Seismic design of plane steel braced frames using equivalent modal damping ratios

Soil Dynamics and Earthquake Engineering 129 (2020) 105947

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Seismic design of plane steel braced frames using equivalent modal damping ratios Nicos A. Kalapodis a, George A. Papagiannopoulos b, * a b

Department of Engineering Science, University of Greenwich, Central Avenue, ME4 4TB, Chatham, UK School of Science and Technology, Hellenic Open University, 26335, Patras, Greece

A R T I C L E I N F O

A B S T R A C T

Keywords: Modal damping ratios Eccentrically braced frames Buckling-restrained braced frames Seismic design

A force-based seismic design method for eccentrically braced and buckling-restrained braced plane steel frames is proposed. The method makes use of equivalent modal damping ratios that play the role of the behavior (strength reduction) factor. A comparison of the proposed design method with the EC 8 design method is presented herein and conclusions are drawn.

1. Introduction

2. Seismic analysis of the EBFs and BRBFs considered

The seismic design procedure stipulated by current codes e.g., EC 8 [1], is of force-based design one, where the design base shear is deter­ mined by modal synthesis and an acceleration design spectrum obtained by dividing the elastic one by an approximate and mostly empirical behavior (strength reduction) factor. A common value of the behavior factor is employed per mode and seismic displacements are evaluated by means of the equal-displacement rule. An alternative seismic design method where equivalent modal damping ratios substitute the behavior factor, has been proposed in Refs. [2,3] for steel and reinforced concrete moment resisting frames (MRFs), respectively. According to this method, the design base shear is calculated through the use of equiva­ lent modal damping ratios (ξk ) for the first k modes, in conjunction with an elastic design spectrum properly adjusted to high amounts of damping [2,3]. These ξk are functions of modal periods Tk and are given for specific types of seismic motion [2,3] and for soil types A-D [3], following the soil categorization in Ref. [1]. In both works [2,3], ξk are deformation (in terms of interstorey drift ratio IDR) and damage (in terms of member end inelastic rotation) dependent, where the values of IDR and inelastic rotation are set for various seismic performance levels in agreement with [4]. For reasons of brevity, the process of computing the deformation and damage dependent ξk is omitted, but one can find it in Refs. [2,3]. The seismic design method employing ξk is expanded herein to plane eccentrically braced frames (EBFs) and buckling-restrained braced frames (BRBFs).

A total of 42 EBFs and 14 BRBFs are considered, where the braces are in a chevron configuration. The 42 EBFs are separated in three equal groups, i.e. short, intermediate and long, according to the expected deformation pattern of the seismic link [1]. Typical EBF and BRBF are shown in Fig. 1, where the storey height, the bay width and the load (dead and live) on steel beams are also given. In absence of any seismic design procedure for BRBFs in Ref. [1], these braced frames are designed as typical concentrically braced frames for the high ductility class. The design seismic load for the EBFs and BRBFs is calculated using the design spectrum of EC 8 [1] that corresponds to peak ground acceleration (PGA) 0.24 g, soil type B and behaviour factor equal to 4. The EC 8 [1] seismic design of these frames is done with the aid of SAP 2000 [5]. Due to space limitations, several modeling details and chosen steel sections for the EBFs and BRBFs can be found in Refs. [6,7]. Parametric seismic inelastic time-history analyses of the EBFs and BRBFs are then conducted by means of [8] in order to compute the ξk values that correspond to four seismic performance levels (SP), i.e., immediate occupancy (IO), damage limitation (DL), life-safety (LS) and collapse prevention (CP). For the EBFs, the values of IDR and link rotation θlink associated with the three SP levels are stated in Table 1, following [1,4]. In absence of any SP level expressed by IDR and axial ductility μδ of the buckling-restrained braces (BRBs) in Refs. [1,4], the IDR values of Table 1 have been also used for the BRBFs, whereas μδ values are approximated on the basis of those proposed by Ref. [9]. Modeling of EBFs and BRBFs for seismic inelastic time-history analyses

* Corresponding author. E-mail address: [email protected] (G.A. Papagiannopoulos). https://doi.org/10.1016/j.soildyn.2019.105947 Received 24 October 2019; Received in revised form 30 October 2019; Accepted 31 October 2019 Available online 13 November 2019 0267-7261/© 2019 Elsevier Ltd. All rights reserved.

