thissection offersadviceonthegeneral pri nciplesto beappliedwhen prepari ngastr uctural schemefor mult istory steel and composite fr ames. The aim is to establish several structural schemes thatare practicable, sensibly economic, and functional to the changesthat are likely to beencountered asthe overall design develops. Thesection beginsby examining the design procedure and constr uctionconsiderations that are specific to steel gravity fr ames, braced frames, and moment resisting frames,and the design approaches to beadopted for sizing tall buildi ng frames. The potential use of steel-concrete composite material for high-ri se constr uction is then presented. Finally, the design issuesrelated to braced and unbraced composite frames are discussed, and future directions for researcharehighlighted.

12.3.2 Gravity Frames

Gravit yframesrefer to str ucturesthat aredesigned to resist only gravit y loads. Thebasesfor designinggravity framesare asfollows:

1. Thebeam and girder connectionstransfer only verti cal shear reactionswithout developingbending moment that will adversely affect the members and the structureasa whole.

2. Thebeamsmay bedesigned asa simply supported member.

3. Columnsmust befully continuous. Thecolumnsare designed to carry axial loadsonly.Somecodesof practice(e.g., [ 11]) requiret hecolumn to carry nominal momentsduetothe reaction forceat thebeam end, applied at an appropriateeccentr icity.

4. Lateral forces are resisted entirely by bracing frames or by shear walls, lift, or staircaseclosures, through floor diaphragm action.

General Guides

Thefollowing pointsshould beobserved in the design of gravity frames:

1. Providelateral stabil it y to gravit y framing by arranging suitablebraced baysor corewallsdeployed symmetr ically in orthogonal directions, or wherever possible, to resist lateralforces.



27/77

2. Adopt asimplearr angement of slabs, beams, and columnssothat loadscan betransmittedto the foundati onsby the shortest and most direct load paths.

3. Tieall thecolumnseffectively in orthogonal directionsat every story. Thismaybeachievedby the provision of beamsor tiesthat are placed ascloseaspracti cable to the columns.

4. Select aflooringschemethat providesadequatelateral restr aint to thebeamsandadequatediaphragm action to tr ansfer the lateral load to the bracing system.

5. For tall building construction, chooseaprofiled-steel-deckingcompositefl oor constr uc-tion if uninterrupted floor space is required and/or height is at a premium. Asa guide,limit thespan of the floor slab to 2.5to 3.6m; thespan of thesecondary beamsto 6 to 12m; and the span of theprimary beamsto 5 to 7 m. Otherwise, choosea precast or aninsitureinforced concrete floor, limiti ng its span to 5 to 6 m, and thespan of t hebeamsto6 to 8 m approximately.

Structural Layout

In buildi ng constr uction, greater economy can be achieved through a repeti tion of similarlyfabricated components. A regular column grid is lessexpensive than a non-regular gri d for a given

floor area. Orthogonal arrangements of beams and columns, as opposed to skewed arrangements,providemaximum repeti ti onof standard details. In addit ion, greater economiescan beachievedwhenthecolumn gridsin theplan arerectangular in which thesecondary beamsshould span in thelongerdirection and theprimary beamsin theshorter, asshown in Figures12.22aand b. Thi sarrangementreducesthe number of beam-to-beam connectionsand the number of individual members per unitarea of thesupported floor [52].

In gravit y frames, the beamsare assumed to besimply supported between columns. Theeffectivebeam span to depth ratio (L/D) is about 12to 15 for steel beamsand 18 to 22 forcomposite beams.The design of the beam isoften dependent on the appli ed load, the typeof beam system employed,and the restri ctions on structural floor depth. The floor-to-floor height in a multistory buildingis influenced by the restr ictionson overall building height and the requirements for services aboveand/or below thefloor slab. Naturally, flooringsystemsinvolving theuseof str uctural steel membersthat act compositely with the concrete slab achievethe longest spans(seeSection12.2.5).

AnalysisandDesign

Theanalysis and design of asimplebraced framemust recognizethe following points:

1. Themembers intersecting at a joint are pin connected.

2. Thecolumnsarenot subjected to any direct moment transferred through theconnection(nominal moments dueto eccentr icity of the beam reaction forces may beconsidered).Thedesign axial forcein thecolumn is predominately governed by floor loading and thetr ibutary areas.

3. The str ucture is statically determinate. The internal forces and moments are thereforedetermined from a consideration of statics.

4. Gravity frames must be attached to a bracing system so as to provide lateral stabil it y tothe part of thestr uctureresisting gravit y load. Theframecan bedesigned asanon-swayframeand the second-order moments associated with framedrift can bei gnored.

5. Theleaning column effects dueto column sidesway must beconsidered in the design ofthe fr amesthat are part icipati ng in sidesway resistance.

Sincethe beamsare designed assimply supported between their supports, the bending momentsand shear forcesare independent of beam size. Therefore, initi al sizing of beamsisastr aightforward



28/77

FIGURE 12.22: ( a) Retangular grid layout and (b) preferred and non-preferred gri d layout.

task. Beam or girder members supporting morethan 40m2 of fl oor at onestory should bedesignedfor a reduced liveload in accordancewith ASCE[ 6].

Most conventional types of floor slab constr uction will provide adequate lateral restraint to thecompression flange of the beam. Consequently, the beams may be designed as laterally restr ainedbeamswithout themoment resistancebeing reduced by lateral-torsional buckling.

Under the service loading, the total centr al deflection of the beam or the deflection of the beamdue to unfactored l ive load ( with proper precambering for dead load) should satisfy the deflecti onlimitsasgiven in Table12.2.

In some occasions, it may be necessary to check the dynamic sensitivit y of the beams. Whenassessing thedeflection and dynamic sensit ivity of secondary beams, thedeflection of thesupportingbeams must also be included. Whether it is the strength, deflection, or dynamic sensit ivity thatcontrols the design will depend on the span-to-depth r atio of the beam. Figure 12.18 gives typicalspan rangesfor beamsin officebuildingsfor which the design would beoptimi zed for strength andserviceability. For beamswith their span lengthsexceedingthoseshown in Figure12.18, serviceabilitylimits dueto deflection and vibration will most likely bethegoverning criteria for design.

Therequired axial forcesin thecolumnscan bederived from thecumulativereaction forcesfromthosebeamsthat frameinto thecolumns. Liveload reduction should beconsidered in thedesign ofcolumnsin a multistory frame[ 6]. If theframeis braced against sidesway, the column nodepoint sare prevented from lateral t ranslation. A conservativeesti mate of column effective length, K L, forbuckling considerations is 1.0L, where Lis the story height. However, in cases where the columnsabove and below t he story under consideration are underutil ized in terms of load resistance, therestr aini ngeffectsoffered by thesemembersmay result in an effectivelength of lessthan 1.0L for the



29/77

TABLE12.2 Recommended Deflection Limits for Steel Building

Frames

Beam deflecti onsfrom unfactored imposed loads

Beamscarrying plaster or brittl efinish Span/360(with maximumof 1/4to 1 in.)Other beams Span/240

Columnsdeflectionsf rom unfactored imposed and wind loads

Column in singlestory frames Height/300Column in multistory frames Height of story/300For column supportingcladding Height of story/500

which is sensit ive to largemovement

Framedri ft under 50years wind load

Framedrift Frame height/450 fr ame height/600

column under consideration. Such a situati on ari ses where the column is continuousthrough the

restr aint points and the columnsaboveand/or below the restr aint points are of different length.

An example of such cases is the continuous column shown in Figure 12.23in which ColumnAB is longer than Column BC and hence Column AB is restrained by Column BC at the restr aintpoint B. A buckling analysis shows that the critical buckling load for the continuous column isPcr= 5.89EI/L2, which gives rise to an effective length factor ofK= 0.862 for Column AB andK= 1.294for Column BC. Column BChasalarger effectivelength factor becauseit providesrestr aintto Column AB, whereas Column AB has a smaller effectivelength factor because it is restr ained byColumn BC during buckling. Figure12.24 summari esthe reductionsin effectivelength which maybe considered for columns in a frame with different story heights having various values of a/Lratios[52].

