Seismic design of space steel frames using advanced methods of analysis

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Soil Dynamics and Earthquake Engineering 29 (2009) 194–218 www.elsevier.com/locate/soildyn

Seismic design of space steel frames using advanced methods of analysis A.A. Vasilopoulosa, D.E. Beskosa,b, a

Department of Civil Engineering, University of Patras, GR-26500 Patras, Greece Office of Theoretical and Applied Mechanics, Academy of Athens, 4 Soranou Efessiou, GR-11527 Athens, Greece

b

Received 16 March 2006; received in revised form 3 December 2007; accepted 14 December 2007

Abstract A rational and efficient seismic design methodology for regular space steel frames using an advanced time domain finite element method of analysis that takes into account geometrical and material nonlinearities is presented. Seismic loads are applied in the form of Eurocode 8 spectrum compatible real accelerograms along the two horizontal directions of the frame. The iterative design procedure starts with assumed member sections, continues with the response checks for the damage, ultimate and service limit states and ends with the adjustment of member sizes so as all the response checks to be satisfied for all limit states. Thus, the proposed design method deals with nonlinearities and member interactions at the global level and consequently separate member capacity checks through the interaction equations of Eurocode 3 or the usage of the conservative and crude q-factor of Eurocode 8 are not required. Two numerical examples dealing with the design of (a) a space three storey steel frame with one bay in both horizontal directions and (b) a space seven storey steel frame with two and three bays along its two horizontal directions are presented to illustrate the method and demonstrate its advantages. r 2007 Elsevier Ltd. All rights reserved. Keywords: Seismic design; Space steel frames; Finite element method; Advanced analysis methods; Inelastic dynamic analysis; Eurocode 8; Eurocode 3

1. Introduction Present codes for the static design of steel building frames, such as Eurocode 3 (EC3) [1] or the American Load and Resistance Factor Design Code (LRFD) [2], are characterized by two important shortcomings. The first is that they do not consider the interaction of strength and stability between the members and the whole structure in a direct manner but in an indirect one through the effective length concept. The second and most important shortcoming is that the global analysis of the structure for determining internal member design forces is first order elastic and inelasticity is introduced indirectly through the interaction equations, which check the strength of every member separately. Use of advanced methods of analysis taking into account geometric and material nonlinearities can eliminate the above shortcomings and lead to a design procedure capable Corresponding author. Tel.: +30 2610 996559; fax: +30 2610 996579.

E-mail address: [email protected] (D.E. Beskos). 0267-7261/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.soildyn.2007.12.004

of sufficiently capturing the limit state strength and stability of a structure and its members so that separate member capacity strength checks are not necessary. Chen and Kim [3] and Kim and Chen [4] have been successful over the past 10 years or so in developing such a design procedure for plane steel frames with the aid of advanced analysis on the basis of the American LRFD [2] steel design code. This idea of designing plane steel frames with the aid of advanced analysis has been extended by the present authors from the static to the seismic case [5]. A comprehensive literature review on the subject of design (static or dynamic/seismic) of plane steel frames using advanced analysis as well as on methods of analysis taking into account geometrical and material nonlinearities can also be found in [5]. In this work, the idea of seismic design of plane steel frames using advanced analysis [5] is extended to the space steel frames case. Static design of space steel structures using advanced methods of analysis has been reported by Ziemian et al. [6] and Kim et al. [7,8]. Seismic design of space reinforced concrete frames using advanced analysis

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has been proposed by Kappos and Panagopoulos [9]. They use both static-pushover and dynamic inelastic analyses. In both Refs. [5,9] as well as in the present work, use of advanced methods of analysis for seismic design of frameworks is combined with the idea of performancebased design [10]. Modern seismic codes, such as the American UBC [11] and Eurocode 8 (EC8) [12], in addition to the standard static or spectral dynamic elastic analyses, permit the use of inelastic time history analysis for design purposes, while the NEHRP [13] guidelines provide details for its practical application. A literature survey shows that, even though there are some works devoted to the seismic design of plane steel frames using advanced analysis which are mentioned in [5], there are no such works dealing with the case of space steel frames. The present work comes to fill this literature gap. When one deals with steel frames he has to consider both material nonlinearities (inelasticity, residual stresses) and geometric nonlinearities (Pd and PD effects, imperfections). All these phenomena have already been considered in connection with the static design of steel frames done with the aid of advanced methods of analysis [3,4,6–8]. These are extended here to the dynamic (seismic) design case on the basis of information already existing in the literature and dealing with dynamic inelastic analysis of space steel frames undergoing small or large deformations. Thus, one can mention here the following works dealing with static or dynamic analysis of space frames mainly made of steel: those on static geometrically nonlinear elastic analysis [14–19], those on static elastoplastic analysis [20–23], those on static geometrically and materially nonlinear analysis [24–26], those on dynamic geometrically and materially nonlinear analysis [27,28] and those on seismic nonlinear analysis [29–31]. Summarizing, the present work develops a seismic design procedure for regular space steel moment resisting frames, which is accomplished with the aid of advanced methods of analysis (dynamic inelastic analysis including geometric nonlinearities) in conjunction with three limit states of performance: life safety (LS) or damage limit state (DLS), collapse prevention (CP) or ultimate limit state (ULS) and immediate occupancy (IO) or serviceability limit state (SLS). Thus, an effort is made to develop a design procedure that captures, in a practical and simple way, all aspects of structural behavior in the framework of Eurocodes (EC3) [1] and (EC8) [12] by appropriately modifying the computer program DRAIN-3DX [32]. Thus, the main limitations of EC3 [1] and EC8 [12] codes, i.e., those mentioned at the beginning of this introduction and the employment of the crude behavior factor q, respectively, are eliminated with the use of the proposed method of seismic design. One can think of this work as an extension of the previous work of the authors [5] on seismic design of steel frames using advanced analysis from the case of plane frames to that of regular space frames. This extension

195

involves seismic loads acting along two orthogonal directions with one or two different magnitude combinations. In regular buildings there is no any real eccentricity between the stiffness center and the center of mass at every floor and hence no torsional effects on the response are developed. However, in conformity with current code provisions (e.g., EC8 [12], UBC [11]), accidental eccentricity equal to 5% of the corresponding plan dimension at each floor is taken into account in the present work. Furthermore, the concept of capacity design (strong columns—weak beams) is also adopted in the proposed design method. 2. Suggested seismic analysis and design procedure In the following, the most important aspects of the proposed seismic design of steel structures based on advanced methods of analysis are briefly described. 2.1. Selection of the earthquake loading and its application The seismic loading considered here is taken to be in the form of at least three actual seismic accelerograms appropriately modified so that for each one of them their elastic spectrum to be compatible with the elastic spectra corresponding to the three performance levels of DLS, ULS and SLS considered here. These elastic spectra are obtained from the elastic spectrum of EC8 [12] as described in [5]. The aforementioned spectrum compatible seismic motions are applied onto the space frames either by following the well known design combinations x+0.3z and z+0.3x, where x and z define the two horizontal directions and y the vertical one of the building, in accordance with DRAIN-3DX [32], or by using directly their two horizontal components in turn along the x and z directions. Due to the regularity of the building, only accidental eccentricities are taken into account by displacing the mass center at each floor from its nominal location by 70.05Li in each horizontal direction, where Li is the floor dimension perpendicular to the direction of the seismic motion. 2.2. Inelastic modeling of members A simple and efficient approach for representing inelasticity in steel frames is the refined elastic–plastic hinge method [3–5], which assumes that the plastic hinge stiffness degradation is gradual. In DRAIN-3DX [32] analysis program, which is adopted in this work, all the structural members are modeled using the fiber beamcolumn element E15. This element employs the concentrated plasticity concept in conjunction with the fiber model at every plastic hinge. Thus, the material hardening, the axial force-bending interaction and the resulting gradual section stiffness degradation are taken into account accurately as in the refined elastic–plastic hinge model. The only modification implemented in

