Direct displacement-based seismic design of steel eccentrically braced frame structures

Bull Earthquake Eng (2013) 11:21972231DOI 10.1007/s10518-013-9486-8

ORIGINAL RESEARCH PAPER

Direct displacement-based seismic design of steeleccentrically braced frame structures

Timothy John Sullivan

Received: 24 January 2013 / Accepted: 2 July 2013 / Published online: 13 July 2013 Springer Science+Business Media Dordrecht 2013

Abstract This paper details a direct displacement-based design procedure for steel eccentri-cally braced frame (EBF) structures and gauges its performance by examining the non-lineardynamic response of a series of case study EBF structures designed using the procedure. Ana-lytical expressions are developed for the storey drift at yield and for the storey drift capacityof EBFs, recognising that in addition to link beam deformations, the brace and column axialdeformations can provide important contributions to storey drift components. Case studydesign results indicate that the ductility capacity of EBF systems will tend to be relativelylow, despite the large local ductility capacity offered by well detailed links. In addition, itis found that while the ductility capacity of EBF systems will tend to reduce with height,this is not necessarily negative for seismic performance since the displacement capacity fortaller EBF systems will tend to be large. To gauge the performance of the proposed DBDmethodology, analytical models of the case study design solutions are subject to non-lineartime-history analyses with a set of spectrum-compatible accelerograms. The average dis-placements and drifts obtained from the NLTH analyses are shown to align well with designvalues, confirming that the new methodology could provide an effective tool for the seismicdesign of EBF systems.

Keywords Eccentrically braced frame Displacement-based design EBF Yield drift EBF drift capacity

1 Introduction

Steel eccentrically braced frame (EBF) structures, such as that sketched in Fig. 1, wereproposed in the late seventies (Roeder and Popov 1977) as a ductile structural system suitable

T. J. Sullivan (B)Department of Civil Engineering and Architecture, University of Pavia, Pavia, Italye-mail: [email protected]

T. J. SullivanEuropean Centre for Training and Research in Earthquake Engineering (EUCENTRE), Pavia, Italy

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2198 Bull Earthquake Eng (2013) 11:21972231

Links yield under intense seismic shaking

Lb

e

hs

1

2

3

4

(a) (b)

Fig. 1 a Steel eccentrically-braced frame with centrally located links, and b expected plastic mechanismunder intense seismic shaking

for use in regions of high seismicity. By selecting relatively short links the EBF systems tendto be relatively stiff, which is advantageous for the control of serviceability drift limits. Underintense shaking the links are designed to yield (either in shear, flexure or a combination ofthe two depending on the link length) and dissipate energy. A large number of experimentaltests of EBFs and their shear links were conducted in the 1980s (Roeder and Popov 1977;Hjelmstad and Popov 1983; Popov and Malley 1983; Kasai and Popov 1986; Richles andPopov 1987; Engelhardt and Popov 1989) and demonstrated that well-detailed EBFs possessgood ductility capacity, capable of sustaining large inelastic deformation demands.

Current codes (ASCE7-10 2010; CEN 2005; NZS1170.5 2004) specify the use of eitherthe equivalent lateral-force method or the modal response spectrum method for the seismicdesign of steel EBF structures. However, it has been demonstrated (Priestley 1993; Priestleyet al. 2007) that force-based design methods possess a number of fundamental shortcomingssuch as the use of force-reduction (behaviour) factors that are set without explicitly evaluationof ductility demands, and the use of elastic analysis to estimate inelastic force distributions inmixed structural systems (see Priestley et al. 2007, for details). To overcome such limitationswith force-based design methods, a large number of displacement-based design methodshave been proposed (see Sullivan et al. 2003, for a review of various methods).

The most developed DBD methodology is the Direct DBD procedure which has beenpublished as a text by Priestley et al. (2007) and in model-code format Sullivan et al. (2012).Existing guidelines for Direct DBD have been extensively developed and tested for RCstructures (Pettinga and Priestley 2005; Sullivan et al. 2005, 2006). In contrast, developmentshave been relatively limited for steel structures with most attention to date focussing on steelconcentrically-braced frame structures (Della Corte et al. 2008; Goggins and Sullivan 2009;Della Corte et al. 2010; Wijesundara et al. 2009, 2011; Nascimbene et al. 2011; Grande andRasulo 2013) and steel moment-resisting frame structures (Sullivan et al. 2011). Recently,it has been shown (Sullivan 2013) that design base shears obtained for RC frame structuresfrom DDBD can range from one half to four times those obtained from the equivalent lateralforce method currently specified in international codes. Given the general limitations of FBDmethods, this paper proposes a Direct DBD procedure for steel EBF structures and gaugesthe performance of the methodology through non-linear time-history analysis of a numberof case study EBF buildings.

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Bull Earthquake Eng (2013) 11:21972231 2199

me FuF Fn rKi

H e Ki Ke

y d(a) (b)

0 2 4 6Displacement Ductility

0

0.05

0.1

0.15

0.2

0.25

Dam

ping

R

atio

,

0 1 2 3 4 5Period (seconds)

0

0.1

0.2

0.3

0.4

0.5

Disp

lace

men

t (m)

=0.05

=0.10=0.15=0.20=0.30

d

Te

Elasto-PlasticSteel Frame

Concrete Frame

Hybrid Prestress

(c) (d)

Concrete Bridge

Fig. 2 Conceptual overview of the Direct DBD procedure (reproduced with permission from Priestley et al.2007). a SDOF simulation. b Effective stiffness Ke. c Equivalent damping versus ductility. d Design displace-ment spectra

2 Fundamentals of Direct DBD

As shown in Fig. 2a, the Direct DBD approach is based on the substitute-structure approachof Shibata and Sozen (1976) and Gulkan and Sozen (1974), representing a multi-degree-of-freedom structure with an equivalent single-degree-of-freedom (SDOF) system (Fig. 2a) thatis characterised by a secant or effective stiffness, Ke, at the target displacement level (Fig. 2b).The target (design) displacement, d , is set by the engineer considering performance criteriafor the building, such as storey drift limits for non-structural elements or plastic deformationlimits for structural elements, as will be illustrated for EBF systems later in this paper.

In order to account for the fact that the response will be non-linear under intense seismicexcitation, the substitute structure is also characterised by an equivalent viscous dampingvalue, eq, that is a function of the ductility demand, as shown in Fig. 2c. The design dis-placement spectrum is then scaled to the equivalent viscous damping value and a requiredeffective period, Te, is identified by reading in with the design displacement, as shown inFig. 2d. With the required effective period known, the required effective stiffness is computedusing Eq. 1:

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Ke = 42 meT 2e(1)

where me is the effective mass of structure. For cases in which p-delta effects are not signif-icant, the design base shear is then obtained as the product of the effective stiffness and thedesign displacement, d , (see Fig. 2b):

Vb = Ked (2)The design base shear given by Eq. 2 can then be distributed as a set of equivalent lateralforces over the height of the building according to:

Fi = miimii

Vb (3)

where i is the design displacement and mi is the seismic mass of level i . For RC framestructures Priestley et al. (2007) recommend that Eq. 3 be modified such that 10 % of thedesign base shear is lumped at roof level with the remainder distributed as per Eq. 3.

The forces given by Eq. 3 can be used to identify the required strengths of plastic hingezones. Capacity design procedures should then be followed to ensure that the intended plasticmechanism can be developed and sustained for high levels of seismic intensity.

From this short description it will be apparent that the Direct DBD procedure is relativelysimple. Any complexity that exists lies principally in the identification of the design dis-placement profile, i , and system ductility demand (which is required for computation ofthe equivalent viscous damping). Note that the identification of the equivalent SDOF systemcharacteristics of design displacement, d , effective mass, me, and effective height, He, allrely on knowledge of the design displacement profile, as shown by Eqs. (4)(6):

d =

mi2i

mii(4)

me =

miid

(5)

He =

mii hi

mii(6)

where i is the design displacement, mi is the seismic mass, and hi is the height of level i .In the next section the means of extending the Direct DBD approach to EBF structures

will be provided with reference to various parts of the general procedure just described.

3 Extending the Direct DBD approach to EBF structures

As explained in the previous section, in order to undertake the Direct DBD of EBF systems onerequires (1) knowledge of the deformation capacity of EBFs for different performance limitstates and the displaced shape of EBFs at the development of these deformation limits, (2) anestimate of the displacement ductility demand associated with the design displacement, whichin turn requires an estimate of the yield displacement of the EBF system, and (3) an expressionfor the equivalent viscous damping of the EBF as a function of the ductility demand. Thefollowing sub-sections explain how each of these parameters can be obtained and also discusshow one should consider higher-mode effects, the internal strength distribution, and capacitydesign requirements. At the end of the section an overview of the final design approach isprovided. The guidelines provided here are for EBF structures with centrally located links,

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Table 1 Proposed chord rotation values for different performance limit states

Performance limit state

No damage Repairable damage No collapse

EBF links with e 1.6 Mp/Vp y y + 0.08 y + 0.10EBF links with e 3.0 Mp/Vp y y + 0.02 y + 0.025e, link length; y, link chord rotation at yield; Mp, plastic flexural strength; Vp, shear strength of the linksection

of the type shown earlier in Fig. 1. However, extension to other forms of EBF, such as thosewith links adjacent to columns, could follow a similar approach to that presented here.

