Eurocode conforming design of BRBF – Part II: Design procedure evaluation

Eurocode conforming design of BRBF – Part II: Design procedure evaluation

Journal of Constructional Steel Research 135 (2017) 253–264 Contents lists available at ScienceDirect Journal of Constructional Steel Research journ...

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Journal of Constructional Steel Research 135 (2017) 253–264

Contents lists available at ScienceDirect

Journal of Constructional Steel Research journal homepage: www.elsevier.com/locate/jcsr

Eurocode conforming design of BRBF – Part II: Design procedure evaluation ⁎

Ádám Zsarnóczay , László Gergely Vigh

MARK

Budapest University of Technology and Economics, Department of Structural Engineering, H-1111 Budapest, Műegyetem rkp. 3, Hungary

A R T I C L E I N F O

A B S T R A C T

Keywords: Buckling restrained braced frames Eurocode 8 Design procedure Seismic performance assessment FEMA P-695

Buckling Restrained Braces (BRB) are innovative displacement dependent devices having balanced hysteretic behavior. Frames equipped with such devices are known as Buckling Restrained Braced Frames (BRBF). Application of BRB elements in Europe is seriously limited by the lack of a standardized European design procedure for BRBF. In the first part of this paper the authors propose a Eurocode conforming design procedure for BRBF and provide the seismic design parameters and capacity design rules by enhancing Eurocode 8 specifications on steel Concentrically Braced Frames. This second part introduces 24 BRBF archetypes designed with the proposed procedure. Their performance under seismic excitation is investigated by nonlinear dynamic analysis in an environment developed by the authors through improvement of the methodology in FEMA P-695. Feasibility of the proposed design procedure is evaluated as a function of conditional failure probability of the archetype buildings. The presented assessment required detailed experimental studies on BRB behavior and development of a novel material model that can simulate complex nonlinear hardening under irregular cyclic loading. The material model is implemented in the OpenSees finite element code. A software environment was developed for automatic evaluation of braced frames using OpenSees. The sufficiently low collapse probability of BRBF archetypes confirms the applicability of the proposed design procedure in Europe.

1. Introduction Buckling Restrained Braced Frames (BRBF) are first generation dissipative structural solutions similarly to Special Concentrically Braced Frames (SCBF) and Special Moment Resisting Frames (SMF) [1]. Design procedures for BRBF were developed only recently in the USA [2] and few, if any of existing structures have been subjected to actual earthquakes of high intensity. The European Eurocode standards [3] do not regulate the design of BRBF and the few BRBF in Europe were designed using various procedures and built only recently. Numerous laboratory tests have been performed on individual BRB elements (e.g. [4,5,6]), but the number of full-frame tests is very limited (e.g. [7]). Behavior factors (identical to response modification factors in the US) for design of dissipative structural systems are often a result of subjective comparison of structural behavior rather than quantitative analysis of structural performance. Performance of a dissipative structural system at high seismic intensities beyond the design intensity is just as much dependent on the behavior of non-dissipative members as it is on the behavior of dissipative elements. Consequently, performance of any BRBF design procedure needs to be evaluated quantitatively,



through advanced analysis that can consider the inelastic response of all key members of the braced frame. The authors proposed a BRBF design procedure in the first part of this paper that conforms with existing EC8 regulations. This paper provides details about the verification of the proposed design procedure and explains how the performance of the procedure was evaluated in a quantitative manner. 1.1. Background of design procedure evaluation The conventional qualitative approach to design procedure development uses incremental improvements based on a combination of experience with existing structures, numerical results and engineering judgement. That approach was justified in the past for structural systems with limited complexity and a sufficient number of existing structures (and collapsed structures) available. Design procedures of recent structural solutions with complex nonlinear behavior are more difficult to assess qualitatively. The Eurocode 0 standard defines target structural reliability measures through annual collapse probabilities. Numerical assessment of the

Corresponding author. E-mail addresses: [email protected] (Á. Zsarnóczay), [email protected] (L.G. Vigh).

http://dx.doi.org/10.1016/j.jcsr.2017.04.013 Received 19 May 2016; Received in revised form 12 April 2017; Accepted 13 April 2017 0143-974X/ © 2017 Elsevier Ltd. All rights reserved.