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DRAIN-3DX [32], is the tangent modulus concept to capture the residual stress effect along the member [5]. The degradation of the flexural strength and stiffness caused by lateral torsional buckling can also be taken into account in this modeling in an approximate manner by following [33] as explained in the next section. According to EC3 [1], when plastic analyses are employed, member cross sections should be capable of developing the full plastic moment capacity and sustaining large hinge rotations before the onset of local buckling. Only class 1 sections (EC3 [1]) satisfy these requirements and thus, only these sections are used in this work. 2.3. Nonlinear geometric effects

where ssd is the design stress, spl is the stress at first yield, su is the ultimate normal stress and / S denote the McCauley bruckets. (2) Storey damage index Ids, which can be obtained by [37] , nb nb X X I ds ¼ I 2dm I dm , (3) 1

where nb is the number of frame bays and Idm are defined in Eq. (1). (3) Global damage index Idg, which is a damage index of the whole structure and can be obtained as [37] , m m X X 2 I dg ¼ I dm I dm . (4) i¼1

Geometric nonlinearities in usual steel frame analysis include Pd and PD effects as well as imperfection effects. DRAIN-3DX [32] accounts for the Pd effect in an approximate manner through the geometric stiffness coefficients. It has been found by Beskos [34] that, in general, two finite elements per member can provide satisfactory results. Incidentally, two finite elements per member are also satisfactory with respect to the lumping of the mass modeling [35]. DRAIN-3DX [32] takes this PD effect into account approximately in a tall building by adding to the elastic stiffness of the columns a geometric stiffness based on the axial force in the columns under gravity load alone, as explained in [36]. In this work imperfections are taken into account by the simple, yet effective, approach described in Chen and Kim [3], which simply reduces the tangent modulus by a reduction factor of 0.85. 2.4. Seismic damage index It is well known that damage prediction of a structure under seismic motion is very valuable in assessing its strength [37–39]. The quantification of damage is accomplished by the use of the damage index, which provides a comprehensive design measure in the framework of the SLS, DLS and the ULS of the structure. The damage index can be determined at member level, storey level or global structural level. All the necessary modifications, concerning the damage estimation have been implemented in the analysis software DRAIN-3DX [32]. The following types of damage indices are considered here: (1) Member damage index Idm, which is calculated at a plastic hinge consisting of nf fibers and has the form [38] , nf nf X X I dm ¼ I 2df I df , (1) 1

1

where Idf is the fiber damage index defined as [38] I df ¼ hssd  spl i=ðsu  spl Þ,

(2)

1

i¼1

3. Advanced analysis implementation The matrix equation of equilibrium for a beam-column with its two potential elastic–plastic hinges at its ends connects the load vector {F} of nodal bending moments and axial and shear forces with the deformation vector {d} of nodal displacements and rotations as fF g ¼ ð½K f   N½K g Þfdg,

(5)

where [Kf] is the flexural stiffness matrix, [Kg] the geometric stiffness matrix and N the axial load with + for tension and for compression. Matrix [Kg] is a function of only the length L, while [Kf] a function of the moment of inertias Iy, Iz, length L, tangent modulus Et, shear modulus G, torsional modulus J and two scalar parameters n at the two ends of the element allowing for gradual inelastic stiffness reduction there. The tangent modulus Et is used to account for gradual yielding effects due to residual stresses along the length of members under axial loads between two plastic hinges and

Table 1 Allowable response limits for DLS (PGA ¼ 0.30 g) 1. Relative storey drifts should be p1.5% of storey height h 2. Damage indices D at member, storey and global level should be p20% 3. Plastic rotations ypl at member ends should be p6yy, where yy is the rotation at first yielding, which equals WplfyLb/6EIb and WplfyLc(1N/Ny)/ 6EIc for beams (b) and columns (c), respectively 4. Plastic hinge formation only in beams (capacity design)

Table 2 Allowable response limits for ULS (PGA ¼ 0.45 g) 1. Relative storey drifts should be p3.0% of storey height h 2. Damage indices D at member, storey and total level should be p50% 3. Plastic rotations ypl at member ends should be p8yy, where yy has been defined in Table 1 4. Plastic hinge formation in beams and columns without collapse

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Fig. 1. Geometry, finite element numbering and section selection (frame A) of the three storey frame.

2

3

Storey Level

Storey Level

3

Bingol Friuli LomaP

1

1 0.01

0.02

0.03

0.04

0. 00 2 0. 00 3 0. 00 4 0. 00 5 0. 00 6 0. 00 7 0. 00 8 0. 00 9 0. 01 0 0. 01 1 0. 01 2

0.00

2

Bingol Friuli LomaP

X-Drifts (m)

Z-Drifts Frame A Fig. 2. Seismic storey drifts (x+0.3z/DLS).

has the form [3] E t ¼ 1:0E

  E t ¼ 4 NNsdy E 1  NNy

for N sd p0:5N y ; for N sd 40:5N y ;

(6)

where Ny ¼ Afy is the axial load at yield elastic modulus. A further reduction of modulus by multiplying Et by 0.85 is made for imperfections in a very simple, yet manner [3].

and E the the elastic to account satisfactory

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14.00 12.00

12.00

10.00 Total Damage (%)

Bingol Friuli LomaP

8.00 6.00 4.00 2.00 0.00 -2.00 -500

8.00 6.00

Impo Parko

4.00 2.00 0.00

0

500 1000 1500 2000 2500 3000 3500 4000 4500

0

500

1000

1500

2000

Time step (0.01sec)

Time step (0.01sec)

Frame A

Frame D

14.00 12.00 10.00 Total Damage

Total Damage

10.00

8.00 6.00 Bingol Friuli LomaP

4.00 2.00 0.00 -2.00 -500

0

500 1000 1500 2000 2500 3000 3500 4000 4500 Time step (0.01sec) Frame D

Fig. 3. Seismic total damage index (x+0.3z/DLS).

Fig. 4. Plastic hinge formation from seismic excitations (x+0.3z/DLS).