FIGURE 12.23: Buckling of a continuouscolumn with intermediaterestrain.



30/77

FIGURE 12.24: Effectivelength factors of continuousbraced columns.

SimpleShearConnections

Simpleshear connecti onsshould bedesigned and detailedto allow freerotation and to preventexcessive tr ansfer of moment between the beams and columns. Such connections should complywith the classifi cati on requirement for a nominally pinned connection in terms of both strengthand sti ffness. A computer program for connection classifi cation hasbeen madeavail ablei n abook byChen et al. [15], and their design implicationsfor semi-r igid framesarediscussed in Liew et al. [47].

Simple connections are designed to resist vert ical shear at the beam end. Depending on theconnection details adopted, it may also be necessary to consider an addit ional bending momentresulti ng from the eccentri city of the bolt line from the supporting face. Often the fabricator istold to design connectionsbased on thebeam end reaction for one-half uniformed distr ibuted load(UDL). Unlesstheconcentr ated load isl ocated very near to thebeam end, UDL reactionsaregenerallyconservative. Because of the largereaction, the connection becomesvery strong which may requirea large number of bolts. Thus, it would be a good practice to design the connectionsfor theactualforcesused in thedesign of thebeam. Theengineer should givethedesign shear forcefor every beamto the steel fabricator so that a more reali stic connecti on can be designed, instead of requiring allconnections to develop the shear capacity of the beam. Figure12.25 showsthe typi cal connectionsthat can be designed as simple connections. When the beam reaction i s known, capacity tables

developed for simplestandard connectionscan beused for detaili ng such connections[2].

12.3.3 BracingSystems

The main purpose of a bracing system is to provide the lateral stabili ty to the entire structure. Ithas to be designed to resist all possible kinds of lateral loading due to external forces, e.g., windforces, earthquakeforces, and leaning forces from the gravit y frames. Thewind or the equivalent



31/77

FIGURE12.25: Typical beam-to-column connectionsto beconsidered asshear connections.

eart hquake forces on the str ucture, whichever are greater, should be assessed and divi ded into thenumber of bracing baysresisting the lateral forcesin each direction.

Structural Forms

Steel braced systemsareoften in aform of avert ical tr usswhichbehaveslikecanti lever elementsunder lateral loads developing tension and compression in the column chords. Shear forces areresisted by the bracing members. The tr uss diagonalization may take various forms, as shown inFigure12.26. Thedesign of such str ucturesmust takeinto account themanner in which the framesare erected, the distributi on of lateral forces, and their sidesway resistance.

In thesinglebraced forms, whereasinglediagonal braceisused (Figure12.26a), it must becapableof resisting both tensile and compressive axial forces caused by the alternate wind load. Hollowsections may be used for the diagonal braces as they are stronger in compression. In the designof diagonal braces, gravit y forces may tend to dominate the axial forces in the members and dueconsideration must begiven in the design of such members. It is recommended that theslendernessratio of the bracing member(L/r)not be greater than 200 to prevent the self-weight deflecti on ofthe bracelimit ing itscompressiveresistance.



32/77

FIGURE12.26: (a) Diagonal bracing, (b) cross-bracing, (c) K-bracing, and (d) eccentri c bracing.

In across-braced system (Figure12.26b), the bracemembers areusually designed to resist tensiononly. Consequently, light sections such as str uctural angles and channels, or ti e rods can be usedto provide a very sti ff overall str uctural response. The advantage of the cross-braced system is thatthe beamsare not subjected to significant axial force, asthe lateral forcesare mostly taken up by thebracing members.

The K tr usses are common sincethe diagonals do not part icipate extensively in carr ying columnload, and can thusbedesigned for wind axial forceswi thout gravity axial forcebeing considered asamajor contributi on. A K-braced frameis moreefficient i n preventing sidesway than a cross-bracedfr ame for equal steel areas of braced members used. Thistypeof system is preferred for longer baywidth because of the shorter length of the braces. A K-braced frameis found to bemore efficient i fthe apexesof all thebracesare pointing in theupward direction (Figure 12.26c).

For eccentr ically braced frames, thecenter lineof thebraceisposit ioned eccentr ically to thebeam-column joint, asshown in Figure12.26d. Thesystem reli es, in part, on flexureof the short segmentof the beam between the brace-beam j oint and the beam-column joint. Theforcesin the bracesaretr ansmitted to the column through shear and bending of the short beam segment. This particulararr angement provides a more flexible overall response. Nevertheless, it is more effective againstseismic loading becausei t allowsfor energy dissipation dueto flexural and shear yieldingof theshortbeam segment.

Drift Assessment

The story drift of a single story diagonally braced frame, asshown in Figure 12.27,can beapproximated by thefollowing equation:

= s+f

= H L3

d

AdEL2+ H h

3

AcEL2 (12.3)



33/77

where

= inter-story drifts = story drift dueto shear componentf

= story drift dueto flexural component

Ac = area of the chordAd = area of the diagonal braceE = modulusof elasticit yH = horizontal forcein thestoryh = story heightL = length of braced bayLd = length of the diagonal brace

FIGURE12.27: Lateral displacement of adiagonally braced frame.

Theshear componentsin Equation 12.3 is caused mainly by thestr aini ngof thediagonal brace.The deformati on associated with girder compression has been neglected i n the calculati on ofsbecause the axial sti ffness of the girder is very much larger than the sti ffness of the brace. Theelongation of the diagonal bracesgivesr iseto shear deformation of theframe, which is afunction ofthebracelength, Ld, and theangle of thebrace(Ld/L). A shorter bracelength with asmaller braceangle will producea lower story drift.

Theflexural component of theframedri ft isdueto tension and compression of thewindward andleeward columns. The extension of the windward column and shortening of the leeward columncauseflexural deformation of theframe, which is a function of thearea of thecolumn and the ratioof the height-to-bay length (h/L). For a slender bracing frame with a largeh/Lratio, the flexuralcomponent can contribute significantly to the overall story drift.

A low-ri sebraced framedeflects predominately in shear modewhilehigh-risebraced framestendto deflect morein flexural mode.

DesignConsiderations

Frames wit h braces connecting columns may obstr uct locati ons of access openings such aswindows and doors; thus, they should be placed where such access is not required, e.g., aroundelevators and service and stair wells. The location of the bracing systems within the structure will



34/77

infl uencethe efficiency with which the lateral forcescan beresisted. Themost appropriateposit ionfor thebracing systemsi sat the periphery of the building (Figure12.28a) because thisarrangementprovides greater torsional resistance. Bracing fr ames should be situated where the center of lateralresistance is approximately equal to the center of shear resultant on t he plan. Where this is not

possible, torsional forces will be induced, and they must be considered when calculating the loadcarr ied by each braced system.

FIGURE12.28: Locati ons of bracing systems: (a) exterior braced frames, ( b) internal braced core,and (c) bracing arrangements to beavoided.

When core braced systems are used, they are normally located i n the center of the building (Fig-ure12.28b). Thetorsional stability ist hen provided bythetorsional rigidit y of thecorebrace. For tallbuilding frames, aminimum of threebraced bentsarerequired to providetransitional and torsionalstabil it y. These bents should becarefully arranged so that their planesof action do not meet at onepoint so asto form acenter of rotation. Thebracing arrangement shown in Figure12.28c should beavoided.



35/77

Theflexibil it y of different bracing systemsmust betaken i nto account i n theanalysis becausethesti ffer braces will attract a larger share of t he applied lateral load. For tall and slender frames, t hebracing system itself can be a sway frame, and a second-order analysis is required to evaluate therequired forcesfor ultimatestrength and serviceabili ty checks.

