Direct Displacement-Based Design of Frame-Wall Structures-Sullivan,Priestley,Calvi-2006

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Journal of Earthquake EngineeringVol. 10, Special Issue 1 (2006) 91124c Imperial College Press

DIRECT DISPLACEMENT-BASED DESIGN OFFRAME-WALL STRUCTURES

T. J. SULLIVAN, M. J. N. PRIESTLEY and G. M. CALVI

ROSE School, University of Pavia, Via Ferrata 1, 27100 Pavia, Italy

A direct displacement-based design (DBD) procedure for structures that comprise bothframes and walls is presented in this paper. Within the new procedure, strength pro-portions between walls and frames are assigned and are used to establish the designdisplacement prole before any analysis has taken place. Knowledge of the displacementprole and recommendations for the combination of frame and wall damping compo-nents enables representation of the structure as an equivalent single-degree of freedomsystem. The Direct DBD process is then utilised to set the required strength level whichis proportioned to the structure in line with the initial strength assignments. To test thedesign methodology, two sets of 4-, 8,- 12-, 16- and 20-storey reinforced concrete struc-tures are designed. The rst set considers frame-wall structures in which the frames areparallel to the walls and the second considers structures in which link-beams connectfrom the frames directly onto the ends of the walls. A suite of time-history analyses areconducted to validate the methodology, which is seen to perform excellently.

Keywords: Displacement-based design; frame wall; dual system; seismic design.

1. Introduction

This paper presents guidelines for the direct displacement-based design (DBD) ofstructures that utilise both frames and walls to resist earthquake actions in parallel.There is a need for a design methodology that is applicable to this particular formof structure, commonly known as a frame-wall structure or dual system structure,because the dynamic behaviour of dual systems is considerably dierent from pureframe or wall structures for which many design recommendations already exist.Such dierences in dynamic behaviour are attributed principally to the interactionthat takes place between the frames and walls, which is not well accounted for incurrent design practice.

A further motivation for this work stems from the consideration that the com-bined structural form is a very ecient and convenient way to resist earthquakeactions that is not currently being widely exploited. It can be argued from bothstructural and aesthetic points of view that the combination of frames and wallspresents considerable advantage over structures formed purely out of frames orwalls. The sti nature of cantilever reinforced concrete (RC) walls means that theyare naturally suited to control storey-drifts in the lower levels of buildings. In con-trast, frames typically restrain deformation in upper storeys but moreover, they

91

April 28, 2006 13:47 WSPC/124-JEE 00274

92 T. J. Sullivan, M. J. N. Priestley & G. M. Calvi

oer signicant energy dissipation up the height of the building which reducesthe total displacements that a building experiences. From aesthetic and functionalpoints of view, frames enable large open spaces within minimum constraints onusage. On the other hand, walls are an attractive means of forming stair wells andlift shafts in a building, while at boundary lines they are commonly used to providere resistance between buildings.

2. Challenges for the Direct DBD of Frame-Walls

An ideal seismic design procedure will establish the minimum basic strength of astructure sucient to ensure pre-dened performance criteria for the building aresatised at the design ground motion intensity, with a minimum of eort. Previouswork by Sullivan et al. [2005] investigated a trial methodology which providedencouraging results when applied to regular frame-wall structures in which theframes were parallel to the walls. The research identied that the following twotasks were required to improve the accuracy of the methodology and thereby enablethe Direct DBD [Priestley, 2003] approach to be used for frame-wall structures:

Development of an expression for the displaced shape of frame-wall structures atmaximum response, to enable equivalent SDOF characteristics to be established.

Development of an expression for the equivalent SDOF system ductility or equiv-alent viscous damping that takes into account the frame-wall interaction.

Sullivan et al. [2005] proposed that the design displacement prole be set as a func-tion of the moment prole in the walls, using proportions of strength assigned atthe start of the design procedure. There is experimental evidence that supports thevalidity of this approach as reported by Sullivan et al. [2004]. Another recommen-dation made by Sullivan et al. [2005] was that the equivalent SDOF system viscousdamping could be obtained by factoring the individual frame and wall componentsby the proportions of overturning they resist.

The challenge in this paper is therefore to nalise the design procedure proposedby Sullivan et al. [2005] and to verify its accuracy through examination of a rangeof case study structures.

3. Description of the New Design Procedure

The various steps of the seismic design procedure for frame-wall structures areshown as a owchart in Fig. 1. The rst set of steps aims to develop an equiva-lent SDOF representation of the MDOF structure. This is achieved by assigningstrength proportions and subsequently using the moment prole in the walls toset a design displaced shape. With knowledge of the displacement prole, variousequivalent SDOF properties of the structure are obtained. The second importantset of steps in the design aim to determine the required eective period and then

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Direct Displacement-Based Design of Frame-Wall Structures 93

Assign strength proportions to frames and walls.

Determine wall inflection height, hinf. Determine yield displacements of walls and yield drift of frames.

Calculate design displacement profile.

Determine effective height, he, effective mass, me, and design displacement, d.

Plot displacement spectra at system damping level and use design displacement to obtain required effective period, Te. Check, Te = Te,trial?

Determine equivalent viscous damping values for frames and walls.

Determine effective stiffness and design base shear, Vb = K e d.

Obtain beam & column strengths by factoring strength

proportions by base shear.

Reduce drift limit.

Calculate the ductility demands on the frames and walls. Are ductility demands excessive?

NO

YES

Use proportions of overturning moment resisted by the frames and walls to factor damping values & obtain an equivalent system damping value sys.

Choose a trial effective period,

Te,trial. Reset

Te,trial = Te

YES

Distribute base shear up height in proportion to displacements of masses. Subtract frame shears from total shears to obtain wall shears & thereby moments.

Perform capacity design with allowance for higher mode effects, to obtain design strengths in non-yielding elements and design shears in frames and walls.

NO

Fig. 1. Flowchart of recommended design procedure for frames-wall structures.

stiness using the substitute structure approach [Gulkan and Sozen, 1974; Shibataand Sozen, 1976]. The design base shear is obtained through multiplication of thenecessary eective stiness by the design displacement and the strength of individ-ual structural elements is set taking care to ensure that initial strength assignmentsare maintained. Each of these phases is described in more detail in the sub-sectionsthat follow.

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3.1.1. Assignment of strength proportions to establish the wallinection height

In order to develop a suciently accurate SDOF representation of the frame-wallstructure, strength proportions are assigned at the very start of the design proce-dure. This involves setting the proportion of base shear or overturning resistanceoered by the frames and walls, in addition to the relative strength distributionof yielding elements (beams and ground storey columns) within the frames. Asmentioned above, by assigning these strength proportions the shear and momentprole in the walls can be established and this then enables determination of theinection height. Figure 2 locates the inection height for a frame-wall structure inwhich the frames and walls resist the total base shear in equal proportions and theframes provide a constant shear resistance over their height. The inection heightis of particular interest as it will be used to form the design displacement prole.

Note that the proportions of strength assigned at this stage of the design processare related to the forces expected at formation of a 1st mode plastic mechanism.They should not be confused with the proportions of force that are expected todevelop at maximum response. The maximum forces are aected by overstrengthand higher mode eects and are established following DBD as part of a capacitydesign procedure.

The storey shear above the base of the walls cannot be obtained directly from thedesign base shear since the walls remain elastic above the ground storey and upperstorey shears will depend on the proportion of shear carried by the frames. As such,wall shears are obtained as the dierence between the total shear and the frameshear as shown in Eq. (1). Recall that the frame storey shear can be determinedsince it is dependent only on the strength of the beams up the building height.

Vi,wallVb

=Vi,total

Vb Vi,frame

Vb, (1)

where Vb is the total base shear, Vi,wall is the wall shear at level i, Vi,total is thetotal shear at level i, and Vi,frame is the frame shear at level i.

Total shear(solid line)

Frame shear (dashed line)

Wall shear (shaded area)

MWall

Wall BMDFrame

overturning

Wall inf lection height, hinf

MOMENTS SHEARS BEAM-SWAY MECHANISM 1.0Vb0.5Vb

Fig. 2. Use of frame-wall strength proportions to locate inection height in walls.