N.A. Kalapodis and G.A. Papagiannopoulos

Soil Dynamics and Earthquake Engineering 129 (2020) 105947

Fig. 1. EBF (left) and BRBF (right) with chevron bracings.

the seismic motions accounting for soil types B and D and several viscous damping ratios are shown in Fig. 2. More data regarding the seismic motions and their acceleration spectra for various viscous damping ra­ tios are given in Refs. [6,7].

Table 1 Seismic performance (SP) levels for EBFs and BRBFs. SP level

IDR

μδ

θlink (rad)

SP1-IO SP2-DL SP3-LS SP4-CS

0.004 0.013 0.022 0.032

1.0 3.5 6.3 Not provided

0.0 Not provided 0.02 (long links), 0.08 (short links) Not provided

3. Design equations of ξk After computing ξk for each EBF and BRBF, regression analyses on the lowest ξk values with period Τ are then performed in order to construct simple formulae (design equations) to be used for seismic design purposes [2,3]. These design equations essentially correspond to the SP levels of Table 1 and to soil types A-D [1]. Tables 2–10 provide the

is described in detail in Refs. [6,7]. Each EBF and BRBF is subjected to 100 seismic motions recorded in far-field and at soil conditions classified as A-D in agreement with [1]. The mean acceleration spectra derived by

Fig. 2. Mean acceleration spectra for soil type B (a) and D (b) and several viscous damping ratios. Table 2 ξk (%) for EBFs with long links, soil type B. SP level

Mode 1

Mode 2

Mode 3

Mode 4

SP1-IO

ξ1¼ 0.14Τþ1.45 (0.30 � T � 1.70)

ξ3¼ 0.48Τþ0.43 (0.14 � Τ � 0.35)

ξ4¼

SP2-DL

ξ1¼

15.11Τ þ38.14 (0.30 � Τ � 1.70)

ξ1¼

12.23Τ þ71.16 (0.30 � Τ � 1.70)

ξ3¼ 104.29Τþ38.00 (0.23 � Τ � 0.30) ξ3¼ 18.00Τþ1.31 (0.30 � Τ � 0.35) ξ3¼ 100.00 (0.14 � T � 0.35)

ξ4¼ 100.00 (0.09 � Τ � 0.24)

SP3-LS

ξ2¼ 7.75Τþ2.55 (0.10 � Τ � 0.25) ξ2¼ 1.14Τþ0.33 (0.25 � Τ � 0.60) ξ2¼ 30.56Τ þ22.39 (0.35 � Τ � 0.50) ξ2¼ 29.00Τ 7.39 (0.50 � Τ � 0.60) ξ2¼ 100.00 (0.10 � Τ � 0.60)

Mode 2

Mode 3

Mode 4

ξ2¼ 43.30Τþ9.47 (0.09 � Τ � 0.20) ξ2¼ 0.81 (0.20 � Τ � 0.55) ξ2¼ 30.56Τ þ21.30 (0.36 � Τ � 0.55)

ξ3¼ ξ3¼ ξ3¼

ξ2¼ 100.00 (0.09 � T � 0.55)

ξ3¼ 100.00 (0.11 � T � 0.30)

1.35Τþ1.13 (0.09 � Τ � 0.24)

ξ4¼ 100.00 (0.09 � Τ � 0.24)

Table 3 ξk (%) for EBFs with short links, soil type B. SP level

Mode 1

SP1-IO

ξ1¼

0.14Τþ1.67 (0.22 � T � 1.70)

SP2-DL

ξ1¼ ξ1¼ ξ1¼

26.16Τ þ41.28 (0.22 � Τ � 1.10) 0.83Τþ13.43 (1.10 � Τ � 1.70) 13.70Τþ66.88 (0.22 � T � 1.70)

SP3-LS

2

60.00Τþ10.30 (0.11 � Τ � 0.16) 0.71Τþ0.81 (0.16 � Τ � 0.30) 25.00Τ þ14.00 (0.20 � Τ � 0.30)