3.1 Link deformation capacity

The design of steel EBF systems should consider the performance requirements of bothstructural and non-structural elements. Non-structural drift requirements should be taken asfor other building systems and as such, serviceability drift limits could range between 0.5and 1.0 % depending on the type of non-structural element (refer Eurocode 8) whereas driftlimits for a repairable damage limit state might be in the order of 2.02.5 % in line with U.S(ASCE7-10 2010; NZS1170.5 2004) Standards. For what regards the structural performancerequirements of an EBF, Table 1 proposes values for the link chord rotation (shown in Fig. 1as i for link i) at three different performance limit states. For well detailed intermediatelink lengths, one could assume that the deformation capacity could be linearly interpolatedbetween the limits shown in Table 1, as is suggested in EC8.

The no damage (serviceability) limit state value has been set to avoid yield of the linkand later sections of this paper will illustrate how the yield rotation can be evaluated. Theplastic deformation limits shown in Table 1 are based on the recommendations of Engelhardtand Popov (1989) which also appear to agree with the limits indicated in Eurocode 8. Therotation limits for long links are only applicable for links located in the mid-span of a beamand should not be applied for links adjacent to columns. The values of rotation capacityexperimentally recorded by Engelhardt and Popov (1989) initially included both elastic andinelastic rotation components of the links, but not elastic deformation components due tobraces or columns. However, the elastic deformation components were then removed (seep. 28 of Engelhardt and Popov 1989) such that the actual plastic rotation angle was obtained.The plastic rotation obtained in this way included contributions of inelastic link deformationsas well as inelastic link end rotations caused by yielding or inelastic buckling of beam sectionsoutside the link.

The chord rotation limits in Table 1 are quite similar for the repairable damage andno collapse limit states. The approach adopted in this paper is the same as that adoptedfor the DBD of other systems (see Priestley et al. 2007) in which the no collapse limitstate indicates deformation limits that are close to mean values of ultimate deformationcapacity, whereas the repairable damage limit state is essentially a conservative estimate ofthe average ultimate deformation capacity. Along these lines, note that the 0.02 rad suggestedfor long links was recommended by Engelhardt and Popov (1989) as a conservative limitfor the ultimate deformation capacity that actually appeared to be around 0.025 rad. Clearly,such limits could be revised in the light of relevant experimental evidence or if alternativedefinitions of key limit states were adopted.

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Failure of links tested in Berkeley in the 1980s was typically in the form of local shearbuckling of panel zones at link ends, ultimately leading to fracture due to excessive localplastic deformations. More recently, testing of shear links made of high strength materials hasindicated (Okazaki and Engelhardt 2007) that a different type of failure may occur, with webfracture occurring before local buckling. Considering these results, Okazaki and Engelhardt(2007) suggest that this new type of behaviour may be because of differences in the weldingprocesses and stiffener details compared to those implemented in the 1980s. In the testsreported by Okazaki and Engelhardt (2007) the new type of failure mode led to a reducedshear link deformation capacity, below the standard requirement of 0.08 rad plastic rotationcapacity under a conventional cyclic loading history. However, a new testing protocol waspurposely developed for shear links by Richards and Uang (2006) which involved applicationof a smaller number of large-amplitude deformation cycles and a larger number of small-amplitude deformation cycles. Even though modern links tested with the new protocol stilltended to fail by web fracture before buckling, the links were able to sustain 0.08 rad plasticrotation demands.

While the previous paragraph suggests that some uncertainty may remain as to the appro-priate deformation capacity of EBF links, an advantage of the Direct DBD method is that itis easily adaptable to different deformation limits. As such, if future research suggests thatthe deformation limits in Table 1 should be updated, this can be done without modifying thedesign procedure developed in this work.

3.2 Storey yield drift

In order to estimate the EBF deformation capacity, the local deformation limits presented inthe previous section need to be related to global deformation values. The first step in doingthis is therefore to identify the storey drift at yield of an EBF as this will be useful for designto the no-damage limit state. Yield drift expressions will also be useful for the estimation ofductility demands and equivalent viscous damping of the EBF system.

Figure 3 depicts the deformed shape of an EBF at yield. It is proposed that the storeyyield drift can be estimated with account for the following three deformation components:(1) beam (including link) deformations, (2) brace axial deformations, and (3) column axialdeformations. It is also recognised that axial deformations of the beams may increase thestorey drift at yield, but the effect should be relatively limited and is difficult to estimate sincethe axial stiffness will tend to benefit from the surrounding floor slab. Accurate evaluation ofthe elastic deformations at yield should be undertaken using second-order analyses. However,for design purposes it is argued that the beam, brace and column drift components can beevaluated separately and then added together. In line with this, the following sub-sectionsexplain how each of these elastic storey drift components can be estimated.

3.2.1 Elastic storey drift component associated with beam deformations

In order to estimate the elastic storey drift component associated with flexural and sheardeformations of the beams (including link segments), the beam is first idealised as beingsimply supported between the column and the link centre. Due to symmetry, considerationof one side of the EBF is sufficient and as such, the beam is idealised as shown in Fig. 4,where the force P can be considered equivalent to the vertical component of the internal forcebeing transmitted between the beam and the tension brace. As will be shown in the passagethat follows, the vertical displacement of the beam due to the force P is obtained and simplegeometric relations are then used to determine the equivalent storey drift component.

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hs

Lb

2

vLbr

e

brace elongates

brace shortens

columns deform axially

beams deform in flexure and shear

= storey drift

Pinned beam-column connections

assumed

Fig. 3 Illustration of main elastic deformation components for an EBF

Fig. 4 Idealisation of the bracedbeam as a simply supported beamsubject to a point load, P

P

(Lb e)/2 e/2

The magnitude of the force P that is of interest here is that which causes the link to yield.For the case of a short link which yields in shear at a stress equal to 0.577 Fy (where Fy isthe yield stress of the steel), the corresponding force P is:

P = 0.577 Fy Av(

1 + eLb e

)

(7)

where Av is the shear area of the beam section and e and Lb are defined in Fig. 3.Subsequently, the vertical displacement of the beam, v , at the point P (which corresponds

to the brace connection point as illustrated in Fig. 3) is given by:

v = 0.577 Fy Av(

e2(Lb e)24 E I

+ e2 G Av

)

(8)

where E is the elastic modulus, G is the shear modulus, I is the second moment of inertiaof the beam and all the other symbols have been defined earlier. Note that the term on theleft within the brackets provides the displacement component due to flexural deformationswhereas the term on the right gives that due to shear.

For long links, the force P should be related to the plastic bending strength of the link,Mp , such that the vertical displacement is given by:

v = Mp(

e(Lb e)12 E I

+ 1G Av

)

(9)

Subsequently, the storey drift component at yield due to beam deflection can be approximatedby considering that the vertical deflection of the beam, v , over a distance (Lb e)/2, resultsin a rigid body rotation (in which the columns and braces can be considered rigid since theirdeformation components are considered separately) of the EBF system equal to:

link,i = 2 v,iLb ei (10)

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where the term link,i is the storey drift component at level i due to the beam deformation inflexure and shear. Note that the beam displacement and link length should be computed foreach level i .

3.2.2 Elastic storey drift component associated with brace axial deformations

The elastic storey drift component due to brace axial deformations can be derived by assuminga brace axial strain, br , of:

br = kbry (11)where y is the yield stain of the steel and kbr is the brace strain ratio, which can also beconsidered as the ratio of the axial force in the brace obtained from displacement-basedseismic design, NE,br,i , to the brace section resistance, NRs,br,i , at level i :

kbr,i = NE,br,iNRs,br,i (12)

Examining Eqs. (11) and (12), it is clear that the brace strain ratio is a design choice, asit depends on the final section size provided to the EBF. A procedure for dealing with thiswithin DBD will be explained in Sect. 3.9.

Knowing that the brace is inclined at an angle (see Fig. 3), once the brace axial strainis known it then follows that the storey drift, br,i , due to brace axial deformations can becomputed as:

br,i = 2kbr,iysin 2i

(13)

where the brace angle, i , and the strain ratio, kbr,i , for each storey i are used to compute thestorey drift component for that storey.

3.2.3 Elastic storey drift component associated with column axial deformations

The axial forces and deformations of columns will vary during an earthquake as seismicdemands on various modes of vibration change during ground shaking. However, at peakdisplacement response it will be assumed in Direct DBD that a full mechanism has formed upthe height of the EBF. In such a case, the seismic components of axial force on the columnscan be derived from equilibrium. Figure 5 illustrates the internal force distribution that woulddevelop in a 5-storey EBF that is pushed by an equivalent set of lateral forces from left toright. The link shear forces, Vlink , can be obtained directly from the resistance of the links,as will be illustrated in later sections. It is apparent from Fig. 5 that the overturning actionprovided by the earthquake motion puts the columns on the right in compression and thecolumns on the left in tension. However, at the top level the column axial forces due toearthquake change sign (because the role of the columns at the top storey is to restrain thebeams, not the braces), with the column on the right being subject to tension and the columnon the left being subject to compression.