Lateral loadsproduce transverseshears, overt urning moments, and sidesway. The sti ffness andstrength demands on the lateral system increase dramati cally wit h height. The shear i ncreases li n-early, the overturning moment as a second power, and the sway as a fourth power of t he height ofthe building. Therefore, apart from providing the strength to resist lateral shear and overturningmoments, the dominant design consideration (especially for tall building) is to develop adequatelateral stif fnessto control sway.

For serviceabilit yveri ficati on,it requiresthat both theinter-storydriftsand thelateral deflectionsofthestructureasawholemust belimited. Theli mitsdependonthesensit ivity of thestructural elementsto shear deformations. Recommended limits for t ypical multi stor y frames are given in Table 12.2.When considering the ultimate limit state, the bracing system must be capable of transmitting thefactored lateral loads safely down to the foundations. Braced bays should be effective throughoutthe full height of thebuilding. If i t is essential for bracing to bediscontinuousat onelevel, provision

must bemadeto transfer the forcesto other braced bays. Where thisis not possible, torsional forcesmay bei nduced, and they should beallowed for in the design (seeSection12.2.6).

The design of the internal bracing members in a steel bracing system is simi lar to the design oflatticetrusses. Thehorizontal member in a latticed bracing system servesalso asa floor beam. Thismember will be subjected to bending dueto gravity loadsand axial compression dueto wind. Thecolumnsmust be designed for additi onal forcesdueto leaning column effects from adjacent gravityframes. Theresistanceof the members should therefore bechecked asa beam-column based on theappropriateload combinations.

Figure12.29 showsan exampleof abuildi ngthat illustr atesthel ocati onsof vert ical braced tr ussesprovided at the four corners to achieve lateral stabil it y. Diaphragm action is provided by 130 mmli ghtweight aggregate concrete slab which actscompositely wit h metal decking and floor beams. Thefloor beam-to-column connectionsaredesigned to resist shear forceonly asshown in the figure.

FIGURE 12.29: Simplebuildi ng framewith vert ical braced trusseslocated at the corners.



36/77

12.3.4 Moment-ResistingFrames

In caseswherebracing systemswould disturb thefunctioningof thebuilding, ri gidly jointed momentresistingframescan beused to providel ateral stability to thebuilding, asi llustrated in Figure12.30a.The efficiency of development of lateral stiffness is dependent on bay span, number of bays in theframe, number of frames, and theavailabledepth in thefloorsfor theframegirders. For buildingwithheightsnot morethan threetimestheplan dimension, themoment framesystem isan efficient form.Bay dimensionsin the range of 6 to 9 m and structural height up to 20 to 30 storiesare commonlyused. However, as the building height i ncreases, deeper girders are required to control dri ft ; thus,the design becomesuneconomical.

FIGURE 12.30: Sidesway resistanceof a ri gid unbraced frame.

When arigid unbraced frameis subjected to lateral load, the horizontal shear in astory is resistedpredominantly by the bending of columns and beams. These deformati ons cause the frame todeform in ashear mode. The design of theseframesis controlled, therefore, by the bending sti ffnessof individual members. The deeper the member, the more efficiently the bending sti ffness can bedeveloped. A small part of the frame sidesway i s caused by the overtur ning of the entire frameresulting in shortening and elongation of thecolumnsat opposite sidesof theframe. For unbracedrigid framesup to 20 to 30 stories, theoverturning moment contributesfor about 10 to 20% of thetotal sway, whereas shear racking accounts for the remaini ng 80 to 90% (Figure 12.30b). However,the story drift dueto overall bending tendsto increase with height, while that due to shear rackingtendsto decrease.

Drift Assessment

Sinceshear racking accountsfor most of the lateral sway, the design of such framesshould bedirected towards minimizing the sidesway dueto shear. Theshear displacement in atypical storyin a multistory frame, asshown in Figure12.31, can beapproximated by the equation:

i=Vi h

2

i

12E

1

(Ici / hi )+ 1

(Igi /Li )

(12.4)



37/77

where

i = the shear deflection of the i-th storyE = modulusof elasticityIc, Ig

= second moment of areafor columns and girders, respectively

hi = height of the i-th storyLi = length of girder in the i-th storyVi = total horizontal shear forcein the i-th story(Ici / hi )= sum of thecolumn stiffnessi n thei-th story(Igi /Li ) = sum of the girder stiffnessin the i-th story

FIGURE12.31: Story drift dueto (a) bending of columnsand (b) bending of girders.

Examination of Equation 12.4showsthat sidesway deflection caused bystory shear isinfluenced bythesum of thecolumn and beam stiffnessi n astory. Sincefor multistory construction, span lengthsare generally larger than t hestory height, the moment of i nertia of t hegirders needsto belarger tomatch the column sti ffness, as both of these members contribute equally to the story drift. As thebeam span increases, considerably deeper beam sectionswill berequired to control framedrift.

Sincet hegravity forcesi n columnsarecumulative, larger column sizesareneeded in lower stori esasthe frameheight increases. Similarly, story shear forcesarecumulati veand, therefore, larger beampropert iesin l ower stori es are required to control lateral drift. Because of li mitati ons in availabledepth, heavier beam memberswi ll need to beprovided at lower floors. Thisisthemajor shortcomingof unbraced framesbecauseconsiderable premium for steel weight isrequired to control lateral driftasbuil ding height increases.

Apart from the beam span, height-to-width r atiosof the building play an i mportant role in thedesign of such str uctures. Wider buildingframesallow alarger number of bays(i .e., larger valuesfor



38/77

story summation terms (Ici / hi )and(Igi /Li )in Equation12.4)with consequent reduction inframedrift. Moment frameswith closed spaced columnsthat areconnected by deep beamsare veryeffectivein resisting sidesway. This kind of framing system is suitable for use in the exterior planesof thebuilding.

MomentConnections

Full y welded moment j oint s are expensive to fabricate. To mini mizelabor cost and to speedup site erecti on, field bolting instead of field welding should be used. Figure 12.32shows severaltypes of bolted or welded moment connections that are used in practi ce. Beam-to-column flangeconnectionscan beshop-fabricated bywelding of abeam stub to an end plateor directly to acolumn.Thebeam can then beerected by field bolting theend plate to thecolumn flangesor splicing beams(Figures 12.32c and d).

FIGURE12.32: Rigid connections: (a) bolted and welded connection with doubler plate, (b) boltedand welded connection with diagonal sti ffener, (c) bolted end-plateconnection, and (d) beam-stubwelded to column.

An addit ional parameter to be considered i n the design of columns of an unbraced frameis thepanel zone between the column and the tr ansversefr aming beams. When an unbraced frame issubjected to l ateral load, addit ional shear forces are induced in the column web panel as shown in



39/77

Figure 12.33. The shear force is induced by the unbalanced moments from the adjoini ng beamscausing the joint panel to deform i n shear. The deformation is attr ibuted to the large flexibili tyof the unsti ffened column web. To prevent shear deformation so as to maintain the moment jointassumpti on asassumed in theglobal analysis, it may benecessary to sti ffen thepanel zoneusingeit her

adoubler plateor adiagonal stiffener asshown in thejoint detailsin Figures12.32aand b. Otherwi se,a heavier column with a larger web area is required to prevent excessiveshear deformation, and thisis often the preferred method assti ffeners and doublers can add significant costs to fabri cation.

FIGURE12.33: Forcesacting on apanel joint: (a) balanced moment dueto gravit y load and (b) un-balanced moment dueto lateral load.

The engineer should not specify full strength moment connections unless they are required forducti le frame design for high seismic loads. For wi nd loads and for conventi onal moment frameswhere beams and columns are sized for sti ffness (drift control) instead of strength, full strengthmoment connections are not required. Even so, many designers will specify full strength momentconnecti ons, adding to the cost of fabrication. Designing for actual loadshasthepotential to reducecolumn weight or the sti ffener and doubler plate requirements.

