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Journal of Earthquake Engineering Vol. 10, Special Issue 1 (2006) 91–124 c Imperial College Press


DIRECT DISPLACEMENT-BASED DESIGN OF FRAME-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 both frames and walls is presented in this paper. Within the new procedure, strength proportions between walls and frames are assigned and are used to establish the design displacement profile before any analysis has taken place. Knowledge of the displacement profile and recommendations for the combination of frame and wall damping components enables representation of the structure as an equivalent single-degree of freedom system. The Direct DBD process is then utilised to set the required strength level which is proportioned to the structure in line with the initial strength assignments. To test the design methodology, two sets of 4-, 8,- 12-, 16- and 20-storey reinforced concrete structures are designed. The first set considers frame-wall structures in which the frames are parallel to the walls and the second considers structures in which link-beams connect from the frames directly onto the ends of the walls. A suite of time-history analyses are conducted 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) of structures 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 form of structure, commonly known as a frame-wall structure or dual system structure, because the dynamic behaviour of dual systems is considerably different from pure frame or wall structures for which many design recommendations already exist. Such differences in dynamic behaviour are attributed principally to the interaction that takes place between the frames and walls, which is not well accounted for in current design practice. A further motivation for this work stems from the consideration that the combined structural form is a very efficient and convenient way to resist earthquake actions that is not currently being widely exploited. It can be argued from both structural and aesthetic points of view that the combination of frames and walls presents considerable advantage over structures formed purely out of frames or walls. The stiff nature of cantilever reinforced concrete (RC) walls means that they are naturally suited to control storey-drifts in the lower levels of buildings. In contrast, frames typically restrain deformation in upper storeys but moreover, they 91

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offer significant energy dissipation up the height of the building which reduces the total displacements that a building experiences. From aesthetic and functional points of view, frames enable large open spaces within minimum constraints on usage. On the other hand, walls are an attractive means of forming stair wells and lift shafts in a building, while at boundary lines they are commonly used to provide fire resistance between buildings.

2. Challenges for the Direct DBD of Frame-Walls An ideal seismic design procedure will establish the minimum basic strength of a structure sufficient to ensure pre-defined performance criteria for the building are satisfied at the design ground motion intensity, with a minimum of effort. Previous work by Sullivan et al. [2005] investigated a trial methodology which provided encouraging results when applied to regular frame-wall structures in which the frames were parallel to the walls. The research identified that the following two tasks were required to improve the accuracy of the methodology and thereby enable the Direct DBD [Priestley, 2003] approach to be used for frame-wall structures: • Development of an expression for the displaced shape of frame-wall structures at maximum response, to enable equivalent SDOF characteristics to be established. • Development of an expression for the equivalent SDOF system ductility or equivalent viscous damping that takes into account the frame-wall interaction. Sullivan et al. [2005] proposed that the design displacement profile be set as a function of the moment profile in the walls, using proportions of strength assigned at the start of the design procedure. There is experimental evidence that supports the validity of this approach as reported by Sullivan et al. [2004]. Another recommendation made by Sullivan et al. [2005] was that the equivalent SDOF system viscous damping could be obtained by factoring the individual frame and wall components by the proportions of overturning they resist. The challenge in this paper is therefore to finalise the design procedure proposed by Sullivan et al. [2005] and to verify its accuracy through examination of a range of case study structures.

3. Description of the New Design Procedure The various steps of the seismic design procedure for frame-wall structures are shown as a flowchart in Fig. 1. The first set of steps aims to develop an equivalent SDOF representation of the MDOF structure. This is achieved by assigning strength proportions and subsequently using the moment profile in the walls to set a design displaced shape. With knowledge of the displacement profile, various equivalent SDOF properties of the structure are obtained. The second important set of steps in the design aim to determine the required effective period and then

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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. Reduce drift limit.

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

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

Choose a trial effective period, Te,trial.

NO Determine equivalent viscous damping values for frames and walls.

Reset Te,trial = Te

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

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

NO

Determine effective stiffness and design base shear, Vb = Ke ∆d.

Obtain beam & column strengths by factoring strength proportions by base shear.

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. Fig. 1.

Flowchart of recommended design procedure for frames-wall structures.

stiffness using the substitute structure approach [Gulkan and Sozen, 1974; Shibata and Sozen, 1976]. The design base shear is obtained through multiplication of the necessary effective stiffness by the design displacement and the strength of individual structural elements is set taking care to ensure that initial strength assignments are maintained. Each of these phases is described in more detail in the sub-sections that follow.

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3.1.1. Assignment of strength proportions to establish the wall inflection height In order to develop a sufficiently accurate SDOF representation of the frame-wall structure, strength proportions are assigned at the very start of the design procedure. This involves setting the proportion of base shear or overturning resistance offered by the frames and walls, in addition to the relative strength distribution of yielding elements (beams and ground storey columns) within the frames. As mentioned above, by assigning these strength proportions the shear and moment profile in the walls can be established and this then enables determination of the inflection height. Figure 2 locates the inflection height for a frame-wall structure in which the frames and walls resist the total base shear in equal proportions and the frames provide a constant shear resistance over their height. The inflection height is of particular interest as it will be used to form the design displacement profile. Note that the proportions of strength assigned at this stage of the design process are 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 to develop at maximum response. The maximum forces are affected by overstrength and higher mode effects and are established following DBD as part of a capacity design procedure. The storey shear above the base of the walls cannot be obtained directly from the design base shear since the walls remain elastic above the ground storey and upper storey shears will depend on the proportion of shear carried by the frames. As such, wall shears are obtained as the difference between the total shear and the frame shear as shown in Eq. (1). Recall that the frame storey shear can be determined since it is dependent only on the strength of the beams up the building height. Vi,total Vi,frame Vi,wall = − , Vb Vb Vb

(1)

where Vb is the total base shear, Vi,wall is the wall shear at level i, Vi,total is the total shear at level i, and Vi,frame is the frame shear at level i. Frame shear (dashed line) Frame overturning

Total shear (solid line) Wall shear (shaded area)

Wall inf lection height, h inf

MWall 0.5Vb BEAM-SWAY MECHANISM

Fig. 2.

SHEARS

Wall BMD

1.0Vb MOMENTS

Use of frame-wall strength proportions to locate inflection height in walls.

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For the purpose of establishing the inflection height, a triangular distribution of 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 of the base shear as shown in Eq. (2). i (i − 1) Vi,total , =1− Vb n (n + 1)

(2)

where Vi,total is the total shear at level i, Vb is the total base shear, and n is the total 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 this shear proportion, the relative strength distribution of yielding elements within the frames is used. Although the designer is free to choose any strength distribution they prefer, it is proposed that the use of beams of equal strength up the height of the structure is advantageous for design and construction. Assuming that beam moments are carried equally by columns above and below a beam-column joint, the frame storey shear is obtained as a function of the beam strength using Eq. (3).