The right side of Fig. 5 shows how the axial displacement (shortening and elongation)of columns over the lower four storeys, cols,4, will result in a rigid-body rotation of level5. This rigid body rotation correlates to an equivalent increase in the apparent storey driftat yield and therefore the column axial deformations below a given level can contribute animportant storey drift component, particularly for taller EBF systems. Subsequently, Eq. (14)

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Ncol,5 = -Vbeam,5

Ncol,4 = Vlink,5 -Vbeam,4

Nbrace,5 = (Vlink5+Vbeam5)/sin5

Nbrace,4 = (Vlink4+Vbeam4)/sin4

Ncol,3 = Vlink,4 + Vlink,5 -Vbeam,3Nbrace,3 = (Vlink3+Vbeam3)/sin3

Vlink5 Vbeam5

Vlink4

Vlink3

Vlink2

Vlink1

TCTC

CCTT

CCTT

CCTT

CCTT Ncol,1 = (Vlink,2 ++ Vlink,n ) -Vbeam,1Nbrace,1 = (Vlink1+Vbeam1)/sin1

cols,4

Fig. 5 Internal force distribution in a 5-storey EBF (left) and 5th storey elastic drift component due to axialdeformations of columns, cols,4, from the four levels below (right)

is proposed to compute the elastic component of drift, cols,i , for storey i due to column axialdeformations below the storey.

cols,i = col (hi hs)Lb/2 =2kcols,i1y (hi hs)

Lb(14)

where hi is the height to the top of storey i, hs is the inter-storey height at level i , and colis the average seismically-induced strain in the columns below storey i . As shown on theright side of Eq. (14), the average strain in the columns can be taken as the yield strain, y ,multiplied by the column strain ratio, kcols,i1, which is given by:

kcols,i1 =[

NE,col1NRs,col1

+ NE,col2NRs,col2

+ + NE,col,i1NRs,col,i1

]

= 1i 1

j=i1

j=1

NE,col, jNRs,col, j

(15)

and is simply summing, for each storey below level i , the ratio of the displacement-basedseismic design axial force in the column, NE,col , to the column section resistance, NRs,col. Itwill be apparent from Eq. (15) that the column strain ratio is a design choice since it dependson the selected column sizes and how much reserve capacity they have against seismicloading. Note that a typical value for kcols,i1 will be in the order of 0.25 for columns withEuropean open flanged (HE) sections.

In addition to the storey drift caused by underlying column axial deformations, one couldalso expect an additional drift contribution from axial deformations of the columns at theheight of the storey in question (e.g. the columns at level 5 for the frame on the right ofFig. 5). However, as such axial deformations do not cause a rigid-body rotation they are noteasily incorporated within Eq. (14), and because this additional drift contribution is likely tobe negligible, the proposal is to ignore it during design. Results of NLTH analyses in latersections will indicate that such a simplification appears to be reasonable.

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3.2.4 Total storey drift at yield

Summing the contributions detailed in the previous sub-sections, the total storey drift at yieldof an EBF can be approximated as:

y,i = link,i + br,i + cols,i = 2v,iLb ei +2kbr,iysin 2br,i

+ 2kcols,i1y (hi hs)Lb

(16)

where the symbols have been defined earlier with reference to Eqs. (10), (13) and (14) andwhere the vertical displacement of the beam, v,i , should be evaluated using Eqs. (8) or (9)in the case of short or long links respectively.

Applying this procedure to a single storey EBF system where column axial deformationswere negligible and beam axial deformations were constrained to zero (rigid diaphragmbehaviour) it was found that the above approach estimated the storey yield drift obtainedfrom pushover analyses very accurately (0.314 % obtained from pushover versus 0.315 %estimated from above approach).

3.3 Total storey drift capacity

As was introduced earlier in Sect. 3.1, the total storey drift capacity for a given performancelevel should be taken as the smaller of the drift capacity for structural and non-structuralelements. For structural elements, the total storey drift capacity, c,str,i , is obtained for leveli by summing the storey yield drift from Eq. (16) with the plastic storey drift capacity, p,i ,as shown:

c,str,i = y,i + p,i (17)The plastic storey drift capacity of level i should be found from:

i p = ei p,link,iLb =ei

(ls,link,i y,link,i

)

Lb(18)

where p,link,i is the allowable plastic chord rotation of the link for the intended performancelimit state. This relationship for the plastic deformation capacity, presented previously byEngelhardt and Popov (1989) amongst others, is valid for small rotation angles (within therange indicated in Table 1) and is based on the simple geometrical relationship obtainedconsidering a rigid-body rotation of the column-brace-beam assemblage. As shown on theright of Eq. (18), the plastic chord rotation can be computed as the total chord rotation,ls,link,i , taken from Table 1 for the design limit state, minus the link chord rotation at yield.Note also that because the total chord rotation limits in Table 1 are all expressed as a constantplus the chord rotation at yield, one does not actually need to compute the chord rotation atyield since it cancels within Eq. (18).

The expressions for the storey drift capacity developed in this and the preceding sectionsare considered to be a valuable contribution not only for displacement-based seismic designbut also for seismic assessment in general. As will be shown later in case study applications,the elastic storey drift component in Eq. (17) will often be of similar magnitude to the plasticstorey drift component, even when the maximum allowable plastic rotation limits in Table 1are adopted. This point is made to emphasize the importance of considering not only theplastic storey drift component in EBFs but also the elastic storey drift component [fromEq. (16)].

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3.4 Design displacement profile

With the storey drift capacity now developed, one requires an expression for the designdisplacement profile in order to be able to relate the local storey drift capacities to globaldisplacement limits. At first, one might consider it appropriate to utilise a non-linear designdisplacement profile typically used for steel moment resisting frames. However, it is evidentthat for EBFs, the column axial deformations discussed in Sect. 3.2.3 will tend to let theframes deform with more of a cantilever type profile, a tendency that will be counteractedby the shear restraint offered by the EBFs.

Results of shake table testing undertaken on a steel EBF structure and reported by Whit-taker et al. (1987) and plotted in Fig. 6, indicated a linear displacement profile at low inten-sities, tending towards a concave profile at large intensities. This behaviour would appearto support the notion that the elastic column axial deformations counteract the shear-typedeformations of the brace system to generate to relatively linear displacement profiles at lowintensities. At high intensities, however, the displaced shape tends to become more non-linearas the plastic deformation demands concentrate at the links. For displacement-based designof EBFs, therefore, the limit state displacement profile, i,ls , is proposed as:

i,ls = chi for c y (19a)i,ls = yhi +

(c y

)hi (2Hn hi )

(2Hn h1) for c > y (19b)

where hi is the height of level i above the base, Hn is the total building height, h1 is theheight of the 1st storey, y is the minimum storey yield drift over the height of the structureand c is the critical storey drift limit. The critical storey drift limit, c, should be taken asthe minimum value of the drift limit for non-structural elements or structural elements (c,strfrom Eq. 17) over the height of the EBF structure. Even though Eq. (19) would appear tosuggest that peak storey drifts are expected at the base storey of EBFs, by specifying the useof the minimum drift capacity over the height of the EBF the expression is providing someallowance for the possibility that peak drifts in upper levels are also high. High drift demandscould be expected in upper levels because of higher mode effects, as will be discussed furtherin Sect. 3.5, and non-uniform strength distributions, as will be discussed further in Sect. 3.6.

To gauge the accuracy of Eq. (19) it is compared in Fig. 6a with the displacement profilesobtained from the shake table testing conducted by Whittaker et al. (1987). To computethe displacement profile from Eq. (19) for the test structure, the base shear versus storeydrift loops reported by Whittaker et al. (1987) were used to interpret a yield drift for theEBF of approximately 0.25 %. The critical storey drift was then varied to give the same firstfloor displacement as the test structure using the relevant form of Eq. (19) (i.e. Eq. 19a forthe low intensity level of 0.08 g and Eq. 19b for the higher intensities of 0.27 and 0.66 g).Similarly, out of interest, the elastic 1st mode shape was also scaled to give the same firstfloor displacements as the test structure and the resulting displacement profiles are shown inFig. 6b.

The results presented in Fig. 6 indicate that Eq. (19) provides a relatively good match tothe observed displaced shapes. At large inelastic demands the equation appears to overes-timate displacements in the upper floors but significantly less so than displacement profilesassociated with the elastic first mode shape. Given these results, the proposal of Eq. (19) isjustified, even if future research could look to develop more accurate expressions by makingcomparisons with the results of additional shake table tests or possibly obtaining displacementprofiles from instrumented buildings recorded subject to earthquakes.

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Fig. 6 Comparison of displacement profiles obtained from shake table testing of a 6-storey EBF structureby Whittaker et al. (1987) with a the displacement profiles obtained from Eq. (19) and b displacement profilesassociated with the elastic first mode shape

3.5 Accounting for higher mode effects

Direct DBD is a design procedure that is based on controlling the response of the fundamentalmode of vibration. Nevertheless, higher mode effects are considered in a simplified fashionin the following two ways: (1) by reducing the first mode design displacement profile toaccount for the additional displacements and drifts caused by higher mode effects, and (2)by adopting equivalent lateral force distributions that aim to protect against the developmentof non-uniform displacement demands generated by higher modes. Equivalent lateral forcedistributions will be discussed in Sect. 3.7 whereas the reduction that can be made to thedesign displacement profile is explained here.

The left side of Fig. 7a sketches three different displacement profiles that might developin a structure around the time at which it reaches its peak response. The thicker line that isrelatively linear is intended to represent a first mode displaced shape and therefore it is denotedas n,1. The other two displacement profiles are intended to illustrate how the displacementswould change during a full oscillation of the second mode of vibration. In one instance, thedisplacements at the top of the structure increase while the displacements over the lowerstoreys decrease. In the other instance, the displacements at the top of the structure decreasewhile the displacements over the lower storeys increase. Note that Gardiner et al. (2013)measured post-earthquake permanent building offsets in a multi-storey EBF structure thatsuggested this type of second mode response occurred during the Christchurch earthquake.The resulting effect of higher mode actions is that the total storey drift demands over theheight of the building will tend to be increased, as is indicated in Fig. 7b.