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For the purpose of establishing the inection height, a triangular distributionof the fundamental mode inertia forces up the height of the structure is assumed.This approximation enables the total storey shear to be obtained as a function ofthe base shear as shown in Eq. (2).

Vi,totalVb

= 1 in

(i 1)(n + 1)

, (2)

where Vi,total is the total shear at level i, Vb is the total base shear, and n is thetotal number of storeys in the building.

As Eq. (2) provides the distribution of total storey shear up the building height,the only unknown of Eq. (1) is the frame storey shear distribution. To obtain thisshear proportion, the relative strength distribution of yielding elements within theframes is used.

Although the designer is free to choose any strength distribution they prefer, itis proposed that the use of beams of equal strength up the height of the structureis advantageous for design and construction. Assuming that beam moments arecarried equally by columns above and below a beam-column joint, the frame storeyshear is obtained as a function of the beam strength using Eq. (3).

Vi,frame =(

Mb,i +

Mb,i1)2(hi hi1) =

Mb,i

hcol, (3)

where Vi,frame is the frame shear at level i, Mb,i are the beam strengths at level i,and hcol, is the inter-storey height. Although the beam strengths are not actuallyknown to begin with, Eq. (3) is useful as it indicates that provided beams of equalstrength are to be used then the frame storey shear is constant up the buildingheight. Consequently, if 40% of the base shear is being carried by the frames, this40%Vb will be carried up the entire height of the frame. As such, the shear propor-tion carried by the frame can be substituted into Eq. (1) and the wall shears andbending calculated, all as a function of the design base shear.

A perfectly constant shear up the height of the frame requires that the sumsof the base column strengths and roof beam strengths are both equal to half thesum of the intermediate level beam strengths. If roof level beams are assignedstrength equal to those on other stories, then the frame shear at roof level shouldbe considered to be 50% greater than that at other levels. Larger base columnstrengths will also imply larger ground storey shears, with the column inectionheight shifting above 0.5hcol.

The storey shear and consequently the moment in the walls are used to establishthe inection height in the walls, hinf , where the moment and curvature is zero. Thisinection height will be used to nd the displacements of the structure at yield ofthe walls and to develop the design displacement prole, as detailed in the nextsubsections.

Other important design quantities that should be obtained from the strengthassignments are the proportion of overturning resisted by the frames and wallsrespectively. The proportions of overturning can be obtained directly from the shear

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prole up the height of the structures. These overturning proportions are used laterin the design procedure for denition of the system damping and for adjustment ofthe design drift to allow for higher modes.

3.1.2. Yield deformations of the walls and frames

As the walls tend to control the response of frame-wall structures, the wall yieldcurvature and displacements at yield are important for the development of thedesign displacement prole. The frame yield displacement, or yield storey drift, isalso important to the design process as it is used to provide an indication of theenergy absorbed through hysteretic response of the frame.

The yield curvature of the walls, yWall, is rstly obtained using Eq. (4)[Priestley, 2003].

yWall =2yLw

, (4)

where y is the yield strain of the longitudinal reinforcement in the wall and Lw isthe wall length.

The displacement prole of the structure at yield of the wall, i,y , can thenbe established using the wall yield curvature, inection height and storey height inaccordance with Eqs. (5a) and (5b).

iy =yWallhinfhi

2 yWallh

2inf

6for hi hinf , (5a)

iy =yWallh

2i

2 yWallh

3i

6hinffor hi < hinf . (5b)

The frame yield drift, yframe, used to estimate the ductility and equivalent viscousdamping of the frames, is obtained in accordance with Eq. (6) [Priestley, 2003]:

yframe =0.5lby

hb, (6)

where lb is the average beam length, y is the yield strain of beam longitudinalreinforcement and hb is the average depth of the beams at the level of interest.

3.1.3. Design displacement prole and equivalent SDOF characteristics

The design displacement prole is developed using the various values obtained inthe preceding subsections, together with the design storey drift, as shown in Eq. (7).

i = iy +(d yWallhinf2

) hi, (7)

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where i is the design displacement for level i, i,y is the displacement of leveli at yield of the walls, d is the design storey drift, yWall is the yield curvatureof the walls, hinf is the inection height, and hi is the height at level i. Note thatthe design storey drift can be initially taken as the code limit for non-structuraldamage, reduced to allow for higher mode eects in accordance with Eq. (8).

d = d,limit

[1 (N 5)

100

(MOT,frameMOT,total

+ 0.25)]

d,limit, (8)

where N is the number of stories, MOT,frame is the overturning resistance of theframe and MOT,total is the total overturning resistance of the structure. This approx-imate equation was proposed after reviewing the results of initial trial case studies[refer to Sullivan, 2005]. As mentioned earlier, the ratio of frame to total overturningresistance can be obtained in terms of the base shear using the strength assignmentsmade at the start of the design procedure. The design drift given by Eq. (8) maybe reduced further if it is found that inelastic demands on the structure are likelyto be excessive. Alternatively, the critical value of storey drift can be determinedbefore the design displacement prole is developed.

With knowledge of the displacement prole at maximum response; i, the seis-mic masses; mi, and storey heights; hi, the equivalent SDOF design displacement;d, eective mass; me, and eective height; he, can be calculated as shown inEqs. (9) to (11) [Priestley, 2003] respectively.

d =n

i=1 (mi2i )n

i=1 (mii), (9)

me =n

i=1 (mii)d

, (10)

he =n

i=1 (miihi)ni=1 (mii)

. (11)

3.1.4. Design ductility values, eective period and equivalent viscous damping

The only other substitute structure characteristic required for Direct DBD[Priestley and Kowalsky, 2000] is the equivalent viscous damping. This is a functionof ductility and according to recent recommendations by [Blandon and Priestley,2005] and [Grant et al., 2005], the eective period.

The ductility demands on the walls for use within this equivalent viscous damp-ing approach should be calculated using displacement at the eective height. Thewall ductility demand, wall, is therefore simply the design displacement dividedby the yield displacement of the walls at the eective height, as shown in Eq. (12).

wall =d

he,y, (12)

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where d is the design displacement (from Eq. (9)) and he,y is the yield displace-ment of the wall at the eective height (obtained substituting the eective heightinto the appropriate version of Eq. (5)).

The displacement ductility demands on the frames at each level up the heightof the structure can be obtained using the storey drifts as shown in Eq. (13).

frame,i =(i i1hi hi1

)1

yframe, (13)

where i, i1, hi, and hi1, are the displacements and heights at level i and leveli 1 respectively, frame,i is the frame ductility at level i, and yframe is the yielddrift of the frame (from Eq. (6)). When beams of equal strength are used up theheight of the structure, the ductility obtained from Eq. (13) for each storey can beaveraged to give the frame displacement ductility demand.

Before proceeding with calculations of the equivalent viscous damping, it isnecessary to check that the ductility demands are sustainable. Ductility demandson frames are typically not critical as the walls tend to have smaller yield curvaturesand yield displacements. For frame-wall structures in which frames are parallel towalls, ductility demands will be fairly low and can typically be detailed for relativelyeasily. However, when link-beams connect between frames and walls then these link-beams are likely to be subject to higher curvatures than other beams and should bechecked separately, as is discussed Sec. 4 where the procedure is applied to variouscase study structures.

Although the wall displacement ductility demand indicated by Eq. (12) is appro-priate for estimation of the equivalent viscous damping, it is not a good representa-tion of the inelastic deformation that the walls must undergo. A more appropriateparameter is the wall curvature ductility, wall, which can be obtained in accor-dance with Eq. (14).

wall = 1 +1

LpyWall

(d yWallhinf2

), (14)

where Lp is the wall plastic hinge length, d is the design storey drift, yWall isthe yield curvature of the walls and hinf is the inection height. Note that becausethe curvature ductility demand is a function of the inection height and not thetotal height, inelastic deformation demands in walls of frame-wall structures willtypically be larger than those in plane wall structures.

The wall plastic hinge lengths to be used within Eq. (14) are taken as theminimum of Eqs. (15a) and (15b).