Mb,i ( Mb,i + Mb,i−1 ) = , (3) Vi,f rame = 2(hi − hi−1 ) hcol where Vi,f rame 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 actually known to begin with, Eq. (3) is useful as it indicates that provided beams of equal strength are to be used then the frame storey shear is constant up the building height. Consequently, if 40% of the base shear is being carried by the frames, this 40%Vb will be carried up the entire height of the frame. As such, the shear proportion carried by the frame can be substituted into Eq. (1) and the wall shears and bending calculated, all as a function of the design base shear. A perfectly constant shear up the height of the frame requires that the sums of the base column strengths and roof beam strengths are both equal to half the sum of the intermediate level beam strengths. If roof level beams are assigned strength equal to those on other stories, then the frame shear at roof level should be considered to be 50% greater than that at other levels. Larger base column strengths will also imply larger ground storey shears, with the column inflection height shifting above 0.5hcol. The storey shear and consequently the moment in the walls are used to establish the inflection height in the walls, hinf , where the moment and curvature is zero. This inflection height will be used to find the displacements of the structure at yield of the walls and to develop the design displacement profile, as detailed in the next subsections. Other important design quantities that should be obtained from the strength assignments are the proportion of overturning resisted by the frames and walls respectively. The proportions of overturning can be obtained directly from the shear

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profile up the height of the structures. These overturning proportions are used later in the design procedure for definition of the system damping and for adjustment of the 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 yield curvature and displacements at yield are important for the development of the design displacement profile. The frame yield displacement, or yield storey drift, is also important to the design process as it is used to provide an indication of the energy absorbed through hysteretic response of the frame. The yield curvature of the walls, φyW all , is firstly obtained using Eq. (4) [Priestley, 2003]. φyW all =

2εy , Lw

(4)

where εy is the yield strain of the longitudinal reinforcement in the wall and Lw is the wall length. The displacement profile of the structure at yield of the wall, ∆i,y , can then be established using the wall yield curvature, inflection height and storey height in accordance with Eqs. (5a) and (5b). ∆iy =

φyW all h2inf φyW all hinf hi − 2 6

∆iy =

φyW all h3i φyW all h2i − 2 6hinf

for hi ≥ hinf ,

for hi < hinf .

(5a) (5b)

The frame yield drift, θyf rame , used to estimate the ductility and equivalent viscous damping of the frames, is obtained in accordance with Eq. (6) [Priestley, 2003]: θyframe =

0.5lb εy , hb

(6)

where lb is the average beam length, εy is the yield strain of beam longitudinal reinforcement and hb is the average depth of the beams at the level of interest. 3.1.3. Design displacement profile and equivalent SDOF characteristics The design displacement profile is developed using the various values obtained in the preceding subsections, together with the design storey drift, as shown in Eq. (7).   φyWall hinf ∆i = ∆iy + θd − · hi , 2

(7)

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where ∆i is the design displacement for level i, ∆i,y is the displacement of level i at yield of the walls, θd is the design storey drift, φyW all is the yield curvature of the walls, hinf is the inflection height, and hi is the height at level i. Note that the design storey drift can be initially taken as the code limit for non-structural damage, reduced to allow for higher mode effects in accordance with Eq. (8).    (N − 5) MOT,f rame + 0.25 ≤ θd,limit , θd = θd,limit 1 − 100 MOT,total

(8)

where N is the number of stories, MOT,f rame is the overturning resistance of the frame and MOT,total is the total overturning resistance of the structure. This approximate 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 overturning resistance can be obtained in terms of the base shear using the strength assignments made at the start of the design procedure. The design drift given by Eq. (8) may be reduced further if it is found that inelastic demands on the structure are likely to be excessive. Alternatively, the critical value of storey drift can be determined before the design displacement profile is developed. With knowledge of the displacement profile at maximum response; ∆i , the seismic masses; mi , and storey heights; hi , the equivalent SDOF design displacement; ∆d , effective mass; me , and effective height; he , can be calculated as shown in Eqs. (9) to (11) [Priestley, 2003] respectively.
n (mi ∆2i ) , ∆d =
i=1 n i=1 (mi ∆i )
n (mi ∆i ) me = i=1 , ∆d
n (mi ∆i hi ) he =
i=1 . n i=1 (mi ∆i )

(9) (10) (11)

3.1.4. Design ductility values, effective 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 function of ductility and according to recent recommendations by [Blandon and Priestley, 2005] and [Grant et al., 2005], the effective period. The ductility demands on the walls for use within this equivalent viscous damping approach should be calculated using displacement at the effective height. The wall ductility demand, µwall , is therefore simply the design displacement divided by the yield displacement of the walls at the effective height, as shown in Eq. (12). µwall =

∆d , ∆he,y

(12)

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where ∆d is the design displacement (from Eq. (9)) and ∆he,y is the yield displacement of the wall at the effective height (obtained substituting the effective height into the appropriate version of Eq. (5)). The displacement ductility demands on the frames at each level up the height of the structure can be obtained using the storey drifts as shown in Eq. (13).   ∆i − ∆i−1 1 , (13) µframe,i = hi − hi−1 θyframe where ∆i , ∆i−1 , hi , and hi−1 , are the displacements and heights at level i and level i − 1 respectively, µf rame,i is the frame ductility at level i, and θyf rame is the yield drift of the frame (from Eq. (6)). When beams of equal strength are used up the height of the structure, the ductility obtained from Eq. (13) for each storey can be averaged to give the frame displacement ductility demand. Before proceeding with calculations of the equivalent viscous damping, it is necessary to check that the ductility demands are sustainable. Ductility demands on frames are typically not critical as the walls tend to have smaller yield curvatures and yield displacements. For frame-wall structures in which frames are parallel to walls, ductility demands will be fairly low and can typically be detailed for relatively easily. However, when link-beams connect between frames and walls then these linkbeams are likely to be subject to higher curvatures than other beams and should be checked separately, as is discussed Sec. 4 where the procedure is applied to various case study structures. Although the wall displacement ductility demand indicated by Eq. (12) is appropriate for estimation of the equivalent viscous damping, it is not a good representation of the inelastic deformation that the walls must undergo. A more appropriate parameter is the wall curvature ductility, µφwall , which can be obtained in accordance with Eq. (14).   1 φyWall hinf µφwall = 1 + θd − , (14) Lp φyWall 2 where Lp is the wall plastic hinge length, θd is the design storey drift, φyW all is the yield curvature of the walls and hinf is the inflection height. Note that because the curvature ductility demand is a function of the inflection height and not the total height, inelastic deformation demands in walls of frame-wall structures will typically be larger than those in plane wall structures. The wall plastic hinge lengths to be used within Eq. (14) are taken as the minimum of Eqs. (15a) and (15b). Lp = 0.022fy db + 0.054hinf ,

(15a)

Lp = 0.2Lw + 0.03hinf ,

(15b)

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

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have been adapted from [Priestley, 2003] with the inflection height substituting the total height. The curvature ductility capacity of a RC wall will depend on the strain limits selected for the concrete in compression (εc ) and longitudinal reinforcement in tension (εs ). For reasonably conservative values of εc = 0.018 and εs = 0.06, Priestley and Kowalsky [1998] found that the ultimate curvature of reinforced concrete walls is well represented by Eq. (16). φu =

