Seismic performance of steel frames equipped with buckling-restrained braces (BRBs) using nonlinear static and dynamic analyses

3.1. Nonlinear static pushover

Nonlinear static analysis is an applied method for evaluation of the seismic behavior of the structures. Despite its limitations, it is widely used in practical methods, because of its simplicity. In this analysis, a lateral load is applied on the structure with a specified load pattern and incremental lateral loading represents the seismic lateral inertial force. In this paper, the nonlinear pushover is used based on the Capacity Spectrum Method (CSM) available in ATC-40 [30].

As shown in Fig. 4, general process of the CSM method including of the derivation of the demand point from the capacity diagram of the structure and the demand spectra curve of the earthquake have been presented. Then, the elastic response (or design) spectrum converted from the pseudo acceleration “$A$” versus the natural period “$T$” format, to the A-D format (the Acceleration-Displacement Response Spectrum (ADRS)), where $D$ is the deformation spectrum ordinate, to obtain the demand diagram. By plotting the demand and capacity diagram together, the performance demand of the structure is determined from the interaction of two diagrams with trial-and-error steps [31-33].

Fig. 4Procedure of determination of demand (Performance) point in the capacity spectrum method (CSM) based on the acceleration-displacement response spectrum (ADRS) format [31]

3.2. Nonlinear dynamic analysis

The nonlinear time-history analysis is one of the comprehensive methods, which considers the effects of the properties of strong ground motion earthquakes and non-linear characteristics of structure. The general approach of dynamic analysis involves the systematic integration of the equations of motion, which is assumed to be independent of the superposition effects for force. In fact, this method shows the behavior of structure in time as systematic procedure. In this study, the constant acceleration method [34] is used for the nonlinear dynamic analysis. In this method, at the time interval $t_i$ to $t_{i+1}$ ($\mathrm{\Delta }t$), acceleration of system will be considered as constant. However, this constant acceleration can be the acceleration at $t_i$ or $t_{i+1}$, or the average value of $t_i$ and $t_{i+1}$ in most case. Therefore, the equilibrium equation in general form is determined as follows Eq. (1):

By replacing the values obtained for mass, damping and stiffness for every step, in Eq. (1) the final equation is converted to Eq. (2):

By calculating the $∆u_i$ in each time step, displacement, velocity and acceleration at the end of the step ($t_{i+1}$) is obtained.

Since time steps have a direct effect on the accuracy of time history analysis, therefore, selection of appropriate time steps is important. By choosing a time interval less than one-tenth of the natural frequency of the structure, the accuracy of the results will be accurate enough. Also, time interval shall be selected such that it can show changes in loading properly. Thus, if changing of loading intensity at some moments is significant, in those moments, time interval should be selected too small to be actual representative of the loading in the calculations. Appropriate time steps in this procedure in order to achieve an exact solution must not be greater than $T/$12, ($T$ is the period of dominant mode of the structure). Also, time step selected must not be greater than the recorded acceleration time step [34]. In this study, time step is considered as 0.005 seconds.

4.1. Building models and frames

In this study, six steel frames including 5, 10, and 15-story CBFs and BRB frames with pinned connections are designed according to AISC building code 2010[20] and SEI/ASCE7-10 [21]. Buildings are located in a high seismic risk zone with the soil type C [21], which has the shear wave velocity between 360 to 760 m/sec². Dead and live load was 570 and 200 kg/m² respectively. Live load for the roof was considered150 kg /m² for all models. Fig. 5 shows the plan and elevation view of the buildings.

The steel buckling-restrained braced buildings have been designed based on the load and resistance factor design (LRFD) in accordance with the provisions of AISC360 [35]. Acceptance criteria and performance levels of models are checked based on the provisions of AISC2010 [20], SEI/ASCE 7-10 [21], and ASCE41-13 [36] has been done.

Fig. 5Elevation view: a) 5, b) 10, c) 15-story models and d) plan view

a) Elevation view – 5 story

b) Elevation view – 10 story

c) Elevation view – 15 story

d) Plan view

4.2. Brace modeling of buildings

Load-deformation capacity curve of the ordinary braces is shown in Fig. 6 and Table 1. Based on the analytical and experimental results, the generalized load-deformation curve shown in Fig. 6, with parameters $a$, $b$, and $c$ as defined in Table 1, is being used for components of steel concentrically braces. The parameter $P$ in Fig. 6, is generalized component load and $\mathrm{\Delta }$ is total elastic and plastic displacement, in elastic and inelastic behavior of steel concentrically braces, respectively [36].

In Table 1, acceptance criteria and modeling parameters of double angle sections for concentrically braced frames in compression is obtained [34].