Lp = 0.022fydb + 0.054hinf, (15a)

Lp = 0.2Lw + 0.03hinf, (15b)

where fy is the yield stress and db the diameter of the longitudinal reinforcementin the wall, Lw is the wall length and hinf is the inection height. These equations

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have been adapted from [Priestley, 2003] with the inection height substituting thetotal height.

The curvature ductility capacity of a RC wall will depend on the strain limitsselected for the concrete in compression (c) and longitudinal reinforcement in ten-sion (s). For reasonably conservative values of c = 0.018 and s = 0.06, Priestleyand Kowalsky [1998] found that the ultimate curvature of reinforced concrete wallsis well represented by Eq. (16).

u =0.072Lw

, (16)

where u is the ultimate curvature and Lw is the wall length. This equation wasshown to be representative of ultimate curvature over a range of axial load ratiosand longitudinal reinforcement contents. Combining Eqs. (16) and (4), it is foundthat the curvature ductility capacity is approximately equal to 0.036/y.

If the checks on ductility indicate that the inelastic deformations associated withthe design drift will be excessive then the design drift must be reduced and thedesign displacement prole re-computed as discussed in the previous sub-section.If the ductility demands are sustainable then the next step in the design procedureis to compute equivalent viscous damping values.

Recent work by Blandon and Priestley [2005] (developed further by Grant et al.[2005]), recommends that the equivalent viscous damping be computed as a functionof the eective period. As this is unknown at the start of the design process, a trialvalue can be used and an iterative design process adopted. A reasonable estimatefor the trial value of the eective period can be obtained from Eq. (17).

Te,trial =N

6

sys , (17)

where N is the total number of stories and sys is the system ductility. Equation (17)is similar in form to a code based equation that uses the height or number of storeysto estimate the initial period. The ductility term accounts for the dierence betweenthe initial and eective periods, neglecting the eect of strain hardening. Given theapproximate nature of Eq. (17) [refer Sullivan, 2005] trial eective period valuesmay be some 30% dierent than the nal eective period, however by using sucha trial value, it will be found that convergence is attained within one or, at most,two iterations.

Having set the trial eective period and established expected ductility values,the frame and wall equivalent viscous damping components are calculated usingEqs. (18) and (19) respectively [Grant et al., 2005].

hyst,wall =951.3

(1 1

0.5wall 0.1rwall

)(1 +

1(Te,trial + 0.85)4

), (18)

hyst,frame =1201.3

(1 1

0.5frame 0.1rframe

)(1 +

1(Te,trial + 0.85)4

), (19)

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where r is the post-elastic stiness coecient, typically taken as 0.05 for new RCstructures. Note that by considering the inuence of the period on the dampingvalues, it could be argued that the period-dependence of the damping values canbe neglected when eective periods are greater than 1.0 s, which is usually the casefor frame-wall structures. The equivalent viscous damping for the frames and wallsis obtained adding the elastic and hysteretic components together and then a valueof damping for the equivalent SDOF system is determined using Eq. (20).

SDOF =Mwall wall + MOT,frame frame

Mwall + MOT,frame, (20)

whereMOT,frame is the overturning resistance of the frames and MWall is the over-turning resistance (exural strength) of the walls. At this point of the design process,all of the substitute structure characteristics have been established and as such, thedisplacement spectrum is developed at the design level of damping. This can bedone using a damping-dependent scaling factor appropriate for the seismologicalcharacteristics of the design region. The Eurocode 8 [CEN, 1998] recommends thatthe value obtained from Eq. (21) be used to scale the elastic spectrum to thedamping level of interest.

=

10/(5 + SDOF ) 0.55, (21)where SDOF is the equivalent viscous damping of the system as given by Eq. (20).The design displacement is then used to read o (or interpolate between knownpoints) the required eective period, Te, as shown in Fig. 3.

The eective period obtained from the Direct DBD process illustrated in Fig. 3 isthen compared to the trial eective period value. If the period values do not match,then the period obtained from Fig. 3 replaces the trial period and the design step

0.00

0.20

0.40

0.60

0.80

1.00

0.0 1.0 2.0 3.0 4.0 5.0

Period (s)

Spec

tral

Dis

pla

cem

ent

at S

DO

F d

amp

ing

(m)

Te

d

Displacement spectrum atsystem damping level.

Fig. 3. Direct displacement based design to obtain the required eective period.

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is repeated. When eective periods nally match, the designer is in a position todetermine the eective stiness and design base shear as outlined next.

3.1.5. Determining the design base shear and member strengths

With the eective period established, the eective stiness, Ke, is determined inaccordance with Eq. (22).

Ke = 42meT 2e

, (22)

where me is the eective mass (from Eq. (10)) and Te is the eective period. Thiseective stiness is then multiplied by the design displacement, d, to obtain thebase shear, Vb, as shown by Eq. (23).

Vb = Ked. (23)

Individual member strengths are then determined maintaining the strength propor-tions assigned at the start of the design process. Note however, that rather thanuse a triangular lateral force distribution, better results are obtained distributingthe base shear up the height of the structure according to Eq. (24).

Fi =miiNi=1 mii

Vb, (24)

where Fi is the portion of base shear applied at level i, mi is the mass at level i,and i the displacement at level i.

This then completes the DBD process. It is evident that there are several stepsto the design procedure, however, the process is simple and does provide excel-lent control of displacements and storey drifts as is demonstrated in the followingsection.

4. Verification of the Design Method

The design method is veried through examination of several case studies. A rangeof frame-wall structures are designed using the new procedure with the aim ofmaintaining storey drift and curvature ductility limits typical of a life-safety per-formance level. The design strengths obtained for each case study are then usedto set the strength of accurate non-linear analytical models which are subject to aseries of time-history analyses using earthquake records compatible with the designspectrum. The success of the design process is gauged by comparing the targetdeformations anticipated during the design phase, with the actual deformations aspredicted by the time-history analyses.

4.1. Description of case study structures

The frame-wall structures shown in plan in Fig. 4 and elevation in Fig. 5 aredesigned using the new procedure. Two types of structure are considered; (i) struc-tures with walls and frames connected only by oor slabs, and (ii) structures with

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RC Walls 8m long w. L-Beams

10m long w/o L-Beams

8m 8m

20m

Structures with & without Link-Beams

56m

8m 8m

EQ Excitation

Direction

EQ Excitation

Direction

Fig. 4. Plan view of frame-wall structures being examined in the verication of the proposedDBD method.

(i) Eight-storey structure without link-beams (ii) Eight-storey structure with link-beams

Fig. 5. Elevation of frame-wall structures; (i) without link-beams and (ii) with link beams, to beexamined as part of nal verication of the DBD method.

link-beams extending from frames directly to the ends of the walls. In order tocomprehensively test the approach, buildings of 4, 8, 12, 16 and 20 storeys areexamined.

These case study structures are regular in layout with a RC frame-wall systembeing used to resist lateral loads acting along the longitudinal axis of the building.In the transverse direction it is assumed that a regular arrangement of RC wallswould be used to resist lateral loads, however, this does not aect the design pro-cedure proposed here for the frame-wall system. The structures are considered ashaving rigid foundations with oor slabs that act as rigid-diaphragms in plane, fullyexible out of plane. It has been shown elsewhere [Sullivan, 2005] that for frame-wall structures of typical layout diaphragm exibility does not require considerationwithin the design procedure.

The concrete and reinforcement material properties assumed for the struc-tures are values that could typically be found in building practice. Values forthe concrete include: (i) f c = 30.0MPa and (ii) Ec = 25 740MPa. The expectedstrengths adopted for the reinforcing steel include: (i) fy = 400MPa and (ii) Es =200 000MPa. For seismic design, material strengths are not factored to dependablestrength levels and instead these values have been taken as the expected strength

April 28, 2006 13:47 WSPC/124-JEE 00274


and stiness characteristics. The seismic weights of individual oors have been esti-mated assuming a concrete density value of 24.5 kN/m3, a superimposed dead-loadof 1.0 kPa, a reduced live-load of 1.0 kPa and a loaded oor area of 1105m2 perlevel. Axial load ratios have been computed using these oor weights factored bythe tributary area of oor supported by the individual elements. Floor weights, axialload ratios and dimensions of individual elements are presented for the structureswithout link-beams in Table 1 and for the structures with link-beams in Table 2.Axial load ratios shown are for the elements at the ground storey of the buildings.