0.072 , Lw

(16)

where φu is the ultimate curvature and Lw is the wall length. This equation was shown to be representative of ultimate curvature over a range of axial load ratios and longitudinal reinforcement contents. Combining Eqs. (16) and (4), it is found that the curvature ductility capacity is approximately equal to 0.036/εy . If the checks on ductility indicate that the inelastic deformations associated with the design drift will be excessive then the design drift must be reduced and the design displacement profile re-computed as discussed in the previous sub-section. If the ductility demands are sustainable then the next step in the design procedure is 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 function of the effective period. As this is unknown at the start of the design process, a trial value can be used and an iterative design process adopted. A reasonable estimate for the trial value of the effective period can be obtained from Eq. (17). Te,trial =

N√ µsys , 6

(17)

where N is the total number of stories and µsys is the system ductility. Equation (17) is similar in form to a code based equation that uses the height or number of storeys to estimate the initial period. The ductility term accounts for the difference between the initial and effective periods, neglecting the effect of strain hardening. Given the approximate nature of Eq. (17) [refer Sullivan, 2005] trial effective period values may be some 30% different than the final effective period, however by using such a trial value, it will be found that convergence is attained within one or, at most, two iterations. Having set the trial effective period and established expected ductility values, the frame and wall equivalent viscous damping components are calculated using Eqs. (18) and (19) respectively [Grant et al., 2005].    95 1 1 1+ , (18) 1 − 0.5 − 0.1rµwall ξhyst,wall = 1.3π µwall (Te,trial + 0.85)4    1 120 1 1 − 0.5 − 0.1rµf rame 1+ ξhyst,f rame = , (19) 1.3π µf rame (Te,trial + 0.85)4

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where r is the post-elastic stiffness coefficient, typically taken as 0.05 for new RC structures. Note that by considering the influence of the period on the damping values, it could be argued that the period-dependence of the damping values can be neglected when effective periods are greater than 1.0 s, which is usually the case for frame-wall structures. The equivalent viscous damping for the frames and walls is obtained adding the elastic and hysteretic components together and then a value of damping for the equivalent SDOF system is determined using Eq. (20). ξSDOF =

Mwall · ξwall + MOT,frame · ξframe , Mwall + MOT,frame

(20)

whereMOT,f rame is the overturning resistance of the frames and MW all is the overturning resistance (flexural strength) of the walls. At this point of the design process, all of the substitute structure characteristics have been established and as such, the displacement spectrum is developed at the design level of damping. This can be done using a damping-dependent scaling factor appropriate for the seismological characteristics of the design region. The Eurocode 8 [CEN, 1998] recommends that the η value obtained from Eq. (21) be used to scale the elastic spectrum to the damping level of interest.  (21) η = 10/(5 + ξSDOF ) ≥ 0.55, where ξSDOF is the equivalent viscous damping of the system as given by Eq. (20). The design displacement is then used to read off (or interpolate between known points) the required effective period, Te , as shown in Fig. 3. The effective period obtained from the Direct DBD process illustrated in Fig. 3 is then compared to the trial effective period value. If the period values do not match, then the period obtained from Fig. 3 replaces the trial period and the design step

Spectral Displacement at ξSDOF damping (m)

1.00 Displacement spectrum at system damping level.

0.80

0.60

∆d

0.40

0.20 Te

0.00 0.0

1.0

2.0

3.0

4.0

5.0

Period (s) Fig. 3.

Direct displacement based design to obtain the required effective period.

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is repeated. When effective periods finally match, the designer is in a position to determine the effective stiffness and design base shear as outlined next. 3.1.5. Determining the design base shear and member strengths With the effective period established, the effective stiffness, Ke , is determined in accordance with Eq. (22). me (22) Ke = 4π 2 2 , Te where me is the effective mass (from Eq. (10)) and Te is the effective period. This effective stiffness is then multiplied by the design displacement, ∆d , to obtain the base shear, Vb , as shown by Eq. (23). Vb = Ke ∆d .

(23)

Individual member strengths are then determined maintaining the strength proportions assigned at the start of the design process. Note however, that rather than use a triangular lateral force distribution, better results are obtained distributing the base shear up the height of the structure according to Eq. (24). mi ∆i Vb , (24) Fi =
N i=1 mi ∆i 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 steps to the design procedure, however, the process is simple and does provide excellent control of displacements and storey drifts as is demonstrated in the following section. 4. Verification of the Design Method The design method is verified through examination of several case studies. A range of frame-wall structures are designed using the new procedure with the aim of maintaining storey drift and curvature ductility limits typical of a life-safety performance level. The design strengths obtained for each case study are then used to set the strength of accurate non-linear analytical models which are subject to a series of time-history analyses using earthquake records compatible with the design spectrum. The success of the design process is gauged by comparing the target deformations anticipated during the design phase, with the actual deformations as predicted 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 are designed using the new procedure. Two types of structure are considered; (i) structures with walls and frames connected only by floor slabs, and (ii) structures with

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Structures with & without Link-Beams 8m

8m

8m

EQ Excitation Direction

RC Walls 8m long w. L-Beams 10m long w/o L-Beams

8m

EQ Excitation 20m

Direction

56m

Fig. 4. Plan view of frame-wall structures being examined in the verification of the proposed DBD 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 be examined as part of final verification of the DBD method.

link-beams extending from frames directly to the ends of the walls. In order to comprehensively test the approach, buildings of 4, 8, 12, 16 and 20 storeys are examined. These case study structures are regular in layout with a RC frame-wall system being 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 walls would be used to resist lateral loads, however, this does not affect the design procedure proposed here for the frame-wall system. The structures are considered as having rigid foundations with floor slabs that act as rigid-diaphragms in plane, fully flexible out of plane. It has been shown elsewhere [Sullivan, 2005] that for framewall structures of typical layout diaphragm flexibility does not require consideration within the design procedure. The concrete and reinforcement material properties assumed for the structures are values that could typically be found in building practice. Values for the concrete include: (i) fc
= 30.0 MPa and (ii) Ec = 25 740 MPa. The expected strengths adopted for the reinforcing steel include: (i) fy = 400 MPa and (ii) Es = 200 000 MPa. For seismic design, material strengths are not factored to dependable strength levels and instead these values have been taken as the expected strength

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and stiffness characteristics. The seismic weights of individual floors have been estimated assuming a concrete density value of 24.5 kN/m3 , a superimposed dead-load of 1.0 kPa, a reduced live-load of 1.0 kPa and a loaded floor area of 1105 m2 per level. Axial load ratios have been computed using these floor weights factored by the tributary area of floor supported by the individual elements. Floor weights, axial load ratios and dimensions of individual elements are presented for the structures without 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 ensure that the design procedure performs adequately for this peculiar form of frame-wall structure. The interaction between the frames and walls of structures with linkbeams is more significant than in the classical form of frame-wall structure in which the frames are parallel to the walls. As the walls deform their ends either lift or drop, depending on whether the bending in the wall puts that part of the wall in compression or tension, as illustrated in Fig. 6. Additional curvatures are imposed on the link beams due to the change in elevation of the wall ends. The magnitude of these curvatures can be gauged taking the shift in elevation of the wall edge and dividing by the beam length, which gives the equivalent chord rotation imposed on the link beams. Table 1. Characteristics of frame-wall structures without link-beams, examined in the verification of the proposed DBD method.