Figure 7c presents a series of higher mode drift reduction factors, , that vary as afunction of the number of storeys. In order to account for this apparent drift amplification inthe Direct DBD of EBF structures, the limit state displacement profile, i,ls, from Eq. (19)should be factored by the higher mode drift reduction factor, , to give the final (first mode)design displacement profile, i,:

i = i,ls (20)

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(a) Displacement Profiles (b) Storey Drift Profiles

.c

n,1Storey drift

limit, c

n,total

1st mode profile

Total deformation

profile

0.0 Number of storeys5 10 15

1.00

0.60

Higher Mode Factor,

(c) Higher mode drift reduction factors proposed for EBF design

Fig. 7 Illustrating the potential influence of higher modes on a displacements and b storey drifts, andc tentative higher mode drift reduction factors for use in Direct DBD (adapted from Sullivan et al. 2012)

As shown in Fig. 7b, the peak drift associated with the first mode displacement profile willbe times that of the total drift, and, if the Direct DBD procedure performs well, the totaldrift demand will equal the critical storey drift limit (indicated by the dashed vertical line inFig. 7b).

Ideally, higher mode drift reduction factors will be set using the results of an extensivenon-linear time-history analysis campaign, as was done by Pettinga and Priestley (2005) forRC moment-resisting frame structures. However, in the absence of the results of such a studyon EBF systems, Fig. 7c includes a trial range of drift reduction factors that are proposedfor EBF systems. For low rise EBFs the drift reduction factor is equal to 1.0, implying thathigher mode effects are assumed to be relatively insignificant. For taller EBF systems, thedrift reduction factors become quite low, with a value of 0.6 proposed for structures of 16storeys and higher. The low values have been set in recognition of the relatively long periodsof the higher modes of vibration in EBF structures. Longer periods of vibration tend to attractgreater spectral displacement demands and as such, higher mode drift demands in EBFs canbe expected to provide a larger proportion of the total drift demand in comparison to otherstructures that possess relatively small higher mode periods of vibration (such as RC wallstructures). As reported in Sect. 4, the drift reduction factors shown in Fig. 7c are used in theapplication of the proposed DBD procedure to a series of EBF structures that range up to 15storeys in height. While the results shown later will suggest that the trial values in Fig. 7c maybe adequate, future research should aim to establish more accurate values and investigate thepossibility of applying the DBD approach to EBF systems that are greater than 15 storeysin height.

3.6 Equivalent viscous damping; displacement reduction factors

As was shown earlier in Fig. 2c, the Direct DBD procedure traditionally utilises ductility-dependent equivalent viscous damping expressions combined with damping-proportionalspectral displacement scaling factors to account for the effects of non-linear behaviour onseismic demands. However, as explained in Pennucci et al. (2011), the accuracy of this two-step procedure is likely to be improved by being condensed into a single step in which inelasticdisplacement spectra (rather than highly damped elastic spectra) are generated directly asa function of the design ductility demand using so-called displacement-reduction factors.

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Fig. 8 a Variation of displacement reduction factors with ductility demand and b use of design displacementspectra, scaled by displacement reduction factors, to identify a design effective period, Te

Accordingly, in this work the following expression is used to obtain a ductility-dependentspectral displacement reduction factor, ,EBF , for steel EBF systems:

,E B F =

(

1 + 1.17 ( 1)1 + e1

)1for 1.0 (21)

where is the design displacement ductility demand. This expression is based on the resultsof numerous NLTH analyses reported by Maley (2011) for systems with bilinear hystereticproperties characterised by a post-yield stiffness of 5 % the initial stiffness. In reality, onecould argue that the hysteretic properties of EBF systems are not exactly bilinear, but it isproposed that for design purposes, Eq. (21) should be sufficient.

In order to construct an inelastic displacement spectrum, Sd,in(Te), using the displacementreduction factor obtained from Eq. (21), one simply needs to multiply the elastic spectradisplacement demand, Sd(T ), by the displacement reduction factor:

Sd,in(Te) = ,EBF Sd(T ) (22)Note that the inelastic spectrum so obtained is that associated with the effective period of thesubstitute structure (i.e. the period associated with the secant stiffness, Ke, from Fig. 2b andthe effective mass, me). Figure 8 shows both the displacement reduction factors obtained fromEq. (21) and the inelastic spectra that are produced through their insertion within Eq. (22).

As shown in Fig. 8a, the displacement reduction factor given by Eq. (21) initially dropsrelatively quickly with increasing ductility demands, but reduces less quickly from ductilitydemand of around 3.0 onwards, tending towards a value of around 0.5 for larger ductilitydemands. These trends are then reflected by the inelastic displacement spectra in Fig. 8b[obtained through application of Eq. (22)], with differences in inelastic displacement demandsreducing with increasing ductility demands.

To use the effective-period inelastic displacement given by Eq. (22), one follows thesame approach that was described in Fig. 2d, entering Fig. 8b with the design displacementdemand, d, and reading off the required effective period for the spectrum that correspondsto the system ductility demand. This process is illustrated in Fig. 8b where, for an arbitrarilyassumed design ductility of 2.0 and design displacement of 0.29 m, one reads off a requiredeffective period of 3.2 s.

In order to establish the system ductility demand for a multi-storey EBF it is recommendedthat the ductility demand, i , at each storey first be computed as the ratio of the design storeydrift demand, i , to the storey drift at yield, y,i , (from Eq. 16) as shown:

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i = iy,i

= i i1y,i (hi hi1) (23)

where i and i1are the design displacements and hi and hi1 are the heights, of floors iand i1 respectively.

The system ductility demand, , for use in Eq. (21) can then be computed using Eq. (24):

=i=n

i=1 i Viii=n

i=1 Vii(24)

where Vi is the shear and, i , the storey drift demand at level i , and n is the total number ofstoreys. Eq. (24) utilizes what is commonly referred to as a work-done approach which Maley(2011) has shown to be reasonably accurate for mixed systems. Note that in Direct DBDthe shear profile will not initially be known but since the shear term appears in both thenumerator and denominator of Eq. (24) only the relative shear proportions are required forthe computation of the system ductility demand, and these can be found as a function of thedesign displacement profile, as explained next.

3.7 Design strength distribution

Once the required effective period has been obtained as per Fig. 8b, and the effective stiffness,Ke, is calculated according to Eq. (1), the Direct DBD base shear, Vb, can be computed as:

Vb = Ked + Cn

i=1 PiiHe

(25)

where Pi is the seismic weight at level i, He is the effective height of the substitute structure asper Eq. (6), C is a constant (that takes on a value of 1.0 for steel structures when megd/Vb Heis greater than 0.05, as per Priestley et al. 2007) to account for P-delta effects, and n is the totalnumber of storeys. It is apparent that this equation varies from that presented in Eq. (2) as aterm has now been introduced to account for P-delta effects, in line with the recommendationsof Priestley et al. (2007).

The design base shear obtained from Eq. (25) should be distributed as a set of equivalentlateral forces over the height of the building to permit the design forces on individual linksto be found. For steel EBF systems the following equation is recommended for the lateralforce, Fi , at level i :

Fi = k miimii

Vb for i < n (26a)

Fi = (1 k) .Vb + k miimii

Vb for i = n (26b)

where k adopts a value of 0.9 for EBF structures of 6 storeys or more and is otherwise equalto 1.0. By adopting values of k in this way, the equivalent lateral force distribution is lumping10 % of the base shear at the roof level of taller systems in order to help mitigate higher modeeffects. The same recommendation is made in Priestley et al. (2007) for tall MRF structuresand similar requirements are specified in the New Zealand loadings standard (NZS1170.5)for the equivalent lateral force design procedure.

By summing the equivalent lateral forces down the height of the building, the total sheardemand at each level is obtained:

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Vi =j=n

j=iFj (27)

The required shear, Vlink,i , and flexural strength, Mlink,i , of each link can then be found usingEqs. (28) and (29) respectively:

Vlink,i = Vi hsLb (28)

Mlink,i = Vlink,i ei2 (29)

where, as per Fig. 3, hs is the storey height, Lb is the EBF length and ei is the link length.In the event that short links are being used it is evident that the link section should be

sized using the shear force obtained from Eq. (28), whereas sections should be sized usingthe moment from Eq. (29) if long links are desired.

When sizing links, the ratio of the storey shear resistance to the design storey shear forceshould be evaluated for each storey. This ratio will be referred to here as the excess-strengthratio. In order to ensure globally distributed dissipation in line with S.6.8.2(7) of EC8, theexcess-strength ratios computed for all storeys should not vary by a factor greater than 1.25over the height of the building. Furthermore, the excess-strength ratio between adjacentstoreys should vary as little as possible, ideally changing by no more than a factor of 1.15.For the case study structures examined in Sect. 4 it will be shown that these requirementscan be satisfied using standard European section sizes. The relative excess-strength ratios areconsidered particularly important for EBF systems as they help to reduce the possibility thatdrift concentrates in a single storey. As will be shown from the results of NLTH analyses inSect. 4, drift demands do indeed tend to concentrate at levels where large differences in theexcess-strength ratio occur from one level to another.

3.8 Capacity design considerations

The Direct DBD procedure is intended to identify the required strength of the design plasticmechanism. Capacity design guidelines should then be followed to identify design forces forthose members and actions not intended to yield. For the case of EBF structures, the mainchallenge for capacity design is therefore to ensure that columns, braces and connectionsremain elastic. Priestley et al. (2007) provide general guidelines for the capacity design ofstructures following the application of Direct DBD, and recommend consideration of (1)overstrength of plastic hinge zones and (2) higher mode effects.

For the case of EBFs, the link overstrength appropriate for capacity design could berelatively high as it should account for the following three factors:

In the same way that the characteristic 5th percentile value of strength will be considerablylower than the expected material strength used in Direct DBD, the maximum (say 95thpercentile) likely strength could be considerably greater.