2500

3000

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Fig. 5. Plastic hinge formation from seismic excitations (x+0.3z/DLS).

3

Bingol Friuli LomaP

2

Storey Level

Storey Level

3

1

Impo Parko

2

1 0.015

0.020

0.025

0.030

0.035

0.040

0.015

0.020

X-Drifts (m)

0.030

0.035

0.040

3

3 Bingol Friuli LomaP

Storey Level

Storey Level

0.025

X-Drifts (m)

2

Impo Parko

2

1

1 0.006

0.008

0.010

0.012

0.014

0.006

0.008

0.010

0.012

Z-Drifts

Z-Drifts

Frame D

Frame D

Fig. 6. Seismic storey drifts (x+0.3z/DLS).

0.014

0.016

0.018

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3

Storey Level

Storey Level

3

2

2

Impo Parko

Bingol Friuli LomaP

1

1

0.000

0.005

0.010

0.015

0.010

0.012

X-Drifts (m)

0.018

0.020

X-Drifts (m)

3

3

2

Storey Level

Storey Level

0.016

0.014

Bingol Friuli LomaP

2

Impo Parko

1

1

0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050

0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.045 0.050

Z-Drifts

Z-Drifts Frame D Fig. 7. Seismic storey drifts (z+0.3x/DLS).

10.00

10.00 8.00

6.00 4.00 Bingol Friuli LomaP

2.00 0.00 -2.00 -500

Total Damage (%)

Total Damage (%)

8.00

6.00 4.00 Impo Parko

2.00 0.00 -2.00

0

0

500 1000 1500 2000 2500 3000 3500 4000 4500 Time step (0.01sec)

500

1000

1500

2000

Time (0.01sec)

Frame D Fig. 8. Seismic total damage index (z+0.3x/DLS).

2500

3000

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The two scalar parameters n at the two ends of the element can be computed in accordance with the formula [3] n¼1

for ap0:5;

(7)

n ¼ 4að1  aÞ for a40:5;

where a is the force-state parameter allowing for the gradual inelastic stiffness reduction at the two element ends (potential plastic hinge positions) due to possible large design shear forces and/or nonnegligible lateral torsional buckling effects. Thus, with the aid of EC3 [1], one can express a in the form a¼

N sd M ysd M zsd þ þ p1:0. N pl:Rd M ply:Rd M plz:Rd

(8)

201

In the above, Nsd is the design axial force, My,sd and Mz,sd are the design bending moments with respect to the y (strong) and z (weak) axes of the member section, respectively, N pl:Rd ¼ Af y =gMo is the axial resistance with A, fy and gMo ¼ 1:1 being the cross sectional area, the yield strength of the steel and the safety factor, respectively, and M ply:Rd and M plz:Rd denote the plastic moment resistances with respect to the y and z axes, respectively, of the member. The moment resistances MplyRd ¼ Wplyfy/gM1 and MplzRd ¼ Wplzfy/gM1 with Wply and Wplz being the plastic section moduli with respect to the y and z axes, respectively, and gM1 ¼ 1.1 the safety factor are appropriately reduced when design shear forces are greater than 50% of the shear force resistance (EC3 [1]). The moment

Fig. 9. Plastic hinge formation from seismic excitations (z+0.3x/DLS).

2

3 Bingol Friuli Impo LomP Parko

Storey Level

Storey Level

3

1 0.030

Bingol Friuli Impo LomP Parko

2

1 0.035

0.040

0.045

0.050

0.055

0.018

0.020

X-Drifts (m) Fig. 10. Seismic storey drifts for frame D (x+0.3z/ULS).

0.022 Z-Drifts (m)

0.024

0.026

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resistance Mply.Rd is also appropriately reduced when the lateral torsional buckling effect due to possible inadequate lateral bracing of the members has to be taken into account. In that case, the above moment resistance in Eq. (8) is replaced by the reduced moment resistance MbyRd, which is computed according to EC3 [1] by M byRd ¼ wLT W ply f y =gM1 ,

(9)

where wLT is the reduction factor for lateral torsional buckling given in EC3 [1] in terms of the corresponding dimensionless slenderness l¯ LT ¼ lLT =pðE=f y Þ0:5 . The slenderness lLT for lateral torsional buckling is also given in EC3 [1] in terms of the geometrical and material properties of the member [1]. According to EC3 [1], when l¯ LT p0:4, the lateral torsional buckling effect can be neglected and Eq. (8) can be used only when the shear force exceeds 50% of the shear force resistance. If both the high shear and the lateral torsional buckling effects are negligible, Eqs. (7) and 18.00

4. Seismic design procedure via advanced analysis

16.00 14.00 Total Damage (%)

(8) are omitted. The above simple, code-based method for taking into account lateral torsional buckling is based on [33]. A better way, consistent with the advanced analysis philosophy, would be to add one more degree of freedom (the warping one) per node in the beam-column element used here in accordance with [40]. However, this would require major changes in DRAIN-3DX [32]. As it was mentioned in the introduction, the proposed seismic design method is associated with three performance levels or limit states: the DLS, ULS and SLS. The performance objectives or allowable limits of the limit states DLS and ULS refer to the seismic response of the structure and are listed in Tables 1 and 2, respectively, with the PGA standing for peak ground acceleration. Only, relative storey drift limits are associated with the SLS (PGA=0.20 g) and these are equal to 0.5% h, where h is the storey height.

12.00 Bingol Friuli Impo LomP Parko

10.00 8.00 6.00 4.00 2.00 0.00 -2.00 0

500

1000

1500

2000

2500

3000

Time step (0.01sec) Fig. 11. Seismic total damage index for frame D (x+0.3z/ULS).

On the basis of the preceding discussion, one can establish a seismic design procedure for regular space steel moment resisting frames consisting of the following steps: Step 1: Types of loads and design load combinations. These are established according to EC3 [1] and EC1 [41]. Step 2: Seismic load selection. This is based on the discussion presented in Section 2.1 for the SLS, DLS and ULS cases. Step 3: Preliminary member sizing. This depends on the experience of the designer or some simplified analysis. For example, beam sections are usually selected by assuming that beams are simply supported and subjected to gravity loads only and column sections on the basis of the overall drift requirements rather than the tedious strength checks of individual columns. The capacity demand design concept (strong columns—weak beams) should be applied in the preliminary member sizing.

Fig. 12. Seismic plastic hinge formation for frame D (x+0.3z/ULS).

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3

Bingol Friuli Impo LomP Parko

2

Storey Level

Storey Level

3

203

1

Bingol Friuli Impo LomP Parko

2

1 0.012

0.014

0.016

0.018

0.020

0.022

0.030

0.035

0.040

X-Drifts (m)

0.045

0.050

0.055

0.060

Z-Drifts (m) Fig. 13. Seismic storey drifts for frame D (z+0.3x/ULS).