ξ4¼

2.22Τþ1.17 (0.12 � Τ � 0.21)

ξ4¼ 100.00 (0.12 � T � 0.21) ξ4¼ 100.00 (0.12 � T � 0.21)

N.A. Kalapodis and G.A. Papagiannopoulos

Soil Dynamics and Earthquake Engineering 129 (2020) 105947

Table 4 ξk (%) for EBFs with long links, soil type C. SP level

Mode 1

Mode 2

Mode 3

Mode 4

SP1-IO

ξ1¼ 0.58Τþ1.00 (0.30 � T � 1.70)

ξ3¼ 1.94Τþ0.22 (0.14 � Τ � 0.35)

ξ4¼

SP2-DL

ξ1 ¼ ξ1 ¼ ξ1 ¼

ξ2¼ 7.59Τþ2.60 (0.10 � Τ � 0.25) ξ2¼ 1.13Τþ0.41 (0.25 � Τ � 0.60) ξ2¼ 16.13Τ þ14.23 (0.35 � Τ � 0.60) ξ2¼ 100.00 (0.10 � Τ � 0.60)

ξ3¼ 100.00 (0.14 � T � 0.35)

ξ4¼ 100.00 (0.09 � Τ � 0.24)

Mode 3

Mode 4

SP3-LS

26.46Τ þ41.64 (0.30 � Τ � 0.90) 6.52Τ þ23.46 (0.90 � Τ � 1.70) 11.43Τ þ63.89 (0.30 � Τ � 1.70)

ξ3¼

53.45Τþ22.60 (0.23 � Τ � 0.35)

1.44Τþ1.25 (0.09 � Τ � 0.24)

ξ4¼ 100.00 (0.09 � Τ � 0.24)

Table 5 ξk (%) for EBFs with short links, soil type C. SP level

Mode 1

Mode 2

SP1-IO

ξ1¼ 3.77Τþ5.00 (0.22 � T � 0.78) ξ1 ¼ 0.11Τþ1.97 (0.78 � T � 1.70) ξξ1¼ 24.18Τ þ37.60 (0.22 � Τ � 1.10) ξ1¼ 1.82Τþ8.99 (1.10 � Τ � 1.70) ξ1¼ 50.00Τþ79.00 (0.22 � T � 0.48) ξ1¼ 10.03Τþ59.79 (0.48 � T � 1.70)

ξ2¼ ξ2 ¼ ξ2¼

34.29Τ þ8.74 (0.09 � Τ � 0.22) 0.61T þ 1.33 (0.22 � Τ � 0.55) 22.22Τ þ17.21 (0.30 � Τ � 0.55)

ξ3¼ ξ3¼ ξ3¼

ξ2¼

33.33Τþ34.33 (0.45 � T � 0.55)

ξ3¼ 100.00 (0.11 � T � 0.30)

SP2-DL SP3-LS

69.17Τþ11.92 (0.10 � Τ � 0.16) 1.07Τþ1.01 (0.16 � Τ � 0.30) 44.00Τþ19.20 (0.20 � Τ � 0.30)

ξ4¼

2.86Τþ1.35 (0.12 � Τ � 0.21)

ξ4¼ 100.00 (0.12 � T � 0.21) ξ4¼ 100.00 (0.12 � T � 0.21)

Table 6 ξk (%) for EBFs with long links, soil type D. SP level

Mode 1

Mode 2

Mode 3

SP1-IO

ξ1¼ 0.14Τþ1.45 (0.30 � T � 1.70)

ξ3¼

SP2-DL SP3-LS

ξ1¼ ξ1¼

ξ2¼ 7.95Τþ2.63 (0.10 � Τ � 0.25) ξ2¼ 0.58Τþ0.49 (0.25 � Τ � 0.60) ξ2¼ 20.48Τ þ17.77 (0.35 � Τ � 0.60) ξ2¼ 100.00 (0.10 � Τ � 0.60)

ξ3¼ 25.02Τ þ14.15 (0.23 � Τ � 0.35) ξ3¼ 100.00 (0.14 � T � 0.35)

ξ4¼ 100.00 (0.09 � Τ � 0.24) ξ4¼ 100.00 (0.09 � Τ � 0.24)