The links may be expected to undergo relatively large inelastic deformation demands andit has been shown from experimental testing that in such cases, hardening of links canbe significant. Such hardening should be accounted for in assessing the link overstrengthfor capacity design.

Links are often sized using lower-bound expressions (from codes) for the shear areaof sections that ignore the possible contribution of flanges or floor slabs to the shearresistance. As such, in addition to consideration of excess-strength that results from

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selection of section sizes that are greater than DBD actions, some estimate for additionalshear resisting mechanisms would seem appropriate.

For what regards higher mode effects, brace actions should be relatively unaffected byhigher modes but some moment demand should be expected in columns, even if they maynot be significant in comparison to the high axial force demands.

A detailed study into suitable capacity design requirements for EBF structures is outsidethe scope of this paper. However, in order to achieve realistic section sizes for case studyapplications reported later in this paper, capacity design actions have been computed for thebraces as a function of the link shear resistances multiplied by an overstrength factor of 1.5.Plastic mechanism forces were also used as capacity design forces for the columns low-risecase study storey structures, but for the taller case study structures it was recognised thatcontemporary yielding of links up the full height was unlikely and therefore, capacity designaxial forces for the taller buildings were taken as 1.5 times the DBD axial forces. Futureresearch should examine the subject of capacity design of steel structures in more detail,taking into consideration observations by Elghazouli (2005) and Della Corte et al. (2012)and also making probabilistic considerations such as those proposed by Victorsson et al.(2011).

3.9 Summary of proposed design approach

The equations developed in the previous subsections now permit a full Direct DBD procedureto be defined for steel EBF systems. This is done through the flowchart provided in Fig. 9.The design procedure is seen to be iterative because yield drifts and drift capacities dependon final section sizes, which are not initially known. As such, the designer must first assumelink sizes and column and brace average strain factors, which are checked towards andend of the procedure and updated if necessary. As the general Direct DBD procedure hasalready been described in Sect. 2 and the new equations within the procedure have just beenexplained in detail in the previous sub-sections, the description of the method is limited tothe flowchart here. However, the appendix to this paper also illustrates the method in detailthrough application to a 5-storey case study design example.

4 Investigating the performance of the procedure

In order to investigate the performance of the new DBD procedure, it is applied to a series ofcase study EBF structures. The design solutions are then used to create accurate non-linearmodels of the structures which are then subject to non-linear time-history (NLTH) analysesusing a set of real accelerograms scaled to an intensity level compatible with that used indesign. The performance of the design method is then gauged by comparing the designdeformations with those recorded in the NLTH analyses.

4.1 Description of the case study structures

Figure 10 shows the case study EBF structures selected for examination in this work, Asshown, the case study buildings consist of regular 1, 5, 10 and 15 storey EBF systems witha uniform storey height of 3.5 m and a uniform EBF bay length of 7 m. The plan viewof the case study structures shows that four EBF systems resist lateral loads in each of theorthogonal response directions. Internal columns are assumed to provide resistance to gravityloads only and are located along the intersection of the 7 m by 7 m grid lines. The RC floor

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Inputs: seismic masses, storey height, bay length, material properties (E, G, Fy), design intensity and performance criteria (e.g. non-structural drift limits and link chord rotation limit from Table 1).

1. Trial Design Decisions: trial section sizes, link lengths, ei; average strain factors, kbr,i (Eq.12) and kcol,i-1 (Eq.15).

2. Set Design Displacement Profile: compute storey yield drifts (Eq.16) and drift capacities (Eq.17) and compare with non-structural drift limit. Use critical

drift limit to define design displacement profile using Eqs.(19) & (20).

3. Identify Equivalent SDOF System Properties: Using the design displacement profile, compute the design displacement (Eq.4), effective mass (Eq.5) and effective height (Eq.6).

4. Scale Design Spectrum: For a unit base shear compute storey shear profile (Eq.27). Subsequently, find storey ductility demands via Eq.(23) and system ductility, , from Eq.(24). Compute displacement

reduction factor by inserting in Eq.(21) and scale the displacement spectrum as per Eq.(22).

5. Identify required effective period and design base shear: Enter scaled displacement spectrum with the design displacement (Figure 8b) to identify required effective period.

Then compute the effective stiffness (Eq.1) and design base shear (Eq.25).

6. Check link sizes: Distribute design base shear (Eq.26) and compute storey shear demands (Eq.27) and link shear demands (Eq.28). Compare link shear demands with link resistances.

Link Sections OK?

7. Size columns and braces: Use capacity design rules to identify design forces for columns and braces. Identify column and brace sections that can resist the demands.

8. Check average strain factors: Compute average strain ratios for braces (Eq. 12) and columns (Eq.15) using DBD design forces and section resistances. Check if strain factors match initial values.

Average Strain Factors OK?

Modify link sizes

Revise kbr,i & kcols,i-1

Member sizing complete

Fig. 9 Flowchart outlining the proposed Direct displacement-based design procedure

system is assumed to be fully flexible out-of-plane and provide rigid diaphragm behaviourin-plane. The seismic weight of each storey is taken as 6,240 kN, except at roof level where5,500 kN is assumed. The steel for the case study structures is European grade S450, witha characteristic yield strength of 450 MPa (for plate thicknesses less than 40 mm). In Direct

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EBF EBF

EBF EBF

EBF

EBF

EBF

EBF

ELEVATION OF EBF SUB-SYSTEMS PLAN VIEW

3 bays at 7m = 21m

5ba

ys a

t 7m

= 3

5m

Constant storey height

hs = 3.5

1 Storey 5 Storeys 10 Storeys 15 Storeys

Roof Level Weight = 5500kN

Typical Storey Weight = 6240kN

Fig. 10 Illustration of the case study EBF structures

DBD, expected material properties should be used and the expected yield strength of theS450 steel is therefore computed as 528 MPa, in line with Badalassi et al. (2011).

4.2 Design criteria and loading

The case study structures are designed for the repairable damage limit state in which the driftlimit for control of damage to non-structural elements is taken as 2.5 % (in line with Priestleyet al. 2007) and the link chord rotation limits shown in Table 1 are adopted for the control ofdemands to structural elements. A design decision is made to use short links for the framesand as a consequence, a plastic chord rotation limit of 8mrad is adopted for the designs.

The case study structures are assumed to lie in a region of high seismicity, with a peakground acceleration (on rock) of ag = 0.3 g for the repairable damage design intensity level.The design spectrum, shown as a bold solid line in Fig. 11, corresponds to the Eurocode 8type 1 spectrum for soil type C, but with a period value TD = 8 s (instead of 2 s) such thatspectral displacement demands increase linearly up to a corner period of 8 s. The 8 s value isadopted for TD because research (Faccioli et al. 2007) has demonstrated that longer periodvalues can be expected in certain regions of Italy and because longer periods are also expectedin other parts of the world, including the U.S. (ASCE7-10 2010), where it is foreseen that themethod may be applied. Figure 11 also includes the spectra for ten accelerograms that willbe used to test the performance of the design solutions via NLTH analyses, reported later inSect. 4.5. The accelerograms, selected by Maley et al. (2012) from the PEER NGA databaseand the 2010 Darfield (N.Z.) earthquake, originate from earthquakes ranging in magnitudefrom 6.2 to 7.6 and were all recorded on sites conforming to Eurocode 8 soil type C.

Using the procedure described in Sect. 3.9 and detailed in the appendix, design solutionsare developed to the point that link, brace and column member section sizes are finalised.Connections are assumed to be pinned but their design is not undertaken because knowledgeof the connection details is not required for the design method verification analyses reportedlater in Sect. 4.5.

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Fig. 11 Comparison of the design acceleration (left) and displacement (right) response spectra with the spectraof ten real records selected and uniformly scaled to be spectrum compatible by Maley et al. (2012)

Table 2 Intermediate design results for the four case study EBF structures

Design parameter 1-Storey 5-Storey 10-Storey 15-Storey

Design displacement d (m) 0.037 0.108 0.250 0.342Effective mass me (T) 141 685 1,332 1,984Effective height He (m) 3.5 12.1 23.2 34.3System design ductility 2.90 2.63 2.05 1.61Displacement reduction factor 0.576 0.600 0.675 0.763Effective period Te (s) 0.49 1.40 2.88 3.48Design base shear Vb (kN) 2,759 5,982 6,894 9,612Base shear normalised by building weight Vb/Wt 49.7 % 19.6 % 11.2 % 10.4 %

4.3 Intermediate design results

Table 2 presents the intermediate design results for the four case study structures. Inter-estingly, in order to limit local ductility demands on links, the design ductility values arerelatively low. It is evident from Sect. 3 that the ductility demand will be a factor of manyparameters, including the link length, local deformation capacity and brace and column sizes.However, the general tendency for system ductility factors to reduce with height should beexpected for other EBF configurations, owing to the increasing importance of the column axialdeformation component (Sect. 3.2.3) for taller buildings and the non-linear design displace-ment profile (Eqs. 3, 4). Consequently, the results in Table 2 would suggest that force-baseddesign reduction (behaviour) factors for EBF systems in current codes are non-conservative,ranging from 4.0 to 6.0.