Wall length (mm) Wall thickness (mm) Beam depth × width (mm) Int. column depth × width (mm) Ext. column depth × width (mm) Inter-storey height (mm) Wall axial load ratio Int. column axial load ratio Ext. column axial load ratio Floor seismic weight (kN)

4 storey

8 storey

12 storey

16 storey

20 storey

8000 350 750 × 450 750 × 600 600 × 600 3600 0.021 0.089 0.060 11400

10 000 350 750 × 450 750 × 600 600 × 600 3600 0.040 0.178 0.120 11800

10 000 350 750 × 450 750 × 600 600 × 600 3600 0.060 0.267 0.180 11800

10 000 350 750 × 550 750 × 600 600 × 600 3600 0.080 0.374 0.251 11900

10 000 350 750 × 550 800 × 650 650 × 650 3600 0.101 0.412 0.276 12000

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

Wall length (mm) Wall thickness (mm) Beam depth × width (mm) Int. column depth × width (mm) Ext. column depth × width (mm) Inter-storey height (mm) Wall axial load ratio Int. column axial load ratio Ext. column axial load ratio Floor seismic weight (kN)

4 storey

8 storey

12 storey

16 storey

20 storey

8000 350 750 × 450 600 × 600 600 × 600 3600 0.021 0.108 0.060 11600

8000 350 750 × 450 600 × 600 600 × 600 3600 0.042 0.217 0.120 11600

8000 350 750 × 450 600 × 600 600 × 600 3600 0.063 0.325 0.180 11600

8000 350 750 × 450 650 × 600 600 × 600 3600 0.084 0.404 0.240 11600

8000 350 750 × 450 750 × 750 650 × 650 3600 0.105 0.386 0.276 11900

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Wall edge lifts Wall edge drops High curvature ductility demands expected at beamwall interface.

Assumed NAD of wall Fig. 6.

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

The walls are also affected by the link-beams since the moment and shear from each beam must be carried by the walls. The link beam moments can change the wall moment profile significantly as will be seen in later sections, whereas the shears may affect the axial load on the walls. For the frame-wall structures shown in Fig. 5 the wall axial loads are not affected by the link-beam shears which apply equal shears (owing to their equal strength) in opposing directions on either side of the wall and therefore cancel each other out. The moments however will need to be accounted for as these tend to sum together at the wall centreline and can reduce the wall inflection height, which in turn affects the design displacement profile. Specific recommendations that account for the influence of link beams will be presented in a later section.

4.2. Design criteria A design storey drift of 2.5% was selected for the design of the case studies. In seismic design codes (e.g. NZS1170.5:2004 [2004]) this storey drift limit is commonly associated with a life-safety performance level. A design spectrum was selected to match a set of accelerograms available as shown in Fig. 7. The design spectrum can be selected in this arbitrary manner for these case studies since the design method should be applicable to any spectral shape and its applicability is not restricted to a particular code. The design displacement spectra at levels of damping greater than 5% were observed to vary by a factor of η, where η is given by Eq. (25). η=

 6/(1 + ξ).

(25)

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1.6

1.0 0.9

Spectral Displacement (m)

5% Damped Acceleration (g)

Direct Displacement-Based Design of Frame-Wall Structures

Avg Sa 5%

0.8 Design Spectrum

0.7 0.6 0.5 0.4 0.3 0.2 0.1

Avg. 5%

1.4

Avg. 10%

1.2

Avg. 15% Avg. 20%

1.0

Sd (5%)

0.8

Sd (10%)

0.6

Sd (15%) Sd (20%)

0.4 0.2 0.0

0.0 0

1

2

3

4

5

0

1

2

3

4

5

6

7

Period(s)

Period (s)

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

For a given increment of damping this equation reduces the displacement spectra significantly more than the more realistic Eurocode 8 [CEN, 1998] equation presented in Eq. (21). It is not proposed that Eq. (25) should be used in place of the Eurocode equation in normal circumstances. However, Eq. (25) does provide the best representation of the accelerograms used in this study and therefore it is used here only in order to obtain the most valid verification of the design procedure. Fig. 7 shows that the factor from Eq. (25) provides good correlation between the design 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 nonstructural items in the buildings. Damage to structural items was controlled by imposing strain limits on the concrete and reinforcing. Ultimate compressive strains of 0.018 for the concrete and 0.06 for the reinforcing steel were deemed appropriate for these case studies. Priestley and Kowalsky [1998] have argued that these strain limits are reasonably conservative estimates for well-confined concrete and wellrestrained reinforcement as results from detailing to the requirements of NZS3101 [1995]. Priestley and Kowalsky [1998] observed that the ultimate curvature ductility of a wall, µφ , is well represented by Eq. (26), in which εy is the yield strain of the longitudinal reinforcement in the wall. µφ =

0.072 . 2εy

(26)

For the material properties being used in these case studies, Eq. (26) suggested that the 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 curvature ductility demands greater than 18.0, the design storey-drift should be reduced until the ductility limit is satisfied. It will be shown that for these case studies the design drift had to be reduced for both of the eight-storey structures in order to satisfy the curvature ductility limit.

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For the taller case studies the design drift was also reduced from the limit of 2.5% in order to control the deformations caused by higher modes. This was done because 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 of the structures, higher modes could increase storey drifts significantly. This was especially evident for taller structures. The reduction in storey drift was made using Eq. (8). Another control on the design of these case studies has been imposed to maintain realistic reinforcement contents and axial load ratios. Column dimensions were set initially to be 600 × 600 mm square. These dimensions were then increased if necessary, to limit axial load ratios (N/fc
Ag ) to a maximum of 0.40. However, column dimensions are also influenced by the necessary strength. In these case studies the building layout was such that the limit on axial load ratio only affected the dimensions of the interior columns of the taller structures with link beams. Axial load ratios on the walls were not of concern in this set of case studies owing to the large area of the walls. Longitudinal reinforcement ratios in the walls were of more importance, and maximum and minimum longitudinal reinforcement ratios were set at 1.6% and 0.3% respectively. For the columns, maximum and minimum longitudinal reinforcement ratios were set at 3.0% and 0.5% respectively, while for the 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 flowchart presented in Fig. 1 has been used to design the case study structures. Rather than describe the design steps therefore, this section identifies the strength assignments that were made for these case studies, provides recommendations specific to the design of frame-wall structures with link-beams, and presents intermediate and final design results. 4.3.1. Strength assignments The proportions of total shear resisted by the frames and the walls were assigned arbitrarily to begin with, however, it was observed that by altering the shear proportions the design could be improved. For example, an initial strength assignment that assumes the frames will carry a large fraction of the lateral load is likely to result in a low inflection height. Having a low inflection height implies that the design displacement and damping is maximised and the minimum possible base shear is obtained. However, a low inflection height will impose large curvature ductility demands on the walls, and if these are excessive then the design storey drift should be reduced or a larger strength proportion assigned to the walls. By assuming a large proportion of the shear is given to the walls then the opposite will occur, the consequences being that the design may not be very efficient, with