Mode 2

Mode 3

Mode 4

ξ2¼ 33.13Τ þ9.48 (0.09 � Τ � 0.25) ξ2¼ 1.33T þ 1.53 (0.25 � Τ � 0.55) ξ2¼ 18.57Τ þ15.21 (0.20 � Τ � 0.55)

ξ3¼

0.50Τþ0.85 (0.10 � Τ � 0.16)

ξ4¼

ξ3¼

20.00Τ þ11.00 (0.20 � Τ � 0.30)

ξ4¼ 100.00 (0.12 � T � 0.21)

15.11Τ þ36.14 (0.30 � Τ � 1.70) 6.47Τþ59.20 (0.30 � Τ � 1.70)

Mode 4

1.46Τþ1.00 (0.14 � Τ � 0.35)

ξ4¼

4.05Τþ1.48 (0.09 � Τ � 0.24)

Table 7 ξk (%) for EBFs with short links, soil type D. SP level

Mode 1

SP1-IO

ξ1¼

SP2-DL

ξ1¼ 18.61Τ þ32.46 (0.22 � Τ � 1.10) ξ1¼ 1.67Τþ10.17 (1.10 � Τ � 1.70) ξ1¼ 55.55Τþ83.30 (0.22 � T � 0.60) ξ1¼ 9.09Τþ55.45 (0.60 � T � 1.70)

SP3-LS

0.14Τþ1.67 (0.22 � T � 1.70)

ξ2¼

5.56Τþ20.06 (0.45 � T � 0.55)

ξ3¼ 100.00 (0.11 � T � 0.30)

2.86Τþ1.35 (0.12 � Τ � 0.21)

ξ4¼ 100.00 (0.12 � T � 0.21)

Table 8 ξk (%) for BRBFs, soil type B. SP level

Mode 1

Mode 2

Mode 3

SP1-IO

ξ1¼ 3.33Τþ2.77 (0.22 � T � 0.50) ξ1¼ 0.40Τþ0.90 (0.50 � T � 1.50) ξ1¼ 38.16Τ þ51.16 (0.22 � Τ � 1.00) ξ1¼ 9.61Τþ3.39 (1.00 � Τ � 1.50) ξ1¼ 28.57Τ þ61.14 (0.22 � Τ � 0.50) ξ1¼ 26.53Τþ88.80 (0.50 � Τ � 1.50)

ξ2¼ 16.67Τþ3.83 (0.11 � Τ � 0.17) ξ2¼ 1.27Τþ1.22 (0.17 � Τ � 0.48) ξ2¼ 54.29Τ þ28.03 (0.25 � Τ � 0.42) ξ2¼ 57.33Τ 16.93 (0.42 � Τ � 0.48) ξ2¼ 100.00 (0.11 � T � 0.48)

ξ3¼

4.60Τþ1.57(0.10 � Τ � 0.27)

ξ4¼

ξ3¼

5.44Τ þ9.97 (0.18 � Τ � 0.27)

ξ4¼ 100.00 (0.11 � T � 0.19)

SP2-DL SP3-LS

ξ3¼ 100.00 (0.10 � T � 0.27)

Mode 4 6.51Τþ2.03(0.11 � Τ � 0.19)

ξ4¼ 100.00 (0.11 � T � 0.19)

Table 9 ξk (%) for BRBFs, soil type C. SP level

Mode 1

Mode 2

Mode 3

SP1-IO

ξ1¼ 9.58Τþ5.60 (0.21 � Τ � 0.48) ξ1¼ 0.39Τþ0.81 (0.48 � Τ � 1.50) ξ1¼ 50.0Τ þ65.00 (0.22 � Τ � 1.00) ξ1¼ 1.82Τþ16.82 (1.00 � Τ � 1.50) ξ1¼ 31.25Τþ84.50 (0.22 � Τ � 1.20) ξ1¼ 2.86Τþ43.57 (1.20 � Τ � 1.50)

ξ2¼ 33.33Τþ7.00 (0.11 � Τ � 0.18) ξ2¼ 1.30Τþ1.24 (0.18 � Τ � 0.48) ξ2¼ 53.14Τ þ27.05 (0.25 � Τ � 0.42) ξ2¼ 30.99Τ þ7.86 (0.42 � Τ � 0.48) ξ2¼ 100.00 (0.11 � T � 0.48)