Case study structures with link-beams are being examined in this work to ensurethat the design procedure performs adequately for this peculiar form of frame-wallstructure. The interaction between the frames and walls of structures with link-beams is more signicant than in the classical form of frame-wall structure in whichthe frames are parallel to the walls. As the walls deform their ends either lift ordrop, depending on whether the bending in the wall puts that part of the wall incompression or tension, as illustrated in Fig. 6. Additional curvatures are imposedon the link beams due to the change in elevation of the wall ends. The magnitudeof these curvatures can be gauged taking the shift in elevation of the wall edge anddividing by the beam length, which gives the equivalent chord rotation imposed onthe link beams.

Table 1. Characteristics of frame-wall structures without link-beams, examined in the veri-cation of the proposed DBD method.

4 storey 8 storey 12 storey 16 storey 20 storey

Wall length (mm) 8000 10 000 10 000 10 000 10 000Wall thickness (mm) 350 350 350 350 350Beam depth width (mm) 750 450 750 450 750 450 750 550 750 550Int. column depthwidth (mm) 750 600 750 600 750 600 750 600 800 650Ext. column depthwidth (mm) 600 600 600 600 600 600 600 600 650 650Inter-storey height (mm) 3600 3600 3600 3600 3600Wall axial load ratio 0.021 0.040 0.060 0.080 0.101Int. column axial load ratio 0.089 0.178 0.267 0.374 0.412Ext. column axial load ratio 0.060 0.120 0.180 0.251 0.276Floor seismic weight (kN) 11400 11800 11800 11900 12000

Table 2. Characteristics of frame-wall structures with link-beams, examined in the vericationof the proposed DBD method.


Wall length (mm) 8000 8000 8000 8000 8000Wall thickness (mm) 350 350 350 350 350Beam depthwidth (mm) 750 450 750 450 750 450 750 450 750 450Int. column depthwidth (mm) 600 600 600 600 600 600 650 600 750 750Ext. column depthwidth (mm) 600 600 600 600 600 600 600 600 650 650Inter-storey height (mm) 3600 3600 3600 3600 3600Wall axial load ratio 0.021 0.042 0.063 0.084 0.105Int. column axial load ratio 0.108 0.217 0.325 0.404 0.386Ext. column axial load ratio 0.060 0.120 0.180 0.240 0.276Floor seismic weight (kN) 11600 11600 11600 11600 11900

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High curvature ductility demands expected at beam-

wall interface.

Assumed NAD of wall

Wall edge lifts

Wall edge drops

Fig. 6. Illustration of high curvature ductility demands expected at ends of link-beams.

The walls are also aected by the link-beams since the moment and shear fromeach beam must be carried by the walls. The link beam moments can change the wallmoment prole signicantly as will be seen in later sections, whereas the shears mayaect the axial load on the walls. For the frame-wall structures shown in Fig. 5 thewall axial loads are not aected by the link-beam shears which apply equal shears(owing to their equal strength) in opposing directions on either side of the wall andtherefore cancel each other out. The moments however will need to be accountedfor as these tend to sum together at the wall centreline and can reduce the wallinection height, which in turn aects the design displacement prole. Specicrecommendations that account for the inuence of link beams will be presented ina later section.

4.2. Design criteria

A design storey drift of 2.5% was selected for the design of the case studies. Inseismic design codes (e.g. NZS1170.5:2004 [2004]) this storey drift limit is commonlyassociated with a life-safety performance level. A design spectrum was selected tomatch a set of accelerograms available as shown in Fig. 7. The design spectrum canbe selected in this arbitrary manner for these case studies since the design methodshould be applicable to any spectral shape and its applicability is not restricted toa particular code.

The design displacement spectra at levels of damping greater than 5% wereobserved to vary by a factor of , where is given by Eq. (25).

=

6/(1 + ). (25)

April 28, 2006 13:47 WSPC/124-JEE 00274


0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 1 2 3 4 5

Period (s)

5% D

amp

ed A

ccel

erat

ion

(g

)

Avg Sa 5%

Design Spectrum

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

0 1 2 3 4 5 6 7

Period(s)S

pec

tral

Dis

pla

cem

ent

(m) Avg. 5%

Avg. 10%

Avg. 15%

Avg. 20%

Sd (5%)

Sd (10%)

Sd (15%)

Sd (20%)

Fig. 7. Case study design acceleration spectrum (left) and displacement spectra (right) at dif-ferent levels of viscous damping, compared with the average spectra of selected accelerograms.

For a given increment of damping this equation reduces the displacement spec-tra signicantly more than the more realistic Eurocode 8 [CEN, 1998] equationpresented in Eq. (21). It is not proposed that Eq. (25) should be used in place ofthe Eurocode equation in normal circumstances. However, Eq. (25) does providethe best representation of the accelerograms used in this study and therefore it isused here only in order to obtain the most valid verication of the design procedure.Fig. 7 shows that the factor from Eq. (25) provides good correlation between thedesign displacement spectra and the accelerograms at damping levels of 10%, 15%and 20%.

The design storey drift limit of 2.5% is intended to control damage of non-structural items in the buildings. Damage to structural items was controlled byimposing strain limits on the concrete and reinforcing. Ultimate compressive strainsof 0.018 for the concrete and 0.06 for the reinforcing steel were deemed appropriatefor these case studies. Priestley and Kowalsky [1998] have argued that these strainlimits are reasonably conservative estimates for well-conned concrete and well-restrained reinforcement as results from detailing to the requirements of NZS3101[1995]. Priestley and Kowalsky [1998] observed that the ultimate curvature ductilityof a wall, , is well represented by Eq. (26), in which y is the yield strain of thelongitudinal reinforcement in the wall.

=0.0722y

. (26)

For the material properties being used in these case studies, Eq. (26) suggested thatthe curvature ductility of the walls should be limited to a value of 18.0. Therefore,if structural deformations associated with the storey-drift limit impose curvatureductility demands greater than 18.0, the design storey-drift should be reduced untilthe ductility limit is satised. It will be shown that for these case studies the designdrift had to be reduced for both of the eight-storey structures in order to satisfythe curvature ductility limit.

April 28, 2006 13:47 WSPC/124-JEE 00274


For the taller case studies the design drift was also reduced from the limit of2.5% in order to control the deformations caused by higher modes. This was donebecause results from the initial set of case studies reported by Sullivan et al. [2006],indicated that despite the fact that the 1st mode controlled the displacements ofthe structures, higher modes could increase storey drifts signicantly. This wasespecially evident for taller structures. The reduction in storey drift was madeusing Eq. (8).

Another control on the design of these case studies has been imposed to main-tain realistic reinforcement contents and axial load ratios. Column dimensions wereset initially to be 600 600mm square. These dimensions were then increased ifnecessary, to limit axial load ratios (N/f cAg) to a maximum of 0.40. However, col-umn dimensions are also inuenced by the necessary strength. In these case studiesthe building layout was such that the limit on axial load ratio only aected thedimensions of the interior columns of the taller structures with link beams. Axialload ratios on the walls were not of concern in this set of case studies owing tothe large area of the walls. Longitudinal reinforcement ratios in the walls were ofmore importance, and maximum and minimum longitudinal reinforcement ratioswere set at 1.6% and 0.3% respectively. For the columns, maximum and minimumlongitudinal reinforcement ratios were set at 3.0% and 0.5% respectively, while forthe beams, tension reinforcement limits of 1.5% and 0.35% were maintained.

4.3. Details of the design

The procedure for the design of frame-wall structures summarised in the owchartpresented in Fig. 1 has been used to design the case study structures. Rather thandescribe the design steps therefore, this section identies the strength assignmentsthat were made for these case studies, provides recommendations specic to thedesign of frame-wall structures with link-beams, and presents intermediate andnal design results.