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expected lifetime collapse probability of structures designed with a new procedure provides a quantitative measure of its merits and its performance. This approach allows comparison of alternative design procedures and a selection methodology that is in line with current European regulations. Credibility of collapse probability assessment heavily depends on the applied evaluation methodology. The applied methodology for this research is an improvement over the recommendations in the FEMA P695 document [8]. FEMA P-695 is developed from the advanced performance assessment framework proposed at the Pacific Earthquake Engineering Research Center [9]. It has been applied recently for the evaluation of several design procedures [10,11,12] and it is considered to provide a sound basis for comparison of different structural systems [13]. The collapse probability corresponding to a design procedure in this paper is described through the seismic performance of representative structural archetypes. Archetypes are designed with the procedure under consideration, thus any modification in the procedure will directly influence the results of its evaluation. Archetype behavior is described by a complex numerical model that shall be calibrated to and verified by experimental results. The limited amount of experimental data, the assumptions in numerical modeling and other sources of uncertainty are considered by the probabilistic framework of the evaluation methodology. Failure modes that cannot be captured by the numerical model are included in the evaluation in a post-processing step. This approach provides an iterative process for design procedure development: key parameters or regulations can be enhanced gradually until all structural archetypes have sufficiently high performance. 1.2. Objectives and scope Archetypes for the performance evaluation were selected to cover a large parameter space, thus provide information about the performance of the design procedure for several typical design cases. The following general assumptions were made regarding BRBF configurations: – BRBF are designed with a concentrically braced frame topology (Fig. 1 provides an example). BRBs are installed in a two-bay Xbrace configuration in an even number of bays (i.e. back-to-back, adjacent bays of “zigzag” diagonal bracing). Although other configurations are not investigated in this research, the authors expect similar performance for all concentrically braced BRBF topologies. Thus, applicability of the results is not limited to the investigated two-bay X-brace configuration. – The rotational stiffness of BRB-braced frame connections is not taken into consideration directly in the numerical models. Our modeling approach provides an accurate representation for ideal pinned BRB elements. The finite rotational stiffness of BRB connections with bolted or welded configuration may result in secondary stresses in the BRB steel core due to frame deformations. This effect shall be taken into account in the numerical BRB element models. The influence of semi-rigid connections on the gusset plate and connection behavior was not investigated. It is assumed that those details of the connection are designed to have strength and ductility capacity superior to that of the buckling restrained brace. – BRB design is typically performed by the manufacturer. It is assumed that the manufacturer performs the necessary verifications to ensure sufficiently high resistance against local failure (e.g. flexural buckling of the unrestrained part of the steel core [14], concrete failure due to jammed steel core under compression [15], etc.) and element level failure modes (e.g. flexural buckling of the BRB element including the casing [16]). The BRB element is assumed to be able to sustain at least 2 load cycles at a deformation amplitude that produces ± 6% strain in its yielding zone. This assumption is in agreement with experimental results on low cycle fatigue performance of BRB elements [17].

Fig. 1. Plan and elevation layout of the example structure.

– BRB elements are assumed to use a conventional concrete casing to provide lateral support for the steel core of the brace. The numerical BRB element model represents such conventional BRBs. If the element characteristics are similar, the results are valid for other types of BRB configuration as well (e.g. steel-only BRBs [18,19]). Extrapolation beyond the presented scope shall be supported by either additional numerical results or several experienced seismic design experts.

2. Methodology FEMA P-695 provides a comprehensive framework to guarantee the advantageous behavior of the designed structures at a sufficiently high seismic intensity level. The methodology applied in this research 254

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ciently high seismic intensity level. We follow the recommendations in FEMA P-695 and perform the evaluation at Sa,L = 1.5 Sa,D, where Sa,D is the design spectral acceleration at T1. 11. Design procedure evaluation: The authors believe that the modifications applied to the original FEMA P-695 procedure improve its accuracy and robustness, but do not necessitate a change in the evaluation methodology. Using the same performance limits allows us to compare results with BRBF to results with other systems in the literature. Similar archetypes are organized into Performance Groups (PG). Average collapse probability at Sa,L shall not exceed 10% for each PG. Single archetypes in a PG may exceed this limit, but none of the archetypes is allowed to have more than 20% probability of collapse at Sa,L. The design procedure is appropriate if these criteria are met by all archetypes and PGs. 12. Reliability index: The basis of safety in the Eurocodes is the probability of failure over the lifetime of the structure [20]. Because reliability indices (β) express such failure probability, they would provide a Eurocode conforming basis of evaluation in the future. Current Eurocodes do not prescribe unique β limits for seismic performance. To provide data for future regulation, such indices shall be calculated and presented for various types of structures.

enhances that methodology at several points to improve robustness and to arrive at results conform with the European design environment. The following are a brief summary of the main steps of the applied methodology: 1. Archetype definition: Identify key parameters of the structural system under consideration (e.g. building height, bay width, braced area, design seismicity, etc.) and select a sufficiently large number of parameter-sets to represent it. Each parameter set corresponds to an archetype. 2. Archetype design: It is important to mimic the expertise and equipment of a practicing engineer in this phase. Structural response is calculated using either the equivalent lateral force method or modal response spectrum analysis. Archetype design is performed using the proposed capacity design rules. 3. Background information: Gather supporting information (preferably experimental data, but numerical analysis results are also considered) to characterize the inelastic cyclic behavior of the structural system as a whole and its components. Lack of information in certain areas shall be considered by conservative assumptions in the following numerical analysis and post-processing. 4. Numerical modeling: Develop an advanced numerical model for each archetype. The model shall be able to describe the nonlinear dynamic response of the structure up to the point of collapse due to pre-defined failure modes. Try to directly model as many failure modes of the system as possible. Identify the failure modes that cannot be described by the numerical model so that they can be considered in post-processing during collapse assessment. 5. Pushover analysis: Perform nonlinear static pushover analysis on each archetype model to evaluate its global ductility (μ). 6. Dynamic analysis: Perform a series of nonlinear dynamic analyses (Incremental Dynamic Analysis – IDA or Multi-Stripe Analysis – MSA) on each archetype model to evaluate its performance at a set of seismic intensities. Assemble a set of ground motion records that appropriately represent the characteristic seismic hazard at each seismic intensity level. The response of every archetype shall be analyzed at several levels of seismic intensity using every single ground motion in the corresponding record set. 7. Fragility curve: Evaluate the empirical fragility curve of the structure. The governing maximum interstory drift ratio is used as a proxy for structural damage. Non-simulated failure modes are considered at this step. Drifts at any given seismic intensity are assumed to follow a censored lognormal distribution. Using the results of dynamic analyses, the probability of exceedance of a predefined drift limit that corresponds to collapse can be calculated for each seismic intensity level. The empirical fragility curve of the structure is determined by fitting a lognormal distribution on the set of collapse probability samples. 8. Spectral shape: Modify the empirical fragility curve to consider the change of the characteristic spectral shape with increasing seismic intensity. This step is required only if the same set of ground motions is used at all seismic intensity levels. Otherwise, an appropriate ground motion set can be applied at each level that can follow the change in spectral shape. The effect of changing spectral shape heavily depends on the expected period elongation of the structure before collapse. Hence, modification of fragility curves is based on the dominant period of vibration (T1) and the global ductility (μ) of the structure. 9. Resistance and model uncertainty: Increase the variance of the lognormal collapse probability distribution to consider the effect of various sources of uncertainty. Use qualitative judgement to determine the uncertainty corresponding to design requirements, available test data and modeling assumptions. Detailed explanation of each source and guidelines for judgement are available in [8]. 10. Archetype performance: Evaluate the seismic performance of the archetypes as the conditional probability of collapse at a suffi-