ξ3¼

SP2-DL SP3-LS

design equations of ξk for EBFs with long or short links, BRBFs and soil types B-D. Due to space limitations, the rest design equations of ξk involving: i) EBFs with long or short links and BRBFs for soil type A; ii) EBFs with intermediate links for soil types A-D, can be found in similar

3.70Τ þ1.48 (0.10 � Τ � 0.27)

ξ3¼ 7.66Τ þ24.41 (0.18 � Τ � 0.23) ξ3¼ 15.38Τþ3.34 (0.23 � Τ � 0.27) ξ3¼ 100.00 (0.10 � T � 0.27)

Mode 4 ξ4¼

8.87Τþ2.13 (0.11 � Τ � 0.19)

ξ4¼ 100.0 (0.11 � T � 0.19) ξ4¼ 100.00 (0.11 � T � 0.19)

table forms in Ref. [7]. It should be noted that values of ξk in excess of 100% have been computed for the collapse prevention (CP) seismic performance level. However, for the CP performance level, according to Refs. [2,6], one has to use ξk ¼ 100% for all k modes. Thus, design 3

N.A. Kalapodis and G.A. Papagiannopoulos

Soil Dynamics and Earthquake Engineering 129 (2020) 105947

Table 10 ξk (%) for BRBFs, soil type D. SP level

Mode 1

Mode 2

Mode 3

Mode 4

SP1-IO

ξ1¼ 1.50 (0.22 � T � 1.50) ξ1¼ 47.37Τ þ61.37 (0.22 � Τ � 1.00) ξ1¼ 3.64Τþ10.36 (1.00 � Τ � 1.50) ξ1¼ 9.68Τþ74.68 (0.22 � Τ � 0.48) ξ1¼ 39.00Τþ98.15 (0.48 � Τ � 1.50)

ξ3¼ 37.77Τ þ5.72 (0.10 � Τ � 0.14) ξ3¼ 3.42Τþ0.30 (0.14 � Τ � 0.27) ξ3¼ 111.11Τ þ29.94 (0.18 � Τ � 0.22) ξ3¼ 66.00Τ-9.02 (0.22 � Τ � 0.27) ξ3¼ 100.00 (0.10 � T � 0.27)

ξ4¼

SP2-DL

ξ2¼ 24.23Τþ6.26 (0.11 � Τ � 0.24) ξ2¼ 1.01Τþ0.21 (0.24 � Τ � 0.48) ξ2¼ 82.14Τ þ37.71 (0.25 � Τ � 0.38) ξ2¼ 26.17Τ 3.44 (0.38 � Τ � 0.48) ξ2¼ 100.00 (0.11 � T � 0.48)

SP3-LS

Table 11 Sections per storey level. Storey 1st 2nd 3rd 4th 5th

9.86Τþ2.28 (0.11 � Τ � 0.19)

ξ4¼ 100.00 (0.11 � T � 0.19) ξ4¼ 100.00 (0.11 � T � 0.19)

Table 12 Sections per storey level.

EC8

Proposed Method

Storey

HEB

IPE

CHS

HEB

IPE

CHS

280 280 260 260 240

330 300 300 300 300

193.7x4.5 193.7x4.5 193.7x4.5 168.3x4.0 168.3x4.0

280 280 260 260 240

330 330 300 300 300

219.1x5.0 193.7x4.5 193.7x4.5 168.3x4.0 168.3x4.0

1st 2nd 3rd 4th 5th

HEB: sections for columns. IPE: sections for beams. CHS: sections for braces.

EC8

Proposed Method

HEB

IPE

BRB

HEB

IPE

BRB

320 320 300 300 260

330 330 330 330 330

A ¼ 27cm2 A ¼ 27cm2 A ¼ 21cm2 A ¼ 17cm2 A ¼ 17cm2

340 340 320 300 260

360 330 300 300 300

A ¼ 34cm2 A ¼ 27cm2 A ¼ 21cm2 A ¼ 17cm2 A ¼ 17cm2

HEB: sections for columns. IPE: sections for beams. BRB: sections for BRBs.