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107

heavily reinforced walls and poorly utilised beams and columns possessing only minimum reinforcement contents. The proportions of base shear assigned to the frames and walls are presented with the intermediate design results later in this section. Equal strengths are assigned to the beams up the height of the frames as this is an attractive solution for construction. However, to avoid spikes in frame storey shear at the top level of the structures, these case studies set the strength of the top storey (roof) beams equal to half that of the other beams. The base column strengths were set to be fractionally larger than the beam strengths to provide an inflection height of 0.6 times the storey height. This design choice was made to provide some protection against column hinging at the top of the first storey. Protection against column hinging is necessary in frame structures to avoid formation of a soft-storey mechanism. However, in frame-wall structures this provision is not necessary because the cantilevering walls will protect against soft-storey mechanisms [refer to Paulay and Goodsir, 1986]. Nevertheless, the large column strength is 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 characteristics that must be allowed for in design. One of the first adjustments that must be made when link-beams exist, is to alter the wall moment profile associated with the 1st 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 beam strengths can be established as a fraction of the total design base shear. For the strength assignments used for these case studies, the sum of the beam strengths,
Mb , at a given level, i, is given by Eq. (27).

Vi,f rame hcol  , Mbi =

1 + dLcol b

(27)

where Vi,f rame is the frame shear (known as a fraction of the total design base shear), hcol is the storey height at level i, dcol is the depth of the columns and Lb is the beam length (between column faces). The beam strengths in this equation refer to the strength at the face of the columns which have been projected to the column centrelines using the dcol on Lb ratio. For simplicity, these case studies neglect the effects of beam-column joints and assume that the beam strengths develop at the column centrelines. This simplification implies that the dcol on Lb term drops out of Eq. (27). The strength of a single beam is obtained using Eq. (28), in which the sum of the beam moments on the floor are divided by the number of beam ends, nbj , that connect to beam-column joints. As the frame shear used in Eq. (27) is equal to

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the sum of the column shears, the number of beam ends that connect to the walls should not be included within nbj .
Mbi Mb = . (28) nbj Since the link-beams will develop the same strength as given by Eq. (27) at the edge of the wall, the moment transferred to the centre of the walls can be obtained from 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 presented in Fig. 8, Eq. (29) is obtained for the moment transferred from a link-beam to the wall centreline.   Lw h

col , (29) MbW all = Vi,f rame 1 + Lb nbj 1 + dcol Lb where Lw is the wall length and nbj is the number of beam ends connecting to beam-column joints per link-beam. For these case studies the dcol on Lb term was neglected for simplicity. The moments transferred to the wall from the link-beams are used to adjust the moment profile as shown for an eight-storey structure in Fig. 9. Using this approach, the moment profile in the walls is known as a proportion of the design base shear. This then allows the inflection height to be determined and the design can proceed as normal. Another stage in the design process in which the inclusion of link-beams needs to be accounted for is in determination of the frame displacement ductility. As mentioned earlier, link-beams undergo larger plastic rotations than other beams at the same level. In order to estimate the ductility demands on the link beams it is worth reviewing how the ductility demands on a standard RC frame are established. For a standard beam-column sub-assemblage, the yield drift for design is obtained using Eq. (30) as recommended by Priestley [2003]. This is an approximate expression developed by assuming that the columns and joints add respectively an additional 40% and 25% of the displacement associated with the beams yielding in Lw/2

M b,wall

Fig. 8.

Lb

Mb

dcol

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

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8 7 6

Level

5 Moments from shears only

4

Adjusted for L-Beam moments

3 2 1

-5.00

0 0.00 5.00 10.00 Wall moments for unit base shear

15.00

Fig. 9. Wall moment profile of eight-storey structure, adjusted to allow for moments transferred from link-beams.

flexure, to the storey deformation. It also assumes that member shear deformations add a further 10% to the yield drift.     lb lb θy,beam = (1.0 + 0.4 + 0.25 + 0.1) ∗ 0.283εy = 0.5εy , (30) hb hb where εy is the yield strain of the longitudinal reinforcement in the beams, hb is the depth of the beams and lb is the beam length. For a beam-wall assemblage it could be assumed that the “column” and joint deformation contributions can be neglected. This would imply that the factor of 0.5 in the yield drift equation of (30) reduces to 0.31. As a link-beam is supported at one end by a stiff wall and at the other end by a column, it is apparent that an average factor of 0.4 can be used to approximate the yield drift of a link-beam, θy,link , as shown in Eq. (31).   lb θy,link = 0.4εy . (31) hb The displacement ductility demands on the link-beams and other bays of the frame can be obtained using Eqs. (30) and (31) respectively, together with the storey drift associated with the design displacement profile. A weighted average ductility value, µf rame,i , for each floor is then obtained in proportion to the number of link-beams, as shown in Eq. (32). µf rame,i =

θD,i θy,link nlink

+

θD,i θy,beam (nb

nb

− nlink )

,

(32)

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where θD,i is the storey drift associated with the design displacement profile at level i, nlink is the number of link-beams in the storey, and nb is the total number of beams on the storey. Equation (32) is valid when beams have equal length and strength. If this is not the case, it would be more appropriate to factor the ductility demands by the beam shears. For these case studies, beams have equal strength and length at each storey and up the full height of the building. Therefore, the frame ductility has been obtained as the average of the storey ductility values obtained using Eq. (32). Having determined the frame ductility, the design proceeds as normal with the equivalent viscous damping determined in the same manner as for the standard frame-wall structures. 4.3.3. Design results Design was only conducted to the point that would allow the strengths of plastic hinge regions to be set. With knowledge of these strength values, accurate nonlinear models of the structures could be developed for verification of the design solutions through time-history analysis as explained later in Sec. 4.4. Intermediate design values for the structures with and without link-beams are presented in Tables 4 and 3 respectively. Note that in the design of these case studies an elastic damping component was first obtained in accordance with the recommendations of Priestley and Grant [2005] and then added to a hysteretic component determined using the recommendations of Blandon and Priestley [2005]. This was because the recommendations of Grant et al. [2005] were not available at the time of this work. For these case studies, it can be seen that desirable design solutions were obtained 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-storey structures the design storey drift had to be reduced to ensure material strain limits were not exceeded. Note that both wall displacement ductility and curvature ductility demands are reported since the former relates more to the equivalent SDOF representation of the structure (being calculated at the effective height) whereas Table 3.