Table 3 Member section selections for the three storey space steel frame Space frame

Member sections

A B C D

HEB280/240/220(columns)–IPE300/270/240(beams) HEB260/240/220(columns)–IPE270/240(beams) HEB240/220/200(columns)–IPE270/240(beams) HEB220/200(columns)–IPE270/240(beams)

Table 5 Member damage indices from seismic excitations (x+0.3z/DLS): only max values are shown Member

Table 4 Member damage indices from seismic excitations (x+0.3z/DLS): only max values are shown Member

Damage start

Three storey frame A Bingol 31 2413 32 2414

i-End damage (%)

j-End damage (%)

11.80 10.60

682

1.30

LomaP 31 32

691 692

3.10 1.49

Step 4: Time history analysis execution for the DLS. Fully nonlinear (involving both material and geometric nonlinearities) time history analyses are executed using three moderate intensity seismic records as described in step 2. Every seismic component pair should be applied twice. One time with its strong component parallel to the strong building axis and a second time with this component perpendicular to that axis. Columns and beams are modeled by two finite elements as explained in

i-End damage (%)

Three storey frame D Bingol 12 2390 15 2737

11.82

Friuli 12 15

595 560

11.83

LomaP 15 16

576 579

11.83 7.67

j-End damage (%)

11.88

11.92

Table 6 Member damage indices from seismic excitations (z+0.3x/DLS): only max values are shown Member

Friuli 31

Damage start

Damage start

Three storey frame D Bingol 1 2628 4 2438 Friuli 4 10 LomaP 13 29

i-End damage (%)

j-End damage (%)

3.00 3.00

389 454

4.54

1458 1472

11.90 7.03

6.46

Section 2.3. Nodal points should be provided where concentrated loads are applied. Uniform loads should be converted into equivalent concentrated loads.

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Step 5: Satisfaction of the DLS criteria. The design criteria of Table 1 should be checked for this performance level to see if they are satisfied or not. Step 6: Adjustment of member sizes for the DLS. If the response values of the analysis are much lower than the allowable values of Table 1 indicating an overdesigned structure, the few members where plastic hinges have been formed remain the same, whereas the members without plastic hinges are replaced by lighter ones. In the extreme case where no plastic hinges have been formed in any of the members, all the members are replaced by lighter ones. In the opposite case, when all or some of the response values of the analysis exceed the allowable values of Table 1 indicating an inadequately designed structure, the few possible members without plastic hinges remain the same, whereas the members containing the first plastic hinges should be replaced with stronger ones. After the first replacement (adjustment) in both aforementioned cases, a new dynamic time history analysis is performed and its response values are again checked to see if they satisfy or not the allowable limits of Table 1. Many such analyses may be necessary to be done in an iterative manner in order to achieve, by proper member adjustments, response values which are close to the allowable limits but do not exceed them. Step 7: Verification for the DLS. Using two more seismic records appropriately scaled for the DLS, as in step 2, additional fully nonlinear time histories analyses are performed, in order to check whether or not, the designed

frame (step 6) conforms again to the DLS design objectives. Step 8: Time history analysis execution for the ULS. At least three seismic records appropriately scaled as in step 2 for the ULS are used for fully nonlinear time history analyses following step 4. Step 9: Satisfaction of the ULS criteria. The design criteria of Table 2 should be checked for this performance level to see if they are satisfied or not.

Table 7 Member damage indices from seismic excitations (x+0.3z/DLS): only max values are shown

Table 9 Member damage indices from seismic excitations (x+0.3z/ULS): only max values are shown

Member

Damage start

i-End damage

j-End damage (%)

16.00 14.00

Total Damage (%)

12.00 10.00 Bingol Friuli Impo LomP Parko

8.00 6.00 4.00 2.00 0.00 -2.00 0

500

Member

Damage start

11.80 10.10

Parko 12 28

11.90 11.90

LomaP 15 16

Member

Damage start

Three storey frame D Impo 1 348 10 439 Parko 10 13

507 559

i-End damage (%)

j-End damage (%)

5.13 10.00 5.40 8.44

2000

2500

3000

Fig. 14. Seismic total damage index for frame D (z+0.3x/ULS).

Three storey frame D Bingol 12 2385 28 2385

Table 8 Member damage indices from seismic excitations (z+0.3x/DLS): only max values are shown

1500

Time step (0.01sec)

Three storey frame D Impo 12 574 28 577 494 502

1000

566 567

i-End damage (%)

j-End damage (%)

16.41 16.21 21.25 21.98

Table 10 Member damage indices from seismic excitations (z+0.3x/ULS): only max values are shown Member

Damage start

i-End damage (%)

Three storey frame D Bingol 10 2706 26 2396 LomaP 13 14

1447 1450

j-End damage (%)

15.65 15.48 23.20 21.44

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Fig. 15. Seismic plastic hinge formation for frame D (z+0.3x/ULS).

Table 11 Design/strength (capacity) ratios of beams and columns of frame D according to EC3/EC8 Member Section strength

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Member strength

Biaxial bending+shear+axial force

Flexural buckling

Lateral buckling

0.173 0.173 0.173 0.122 0.122 0.122 0.116 0.116 0.116 0.006 0.167 0.005 0.144 0.010 0.067

0.383 0.383 0.383 0.219 0.219 0.219 0.183 0.183 0.183 0.013 0.858 0.014 0.794 0.018 0.256

0.402 0.402 0.402 0.229 0.229 0.229 0.195 0.195 0.195

Fig. 16. Member numbering of three storey space frame designed by EC8/ EC3.

Step 10: Time history analysis execution for the SLS and satisfaction of its criteria. The five seismic motions used for the DLS and ULS appropriately scaled for the SLS are used for time history response analysis and the relative storey drifts are checked to see if they satisfy or not the allowable value of 0.5% h, where h is the storey height. It is expected that, structures optimally designed for DLS peak ground acceleration can successfully survive motions with ULS peak ground accelerations [42].

5. Application of the method Two numerical examples dealing with the seismic design of moment resisting space steel frames are presented in this

section in order to illustrate the proposed design method and demonstrate its advantages. 5.1. Three storey-one bay space steel frame A three storey-one bay space steel frame, which has been seismically designed according to EC8 [12] and experimentally verified with respect to that design by Kakaliagos [43] is considered first. Fig. 1a shows the geometry of the structure and its finite element modeling. The Kakaliagos [43] frame has HEB400 columns, IPE400 beams along the z direction for all stories, IPE500 beams along the x direction for the first storey and IPE450 beams along the x direction for the upper two stories. The first storey height equals

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3.40 m, while those of the next two upper storeys equal 3.225 m. The bay opening in both horizondal directions x and z equals to 5.00 m. The beam and column sections of the building are IPE and HEB, respectively, while the steel grade is S275. The modulus of elasticity E and the shear modulus G are equal to 205 and 85.4 GPa, respectively, while the strain hardening equals to 3.00%. The soil class and the damping ratio are assumed to be B and x ¼ 0.05, respectively. Accidental eccentricities are taken into account as described in Section 2.2. The gravity weight gs of structural members (beam-columns) is 78.50 kN/m3, while the concrete slab and secondary beams self weigh Gs is 6.25 kN/m2 and 5.57 kN/m2 for the first storey and the two upper storeys, respectively. The live loads Q for the first storey and the two upper storeys are 2.00 and 1.50 kN/m2, respectively. According to EC8 [12] provisions, the effective seismic mass is defined through the combination Gtot+0.30Q and it is assumed to be concentrated at the mass center of each floor. The diaphragm modeling of each floor is achieved by permitting two horizondal translational and one torsional components of motion with respect to the x, z and y axes, respectively.

