Development of standard dynamic loading protocol for bucklingrestrained braced frames M. Dehghani & R. Tremblay École Polytechnique de Montréal, Montréal, Québec, Canada

ABSTRACT: In this study a dynamic loading protocol is developed for seismic qualification of bucklingrestrained braces in western Canada. Series of multi-storey steel braced frames incorporating bucklingrestrained braces were designed according to the latest edition of Canadian building codes, simulated using refined inelastic behavioural models, and subjected to a set of 20 ground motions compatible with earthquake scenario of western Canada. Key response parameters such as brace deformation and deformation rate histories were extracted and subjected to cycle counting and statistical post-processing. The adequacy of the current Canadian codes for the design of BRBFs is also investigated.

1.1 Buckling-restrained braced frames It is generally accepted that Buckling-Restrained Braced Frames (BRBFs) exhibit better seismic performance than the conventional steel concentrically braced frames. In BRBFs, conventional bracing elements are replaced by braces made of a ductile slender core and an encasing system that prevents buckling of the core. Since lateral buckling of the fuse (core) is prohibited, within the design range of deformations, this bracing element shows almost identical force-deformation response in compressive and tensile loading cycles. This leads to more stable hysteresis behaviour, enhanced element ductility, and better system performance. A special type of this brace element referred to “all-steel BRB” is illustrated in Figure 1.

Figure 1. a) Possible restraining mechanisms in “all-steel BRB” members; b) BRB core.

Recently, the Canadian steel design standard CSA S16-09 (CSA, 2009) has included BRBFs for seismic applications; however, the system can be used provided that satisfactory performance of the bracing members has been demonstrated through qualifi-

cation test program conducted on full-scale prototypes. 1.2 Loading protocols Seismic performance of structural components is widely evaluated by deformation-controlled cyclic tests with regular and simple deformation patterns or loading protocols. In past researches, different loading protocols have been developed for various purposes (Krawinkler, 2009). Some of them have been implemented in codes for the seismic qualification of steel connection assemblies and structural elements such as BRBs. The loading protocol specified in CSA S16-09 is the same as the one given in Appendix T of AISC 341-05 (AISC, 2005). This protocol includes a series of displacement cycles with stepwise incremented ductility ratio amplitudes (Figure 2). This protocol was developed for BRBFs assumed to be located along the west coast of California and designed according to the USA provisions (Sabelli et al., 2003) and may not be suitable for BRBFs designed and built in Canada. Ductility, μ = Δ/Δy

1 INTRODUCTION

10 7.5 5 2.5 0 -2.5 -5 -7.5 -10

Pseudo Time

Figure 2. AISC 341-05 loading protocol for BRBFs.

The AISC protocol is applied in a quasi-static manner and loading (strain) rate effects are therefore ignored. Past studies have shown that high loading rates as those expected under design ground motions can increase the yield strength of structural steels (Bruneau et al., 1998). This increased yield strength may amplify the force demand used for the design of force-controlled elements such as connections, columns and beams. High loading rates may also cause damage concentration and pronounced brittle fracture modes (Fell, 2008; Jones, 1997). The main motivation for this study is the development of a standard dynamic loading protocol that could be used for the testing of BRBFs designed and constructed in Canada. The performance of codeconforming BRBFs is also studied. 1.3 Research Methodology A total of 56 multi-storey BRBFs were designed in accordance with the latest Canadian code provisions. The building structures were then subjected to a set of site-representative historical ground motions through nonlinear dynamic time history analyses. Axial deformation time histories of the most damaged brace elements were extracted and subjected to rainflow cycle counting. Representative brace axial deformation rate was also obtained by means of statistical analysis.

based on 2% probability of exceedance in 50 years, is shown in Figure 5. Buildings were assumed not to have any type of horizontal or vertical irregularities; however, storey shears per frame were amplified by 1.06 to account for accidental torsion effects. The equivalent static force method was adopted for the calculation of the lateral forces. The fundamental periods of the buildings were taken equal to 2.0 times the period obtained from the code empirical formula, as permitted in NBCC 2010. The lateral forces were amplified for P-Delta effects as specified in CSA S16-09. The brace cores were made of steel plates conforming to ASTM A572, grade 50 (Fy = 345 MPa) specifications, a readily available steel plate material in North America. ASTM A992 steel was selected for the wide flange beams and columns. As per the code, the ductility- and overstrength-related force modification factors were set to Rd = 4.0 and Ro = 1.2, respectively, and the buildings were assumed to be of normal importance category with IE = 1.0. Inelastic lateral inter-storey drifts were obtained by multiplying the elastic drifts by Ro × Rd / IE. Code limitation for inter-storey drift (2.5%) was not governed. Brace and column axial flexibilities were included in the calculation of the lateral deformations, assuming the beam axial deformations would be small due to the presence of the concrete floor slabs. 2.3 Capacity-based design of frame components

2 DESIGN OF PROTOTYPE STRUCTURES 2.1 Basic assumptions The prototype structures selected for this study are typical office buildings located on soft rock (site class C) in Victoria, British Columbia, Canada. Victoria was selected for this study because it is exposed to the highest level of seismic hazard (highest deign spectral acceleration) in the western part of Canada. All buildings have 36 × 54 m plan dimensions, with columns spaced 9.0 m c/c in both directions, and uniform storey height of 4.0 m. The buildings have 3, 5 ,7 and 9 storeys and are referred to as Buildings Nos. 3S, 5S, 7S, and 9S, respectively. In each direction, lateral loads are resisted by braced bent located along the building exterior walls. 2.2 Lateral force calculation Structural design was carried out using the National Building Code of Canada (NBCC 2010) (NRCC, 2010) and CSA S16-09 (CSA, 2009). Typical gravity loads for office building applications were assigned so that effective seismic weights of floors and roof were 4.75 and 4.0 kPa, respectively. Live load reduction was considered in the design of the gravity columns and the calculation of P-Delta effects. Uniform Hazard Spectrum for site class C, which is