3. Investigated scenarios Performance Groups describe the ranges of parameters in expected application scenarios for a BRBF system in Europe. Considering the European building stock and existing experience with BRBF in North America and Asia, the following assumptions were made for the archetypes: – All buildings are built for office use with gypsum walls and perimeter BRBF for lateral load resistance (Fig. 1). A distributed dead load of 5 kN/m2 and a live load of 3 kN/m2 is assumed for each floor and the roof. – Beams of the frame are pinned to gravity and braced frame columns. The base of gravity columns is also pinned, while the braced frame columns are fixed at their base. This represents a less favorable configuration compared to pinned braced frame columns and leads to more challenging design problems. Columns are installed in the braced frame in such a position that they experience weak-axis bending under in-plane deformation of the frame. This measure reduces bending moments in the columns from frame deformation, but increases the likelihood of soft story formation. – All buildings in this study are regular in plan and in elevation. Irregularity and corresponding torsional effects are considered through design penalties in EC8. The authors believe that existing EC8 regulations can effectively penalize irregularity in BRBF. Torsional effects generally lead to increased seismic load for the braced frame, but otherwise their design is not affected. Hence, a set of regular archetypes designed for several seismic load levels are sufficient for the evaluation of BRBF design procedure performance. – Two sizes of floor area (4 × 4 bay and 6 × 6 bay) are considered to investigate the influence of seismic mass on the results. Two-bay braced frames are used in both cases. Gravity loads are expected to heavily influence second order effects, such as the global overturning moment. – Two types of seismic environments are considered. The selected Romanian and Turkish sites with moderate and high seismicity have a characteristic peak ground acceleration (PGA) of 0.15 g and 0.4 g, respectively. Design of structures in the moderate seismic region is typically controlled by limitation of second order effects; thus the load bearing capacity of BRB elements cannot be fully utilized and non-dissipative members are designed for significant overstrength. Design of structures in the area of high seismicity is force controlled. BRB and braced frame column sections are considerably larger; their 255

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4. Performance evaluation example

Table 1 Performance Groups used for evaluation of the proposed BRBF design procedure. Group no.

Characteristics Design load level

PG-1 PG-2 PG-3 PG-4 PG-5 PG-6 PG-7 PG-8

Seismic

Low

Medium High Medium High

Archetype 6LH from PG-4 is used here to demonstrate the steps of performance evaluation in detail. Fig. 1 shows a schematic of the structural configuration. The sample structure is a BRBF with 6 stories, a bay size of 4.0 m × 6.0 m and a 4 × 4 bay floor layout. It is in a region of high seismicity characterized by a PGA of 0.4 g. Its detailed seismic design process is explained in the first part of this paper.

Period domain

Gravity

High

4.1. Example structure

Number of archetypes

Short Long Short Long Short Long Short Long

3 3 3 3 3 3 3 3

4.2. Numerical BRB model Numerical analyses were performed in the OpenSees 2.4.6 finite element environment [21]. Appropriate modeling of the cyclic nonlinear response of BRBs is essential for the analysis of BRBF performance. Each BRB is modeled by a combination of a corotational truss element and a nonlinear material model. Typical approaches from the literature use a material model with bilinear kinematic hardening and a 3–5% hardening ratio to describe BRB behavior [22,23]. The authors have shown in a sensitivity study [24] that those bilinear models can simulate the envelope response of the braces under small amplitude cyclic loading, but they overestimate their stiffness and capacity under large deformations. Therefore, a more advanced material model is needed to properly simulate the hardening behavior of BRBs. BRB behavior under uniaxial cyclic loading has been investigated in a large number of experimental studies [4,5,6]. These results provide ample support for the stable hysteretic behavior of the braces and allow us to describe and model their cyclic behavior. A few tests were also performed for subassembly structures [25] and complete frames [7]. These results suggest that the advantageous element-level behavior of the BRB can be efficiently utilized at the structural level and the real behavior can be described by simplified 2D numerical models with high accuracy. Despite the large number of supporting studies, it is important to mention that more results are needed to appropriately describe low cycle fatigue life and cyclic hardening of BRBs under irregular load protocols. A novel numerical material model was developed by Zsarnóczay [26] to provide a tool for BRBF response simulation. The model is implemented in OpenSees as Steel4. It is a versatile material with kinematic and isotropic hardening, an ultimate strength limit, nonsymmetric behavior and load history memory. Low cycle fatigue is taken into account by coupling Steel4 with the Fatigue material. The complex material was calibrated using 15 experimental test results from the Budapest University of Technology and Economics (BME) [4] and the University of California, San Diego (UCSD) [5]. Several tests were completed with special protocols to allow more accurate calibration of isotropic and kinematic hardening parameters. Material parameters used in this research are summarized in Table 3. Note that different parameters are applied for pushover and response history analyses, because the former requires the envelope of the cyclic response. Parameters for pushover analysis are set to model the design backbone curve of the BRB. Although this follows current design practice, a more appropriate solution using the secant stiffness of the envelope curve is suggested in the first part of this paper. Fig. 2 shows a comparison of a Steel4-based result (gray) with results of two alternative models using bilinear materials (red and blue). The blue line corresponds to a material with 0.5% and 2.5% hardening ratio under tension and compression, respectively. These values are a good approximation of the kinematic hardening in BRBs, but the model lacks isotropic hardening, thus it underestimates the load bearing capacity of the braces. The red line corresponds to a typical BRB modeling approach: a symmetric bilinear kinematic hardening material with 4% hardening ratio. Stress in this material at 3% strain exceeds twice its yield strength. BRBs are subjected to large deformations in collapse performance analyses and experimental results suggest

