A Mixed Model Representing Accurately the Inelastic Behavior of Multi-storey Buckling Restrained Braced Frames

Seismic structural fuses such as buckling restrained braces (BRBs) are highly nonlinear members that need a detailed finite element model (FEM) to represent their actual behavior. On the other hand, making detailed model for the whole building from solid or shell elements to be consistent with the model of BRBs would require expensive computational time and large storage space especially when performing nonlinear time history analysis (NTHA). Therefore, a mixed FEM is developed in this research so that the solid elements are used for the core plate and the restraining system only. The non-yielding segments of the BRBs and the conjoined girders and columns are simulated using beam elements with nonlinear material properties while the effect of large deformations is taken during analysis and this was able to represent plastic hinge formation, plastic rotations and residual displacements in such members. A beam-solid transfer mechanism is developed to properly transfer forces between the non-yielding segments of BRB and the inner solid plate. The core plate is connected to the surrounding restraining system with contact elements to simulate the normal and frictional forces generated upon contact based on the penalty algorithm. This model was validated using the data and the results of an experimental work mentioned in the literature where very good agreement was achieved. Thereafter, a rehabilitation study for the SAC 9-storey building was performed. The results showed how the BRBs work as structural fuses during earthquakes and confirmed the ability of the model to represent inelastic behavior of multi-story BRBFs.


Introduction
Recent decades have witnessed continuous developments in design and the addition of new structural elements that have a high ability to dissipate energy during earthquakes to save lives and property and reduce the cost of repairs. One of the latest technologies is the use of structural fuses such as BRBs and steel plate shear walls [1][2][3][4]. The design approach for buildings containing these elements is to focus on the occurrence of yield and plastic deformations in them under the influence of lateral loads. This plastic behavior dissipates energy and improves ductility demands under seismic loads and at the same time these elements are easy to replace if they are damaged [5][6][7][8][9]. In addition to these features, the B Ebtsam Fathy efsadik@zu.edu.eg; efsadik@gmail.com 1 Structural Engineering Department, Zagazig University, Zagazig 44519, Egypt BRBs are distinguished from conventional steel braces by their ability to restrain buckling and thus provide stable hysteresis performance under the influence of earthquake loads [7][8][9][10].
On the other hand, due to the low post-yield stiffness of BRBs, residual deformation occurs under moderate and severe earthquakes. It was found that the post-yield stiffness is highly dependent on the length of yielding core as it decreases with increasing its length [11]. Some innovations have been made to enhance the post-yield stiffness of BRB by using multi-core BRB from various materials including high strain hardening stainless steel and high-performance steel [12] or by using short-length hybrid core connected serially to an elastic robust member [13]. The conventional Buckling restrained brace consists of three main parts. The first is the two ends that are linked to the adjacent beam and column with bolts or welded connections. These two ends are called the projection or non-yielding segments because they are designed to act elastically during earthquakes. They are Fig. 1 a Typical all-steel buckling-restrained braces (BRBs), and b details of cross sections connected to each other by means of the second part of BRB which has a plate of smaller area, as it is designed to yield when subjected to seismic loads equal to or greater than the design basis earthquake (DBE) resulting in energy dissipation and it is called the yielding segment or the core plate. This segment is prevented from buckling by being surrounded by a restraining system, which is the third part of BRB, to insure it reaches the yielding stage, thence provides roughly equal behavior under the compressive and tensile forces. Two types of restraining system are common to use either a concretefilled tube system or an all-steel restraining mechanism [e.g., 7, [14][15][16]. In the latter type, the weak axis of the yielding segment is restricted by two steel restrictive members and in the other direction two shim plates are used and they are connected to the restrictive members with high strength bolts uniformly distributed along their length as shown in Fig. 1. The shim plates are usually a little thicker than the yielding plate to leave clearance around it. This clearance is important to allow the cross section to be expanded when subjected to compressive force due to effect of Poisson's ratio and to reduce the forces transmitted to the restraining system [7,16,17]. Unbonded materials are often used to fill the gap around the core plate to achieve further reduction in the frictional forces between the yielding segment and the restraining system under compressive forces [9,18]. The steel restraining mechanism is distinguished from the concrete-filled tube system by allowing the core plate to be visually inspected after an earthquake, and if damage occurs the restrictive members can be easily dismantled, and the plate can be changed [7,17].
