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DISPLACEMENT CONSIDERATIONS FOR THE SEISMICDESIGN OF TALL RC FRAME-WALL BUILDINGS

Timothy J. Sullivan

Buro Happold, London, United Kingdom

ABSTRACT:

In this paper various displacement considerations for the seismic-design of tall reinforcedconcrete (RC) frame-wall buildings are made. In particular, simplified expressions arepresented to allow the likely displacement ductility demands on frame and wall elements

to be predicted. A 60-storey RC frame-wall case study structure is subject to non-lineartime-history analyses to illustrate that for realistic large magnitude ground motions the

displacement demands are not likely to be excessive and ductility demands will be low.The implications this has for the force-based modal response spectrum analyses are

discussed and the need for designers to make displacement considerations is emphasised.Future work should explore means of better predicting higher mode drifts and capacity

design requirements for tall buildings.

1 INTRODUCTION

With continued urbanisation worldwide, the construction of tall buildings is becoming increasinglycommon. However, there appears to be a lack of international standards that provide specific guidance

on the seismic design of tall buildings. Regulations in US codes such as the UBC97 (ICBO, 1997) doregulate the type of lateral load resisting systems that can be used for buildings above 160ft (approx 12

storeys). However, such guidance is limited and recommendations for the selection of force-reductionor behaviour factors as well as capacity design considerations specific to tall buildings are not

provided. In addition, most code design spectra have been developed for relatively short spectral

periods (up to say 5s) and do not provide clear guidance on long-period spectral demands that will

most affect tall building response. In this paper it will be argued that a number of straightforwarddisplacement considerations should be made for the seismic design and assessment of tall frame-wallstructures. In addition, the displacement ductility demands that are likely to develop in tall frame-wall

buildings will be explored and the consequences this will have for seismic design is discussed.

1.1 Long period spectral displacements

The demands that earthquakes impose at large periods, from say 5s to 20s, are typically not well

reported in codes. This is starting to change however, with codes such as the Eurocode EC8 (CEN2004), the IBC2006 (ICC, 2006) and NEHRP guidelines (FEMA274, 1997) now including some

recommendations for displacement spectra. The general shape of an elastic design displacement

spectrum provided in EC8 is shown in Figure 1. Also included in Figure 1 is the displacementspectrum derived from the acceleration spectrum of the UBC97. The EC8 spectrum is characterized by

a relatively linear increase in displacement demand up until a corner period TD. The displacementdemand is then constant until a period of TE, beyond which point it reduces down to the peak ground

displacement at period TF. In contrast, codes which specify spectral accelerations that continue toreduce in proportion to the inverse of the structural period (such as the UBC97, where Sa = SD1 /T) are

in reality imposing displacement demands that continue to increase with increasing period.

The EC8 displacement spectrum of Figure 1 implies that if a single-degree-of-freedom (SDOF)

structure has a period of greater than TD, then provided its yield displacement is greater than the

displacement associated with TD, the design earthquake will be unable to cause structural damage. TheEC8 periods TD, TE and TF indicated in Figure 1, are dependent on the soil type and earthquakemagnitude (type 1 or type 2 spectra). For a type 1 spectrum (large magnitude events) at a stiff soil site,



2

values of 2s, 6s and 10s are recommended by EC8 for TD, TE and TF respectively. As discussed byPriestley et al. (2007) the EC8 displacement corner period of 2s is considered to be non-conservatively

low for regions of high seismicity.

Figure 1. Form of Eurocode EC8 elastic design displacement spectrum (solid line) compared with

UBC97 displacement spectrum (dashed line).

Rather than considering a constant value for the spectral displacement corner period, the NEHRP

guidelines (FEMA 274) and Faccioli et al. (2003) indicate that the corner period is principally afunction of earthquake magnitude, with larger period earthquakes being characterised by larger cornerperiod values. For an interesting comparison of the two magnitude-dependent expressions for the

corner period presented in FEMA 274 and Faccioli et al. (2003), refer to Priestley et al. (2007).