Despite the low system ductility demands obtained for the taller EBF structures, the designbase shear values for the tall systems are not excessive, being around 10 % of the buildingweight. A traditional force-based design mentality would suggest that this is the result of therelatively long periods of vibration of these structures. While this is true, another arguablymore appropriate way of looking at this trend is to consider the increased displacementcapacity of the taller systems; large yield displacements lead not only to low system ductilitycapacities, but also to large displacement capacities. As such, while one might at first beconcerned that EBF systems are actually characterised by low system ductility capacities,the more important observation should actually be that the systems possess large displacement

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Table 3 Final design results for the 1-storey EBF structure

Level Bracesection

Columnsection

Linksection

Linklength(m)

y,i(%)

i (%) i Vd,i (kN) VR,i/Vd,i

1 HE 160 B HE 160 A HE 160 A 0.60 0.36 1.05 2.90 690 1.06


Level Bracesection

Columnsection

Linksection

Linklength(m)

y,i(%)


5 HE 180 B HE 180 A HE 160 A 0.60 0.48 0.42 0.87 393 1.164 HE 180 B HE 180 A HE 180 B 0.70 0.49 0.60 1.24 792 1.153 HE 200 B HE 200 B HE 200 B 0.70 0.41 0.79 1.94 1,123 1.082 HE 200 B HE 240 B HE 220 B 0.80 0.39 0.98 2.52 1,364 1.091 HE 200 B HE 280 B HE 220 B 0.80 0.32 1.16 3.60 1,495 1.08

capacity and this assists in reducing strength and stiffness requirements. Interestingly, in thecourse of the case study designs it was observed that more efficient designs were obtained notby minimising the yield displacement through the selection of very short links but actuallyby maximising the EBF displacement capacity by using relatively long links (the lengthsof which were limited, however, to be short enough to ensure yield in shear). This pointhighlights the fact that greater importance should be given to displacement capacity ratherthan energy deformation; both are beneficial but displacement capacity is more so.

4.4 Summary results of the design procedure

Final section sizes, storey drifts, storey ductility demands, shear demands and excess-strengthratios are reported in Tables 3, 4, 5 and 6. The section sizes are relatively realistic, beingselected from European standard section tables. Despite having a relatively limited numberof section sizes to choose from, it was always possible to maintain the excess-strength ratiosto within the limits discussed in Sect. 3.7. Also observe that because of the non-linear designdisplacement profile and the tendency for storey yield drifts to be greater towards the top ofthe EBF systems, the design ductility demands over the upper storeys of the 10- and 15-storeybuildings were



Level Bracesection

Columnsection

Linksection

Linklength(m)

y,i(%)


10 HE 200 A HE 180 A HE 200 A 0.80 0.80 0.44 0.56 388 1.029 HE 200 B HE 180 A HE 200 B 0.90 0.84 0.56 0.66 620 1.088 HE 200 B HE 260 A HE 200 B 0.85 0.79 0.67 0.85 839 1.037 HE 200 B HE 260 A HE 220 B 0.90 0.74 0.78 1.05 1,040 1.156 HE 200 B HE 320 B HE 220 B 0.90 0.70 0.89 1.27 1,221 1.045 HE 220 B HE 320 B HE 240 B 1.00 0.65 1.00 1.53 1,379 1.114 HE 220 B HE 500 B HE 240 B 1.00 0.60 1.11 1.86 1,511 1.063 HE 220 B HE 500 B HE 240 B 1.10 0.58 1.22 2.08 1,614 1.012 HE 220 B HE 500 M HE 240 B 1.10 0.53 1.33 2.53 1,686 1.011 HE 220 B HE 500 M HE 260 B 1.20 0.46 1.44 3.10 1,723 1.10


Level Bracesection

Columnsection

Linksection

Linklength(m)

y,i(%)

i (%) i Vd,i(kN)

VR,i/Vd,i

15 HE 260 B HE 180 A HE 280 B 1.00 1.16 0.35 0.30 442 1.1514 HE 260 B HE 180 A HE 300 B 1.00 1.12 0.42 0.38 664 1.0713 HE 260 B HE 260 A HE 320 B 1.00 1.08 0.49 0.46 879 1.0912 HE 260 B HE 320 B HE 320 B 1.00 1.06 0.57 0.54 1,085 1.0411 HE 260 B HE 320 B HE 320 B 1.00 1.02 0.64 0.63 1,281 1.0310 HE 260 B HE 320 M HE 320 B 1.00 0.99 0.71 0.72 1,466 1.039 HE 260 B HE 320 M HE 320 B 1.05 0.97 0.78 0.81 1,638 1.038 HE 260 B HE 450 M HE 320 B 1.15 0.97 0.86 0.89 1,796 1.047 HE 260 B HE 550 M HE 320 B 1.15 0.91 0.93 1.02 1,938 1.096 HE 260 B HE 550 M HE 340 B 1.20 0.85 1.00 1.17 2,065 1.185 HE 260 B HE 650 M HE 340 B 1.30 0.81 1.08 1.33 2,173 1.154 HE 260 B HE 650 343 HE 340 B 1.30 0.72 1.15 1.59 2,263 1.153 HE 260 B HE 650 343 HE 340 B 1.40 0.67 1.22 1.83 2,332 1.142 HE 260 B HE 650 407 HE 340 B 1.40 0.57 1.29 2.26 2,379 1.161 HE 260 B HE 650 407 HE 340 B 1.45 0.49 1.37 2.80 2,403 1.18

then conducting NLTH analyses using the spectrum-compatible accelerograms illustratedearlier in Sect. 4.2, one can gauge the likely performance of the structures in a design eventand compare this to the intended response.

A bi-linear hysteretic model with 5 % post-yield strain hardening is specified for the plastichinges of all the link elements. It is recognised that the link behaviour could be modelledmore accurately, with account for both kinematic and isotropic hardening. However, as theintention of these analyses is only to gauge the system response and given that the use ofa bilinear hysteretic model is consistent with the expression for the displacement reductionfactors adopted in design, the simplified bilinear hysteretic model is considered acceptable

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here. While this modelling approach is considered adequate for gauging the performanceof the trial DBD methodology, it is recognized that there are many uncertainties associatedwith the modelling of EBF structures, and future research should consider how modellingdecisions, and the presence of secondary elements such as the facades and surrounding floorslabs, could impact on the seismic response.

The analyses are conducted using a Newmark integration scheme with an integration time-step of 0.001 s. A lumped mass matrix is adopted. A Rayleigh tangent stiffness-proportionaldamping model is assumed (for reasons given in Priestley et al. 2007) with the equivalent of3 % elastic damping specified on the 1st and 2nd modes of vibration. Floors are assumed tobehave as rigid diaphragms in-plane by constraining the end nodes of columns to move withthe same horizontal displacement. Large displacement analyses are conducted and in orderto reproduce the destabilising effect of gravity loads not acting directly on the columns ofthe EBFs, the gravity loads are lumped onto dummy column elements, modelled in parallelto the EBF and provided with end releases so that no lateral restraint is offered to the frames.Beam-column, brace-beam, brace-column and column base connections are all modelledas perfect pins, in line with the design assumptions. The analyses are conducted for oneexcitation direction only since the structural systems (shown earlier in Fig. 10) are deemedto be independent, the building is symmetric in plan and because the design solutions to beverified have assumed unidirectional response. Future research could investigate appropriatemeans of undertaking DBD for bi-directional excitation and torsional response.

4.6 Non-linear time-history analysis results

The results of the NLTH analyses are presented in this section. The peak displacements andpeak storey drifts recorded for each accelerogram, together with the average of the peakvalues, are shown in Fig. 12 for the 1- and 5-storey buildings and in Fig. 13 for the 10- and15-storey buildings. In addition, for comparison purposes, the DBD 1st mode displacement-profile is shown as a solid line and the design storey drift capacity is shown as a bold solidline.

The results presented in Figs. 12 and 13 permit some valuable observations to be made.Firstly, the average storey drift demands are less than or equal to the design storey driftcapacities, demonstrating that the design solutions have been successful in limiting the peakstorey drift demands to within the design drift capacities. This represents an important findingof the work, suggesting that the new DBD methodology is very promising.

A second observation is that the design displacement profile appears to be relatively wellcaptured by Eq. (19), particularly for the 1-storey, 5-storey and 10-storey EBFs. For the15-storey structure the displaced shape is well predicted over the bottom half of the build-ing but over the upper storeys a relatively linear profile would have been more appropriate.Reviewing Eq. (19) it is clear that a linear displacement profile was not predicted because ofthe relatively high design ductility demands predicted over the lower storeys. It is apparentthat because the upper levels were expected to respond elastically, an improved design dis-placement profile might be obtained by considering the average storey ductility demand overthe building height, rather than the peak demands associated with a single storey. Neverthe-less, given its simplicity and the current lack of a better alternative, the recommendation isto proceed with Eq. (19) which has been found to lead to good results in this study and forfuture research to investigate alternative design displacement profiles.

Modern seismic design requires engineers to control the likelihood of exceeding a givenlimit state. With this in mind, in addition to the average drift demands shown in Figs. 12 and 13,it is worthwhile noting that if one considers the full range of the 1st storey drift demands

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Fig. 12 Comparison of the design displacement (left) and drift (right) profiles with the results of non-lineartime-history analyses for a the 1-storey and b the 5-storey EBF structure

(that tended to be critical), coefficients of variation of 0.43, 0.39, 0.12 and 0.27 are obtainedfor the 1, 5, 10 and 15 storey structures respectively. If this were to be considered within aprobabilistic performance-based design context, it would suggest that one needs to accountfor significant dispersion in demands, particularly for the shorter buildings. However, as thecoefficients of variation reported here were obtained from the results of only 10 earthquakerecords, which were known to be less spectrum compatible in the short period range (seeFig. 11), future research would be required to confidently establish the likely dispersion.