load bearing capacity limits their performance. – The least favorable, but practically relevant soft soil (vs,30 < 180 m/s) that corresponds to class D as per Eurocode 8 is assumed for every scenario. Both design and verification are based on the same design acceleration response spectrum. The type of soil considered shall not influence design performance as long as the target spectrum is properly matched by the set of ground motion records in dynamic analysis. Therefore, similar results are expected for all soil types. – The number of stories range from 2 to 8 among the archetypes to have both short and long period structures among them. Archetypes have a bay geometry of 4.0 m × 6.0 m. The shortest frame with only 2 stories is 8 m tall, while the height of the tallest building is 32 m. The period of vibration and sensitivity to second-order effects increases with the height of the structure. A total of 8 Performance Groups were created. Their characteristics are shown in Table 1. Each PG contains three archetype structures. Archetypes were designed with the procedure proposed in the first part of this paper. Archetypes and their characteristics are summarized in Table 2.

Table 2 Design parameters and dominant vibration periods of the BRBF structural archetypes. PG

PG-1

PG-2

PG-3

PG-4

PG-5

PG-6

PG-7

PG-8

ID

2LM 3LM 4LM 5LM 6LM 8LM 2LH 3LH 4LH 5LH 6LH 8LH 2HM 3HM 4HM 5HM 6HM 8HM 2HH 3HH 4HH 5HH 6HH 8HH

Stories

2 3 4 5 6 8 2 3 4 5 6 8 2 3 4 5 6 8 2 3 4 5 6 8

Floor plan

4 × 4 grid

6 × 6 grid

Seismicity

T1 [s]

Location

PGA

Bucharest

0.15

Istanbul

0.40

Bucharest

0.15

Istanbul

0.40

0.768 1.000 1.150 1.276 1.400 1.627 0.522 0.634 0.732 0.842 0.962 1.297 0.777 1.020 1.151 1.286 1.401 1.620 0.527 0.653 0.761 0.908 1.019 1.391

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Table 3 Proposed general BRB parameters for nonlinear dynamic analysis. Tension

Compression

Es·fSM γm,ov·fy,k

E0 fy

Kinematic hardening 0.5% (4.5%) bk R0 25.0 r1 0.91 (0.01) r2 0.10 (0.15) Isotropic hardening bi bl ρi Ri lyp

3 % − 0.5 % ∙ fA (6.5%) 25.0 0.89 (0.01) 0.02 (0.15)

0.3 % − 0.05 % · fA 0.01% 0.25 + 0.05 · fA 3.0 1.0

0.5 % + 0.1 % · fA 0.04 % − 0.007 % · fA

Ultimate strength limit fu 1.65·fy Ru 2.0 Fatigue m ε0 where fA =

2.40·fy

−0.400 (0.14 + 0.4 (γm , ov − 1.1))/fDM

Ay [mm2] 5000 ; common parameters are at the middle, compression or

tension specific parameters are on either side of the table. Parameters modified in pushover analysis are shown in parentheses. Isotropic hardening is turned off for pushover analyses.

that tensile stress in the material cannot exceed its ultimate strength, not even under cyclic loading. Consequently, the blue material leads to unconservative results. Note the significant difference in collapse probability at Sa,L seismic intensity level (Sa,L = 1.5 Sa,D corresponds to 1.5 on the normalized intensity scale). 4.3. Numerical braced frame model A planar model of a single braced frame was developed for each archetype. The main attributes of the frame model are highlighted in Fig. 3 and explained in the following paragraphs. Columns of the braced frame are modeled by continuous elastic beam elements (elasticBeamColumn). Corotational geometric transformations are applied to properly consider local and global geometric nonlinearity. Plastic hinges (zeroLength elements with Bilin materials that implement the modified Ibarra-Medina-Krawinkler deterioration model [27]) are defined at column ends to capture the inelastic moment-rotation response and describe cyclic degradation of column strength and stiffness due to load cycles with large deformations. Calibration of plastic hinge parameters is based on the work of Lignos and Krawinkler [27] with the additional assumption that columns have similar behavior and at least the same level of ductility under weak-axis bending when compared to strong-axis bending. Interaction between axial force and bending moment at plastic hinges is taken into consideration by an approximate reduction of plastic hinge bending capacity based on the theory of plasticity. The plastic moment resistance is modified with the following fNM normal force-bending interaction factor from Eurocode 3 [28]:

fNM

⎛ Nc Npl,Rd − a ⎞2 =1−⎜ ⎟ ⎝ ⎠ 1−a

if

Nc

Npl,Rd − a > 0,

Fig. 2. Comparison of median IDA responses (a) and adjusted fragility curves (b) corresponding to archetype 4HM for three different approaches to modeling BRB behavior.