Clearly, long period displacement demands are not only affected by the value of the corner period butalso the displacement magnitude at the corner period. For a given magnitude event, the spectral

displacement at the corner period will be related principally to the distance the earthquake occurs from

the site. However, as will be seen later in this work, even considering some of the most demandingaccelerograms on record, long period displacement demands are limited. Given this, the question thatfollows is how do the demands relate to the displacement characteristics of tall buildings?

2 YIELD DISPLACEMENTS OF TALL RC FRAME-WALL STRUCTURES

2.1 Nominal yield curvatures of reinforced concrete sections

In evaluating the yield displacements of a structure one can first consider the response at a sectionallevel. This is perhaps best done for RC structures through the use of moment-curvature analyses. In

moment-curvature analysis, a certain curvature demand is imposed onto a section, strains and stressesare computed, equilibrium is checked and the flexural strength corresponding to the curvature level is

then established. A range of curvature demands are considered in the process such that the flexural

strength versus curvature plot can be developed such as that shown on the right of Figure 2. Bi-linearrepresentations for the moment-curvature response are established by considering limiting strain

values for the concrete in compression and the reinforcement in tension.

By undertaking moment-curvature analyses for a number of reinforced concrete sections, considering

variations in both axial load and longitudinal reinforcement content, Priestley (2003) and Paulay(2002) have shown that the nominal yield curvature of a reinforced concrete member is practicallyindependent of its strength and instead is a function of its depth and longitudinal reinforcement yieldstrain. Based on such moment-curvature analyses, Priestley (2003) reports that the nominal yield

curvature for rectangular RC wall sections, φ y,wall, can be adequately estimated through Eq. (1).

w

y

wall y L

ε φ

2, = (1)

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

UBC97

Spectral

Displacement

Sd

EC8

TD TE TF TC Period



3

To demonstrate the implications of this important finding, Figure 2 shows the moment-curvatureresponse for two different walls of the same length and thickness but with wall 1 possessing roughly

twice the longitudinal reinforcement of wall 2. Note that not only is the nominal yield curvature forboth sections the same, but because the strength of wall 1 is twice that of wall 2, this implies that thecracked stiffness of wall 1 is twice that of wall 2.

Figure 2. Strength and stiffness considerations of reinforced concrete elements.

2.2 Frame yield drifts for tall RC frame-wall structures

A similar expression to Eq.(1) was put-forward by Priestley (2003) for RC beam sections. In addition,

however, by considering the typical proportions of frame deformation caused by beams, columns and

joints, Priestley (2003) proposed Eq.(2) for the yield drift of RC frames.

b

b

y frame yh

Lε θ 5.0, = (2)

where and Lb is the beam length (between column centres) and hb is the beam depth.

Within Eq.(2) the assumption is made that columns will deform to around 75% of their yield curvatureand as such, column flexure makes up approximately 23% of the yield drift estimate. In addition,

based on experience, Priestley (2003) argued that joint deformations cause approximately 14% of theframe yield drift. Priestley showed that yield drift values predicted through Eq. (2) compare well with

experimental results obtained from 46 different beam-column test assemblages.

For tall frame structures however, the columns tend to become very large owing to high gravity loadrequirements. Consequently, it is considered that columns and joints will contribute less to the storey

drift in comparison to normal frame structures for which Eq.(2) was developed. In light of this, Eq. (3)is proposed for the yield drift of tall frame structures whereby it has been assumed that, due to the

presence of massive columns, the column and joint contributions are 75% of those within Eq.(2).

b

b

y frametall yh

Lε θ 45.0_, = (3)

It is pointed out that Eq.(3) has not been calibrated against experimental test results and as such, some

care should be adopted in its application. The expression does have a logical basis however, and

therefore it is expected that a calibrated expression is not likely to vary greatly.