Several studies have examined the performance of BRBs either by experimental tests or numerically by simulating BRBs using the finite element method [e.g., 7,9,10,14,15,[19][20][21][22][23][24][25]. In numerical studies, the modeling of BRBs has taken two tendencies, either by making a detailed FEM or by using simplified models. In the detailed FEM, the entire BRBs and surrounding beams and columns are simulated using solid or shell elements while contact elements are used to mimic contact and Coulomb friction between the core plate and the restraining system in the normal and tangential directions, respectively [e.g., 7-9, 15, 16, 25-27]. This model has proven effective in accurately predicting the behavior of BRBs. However, the frames that have been tested with this model so far have not exceeded two floors. This is because it requires unacceptable computational time and large storage space for the results file, in accordance with the capabilities of ordinary computers available these days. Thus, till now it is difficult to use this model to perform NTHA to design and measure the efficiency of multi-stories buckling restrained braced frames (BRBFs) under the influence of earthquake loads. On the other hand, most of the simplified models of BRBs, are simulated using truss elements with uniform area while the modulus of elasticity and strain hardening parameters of the yielding and non-yielding segments were adjusted to take into account the different area of each part [10,21,[28][29][30]. The girders and columns are simulated in simplified models using beam elements that take into account both material and geometric nonlinearities. This model assists in examining the performance of BRBs in large frame buildings but it cannot simulate the high-mode buckling of the core plate under compressive forces, while such large deformations raise energy dissipation and lead to better performance during earthquakes. In addition, this model of BRBs doesn't include the effect of frictional forces between the core plate and the restraining system that causes a significant increase in the compressive capacity [17,18,31]. Rahnavard [26] made another type of simplified model to reduce analysis time compared to the fully detailed model. In this model, the restraining part was dispensed and its effect is represented using springs uniformly distributed along the core plate, which was simulated using shell elements. The surrounding beam, columns, and non-yielding parts were also simulated using shell elements to be consistent with the model of core plate. The analysis time for this simplified model was 11 h for a two-story, single-bay frame while it was 65 h when using a detailed finite element model for the same frame. However, there are still two issues facing this model, the first is that when using a large number of shell elements, this will happen when studying large frames, the solution time will increase significantly. In addition, this model does not include the effect of friction as in the other simplified models.
As it is important to carry out continuous development and new innovations for structural systems, it is necessary to provide methods for modeling these systems so that they give accurate results and at the same time are executable and do not consume much time during the analysis process so that the designer can use and rely on them easily. Many researchers seek this goal. Hajirasouliha and Doostan [32] propose a simplified analytical model for prediction seismic response of concentrically braced frames. In this model, a multistory frame model is reduced to an equivalent shearbuilding and supplementary springs are introduce to account for flexural displacements in addition to shear displacements. The model saves 96% of analysis time compared to typical frame models as well as it gave better estimates of the nonlinear dynamic response of the framed structures, compared to conventional shear models. Therefore, this paper is concerned with developing a finite element model that represents the inelastic behavior of BRBFs with less computational effort. Since BRB requires a detailed model and nonlinear structural analysis that includes both nonlinear material and large deformation effects to represent their performance, this research develops a model that combines between a detailed model for the core plate and surrounding restraining system, while a simplified model is used for the projection parts and the adjacent frame parts. The contact elements are used to fasten the core plate and the restraining system to simulate the normal and frictional forces caused by the compressive deformations of the core plate. The ability of this model to simulate the actual behavior of a BRBF was validated using the experimental test performed by Tremblay et al. [31]. In addition, its capability to perform a NTHA for multi-storey frames was examined by conducting a seismic rehabilitation study for a pre-Northridge, 9-storey building that was part of the SAC project [33]. The building is located in Seattle, USA, and has steel moment resisting frames (SMRF) as a lateral load resisting system. It was designed according to UBC-94 [34]. The BRBs are added in two bays in the perimeter frames with inverted V-shape. The design process of BRBs was performed according to ASCE/SEI 7-16 [35], ANSI/AISC 341-16 [36] and ANSI/AISC 360-16 [37]. Cases with and without BRBs were evaluated under the influence of various earthquakes that were scaled to simulate the DBE and maximum considered earthquake (MCE) at the building site. Extensive comparisons were made to show the difference in lateral displacement, inter-storey drift ratio, residual displacement, and stresses distributions.