which suggests that flexural buckling of columns is not expected. Consequently, flexural buckling of columns was not modeled directly in numerical analyses of this research. Development of plastic hinges in columns of a story often leads to soft story formation and collapse during numerical analyses. Improvements of column modeling after plastic hinge formation shall be a topic of future research to improve numerical result accuracy. Beams of the braced frame and members of the leaning truss are modeled with truss elements and a perfectly Elastic material. Sufficiently rigid behavior is guaranteed by a Young's modulus of 10,000 GPa. Masses and vertical loading are identical to those applied during linear static analysis of the frame. Table 4 summarizes the modeling status of BRBF components.

4.4. Nonlinear static analysis A pushover analysis needs to be performed to evaluate the global ductility of the structure. The quasi-permanent load combination is used as constant loading and the response of the structure is recorded under gradually increasing lateral loading. The lateral load distribution is influenced by the dominant mode shape and the structural masses as per EC8. The resulting capacity curve (i.e. base shear force - roof drift ratio relationship) for the sample archetype is shown in Fig. 4. Global ductility is defined as the ratio of ultimate roof displacement (δu) to yield roof displacement (δy).

(1)

where Nc is the characteristic value of the normal force in the plastic hinge during the seismic event; a is the ratio of the web area and the total area of the column cross-section. The value of Nc is approximated through pushover analysis as the axial load in the column at the design drift level. Verifications with several archetypes at various seismic intensities confirm that this is a conservative assumption on the magnitude of Nc. Nc values are consistently below Nb,Rd resistances, 257

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Fig. 3. Configuration of the numerical BRBF model in OpenSees.

the seismic hazard. The scatter of individual spectra around the target Sa(T1) represents that uncertainty. Although the geometric mean spectrum of the record set matches the standard design acceleration response spectrum at T1 and in its vicinity, spectral accelerations at other periods are often estimated with large errors. On the one hand short period spectral accelerations corresponding to higher modes of tall structures are greatly overestimated (Fig. 5). This leads to conservative seismic demand for those structures. On the other hand if the geometric mean of the Far Field set is scaled to the plateau of the design spectrum, Sa at long periods will be underestimated. This leads to unconservative demand for short period ductile structures where significant period elongation is expected under seismic excitation. This phenomenon and related issues had been recognized by the authors [29] and an alternative, superior approach is being developed, but it is out of the scope of this research. The widespread acceptance and general conservativism for medium and long period structures led to the application of the Far Field record set and conventional scaling methodology in this research. During IDA, starting from a sufficiently low intensity level, the ground motion record set is gradually scaled to increasing Sa(T1) and the corresponding 44 governing maximum interstory drift ratios (δmax) are calculated at each seismic intensity level. The so-called IDA curves (in gray) in Fig. 6 are created by connecting results from the same ground motion. The end of each curve is highlighted by a black square –

The ultimate roof displacement is often defined as a displacement on the softening part of the curve. A more stringent alternative is applied here: δu corresponds to the displacement at the maximum capacity if it is reached without non-simulated brace or column failure. Otherwise, δu is the maximum displacement at the point of nonsimulated element failure. The yield roof displacement is the displacement at the yielding point of the bilinear representation of structural behavior. The bilinear representation is defined by the initial stiffness and the maximum shear force (Vmax) The sample archetype has a global ductility of μ = 10.13.

4.5. Nonlinear dynamic analysis Incremental Dynamic Analysis (IDA) is performed to describe the governing maximum interstory drift ratio (i.e. the governing value of maximum interstory drift ratios along the height of the structure) as a function of Sa(T1) intensity. The 44 ground motion records of the FarField record set of FEMA P-695 are used for the analysis. Records are always scaled uniformly as a set. Their characteristic Sa(T1) is defined by the geometric mean of individual spectral accelerations at T1. Fig. 5 shows the record set scaled to the Sa,D(T1) = 1.12 g, the design spectral acceleration of the sample archetype at T1 = 0.962 s. The Far-Field set was assembled to appropriately represent the inherent uncertainty in 258

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Table 4 Modeling of BRBF components. Phenomenon

Status Simulated

Buckling Restrained Braces Element level buckling Local buckling Strength degradation due to low-cycle fatigue Element failureb Columns Flexural buckling Excessive end rotations due to bending Strength degradation due to low-cycle fatigue Element failureb Failure due to torsional effects Connections Premature failure or strength degradation

Nonsimulated

Preventeda

X X X X X X X Fig. 6. IDA response curves of the sample archetype. Ordinates show Sa(T1) intensities normalized by the design spectral acceleration Sa,D.