2.3 Wall yield displacements for tall RC frame-wall structures

The deformed shape and yield displacement of a RC frame-wall structure will depend on theproportions of strength assigned to the frames and walls respectively. As reported by Sullivan et al.

(2006), the curvatures that develop in the walls of frame-wall structures tend to control their displacedshape. This is because RC walls are large stiff elements that are designed to remain elastic above their

base. The frames do influence the displaced shape by changing the moment and therefore curvature

Curvature

Moment

Wall 1 Wall 2

Wall 2

Wall 1

φ y

EI1

EI2

Mn

Mn1

Mcr



5

where n is the number of storeys and H is the total building height. Eq.(5) considers floors asdiscretised masses and as such, a point of contra-flexure cannot develop above the penultimate floor

level.

Figure 4 plots Eq. (5) for different numbers of storeys. It can be seen that as the proportion ofoverturning resisted by the frames increases, the point of contra-flexure reduces. For low and medium

rise structures one will typically assign a greater proportion of the overturning resistance to the walls.

However, for high rise structures, walls will typically have such large aspect ratios (height/length) thatit will be difficult to force them to resist a large portion of the overturning. Note also that when thebuilding possesses more than around 15 storeys, the contra-flexure height ratio becomes relatively

independent of the number of storeys.

0.0

0.2

0.4

0.6

0.8

1.0

0.00 0.20 0.40 0.60 0.80 1.00

hcf/H

M f r / M t o t a l n = 5

n = 10

n = 15

n = 20

Figure 4. Wall contra-flexure height ratio as a function of the proportion of frame overturning

resistance.

Figure 4 can be used to obtain the contra-flexure height for a given structure which is then substitutedinto Eq.(4) to obtain the fundamental mode displaced shape of the structure at wall yield. Given the

relatively large proportion of overturning resistance that the frames will resist in tall frame-wallstructures, the displaced shape at peak response will be fairly linear. If a linear displaced shape at

maximum response is assumed, then the effective height of an equivalent SDOF representation of a

tall frame-wall structure can be taken as he=0.675H. This can then be substituted into the appropriateform of Eq. (4) to obtain an estimate of the building displacement at yield of the walls. This is usefulin assessing the likely ductility demands on frame-wall structures as explained in the next section.

3 DUCTILITY AND DRIFT DEMANDS ON TALL RC FRAME-WALL STRUCTURES

Displacement ductility demands on the frames and walls can be estimated rather quickly using a

displacement-based assessment procedure similar to that described by Priestley et al. (2007). For tallframe wall structures it can be conservatively assumed that the building period will lie beyond the

spectrum corner period and therefore the plateau of the elastic design displacement spectrum can betaken as the elastic displacement demand imposed on the structure, as illustrated in Figure 5.

The wall displacement ductility can be estimated by dividing the 1st mode spectral displacement by the

yield displacement at the effective height, from the appropriate form of Eq. (4), as shown in Eq. (6).

wall y

d

hwall

S

e

,

1,

,∆

= µ (6)

The building drift due to the first mode can be approximated by dividing the spectral displacement,S d,1, by the effective height (approximately 0.675H for tall buildings) as shown in Eq. (7).

H

S d

675.0

1,

1 =θ (7)



6

Figure 5. Equivalent SDOF considerations of the likely earthquake demands for tall buildings.

The influence of higher modes on storey drifts, however, is likely to be very significant, particularly ifthe fundamental mode period is well beyond the spectral displacement corner period. Considering the

right side of Figure 5, one can see that if the structure is very flexible, it is conceivable that the 2nd

mode period would begin to approach the corner period TD and therefore the 2nd

mode spectraldisplacement could potentially be the same as the fundamental mode displacement. The aim of future

work should be to consider means of estimating higher mode drifts as a function of the first mode drift

so that a simple displacement-based assessment procedure can be developed. One possibility to beexplored here is to use the characteristic functions for uniform beams, as presented by Young and

Felgar (1949), to develop Eq. (8) as a trial estimate of the 2nd

mode drift component, θ 2,trial.