4.3.1. Strength assignments

The proportions of total shear resisted by the frames and the walls were assignedarbitrarily to begin with, however, it was observed that by altering the shear pro-portions the design could be improved. For example, an initial strength assignmentthat assumes the frames will carry a large fraction of the lateral load is likely toresult in a low inection height. Having a low inection height implies that thedesign displacement and damping is maximised and the minimum possible baseshear is obtained. However, a low inection height will impose large curvatureductility demands on the walls, and if these are excessive then the design storeydrift should be reduced or a larger strength proportion assigned to the walls. Byassuming a large proportion of the shear is given to the walls then the oppositewill occur, the consequences being that the design may not be very ecient, with

April 28, 2006 13:47 WSPC/124-JEE 00274


heavily reinforced walls and poorly utilised beams and columns possessing onlyminimum reinforcement contents. The proportions of base shear assigned to theframes and walls are presented with the intermediate design results later in thissection.

Equal strengths are assigned to the beams up the height of the frames as thisis an attractive solution for construction. However, to avoid spikes in frame storeyshear at the top level of the structures, these case studies set the strength of thetop storey (roof) beams equal to half that of the other beams. The base columnstrengths were set to be fractionally larger than the beam strengths to provide aninection height of 0.6 times the storey height. This design choice was made toprovide some protection against column hinging at the top of the rst storey. Pro-tection against column hinging is necessary in frame structures to avoid formationof a soft-storey mechanism. However, in frame-wall structures this provision is notnecessary because the cantilevering walls will protect against soft-storey mecha-nisms [refer to Paulay and Goodsir, 1986]. Nevertheless, the large column strengthis not unrealistic and therefore this strength assignment was maintained.

4.3.2. Design recommendations for frame-wall structures with link-beams

As stated earlier, frame-wall structures with link-beams possess peculiar character-istics that must be allowed for in design. One of the rst adjustments that must bemade when link-beams exist, is to alter the wall moment prole associated with the1st mode wall shears to account for the moments transferred from the link-beams.Having decided on the strength assignments for the frame-wall system, the beamstrengths can be established as a fraction of the total design base shear. For thestrength assignments used for these case studies, the sum of the beam strengths,

Mb, at a given level, i, is given by Eq. (27).

Mbi =

Vi,framehcol(1 + dcolLb

) , (27)

where Vi,frame is the frame shear (known as a fraction of the total design baseshear), hcol is the storey height at level i, dcol is the depth of the columns and Lb isthe beam length (between column faces). The beam strengths in this equation referto the strength at the face of the columns which have been projected to the columncentrelines using the dcol on Lb ratio. For simplicity, these case studies neglect theeects of beam-column joints and assume that the beam strengths develop at thecolumn centrelines. This simplication implies that the dcol on Lb term drops outof Eq. (27).

The strength of a single beam is obtained using Eq. (28), in which the sum ofthe beam moments on the oor are divided by the number of beam ends, nbj , thatconnect to beam-column joints. As the frame shear used in Eq. (27) is equal to

April 28, 2006 13:47 WSPC/124-JEE 00274


the sum of the column shears, the number of beam ends that connect to the wallsshould not be included within nbj .

Mb =

Mbinbj

. (28)

Since the link-beams will develop the same strength as given by Eq. (27) at theedge of the wall, the moment transferred to the centre of the walls can be obtainedfrom the beam moments and geometry as shown in Fig. 8. Substituting Eq. (27)into Eq. (28) and using the geometry and beam bending moment diagram presentedin Fig. 8, Eq. (29) is obtained for the moment transferred from a link-beam to thewall centreline.

MbWall = Vi,frame

(1 +

LwLb

)hcol

nbj

(1 + dcolLb

) , (29)where Lw is the wall length and nbj is the number of beam ends connecting tobeam-column joints per link-beam. For these case studies the dcol on Lb term wasneglected for simplicity.

The moments transferred to the wall from the link-beams are used to adjustthe moment prole as shown for an eight-storey structure in Fig. 9. Using thisapproach, the moment prole in the walls is known as a proportion of the designbase shear. This then allows the inection height to be determined and the designcan proceed as normal.

Another stage in the design process in which the inclusion of link-beams needsto be accounted for is in determination of the frame displacement ductility. Asmentioned earlier, link-beams undergo larger plastic rotations than other beams atthe same level. In order to estimate the ductility demands on the link beams it isworth reviewing how the ductility demands on a standard RC frame are established.

For a standard beam-column sub-assemblage, the yield drift for design isobtained using Eq. (30) as recommended by Priestley [2003]. This is an approximateexpression developed by assuming that the columns and joints add respectively anadditional 40% and 25% of the displacement associated with the beams yielding in

Lw/2 Lb

Mb,wallMb dcol

Fig. 8. Illustration of bending moment transferred from link-beams to wall centrelines.

April 28, 2006 13:47 WSPC/124-JEE 00274


0

1

2

3

4

5

6

7

8

-5.00 0.00 5.00 10.00 15.00

Wall momentsfor unit base shear

Lev

el Moments fromshears only

Adjusted forL-Beammoments

Fig. 9. Wall moment prole of eight-storey structure, adjusted to allow for moments transferredfrom link-beams.

exure, to the storey deformation. It also assumes that member shear deformationsadd a further 10% to the yield drift.

y,beam = (1.0 + 0.4 + 0.25 + 0.1) 0.283y(

lbhb

)= 0.5y

(lbhb

), (30)

where y is the yield strain of the longitudinal reinforcement in the beams, hb isthe depth of the beams and lb is the beam length.

For a beam-wall assemblage it could be assumed that the column and jointdeformation contributions can be neglected. This would imply that the factor of0.5 in the yield drift equation of (30) reduces to 0.31. As a link-beam is supportedat one end by a sti wall and at the other end by a column, it is apparent thatan average factor of 0.4 can be used to approximate the yield drift of a link-beam,y,link, as shown in Eq. (31).

y,link = 0.4y

(lbhb

). (31)

The displacement ductility demands on the link-beams and other bays of the framecan be obtained using Eqs. (30) and (31) respectively, together with the storey driftassociated with the design displacement prole. A weighted average ductility value,frame,i, for each oor is then obtained in proportion to the number of link-beams,as shown in Eq. (32).

frame,i =D,i

y,linknlink +

D,iy,beam

(nb nlink)nb

, (32)

April 28, 2006 13:47 WSPC/124-JEE 00274


where D,i is the storey drift associated with the design displacement prole at leveli, nlink is the number of link-beams in the storey, and nb is the total number ofbeams on the storey.

Equation (32) is valid when beams have equal length and strength. If this isnot the case, it would be more appropriate to factor the ductility demands by thebeam shears. For these case studies, beams have equal strength and length at eachstorey and up the full height of the building. Therefore, the frame ductility hasbeen obtained as the average of the storey ductility values obtained using Eq. (32).Having determined the frame ductility, the design proceeds as normal with theequivalent viscous damping determined in the same manner as for the standardframe-wall structures.

4.3.3. Design results

Design was only conducted to the point that would allow the strengths of plastichinge regions to be set. With knowledge of these strength values, accurate nonlinearmodels of the structures could be developed for verication of the design solutionsthrough time-history analysis as explained later in Sec. 4.4.

Intermediate design values for the structures with and without link-beams arepresented in Tables 4 and 3 respectively. Note that in the design of these casestudies an elastic damping component was rst obtained in accordance with therecommendations of Priestley and Grant [2005] and then added to a hystereticcomponent determined using the recommendations of Blandon and Priestley [2005].This was because the recommendations of Grant et al. [2005] were not available atthe time of this work.

For these case studies, it can be seen that desirable design solutions wereobtained when the walls were assigned around 60% of the total design base shear.Wall curvature ductility demands were fairly large in general and for the eight-storeystructures the design storey drift had to be reduced to ensure material strain limitswere not exceeded. Note that both wall displacement ductility and curvature duc-tility demands are reported since the former relates more to the equivalent SDOFrepresentation of the structure (being calculated at the eective height) whereas

Table 3. Intermediate design results for the frame-wall structures without link-beams.