Proposed Finite Element Model for the BRBS
The proposed FEM was created using ANSYS 15 software [38] to simulate the all-steel BRBs. The model consists of six components which are the projection, the core plate, the restraining system, the contact elements, the transition parts between the projection and the core plate, and the middle part connections of the restraining system as shown in Fig. 2.
In this model, the non-yielding part (the projection part of BRB) and frame members (girders and columns) are simulated using the beam element (BEAM188). This element is a three-dimensional element based on Timoshenko beam theory [39]. It is defined with two nodes each with six degrees of freedom three translations, and three rotations with the directions of the X-, Y -and Z-axes of the node. The advantages of this element are that it can simulate the occurrence of large strain, large rotation and also take into account the effect of shear deformations, which make it well suited for modeling the projection part of BRB and the frame members. The shim plate, restrictive members and core plate are simulated using solid elements (SOLID185), which defined by eight nodes, each has three degrees of freedom that are translations in the nodal X, Y , and Z directions. This element has many capabilities such as plasticity, large deflection, large strain, stress stiffening and creep. The solid elements representing the core plate and the restraining system are separated from each other, and connected by the contact elements which have the ability to represent the contact and the slide between the nodes, whereas the node to node contact element CONTA178 is used between every two mesh nodes facing each other; therefore, the same mesh size was used for the restrictive members, shim plates and core plate as shown in Fig. 2b. The contact elements are distributed around the inner core in all directions. Therefore, this model can mimic both in-plane and out-of-plane local buckling of the core plate. CONTA178 was chosen for this model because it showed its efficiency in simulating the contact in the detailed FEM of BRBs made by Guo et al. [8]. In addition, it is easier to use and in terms of CPU solution time, it is less expensive compared to other types such as node-to-surface and surface-to-surface contact elements. This element is defined by two nodes, each has three degrees of freedom, which are translations in the nodal X, Y , and Z directions. The element is able to support the formation of compressive force in the direction of contact and frictional force in the tangential direction. The gap size is determined based on the locations of the nodes of the core plate and the surrounding solids while it is set to zero for the contact elements between the shim plates and the restrictive members as they are tied together by bolts and thus no clearance between them. The important part of this model is the transitional part between the beam element and the solid elements that transmit the compressive and tensile forces to the core plate. This part of the model contains a small solid segment in direct contact with the beam element simulating the projection and the solid elements simulating the inner core plate as shown in Fig. 2b. The degrees of freedom of all nodes of this transitional segment are coupled with the node connected to it from the projection part (master node) in all local directions. Where local directions are defined for all elements and nodes of the proposed model of BRB in which the longitudinal axis of BRBs is the local y-axis (Y L ), the axis in-plane of BRB is the local x-axis (X L ) and the axis located out-of-plane is the local z-axis (Z L ) as shown in Fig. 2a. By choosing the node of projection part that connected to transitional segment to be the master node of the coupling, the tensile or compressive deformations created in the projection due to lateral movement of the frame will be uniformly distributed over the starting nodes of the core plate. As for the restraining system, it is governed in its beginnings by the same deformations that occur in the master nodes of projection parts, and this is also done by coupling all the degrees of freedom of their starting nodes with the obverse master node to them. In order to prevent the restraining system from carrying any axial forces, as in the real BRB, the solid elements of the restraining system are separated in the middle as shown in Fig. 2b and contact elements with a large gap used to join the two opposite parts of the solids in the axial direction (Local Direction-Y ). To achieve continuity of the restraining system in the local X and Z directions, the degrees of freedom of the opposite nodes of the separated part are coupled in these directions. This helped make the restraining system act as a one segment able to restrain buckling of core plate in out-of-plane and in-plane directions. The constitutive equations of contact elements and material models that used for the core plates and other parts are explained in the following subsections.