Fig. 3. Design response spectra.

Fig. 4. Design response spectra.

equations of ξk for the CP performance level can be omitted from Tables 2–10.

to a 5-storey chevron BRBF which is designed for the LS seismic per­ formance level employing the design spectrum mentioned above for the case of the 5-storey EBF. Dimensioning of the BRBs follows [9]. There­ fore, according to Ref. [1], the design base shear is found 547.68 kN and the chosen sections are shown in Table 12. The first four period-equivalent damping pairs (Table 8) to be used in the context of

4. Comparison between the ξk and the EC 8 design methods A 5-storey chevron EBF with long links is designed for the LS seismic performance level of Table 1, by making use of both EC 8 [1] and the proposed design method with ξk. According to the seismic design by EC 8 [1] involving the design spectrum for PGA 0.36 g, soil type B and behaviour factor equal to 4, the design base shear is found 376.56 kN and the chosen sections are shown in Table 11. To apply the proposed design method, an initial seismic design of the EBF under study is done using [1] and its first four periods are obtained, i.e., Τ1 ¼ 0.72sec, 2 ¼ 0.25sec, Τ3 ¼ 0.14sec and Τ4 ¼ 0.10sec. Utilizing these periods in Table 2, one obtains the equivalent modal damping ratios: ξ1 ¼ 62.40%, ξ2 ¼ 100%, ξ3 ¼ 100%, ξ4 ¼ 100%. Then, a modified design spectrum is constructed (Fig. 3) using the ordinates of the mean highly damped acceleration spectra (Fig. 2a) that correspond to the aforementioned values of periods and equivalent modal damping ratios ξk. Inserting this modified spectrum in SAP2000 [5] and after performing modal syn­ thesis, the design base shear is determined to be equal to 436 kN and the chosen sections are shown in Table 11. Both the EC 8 [1] and the proposed design methods are now applied

Table 13 Seismic analyses results for the 5-storey EBF. EC 8 method

4

Proposed method with ξk

Seismic motion

Vy (kN)

IDR

θlink (rad)

Vy (kN)

IDR

θlink (rad)

1 2 3 4 5 6 7 8 9 10 Average Design

379.26 378.51 378.47 378.71 379.33 378.28 377.39 377.01 378.60 377.57 378.31 376.56

0.0138 0.0155 0.0087 0.0115 0.0118 0.0135 0.0147 0.0109 0.0132 0.0134 0.0127 0.0129

0.0302 0.0342 0.0183 0.0251 0.0254 0.0291 0.0321 0.0235 0.0287 0.0292 0.0276 0.0197

478.09 475.69 474.50 476.73 477.49 476.86 476.39 473.97 477.44 477.69 476.48 436.00

0.0104 0.0106 0.0099 0.0088 0.0085 0.0124 0.0122 0.0106 0.0127 0.0101 0.0106

0.0222 0.0224 0.0209 0.0185 0.0177 0.0265 0.0261 0.0227 0.0245 0.0215 0.0223

N.A. Kalapodis and G.A. Papagiannopoulos

Soil Dynamics and Earthquake Engineering 129 (2020) 105947

the BRBF. On the other hand, the proposed method provides an average base shear value significantly higher than this of the seismic design, especially for the BRBF. This increase is from the side of safety but leads to a less economical (in terms of the steel sections used) design in comparison to the EC 8 method.

Table 14 Seismic analyses results for the 5-storey BRBF. EC 8 method

Proposed method with ξk

Seismic motion

Vy (kN)

IDR

μδ

Vy (kN)

IDR

μδ

1 2 3 4 5 6 7 8 9 10 Average Design

633.02 679.57 673.28 693.56 688.70 697.84 664.14 696.81 692.79 693.45 681.32 547.68

0.0063 0.0092 0.0084 0.0052 0.0089 0.0075 0.0073 0.0094 0.0089 0.0064 0.0078 0.0063

2.790 3.838 3.582 2.461 3.751 3.272 3.042 3.991 3.777 2.837 3.330

712.66 753.26 746.64 767.52 786.98 715.59 797.84 786.35 753.14 794.17 761.42 573.60