Totally, five natural time history accelerograms taken from [44] are used for time history nonlinear analyses (Bingol, Friuli, Loma Pietra, Imperial Valley and Parkfield with PGA equal to 0.515, 0.375, 0.644, 0.315 and 0.476 g, respectively). The first three are used for the member adjustment procedure, while the second two for the verification phase. These seismic records have been made

Table 12 Member section selection for the seven storey space steel frame Space frame

Member sections

A B

HEB300/280/240(columns)–IPE360/330/300/140(beams) HEB300/280/240(columns)–IPE360/330/300/270/240/ 220(beams) HEB320/300/280(columns)–IPE360/330/300/270/240/ 220(beams) HEB320/300/280(columns)–IPE400/360/330/140(beams) HEB360/340/320/300/280/240(columns)–IPE450/360/ 140(beams) HEB320/300/280(columns)–IPE400/360/330/200/140(beams)

C D E F

Fig. 17. Geometry and member numbering of the seven storey space steel frame.

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frame, which come from the proposed iterative design procedure and represent five different structures from the design point of view. In that table, slashes between numbers separate member sizes for the three stories (from first to third). For the case of only two numbers, the first corresponds to the first two storeys, while the second to the third (top) storey. By looking at Table 3, one can observe that the Kakaliagos [43] frame is overdesigned as its sections are much heavier than those of any of the frames A–D. Consider first frame A (Fig. 1) under the x+0.3z loading combination involving the Bingol, Friuli and Loma Prieta earthquakes. Fig. 2 shows the relative storey drifts along the x and z directions. It is observed that the maximum drift values (2.64 and 2.90 cm for the x direction due to Bingol motion) do not exceed the allowable limits 1.5%  h ¼ 1.5%  340 cm ¼ 5.10 cm and 1.5%  322.5 cm ¼ 4.90 cm, which correspond to the DLS.

7

7

6

6

5

5

Storey Level

Storey Level

compatible to the elastic design spectrum of EC8 [12] for soil class B and PGA ¼ 0.30 g, which is well suited for the DLS. The scaling procedure has been performed using the SRP software [45]. Additionally, for confirmation purposes, the same five natural time history accelerograms are made compatible to the elastic design spectrum corresponding to the ULS as in [5] and are used for new seismic analyses. The damping in all dynamic analyses is equal to 5% for the first two modes and becomes higher for the rest of the modes following the Rayleigh damping rule [46]. The results of the suggested iterative design procedure for the space steel frame of Fig. 1 have been plotted in Figs. 2–13. According to EC8 [12] provisions, two combinations, x+0.3z and z+0.3x, concerning the imposed seismic loading are examined. The x and z horizontal directions of the space frame correspond to the strong and weak axes of the building, respectively. Table 2 includes five alternative member section selections of the space

4 Bingol Friuli LomaP

3

4 3

2

2

1

1

0.000

0.005

0.010

0.015

0.020

207

Bingol Friouli LomaP

0.025

0.000

0.002

0.004

X-Drifts (m)

0.006

0.008

0.010

0.012

0.014

Z-Drifts (m) Frame A Fig. 18. Seismic storey drifts (x+0.3z/DLS).

14.00

7 Bingol Friuli LomP

10.00 8.00 6.00 4.00

5 4 3 2

2.00

1

0.00 -500

Bingol Friuli LomaP

6

Storey Level

Total Damage (%)

12.00

0

500 1000 1500 2000 2500 3000 3500 4000 4500 Time step (0.01sec)

-2

0

2

4

6

8

10

Storey Damage (%)

Fig. 19. Total and storey damage indices for frame A (x+0.3z/DLS).

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14

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The maximum values of the total damage indices (shown in Fig. 3) are equal to DBingol ¼ 11.84%, DFriuli ¼ 1.31% and DLomP ¼ 3.11%, respectively, while the maximum storey damage indices (not shown here) are caused by the Bingol motion and are equal to ds1 ¼ 0.00%, ds2 ¼ 11.84% and ds3 ¼ 0.0% for the first, second and third storeys, respectively. Finally, Table 4 presents maximum values of member damage indices for each seismic case. It is observed that all three kinds of maximum damage indices do not exceed the 20% damage limit for the DLS indicating an insignificant or small damage. Here and throughout the paper numbers under the title ‘‘Damage starting’’ in tables dealing with member damage indices are in 102 s. Plastic hinges are developed only in one beam of the second storey, as shown in Fig. 4. The maximum base shears for the worst seismic case (Loma Pietra) were found to be equal to Vx ¼ 222.81 kN and Vz ¼ 57.24 kN. Analogous results were found for the z+0.3x loading combination [47]. According to all the above results, frame Table 13 Member damage indices from seismic excitations (x+0.3z/DLS): only max values are shown i-End damage (%)

j-End damage (%)

Seven storey frame A Bingol 13 2440/2442 42 2440/2441

15.28 15.33

11.96 13.75

Friuli 13 42

408/410 408/410

11.65 12.61

6.69 10.43

1483/1489 1481/1490

9.25 7.24

3.90 9.53

Member

LomP 42 71

Damage start

A seems to have been designed very conservatively and almost all its members may be replaced by lighter ones. Due to space limitations, the study of the intermediate section selections (frames B and C of Table 2) is omitted here and only the final selection of member sections (frame D of Table 2) is presented and discussed. For more details one can look at [47]. Consider frame D under the x+0.3z loading combination involving the Bingol, Friuli and Loma Prieta earthquakes. For this frame storey drifts (Fig. 6), total damage indices (Fig. 3) and member damage indices (Table 5) do not exceed the DLS limits and plastic hinges are developed only in beams (Fig. 5). It is also observed that frame D complies with the DLS design objectives under the z+0.3x load combination. More specifically, storey drifts (Fig. 7), total damage indices (Fig. 8) and member damage indices (Table 6) do not exceed the DLS limits, while plastic hinge formation takes place only in beams following the capacity design criteria (Fig. 9). A further confirmation, under both load combinations x+0.3z and z+0.3x, of the finally selected member sections (frame D) is done by using two additional scaled seismic accelerograms, namely those of Imperial Valley and Parkfield. Figs. 3 and 6–8 and Tables 7 and 8 exhibit storey drifts, total damage indices and member damage indices. In all cases the response magnitudes are within the allowable design limits of the DLS. It was also found that the maximum ratio ypl/yy in the members of frame D was 3.79, 3.86 and 3.86 for the Bingol, Friuli and Loma Prieta seismic motions, respectively, i.e., smaller than the DLS allowable value of 6. Concerning base shear forces Vx and Vz along the x and z directions, respectively, it was found that for all five motions considered here, these shears have maximum values very close for both load combinations. Indeed, 210 kNpV x p245 kN and 65 kNp V z p73 kN for the x+0.3z load combination and

Fig. 20. Plastic hinge formation in frame A from seismic excitations (x+0.3z/DLS).