H

S d

trial

2,

,2 2=θ (8)

Where S d,2, is the second mode spectral acceleration. Note that this expression does not attempt toaccount for the effects of inelasticity on mode shape or period and requires knowledge of the 2

nd mode

period of the structure to obtain the 2nd mode spectral acceleration. With this in mind, modalsuperposition approaches are likely to give a more accurate estimate of the storey drifts, particularly if

the response is likely to be elastic. However, it is of interest to see whether a simplified approach canprovide approximate values in order that future work can look to develop a simplified displacement-

based assessment procedure that takes account of higher modes.

In order to estimate the maximum storey drifts, a trial application will simply add the 1st and 2ndmode drift components from Eq. (7) and (8). This is proposed in place of the SRSS combination in

order to approximately account for 3rd and 4th (and higher) mode contributions. This maximum

storey drift can then be divided by the frame yield drift of Eq.(3) to evaluate approximate frameductility demands, as shown in Eq. (9).

frametall y

trial

frame

_,

,21

θ

θ θ µ

+= (9)

Clearly, the maximum storey drifts are also a practical means of considering non-structural damage.International codes tend to require designers to limit storey drifts to between 2.0% and 2.5% for a life-

safety event. As such, in the next section, the maximum storey drifts recorded for a tall frame-wallcase study structure will also be considered from a non-structural damage point of view.

For Eq.(6) to Eq.(8) one can iterate to consider the effects of inelasticity through an equivalent viscousdamping approach which will reduce the spectral displacements (Priestley et al. 2007). However, asductility demands are likely to be low in tall buildings, iteration may not be deemed necessary.

he

Equivalent SDOF representation

RC Frames RC Walls

∆d

me

S d ,1

Spectral

Displacement

Building

Period

T1

Increasing period

beyond TD does not

increase Sd1

TD

T2 S d ,2

Potential Earthquake Demands

5%

H



7

4 CASE STUDY: 60 STOREY RC FRAME-WALL STRUCTURE

To investigate and illustrate the likely displacement demands on a tall RC frame-wall structure, the 60

storey case study building shown in plan in Figure 6 below has been examined. The building consistsof a central core wall structure and columns arranged in a regular 7.5m grid and connected by beamsto form moment resisting frames in the two orthogonal directions. Sections sizes are indicated on

Figure 6. The areas of columns and walls were sized to control axial load ratios under gravity loads.

The concrete and reinforcement material properties adopted for the seismic design are values thatcould typically be found in tall building practice. Values for the concrete include: (i) f’ c = 60.0 MPa

and (ii) Ec = 33200 MPa. The expected strengths adopted for the reinforcing steel include: (i) f y = 500

MPa and (ii) Es=200000 MPa.

Figure 6. Plan view of 60-storey (240m) reinforced concrete frame-wall case study structure.

A specific design procedure for assigning strength was not followed and instead beams and walls were

given strength corresponding to reasonable longitudinal reinforcement contents (As /Ag = 1.07% forwalls and 1.70% for beams). As such, the frames resist 73% of the total overturning moment. From

Figure 4, the wall contra-flexure height is estimated at 0.40H = 96m, and the equivalent SDOFdisplacement at wall yield is 2.1m (Eq.(4a)). The yield drift of the frames, according to Eq.(3) is 1.0%.

Cracked section properties were obtained for modal analysis by dividing the member nominal flexuralstrengths by the section yield curvatures obtained from the Priestley (2003) equations. The wall

cracked section stiffness determined at the base section of the wall was assumed constant up the

building height by assuming that despite possible reductions in the amount of cracking there wouldalso be reductions in the wall thickness. Modal analysis gave 1st and 2nd mode periods of 13.4s and3.5s respectively. These are long periods for a 60 storey structure and it is likely that wind

requirements would force the design to be stiffer, but the aim here is to demonstrate that even when atall RC frame-wall structure is very flexible building deformations are not likely to be problematic.