0.0067 0.0063 0.0064 0.0051 0.0060 0.0072 0.0064 0.0058 0.0063 0.0057 0.0062

2.558 2.433 2.736 2.022 2.525 2.752 2.799 2.452 2.653 2.261 2.520

5. Conclusions In conclusion, the proposed design method can be safely used for the seismic design of steel EBFs and BRBFs, offering direct control of seismic deformation and damage but at the expense of heavier steel sections. Declaration of competing interest We have no conflict of interest to declare. All authors certify that have seen and approved the manuscript being submitted. We warrant that the article is the Authors’ original work. We warrant that the article has not received prior publication and is not under consideration for publication elsewhere. On behalf of all CoAuthors, the corresponding Author shall bear full responsibility for the submission.

the proposed method are 0.47sec–74.85%, 0.18sec-100%, 0.10sec-100%, 0.08sec-100% and the thus modified design spectrum is shown in Fig. 4. Inserting this modified spectrum in SAP2000 [5] and after performing modal synthesis, the design base shear is determined to be equal to 573.60 kN and the chosen sections are shown in Table 12. To evaluate both design methods for the EBF and BRBF under study, seismic inelastic time-history analyses are conducted [8], using 10 seismic motions compatible to the elastic design spectrum considered above. The seismic analyses results involve maximum values for IDR, θlink , μδ and are provided in Tables 13 and 14. Base shear values Vy that correspond to the appearance of the first plastic hinge (first yield) and, thus, are directly comparable to the base shear of seismic design, are also given in Tables 13 and 14. The average value of the seismic analyses for IDR, θlink , μδ and Vy is compared with the design values obtained by the two design methods and the limit values of Table 1 for the LS seismic performance level. It should be noted that the proposed method with deformation and damage dependent ξk, automatically satisfies the IDR, θlink, μδ limit values of Table 1, whereas the equal-displacement rule has to be applied according to the EC 8 [1] method in order to compare the design. From the results of Tables 13 and 14, one observes the following: i) the maximum IDR limit is not exceeded by the two seismic design methods. The proposed method provides a lower maximum IDR value in comparison to the EC 8 method, leading to a less economical (in terms of the steel sections used) design; ii) the average θlink is underestimated by both methods but certainly more by the EC 8 method; iii) the average μδ is conservatively estimated by both methods but certainly more by the proposed method; iv) in comparison to the seismic design base shear of EC 8, the average base shear obtained by inelastic seismic analyses is marginally higher for the case of the EBF and quite higher for the case of

Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.soildyn.2019.105947. References [1] EC 8. Design of structures for earthquake resistance - Part 1 : general rules, seismic actions and rules for buildings. Brussels: European Commitee for Standardization; 2009. [2] Papagiannopoulos GA, Beskos DE. Towards a seismic design method for plane steel frames using equivalent modal damping ratios. Soil Dyn Earthq Eng 2010;30: 1106–18. [3] Muho, E.V., Papagiannopoulos, G.A. and Beskos, D.E. Deformation dependent equivalent modal damping ratios for the performance-based seismic design of plane R/C structures, Soil Dyn Earthq Eng, doi.org/10.1016/j.soildyn.2018.08.026. [4] SEAOC Blue Book. Recommended lateral force requirements and commentary. Sacramento: Structural Engineers Association of California; 1999. [5] SAP 2000. Static and dynamic finite element analysis of structures: version 19.0. Berkeley, California: Computers and Structures; 2016. [6] Kalapodis NA, Papagiannopoulos GA, Beskos DE. Modal strength reduction factors for seismic design of plane steel braced frames. J Constr Steel Res 2018;147:549–63. [7] Kalapodis NA. Seismic design of steel plane braced frames with the use of three new methods. Ph.D. Dissertation. Greece: University of Patras; 2017. http://hdl.handle. net/10889/10598. [8] Carr AJ. RUAUMOKO 2D. Canterbury, New Zealand: University of Canterbury; 2007. [9] Bosco M, Marino EM, Rossi PP. Design of steel frames equipped with BRBs in the framework of Eurocode 8. J Constr Steel Res 2015;113:43–57.

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