X X

that is the last intensity level with δmax < 12% using the particular ground motion record. Responses with more than 12% governing maximum interstory drift ratio are neglected, because they are beyond the range of applicability of our numerical models. Great care was taken to configure numerical analyses appropriately to ensure sufficiently accurate results and to avoid convergence problems. Configuration involved selecting sufficiently small time steps (the maximum step size was dt/2 where dt is the time step in the ground motion record) and appropriate convergence criteria (NormDispIncr is used in OpenSees with a tolerance of 10− 8) All results presented in this paper are from completed analyses; there were no numerical convergence issues within the acceptable 12% limit of interstory drift ratios. Consequently, structural collapse in our results is always due to excessive drifts in numerical analysis and never due to non-convergence. The curves in Fig. 6 are made from results of more than 2000 response history analyses. The non-simulated fracture of BRB steel cores is considered by neglecting results if the expected elongation or compression of the brace exceeds 6%. This limit is in line with the initial assumptions of this research. Considering the bay geometry of the archetypes and the core geometry of the designed BRB elements, BRB fracture is expected if δmax > 8% (highlighted by the red dashed line in Fig. 6). Therefore, results with more than 8% governing maximum interstory drift ratio are considered collapses due to BRB failure. If reliable experimental results confirm that BRB elements perform well beyond 6% axial strain, these limits shall be relaxed in future evaluations. Median response (blue curve in Fig. 6) is defined by connecting the median δmax values at each seismic intensity level. To determine the median, the distribution of governing maximum interstory drift ratios is calculated by fitting a censored two parameter lognormal distribution on the available non-collapsed samples. Collapsed cases are considered as censored samples to improve the accuracy of the fit. Improvement is especially apparent at high collapse probabilities. Fig. 7 shows the histogram of non-collapse results and the fitted probability density functions at several seismic intensity levels. The ordinate of the median response at δmax = 8% corresponds to the empirical median collapse capacity (SCT). According to Fig. 6, SCT of the sample archetype equals 1.84 times its design spectral acceleration.

X

a Phenomena in this column are assumed to be prevented by appropriate design and detailing measures. b Although element stiffness and strength are significantly reduced, fracture of elements is not modeled directly. This improves numerical stability and does not introduce additional uncertainty in the results, because element failure is typically immediately followed by structural collapse in the investigated archetypes.

Fig. 4. Capacity curve of the sample archetype with the definition of ultimate and yield roof displacements highlighted. Green and red dashed lines show the level of design base shear force and shear force increased by structural overstrength, respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

4.6. Fragility curve evaluation FEMA P-695 recommends a simple approach to evaluate fragility curves of systems with high ductility: the empirical collapse capacity shall be a lognormally distributed random variable with a median of SCT and a standard deviation of 0.4 for ductile structures. The cumulative distribution function of the random variable is the fragility curve of the structure.

Fig. 5. Response spectra of the Far Field record set scaled to Sa,D(T1) = 1.12 g.

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Our IDA results suggest that the variance of BRBF response is considerably overestimated by a standard deviation of 0.4. Therefore, a more accurate approach is used in this research to assess the conditional collapse probability at each Sa(T1) level. Using the distribution of δmax (Fig. 7), the conditional probability of collapse at intensity Sa(T1)i is determined as P(δmax > 8%| Sa(T1) = Sa(T1)i). The empirical fragility curve is defined by fitting a lognormal distribution on the set of exceedance probability samples (black dots in Fig. 8). The empirical fragility curve is fit to the samples with a nonlinear least squares algorithm. The magnitude of error is assumed identical at each sample and only the samples with less than 80% probability of collapse are used to focus on appropriate fitting for low probabilities of collapse. Note that our results suggest that the empirical fragility curve at high collapse probabilities often does not follow a lognormal distribution. This is especially apparent at the fragility curves of other archetypes. (Those fragility curves are attached as Supplementary material to this paper.) Fig. 9 shows the curves of Fig. 8 in logarithmic space to highlight the magnitude of fitting errors at low collapse probabilities. The typical spectral shape of rare ground motions is more advantageous from a structural performance standpoint than the spectral shape of frequent ground motions. The median of the empirical fragility curve (SCT) needs to be increased to compensate for not taking the change in spectral shape into account during IDA. The so-called Spectral Shape Factor (SSF) from FEMA P-695 is applied in this research to calculate the magnitude of adjustment in SCT. The global ductility (μ) from pushover analysis is used as a proxy for period elongation. Because all investigated BRBF archetypes have global ductility above 8.0, the maximum values of T1 dependent SSF are applied. Table 5 displays these values for two levels of design PGA. The SSF for the sample archetype is 1.45. The SSF is used to scale SCT and get SCT,S. The final step of fragility curve evaluation is the consideration of additional sources of uncertainty. Although uncertainty of the seismic hazard (βRTR) is implicitly considered in the analysis by the variance of spectral accelerations in the Far Field set, other sources of uncertainty need to be included during post-processing by an appropriate increase of the standard deviation of the lognormal distribution of collapse capacity. The following three sources are recommended by FEMA P-695 and are used here:

Fig. 7. Fitting censored two parameter lognormal distributions to maximum interstory drift ratio data. Ordinates show Sa(T1) intensities normalized by the design spectral acceleration Sa,D.

Fig. 8. Evaluation of the adjusted fragility curve for the sample archetype. Abscissae show Sa(T1) intensities normalized by the design spectral acceleration Sa,D.

– Design requirements uncertainty is influenced by the robustness and completeness of the design specifications, with special focus on the probability of occurrence of unanticipated failure modes. Although design requirements are generally considered to be appropriate to safeguard against unanticipated failure modes, it is recognized that design of BRBF connections shall be improved by further research on connection behavior. Based on Table 3-1 in [8], βDR = 0.14 is selected. – Test data uncertainty describes the quality of test data with emphasis on the ability to predict nonlinear response when the structure is subjected to large seismic demands. It is also important to collect sufficient data for the proposal of design criteria that can ensure reliable performance of the designed system. Available test results are considered to give a good description of the general behavior of a BRB element. However, only limited amount of experimental information is available on BRB and BRBF behavior under irregular cyclic loading. These circumstances are considered by using a βTD of 0.22 from Table 3-2 in [8]. – Modeling uncertainty depends on how well the archetypes represent

Fig. 9. Logarithmic plot of the evaluation of the adjusted fragility curve for the sample archetype. Abscissae show Sa(T1) intensities normalized by the design spectral acceleration Sa,D.