Using the displacement spectra of Figure 7 and the periods obtained from modal analysis, one mightexpect a 1

st and 2

nd mode spectral displacements of 1.05m and 0.63m respectively. Combining Eq. (7)

and Eq. (8) the maximum storey drift is estimated at 1.17%. From Eq. (6) and Eq. (8) these values give

displacement ductility demands for the walls and frames of 0.5 (ie. elastic) and 1.17 respectively.

Non-linear time-history analyses of the case study structure were carried out using Ruaumoko (Carr,

2007) using an integration time step of 0.001s. The five accelerograms listed in Table 1 and shown inFigure 7 were used for the analyses. Note that the real records possess an average spectral

displacement corner period of 7.5s. The records were scaled to provide an intensity corresponding to atype 1 earthquake from EC8 with a PGA = 0.4g and soil type C.

Earthquake Excitation

Direction Considered

Four bays

at 7.5m

centres

F R A M E 1

F R A M E 2

F R A M E 3

F R A M E 4

W A L L 1

W A L L 2

850Dx800W

RC beams

1600Dx1200W

RC columns

Two 15.0m long x 0.750m

thick RC walls

200mm thick

RC floors

Floor Area

30.0mx37.5m = 1125m2



8

Table 1. List of real earthquake records used in the non-linear time-history analyses

Ref. Earthquake nameEarthquake

magnitude

Record

length (s)

Time Step

(s)

Scaled

PGA (g)

Scale

factor

R1 Imp. Valley - El Centro 7.1 30.0 0.02 0.44 2.1

R2 Loma Prieta 7.1 39.635 0.005 0.67 2.4

R3 Chi Chi 7.6 89.998 0.005 0.18 2.4

R4 Tabas 7.7 34.98 0.02 0.33 3.7R5 Landers 7.3 79.98 0.02 0.35 1.45

0.0

0.5

1.0

1.5

2.0

2.5

0 5 10 15 20

Period (s)

S p e c t r a l D i s p l a c e m e n t ( m )

R1

R2

R3

R4

R5

Average

EC8 5%

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 5.0 10.0 15.0 20.0

Period (s)

S p e c t r a l D i s p l a c e m e n t ( m )

5% Damped Spectra10% Damped Spectra15% Damped SpectraEC8 5%

EC8 10%EC8 15%

Figure 7. Displacement spectra (left) of the ground motions used in non-linear time-history analyses

of the case study structure and (right) average spectral demands at different damping levels.

In modelling the structures for non-linear time-history analysis, elastic properties were assigned to

elements that are not intended to yield. This infers that appropriate capacity design would haveensured that inelasticity is concentrated only in regions associated with the collapse mechanism.Yielding elements were represented using the Takeda [Otani, 1981] hysteretic model, with 5% post-

yield displacement stiffness. The Rayleigh tangent stiffness damping model was adopted with 5%

damping set on the 1st and 2

nd modes. The plastic hinge lengths associated with the yielding elements

were calculated using the recommendations from Priestley et al. [2007].

Figure 8 presents the maximum recorded storey displacements and storey drifts for the differentearthquake records. It is seen that both the average displacement recorded at the effective height (level

41) of 0.97m and average maximum storey drift over the top half of the building of 1.28% comparereasonably well with the estimates made prior to the analyses. Walls were observed to remain elastic

and frames underwent limited inelastic rotations as expected.

0

10

20

30

40

50

60

0.0 1.0 2.0 3.0

Max Storey Displacement (m)

S t o r e y

0

10

20

30

40

50

60

0 .0 0% 0 .5 0% 1 .0 0% 1 .5 0% 2 .0 0%

Max Storey Drift

S t o r e y

r1

r2

r3

r4

r5

Figure 8. Maximum storey displacements (left) and storey drifts (right) recorded from non-linear

time-history analyses for the 60-storey case study structure.