Interestingly, drift demands are observed to increase significantly at the 7th storey of the15-storey EBF. The reason for this can be found from the design results presented earlier inTable 6. At level 7 the link section sizes reduce from HE340B to HE320B, and these sizesimply that level 6 possesses an excess strength ratio of 1.18 whereas the excess strengthratio at level 7 is only 1.09. Because of the higher relative strength of level 6, storey drift

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Fig. 13 Comparison of the design displacement (left) and drift (right) profiles with the results of non-lineartime-history analyses for a the 10-storey and b the 15-storey EBF structure

demands are more pronounced in the level above. A similar response occurs between the 7thand 8th storeys of the 10-storey EBF for the same reasons. These observations highlight theimportance of maintaining uniform excess strength ratios, within the limits recommended inSect. 3.7.

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The displacement and drifts for the single storey EBF are considerably smaller thanthe design limit and one might be concerned from such results that the DBD approachis too conservative for short buildings. The fact that the peak time-history displacementswere around 60 % of the design displacement can be attributed to two factors. Firstly, bycomparing the spectra of the accelerograms with the design spectrum in Fig. 11, it can beobserved that the accelerograms were not very compatible in the short period range andimpose on average only around 75 % the design intensity level, and therefore one couldhave anticipated peak displacements of around 75 % of the design deformation for this casestudy structure. Secondly, as explained in Priestley et al. (2007) and Sullivan (2011), theempirical expressions for equivalent viscous damping or displacement reduction factors tendto be conservative in the short-period range. To overcome such conservatism one couldadopt period-dependent damping expressions such as those proposed by Grant et al. (2005).However, as recommended by Priestley et al. (2007), because the use of period-independentequivalent viscous damping (or displacement-reduction factor) expressions are simpler toapply and are not excessively conservative, they appear to remain the most rational choicefor design.

5 Conclusions

This paper has described the basis of a new Direct displacement-based design methodologyfor steel EBF structures. The methodology has been applied to regular case study EBF struc-tures ranging from 1- to 15-storeys in height. The design results indicate that the ductilitycapacity of EBF systems will tend to be relatively low despite the large local ductility capac-ity offered by ductile links. In addition, it has been found that while the ductility capacityof EBF systems will tend to reduce with height, this is not necessarily negative for seis-mic performance since the displacement capacity for taller EBF systems will tend to belarge.

Analytical expressions have been developed for the storey drift at yield and for the storeydrift capacity. It has been shown that in addition to link beam deformations, the braceand column axial deformations can provide important contributions to the storey drift atyield and consequently, to the storey drift capacity. The new expressions for storey drift areexpected to be useful not only for DBD but also for seismic assessment of existing EBFstructures.

To gauge the performance of the proposed DBD methodology, accurate analytical modelsof the design solutions have been subject to non-linear time-history analyses with a set ofspectrum-compatible accelerograms. The average displacements and drifts obtained fromthe NLTH analyses were found to align well with design values and therefore it is concludedthat the new methodology is very promising. Future research should investigate whether animproved expression for the design displacement profile can be identified and should test themethodology for a wider range of EBF configurations and for different design limit states.

Acknowledgments The author wishes to acknowledge funding that supported this work; Fondi Anteno diRicerca 2012. The author also wishes to thank Gaetano Della Corte and Jose Miguel Castro for frequent,illuminating discussions related to the design of steel structures that assisted greatly in the realisation of thispaper.

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Appendix 1: Illustrating the design procedure for the 5-storey structure

In order to illustrate how the design procedure is applied, this appendix presents the detailedsolution for the 5-storey EBF structure. In line with the flowchart of Fig. 9, in addition to thematerial properties and design criteria identified in Sect. 4.1, one must identify trial sectionsizes. For this case study practical section sizes are selected as shown in Table 7, with only twodifferent section types initially proposed for columns, braces and links. In addition, desirablelink lengths are set, and as shown Table 7, the trial values imply that the links are shortaccording to the EC8 classification procedure.

The next step in the design is to compute the yield drift and drift capacity for each storey.In order to do this, the vertical deflection of the beams at yield is first computed using Eq. (8)and then inserted into Eq. (10) to obtain the yield drift component due to shear and bendingdeformations of the beams and links. The results so obtained are shown in the second andthird columns of Table 8. In line with the recommendations of Sects. 3.2.2 and 3.2.3, averagestrain values of kbr,i and k,cols,i1 are assumed for the braces and columns. As shown inTable 8, initial values of 0.25 are assumed for both factors. Eqs. (13) and (14) are then usedto compute the yield drift components due to brace and column deformations, respectively.As per Eq. (16), the three drift components are then summed to obtain the total yield driftand the results are shown in Table 8. Subsequently, the plastic drift capacity of each storey iscomputed by inserting the repairable damage limit state chord rotation for short links fromTable 1 into Eq. (18). The plastic drift is then summed, as per Eq. (17), with the yield drift toobtain the total storey drift capacity for structural elements and the results are shown in thelast column of Table 8. As the values for structural storey drift capacity are all greater thanthe storey drift limit of 2.5% for non-structural elements, the structural storey drift limits inTable 8 are identified as critical.

The minimum storey yield drift from Table 8 is found to be 0.31 % and the minimumdrift capacity is 1.22 %. These two values are then inserted into Eq. (19b) in order to obtainthe displacement profile for the selected performance limit state, which is shown in Table 9.At this stage a higher mode drift amplification factor should be obtained from Fig. 7c andas the building is only five storeys high, the drift amplification factor is = 1.0, suchthat computation of the final design displacement profile using Eq. (20) leads to the samedisplacements as the limit state displacement profile. This displacement profile is used toobtain the first-mode design drift profile, i , reported in column 4 of Table 9. The ductilitydemands at each storey are then found using Eq. (23) and the results are shown in the centralcolumn of Table 9.

Table 7 Initial section sizes, link lengths and brace angles for the 5-storey EBF

Level Height Initial section sizes Link length(m)

Link lengthclassification

Brace angle, ()

Column Brace Link

5 17.5 HE 200 B HE 200 B HE 200 B 0.70 Short 48.04 14.0 HE 200 B HE 200 B HE 200 B 0.70 Short 48.03 10.5 HE 260 B HE 200 B HE 200 B 0.70 Short 48.02 7.0 HE 260 B HE 220 B HE 220 B 0.80 Short 48.51 3.5 HE 260 B HE 220 B HE 220 B 0.80 Short 48.5

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Table 8 Initial estimates of the yield drift and total structural drift capacity for the 5-storey EBF

Level v(Eq. 8)

link(Eq. 10)(%)

kbresti-mate

br(Eq. 13)(%)

kcols,i1estimate

cols(Eq. 14)(%)

y(Eq. 16)(%)

c,str(Eq. 17)(%)

5 0.0069 0.22 0.25 0.07 0.25 0.26 0.55 1.354 0.0069 0.22 0.25 0.07 0.25 0.19 0.48 1.283 0.0069 0.22 0.25 0.07 0.25 0.13 0.42 1.222 0.0074 0.24 0.25 0.07 0.25 0.06 0.38 1.291 0.0074 0.24 0.25 0.07 0 0.00 0.31 1.23

Table 9 Intermediate design results for the 5-storey EBF structure

Level i,ls(Eq. 19)

i(Eq. 20)

Driftprofile,i (%)

Ductility,i (Eq. 23)

Storeymass, mi(T per EBF)

mi i mi 2i mi i hi

5 0.143 0.143 0.41 0.75 140 20.0 2.86 350.24 0.128 0.128 0.61 1.27 159 20.4 2.62 285.73 0.107 0.107 0.82 1.94 159 17.0 1.82 178.42 0.078 0.078 1.02 2.71 159 12.5 0.97 87.21 0.043 0.043 1.22 3.92 159 6.8 0.29 23.8

76.6 8.6 925.2

With the design displacement now known, the substitute structure characteristics shouldbe found using Eqs. (4)(6). As such, an additional four columns are added onto Table 9 topermit the calculation of the numerators and denominators to be inserted into Eqs. (4)(6),which leads to the preliminary substitute structure characteristics shown below:

d =

mi2i

mii= 8.6

76.6= 0.112 m (30)

me =

miid

= 76.60.112

= 686.7 T (31)

He =

mii hi

mii= 925.2

76.6= 12.07 m (32)

In order to obtain the design base shear for the EBF the only remaining substitute structurecharacteristic to be computed is the displacement reduction factor via Eq. (21). This in turnrequires an estimate of the system ductility demand, which, as explained in Sect. 3.6, requiresknowledge of the relative shear proportions over the height of the frame. To establish these,an equivalent lateral force distribution is identified for a unit base shear using Eq. (26) andthe subsequent shear profile is computed through Eq. (27). The resulting equivalent lateralforce distribution and shear profile are shown in Table 10.

In line with Eq. (24), the products of the storey shear proportions and design storey driftsare then used to compute the system ductility demand as shown in Eq. (33). The values ofthe numerator and denominator in Eq. (33) come from the sum of the columns on the rightside of Table 10.