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

% base shear assigned to walls Inflection height Design storey drift Design displacement Wall displacement ductility Average frame ductility System ductility System damping Effective mass Effective period

4 storey

8 storey

12 storey

16 storey

20 storey

60% 14.4 2.09% 0.211 9.52 1.84 6.27 15.1 3806 1.47

60% 21.6 2.23% 0.430 7.43 1.99 5.00 15.0 7479 2.70

60% 30.3 2.37% 0.664 5.39 2.09 3.86 14.8 10 932 4.17

50% 33.5 2.27% 0.840 4.31 2.01 2.97 13.9 14 510 5.39

45% 36.6 2.16% 0.995 3.59 1.91 2.51 13.2 18 121 6.95

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111

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

% base shear assigned to walls Inflection height Design storey drift Design displacement Wall displacement ductility Average frame ductility System ductility System damping Effective mass Effective period

4 storey

8 storey

12 storey

16 storey

20 storey

70% 13.3 2.03% 0.207 9.62 1.81 6.35 16.0 3877 1.48

70% 20.7 2.41% 0.463 6.50 2.14 4.74 16.1 7334 2.99

60% 23.0 2.34% 0.669 5.12 2.10 3.58 15.7 10 788 4.34

50% 17.4 2.21% 0.855 5.78 2.03 3.22 15.4 14 409 5.95

55% 29.5 2.13% 0.993 3.35 1.90 2.70 14.6 18 033 7.58

the latter better reflects the amount of nonlinear deformation the wall would have to undergo. Table 4 indicates that frame displacement ductility demands were fairly low, however, even with low ductility demands it was anticipated that the frames would provide a significant amount of hysteretic energy dissipation. The effective period values shown in Table 4 are long, however they lie within the spectrum compatible range of the accelerograms, suggesting that the time-history analyses to be presented in later sections will provide a good test of the design solutions. Final design strengths and longitudinal reinforcement contents for the structures with and without link-beams are presented in Tables 6 and 5 respectively. Reinforcement contents for the walls and columns were obtained using axial loads associated with the gravity actions only. In reality exterior columns would be subject to a significant variation in axial load during seismic response, however, differences in compression from one side of the building to the other suggest that the actual strength of the columns should be equivalent to the sum of the strengths considering gravity loads only. An interesting observation to be taken from these design results is that the base shear for the buildings of considerably different height is relatively constant. This is attributed to the fact that the seismic weight per floor, the length of the walls and the depths of the beams did not change for the different 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) Wall strength (kNm) Wall long. reinforcement % Beam strength (kNm) Beam reinforcement % Ext. column strength (kNm) Int. column strength (kNm) Ext. col. long. reinforcement % Int. col. long. reinforcement %

4 storey

8 storey

14 691 31 949 0.65% 1014.93 1.23% 767 1533 1.84% 2.23%

17 429 72 827 0.95% 1195.67 1.50% 907 1814 1.91% 2.44%

12 storey 16 475 109 235 1.55% 1124.75 1.42% 859 1717 1.43% 1.88%

16 storey 16 565 117 367 1.61% 1419.52 1.38% 1087 2175 1.99% 2.87%

20 storey 14 735 114 931 1.42% 1395.31 1.36% 1072 2145 0.91% 1.66%

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Final design strengths for the frame-wall structures with link-beams. 4 storey

Base shear (kN) Wall strength (kNm) Wall long. reinforcement % Beam strength (kNm) Beam reinforcement % Ext. column strength (kNm) Int. column strength (kNm) Ext. col. long. reinforcement % Int. col. long. reinforcement %

14 490 31 018 0.63% 530.51 0.62% 454 909 0.79% 1.85%

8 storey 14 980 63 528 1.46% 529.56 0.62% 466 931 0.50% 1.36%

12 storey

16 storey

20 storey

15 120 66 696 1.42% 715.63 0.89% 630 1260 0.53% 2.22%

13 742 39 261 0.42% 915.28 1.11% 716 1431 0.52% 2.14%

12 292 72 981 1.38% 742.48 0.71% 586 1172 0.50% 0.50%

drift were fairly constant. On the contrary, the target displacement (and therefore effective period) as well as the effective mass were almost linearly dependent on height. Since this implies that the effective stiffness is inversely proportional to the height, and the base shear is simply the product of the effective stiffness and the design displacement, it is clear why fairly constant base shears were obtained. The fact that the base shear may depend only on the floor mass and section dimensions suggests that the design procedure could be significantly simplified without the loss of significant accuracy. This is an item for future research. Another important observation to be gleamed from the results in Tables 5 and 6 is that the longitudinal reinforcement contents are all within the minimum and maximum limits specified as part of the design criteria. The reinforcement contents on individual elements were seen to be sensitive to the strength assignments and axial loads, and this point was used to make the design solution for each of the structures more efficient. The fact that reasonable reinforcement contents have been stipulated 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 the case studies were constructed in which the strengths of the beams and walls were selected to match the design values obtained using the design methodology. The models were subjected to seven artificial acceleration records which had been used to construct the design spectrum. The displacement spectra of the seven records are shown at viscous damping levels of 5% and 15% in Fig. 10. The records can be expected to test the design solutions well as they possess relatively small scatter over a large range of periods. Of the seven accelerograms selected, six were artificial records. Record A5 is the North-South component of the 1978 Tabas earthquake, recorded at the Boshrooy station. 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 linear displacement spectrum up to a period of 5 s. Given this modification, none of the accelerograms can be considered as real earthquake records. Criticisms directed

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1.2

1.6

Record A1 Record A2

1.2

Record A3 Record A4 Record A5

0.8

Record A6 Record A7

0.4

15% Damped Displacement (m)

5% Damped Displacement (m)

2.0

113

1.0 0.8 0.6 0.4 0.2 0.0

0.0 0

1

2

3

4

5

6

7

Period (s)

0

1

2

3

4

5

6

7

Period (s)

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

towards the use of artificial accelerograms in seismic analyses usually focus on the different phase content and duration of actual earthquakes compared to artificial records. However, the frame-wall structures being designed in this study are not affected by duration owing to the fact that their strength does not degrade provided that the design displacement is not significantly exceeded. Little is known about the influence of the phase content on structural response. However, no evidence has been found to suggest that any differences in phase content affect structural response. Furthermore, time-history analysis of structures using both artificial and real records that possess similar demand spectra, have indicated that frame-wall structures respond similarly using either artificial or real records. For these reasons it was considered that the use of artificial accelerograms was acceptable for these case studies. The success of the new methodology can be measured by comparing the actual displacement response for the design level earthquake with the target displacement profile selected in the design. If the analysis and target displacements and storey drifts match, then the intended level of damage occurred and it can be concluded that 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 stiffness to account for cracking) were assigned to elements that are not intended to yield. This implies that appropriate capacity design would have ensured that inelasticity is concentrated only in regions associated with the collapse mechanism. An off-shoot of this modelling technique is that the analyses can be used to test capacity design guidelines which are reported elsewhere [Sullivan, 2005]. These case studies model the floors as rigid links fully flexible out of plane and P-delta effects are not considered since no attempt to account for these effects was made in the design. Beams, columns and walls were modelled using 2-hinge Giberson beam elements [Carr, 2004]. It is evident that the beams and the walls of the structures