Constitutive Equations of Contact Elements
ANSYS [38] offers alternative algorithms, the Lagrange multiplier, the penalty method and the augmented Lagrange method, to choose from them the appropriate type to represent the contact and frictional forces in the CONTAC178 element. Among these algorithms, the penalty method was chosen for the proposed model because it is the easiest and least expensive in the computational time, which made it able to implement large models, whether two-dimensional or three-dimensional, compared to other algorithms [40,41]. The disadvantage of this method is that it allows some amount of penetration to occur, but this can be controlled by using appropriate large contact stiffness that reduces this penetration so that its value becomes minimal and thus will not affect the accuracy of the results. To achieve this goal, ANSYS [38] automatically gives an appropriate value for the normal contact stiffness (K n ) of the CONTAC178 element based on the modulus of elasticity and the size of the underlying elements. Thus, when the size of the gap embedded in the CONTAC 178 element becomes equal to zero due to a local buckling occurring in the core plate under the influence of the compressive forces, the inner plate begins to come into contact with the restraining system generating contact and frictional forces as shown in Fig. 3. In this case, based on the penalty method, the normal contact force becomes equal to the normal contact stiffness (K n ) multiplied by the gap size (U n ) as illustrated in Fig. 3b and Eq. (1). Concerning the frictional force (F s ), it has two phases, sticking or sliding, as shown in Fig. 3c. In the sticking phase, the frictional force is proportional to the contact slip distance in the tangential direction (U s ) as long as this force is less than the normal contact force (F n ) multiplied by the coefficient of friction (μ). Otherwise, slip occurs and the friction force becomes equal to μF n , Eq. (2).

Constitutive Models for the Materials
As a result of the design of the core plate to yield during earthquake or under the influence of cyclic loading, plastic strain accumulates in it with the loading time. To simulate this ratchetting behavior, a combination between Voce law nonlinear isotropic hardening plasticity and Chaboche nonlinear kinematic hardening plasticity was chosen as a constitutive model for the material of core plate. In this model, the isotropic hardening plasticity estimates the change of the yield surface size according to the Voce's exponential formula [42,43], Eq. 3. While the nonlinear kinematic hardening plasticity estimates the transfer of the yield surface in the stress space via the back stress, α which is computed using Chaboche [44,45], Eqs. (4)- (5).
where σ y and σ 0 , are the current and the initial yield strength. R ∞ , and b, are material properties, where R ∞ represents the saturation level, while b is the hardening parameter that governs the saturation rate of the exponential term.ε pl is the equivalent plastic strain. Back stress α is the superposition of a number (n) of kinematic models, where C i and γ i are the material constants that define each. The parameters R ∞ , b, C i and γ i should be specified so that the model conforms to the behavior of the material as much as possible. One way to do this is to obtain data from experimental stabilized strain and stress-controlled ratchetting tests and use these data together with the curve-fitting method to determine these parameters that reduce the error between the data and the model predictions. In this paper, the material parameters were determined based on the experimental test conducted by Tremblay et al. [31] and from the supplementary analytical research done by Korzekwa and Tremblay [27], whereas this experimental test is simulated in this paper to validate the proposed FEM.
On the other side, the other components of BRBF such as, projection parts, beams and columns, will not be subject to such large cumulative plastic strain according to the design requirements. Therefore, the bilinear kinematic hardening model (BKIN) with the von Mises yield criterion was chosen to simulate the behavior of these members. In this model, the yield surface translation vector {α} BKIN is computed by the following Equations, ANSYS [38].
where G, E, E T , ε sh , ε sh , ε pl , and k are the shear modulus, modulus of elasticity, tangent modulus, shift strain, change in shift strain, change in equivalent plastic strain, and number of load-step, respectively. The shift strain {ε sh } is initially zero then changes with the next plastic straining resulting from the loading history.