If the panel zone is sti ffened to prevent inelastic shear deformation, the conventional structuralanalysis based on the member center-line dimension wil l generally overesti mate the framedisplace-ment. If thebeam-column joint sizesarerelatively small compared to themember spans, theincreasein framesti ffnessusing member center-l ine dimension will beoffset by the increasein framedeflec-tion dueto panel-joint shear deformation. If the joint sizesare large, a morerigoroussecond-order

analysis, whi ch considers panel zone defor mations, may be required for an accurate assessment oftheframeresponse[43].

AnalysisandDesignofUnbracedFrames

Mult istory moment frames are statically indeterminate. The required design forces can bedetermined using either: (1) elasti c analysisor (2) plasti c analysis. While elasti c methods of analysiscan beused for all kind of steel sections, plasti canalysis is only appli cable for frameswhosemembers



40/77

are of plastic sections so as to enable the development of plastic hinges and to allow for inelasticredistr ibution of forces.

First-order elastic analysis can beused only in the following cases:

1. Where the frameis braced and not subjected to sidesway.

2. Where an indirect allowance for second-order effects is made through the use of mo-ment amplification factors and/or thecolumn effectivelength. Eurocode3 requiresonlysecond-order moment or effectivelength factor to beused in the beam-column capacit ychecks. However, column and frameimperfectionsneed to bemodeled explicit ly in theanalysis. I n AISC LRFD [3], both factorsneed to becomputed for checking thememberstrength and stabil it y, and theanalysisisbased onstr uctureswithout ini tial imperfections.

The fir st- order elasti c analysis is a convenient approach. Most design offices possess computersoftware capable of performing this method of analysis on large and highly indeterminate str uc-tures. Asan alternative, hand calculations can beperformed on appropriate sub-frames withi n thestructure(seeFigure12.34) compri singasignificantly reduced number of members. However, whenconducting the analysis of an isolated sub-frameit is important that:

1. the sub-frameis indeed representativeof the str ucture asa whole2. theselected boundary conditi onsareappropriate

3. account is taken of the possible interaction effectsbetween adjacent sub-frames

4. allowanceis madefor second-order effects through the useof column effectivelength ormoment amplificati on factors

FIGURE 12.34: Sub-frameanalysis for gravity loads.

Plasti c analysis generally requires more sophisticated computer programs, which enable second-order effectsto betaken intoaccount. Computer softwareisnowavailablethroughrecent publicationsmade available by Chen and Toma [ 14] and Chen et al. [ 15]. For building structures in which therequired rotations are not calculated, all members containi ng plastic hingesmust haveplastic cross-sections.



41/77

A basic procedurefor the design of an unbraced frameis asfollows:

1. Obtain approximate member size based on gravity load analysis of sub-frames shownin Figure12.34. If sidesway deflection is likely to control (e.g., slender fr ames) useEquation 12.4 to esti mate the member sizes.

2. Determine wind momentsfrom theanalysis of theentireframesubjected to lateral load.A simple portal wind analysis may beused in lieu of thecomputer analysis.

3. Check member capacit y for the combined effects of factored lateral load plus gravityloads.

4. Check beam deflection and framedrift .

5. Redesign the members and perform final analysis/design check ( a second-order elasti canalysisispreferableat thefinal stage).

The need to repeat the analysis to correspond to changed section sizes is unavoidable for highlyredundant frames. Iteration of Steps1 to 5 gives results that wi ll convergeto an economical designsati sfying the vari ousdesign constr aintsimposed on the analysis.

12.3.5 Tall BuildingFramingSystems

The following subsections discuss four classical systems that have been adopted for tall buildingconstr uctions, namely (1) core braced, (2) moment-tr uss, (3) outriggle and belt, and (4) tube. Tallframesthat utilizecanti lever action will havehigher efficiencies, but the overall structural efficiencydependson theheight-to-width ratio. Interactivesystemsinvolvingmoment frameandvert ical tr ussor coreare effectiveup to 40stori esand represent most buil ding forms for tall str uctures. Outriggertr uss and belt tr uss help to further enhance the lateral sti ffness by engaging the exterior frameswith the corebracesto develop canti lever actions. Exterior framed tube systemswith closely spacedexterior columns connected by deep girders mobilize the three-dimensional acti on to resist lateraland torsional forces. Bundled tubes improve the efficiency of exterior frame tubes by providinginternal sti ffening to the exteri or tube concept. Finally, by providing diagonal bracesto the exteriorframework, asuperframeisformed and can beused for ultr a-tall magastr uctures.

CoreBracedSystems

This type of structural system relies entirely on the internal core for lateral load resistance.The basic concept is to provide internal shear wall core to resist the lateral forces (Figure 12.35).The surrounding steel framing is designed to carr y gravity load only if simple framing is adopted.Otherwise, a rigid framing surrounding the core will enhance the overall lateral-forceresistance ofthe str ucture. Thesteel beamscan besimply connected to thecorewallsusing atypical corbel detail,or by bearing in a wall pocket or by shear plate embedded in the core wall through studs. If rigidconnection is required, the steel beams should be ri gidly connected to steel columns embedded inthecorewall. Rigid framing surroundingthecoresispart icularly useful in high seismic areas, and forvery tall buildingsthat tend to attract stronger wind loads. They act asmoment fr amesand provideresistanceto somepart of the lateral loadsby engaging the corewallsin the buildi ng.

The core generally provides all torsional and flexural ri gidity and strength with no participationfrom thesteel system. Conceptually, thecoresystem should betreatedasacantilever wall system wit hpunched openingsfor access. Thefloor-framing should bearranged in such away that it distr ibutesenough gravit y loads to the core walls so that their design is controlled by compressive stresseseven under wind loads. The geometr ic location of the core should be selected so asto mini mizeeccentri citiesfor lateral load. The core wallsneed to haveadequate torsional resistance for possibleasymmetr y of the coresystem where the center of the resultant shear load is acting at an eccentr icit yfrom the center of thelateral-forceresistance.



42/77

FIGURE12.35: Core-braceframe. (a) Internal corewallswit h simpleexteri or framing and (b) beam-to-wall and beam-to-exterior column connections.

A simple canti lever model should be adequate to analyze a core wall str ucture. However, i f thestr uctural form is a tube with openingsfor access, it may be necessary to perform a more accurateanalysisto includetheeffect of openings. Thewallscan beanalyzed byafini teelement analysis usingthin-walled plateelements. An analysisof thistypemay also berequired to evaluatetorsional stresseswhen the vert ical profileof the core-wall assembly is asymmetr ical.

Theconcretecorewallscan beconstr ucted using slip-form techniques, wherethe corewallscouldbe advanced several floors (typically 4 to 6 story) ahead of the exterior steel framing. A core wallsystem represents an efficient type of str uctural system up to a cert ain height premium because ofits cantil ever action. However, when it i sused alone, the massivenessof the wall str ucture increaseswith height, thereby inhabit ing the freeplanning of interior spaces, especially in the core. Thespaceoccupied by the shear wallsleadsto lossof overall floor areaefficiency, ascompared to atubesystemwhich could otherwisebeused.

In commercial buildingswherefl oor spaceisvaluable, thelargeareataken up byaconcretecolumncan be reduced by the use of an embedded steel column to resist the extreme loadsencountered intall buildings. Sometimes, particularly at thebottom open floorsof ahigh risestructure wherelargeopen lobbiesor atr iumsareut il ized aspart of thearchitectural design, aheavyembedded steel secti onaspart of a composite column is necessary to resist high load and dueto the largeunbraced length.A heavy steel section in a composite column is often util ized where the column size is restrictedarchitecturally and where reinforcing steel percentages would otherwise exceed the maximum codeallowed valuesfor the deign of reinforced concrete columns.