% base shear assigned to walls 60% 60% 60% 50% 45%Inection height 14.4 21.6 30.3 33.5 36.6Design storey drift 2.09% 2.23% 2.37% 2.27% 2.16%Design displacement 0.211 0.430 0.664 0.840 0.995Wall displacement ductility 9.52 7.43 5.39 4.31 3.59Average frame ductility 1.84 1.99 2.09 2.01 1.91System ductility 6.27 5.00 3.86 2.97 2.51System damping 15.1 15.0 14.8 13.9 13.2Eective mass 3806 7479 10 932 14 510 18 121Eective period 1.47 2.70 4.17 5.39 6.95

April 28, 2006 13:47 WSPC/124-JEE 00274


Table 4. Intermediate design results for the frame-wall structures with link-beams.


% base shear assigned to walls 70% 70% 60% 50% 55%Inection height 13.3 20.7 23.0 17.4 29.5Design storey drift 2.03% 2.41% 2.34% 2.21% 2.13%Design displacement 0.207 0.463 0.669 0.855 0.993Wall displacement ductility 9.62 6.50 5.12 5.78 3.35Average frame ductility 1.81 2.14 2.10 2.03 1.90System ductility 6.35 4.74 3.58 3.22 2.70System damping 16.0 16.1 15.7 15.4 14.6Eective mass 3877 7334 10 788 14 409 18 033Eective period 1.48 2.99 4.34 5.95 7.58

the latter better reects the amount of nonlinear deformation the wall would haveto undergo.

Table 4 indicates that frame displacement ductility demands were fairly low,however, even with low ductility demands it was anticipated that the frames wouldprovide a signicant amount of hysteretic energy dissipation. The eective periodvalues shown in Table 4 are long, however they lie within the spectrum compat-ible range of the accelerograms, suggesting that the time-history analyses to bepresented in later sections will provide a good test of the design solutions.

Final design strengths and longitudinal reinforcement contents for the struc-tures with and without link-beams are presented in Tables 6 and 5 respectively.Reinforcement contents for the walls and columns were obtained using axial loadsassociated with the gravity actions only. In reality exterior columns would be subjectto a signicant variation in axial load during seismic response, however, dierencesin compression from one side of the building to the other suggest that the actualstrength of the columns should be equivalent to the sum of the strengths consideringgravity loads only.

An interesting observation to be taken from these design results is that the baseshear for the buildings of considerably dierent height is relatively constant. Thisis attributed to the fact that the seismic weight per oor, the length of the wallsand the depths of the beams did not change for the dierent height structures.Constant section dimensions implied that the system damping values for the design

Table 5. Final design strengths for the frame-wall structures without link-beams.


Base shear (kN) 14 691 17 429 16 475 16 565 14 735Wall strength (kNm) 31 949 72 827 109 235 117 367 114 931Wall long. reinforcement % 0.65% 0.95% 1.55% 1.61% 1.42%Beam strength (kNm) 1014.93 1195.67 1124.75 1419.52 1395.31Beam reinforcement % 1.23% 1.50% 1.42% 1.38% 1.36%Ext. column strength (kNm) 767 907 859 1087 1072Int. column strength (kNm) 1533 1814 1717 2175 2145Ext. col. long. reinforcement % 1.84% 1.91% 1.43% 1.99% 0.91%Int. col. long. reinforcement % 2.23% 2.44% 1.88% 2.87% 1.66%

April 28, 2006 13:47 WSPC/124-JEE 00274


Table 6. Final design strengths for the frame-wall structures with link-beams.


Base shear (kN) 14 490 14 980 15 120 13 742 12 292Wall strength (kNm) 31 018 63 528 66 696 39 261 72 981Wall long. reinforcement % 0.63% 1.46% 1.42% 0.42% 1.38%Beam strength (kNm) 530.51 529.56 715.63 915.28 742.48Beam reinforcement % 0.62% 0.62% 0.89% 1.11% 0.71%Ext. column strength (kNm) 454 466 630 716 586Int. column strength (kNm) 909 931 1260 1431 1172Ext. col. long. reinforcement % 0.79% 0.50% 0.53% 0.52% 0.50%Int. col. long. reinforcement % 1.85% 1.36% 2.22% 2.14% 0.50%

drift were fairly constant. On the contrary, the target displacement (and thereforeeective period) as well as the eective mass were almost linearly dependent onheight. Since this implies that the eective stiness is inversely proportional to theheight, and the base shear is simply the product of the eective stiness and thedesign displacement, it is clear why fairly constant base shears were obtained.

The fact that the base shear may depend only on the oor mass and sectiondimensions suggests that the design procedure could be signicantly simplied with-out the loss of signicant accuracy. This is an item for future research.

Another important observation to be gleamed from the results in Tables 5 and 6is that the longitudinal reinforcement contents are all within the minimum andmaximum limits specied as part of the design criteria. The reinforcement contentson individual elements were seen to be sensitive to the strength assignments andaxial loads, and this point was used to make the design solution for each of thestructures more ecient. The fact that reasonable reinforcement contents have beenstipulated indicates that the design solutions are all realistic.

4.4. Design verification procedure

Nonlinear time-history analyses have been performed using the program Ruaumoko[Carr, 2004] to assess the performance of the proposed methodology. Models of thecase studies were constructed in which the strengths of the beams and walls wereselected to match the design values obtained using the design methodology. Themodels were subjected to seven articial acceleration records which had been usedto construct the design spectrum. The displacement spectra of the seven recordsare shown at viscous damping levels of 5% and 15% in Fig. 10. The records can beexpected to test the design solutions well as they possess relatively small scatterover a large range of periods.

Of the seven accelerograms selected, six were articial records. Record A5 is theNorth-South component of the 1978 Tabas earthquake, recorded at the Boshrooystation. The record was scaled in both magnitude (by a factor of 3.5) and duration(by a factor of two) in order to represent a large earthquake that causes a lineardisplacement spectrum up to a period of 5 s. Given this modication, none of theaccelerograms can be considered as real earthquake records. Criticisms directed

April 28, 2006 13:47 WSPC/124-JEE 00274


0.0

0.4

0.8

1.2

1.6

2.0

0 1 32 4 5 6 7

Period (s)

5% D

amp

ed D

isp

lace

men

t (m

)

Record A1

Record A2

Record A3

Record A4

Record A5

Record A6

Record A7

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 1 2 3 4 5 6 7

Period (s)

15%

Dam

ped

Dis

pla

cem

ent

(m)

Fig. 10. Displacement spectra of the seven accelerograms at 5% (left) and 15% (right) viscousdamping.

towards the use of articial accelerograms in seismic analyses usually focus on thedierent phase content and duration of actual earthquakes compared to articialrecords. However, the frame-wall structures being designed in this study are notaected by duration owing to the fact that their strength does not degrade providedthat the design displacement is not signicantly exceeded. Little is known aboutthe inuence of the phase content on structural response. However, no evidencehas been found to suggest that any dierences in phase content aect structuralresponse. Furthermore, time-history analysis of structures using both articial andreal records that possess similar demand spectra, have indicated that frame-wallstructures respond similarly using either articial or real records. For these reasonsit was considered that the use of articial accelerograms was acceptable for thesecase studies.

The success of the new methodology can be measured by comparing the actualdisplacement response for the design level earthquake with the target displacementprole selected in the design. If the analysis and target displacements and storeydrifts match, then the intended level of damage occurred and it can be concludedthat the objective of the design approach has been met.

4.4.1. Modelling structures for time-history analysis

In modelling the structures elastic properties (with reduced stiness to account forcracking) were assigned to elements that are not intended to yield. This impliesthat appropriate capacity design would have ensured that inelasticity is concen-trated only in regions associated with the collapse mechanism. An o-shoot of thismodelling technique is that the analyses can be used to test capacity design guide-lines which are reported elsewhere [Sullivan, 2005]. These case studies model theoors as rigid links fully exible out of plane and P-delta eects are not consideredsince no attempt to account for these eects was made in the design.

Beams, columns and walls were modelled using 2-hinge Giberson beam ele-ments [Carr, 2004]. It is evident that the beams and the walls of the structures

April 28, 2006 13:47 WSPC/124-JEE 00274


without link beams are likely to carry a constant axial load throughout the seis-mic response. Therefore the beams and walls of these structures could be modelledwithout the use of beam-column elements which account for changes in strengthdue to variations in axial load. The interior columns and the walls of the struc-tures with link-beams were also subject to fairly constant axial loads as they areanked by beams of equal strength and length on either side. However, the exteriorcolumns are subject to varying axial loads during the seismic response and thereforemodelling these members as beam-elements is not accurate. Nevertheless providedthat the gravity load in these columns is well below balance-point axial load thenthe strength discrepancy in the compressive column roughly balances that of thetension column. In addition, the exterior column strengths form a very small por-tion of the total overturning resistance of the frame-wall structures. As such, theuse of beam-elements for all members was deemed acceptable for the vericationstudies.