As required by CSA S16-09, a capacity-based design approach was used for the design of the brace connections, columns, and beams. In this method, maximum probable forces that can be generated by yielding of the brace members were calculated first. These brace forces depend on the anticipated core strains that can be estimated from the maximum expected inelastic storey drifts and kinematics of the braced frame storeys (see Equation 2). Approximately, the maximum tensile and compressive axial loads in brace elements can be obtained by adjustment of the core yield strength for cyclic strain hardening by using:

Tmax ω R y Fy Ac

(1.a)

Pmax ω β R y Fy Ac

(1.b)

where ω = the (tension) strain hardening adjustment factor; β = is the compressive strength adjustment factor. Both ω and β depend on the axial deformation experienced by the brace (see Figure 3). In this study ω and β were assumed to be 1.25 and 1.10, respectively; Ry = the ratio of the probable to nominal steel yield strengths, which was taken as 1.12 in this study; and Ac = the cross-sectional area of the core. According to CSA S16-09, the calculated inelastic storey drift has to be amplified by 2.0 to determine

the brace axial strain and the resulting maximum axial loads. This amplification factor is meant to cover the uncertainties observed in maximum inter-storey drifts of prototypes studied by (Sabelli et al., 2003). Core strains, εc, can be calculated from interstorey drift ratios computed in the analyses assuming that the projected segments at the brace ends remain essentially elastic and the plastic strain is uniformly distributed along the core yielding length. Provided that the same grade of steel is used for the core and projected areas, core strain can be estimated from: 2.0 Ro Rd sin 2ψ εc θ ηωε y 1 γ γ IE 2 2.0 Ro Rd sin 2ψ θ γ IE 2

(2)

1.5 1 0.5 0 -0.5 -1 -1.5

2.4 Optimization of frame configuration An optimization study was conducted to determine the most effective BRBF configurations. The diagonal and chevron bracing configurations shown in Figure 4a were designed for 3-, 5-, 7- and 9-storey structures. The steel tonnage required for the beams and columns of the braced frames were normalized with respect to the minimum amount of steel required for carrying gravity loads only. According to this study, configuration C6 was found to be the optimum patterns for the chevron braced frames (Figure 4). This study revealed that column sizes are highly affected by the bracing pattern and that the most direct or shortest lateral load path resulted in the most effective configuration. It is recognized that this simple steel weight comparison may not directly translate into cost effectiveness. Nevertheless, it was used in this study to select the structural systems considered in the development of the loading protocol. a)

b) Mean of Normalized Steel Weight

Normalized Force, P/Py

where θ = inter-storey drift under factored loads; ψ = brace angle with respect to the horizontal plane; η = ratio of the core cross-section area to the cross sectional area of the projected segment; εy = yield strain of steel; and γ = ratio of the yielding length to the brace total length. Based on the geometry of the braced frames studied and typical BRB design, γ was taken equal to 0.7 and 0.55 for the diagonal and chevron bracing configurations, respectively, assuming that the lengths of the end connections does not vary with the brace size. The parameter η was also taken as 1/4. An axial load-deformation backbone equation is required for the estimation of the maximum axial loads generated by the braces based on the core strain. In this study, a simplified tri-linear forcedeformation backbone was adopted (see Figure 3). Accuracy of this backbone equation was confirmed using cyclic analysis of a refined and calibrated inelastic brace model.

designed for the combination of seismic induced axial forces and bending moments resulting from gravity loads. The beams were assumed to be laterally restrained by means of the floor slabs.

-15

-10

-5 0 5 Ductility, μ = Δ/Δy

10

15

C1

C2

C3

C4

C5

C6

2.5 2 1.5

Girders

1

Columns

0.5

Total

0 C1

C2

C3

C4

C5

C6

Configuration

Figure 3. Normalized force-deformation backbone used in this study.

Figure 4. a) Configuration used in optimization study; b) Results of configuration optimization of V or Inverted-V type BRBFs.

The maximum computed brace forces were then applied as loads to the rest of the structural members such that these members are designed for the maximum anticipated forces. As per CSA S16-09, the columns were also designed for a combination of factored axial loads and additional accidental bending moment equal to 20% of the plastic moment of the column cross-section. The same column size was used for two consecutive storeys. The beams were

Preliminary nonlinear time history analyses on series of equivalent diagonal and chevron (V or Inverted-V) BRBFs also showed that chevron frames generally undergo higher ductility demands. Based on the optimization study and these preliminary dynamic analyses, it was therefore decided to limit the strain demand assessment to the C6 bracing configuration (Double storey X or split-X configuration).

3.1 Structural Modeling The analyses were performed using the SAP2000 program. The axial force-deformation response of the BRB yielding segments was modeled using the simplified Bouc-Wen hysteresis law (CSI, 2010). This model is appropriate for reproducing the response of ductile elements exhibiting cyclically stable hysteretic properties. BRB cores exhibit an axial strength that does not deteriorate as well as cyclic hardening, and represent a good example of such ductile elements. Moreover, Bauschinger effect can be simulated with this behavioural model. In this model, the restoring force at a given time can be obtained from the following equation:

F (t ) a k ut 1 a k zt

(3)

where a = ratio of the post-yielding stiffness to the initial elastic stiffness; k = initial elastic stiffness; u(t) = deformation time history, z(t) = hysteresis parameter obtained by solving the following explicit differential equations:

n k u(t ) 1 z P z t y k u (t ) Py

if

u (t ) z t 0

(4) Otherwise

where Py = brace yield strength, n = yield exponent that controls the sharpness of the transition between initial stiffness and post-yield stiffness. Curve fitting conducted on the actual test results showed that n = 2.0 and a = 0.025 are appropriate values. Since cyclic hardening cannot be simulated by this model, system yield force, Py, should be adjusted based on the maximum expected deformation and post-yield stiffness ratio (a). In this study, the yield strength was multiplied by an adjustment factor 1.24 assuming that the brace elements will experience deformations corresponding to a ductility of 10. This factor is then re-adjusted based on the maximum ductility ratio observed in each analysis case. The flexural behaviour of semi-rigid brace end connections was also modeled using non-linear inelastic rotational springs. Rigid elements were implemented to model the projected areas of the brace assemblies and the gusset plates. The remaining structural elements were modeled using elastic linear elements. Columns were assumed to be continuous but rotationally unrestrained at their bases. Since the flexural stiffness resulting from the continuity of the columns may reduce brace demand (after yielding of braces), columns were orientated so that weak axis bending developed in the plane of the frames. Beams were pin-connected to the columns, reflecting current practice in Canada. Lean-on gravity columns were designed and included in the models to account

for total P-Delta effects in the analyses. These columns were rigidly laterally linked to the braced frame columns at every level using diaphragm constraints. 3.2 Ground motion selection and scaling A set of 20 historical ground motions compatible with the predicted earthquake scenario in Victoria were selected from the PEER Strong Ground Motion Database (PEER, 2005) and utilized in this study. The basic selection criteria were: Magnitude (Mw ≥ 6.5), Hypocentral distance (10 ≤ R ≤ 30 km), Closest distance to fault (D ≥ 10 km), Shear wave velocity in top 30 meters of soil (250 ≤ Vs30 ≤ 750 m/s). Location of the events was restricted to California due to geological similarities with the structure site. Despite all the applied filters, 90 compatible records were found. Additional filters then were applied to reduce the size of the ensemble. Those were related to basic ground motion properties such as frequency content index, significant duration, and Arias Intensity Index, as calculated using the SeismoSpect software (SeismoSoft, 2010). Based on statistical distribution of those parameters, the ground motions were ranked and the first 20 records were selected. Recent studies showed that the scaling to match the spectral shape (either at the building fundamental period or over at a range of periods) are too conservative and may lead to overestimation of structural responses. In view of this, a different linear amplitude scaling method independent of the fundamental period of the structures was developed and adopted in this study. In this method, referred to as the Least Moving Average method, the ratio of the code spectral ordinate (STarget) to the 5% damped spectral acceleration of the record (SGM) is calculated at every period between 0.2 and 4.0 s. The average (STarget / SGM) ratio over a period range from 0.5Ti to 1.5Ti is then determined at every period Ti varying from 0.40 to 2.67 s, and the minimum average ratio is used as the scaling factor. This method, which leads to matching in the region of the spectrum where SGM values are closest to STarget, was found to generally give lower scaling factors than scaling at a predefined period or over a given period range. Design UHS for Victoria (Target) and mean and median spectral accelerations for all 20 records are shown in Figure 5. 1.5 Spect. Acc., Sa, g

3 MODELING AND ANALYSIS

Target Mean Median

1 0.5 0 0

1

2 Period, s

Figure 5. Results of ground motion scaling.

3

4

3.3 Nonlinear time history analyses 2D models of the structures were subjected to the selected and scaled ground motions. Since accidental torsion effects were included in the design of the prototype structures, the records were further amplified by 1.06 to achieve consistency between the demand and the design assumptions. Nonlinear dynamic analyses were conducted using NewmarkBeta integration technique (Gamma = 0.5; Beta = 0.25). Rayleigh damping was specified based on 5% of critical damping in all vibration modes. In total, 80 dynamic analyses were carried out and 1920 brace deformation histories were generated. Time histories of response parameters such as brace axial deformations, inter-storey drifts, storey velocities and accelerations, and columns' internal actions were also obtained. Four (4) nonlinear static (pushover) analyses with inverted triangular force pattern were also performed to compare dynamic and static responses. In these analyses, the roof level was pushed up to a target displacement corresponding to 2.5% drift ratio.

For brace-level performance evaluation, the maximum brace ductility-ratio (μmax = max |U / Uy|); maximum ductility-rate ( μ max = max |dμ / dt|); cumulative ductility (Σμ = Σ|μt + Δt – μt|); and normalized hysteretic energy (EHys.N = Σ(μ × P / Py)) were calculated as the damage indices, where U is the brace core deformation. According to these damage indices, it was found that, in all cases, braces at the uppermost storey were the most damaged ones, probably as a consequence of higher mode effects. Table 1 shows the statistics of these damage indices. The maximum ductility ratio is equal to 5.5, which is lower than the values obtained in past studies. The cumulative ductility, which is widely used for performance evaluation of BRBFs, is about 40% of the value proposed in AISC 341-05 Appendix T. The maximum normalized hysteretic energy is also approximately as low as 6% of the hysteretic energy imposed by the AISC protocol. Figure 7 shows ductility ratio histories of the brace elements with the maximum ductility amplitude and maximum cumulative ductility. These elements are selected in the subsequent steps of this study.

3.4 Performance of the analysed models

Storey

7 6 5 4 3 2 1 0.0%

Nonlinear Time History Pushover Design Drifts

2.0% 4.0% 6.0% Inter-storey Drift Ratio

8.0%

Figure 6. Distribution of inter-storey drifts for Prototype 7S.

Table 1. Descriptive statistics of the damage indices. Damage Index

Max.