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without link beams are likely to carry a constant axial load throughout the seismic response. Therefore the beams and walls of these structures could be modelled without the use of beam-column elements which account for changes in strength due to variations in axial load. The interior columns and the walls of the structures with link-beams were also subject to fairly constant axial loads as they are flanked by beams of equal strength and length on either side. However, the exterior columns are subject to varying axial loads during the seismic response and therefore modelling these members as beam-elements is not accurate. Nevertheless provided that the gravity load in these columns is well below balance-point axial load then the strength discrepancy in the compressive column roughly balances that of the tension column. In addition, the exterior column strengths form a very small portion of the total overturning resistance of the frame-wall structures. As such, the use of beam-elements for all members was deemed acceptable for the verification studies. Rigid elements were used to model the connections between the walls and linkbeams as shown in Fig. 11. When reinforced concrete walls are deformed in flexure to the extent that they crack and later yield, the position of their neutral axis depth shifts. Since a shift in neutral axis depth equates to a shift of the centre of rotation in a section, the use of Giberson-beam elements up the centre of the walls may be inaccurate. In particular, if a wall rotates about some point other its centre, then this implies one side of the wall lifts or drops more than the other. This in turn would imply that the curvature ductility demands on the beams would not be well captured. However, a separate study [Sullivan, 2005] of two fibre-element models analysed in SeismoStruct [SeismoSoft, 2004] has shown that the curvature ductility demands are only different over the lower stories of the building. Furthermore, the overall difference in energy dissipation between a model with shifting neutral-axis depth and a model with constant neutral-axis depth is not significant for these case study structures because the discrepancies in beam curvature on either side of the walls tend to cancel each other out.

Fig. 11.

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

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The hysteretic behaviour of the concrete structures was represented using the Takeda model [Otani, 1981], with 5% post-yield displacement stiffness and the unloading model of Emori and Schonbrich [1978]. Parameters for the Emori and Schonbrich model included an unloading stiffness factor of 0.5 for walls and columns and 0.25 for beams, together with a reloading stiffness factor of 0.0 and a reloading power factor of 1.0 which were used for all the elements. Refer to the Ruaumoko manual [Carr, 2004] for further details. The plastic hinge lengths associated with the yielding elements were calculated using the recommendations from Paulay and Priestley [1992]. The models developed in Ruaumoko use effective section properties up until yield, obtained by taking the design strength and dividing by the yield curvature. Approximations for yield curvature were obtained from expressions provided by Priestley [2003]. The effective stiffness for the ground storey columns at yield was approximated using the strength under axial load from gravity only, divided by the yield curvature. The elastic columns above the ground floor were modelled with the same initial stiffness. Elastic damping was modelled for the structures using tangent stiffness Rayleigh damping with a 1st mode damping value set to provide the effect of 5% tangent stiffness damping for the MDOF structure, as recommended by Priestley and Grant [2006]. Priestley and Grant [2006] provide an expression for this value that considers the stiffness and mass proportional components of the Rayleigh damping equation. By specifying the same damping value at two different frequencies, where the higher frequency is κ times the lower frequency, then the damping in the 1st mode attributed to stiffness proportional damping, ξsp , is given by Eq. (33). In addition, the damping attributed to mass proportional damping, ξmp , in the 1st mode is given by Eq. (34). 1 , κ+1 κ . = κ+1

ξsp =

(33)

ξmp

(34)

Having established the proportions of mass and stiffness proportional damping, Priestley and Grant [2006] recommend that for time-history analyses using Rayleigh tangent stiffness damping, the damping on the 1st mode, ξ1st , should be set using Eq. (35). ξ1st = (ξmp µ−0.75 + ξsp ) × 5%,

(35)

where µ is the displacement ductility of the equivalent SDOF system. The initial periods of the case-study structures were obtained from eigen-value analysis, and the ratio between the 1st and 2nd mode frequencies were calculated to give κ. These frequency ratios were then used together with the system displacement ductility values to obtain the 1st mode elastic damping values for time-history analysis shown in Table 7.

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T2 (s)

κ

ξ1st

4 storey

Without L-beams With L-beams

0.656 0.674

0.129 0.133

5.07 5.08

1.88 1.87

8 storey

Without L-beams With L-beams

1.324 1.539

0.266 0.311

4.98 4.95

2.08 2.14

12 storey

Without L-beams With L-beams

2.216 2.570

0.452 0.587

4.90 4.38

2.35 2.49

16 storey

Without L-beams With L-beams

3.234 3.814

0.708 1.038

4.57 3.67

2.71 2.70

20 storey

Without L-beams With L-beams

4.502 4.835

1.039 1.270

4.33 3.81

2.97 2.92

The dynamic equation of equilibrium is integrated by the unconditionally stable implicit Newmark Constant Average Acceleration (Newmark β = 0.25) method [Chopra, 2000]. The time-step for this form of integration method should be less than 0.1 of the period of the highest mode of free vibration that contributes significantly to the response of the building. Consequently, for these case studies a time step of 0.005 s has been adopted. The results of the analyses, presented in the following 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 accelerograms and the results have been processed to obtain displacements, shears, moments and storey drifts. Results examined in this paper focus on the performance of the proposed methodology with respect to its ability to control drifts and as such, only displacements and drifts are included. In [Sullivan, 2005], the shears and moments developed in the structures during the time history analyses are presented and used to verify the performance of new capacity design recommendations.

4.5.1. Review of maximum recorded displacements Figures 12, 13 and 14 present the maximum floor displacements recorded during time-history analysis using the seven different accelerograms. These are compared with the target displacement profile associated with the design drift limit for the various height structures. The scatter in the results obtained for the seven different records is in general quite small. This indicates that the equivalent viscous damping approach was able to maintain relatively uniform inelastic demands for the accelerograms used. For the taller structures, record A4 tends to impose large displacements which are foreseeable if the displacement spectra in Fig. 10 are examined closely. The target

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1.00

0.90

0.90

0.80

117

0.80 A1 A2

0.60

A3

0.50

A4 A5

0.40

A6

0.30

Relative Height (hi/H)

Relative Height (hi/H)

Target

0.70

A7

0.70 0.60 0.50 0.40 0.30

0.20

0.20

0.10

0.10

0.00 0.0

0.1

0.2

0.3

0.00 0.0

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Displacement (m)

0.1

0.2

0.3

0.4

Displacement (m)

1.00

1.00

0.90

0.90

0.80

0.80 Target

0.70

A1

0.60

A2 A3

0.50

A4

0.40

A5 A6

0.30

Relative Height (hi/H)

Relative Height (hi/H)

(i)

0.70 0.60 0.50 0.40 0.30

A7

0.20

0.20

0.10

0.10

0.00

0.00 0

0.2

0.4

0.6

0

0.8

0.2

0.4

0.6

0.8

Displacement (m)

Displacement (m)

(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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1.00

0.90

0.90 0.80 Target

0.70

A1 A2

0.60

A3

0.50

A4 A5

0.40

A6

0.30

Relative Height (hi/H)

Relative Height (hi/H)

0.80

0.70 0.60 0.50 0.40 0.30

A7

0.20

0.20

0.10

0.10 0.00

0.00 0.0

0.5

1.0

0.0

1.5

0.5

1.0

1.5

Displacement (m)