43/77

Moment-TrussSystems

Vertical shear trusseslocated around the inner corein oneor both directionscan becombinedwith perimeter moment-resisting frames in the facade of a building to form an efficient structure

for lateral load resistance. An example of abuil ding consisting of moment fr ameswit h shear tr usseslocated at the center of the building is shown in Figure 12.36a. For the vert ical trusses arr anged inthe North-South direction, either theK- or X-form of bracing is acceptable sinceaccessto li ft-shaftsis not required. However, K tr ussesare often preferred becausein the caseof X or single braceformbracings, theinfl uenceof gravity loadsisr ather significant. In theEast-West direction, only theKneebracing is effectivei n resisting lateral load.

In somecases, internalbracingcan beprovidedusingconcreteshear wallsasshownin Figure12.36b.Theinternal corewalls substitute thesteel trussesi n K, X, or asinglebraceform which may interferewith openingsthat provide accessto, for example, elevators.

FIGURE 12.36: (a) Moment-frameswith internal braced tr usses. (b) Moment-frameswith internalcorewalls.



44/77


45/77

FIGURE12.38: Behavior of fr ames subjected to l ateral load: (a) i ndependent behavior and ( b) in-teractivebehavior.

Figure 12.39shows a typical example of such a system. The outri gger t russ leads the wind forcesof the core truss to the exterior columns providing cantilever behavior of the total frame system.Thebelt tr ussin the facadeimprovesthe canti lever participati on of the exterior fr ameand createsathree-dimensional fr amebehavior.

FIGURE12.39: Outr igger and belt-trusssystem.

Figure 12.40 showsa schemati c diagram that demonstr atesthe sway characteristi c of the overallbuilding under lateral load. Deflection is significantly reduced by theintroduction of t heoutrigger-belt trusses. Two kinds of sti ffening effects can beobserved; onei srelated to the part icipati on of theexternal columns together with the internal core to act in a canti lever mode; the other is related to



46/77

the sti ffening of the external facade frame by the belt t russ to act as a three-dimensional tube. Theoverall sti ffnesscan beincreased up to 25% ascompared to theshear tr ussand framesystem withoutsuch outrigger-belt t russes.

FIGURE12.40: Improvement of lateral sti ffnessusing outr igger-belt tr usssystem.

The efficiency of this system is related to the number of trussed levels and the depth of the truss.In some cases the outri gger and belt tr usses have a depth of t wo or more floors. They are locatedin services floors where there are no requirements for wide open spaces. These tr usses are oftenpleasingly integrated into the architectural conception of the facade.

FrameTubeSystems

Figure12.41 showsa typical frame tube system, which consistsof a frametube at the exteri orof the building and gravity steel framing at the interior. Theframed tube is constructed from widecolumnsplaced at closecentersconnected by deep beamscreati ng a punched wall appearance. The

exterior frametube structure resists all lateral loadsof wind or earthquakewhereasthe gravity steelframing in theinterior resistsonly itsshareof gravity loads. Thebehavior of t heexterior framet ubeis simil ar to a hollow perforated tube. Theovertur ning moment under the action of lateral load isresisted by compression and tension of the leedward and windward columns, which are called theflange columns. The shear i s resisted by bending of the columns and beams at the two sides of thebuilding parallel to the direction of thelateral load, which arecalled the web frames.

Deepening on the shear rigidity of theframetube, theremay exist ashear lagacrosst hewindwardand leeward sidesof the tube. Asa result of this, not all t heflange columns resist the sameamount



47/77

FIGURE 12.41: Composite tubular system.

of axial force. An approximate approach is to assume an equivalent column model as shown inFigure12.42.In thecalculation of thelateral deflection of theframetubei t isassumed that only theequivalent flangecolumnson the windward and leeward sidesof the tubeand the web frameswouldcontribute to themoment of inertia of thetube.

Theuseof an exterior framed tubehasthreedisti nct advantages: (1) it developshigh ri gidity andstrength for torsional and lateralload resistance because the str uctural components are effectivelyplaced at the exterior of thebuilding forming athree-dimensional closed section; (2) massivenessofthe frametube system eliminatespotenti al upli ft difficultiesand producesbetter dynamic behavior;and ( 3) the use of gravity steel framing in the interior has the advantages of flexibili ty and enablesrapid constr uction. If acompositefloor with metal decking isused, electrical and mechanical servicescan beincorporated in thefloor zone.

Compositecolumnsare frequently used in the perimeter of the buildi ng where the closely spacedcolumns work in conjunction with the spandrel beam (either steel or concrete) to form a three-



48/77

FIGURE12.42: Equivalent column model for frametube.

dimensional cantil ever tuberather than an assembly of two-dimensional planefr ames. Theexteriorframetube significantly enhancesthe str uctural efficiency in resisting lateral loadsand thus reducestheshear wall requirements. However, in caseswhereahigher magnitudeof lateral sti ffnessisrequired(such as for very tall buildings), internal wall cores and interi or columns with floor fr aming can beadded to transform the system i nto a tube-in-tube system. The concrete core may be str ategicallylocated to recapture elevator spaceand to providetr ansmission of mechanical ducts from shaftsandmechanical rooms.

12.3.6 Steel-ConcreteCompositeSystems

Steel-concrete composite constructi on hasgained wideacceptanceasan alternativeto pure steel andpureconcrete construction. Compositebuildi ngsystemscan bebroadly categorized into two forms:one uti lizes the core-braced system by means of interior shear walls, and the other utilizes exteriorframing to form a tubefor lateral load resistance. Combining thesetwo structural formswill enabletaller buildingsto beconstructed.

BracedCompositeFramesSubjectedtoGravity Loads

For composite frames resisting gravit y load only, the beam-to-column connectionsbehaveaspinnedbeforet heplacement of concrete. Duri ngconstructi on, thebeam isdesigned to resist concretedead load and the construction load (to bet reated astemporary live load). At t hecomposite stage,

the composite strength and sti ffness of the beam should be uti lized to resist the full design loads.For gravit y frames consisting of bare steel columns and composite beams, there is now sufficientknowledgeavailable for the designer to usecompositeacti on in the structural element aswell asthesemi-rigid compositejointsto increasedesign choices, leading to moreeconomical soluti ons[38, 39].

Figures12.43aand b showthetypical beam-to-column connections, oneusing afl ushed end-platebolted to the column flangeand the other using a bottom angle with doubleweb cleats. Compositeaction i n the joint is acquired based on the tensile forces developed in the rebars acting with thebalancing compression forcestr ansmitted by the lower port ion of the steel section that bears against



49/77

the column flange to form a couple. Properly designed and detailed composite connections arecapable of providing moment resistanceup to the hogging resistanceof the connecting members.

FIGURE12.43: Compositebeam-to-column connectionswith (a) flushed end plateand (b) seat anddouble web angles.

In designing the connections, slab reinforcements placed within 7 column flange widths are as-sumed to be effective in resisting the hogging moment. Reinforcement steels that fall outside thiswidth should not be considered in calculating the resisting moment of the connection. The con-



50/77

nections to edge columns should be carefully detailed to ensure adequate anchorage of re-bars.Otherwise, they shall be designed and detailed as simply supported. In a braced frame, a momentconnection to theexterior column will increaset hemomentsin thecolumn, resulting in an increaseof column size. Although themoment connectionsrestr ain the column from buckling by reducing

theeffectivelength, thisisgenerally not adequateto offset thestrength requiredto resist thismoment.The moment of inertia of the composite beamIcpmay be esti mated using a weighted average of

moment of inertia in thepositivemoment region (Ip) and negativemoment region (In). For interiorspans, approximately 60% of the span is experiencing posit ivemoment; it is suggested that [ 37] :

Icp= 0.6Ip+0.4In (12.5)

where Ip is the lower bound moment of i nertia for positive moment and In is the lower boundmoment of inertia for negativemoment. However, if the connectionsat both endsof thebeam aredesigned and detailed for a simply supported beam, the beam will bend in single curvature underthe action of gravity loads, and Ip should beused throughout.