Mean

Median

STD

μmax

5.5

2.0

1.8

0.8

(dμ/dt)max

30.3

11.4

10.3

3.1

(Σμ)max

72

41

39

7

(EHys.N)max

9.5

2.0

1.6

1.2

Ductility Ratio, μ

The performance of the analysed systems was evaluated at both the system and brace levels. Systemlevel performance indicators included the maximum inter-storey drifts, storey velocities and accelerations, and base shear ratios. In the most extreme cases (Prototypes Nos. 3S & 7S), inter-storey drifts were about 1%, which is lower than the design drifts and previously reported values in the literature (Fahnestock et al., 2007; Sabelli et al., 2003). These lower drift values can be attributed to the ground motion scaling methodology adopted in this study. Maximum storey accelerations were bounded to 0.47 g (Prototype 3S). In Figure 6, the design storey drifts for Prototype 7S are compared to the results of the pushover analysis and the maximum inter-storey drifts among all 20 ground motion records from nonlinear dynamic analysis. The push-over analysis reveals significant concentration of deformation demand along the building height. As shown, no such soft storey response was observed in the nonlinear dynamic analyses and the computed inter-storey drifts are smaller than the pushover or design values..

4 2 0 Brace with (Σμ)max Brace with μmax

-2 -4 -6 0

10

20

30

40

50

60

Time, s.

Figure 7. Typical ductility ratio time histories.

4 DEVELOPMENT OF LOADING PROTOCOL 4.1 Rainflow cycle counting The rainflow cycle counting algorithm is widely used for fatigue life assessment of mechanical and structural components subjected to irregular nonconstant and complex loading histories. In this relatively simple method an irregular loading signal is converted into almost equivalent regular one with defined stress or strain amplitudes. With respect to the four previously defined damage indices, the four most damaged brace elements were selected for cycle counting. The results of cy-

Number of full cycles

cle counting were then simplified by neglecting mean stress effect. Figure 8 shows the distribution of ductility ratio amplitude. By neglecting ductility ratios less than 0.75, and sorting the rest of data in ascending format, four loading protocols were obtained. Then, a standard loading protocol is defined by taking the average of number of cycles and the maximum of the ductility ratio amplitudes (see Figure 9). 10

8

8 6

4

5

4

3

2

1

2

ACKNOWLEDGEMENTS

1

1

2.25

2.5

0 0.5

0.75

1

1.25

1.5

1.75

2

Ductility Ratio Amplitude, μ

Figure 8. Distribution of ductility ratio amplitudes.

Ductility Ratio, μ

By reducing ductility ratio signals to only-peak and valley data and removing waves with amplitude less than 0.75, the statistical distribution of ductilityrate was obtained for each brace time history signals. The „mean + one standard deviation‟ and the maximum of the rates were about 10 and 17 ductility per second, respectively. The duration over which 90% of the energy is dissipated in a brace can also be a criterion to convert a quasi-static protocol into a dynamic one. Using this parameter gives a total testing duration of 7.5 seconds (on average). A sine wave signal with a constant average loading rate of 10 ductility per second can be used throughout the 7.5 second duration. This loading signal is illustrated in Figure 9. Total number of cycles is 13. 3.5 2.5 1.5 0.5 -0.5 -1.5 -2.5 -3.5

5 cycles

0

2 cycles

6 cycles

2.5

5

7.5

Time, s

Figure 9. Dynamic loading protocol with a defined duration.

5

cation factor of 2.0 for capacity design of frame (as proposed by code) seems to be conservative; All brace damage indices are lower than criteria currently used for seismic qualification; A dynamic loading protocol with a representative loading rate of 10 ductility per second was developed, The results presented herein are still preliminary. The response of the structures needs to be examined under additional record sets. Processing analysis will also be refined in subsequent phases of the research.

CONCLUSIONS

A series of 3- to 9-storey prototype BRB frames were designed according to Canadian codes and subjected to a set of 20 seismic records through nonlinear dynamic analysis. Followings are the results: Double storey X (split-X) bracing configuration results in minimum steel tonnage; Inelastic inter-storey drifts are lower than the design values as well as values reported in past studies; accordingly, brace deformation amplifi-

The financial support provided by the Natural Sciences and Engineering Research Council of Canada (NSERC) for the Canadian Seismic Research Network (CSRN) is acknowledged. REFERENCES AISC 2005. Seismic Provisions for Structural Steel Buildings (Including Supplement No. 1). Chicago, IL: American Institute of Steel Construction, Inc. Bruneau, M., Uang, C.-M., & Whittaker, A. 1998. Ductile design of steel structures. New York: McGraw-Hill. CSA 2009. Limit States Design of Steel Structures. Mississauga, Ontario, Canada: Canadian Standards Association. CSI 2010. CSI Analysis Reference Manual For SAP2000®, ETABS®, SAFE® and CSiBridge™. Berkeley, California, USA: Computers and Structures, Inc. Fahnestock, L. A., Sause, R., & Ricles, J. M. 2007. Seismic Response and Performance of Buckling-Restrained Braced Frames. Journal of Structural Engineering, 133(9): 11951204. Fell, B. 2008. Large-scale testing and simulation of earthquake-induced ultra low cycle fatigue in bracing members subjected to cyclic inelastic buckling. PhD Thesis, University of California, Davis, CA. Jones, N. 1997. Structural Impact: Cambridge University Press. Krawinkler, H. 2009. Loading Histories for Cyclic Tests in Support of Performance Assessment of Structural Components. Paper presented at the The 3rd International Conference on Advances in Experimental Structural Engineering, San Francisco, California. NRCC 2010. National Building Code of Canada. Ottawa, ON.: National Research Council Canada, Institute for Research in Construction. PEER 2005. PEER Strong Ground Motion Database. 2005. Retrieved 27/05/2011 from http://peer.berkeley.edu/nga/ Sabelli, R., Mahin, S., & Chang, C. 2003. Seismic demands on steel braced frame buildings with buckling-restrained braces. Engineering Structures, 25(5), 655-666. SeismoSoft 2010. SeismoSpect (Version 1.0.2).