Table 5 Spectral Shape Factors for structures with very high ductility (μ > 8) [8]. T1 [s]

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

PGA < 0.4 PGA ≥ 0.4

1.14 1.33

1.16 1.36

1.18 1.38

1.20 1.41

1.22 1.44

1.25 1.46

1.27 1.49

1.30 1.52

1.32 1.55

1.35 1.58

1.37 1.61

260

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the structural system and the ability of structural models to capture the collapse behavior of the structures. Because the number of archetypes is not sufficiently large to cover all possible BRBF representations with high confidence, a βMDL of 0.19 is applied from Table 3-3 in [8]. The total standard deviation βTOT of the collapse intensity distribution is calculated as the square root of sum of squares of individual β values. The adjusted fragility curve of the sample archetype has βTOT = 0.415, while the standard deviation of the empirical fragility curve is only βRTR = 0.261. The adjusted fragility curve of the sample archetype is shown in Figs. 8 and 9 by the red line. 4.7. Reliability analysis Calculation of the probability of failure over the lifetime of the structure gives a more complete and robust approach to performance assessment that is conforming with the concept of reliability in the Eurocode standards. Collapse probability evaluation over a time period requires a hazard curve that describes the probability of occurrence of earthquakes with different spectral accelerations within a given timeframe. Hazard curves are influenced by the characteristics of local seismic sources, thus they are site dependent in general. Because they describe spectral acceleration intensities, the curves are also period dependent and they are influenced by the soil characteristics at the site. Data points of median hazard curves are calculated using the open source OpenQuake probabilistic seismic hazard calculation environment [30] with the seismic sources and calculation logic defined by seismic experts in the framework of the European SHARE project [29]. Detailed hazard curves are evaluated for rock (vs,30 > 800 m/s) and verified by comparison with data from the European Facility for Earthquake Hazard and Risk (EFEHR) [31] using their web-based application. Soil amplification effects from the ground motion attenuation functions in probabilistic hazard assessment are generally less severe than the effect of soil amplification factors in the EC8-1 standard design spectra. In order to avoid unconservative underestimation of the seismic hazard, the fSA soil amplification factors are used in this research. The archetype-specific fSA is defined as the ratio of the design EC8 standard spectra and the 475 year return period Uniform Hazard Spectra (UHS) at T1 vibration period. This approach is considered appropriate to provide sample hazard curves for this research, but we believe that a more sophisticated methodology shall be used to model soil amplification in a future standardized reliability assessment. Research on this topic is in progress by the authors. Fig. 10 compares UHS to standard design spectra on rock at the two sites selected to represent moderate and high seismicity: Bucharest and Istanbul. Note the good agreement between the spectral shape at all, but the shortest periods of vibration. The agreement between UHS and design spectra suggests that the selected hazard curves represent the design seismicity appropriately. Fig. 8 displays the hazard survival function (SF) with an orange line based on the assumption of lognormal distribution of characteristic spectral acceleration intensities of seismic events. (Individual results of hazard calculation are shown in Fig. 8 by purple dots.) The probability of structural failure from seismic events at different intensities over the lifetime of the structure (50 years is assumed) can be expressed as the product of the hazard likelihood at the site and collapse probability of the structure. The resulting curve in Fig. 11 is the product of the probability density function of the hazard and the cumulative density function of fragility. It highlights the most perilous range of spectral intensities for the structure and site under consideration. The total probability of collapse is calculated by numerically integrating the collapse likelihood curve over the entire spectral acceleration domain. Integration yields 1.997% collapse probability for the sample archetype. The β reliability index expresses the distance

Fig. 10. Uniform Hazard Spectra and EC8 Type I design spectra on rock at Bucharest, a site of moderate seismicity (a) and Istanbul, a site of high seismicity (b).

Fig. 11. Normalized collapse likelihood at various spectral intensity levels. Abscissae show Sa(T1) intensities normalized by the design spectral acceleration SD.

of the probability of failure on the quantile function from the mean of the standard normal distribution in units of variance. The 1.997% collapse probability corresponds to a β of 2.05. 5. Results and discussion The procedure presented in the previous section was performed for all archetypes in Table 2. Governing parameters of the designs and 261

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Table 6 Design summary for archetypes and Performance Groups.

Note: ηDLC is the maximum story drift over displacement limits ratio as per the damage limitation criteria of EC8; ηBRB is the maximum ratio of BRB axial force and brace capacity; Ωmax/ Ωmin is the maximum variation of BRB overstrength along the height; γRd is the structural overstrength factor; θpl,avg and θpl,max are the average and the maximum of plastic stability indexes along the height, respectively.