Rigid elements were used to model the connections between the walls and link-beams as shown in Fig. 11. When reinforced concrete walls are deformed in exureto the extent that they crack and later yield, the position of their neutral axis depthshifts. Since a shift in neutral axis depth equates to a shift of the centre of rotationin a section, the use of Giberson-beam elements up the centre of the walls may beinaccurate. In particular, if a wall rotates about some point other its centre, thenthis implies one side of the wall lifts or drops more than the other. This in turnwould imply that the curvature ductility demands on the beams would not be wellcaptured. However, a separate study [Sullivan, 2005] of two bre-element modelsanalysed in SeismoStruct [SeismoSoft, 2004] has shown that the curvature ductilitydemands are only dierent over the lower stories of the building. Furthermore, theoverall dierence in energy dissipation between a model with shifting neutral-axisdepth and a model with constant neutral-axis depth is not signicant for these casestudy structures because the discrepancies in beam curvature on either side of thewalls tend to cancel each other out.

Fig. 11. Illustration of model used to represent eight-storey frame-wall structure with link-beams.

April 28, 2006 13:47 WSPC/124-JEE 00274


The hysteretic behaviour of the concrete structures was represented using theTakeda model [Otani, 1981], with 5% post-yield displacement stiness and theunloading model of Emori and Schonbrich [1978]. Parameters for the Emori andSchonbrich model included an unloading stiness factor of 0.5 for walls and columnsand 0.25 for beams, together with a reloading stiness factor of 0.0 and a reloadingpower factor of 1.0 which were used for all the elements. Refer to the Ruaumokomanual [Carr, 2004] for further details. The plastic hinge lengths associated withthe yielding elements were calculated using the recommendations from Paulay andPriestley [1992].

The models developed in Ruaumoko use eective section properties up untilyield, obtained by taking the design strength and dividing by the yield curvature.Approximations for yield curvature were obtained from expressions provided byPriestley [2003]. The eective stiness for the ground storey columns at yield wasapproximated using the strength under axial load from gravity only, divided by theyield curvature. The elastic columns above the ground oor were modelled with thesame initial stiness.

Elastic damping was modelled for the structures using tangent stiness Rayleighdamping with a 1st mode damping value set to provide the eect of 5% tangent sti-ness damping for the MDOF structure, as recommended by Priestley and Grant[2006]. Priestley and Grant [2006] provide an expression for this value that con-siders the stiness and mass proportional components of the Rayleigh dampingequation. By specifying the same damping value at two dierent frequencies, wherethe higher frequency is times the lower frequency, then the damping in the 1stmode attributed to stiness proportional damping, sp, is given by Eq. (33). Inaddition, the damping attributed to mass proportional damping, mp, in the 1stmode is given by Eq. (34).

sp =1

+ 1, (33)

mp =

+ 1. (34)

Having established the proportions of mass and stiness proportional damping,Priestley and Grant [2006] recommend that for time-history analyses using Rayleightangent stiness damping, the damping on the 1st mode, 1st, should be set usingEq. (35).

1st = (mp0.75 + sp) 5%, (35)where is the displacement ductility of the equivalent SDOF system.

The initial periods of the case-study structures were obtained from eigen-valueanalysis, and the ratio between the 1st and 2nd mode frequencies were calculated togive . These frequency ratios were then used together with the system displacementductility values to obtain the 1st mode elastic damping values for time-historyanalysis shown in Table 7.

April 28, 2006 13:47 WSPC/124-JEE 00274


Table 7. 1st mode elastic viscous damping values fortime-history analysis.

T1 (s) T2 (s) 1st

4 storey Without L-beams 0.656 0.129 5.07 1.88With L-beams 0.674 0.133 5.08 1.87





The dynamic equation of equilibrium is integrated by the unconditionally sta-ble implicit Newmark Constant Average Acceleration (Newmark = 0.25) method[Chopra, 2000]. The time-step for this form of integration method should be lessthan 0.1 of the period of the highest mode of free vibration that contributes sig-nicantly to the response of the building. Consequently, for these case studies atime step of 0.005 s has been adopted. The results of the analyses, presented in thefollowing section, were output from Ruaumoko for post-processing every 0.01 s.

4.5. Results of time-history analyses

The two groups of case studies have been analysed under the suite of accelerogramsand the results have been processed to obtain displacements, shears, moments andstorey drifts. Results examined in this paper focus on the performance of the pro-posed methodology with respect to its ability to control drifts and as such, onlydisplacements and drifts are included. In [Sullivan, 2005], the shears and momentsdeveloped in the structures during the time history analyses are presented and usedto verify the performance of new capacity design recommendations.

4.5.1. Review of maximum recorded displacements

Figures 12, 13 and 14 present the maximum oor displacements recorded duringtime-history analysis using the seven dierent accelerograms. These are comparedwith the target displacement prole associated with the design drift limit for thevarious height structures.

The scatter in the results obtained for the seven dierent records is in generalquite small. This indicates that the equivalent viscous damping approach was ableto maintain relatively uniform inelastic demands for the accelerograms used. Forthe taller structures, record A4 tends to impose large displacements which areforeseeable if the displacement spectra in Fig. 10 are examined closely. The target

April 28, 2006 13:47 WSPC/124-JEE 00274


Target

A1

A2

A3

A4

A5

A6

A7

0.0 0.1 0.2 0.3

Displacement (m)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.0 0.1 0.2 0.3 0.4

Displacement (m)

Rel

ativ

e H

eig

ht

(hi/H

)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.4R

elat

ive

Hei

gh

t (h

i/H)

(i)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0 0.2 0.4 0.6 0.8

Displacement (m)

Rel

ativ

e H

eig

ht

(hi/H

)

0.00

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1.00

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Displacement (m)

Rel

ativ

e H

eig

ht

(hi/H

)

Target

A1

A2

A3

A4

A5

A6

A7

(ii)

Fig. 12. Maximum recorded displacements compared with target displacements for the(i) four-storey and (ii) eight-storey structures with (right) and without (left) link-beams.

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0.00

0.10

0.20

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Displacement (m)

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ativ

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eig

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)

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Displacement (m)

Rel

ativ

e H

eig

ht

(hi/H

)

Target

A1

A2

A3

A4

A5

A6

A7

(i)

0.00

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0.20

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Displacement (m)

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ht

(hi/H

)

0.00

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Rel

ativ

e H

eig

ht

(hi/H

)

Target

A1

A2

A3

A4

A5

A6

A7

(ii)

Fig. 13. Maximum recorded displacements compared with target displacements for the(i) 12-storey (ii) 16-storey structures with (right) and without (left) link-beams.

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0.00

0.10

0.20

0.30

0.40

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1.00

0.0 0.5 1.0 1.5

Displacement (m)

Rel

ativ

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eig

ht

(hi/H

)

Target

A1

A2

A3

A4

A5

A6

A7

0.00

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Displacement (m)

Rel

ativ

e H

eig

ht

(hi/H

)

Fig. 14. Maximum recorded displacements compared with target displacements for the 20-storeystructures with (right) and without (left) link-beams.

displacement prole (shown dashed) lies either in the centre or conservatively tothe right of the recorded displacements suggesting that the design method hasworked well.

The excellent correlation between the recorded and anticipated displacements isvery convincing. However, the ability of the design method to control the damagethat the structures are subject to will be better gauged by comparison of the designstorey drift with the average maximum recorded storey drift.

4.5.2. Review of maximum storey drifts

Figures 15, 16 and 17 present the average of the maximum storey drifts recordedduring time-history analysis using the seven dierent accelerograms. These are com-pared with the design drift prole associated with the various height structures. Alsoshown for the 12, 16 and 20 storey structures is a dashed line that represents thelimiting drift. The design drift is less than the drift limits in these cases because ofthe adjustment made to account for the eects higher modes have on storey drifts.