Fig. 6Generalized force-deformation relation for steel concentrically braces [34]

As for the BRBs, the modeling parameters and acceptance criteria are obtained using ASCE41-13 provisions [36] (Table 2). To consider the criteria for dynamic analysis, the tri-linear idealization model from studies of Sahoo and Chao [26] is used. Fig. 7 shows the verification analytical modeling and experimental data. As can be seen from the figure, the hysteresis loops of BRB show the suitable kinematic and isotropic hardening behavior. Based on the calibration of Sahoo and Chao’s models [26], the buckling-restrained braces were modeled using with tri-linear hysteretic behavior, consisting of degradation in stiffness and strength components (Fig. 8).

Fig. 7a) Components of BRB, b) verification of hysteresis model of BRBs in PERFORM-3D

a)

b)

The components Properties of BRBs were selected from studies of Sahoo and Chao [9] in PERFORM-3D [37]. As shown in Fig. 7(a), the elastic zones (transition) in the start and end of segments of BRBs were considered as 1.6 and 2.2 times of the sectional area of the BRBs’ core, respectively. Also, the length of elastic zone (transition in the start and end of segments of BRBs) have been considered as 0.06 and 0.24 times the total length of BRB [9]. The post-yield stiffness of BRBs in tension can be different from that in compression depending on the type of outer casing and confining material used for lateral support to brace core [38-40]. In this study, the post-yield stiffness of core segments in tension and compression was considered as 3 % of their initial stiffness [9]. Both isotropic and kinematic hardening characteristics were considered in the modeling of Force-Deformation response of BRBs [9]. Various hardening parameters were obtained by comparing the hysteretic response of a typical BRB from PERFORM-3D [37] with the component test results [38] as shown in Fig. 7(b).

The tri-linear hysteresis BRB model adopts well with component tests and used in Perform 3D [37]. Therefore, for modeling of inelastic BRB elements, the tri-linear Force-Deformation relationship is used. This model is consisting of the three main components including elastic, yielding and hardening region. Also, it is capable to consider strain hardening behavior, which involves both isotropic and kinematic hardening properties. Therefore, this model has been applied for modeling of BRBs in this study. Fig. 8 shows the general tri-linear force-displacement relationship in BRB modeling. In order to define the tri-linear model for a BRB model as shown in Fig. 8, it would be required the thirteen parameters. The measure of these parameters depend on interior steel core area, length between pin-to-pin connection, length of the interior steel core, modulus of elasticity and yield strength of steel. The parameters that related to the Strain-Stress relationship are based on the property of the BRB yielding core in the hysteresis model of BRBs (Fig. 7(b)) including of the yielding force ($Fy$), force for the first loading cycle ($Fu0$); strength after full hardening ($Fuh$); the stiffness before yielding ($K0$); the post-yielding stiffness ($Kf$) [26]).

For modeling of Buckling Restrained Braces (BRBs), three types of buckling restrained brace with rectangular sections are used in different stories (Table 3). Acceptance criteria and modeling parameters of BRB have been selected based on the results of Sahoo and Chao [26] and ASCE41-13 [36].

Fig. 8Tri-linear idealization model for load-deformation capacity curve for buckling retrained braces (BRB) [26]

Beams, columns, and connections in the bracing bay should be designed for the maximum force to be applied by the braces. This maximum force should be achieved in brace deformation corresponding to the relative displacement of the story design. To calculate buckling brace stiffness, the length of the yielding core has been considered, as 70 % of the total length of BRB, which is verified in many practical purposes [16].

7.1. Inter-story drifts in nonlinear static analysis

As shown in Figs. 11-13, for the mid- and high-rise models, inter-story drift ratio of CBFs are more than BRBFs except for top stories. The Maximum inter-story drift demand in BRBFs is shifted from lower to upper stories, whereas in CBFs shows an opposite trend. Also, the inter-story drifts in some stories are over than the values provided by FEMA 356 [39] as 1.0 % for the BSE-1 hazard level. With regard to Life Safety (LS) performance level, the drift value of 1.5 % is selected for BSE-1 hazard level.

7.2. Base shear corresponding to the target displacement

As shown in Fig. 14, the base shear in the target displacement for CBFs is more than BRBFs. For uniform loading pattern, the value of base shear in BRB frames of 5, 10 and 15-story is 2.06 %, 21.4 % and 31 % lower than the value of CBFs, respectively and for triangular loading pattern is 10.9 %, 16.7 % and 26 %. In fact, high levels of ductility and energy dissipation due to nonlinear behavior of the BRB reduces the seismic demands on the structure including its base shear.