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intense seismic excitations, frame D develops repairable damage. The member damage indices for both seismic combinations are presented in Tables 9 and 10 with values well below the limit of 50.00%. Concerning the plastic hinge formation, the capacity design criterion is not satisfied in very few cases where hinges develop not only in beams but also in columns (Figs. 12 and 15), but this is acceptable as long as there is no collapse. For comparison purposes, the commercial design software SAP 2000 [48] based on EC8 [12] and EC3 [1] codes is used for the seismic analysis and design of the frames considered here. Thus, first order elastic spectrum analyses using the EC8 acceleration design spectrum for soil class B, PGA ¼ 0.30 g, q ¼ 4 and damping 5% are performed for all the frames of Table 3 until the best solution (frame D) is again found. Table 11 provides the values of the

7

7

6

6

5

5

Storey Level

Storey Level

100 kNpV x p113 kN and 150 kNpV z p175 kN for the z+0.3x load combination. From the above, it is apparent that the section selection corresponding to frame D represents the best solution for the DLS since all the design criteria (performance objectives) of this limit state are satisfied. It was also found that frame D satisfies the drift criteria of the SLS. An additional verification is made by performing fully nonlinear time history analyses of frame D using seismic records compatible with the design spectrum of the ULS. Thus, the storey drifts in Figs. 10 and 13 do not exceed the allowable ULS limits of 3%  h=3%  3.225=9.675 cm and 3%  3.40=10.20 cm for both seismic combinations x+0.3z and z+0.3x. The total damage indices for both seismic combinations (Figs. 11 and 14) approach from below the value of 20%, meaning that even in highly

4 Bingol Friuli LomaP

3

3 2

1

1 0.002

0.004

0.006

0.008

0.010

0.012

Bingol Friouli LomaP

4

2

0.000

209

0.01

0.00

0.02

X-Drifts (m)

0.03

0.04

0.05

Z-Drifts (m) Frame A Fig. 21. Seismic storey drifts (z+0.3x/DLS).

14.00 7

Bingol Friuli LomP

8.00 6.00 4.00

5 4 3

2.00

2

0.00

1

-500

Bingol Friuli LomaP

6

10.00 Storey Level

Total Damage (%)

12.00

0

500 1000 1500 2000 2500 3000 3500 4000 4500 Time step (0.01sec)

0

2

4

6

8

Storey Damage (%)

Fig. 22. Total and storey damage indices for frame A (z+0.3x/DLS).

10

12

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210

Table 14 Member damage indices from seismic excitations (z+0.3x/DLS):Only max values are shown i-End damage (%)

j-End damage (%)

Seven storey frame A Bingol 132 2462/2464 142 2391/2393

11.85 10.28

8.86 11.88

Friuli 161 164

412/413 417/419

11.95 12.02

11.87 12.02

976/982 1157/1157

11.91 11.89

11.80 11.90

Member

LomP 132 164

Damage start

design/strength capacity ratios of EC3 [1] under various states of deformation corresponding to members of frame D with numbering that shown in Fig. 16. It is observed that frame D satisfies all the code-based design requirements as the capacity ratios approach 1 without exceeding it. Frame D was finally analyzed for the seismic motions and the limit states considered here by taking the two scaled horizontal components of every motion and its response values were found to be slightly higher than those for the x+0.3z and z+0.3x load combination rule case without, however, leading to heavier sections than those of frame D, which is in agreement with the code-based design. Thus, it was concluded that the x+0.3z and z+0.3x combination rule was an acceptable way of considering seismic loading. Reyes-Salazar et al. [49] made a comparison between the seismic components method (exact) and the 30% load

7

7

6

6

5

5

4 3

Storey Level

Storey Level

Fig. 23. Plastic hinge formation in frame A from seismic excitations (z+0.3x/DLS).

Bingol Friuli LomP

4 3

2

2

1

1 0.006 0.008 0.010 0.012 0.014 0.016 0.018 0.020 0.022

Bingol Friuli LomP

0.000

0.002

X-Drifts (m)

0.004

0.006

Z-Drifts (m) Frame F Fig. 24. Seismic storey drifts (x+0.3z/DLS).

0.008

0.010

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combination rule and found out that, even though base shear values as computed by the 30% rule exhibit a very small error, axial loads of columns are underestimated by this rule by as much as 15%. However, their study does not assume diaphragm action as the present one and applies the two actions of the 30% rule separately and not simultaneously as in here. This indicates that more work is needed on this subject in order to reach definite conclusions. 5.2. Seven storey space steel frame with two and three bays along its two horizontal directions The geometry of the structure and the numbering of its members are shown in Fig. 17. The x and z horizontal directions correspond to the weak and strong structural axes, respectively. The first storey height equals to 3.60 m, while the next six upper storey heights are equal to 3.20 m.

The bay openings in horizontal directions x and z are equal to 5.00 and 4.00 m, respectively. The beam and column sections of the building are IPE and HEB with S275 and S355 grade steel, respectively. The modulus of elasticity E and the shear modulus G are equal to 205 and 85.4 GPa, respectively, and the strain hardening equals to 3.00%. The gravity weight gs of structural members (beam-columns) is 78.50 kN/m3, while the concrete slab, the interior and exterior walls and secondary beams self weigh Gs is 9.00 kN/m2. The live loads Q are 2.00 kN/m2. According to EC8 [12], the effective seismic mass is defined through the combination Gtot+0.3Q and is implemented in the mass center of each floor. The soil class is B and the damping ratio x ¼ 0.05.The diaphragm behavior of each floor and the accidental eccentricity follow the modelization of the previous example. Table 12 includes six alternative member section selections for the space frame considered, which come from the

14.00

7 Bingol Friuli LomP

10.00 8.00 6.00 4.00

5 4 3

2.00

2

0.00

1

-500

0

Bingol Friuli LomaP

6 Storey Level

Total Damage (%)

12.00

500 1000 1500 2000 2500 3000 3500 4000 4500 Time step (0.01sec)

211

-2

0

2

4

6

8

Storey Damage (%)

Fig. 25. Total and storey damage indices for frame F (x+0.3z/DLS).

Fig. 26. Plastic hinge formation in frame F from seismic excitations (x+0.3z/DLS).