ABSTRACT: In this study a dynamic loading protocol is developed for seismic qualification of bucklingrestrained braces in western Canada. Series of multi-storey steel braced frames incorporating bucklingrestrained braces were designed according to the latest edition of Canadian building codes, simulated using refined inelastic behavioural models, and subjected to a set of 20 ground motions compatible with earthquake scenario of western Canada. Key response parameters such as brace deformation and deformation rate histories were extracted and subjected to cycle counting and statistical post-processing. The adequacy of the current Canadian codes for the design of BRBFs is also investigated.

1.1 Buckling-restrained braced frames It is generally accepted that Buckling-Restrained Braced Frames (BRBFs) exhibit better seismic performance than the conventional steel concentrically braced frames. In BRBFs, conventional bracing elements are replaced by braces made of a ductile slender core and an encasing system that prevents buckling of the core. Since lateral buckling of the fuse (core) is prohibited, within the design range of deformations, this bracing element shows almost identical force-deformation response in compressive and tensile loading cycles. This leads to more stable hysteresis behaviour, enhanced element ductility, and better system performance. A special type of this brace element referred to “all-steel BRB” is illustrated in Figure 1.

Figure 1. a) Possible restraining mechanisms in “all-steel BRB” members; b) BRB core.

Recently, the Canadian steel design standard CSA S16-09 (CSA, 2009) has included BRBFs for seismic applications; however, the system can be used provided that satisfactory performance of the bracing members has been demonstrated through qualifi-

cation test program conducted on full-scale prototypes. 1.2 Loading protocols Seismic performance of structural components is widely evaluated by deformation-controlled cyclic tests with regular and simple deformation patterns or loading protocols. In past researches, different loading protocols have been developed for various purposes (Krawinkler, 2009). Some of them have been implemented in codes for the seismic qualification of steel connection assemblies and structural elements such as BRBs. The loading protocol specified in CSA S16-09 is the same as the one given in Appendix T of AISC 341-05 (AISC, 2005). This protocol includes a series of displacement cycles with stepwise incremented ductility ratio amplitudes (Figure 2). This protocol was developed for BRBFs assumed to be located along the west coast of California and designed according to the USA provisions (Sabelli et al., 2003) and may not be suitable for BRBFs designed and built in Canada. Ductility, μ = Δ/Δy

1 INTRODUCTION

10 7.5 5 2.5 0 -2.5 -5 -7.5 -10

Pseudo Time

Figure 2. AISC 341-05 loading protocol for BRBFs.

The AISC protocol is applied in a quasi-static manner and loading (strain) rate effects are therefore ignored. Past studies have shown that high loading rates as those expected under design ground motions can increase the yield strength of structural steels (Bruneau et al., 1998). This increased yield strength may amplify the force demand used for the design of force-controlled elements such as connections, columns and beams. High loading rates may also cause damage concentration and pronounced brittle fracture modes (Fell, 2008; Jones, 1997). The main motivation for this study is the development of a standard dynamic loading protocol that could be used for the testing of BRBFs designed and constructed in Canada. The performance of codeconforming BRBFs is also studied. 1.3 Research Methodology A total of 56 multi-storey BRBFs were designed in accordance with the latest Canadian code provisions. The building structures were then subjected to a set of site-representative historical ground motions through nonlinear dynamic time history analyses. Axial deformation time histories of the most damaged brace elements were extracted and subjected to rainflow cycle counting. Representative brace axial deformation rate was also obtained by means of statistical analysis.

based on 2% probability of exceedance in 50 years, is shown in Figure 5. Buildings were assumed not to have any type of horizontal or vertical irregularities; however, storey shears per frame were amplified by 1.06 to account for accidental torsion effects. The equivalent static force method was adopted for the calculation of the lateral forces. The fundamental periods of the buildings were taken equal to 2.0 times the period obtained from the code empirical formula, as permitted in NBCC 2010. The lateral forces were amplified for P-Delta effects as specified in CSA S16-09. The brace cores were made of steel plates conforming to ASTM A572, grade 50 (Fy = 345 MPa) specifications, a readily available steel plate material in North America. ASTM A992 steel was selected for the wide flange beams and columns. As per the code, the ductility- and overstrength-related force modification factors were set to Rd = 4.0 and Ro = 1.2, respectively, and the buildings were assumed to be of normal importance category with IE = 1.0. Inelastic lateral inter-storey drifts were obtained by multiplying the elastic drifts by Ro × Rd / IE. Code limitation for inter-storey drift (2.5%) was not governed. Brace and column axial flexibilities were included in the calculation of the lateral deformations, assuming the beam axial deformations would be small due to the presence of the concrete floor slabs. 2.3 Capacity-based design of frame components

2 DESIGN OF PROTOTYPE STRUCTURES 2.1 Basic assumptions The prototype structures selected for this study are typical office buildings located on soft rock (site class C) in Victoria, British Columbia, Canada. Victoria was selected for this study because it is exposed to the highest level of seismic hazard (highest deign spectral acceleration) in the western part of Canada. All buildings have 36 × 54 m plan dimensions, with columns spaced 9.0 m c/c in both directions, and uniform storey height of 4.0 m. The buildings have 3, 5 ,7 and 9 storeys and are referred to as Buildings Nos. 3S, 5S, 7S, and 9S, respectively. In each direction, lateral loads are resisted by braced bent located along the building exterior walls. 2.2 Lateral force calculation Structural design was carried out using the National Building Code of Canada (NBCC 2010) (NRCC, 2010) and CSA S16-09 (CSA, 2009). Typical gravity loads for office building applications were assigned so that effective seismic weights of floors and roof were 4.75 and 4.0 kPa, respectively. Live load reduction was considered in the design of the gravity columns and the calculation of P-Delta effects. Uniform Hazard Spectrum for site class C, which is