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5 IMPLICATIONS AND CONCLUSIONS

The considerations made in this paper have a number of implications for the design of tall frame-wall

buildings. Clearly, because the work indicates that even large magnitude events are only likely toimpose limited displacement ductility demands on tall frame-wall buildings, the use of large behaviourfactors for modal response spectrum analysis is inappropriate. On the other hand, it has been shown

that once the fundamental period lies beyond the spectral displacement corner period, reducing the

strength and stiffness of the main lateral load resisting system is not likely to lead to unacceptabledisplacements and deformations, provided that adequate capacity design procedures have been

adopted. To this extent, code (UBC97) prescribed capacity design wall shear forces are likely to be

underestimated when response spectrum analysis is adopted with large behaviour factors. Foralternative guidelines on the capacity design of frame-wall structures refer to Sullivan et al (2006).

An important conclusion of this work is that seismic assessment and design should include

displacement considerations. Such considerations may enable reductions in detailing requirements asductility demands will often be low, and should lead to more appropriate distributions of strength.

Future work should look to study capacity design requirements and further explore the role that higher

modes have on storey drifts of tall frame-wall buildings. It is noted that the analyses conducted in thiswork have been small displacement analyses that did not consider p-delta effects. The effect of this

and other modelling approximations should be explored in future work. Finally, it is pointed out that

the study has not considered displacement demands and capacities of tall buildings that traverse faultsas the permanent displacement of the ground would be expected to permanently deform the building.

ACKNOWLEDGEMENTS:

The author would like to thank Dr Jamie Goggins for his useful comments on the paper and Dr J.

Didier Pettinga for providing the accelerograms used for the non-linear time-history analyses.

REFERENCES:

Carr, A. J. 2004. Ruaumoko3D – A program for Inelastic Time-History Analysis. Department of CivilEngineering, University of Canterbury, New Zealand.

CEN 1998. Eurocode EC8 - Design of structures for earthquake resistance - Part 1: General rules, seismicactions and rules for buildings, prEN-1998-1, Comite Europeen de Normalization, Brussels, Belgium.

Faccioli, E., Paolucci, R., Rey, J. 2004. Displacement Spectra for Long Periods. Earthqauke Spectra, 20(2).

FEMA 274, 1997. Federal Emergency Management Agency, NEHRP Commentary on the Guidelines for theSeismic Rehabilitation of Buildings: FEMA 274. Washington, USA.

ICBO, 1997. Uniform Building Code, International Conference of Building Officials, Whittier, California, USA.

ICC 2006. International Code Council, International Building Code, Thomas Delmar Learning, USA.

Otani, S. 1981. “Hysteresis models of reinforced concrete for earthquake response analysis” Journal of theFaculty of Engineering, University of Tokyo, Vol. XXXVI, No.2, pp125-159.

Paulay, T. 2003. Seismic displacement capacity of ductile reinforced concrete building systems. Bulletin of New Zealand Society for Earthquake Engineering, 36(1).

Paulay, T. 2002. A displacement-focussed seismic design of mixed building systems. Earthquake Spectra, 18(4).

Priestley, M.J.N. 2003. Myths and Fallacies in Earthquake Engineering, Revisited, The Mallet Milne Lecture,IUSS Press, Pavia, Italy.

Priestley, M.J.N., Calvi, G.M., Kowalsky, M.J. 2007. Displacement Based Seismic Design of Structures, IUSSPress, Pavia, Italy. 720pages (www.iusspress.it).

Sullivan, T.J., Priestley, M.J.N., Calvi, G.M. 2006. Seismic Design of Frame-Wall Structures. Research Report ROSE - 2006/02, IUSS press (www.iusspress.it).

Young, D., Felgar, Jr. R.P., 1949. Tables of characteristic functions representing normal modes of vibration of auniform beam. The University of Texas publication No.4913, July 1, 1949.

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