UnbracedCompositeFrames

If reinforcements areprovided in the concreteencasement, compositedesign of membersmaybe uti lized for strength and sti ffness assessment of the overall str ucture. The composite bendingsti ffnessof t hegirder incorporating the slab may beutil ized to reduce steel premium in controllingdrift for high-riseframedesign.

Oneapproach isto use the composite beamsaspart of the frame. Sincethe slab element alreadyexists, thecompositefl exural sti ffnessof thebeamscan beutil ized with thesteel beam aloneresistingall negati vemoments. For an unbraced framesubjectedto gravit yand lateral loads, thebeam typicallybendsin double curvature with a negativemoment at one end of the beam and a posit ive momentat theother end. The concrete is assumed to beineffectivein tension; therefore, only the steel beamsti ffnesson the negativemoment region and the compositesti ffnesson the posit ivemoment regioncan beutil ized for frame action. The frame analysis can beperformed with a variable moment ofinertiafor thebeams(seeFigure12.44). Further research is sti ll needed in order to providetangibleguidancefor design.

If semi-rigid composite joints are used in unbraced frames, the flexibili ty of the connectionswil lcontributeto additional drift over that of afullyrigid frame. In general, semi-rigid connectionsdonotrequire the column sizeto beincreased significantly over an equivalent rigid frame. This is becausethe design of frames with semi-rigid composite joints takes advantage of the additional sti ffness inthe beams provided by the composite action. The increase in beam stiffness would part ially offsetthe additional flexibili ty introduced by the semi-rigid connections.

Thestor yshear displacement in an unbraced framecan beesti matedusingamodifiedexpressionfrom Equation12.4 to account for the connection flexibili ty:

i=Vi h

2

i

12E

1

(Ici / hi )+ 1

(Icpi/Li )

+ Vi h

2

i

Kcon(12.6)

wherei = the shear deflection of the i-th storyE = modulusof elasticityIc = moment of inertia for columnsIg = moment of inert ia of compositegirder based on weighted averagemethodhi = height of the i-th storyLi = length of girder in the i-th storyVi = total horizontal shear forcein the i-th story



51/77

FIGURE 12.44: Compositeunbraced frames: (a) story loadsand idealization, (b) bending momentdiagrams, and (c) compositebeam sti ffness.

(Ici / hi ) = sum of thecolumn stiffnessi n thei-th story(Igi /Li ) = sum of the girder stiffnessin the i-th storyKcon = sum of theconnection rotational sti ffnessin the i-th story

Further research isrequired to assessthe performance of vari oustypesof composite connectionsused in building construction. Issuesrelated to accuratemodeling of effectivesti ffnessof compositemembers and joints in unbraced framesfor thecomputation of second-order effects and driftsneedto beaddressed.

12.4 WindEffectsonBuildings

12.4.1 Introduction

With the development of l ightweight high strength materials, the recent trend is to build tall andslender buildings. Thedesign of such buildingsi n non-seismic areasisoften governed bytheneed tolimit thewind-i nduced accelerations and drift to acceptable levels for human comfort and integrity



52/77

of non-structural components, respectively. Thus, to check for servi ceabil it y of tall buildings, thepeak resultant horizontal acceleration and displacement dueto thecombination of alongwind, acrosswind, and torsional loadsare required. As an approximateestimati on, the peak effects dueto alongwind, across wind, and torsional responses may be determined individually and then combined

vectorally. A reduction factor of .8 may beused on the combined value to account for the fact thatin general theindivi dual peaksdo not occur simult aneously. If the calculated combined effect islessthan any of the individual effects, then the latter should beconsidered for thedesign.

The effects of acceleration on human comfort is given in Table 12.3. The factors affecting thehuman responseare:

1. Period of buildi ngtolerenceto acceleration tendsto increasewith period.

2. Women are moresensit ivethan men.

3. Children are moresensit ivethan adults.

4. Perception increasesasyou go from sittingon thefloor, to sittingon achair, to standing.

5. Perception threshold level decreaseswith prior knowledgethat motion will occur.

6. Human body is moresensitiveto fore-and-aft motion than to side-to-side motion.7. Perception threshold is higher while walking than standing.

8. Visual cuevery sensit iveto rotation of the building relativeto fixed landmarks outside.

9. Acoustic cuebuildingsmakesoundswhileswaying dueto rubbing of contact surfaces.Thesesounds, and soundsof thewind whistli ng, focustheattention on building motioneven before motion is perceived, and thus lower the perception threshold.

10. Theresultant t ranslational acceleration due to the combinati on of longtitudinal, lateral,and torsional motionscauseshuman discomfort. In addition, angular (torsional) motionappears to bemorenoticeable.

TABLE12.3 Acceleration Limits

for Different Percepti on LevelsPerception Accelerationli mits

Imperceptible a


53/77

Sincethe tolerable accelerati on levels increase wit h peri od of buildi ng, the recommended designstandard for peak acceleration for 10-year wind in commercial and residential buildingsisasdepictedin Figure 12.45 [28]. Lower acceleration levels are used for residential buildings for the following

FIGURE 12.45: Design standard on peak acceleration for a 10-year return period.

reasons:

1. Residential buildingsareoccupied for longer hoursof theday and night and arethereforemoreli kely to experiencethedesign wind storm.

2. People are less sensit ive to motion when they are occupied with their work than whenthey relax at home.

3. Peopleare moretolerant of their work environment than of their homeenvironment.

4. Occupancytur nover ratesarehigher in commercial buildi ngsthan in residenti al buildings.

5. People can beevacuated easil y from commercial buildingsthan residential buildi ngsinthe event of a peak storm.

The effects of excessive deflection on building components is described in Table 12.4. Thus, theallowable drift, defined as the resultant peak displacement at the top of the building divided by theheight of the building, is generatly taken to bein therange1/450 to 1/600.

Figure12.46depictsschemati cally theprocedureof esti mati ngthewind-induced accelerati onsanddisplacements in a buildi ng. The steps involved in this design procedure are described below withnumerical examplesfor situationswhere the motion of thebuilding doesnot affect theloadsacti ngon the building. Finally, the situationswhen awind tunnel study is required are listed at theend ofthissection.

FIGURE12.46: Schemati c diagram for wi nd resistant design of str uctures.



54/77

12.4.2 CharacteristicsofWind

MeanWindSpeed

Thevelocity of wind (wind speed) at great heightsabovethe ground is constant and it is called

the gradient windspeed. As shown in Figure 12.47,closer to the ground surface the wind speed isaffected by fr ictional forces caused by the terrain and thus there is a boundary layerwithin whichthe wind speed varies from zero to the gradient wind speed. The thickness of t he boundary layer(gradient height)depends on the ground roughness. For example, the gradient height is 457 m forlargecities, 366 m for suburbs, 274m for open terrain, and 213 m for open sea.

FIGURE 12.47: Mean wind profilesfor different terrains.

Thevelocit y of wind averaged over 1 h is called the hourly mean wind speed,U. Themean windvelocity profilewithin the atmospheric boundary layer is described by a power law

U(z)=U (zref)

z

zref

(12.7)

in which U(z) is the mean wind speed at height z above the ground, zref is the reference heightnormally taken to be10m, and isthe power law exponent.

An alternativedescription of themean wind velocit y is by the logarithmic law, namely,

U(z)= 1K

uln

zdzo

(12.8)

in whichuisthefriction velocity,k is the von Karmonsconstant equal to 0.4, zois the roughness

length, and disthe height of zero-planeabovethe ground wherethe velocity is zero. Generally, zeroplaneis about 1 or 2 m below the average height of buildings and trees providing the roughness.Typical valuesof, zo, and dare given in Table12.5 [4, 21].

Theroughnessaffects both the thicknessof theboundary layer and the power law exponent. Thethi cknessof theboundarylayer and thepower lawexponent increasewith theroughnessof thesurface.Consequently the velocit y at any height decreases as the surfaceroughness increases. However, thegradient velocity will bethe samefor all surfaces. Thus, if thevelocity of wind for aparticular terr ainis known, using Equation 12.7 and Table 12.5, the velocity at someother terrain can becomputed.