As required by CSA S16-09, a capacity-based design approach was used for the design of the brace connections, columns, and beams. In this method, maximum probable forces that can be generated by yielding of the brace members were calculated first. These brace forces depend on the anticipated core strains that can be estimated from the maximum expected inelastic storey drifts and kinematics of the braced frame storeys (see Equation 2). Approximately, the maximum tensile and compressive axial loads in brace elements can be obtained by adjustment of the core yield strength for cyclic strain hardening by using:

Tmax ω R y Fy Ac

(1.a)

Pmax ω β R y Fy Ac

(1.b)

where ω = the (tension) strain hardening adjustment factor; β = is the compressive strength adjustment factor. Both ω and β depend on the axial deformation experienced by the brace (see Figure 3). In this study ω and β were assumed to be 1.25 and 1.10, respectively; Ry = the ratio of the probable to nominal steel yield strengths, which was taken as 1.12 in this study; and Ac = the cross-sectional area of the core. According to CSA S16-09, the calculated inelastic storey drift has to be amplified by 2.0 to determine

the brace axial strain and the resulting maximum axial loads. This amplification factor is meant to cover the uncertainties observed in maximum inter-storey drifts of prototypes studied by (Sabelli et al., 2003). Core strains, εc, can be calculated from interstorey drift ratios computed in the analyses assuming that the projected segments at the brace ends remain essentially elastic and the plastic strain is uniformly distributed along the core yielding length. Provided that the same grade of steel is used for the core and projected areas, core strain can be estimated from: 2.0 Ro Rd sin 2ψ εc θ ηωε y 1 γ γ IE 2 2.0 Ro Rd sin 2ψ θ γ IE 2

(2)

1.5 1 0.5 0 -0.5 -1 -1.5

2.4 Optimization of frame configuration An optimization study was conducted to determine the most effective BRBF configurations. The diagonal and chevron bracing configurations shown in Figure 4a were designed for 3-, 5-, 7- and 9-storey structures. The steel tonnage required for the beams and columns of the braced frames were normalized with respect to the minimum amount of steel required for carrying gravity loads only. According to this study, configuration C6 was found to be the optimum patterns for the chevron braced frames (Figure 4). This study revealed that column sizes are highly affected by the bracing pattern and that the most direct or shortest lateral load path resulted in the most effective configuration. It is recognized that this simple steel weight comparison may not directly translate into cost effectiveness. Nevertheless, it was used in this study to select the structural systems considered in the development of the loading protocol. a)

b) Mean of Normalized Steel Weight

Normalized Force, P/Py

where θ = inter-storey drift under factored loads; ψ = brace angle with respect to the horizontal plane; η = ratio of the core cross-section area to the cross sectional area of the projected segment; εy = yield strain of steel; and γ = ratio of the yielding length to the brace total length. Based on the geometry of the braced frames studied and typical BRB design, γ was taken equal to 0.7 and 0.55 for the diagonal and chevron bracing configurations, respectively, assuming that the lengths of the end connections does not vary with the brace size. The parameter η was also taken as 1/4. An axial load-deformation backbone equation is required for the estimation of the maximum axial loads generated by the braces based on the core strain. In this study, a simplified tri-linear forcedeformation backbone was adopted (see Figure 3). Accuracy of this backbone equation was confirmed using cyclic analysis of a refined and calibrated inelastic brace model.

designed for the combination of seismic induced axial forces and bending moments resulting from gravity loads. The beams were assumed to be laterally restrained by means of the floor slabs.

-15

-10

-5 0 5 Ductility, μ = Δ/Δy

10

15

C1

C2

C3

C4

C5

C6

2.5 2 1.5

Girders

1

Columns

0.5

Total

0 C1

C2

C3

C4

C5

C6

Configuration

Figure 3. Normalized force-deformation backbone used in this study.

Figure 4. a) Configuration used in optimization study; b) Results of configuration optimization of V or Inverted-V type BRBFs.

The maximum computed brace forces were then applied as loads to the rest of the structural members such that these members are designed for the maximum anticipated forces. As per CSA S16-09, the columns were also designed for a combination of factored axial loads and additional accidental bending moment equal to 20% of the plastic moment of the column cross-section. The same column size was used for two consecutive storeys. The beams were

Preliminary nonlinear time history analyses on series of equivalent diagonal and chevron (V or Inverted-V) BRBFs also showed that chevron frames generally undergo higher ductility demands. Based on the optimization study and these preliminary dynamic analyses, it was therefore decided to limit the strain demand assessment to the C6 bracing configuration (Double storey X or split-X configuration).

3.1 Structural Modeling The analyses were performed using the SAP2000 program. The axial force-deformation response of the BRB yielding segments was modeled using the simplified Bouc-Wen hysteresis law (CSI, 2010). This model is appropriate for reproducing the response of ductile elements exhibiting cyclically stable hysteretic properties. BRB cores exhibit an axial strength that does not deteriorate as well as cyclic hardening, and represent a good example of such ductile elements. Moreover, Bauschinger effect can be simulated with this behavioural model. In this model, the restoring force at a given time can be obtained from the following equation:

F (t ) a k ut 1 a k zt

(3)

where a = ratio of the post-yielding stiffness to the initial elastic stiffness; k = initial elastic stiffness; u(t) = deformation time history, z(t) = hysteresis parameter obtained by solving the following explicit differential equations:

n k u(t ) 1 z P z t y k u (t ) Py

if

u (t ) z t 0

(4) Otherwise

where Py = brace yield strength, n = yield exponent that controls the sharpness of the transition between initial stiffness and post-yield stiffness. Curve fitting conducted on the actual test results showed that n = 2.0 and a = 0.025 are appropriate values. Since cyclic hardening cannot be simulated by this model, system yield force, Py, should be adjusted based on the maximum expected deformation and post-yield stiffness ratio (a). In this study, the yield strength was multiplied by an adjustment factor 1.24 assuming that the brace elements will experience deformations corresponding to a ductility of 10. This factor is then re-adjusted based on the maximum ductility ratio observed in each analysis case. The flexural behaviour of semi-rigid brace end connections was also modeled using non-linear inelastic rotational springs. Rigid elements were implemented to model the projected areas of the brace assemblies and the gusset plates. The remaining structural elements were modeled using elastic linear elements. Columns were assumed to be continuous but rotationally unrestrained at their bases. Since the flexural stiffness resulting from the continuity of the columns may reduce brace demand (after yielding of braces), columns were orientated so that weak axis bending developed in the plane of the frames. Beams were pin-connected to the columns, reflecting current practice in Canada. Lean-on gravity columns were designed and included in the models to account

for total P-Delta effects in the analyses. These columns were rigidly laterally linked to the braced frame columns at every level using diaphragm constraints. 3.2 Ground motion selection and scaling A set of 20 historical ground motions compatible with the predicted earthquake scenario in Victoria were selected from the PEER Strong Ground Motion Database (PEER, 2005) and utilized in this study. The basic selection criteria were: Magnitude (Mw ≥ 6.5), Hypocentral distance (10 ≤ R ≤ 30 km), Closest distance to fault (D ≥ 10 km), Shear wave velocity in top 30 meters of soil (250 ≤ Vs30 ≤ 750 m/s). Location of the events was restricted to California due to geological similarities with the structure site. Despite all the applied filters, 90 compatible records were found. Additional filters then were applied to reduce the size of the ensemble. Those were related to basic ground motion properties such as frequency content index, significant duration, and Arias Intensity Index, as calculated using the SeismoSpect software (SeismoSoft, 2010). Based on statistical distribution of those parameters, the ground motions were ranked and the first 20 records were selected. Recent studies showed that the scaling to match the spectral shape (either at the building fundamental period or over at a range of periods) are too conservative and may lead to overestimation of structural responses. In view of this, a different linear amplitude scaling method independent of the fundamental period of the structures was developed and adopted in this study. In this method, referred to as the Least Moving Average method, the ratio of the code spectral ordinate (STarget) to the 5% damped spectral acceleration of the record (SGM) is calculated at every period between 0.2 and 4.0 s. The average (STarget / SGM) ratio over a period range from 0.5Ti to 1.5Ti is then determined at every period Ti varying from 0.40 to 2.67 s, and the minimum average ratio is used as the scaling factor. This method, which leads to matching in the region of the spectrum where SGM values are closest to STarget, was found to generally give lower scaling factors than scaling at a predefined period or over a given period range. Design UHS for Victoria (Target) and mean and median spectral accelerations for all 20 records are shown in Figure 5. 1.5 Spect. Acc., Sa, g

3 MODELING AND ANALYSIS

Target Mean Median

1 0.5 0 0

1

2 Period, s

Figure 5. Results of ground motion scaling.

3

4

3.3 Nonlinear time history analyses 2D models of the structures were subjected to the selected and scaled ground motions. Since accidental torsion effects were included in the design of the prototype structures, the records were further amplified by 1.06 to achieve consistency between the demand and the design assumptions. Nonlinear dynamic analyses were conducted using NewmarkBeta integration technique (Gamma = 0.5; Beta = 0.25). Rayleigh damping was specified based on 5% of critical damping in all vibration modes. In total, 80 dynamic analyses were carried out and 1920 brace deformation histories were generated. Time histories of response parameters such as brace axial deformations, inter-storey drifts, storey velocities and accelerations, and columns' internal actions were also obtained. Four (4) nonlinear static (pushover) analyses with inverted triangular force pattern were also performed to compare dynamic and static responses. In these analyses, the roof level was pushed up to a target displacement corresponding to 2.5% drift ratio.

For brace-level performance evaluation, the maximum brace ductility-ratio (μmax = max |U / Uy|); maximum ductility-rate ( μ max = max |dμ / dt|); cumulative ductility (Σμ = Σ|μt + Δt – μt|); and normalized hysteretic energy (EHys.N = Σ(μ × P / Py)) were calculated as the damage indices, where U is the brace core deformation. According to these damage indices, it was found that, in all cases, braces at the uppermost storey were the most damaged ones, probably as a consequence of higher mode effects. Table 1 shows the statistics of these damage indices. The maximum ductility ratio is equal to 5.5, which is lower than the values obtained in past studies. The cumulative ductility, which is widely used for performance evaluation of BRBFs, is about 40% of the value proposed in AISC 341-05 Appendix T. The maximum normalized hysteretic energy is also approximately as low as 6% of the hysteretic energy imposed by the AISC protocol. Figure 7 shows ductility ratio histories of the brace elements with the maximum ductility amplitude and maximum cumulative ductility. These elements are selected in the subsequent steps of this study.

3.4 Performance of the analysed models

Storey

7 6 5 4 3 2 1 0.0%

Nonlinear Time History Pushover Design Drifts

2.0% 4.0% 6.0% Inter-storey Drift Ratio

8.0%

Figure 6. Distribution of inter-storey drifts for Prototype 7S.

Table 1. Descriptive statistics of the damage indices. Damage Index

Max.