Displacement (m)

(i)

1.00

1.00

0.90

0.90 0.80 Target

0.70

A1

0.60

A2 A3

0.50

A4 A5

0.40

A6

0.30

A7

Relative Height (hi/H)

Relative Height (hi/H)

0.80

0.70 0.60 0.50 0.40 0.30

0.20

0.20

0.10

0.10

0.00 0.0

0.00 0.5

1.0

1.5

0.0

Displacement (m)

0.5

1.0

1.5

Displacement (m)

(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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1.00

0.90

0.90

0.80

0.80 0.70

Target A1

0.60

A2 A3

0.50

A4 A5

0.40

A6 A7

0.30

Relative Height (hi/H)

Relative Height (hi/H)

0.70

0.60 0.50 0.40 0.30

0.20

0.20

0.10

0.10

0.00

119

0.00

0.0

0.5

1.0

1.5

Displacement (m)

0.0

0.5

1.0

1.5

Displacement (m)

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

displacement profile (shown dashed) lies either in the centre or conservatively to the right of the recorded displacements suggesting that the design method has worked well. The excellent correlation between the recorded and anticipated displacements is very convincing. However, the ability of the design method to control the damage that the structures are subject to will be better gauged by comparison of the design storey 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 recorded during time-history analysis using the seven different accelerograms. These are compared with the design drift profile associated with the various height structures. Also shown for the 12, 16 and 20 storey structures is a dashed line that represents the limiting drift. The design drift is less than the drift limits in these cases because of the adjustment made to account for the effects higher modes have on storey drifts. The results of the time-history analyses indicate that the design approach has been very successful in limiting the storey drifts. Most encouragingly, the design drift profile again provides an excellent match to the average maximum drift profile. As would be expected, the storey drifts are correlated closely with the displacements and therefore similar trends are observed.

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4

3

3

1st mode Target

2

Level

Level

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2

T-history Average

1

1

0 0.0%

1.0%

2.0%

0 0.0%

3.0%

1.0%

2.0%

3.0%

Storey drift

Storey drift

(i)

8

8

7

7

6

6

1st mode Target

4

T-history Average

5 Level

Level

5

4

3

3

2

2

1

1

0 0.0%

1.0%

2.0%

0 0.0%

3.0%

Storey drift

1.0%

2.0%

3.0%

Storey drift

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

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12

10

10

8

1st mode Target T-history Average

6

Drift limit

Level

Level

8

6

4

4

2

2

0 0.0%

1.0%

2.0%

121

0 0.0%

3.0%

1.0%

2.0%

3.0%

Storey drift

Storey drift

16

16

14

14

12

12

10

1st mode Target

8

T-history Average Drift limit

6

10 Level

Level

(i)

8 6

4

4

2

2

0 0.0%

1.0%

2.0%

0 0.0%

3.0%

Storey drift

1.0%

2.0%

3.0%

Storey drift

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

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20

18

18

16

16

14

14 1st mode Target

10

T-history Average

8

Drift limit

12 Level

Level

12

10 8

6

6

4

4

2

2

0 0.0%

1.0%

2.0%

Storey drift

3.0%

0 0.0%

1.0%

2.0%

3.0%

Storey drift

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

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

5. Conclusions In conclusion, these case studies have clearly illustrated that the new design procedure for frame-wall structures provides excellent control of storey drifts and displacements for buildings of up to 20 storeys in height. The interaction that takes place between frame and wall elements has been successfully accounted for and it has been shown that the approach works well when structures with or without link-beams are considered. The recommendations made for prediction of the displacement profile are considered 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 elastic stiffness of wall elements. As such, future work could aim to reduce the uncertainty associated with this part of the methodology. Future work should verify that three-dimensional effects do not jeopardise the ability of the method. Testing conducted at UC Berkeley indicated that 3dimensional effects tend to increase the overturning resistance of a structure. This implies that such effects should not require any changes to the DBD process but may need to be accounted for during capacity design. Maximum forces that develop in the structures and recommendations for their capacity design are reported elsewhere [Sullivan, 2005]. Such capacity design guidelines, together with the DBD process presented here, complete a set of recommendations that provide designers with a simple, rational and effective means of conducting seismic design of framewall structures.

References Blandon, C. A. and Priestley, M. J. N. [2005] “Equivalent viscous damping equations for direct displacement based design,” Journal of Earthquake Engineering 9 (Special Issue 2), 257–278. Carr, A. J. [2004] Ruaumoko3D — A Program for Inelastic Time-History Analysis, Department 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 Normalization, 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 structures for strong motion earthquakes,” Civil Engineering Studies, Structural Research Series, No. 434, University of Illinois, Urbana, Illinois, USA. Grant, D. N., Blandon, C. A. and Priestley, M. J. N. [2005] Modelling Inelastic Response in 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 to earthquake motions,” ACI Journal 71(12), 604–610. 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 Structures, and Part 2 — Commentary (Standards New Zealand, Wellington). Otani, S. [1981] “Hysteresis models of reinforced concrete for earthquake response analysis,” Journal of the Faculty of Engineering, University of Tokyo XXXVI(2), 125–159. Paulay, T. and Goodsir, W. J. [1986] “The capacity design of reinforced concrete hybrid structures for multistorey buildings,” Bulletin of NZ National Society for Earthquake Engineering, New Zealand 19(1), 1–17. Paulay, T. and Priestley, M. J. N. [1992] Seismic Design of Reinforced Concrete and Masonry Buildings (John Wiley & Sons, Inc., New York). Priestley, M. J. N. [2003] Myths and Fallacies in Earthquake Engineering, Revisited, The Mallet 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 cantilever structural walls,” Bulletin of the New Zealand National Society for Earthquake Engineering, New Zealand National Society for Earthquake Engineering, Silverstream 31(2). Priestley, M. J. N. and Kowalsky, M. J. [2000] “Direct displacement-based seismic design of concrete buildings,” Bulletin, NZ National Society for Earthquake Engineering, New Zealand 33(4), 421–444. Priestley, M. J. N. and Paulay, T. [2002] “What is the stiffness of reinforced concrete walls,” SESOC Journal, Structural Engineering Society of New Zealand 15(1), 30–34. SeismoSoft [2004] “SeismoStruct — A computer program for static and dynamic nonlinear analysis framed structures,” Available on-line from URL: http//www.seismosoft.com. Shibata, A. and Sozen, M. A. [1976] “Substitute structure method for seismic design in reinforced concrete,” Journal of the Structural Division, ASCE 102(ST1). Sullivan, T. J. [2005] Seismic Design of Frame-Wall Structures, PhD thesis, ROSE school, Universit` a degli studi di Pavia, Italy. Sullivan, T. J., Priestley, M. J. N. and Calvi, G. M. [2004] “Displacement shapes of framewall structures for direct displacement based design,” Proceedings of Japan-Europe 5th 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 innovative seismic design procedure for frame-wall structures,” Journal of Earthquake Engineering 9 (Special Issue 2), 279–307.