Fig. 11Maximum inter-story drift ratio (%) for 5-story models under uniform (UN) and triangular (TR) loading

a) Uniform (UN)

b) Triangular (TR)

Fig. 12Maximum inter-story drift ratio (%) for 10-story models under uniform (UN) and triangular (TR) loading

a) Uniform (UN)

b) Triangular (TR)

Fig. 13Maximum inter-story drift ratio (%) for 15-story models under uniform (UN) and triangular (TR) loading

a) Uniform (UN)

b) Triangular (TR)

Fig. 14Comparison of seismic base shear in Nonlinear static analysis between CBF and BRBF models under uniform (UN) and triangular (TR) loading pattern

a) Uniform (UN)

b) Triangular(TR)

7.3. Performance-based evaluation

The BRBFs have provided the acceptance criteria of Life Safety (LS) and Collapse Prevention (CP) performance levels under different loading patterns based on ASCE41-13 provisions [32]. However, CBFs in any of the models fail to satisfy these performance levels.

Fig. 15 shows the performance-based evaluation for Life Safety (LS) and Collapse Prevention (CP) levels in 10-story model under triangular loading patterns. The results have been presented based on the resistance criteria of BRBs. Therefore, the ratio of the seismic demand axial force to capacity have been mentioned on the specified elements.

The results show that the frame exhibits plastic hinge, which leads to instability and permanent deformation, at the seventh and tenth story of CBF models under triangular loading pattern.

Fig. 15Life Safety (LS) and Collapse Prevention (CP) performance-based evaluation for 10-Story BRBF and CBF models under triangular loading pattern (TR)

a) Life safety (LS)

b) Collapse Prevention (CP)

8.1. Energy dissipation

The amount of energy, which is dissipated by structure during an earthquake, determines the seismic behavior of structure. In a structure with appropriate nonlinear behavior, dissipation of energy occurs by viscous damping in the range of linear region and by the plastic properties of the members in the ranges of non-linear behavior. Fig. 16 shows the average of the energy dissipation in BRBF models under near-fault and far-field ground motions.

Fig. 16Mean of energy dissipation for BRBF models under near-fault ground motions and far-field ground motions

a) Near-fault ground motions

b) Far-field ground motions

The Studied BRBF models under near-fault records, show more ductility and high energy dissipation. Therefore, differences in the amount of the energy dissipation for near-fault ground motions are more than those of far-field. In addition, the mechanism of energy dissipation for all models under near and far-field records are more due to some attenuation within the nonlinear behavior of structures.

8.2. Residual drift response under near-fault and far-field records

Residual Drift Ratio (RDR) estimates as the ratio of the inter-story displacement at various story levels in the last step of all ground motions. Figs. 17-19 show that near-fault ground motions have more effects on the upper stories. Therefore, RDR in near-fault excitations is greater than far-field. Thus, the Residual Drift Ratio at near-fault records in CBFs models, for most stories, especially in the first stories is more than BRBFs. Although the Residual Drift Ratio for BRBFs shifts from the lower to the upper stories, in the CBFs, this ratio changes from the lower to the upper stories.

Fig. 17Residual drift ratio (RDR) response of 5-story BRBF and CBF models under earthquakes

Fig. 18Residual drift ratio (RDR) response of 10-story BRBF and CBF models under earthquakes

Fig. 19Residual drift ratio (RDR) response of 15-story BRBF and CBF models under earthquakes

The 5-story BRBFs under near-fault records show larger residual drift ratio response, compared to the far-field ground motions and the maximum value of mean residual drift ratio of 0.097 %.

The comparison of low-rise CBFs and BFBFs under near and far field ground motions indicate the small changes of RDR, whereas for mid-rise structures, the differences have been slightly increased.

The mean plus standard deviation value of RDR response of the low rise of BRBFs under earthquakes is about 0.15 %; whereas the corresponding values for the BRBFs varied in the range of 0.11-0.16 % and this result in CBF models is about 0.19 % which changed from 0.14 % to 0.20 %. In the mid-rise of BRBFs, the mean results for 10 and 15 stories have changed from 0.20 % to 0.27 % whereas in the mid-rise of CBFs, the average responses have value of 0.52 % and 0.48 % for 10 and 15 stories models, respectively.

The maximum residual drifts ratio for the low-rise buildings in BRBFs and CBFs is concentrated at the intermediate (2nd-4th) stories. For mid-rise BRBFs, the maximum RDR values have been obtained in the upper stories (7th-9th in the 10-story and 11th -13th in the 15-story frames) and in CBFs models, these values have been centralized at the intermediate stories.

8.3. Base shear under near-fault and far- field records

Maximum base shear values are shown in the Fig. 20, for 5-, 10-, and 15-story models, under near-fault and far-field records. It can be concluded that the values of base shear in the CBF models is greater than BRBFs in most stories, especially first ones, while in 5 story frames, the difference between the two systems was small, and in the far-field records for all models, the difference was negligible. The reduction of base shear in the BRRBFs tends to minimize the extra costs of foundation and diaphragm strengthening.

Fig. 20Comparison of the maximum of base shear in the 5, 10 and 15-story CBF and BRBF models

a) 5 story

b) 10 story

c) 15 story