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12

14

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Table 15 Member damage indices from seismic excitations (x+0.3z/DLS): only max values are shown Member

i-End damage (%)

Damage start

j-End damage (%)

Seven storey frame F Bingol 0 Friuli 13 42

408 411

6.82 1.93

1469/1476 1471

11.80 10.90

5.05

7

7

6

6

5

5 Storey Level

Storey Level

LomP 42 71

0

iterative design procedure and represent six different structures from the design point of view. Frame A has HEB300, HEB280 and HEB240 columns and IPE360, IPE330 and IPE300 beams along the weak axis at the first three, intermediate two and upper two storeys, respectively and IPE140 beams along its height with respect to the strong axis. Frame B has the same column and weak axis beam arrangement with frame A, while its strong axis beams are IPE270, IPE240 and IPE220 at the first three, intermediate two and upper two storeys, respectively. Frame C has the same beams with frame B and HEB320, HEB300 and HEB280 columns at the first three, intermediate two and upper two storeys, respectively. Frame D has the same columns with frame C, weak axis beams of IPE400, IPE360, IPE330 and strong axis beams of IPE140,

4 Bingol Friuli LomP

3

4

2

2

1

1 0.002

0.003

0.004

0.005

Bingol Friuli LomP

3

0.006

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040

X-Drifts (m)

Z-Drifts (m) Frame F Fig. 27. Seismic storey drifts (z+0.3x/DLS).

16.00

7 6

12.00 10.00

Bingol Friuli LomP

8.00 6.00

Storey Level

Total Damage (%)

14.00

5 4 Bingol Friuli LomaP

3

4.00 2

2.00

1

0.00 -500

0

500 1000 1500 2000 2500 3000 3500 4000 4500 Time step (0.01sec)

0

2

4

6

8

10

Storey Damage (%)

Fig. 28. Total and storey damage indices for frame F (z+0.3x/DLS).

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16

ARTICLE IN PRESS A.A. Vasilopoulos, D.E. Beskos / Soil Dynamics and Earthquake Engineering 29 (2009) 194–218

following the same arrangement with frame A. Frame E has corner columns HEB280 and HEB240 at the first four and upper three storeys, respectively, intermediate strong and weak axes perimetric columns of HEB340, HEB320 and HEB300 at the first three, intermediate two and upper two storeys, respectively and interior columns of HEB360, HEB340 and HEB320, following the same storey arrangement as in frame A. Finally, frame F has the same member sections with frame D except from the strong axis beams, which have IPE200 and IPE140 sections for the first four and upper three storeys, respectively. Due to space limitations, only results concerning frames A and F are presented here. Results for the other frames of Table 12 can be found elsewhere [47]. Consider first frame A under the x+0.3 z load combination for the Bingol, Friuli and Loma Prieta earthquakes and the DLS. For this case, one can observe from Fig. 18 that all the storeys develop lateral drifts lower than the allowable limit of 1.50%  h ¼ 0.15  320 ¼ 4.80 cm. Fig. 19 shows that the maximum total and storey damage indices are caused by the Bingol earthquake and have values of 12.33% and 11.68%, respectively, i.e., lower than the limit of 20%. The member damage indices presented in Table 13 have values which also do not exceed the limit of 20%, while, as shown in Fig. 20, plastic hinges have been formed only in beams. Consider now frame A under the z+0.3x seismic load combination. For this case, one can observe from Fig. 21 that all the storeys develop lateral drifts lower than the allowable limit of 4.80 cm except the sixth and seventh storeys where their z direction drifts are equal to 5.30 and 5.05 cm (Friuli seismic case). The maximum total and storey damage indices of Fig. 22 as well as the member damage indices of Table 14 do not exceed the 12.00% value. Finally, Fig. 23 shows that plastic hinges have been formed only at the weak axis IPE140 beams of the higher storeys. In conclusion, the only design objectives of frame A which do not comply with the DLS are the upper storey

213

drifts. Thus, using the proposed design method one should proceed to check the adequacy of the other section selections (frames) of Table 12. Due to space limitations, only the final (best) section selection (frame F) is presented below.

Table 16 Member damage indices from seismic excitations (z+0.3x/DLS): only max values are shown i-End damage (%)

j-End damage (%)

Seven storey frame F Bingol 26 2578/2580 132 2438/2440

11.34 15.45

11.96 15.34

Friuli 103 132

381/381 385/385

11.94 15.51

11.90 15.40

1457/1458 1464/1464

15.19 15.57

14.85 15.51

Member

LomP 132 161

Damage start

Table 17 Member ratios ypl/yy (x+0.3z/DLS): only max values are shown i-End ypl/yyield

j-End ypl/yyield

Frame F/Loma 42 606 71 1477

4.87 4.73

3.91 3.65

Frame F/Impo 42 71

396 398

4.76 4.64

3.82 3.58

Frame F/Parko 74 100

667 668

3.85 4.26

3.32 3.47

Member

ypl starting

Fig. 29. Plastic hinge formation in frame F from seismic excitations (z+0.3x/DLS).

ARTICLE IN PRESS A.A. Vasilopoulos, D.E. Beskos / Soil Dynamics and Earthquake Engineering 29 (2009) 194–218

214

(Fig. 29). In conclusion, frame F satisfies all the design criteria of the proposed design methodology corresponding to the DLS. A confirmation of the finally selected member sections (frame F) was done by using the two scaled seismic accelerograms of Imperial Valley and Parkfield and it was found that all the response magnitudes comply with the DLS performance design objectives. Details of this procedure can be found in [47]. Table 17 provides the maximum member ratios ypl =yy for three of the five seismic motions (worst possible cases) of the DLS and shows that the limit value of 6 is nowhere exceeded. Numbers under the title ‘‘ypl starting’’ in Table 17 are in 102 s. Concerning base shear forces Vx and Vz along the x and z directions, respectively, it was found that for all five motions considered here, these shears have maximum values very close for both load combinations. Indeed,

7

7

6

6

5

5 Storey Level

Storey Level

Looking at Figs. 24–26 and Table 15 for the x+0.3z and Figs. 27–29 and Table 16 for the z+0.3x combination, one can observe the following concerning frame F under the Bingol, Friuli and Loma Prieta earthquakes and the DLS. For the x+0.3z load combination the relative storey drifts do not exceed the 4.80 cm allowable limit, the maximum total and storey damage indices of dBingol ¼ 12.35%, and dLomP ¼ 10.00% as well as the member damage indices do not exceed the allowable limit of 20%. Finally, plastic hinges are developed only in the beams of the frame, satisfying the capacity design criteria. Under the z+0.3x combination, storey drifts reach a maximum value (Friuli case) of 3.80o4.80 cm, total damage indices reach a maximum value (Friuli case) of 15.36%o20%, while storey and member damage indices also do not exceed the allowable limit of 20%. Finally, plastic hinges have been developed only in the z-direction beams of the frame

4 Bingol Friuli LomaP

3

4

2

2

1

1

0.005

0.010

0.020

0.015

0.025

0.030

0.035

Bingol Friouli LomaP

3

0.000

0.005

0.010

X-Drifts (m)

0.015

0.020

0.025

0.030

0.035

Z-Drifts (m) Frame F Fig. 30. Seismic storey drifts (x+0.3z/ULS).