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Table 10 Computations used for determination of the system ductility and P-delta component

Level Fi (for unitVb) (Eq. 26)

Vi (for unitVb) (Eq. 27)

Vi i Vi i i Pi i

0.261 0.261 0.261 0.0011 0.0011 1960.266 0.266 0.527 0.0032 0.0041 2000.222 0.222 0.749 0.0061 0.0119 1670.162 0.162 0.911 0.0093 0.0251 1220.089 0.089 1.000 0.0122 0.0478 67

0.0319 0.0900 752

=i=n

i=1 i Viii=n

i=1 Vii= 0.0900

0.0319= 2.82 (33)

The system ductility is then used to compute the displacement reduction factor from Eq. (21):

,E B F =(

1 + 1.17 ( 1)1 + e1

)1=

(

1 + 1.17 (2.82 1)1 + e2.821

)1= 0.58 (34)

and the design displacement spectrum is subsequently scaled by 0.58 according to Eq. (22).Using the design displacement of d = 0.112 m, the required effective period is then readoff the scaled design displacement spectrum (in a manner similar to that shown in Fig. 8b)which leads to a required effective period of Te = 1.49 s. The required effective stiffness isthen calculated from Eq. (1) as:

Ke = 42 meT 2e= 42 686.7

1.492= 12,206 kN/m (35)

Initially ignoring P-Delta effects, the design base shear (Eq. 2) is then:

Vb = Ked = 12,206 0.112 = 1,362 kN (36)One can then consider the P-delta stability coefficient for the system:

P = megdVb He =686.7 9.81 0.112

1,362 12.07 = 0.048 (37)

And as the stability coefficient is less than 0.05, the design base shear does not need tobe amplified to account for P-Delta effects using Eq. (25). The design base shear is nowdistributed as a set of equivalent lateral forces in accordance with Eq. (26) and the designstorey shear and the link design shear forces are then found through Eqs. (27) and (28)respectively. The resulting design actions are reported in Table 11.

Table 11 also presents the design shear resistances for the trial link sizes. To obtain theseresistance values, the link yield strength is first computed using Eq. (38) from Eurocode8 [7]:

Vy,link =(

Fy/

3)

tw(h t f

) (38)

where Fy is the steel yield strength, h is the link section depth and tw and t f are the web andflange thicknesses respectively. The design strength obtained from DDBD is that required

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Table 11 Design shear forces and link resistance ratios for the initial solution of the 5-storey EBF

Level Fi (kN)(Eq. 26)

Vi (kN)(Eq. 27)

VE,link(kN)(Eq. 28)

Linksection

p,i(Eq. 40)(%)

VR,link(kN)(Eq. 39)

VR,link/VE,link

5 355.7 356 178 HE 200 B 0.0 381 2.144 362.8 718 359 HE 200 B 1.3 541 1.503 302.0 1,020 510 HE 200 B 4.0 608 1.192 221.3 1,242 621 HE 220 B 5.6 757 1.221 120.6 1,362 681 HE 220 B 7.9 826 1.21

at the development of the design displacement (or limit state). As such, the likely resistanceconsidered during design should be that expected at the same level of deformation demand,with account for hardening if required. In line with this, assuming that the ultimate strength ofbraces is 1.5 times the yield strength, the design link resistances in Table 11 (and subsequenttables) have been computed as:

VR,link,i = iy,i

Vy,link,i for < 1.0 (39a)

VR,link,i = Vy,link,i(

1 + 0.5 p,ip,u

)

for 1.0 (39b)

where p,u is the ultimate chord rotation capacity of the link (taken here to be 10 % for shortlinks as per Table 1) and p,i is the expected plastic chord rotation demand which can beestimated from:

p,i =(i y,i

) Lbei

(40)

With the design resistances known, the excess-strength ratios (VR,link/VE,link) at each storeyare computed and reported in Table 11.

Over the lower three storeys the assumed link sizes are quite close to the design loads, withexcess-strength ratios of around 1.2. However, the excess-strength ratios are unacceptablyhigh in the upper storeys, with the top storey value of 2.14 implying that the excess-strengthratios computed for all storeys vary by a factor of close to 2.0, significantly greater than therecommended limit of 1.25 (refer Sect. 3.7).

Given the results presented above, the first change that will be made to the trial designsolution is to reduce the section size and length of the links at the top two storeys. The revisedlink sizes are shown in Table 12.

At this stage the design procedure is repeated, with the main difference being that thestorey yield drifts and drift capacities are revised to account for the new link sizes. Table 12reports the new drift estimates, and note that the column and brace drift components inTable 12 have been computed using the same average strain values (kbr,i and kcols,i1) as inTable 8.

For the revised link sizes the system design displacement remains constant at 0.105 mbut the design base shear increases slightly to 1,367 kN owing to the change in ductilitydemands at the fourth storey. This design base shear is then distributed up the height andthe new design shear forces shown in Table 13 are used to check the link strengths, asshown.

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Table 12 Revised link sizes and estimated drift values for the 5-storey EBF

Level Linksection

LinkLength(m)

v(Eq. 8)

link(Eq. 10)(%)

br(Eq. 13)(%)

cols(Eq. 14)(%)

y(Eq. 16)(%)

c,str(Eq. 17)(%)

5 HE 160 A 0.60 0.0084 0.264 0.07 0.26 0.59 1.284 HE 180 B 0.70 0.0084 0.265 0.07 0.19 0.53 1.333 HE 200 B 0.70 0.0069 0.219 0.07 0.13 0.42 1.222 HE 220 B 0.80 0.0074 0.238 0.07 0.06 0.38 1.291 HE 220 B 0.80 0.0074 0.238 0.07 0.00 0.31 1.23

Table 13 Revised design shear forces for the 5-storey EBF

Level Fi (kN) Vi(kN)

VE,link(kN)

Linksection

VR,link(kN)

VR,link/VE,link

5 356.8 357 178 HE 160 A 182 1.024 364.0 721 360 HE 180 B 448 1.243 303.0 1,024 512 HE 200 B 608 1.192 222.1 1,246 623 HE 220 B 757 1.211 121.0 1,367 683 HE 220 B 826 1.21

Table 14 Design results for the 5 storey EBF columns and braces

Level Columns Braces

Axial forceNE,col,i(kN)

Columnsection

ResistanceNRd,col,i(kN)

AxialforceNE,br,i(kN)

Bracesection

ResistanceNRd,br,i(kN)

5 221 HE180 A 817 581 HE 180 B 9774 713 HE180 A 817 964 HE 180 B 9773 1,553 HE200 B 1,615 1,138 HE 200 B 1,3752 2,493 HE240 B 2,658 1,337 HE 200 B 1,3751 3,587 HE280 B 3,816 1,337 HE 200 B 1,375

The results in Table 13 indicate that the new link sizes are quite good; the excess-strengthratios vary by less than 1.25 over the height of the structure and even though slightlysmaller sections might be possible, the sections are considered suitable for this phase of thedesign.

The next step is to compute capacity design actions, size the columns and braces andthen update the average strain values (kbr,i and kcols,i1) initially assumed in Table 8.For the 5-storey case study structure the capacity design actions are obtained by multi-plying the internal actions associated with the link shear resistances by a factor of 1.5, asexplained in Sect. 3.8. This leads to the capacity design axial forces shown in Table 14for the columns and braces. Note that for the columns, the capacity design axial forcesshown in Table 14 include axial loads obtained from gravity actions expected during theearthquake.

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Table 15 Design results for the 5 storey EBF columns and braces

Level Columns Braces

kcols,i1esti-matedrev. 0

kcols,i1rev. 0(Eq. 15)

kcols,i1esti-matedrev. 1

kcols,i1rev. 1(Eq. 15)

kbr,iesti-matedrev. 0

kbr,irev. 0(Eq. 12)

kbr,iesti-matedrev.1

kbr,irev. 1(Eq. 12)

5 0.25 0.167 0.167 0.181 0.25 0.078 0.078 0.1024 0.25 0.204 0.204 0.215 0.25 0.194 0.194 0.1983 0.25 0.238 0.238 0.247 0.25 0.220 0.220 0.2192 0.25 0.272 0.272 0.279 0.25 0.277 0.277 0.2701 0.0 0.000 0.000 0.000 0.25 0.302 0.302 0.293

The initial column and brace sizes shown in Table 7 are checked against the designactions and more efficient section sizes are subsequently selected. Table 14 reports the finalcolumn and brace sizes proposed, together with their design values of resistance calculatedin line with the standard recommendations provided in Eurocode 3 [38]. With the columnand brace sections now established, the average strain values for the braces and columns(kbr,i and kcols,i1) are computed using Eqs. (12) and (15), respectively. The computedvalues are compared with initial estimates in Table 15, which also includes new averagestrain values for the next iteration of design. Note that the rev.1 average strain values pro-posed for the braces do not exactly match the computed values because as soon as thenew average column strain values are imported into the excel design sheet then new aver-age strain values are computed (again via Eq. 12) and these are taken for the brace rev.1values.

The revised average strain values are used to revise the storey yield drifts and drift capac-ities (through Eqs. 13, 14, 16 and 17), leading to a design displacement of d = 0.106 m,which is smaller than the previous values reflecting the fact that the average strain values inthe columns and braces tended to be smaller than the initial estimates. The system ductilityis then recomputed as before and is also found to reduce, to a value of = 2.58. A neweffective period is determined; Te = 1.36 s, which leads to a slightly increased design baseshear of Vb = 1,543 kN. New design actions for the links, braces and columns are subse-quently found and the average strain values shown in Table 15 as rev.1 are obtained. Theresults in Table 15 show that reasonably good convergence has been obtained after just asingle iteration. However, by undertaking an additional iteration using estimated rev.2 valuesset equal to the calculated rev.1 values in Table 15, convergence to within 1 % is obtainedat all floors and the final design base shear is found to be Vb = 1,495 kN. Distributing thisvalue over the height of the building, the link sizes are checked again in Table 16 and it can beseen that the link sizes are acceptable, with excess-strength ratios that vary by no more than10 %. In addition, capacity design axial forces in the braces and columns are checked againstresistances

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Direct displacement-based seismic design of steel eccentrically braced frame structures