55/77

TABLE12.5 Typical Valuesof Terrain

Parameterszo, , and dzo d(m)

City centers .7 .33 15 to 25Subur ban t er rai n .3 .22 5 t o 10

Open terrain .03 .14 0Open sea .003 .10 0

Turbulence

Thevariation of wind velocity with timei sshown in Figure12.48. Theeddiesgenerated by theaction of wind blowing over obstaclescausethe turbulence. In general, the velocity of wind may be

FIGURE 12.48: Variation of longit udinal component of turbulent wind with time.

represented in a vector form as

U(z,t)=U(z)i+u(z, t)i+v(z, t)j+w(z,t)k (12.9)

where u,v, andw are the fluctuating components of the gust inx ,y ,z (longitudinal, lateral, andvertical axes) asshown in Figure12.49 andU(z)ist hemean wind alongthexaxis. Thefluctuatingcomponent along the mean wind direction,u, is the largest and is therefore the most i mportantfor the vertical structures such as tall buildingswhich are flexible in the along wind direction. Thevertical componentw is import ant for horizontal structuresthat are flexible vert ically, such aslongspan bri dges.

An overall measure of the intensit y of tur bulence is given by the root mean square value (r.m.s).Thus, for thelongitudinal component of the turbulence

u(z)= 1

To

Too

{u(z,t)2}dt1/2

(12.10)

whereTois the averaging period. For thestatistical propert iesof thewind to beindependent on thepart of the record that isbeing used, Tois taken to be1 h. Thus, thefl uctuating wind is a stationaryrandom function over 1h.



56/77

FIGURE 12.49: Velocity components of turbulent wind.

Thevalueofu(z)divided by the mean velocityU(z)is called the turbulenceintensity

Iu(z)= u(z)U(z)

(12.11)

which increaseswith ground roughnessand decreaseswith height.Thevarianceof longitudinal tur bulencecan bedetermined from

2u= u2 (12.12)

whereuis the fri ction velocit y determined from Equation12.8 and which isindependent of theheight isgiven in Table12.6 for vari ousroughnesslengths.

TABLE12.6 Valuesof for Various

RoughnessLengthszo(m) .005 0.7 .30 1.0 2.5

6.5 6.0 5.25 4.85 4.0

Integral ScalesofTurbulence

Thefl uctuation of wind velocity at a point is dueto eddiestransported by the mean wind U.Each eddy may beconsidered to becausing a periodic fluctuati on at that point with a frequency n.

Theaveragesizeof the tur bulent eddiesaremeasured by integral length scales.For eddiesassociatedwith longitudinal velocity fluctuation, u, the integral length scales are Lxu, Lyu, andL

zu describing

the sizeof the eddies in longitudinal, lateral, and vert ical directions, respectively. IfLyu andL

zuare

comparableto the dimension of thestructure normal to the wind, then theeddieswill envelopethestr ucture and give rise to well-correlated pressures and thus the effect i s significant. On the otherhand, ifL

yu andL

zu are small, t hen the eddies produce uncorrelated pressures at various parts of

the str ucture and the overall effect of the longitudinal tur bulence will be small. Thus, the dynamicloading on astr ucturedependson the sizeof eddies.



57/77

SpectrumofTurbulence

Thefrequencycontent of theturbulencei srepresented bythepower spectrum, whichindicatesthepower or kineti cenergyper unit ti meassociatedwi th eddiesofdifferent frequencies. An expressionfor the power spectrum is [60] :

nSu(z, n)

u2= 200f

(1+50f )5/3 (12.13)

where f= nz/U(z)is the reduced frequency. A typical spectrum of wind turbulence is shown inFigure12.50. Thespectr um hasa peak value at a very low frequency around .04 Hz. As the typicalrange for t he fundamental frequency of a tall building is .1 to 1 Hz, the buildings are affected byhigh-frequncy small eddiescharacterizing the decending part of the power spectrum.

FIGURE12.50: Power spectrum of longitudinal t urbulence.

Cross-SpectrumofTurbulence

The cross-spectrum of two continuous recordsis a measure of the degree to which the tworecordsarecorrelated. If the recordsaretaken at t wo points, M1and M2, separated by adistance, r,then the cross-spectr um of longitudinal tur bulent component is defined as

Su1u2(r, n)=Scu1u2(r, n)+iSqu1u2(r, n) (12.14)

where the real and imaginary parts of t he cross-spectrum are known as the co-spectrum and thequadrature spectrum, respectively. However, the latter i s small enough to be neglected. Thus, theco-spectrum may beexpressed non-dimensionally asthe coherenceand is given by

2(r, n)=[Su1u2(r, n)]2

Su1(n)Su2(n)(12.15)

whereSu1(n)and Su2(n)are the longitudinal velocity spectr a atM1and M2, respectively.Thesquare root of the coherenceis given by the following expression [19]:

(r,n)=ef (12.16)where

f= n[c2z (z1z2)2 +c2y (y1y2)2]1/2

1/2[U (z1)+U (z2)](12.17)



58/77

in which y1, z1and y2, z2arethecoordinatesof points M1and M2. Thelinejoining M1and M2isassumed to beperpendicular to the directi on of the mean wind. The suggested values ofcy andczfor the engineering calculation are 16and 10, respectively [62].

12.4.3 WindInducedDynamic Forces

ForcesDuetoUniformFlow

A bluff body in a two-dimensional flow, asshown in Figure 12.51, is subjected to a nett forcein thedirection of flow (drag force), and a forceperpendicular to the flow (li ft force). Furthermore,when the resultant force is eccentr ic to the elastic center, the body will be subjected to torsionalmoment. For uni form flow these forces and moment per unit height of the object aredetermined

FIGURE 12.51: Dragand li ft forcesand torsional moment on a bluff body.

from

FD = 12

CDBU2 (12.18)

FL = 12

CLBU2 (12.19)

T = 12

CTB2U2 (12.20)

whereUis the mean velocity of the wind, is the densit y of air,CD andCLare the drag and liftcoefficients, CTist hemoment coefficient, and Bis thecharacteri stic length of the object such astheprojected length normal to theflow.

The drag coefficient for a rectangular building in the plan is shown in Figure 12.52for variousdepth-to-breadth ratios[5]. Theshear layersoriginati ngfrom theseparation pointsat thewindward

cornerssurround aregion known ast hewake.For elongated sections, the stream li nesthat separateat the windward corners reattach to the body to form a narrower wake. This is attributed to t hereduction in thedragfor larger aspect ratios. For cylindrical buildingsin theplan, thedragcoefficientis dependent on Reynolds number asindicated in Figure12.53.

Unlikethedragforce, theli ft forceand torsional moment donot haveamean valuefor asymmetricobject with a symmetric flow around it, ast hesymmetrical distr ibution of mean forcesacting in theacrosswind direction cannot cause a net force. If thedirection of wind is not parallel to the axesofsymmetriesor if theobject isasymmetrical, then therewil l beamean lift forceand torsional moment.



59/77

FIGURE12.52: Dragcoefficient for aretangular section with different aspect ratios.

FIGURE 12.53: Effectsof Reynolds number on dragcoefficient of a circular cylinder.