The results of the time-history analyses indicate that the design approach hasbeen very successful in limiting the storey drifts. Most encouragingly, the designdrift prole again provides an excellent match to the average maximum drift prole.As would be expected, the storey drifts are correlated closely with the displacementsand therefore similar trends are observed.

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0

1

2

3

4

0.0% 1.0% 2.0% 3.0%

Storey drift

Lev

el

1st modeTarget

T-historyAverage

0

1

2

3

4

0.0% 1.0% 2.0% 3.0%

Storey drift

Lev

el

(i)

0

1

2

3

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6

7

8

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Storey drift

Lev

el

0

1

2

3

4

5

6

7

8

0.0% 1.0% 2.0% 3.0%

Storey drift

Lev

el

1st modeTarget

T-historyAverage

(ii)

Fig. 15. Average of maximum recorded storey drifts compared with the target drift prole forthe (i) four-storey and (ii) eight-storey structures with (right) and without (left) link-beams.

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0

2

4

6

8

10

12

0.0% 1.0% 2.0% 3.0%

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el

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el

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Storey drift

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el

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10

12

14

16

0.0% 1.0% 2.0% 3.0%

Storey drift

Lev

el

1st modeTarget

T-historyAverage

Drift limit

(ii)

Fig. 16. Average of maximum recorded storey drifts compared with the target drift prole forthe (i) 12-storey and (ii) 16-storey structures with (right) and without (left) link-beams.

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0

2

4

6

8

10

12

14

16

18

20

0.0% 1.0% 2.0% 3.0%

Storey drift

Lev

el

0

2

4

6

8

10

12

14

16

18

20

0.0% 1.0% 2.0% 3.0%

Storey drift

Lev

el

1st modeTarget

T-historyAverage

Drift limit

Fig. 17. Average of maximum recorded storey drifts compared with the target drift prole forthe 20-storey structures with (right) and without (left) link-beams.

The adjustment for higher modes is most easily assessed through review ofthe displacements and storey drifts for the 12- and 16-storey structures with link-beams and the 20-storey structure without link-beams. For these structures, therecorded and predicted displacement proles were closely matched and therefore anydierences in storey drifts will highlight the eects of higher modes. Consequently,it is clear that the higher mode adjustment that was made during design usingEq. (8) has performed satisfactorily, especially for the 16-storey structure. For the20-storey structure with link-beams, drifts are greater than desired over the topstoreys, suggesting that the higher mode reduction factor should have been greaterand as such, future research could aim to improve Eq. (8).

5. Conclusions

In conclusion, these case studies have clearly illustrated that the new design pro-cedure for frame-wall structures provides excellent control of storey drifts and dis-placements for buildings of up to 20 storeys in height. The interaction that takesplace between frame and wall elements has been successfully accounted for andit has been shown that the approach works well when structures with or withoutlink-beams are considered.

The recommendations made for prediction of the displacement prole are con-sidered valid for structures possessing RC walls with aspect ratio greater than three.If the method were to be applied to structures possessing walls with aspect ratio

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less than three, some account for shear deformations should be made. Furthermore,it has been necessary to make simplifying assumptions as to the cracked elasticstiness of wall elements. As such, future work could aim to reduce the uncertaintyassociated with this part of the methodology.

Future work should verify that three-dimensional eects do not jeopardisethe ability of the method. Testing conducted at UC Berkeley indicated that 3-dimensional eects tend to increase the overturning resistance of a structure. Thisimplies that such eects should not require any changes to the DBD process butmay need to be accounted for during capacity design. Maximum forces that developin the structures and recommendations for their capacity design are reported else-where [Sullivan, 2005]. Such capacity design guidelines, together with the DBDprocess presented here, complete a set of recommendations that provide designerswith a simple, rational and eective means of conducting seismic design of frame-wall structures.

References

Blandon, C. A. and Priestley, M. J. N. [2005] Equivalent viscous damping equationsfor direct displacement based design, Journal of Earthquake Engineering 9 (SpecialIssue 2), 257278.

Carr, A. J. [2004] Ruaumoko3D A Program for Inelastic Time-History Analysis, Depart-ment of Civil Engineering, University of Canterbury, New Zealand.

CEN [1998] Eurocode 8 Design Provisions for Earthquake Resistance of Structures,prEN-1998-1:200X, Revised Final PT Draft (preStage 49), Comite Europeen de Nor-malization, Brussels, Belgium.

Chopra, A. K. [2000] Dynamics of Structures (Pearson Education, USA).Emori, K. and Schonbrich, W. C. [1978] Analysis of reinforced concrete frame-wall struc-

tures for strong motion earthquakes, Civil Engineering Studies, Structural ResearchSeries, No. 434, University of Illinois, Urbana, Illinois, USA.

Grant, D. N., Blandon, C. A. and Priestley, M. J. N. [2005] Modelling Inelastic Responsein Direct-Displacement Based Design, Research Report ROSE 2005/3, IUSS Press,Pavia, Italy.

Gulkan, P. and Sozen, M. [1974] Inelastic response of reinforced concrete structures toearthquake motions, ACI Journal 71(12), 604610.

NZS1170.5:2004 [2004] Structural Design Actions Part 5: Earthquake Actions New Zealand (Standards New Zealand, Wellington, N.Z).

NZS3101 [1995] Concrete Structures Standard. Part 1 The Design of Concrete Struc-tures, and Part 2 Commentary (Standards New Zealand, Wellington).

Otani, S. [1981] Hysteresis models of reinforced concrete for earthquake response analy-sis, Journal of the Faculty of Engineering, University of TokyoXXXVI(2), 125159.

Paulay, T. and Goodsir, W. J. [1986] The capacity design of reinforced concrete hybridstructures for multistorey buildings, Bulletin of NZ National Society for EarthquakeEngineering, New Zealand 19(1), 117.

Paulay, T. and Priestley, M. J. N. [1992] Seismic Design of Reinforced Concrete andMasonry Buildings (John Wiley & Sons, Inc., New York).

Priestley, M. J. N. [2003] Myths and Fallacies in Earthquake Engineering, Revisited, TheMallet Milne Lecture (IUSS Press, Pavia, Italy).

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Priestley, M. J. N. and Grant, D. N. [2006] Viscous damping for analysis and design,Journal of Earthquake Engineering, Special Edition, printing in progress.

Priestley M. J. N. and Kowalsky M. J. [1998] Aspects of drift and ductility capacity of can-tilever structural walls, Bulletin of the New Zealand National Society for EarthquakeEngineering, New Zealand National Society for Earthquake Engineering, Silverstream31(2).

Priestley, M. J. N. and Kowalsky, M. J. [2000] Direct displacement-based seismic design ofconcrete buildings, Bulletin, NZ National Society for Earthquake Engineering, NewZealand 33(4), 421444.

Priestley, M. J. N. and Paulay, T. [2002] What is the stiness of reinforced con-crete walls, SESOC Journal, Structural Engineering Society of New Zealand 15(1),3034.

SeismoSoft [2004] SeismoStruct A computer program for static and dynamic nonlinearanalysis framed structures, Available on-line from URL: http//www.seismosoft.com.

Shibata, A. and Sozen, M. A. [1976] Substitute structure method for seismic design inreinforced concrete, Journal of the Structural Division, ASCE 102(ST1).

Sullivan, T. J. [2005] Seismic Design of Frame-Wall Structures, PhD thesis, ROSE school,Universita` degli studi di Pavia, Italy.

Sullivan, T. J., Priestley, M. J. N. and Calvi, G. M. [2004] Displacement shapes of frame-wall structures for direct displacement based design, Proceedings of Japan-Europe5th Workshop on Implications of Recent Earthquakes on Seismic Risk, Bristol.

Sullivan, T. J., Priestley, M. J. N. and Calvi, G. M. [2005] Development of an innovativeseismic design procedure for frame-wall structures, Journal of Earthquake Engineer-ing 9 (Special Issue 2), 279307.

Documents

Direct Displacement-Based Design of Frame-Wall Structures-Sullivan,Priestley,Calvi-2006