7

6

6

5

5

Storey Level

Storey Level

7

4 3

Bingol Friuli LomaP

4 3

2

2

1

1

0.000

0.002

0.004

0.006

0.008

0.010

Bingol Friouli LomaP

0.00

0.02

X-Drifts (m)

0.04

0.06 Z-Drifts (m)

Frame F

Fig. 31. Seismic storey drifts (z+0.3x/ULS).

0.08

0.10

0.12

ARTICLE IN PRESS A.A. Vasilopoulos, D.E. Beskos / Soil Dynamics and Earthquake Engineering 29 (2009) 194–218

1100 kNpV x p1400 kN and 240 kNpV z p250 kN for the x+0.3z load combination and 460 kNpV x p530 kN and 333 kNpV z p450 kN for the z+0.3x load combination. An additional verification of the frame F is made by performing fully inelastic time history analyses using seismic records compatible with the ULS. Some of the results are shown in Figs. 30–34, while the rest can be found in [47]. Thus, storey drifts for the x+0.3z combination plotted in Fig. 30 appear not to exceed the allowable ULS limit of 3%  h ¼ 3%  320 ¼ 9.60 cm. On the contrary, as shown in Fig. 31, the load combination z+0.3x gives storey drifts slightly higher (10.00 cm) than this ULS limit, which, however, can be acceptable. Concerning the total damage indices (Figs. 32 and 33),

215

their values do not exceed the allowable limit of 50.00% for both the x+0.3z and the z+0.3x seismic combination. The storey (Figs. 32 and 33) and member damage indices (not shown here) also present maximum values not exceeding the 50.00% limit. Fig. 34 shows that plastic hinges develop not only in beams but also in columns, especially in the first floor. Nevertheless, the performance of frame F is considered, in general, to be satisfactory. The best section selection (frame F) has also been verified on the basis of all five previously used seismic records scaled to match the spectrum intensities of the elastic pseudovelocity spectra for the DLS [5,9]. Finally, for comparison purposes, the commercial design software SAP 2000 [48] is also used for the seismic design

16.00 7

14.00

6

10.00

5

8.00 Bingol Friuli LomP Impo Parko

6.00 4.00 2.00

Storey Level

Total Damage (%)

12.00

4 3 Bingol Friuli LomaP

2

0.00

1

-2.00 0

500

1000

1500

2000

2500

3000

-2

0

2

Time (0.01sec)

4

6

8

10

12

14

16

18

Storey Damage (%) Frame F

Fig. 32. Total and storey seismic damage indices (x+0.3z/ULS).

20.00 7

18.00

6

Bingol Friuli LomP

14.00 5

12.00 10.00

Storey Level

Total Damage (%)

16.00

Bingol Friuli LomP Impo Parko

8.00 6.00 4.00

4 3 2

2.00 0.00

1

-2.00 0

500

1000

1500

2000

2500

3000

6

8

10

Time (0.01sec)

12

14

Storey Damage (%) Frame F

Fig. 33. Total and storey seismic damage indices (z+0.3x/ULS).

16

18

20

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216

Fig. 34. Plastic hinge formation from seismic excitations (z+0.3x/ULS). Table 18 Design/strength (capacity) ratios of columns of frame F according to EC3/ EC8

Table 19 Design/strength (capacity) ratios of beams of frame F according to EC3/ EC8

Member Section strength

Member Section strength

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

Member strength

Biaxial bending+shear+axial force

Flexural buckling

Lateral buckling

0.674 0.796 0.791 0.796 0.923 0.918 0.341 0.464 0.462 0.464 0.565 0.563 0.316 0.457 0.456 0.406 0.660 0.659 0.492 0.585 0.580 0.558 0.638 0.632

0.917 0.987 0.967 1.105 1.039 1.154 0.513 0.714 0.712 0.799 0.996 0.992 0.468 0.676 0.674 0.685 0.991 0.989 0.588 0.745 0.739 0.724 0.916 0.910

0.927

0.516 0.717 0.715 0.800 0.994 0.990 0.472 0.681 0.679 0.690

0.591 0.750 0.744 0.730 0.924 0.918

of the frames considered here in accordance with EC8 [12] and EC3 [1] codes. Tables 18 and 19 provide the values of the design/strength (capacity) ratios of EC3 [1] under various states of deformation corresponding to columns and beams of frame F with numbering that shown in Fig. 35. It is observed that frame F represents again the best design as the capacity ratios approach 1 without

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Member strength

Biaxial bending+shear+axial force

Flexural buckling

Lateral buckling

0.445 0.332 0.257 0.200 0.798 0.537 0.524 0.396 0.809 0.520 0.515 0.371 0.239 0.213 0.160 0.136 1.133 1.198 1.281 1.522

0.876 0.737 0.447 0.375 0.949 0.786 0.717 0.585 0.960 0.829 0.712 0.564 0.734 0.624 0.357 0.295 0.876 0.844 0.637 0.503

– – – – – – – – – – – – – – – – – – – –

exceeding it. Frame F was finally analyzed for the seismic motions and the limit states considered here by taking the two scaled horizontal components of every motion and its response values were found to be slightly higher than those for the x+0.3z and z+0.3x load combination rule case without, however, leading to heavier sections than those of frame F, which is in agreement with the code-based design. Thus, on the basis of these findings and the remarks made at the end of the previous example, it was concluded that the x+0.3z and z+0.3x combination rule was an acceptable way of considering seismic loading.

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217

Acknowledgment The authors would like to thank Miss M. Dimitriadi for her excellent typing of the manuscript. References

Fig. 35. Member numbering of seven storey space frame designed by EC8/EC3.

6. Conclusions A seismic design methodology for regular space steel frames using an advanced dynamic finite element method taking into account material and geometric nonlinearities has been proposed. The advantages of this design methodology are the following: 1. It analyzes the whole space frame nonlinearities (material and geometric) and captures its limit state of strength and stability in a direct manner so that the use of the crude behavior factor of EC8 and the member interaction equations and effective length concept of EC3 for approximately and indirectly taking into account inelasticity and stability are avoided. 2. It establishes a performance-based seismic design approach as it checks the satisfaction of its performance objectives in terms of drifts, plastic rotations, total, storey and member damage and plastic hinge formation pattern for the damage, the ultimate and the serviceability limit state. 3. Even though it represents an extension of the 2D case to the 3D case one, this is by no means straightforward. Indeed, real spectrum compatible accelerograms are employed for every limit state along both horizontal x and z axes of the space frame on the basis of the x+0.3z and z+0.3x seismic load combinations. Accidental eccentricities are also taken into account. 4. Member sizes determined by the proposed method are close to those determined by the EC3/EC8 method because the advanced analysis is calibrated against the beam-column interaction equations of EC3 and the EC8 design spectrum. As a result, the proposed method provides at least a more rational and efficient alternative to EC3/EC8 design method.

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