However, dueto vortex shedding, fluctuating li ft forceand torsional moment will bepresent in bothsymmetr ic and non-symmetr ic str uctures. Figure 12.54 showsthe mechanism of vortex shedding.Near the separation zones, strong shear stresses impart rotational motions to the flui d part icles.Thus, discrete vortices are produced in t he separation layers. These vortices are shed alternati velyfrom thesidesof theobject. Theasymmetr ic pressuredistr ibution created bythevorticesaroundthecross-secti on leadsto an alternating tr ansverse force(lif t force) on the object. The vortex sheddingfrequencyin Hz, ns , isrelated to anon-dimensional parameter called theStrouhal number, S, definedas

S=ns B/U (12.21)



60/77

FIGURE12.54: Vortex shedding in the wakeof abluff body.

whereUisthemean wind speed and Bi sthewidth of theobject normal to thewind. For objectswithrounded profiles such as circular cylinders, the Strouhal number varies with the Reynoldsnumber Re defined as

Re=UB/ (12.22)whereis the density of air and is the dynamic viscosit y of theair. Thevortex shedding becomesrandom in thetransition region of4105 < Re < 3106 wheretheboundary layer at thesurfaceofthe cylinder changesfrom laminar to turbulent. Outside thist ransit ion range, the vortex shedding isregular producing aperiodic li ft force. For cross-sectionswi th sharp corners, theStrouhal number isindependent of the Reynoldsnumber. Thevariation of the Strouhal number with length-to-breadthratio of arectangular cross-secti on is shown in Figure12.55.

FIGURE12.55: Strouhal number for a rectangular section.

ForcesDuetoTurbulentFlow

If thewind is turbulent, then thevelocity of the wind in thealong wind direction i sdescribedasfollows:U(t)=U+u(t) (12.23)

whereUis the mean wind andu(t) is the turbulent component in the along wind direction. Theti medependent dragforceper unit height is obtained from Equation 12.18 by replacing Uby U(t).Astheratio u(t)/Uis small, the ti medependent dragforcecan beexpressed as

fD(t)=fD+fD (t) (12.24)



61/77

wherefDand fDarethe mean and the fluctuating partsof thedrag forceper unit height which aregiven by

fD

=

1

2

U2CD B (12.25)

fD = U uCD B (12.26)Thespectral density of the fluctuating part of the drag force is obtained from the Fourier trans-

formation of theauto correlation function as

Sf D (n)=2U2B2C2D Su(n) (12.27)where Su(n) is the spectral densit y of the turbulent velocity, which may be obtained from Equa-tion 12.13.

In practice, the presence of the structure distorts the turbulent flow, particularly the small high-frequency eddies. A correcti on factor known astheaerodynamic admittance function (n)may beintroduced [17] to account for theseeffects. Thefollowing emprical formulahasbeen suggested for(n)[ 62]:

(n)= 11+

2n

A

U(z)

4/3 (12.28)whereA isthefrontal areaof thestructure. Nowwith theintroductionof theaerodynamicadmittancefunction, Equation 12.27 may berewritten as

Sf D(n)=2U2B2C2D2(n)Su(n) (12.29)It is evident from Equation12.26that the fluctuating drag force varies linearly with the turbu-

lence. Thus, largei ntegral length scale and high turbulent intensit ieswill causestrongbuffeting andconsequently increase the along wind response of the str ucture. However, the regularity of vortexshedding is affected by the presence of turbulence in the along wind and, hence, the across windmotion and torsional motion dueto vortex shedding decreaseasthe level of tur bulenceincreases.

12.4.4 ResponseDuetoAlongWind

Tall slender buildi ngs, where the breadth of the structure is small compared to the height, can beidealized as a line-li ke structure as shown in Figure12.56. Modeling the building as a continuoussystem, thegoverningequation of motion for alongwind displacementx(z,t) can bewritten as[ 29]:

m(z)x(z,t)+c(z)x(z,t)+EI (z)x(z,t)GA(z)x (z,t)=f (z,t) (12.30)wherem, c, El , and GAare, respectively, the mass, damping coefficient, flexural ri gidity, and shearrigidit y per unit height. Furthermore,f (z, t)is the fluctuating wind load per unit height given inEquation 12.26.

Expressing the displacement in terms of the normal coordinates,

x(z,t)=N

i=1i (z)qi (t) (12.31)

wherei isthe i-th vibration modeshapeand qi isthe i-th normal coordinate. Using the orthogo-nality conditionsofmode shapes, Equation 12.30 can beexpressed as[ 9]

miqi+ciqi+kiqi= pi i= 1 to N (12.32)



62/77

FIGURE 12.56: Typical deflection modeof a shear wall-framebuilding.

wheremi , ci, k

i, and p

i arethegenerali zed mass, damping, sti ffness, and forcein the i-th modeof

vibration. Thegeneralized massand forcearedetermined from

mi = H

o

m(z)2i(z)dz (12.33)

pi = H

o

f(z,t)i (z)dz

= CDB H

o

U(z)u(z, t)i (z)dz (12.34)

Equation 12.32 consistsof aset of uncoupled equati ons, eachrepresenti ngasingledegreeof freedomsystem. Using therandom vibration theory [54], thepower spectr um of theresponsei n each normal

coordinate is given by

Sqi (n)=|Hi (n)|2 Spi (n)1

(ki)2 (12.35)

where

|Hi (n)|= 11

nni

22 +42i nni21/2 (12.36)

andki= 42n2i mi (12.37)

in which niand iarethefrequency and dampingratio in thei-th mode. Thespectral densit y of thegenerali zed forcetakestheform

Spi

(n) = 2C2D B22(n) H0

H0

U (z1)U (z2)Su1u2(r, n)i (z1)i (z2)dz1d z2 (12.38)

where Su1u2(r, n)is the cross-spectral density defined in Equati on12.14 withr being the distancebetween the coordinatesz1and z2. In Equation 12.38, the aerodynamic admittancehasbeen incor-porated to account for t hedistorti on caused by the structureto the turbulent velocity.



63/77

In view of Equation12.15, Equation 12.38 maybeexpressed as

Spi

(n) = 2C2D B22(n) H

0 H

0

i (z1)i (z2)U (z1)U (z2)Su1(n)

Su2(n) (r, n)dz1dz2 (12.39)

where (r, n)is the square root of the coherence given in Equation12.16, andSu(n)is the spectr aldensit y of the turbulent velocity.

Thevarianceof the i-th normal coordinate is obtained from

2qi=0

Sqi (n)dn= 1ki2

0

|Hi(n)|2 Spi

(n)dn (12.40)

The calculation of the above integral is very much simplified by observing the plot of the twocomponents of the integrant shown in Figure 12.57. The mechanical admittance function is eit her1.0 or 0 for most of thefrequencyr ange. However, over arelatively small rangeof frequenciesaroundthenatural frequency of thesystem, it attainsvery highvaluesif thedampingissmall. Asaresult, theintegrant takesthe shapeshown in Figure12.57c. It hasa sharp spike around the natural frequency

FIGURE12.57: Schemati c diagram for computation of response.

of the system. The broad hump is governed by the shapeof the tur bulent velocity spectrum whichis modified slightly by theaerodynamic admittancefunction. The areaunder thebroad hump is thebroad band or non-resonant response, whereast heareain the vicinity of the natural frequency gives



64/77

thenarrow band orresonant response. Thus, Equati on 12.40 can berewrit ten as

2qi= 1

(ki)2

nino

Spi

(n)dn+ ni4i

Spi

(ni )

=Bqi+2Dqi (12.41)

in which Bqiand Dqiarethe non-resonati ng and resonati ng root mean squareresponseof the i-thnormal coordinate. Astheresponsesdueto variousmodesof vibration arestatisti cally uncorrelated,the responseof thesystem is given by

2x (z)=N

i=12i(z)

2

Bqi+

Ni=1

2i(z)2

Dqi (12.42)

which givesthe vari anceand, hence, the root mean square displacement at vari ousheights.Thetotal displacement is obtained by including the static deflection dueto the mean drag load,

which is determined conveniently asfollows:In view of Equation12.25, the mean generalized forceisgiven by

fi = H

o

12

CDU2(z)Bi (z)dz

= 12

CD B

Ho

U2(z)i (z)dz (12.43)

Then the mean displacement is determined from

x(z)=N

i=1i (z)

fi(2 ni )2m

i

Documents

12-Multi Story Frame Structures