structural overstrength calculation, especially the ones on the consideration of P-Δ effects. Design of archetypes for high seismicity was governed by BRB strength checks. The eight-story archetype with high braced mass (i.e. large braced area corresponds to each braced frame) (8HH) is the only structure at a region of high seismicity that approached the DLC deformation limit of EC8-1. Taller structures would probably become controlled by such deformation limits, thus the braced mass needs to be reduced for such structures to reach economical designs. Structures with low braced mass have quite uniform performance across all heights. Structures with large braced mass in a region of high seismicity once again highlight the problems in EC8 related to second order effects. On the one hand BRBF in moderate seismicity require large braces to fit the design within the θpl = 0.2 limit of EC8-1. Because BRB capacity in those frames is only partially utilized, overstrength factors as high as 3.4 are applied in capacity design. On the other hand θpl for several of the investigated BRBF archetypes in high seismicity is at the vicinity of 0.1. If θpl < 0.1 second order effects need not be considered as per EC8-1. The authors believe that this leads to undersized BRBF elements and inferior seismic performance for such structures (a good example is archetype 5HH with θpl = 0.1 where P-Δ effects were not considered). We suggest consideration of second order effects for θpl in the range of] 0; 0.2] through an enhanced set of rules that reduce the amplification of internal forces for highly P-Δ sensitive structures. Such modifications in EC8 require further research that is out of the scope of this paper. Performance of the archetypes is evaluated at the Sa,L = 1.5 Sa,D spectral acceleration level. The conditional collapse probability of all archetypes at Sa,L is less than 20%. The collapse probability of the

main structural characteristics are summarized in Table 6. Main results of the analyses are summarized in Table 7. Detailed results and design parameters of all archetypes are attached as Supplementary Material to this paper. The standard deviation of empirical fragility curves (βRTR) is in the range of 0.22–0.37. These values suggest smaller variation in the response of BRBF than the 0.4 value suggested by FEMA P-695 developers. Further investigation is recommended through the analysis of other types of BRBFs and preferably other types of steel dissipative braced frames to understand the applicability limits of the 0.4 standard deviation. Until these limits are identified, it is recommended for other researchers to use the method applied in this paper for empirical fragility curve evaluation. Collapse probabilities are generally smaller for archetypes at a region of moderate seismicity than for those at a site of high seismicity. Design of the former group of archetypes is governed by displacement limits to avoid excessive second order effects. These limits lead to oversized BRB elements and structural overstrength factors (γRd) beyond 2.0 for buildings with more than 4 stories regardless of the braced seismic mass (see Table 6 for details). A high γRd reduces the economy of such designs because it leads to an effective behavior factor (i.e. the ratio of the behavior factor and the structural overstrength factor - a good indicator of the expected cost of non-dissipative elements) as low as 2.0. The lowest probabilities of collapse are experienced among these tall structures. This suggests that design of such buildings yields conservative results. The authors believe that this is not a BRBF-specific issue, but a general problem with steel frame design in the EC8-1 standard. Economy of such frames in regions of moderate seismicity could be improved by enhancing the rules of 262

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Table 7 Result summary for archetypes and Performance Groups.

Note: SD, SCT, SCT,S are the design, collapse, and scaled collapse seismic intensity at T1 period, respectively; βRTR is the standard deviation of the empirical fragility curve from IDA results; SSF is the Spectral Shape Factor, βTOT is the adjusted standard deviation of the fragility curves considering additional uncertainty sources; PC | Sa,L is the probability of collapse conditioned on Sa,L = 1.5 SD; PC is the probability of collapse over 50 years; β is the reliability index.

– Reliability indices are evaluated to facilitate the standardization of reliability limits for the seismic hazard. Such limits would make seismic performance assessment conform with the Eurocode concept of regulating collapse probability over the pre-defined lifetime of a structure.

majority of Performance Groups is below 10%; the result of PG-7 is also considered practically acceptable. These results confirm the high performance of BRBF designed with the procedure proposed in the first part of this paper. 6. Concluding remarks

24 BRBF archetypes were designed with the proposed procedure. A detailed numerical model was developed for each archetype. Special attention was paid to BRB element modeling. A new numerical material model was developed and implemented in the OpenSees finite element code that can capture the nonlinear cyclic response of BRBs with more efficiency and accuracy than other solutions in the literature. Several failure modes of braced frame columns were considered to improve numerical model reliability at seismic intensities beyond the design level. The probabilistic seismic performance assessment presented in this paper confirms that all archetypes pass the acceptance criteria on collapse probability. This is considered strong support for the European application of the BRBF design procedure proposed in the first part of this paper. Performance of archetypes in moderate seismicity suggests conservativism in the capacity design of EC8-1, especially concerning the consideration of second-order effects. Further research is recommended to investigate this phenomenon in detail.

The authors present a Eurocode conforming design procedure for Buckling Restrained Braced Frames in the first part of this paper. The procedure is proposed to become part of the next revision of the Eurocode 8 standard. It is developed as an extension of existing regulations for steel Concentrically Braced Frames introducing Buckling Restrained Braces as a special type of concentric bracing. If adopted, the proposed procedure will allow European engineers to design BRBF within the framework of the widely accepted Eurocode standards. The second part of this paper focuses on the evaluation of the proposed design procedure. The authors developed a seismic performance assessment environment for this purpose. The environment is based on the recommendations of the FEMA P-695 document and enhances them at the following steps: – Global ductility evaluation uses the maximum point and the initial slope of the capacity curve rather than considering the softening part of the curve. – Empirical fragility curves are defined using collapse probability estimates at all investigated spectral acceleration levels rather than using only the median collapse intensity. – Record-to-record variability is determined by fitting a lognormal distribution to empirical fragility curve points rather than using an approximate value based on data from other types of structures.

Acknowledgements The work reported in the paper has been developed in the framework of the “Talent care and cultivation in the scientific workshops of BME” project. This project is supported by the TÁMOP-4.2.2.B-10/12010-0009 and the OTKA K 115673 grants. This paper was also 263

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