Mean

Median

STD

μmax

5.5

2.0

1.8

0.8

(dμ/dt)max

30.3

11.4

10.3

3.1

(Σμ)max

72

41

39

7

(EHys.N)max

9.5

2.0

1.6

1.2

Ductility Ratio, μ

The performance of the analysed systems was evaluated at both the system and brace levels. Systemlevel performance indicators included the maximum inter-storey drifts, storey velocities and accelerations, and base shear ratios. In the most extreme cases (Prototypes Nos. 3S & 7S), inter-storey drifts were about 1%, which is lower than the design drifts and previously reported values in the literature (Fahnestock et al., 2007; Sabelli et al., 2003). These lower drift values can be attributed to the ground motion scaling methodology adopted in this study. Maximum storey accelerations were bounded to 0.47 g (Prototype 3S). In Figure 6, the design storey drifts for Prototype 7S are compared to the results of the pushover analysis and the maximum inter-storey drifts among all 20 ground motion records from nonlinear dynamic analysis. The push-over analysis reveals significant concentration of deformation demand along the building height. As shown, no such soft storey response was observed in the nonlinear dynamic analyses and the computed inter-storey drifts are smaller than the pushover or design values..

4 2 0 Brace with (Σμ)max Brace with μmax

-2 -4 -6 0

10

20

30

40

50

60

Time, s.

Figure 7. Typical ductility ratio time histories.

4 DEVELOPMENT OF LOADING PROTOCOL 4.1 Rainflow cycle counting The rainflow cycle counting algorithm is widely used for fatigue life assessment of mechanical and structural components subjected to irregular nonconstant and complex loading histories. In this relatively simple method an irregular loading signal is converted into almost equivalent regular one with defined stress or strain amplitudes. With respect to the four previously defined damage indices, the four most damaged brace elements were selected for cycle counting. The results of cy-

Number of full cycles

cle counting were then simplified by neglecting mean stress effect. Figure 8 shows the distribution of ductility ratio amplitude. By neglecting ductility ratios less than 0.75, and sorting the rest of data in ascending format, four loading protocols were obtained. Then, a standard loading protocol is defined by taking the average of number of cycles and the maximum of the ductility ratio amplitudes (see Figure 9). 10

8

8 6

4

5

4

3

2

1

2

ACKNOWLEDGEMENTS

1

1

2.25

2.5

0 0.5

0.75

1

1.25

1.5

1.75

2

Ductility Ratio Amplitude, μ

Figure 8. Distribution of ductility ratio amplitudes.

Ductility Ratio, μ

By reducing ductility ratio signals to only-peak and valley data and removing waves with amplitude less than 0.75, the statistical distribution of ductilityrate was obtained for each brace time history signals. The „mean + one standard deviation‟ and the maximum of the rates were about 10 and 17 ductility per second, respectively. The duration over which 90% of the energy is dissipated in a brace can also be a criterion to convert a quasi-static protocol into a dynamic one. Using this parameter gives a total testing duration of 7.5 seconds (on average). A sine wave signal with a constant average loading rate of 10 ductility per second can be used throughout the 7.5 second duration. This loading signal is illustrated in Figure 9. Total number of cycles is 13. 3.5 2.5 1.5 0.5 -0.5 -1.5 -2.5 -3.5

5 cycles

0

2 cycles

6 cycles

2.5

5

7.5

Time, s

Figure 9. Dynamic loading protocol with a defined duration.

5

cation factor of 2.0 for capacity design of frame (as proposed by code) seems to be conservative; All brace damage indices are lower than criteria currently used for seismic qualification; A dynamic loading protocol with a representative loading rate of 10 ductility per second was developed, The results presented herein are still preliminary. The response of the structures needs to be examined under additional record sets. Processing analysis will also be refined in subsequent phases of the research.

CONCLUSIONS

A series of 3- to 9-storey prototype BRB frames were designed according to Canadian codes and subjected to a set of 20 seismic records through nonlinear dynamic analysis. Followings are the results: Double storey X (split-X) bracing configuration results in minimum steel tonnage; Inelastic inter-storey drifts are lower than the design values as well as values reported in past studies; accordingly, brace deformation amplifi-

The financial support provided by the Natural Sciences and Engineering Research Council of Canada (NSERC) for the Canadian Seismic Research Network (CSRN) is acknowledged. REFERENCES AISC 2005. Seismic Provisions for Structural Steel Buildings (Including Supplement No. 1). Chicago, IL: American Institute of Steel Construction, Inc. Bruneau, M., Uang, C.-M., & Whittaker, A. 1998. Ductile design of steel structures. New York: McGraw-Hill. CSA 2009. Limit States Design of Steel Structures. Mississauga, Ontario, Canada: Canadian Standards Association. CSI 2010. CSI Analysis Reference Manual For SAP2000®, ETABS®, SAFE® and CSiBridge™. Berkeley, California, USA: Computers and Structures, Inc. Fahnestock, L. A., Sause, R., & Ricles, J. M. 2007. Seismic Response and Performance of Buckling-Restrained Braced Frames. Journal of Structural Engineering, 133(9): 11951204. Fell, B. 2008. Large-scale testing and simulation of earthquake-induced ultra low cycle fatigue in bracing members subjected to cyclic inelastic buckling. PhD Thesis, University of California, Davis, CA. Jones, N. 1997. Structural Impact: Cambridge University Press. Krawinkler, H. 2009. Loading Histories for Cyclic Tests in Support of Performance Assessment of Structural Components. Paper presented at the The 3rd International Conference on Advances in Experimental Structural Engineering, San Francisco, California. NRCC 2010. National Building Code of Canada. Ottawa, ON.: National Research Council Canada, Institute for Research in Construction. PEER 2005. PEER Strong Ground Motion Database. 2005. Retrieved 27/05/2011 from http://peer.berkeley.edu/nga/ Sabelli, R., Mahin, S., & Chang, C. 2003. Seismic demands on steel braced frame buildings with buckling-restrained braces. Engineering Structures, 25(5), 655-666. SeismoSoft 2010. SeismoSpect (Version 1.0.2).