Verification Study
To verify the efficiency of the proposed approach, the developed FEM was used to simulate the experimental test of a full scale one storey frame with a diagonal BRB (specimen S2-1) performed by Tremblay et al. [31], and the results were compared with the experimental ones in this section. Figure 4a illustrates the dimensions of the tested frame, the cross sections of the frame members, connections and the loading direction. In the specimen S2-1, the cross-sectional area of the core plate was 125 mm × 12.7 mm, while its length was 1001 mm as illustrated in Fig. 4c. The non-yielding segments have a length 1441 mm while its cross-sectional area is equal to 2.4 times the area of the core plate to ensure elastic behavior even if the stress of the inner plate reaches its ultimate value. The length of the transitional segments between the projection and inner core was 200 mm. the restraining system on each side has a 270 mm x 10 mm guide plate welded to an ASTM A500 127 × 127 × 4.8 HSS tube. In the proposed FEM, this restraining system was replaced with a solid section whose thickness was calculated to achieve stiffness equivalent to that of the HSS tube and the attached guide plate. The shim plates were inserted on either side of the inner plate leaving a clearance of 0.58 mm around the core plate as illustrated in Fig. 4c. Canadian Standards steel material G40.21-350WT was used, for which coupon tests were performed by Tremblay et al. [31] to determine its basic properties. The measured yield and ultimate stress were 370 and 492 MPa, respectively. The material properties E, R ∞ , b, C 1 and γ 1 were equal to 200,000 MPa, 110 MPa, 4, 8000 MPa, and 75, respectively. Poisson's ratio was taken as 0.3 while the coefficient of friction μ was taken equal to 0.1 as adopted by Chou, and Chen [25].The quasi-static cyclic test with incremental displacements (Protocol H1) shown in Fig. 4b was carried out by a high-performance dynamic actuator has 1500 kN capacity. This protocol is based on the storey drift at yield y , where it contains six cycles with equals ± 1.0 y , four cycles with equals ± 2.5 y , four cycles with equals ± 5 y and two cycles with equals ± 7.5 y . The four additional cycles shown in Fig. 4b were used by Tremblay et al. [31] on other specimens of BRBs made of concrete filled steel tube. Through the test, the shear force and drift were recorded. This experimental test is simulated in this study using two types of finite element models, a detailed model and the proposed mixed model, which are shown in Fig. 5. This is done to compare the results, analysis time, and storage space for each FEM to find out the efficiency of the mixed FEM and its ability to reduce computational efforts. In the detailed model, the beam, columns, non-yielding part of BRB, core plate and restraining system are simulated using solid elements (SOLID185) while contact elements (CONTA178) are used to connect the core plate and restraining system. The constitutive models of the materials and contact element approach were the same as those used in developed mixed FEM. Figure 6 shows the hysteretic load-deformation curve of specimen S2-1 measured in the experimental test conducted Fig. 4 a The full scale frame tested by Tremblay et al. [31], b load protocol and c specifications of specimen S2-1 for BRB brace   [31] and the data listed in Korzekwa and Tremblay [27] while at least three models should be used to reach perfect ratchetting prediction [46]. Figure 7 shows the local buckling occurred in the inner plate of proposed mixed FEM when the storey drift is equal to − 7.5 y and − 5 y , where it is clear that when the compressive forces increase, the local buckling evolved to higher buckling mode. This figure indicates the success of the beam-solid transfer mechanism used in the proposed FEM in properly transmitting and distributing loads on the inner plate which is also shown in the animation video (Online resource 1). Moreover, in the experimental test, a tensile fracture occurred in the second cycle of the loading history with storey drift equals to ± 7.5 y due to the accumulation of plastic tensile strain in the middle part of the core plate as shown in Fig. 8c. This is followed by a sudden loss of strength as shown in Fig. 6a. In the proposed FEM, the failure is judged to have occurred if the stress in any element reaches its ultimate value. This happened in the core plate at the same loading cycle in which the experimental test collapsed as shown in Fig. 6b. Figure 8a, b and (Online Resource 2) show the results of proposed FEM upon failure where the tensile stress reaches its ultimate value and the plastic strain is concentrated in the center of core plate, which is identical to the failure mode in the experimental test. These results confirmed the accuracy of proposed FEM in predicting the actual behavior of BRBF.

Results of Verification Study
Another important feature of this model is that it was able to significantly reduce the computational effort compared to the detailed model, it required 3 h to perform the analysis on a personal laptop with 2.4 GHz core i7 and 12 GB RAM, while the detailed model needed 34.5 h to perform the analysis on the same device. Moreover, the required storage areas were 3.81 and 33.1 GB for the developed FEM and the detailed model, respectively, as illustrated in Table 2. This

Seismic Analysis for Multi Storey BRBF
In this section, a seismic rehabilitation study is carried out using the BRBs to retrofit a pre-Northridge, 9-storey building designed in the SAC project [33]. This is done to verify the ability of the proposed FEM to perform a NTHA to large frames containing multiple BRBs, and at the same time evaluate the effect of the BRBs, as structural fuse members, on the seismic response of buildings. The building is designed in accordance with UBC-94 [34] in which plastic hinges were allowed to be formed into the girder at the faces of the columns to dissipate energy. After the Northridge earthquake, this design mechanism changed because the presence of the plastic hinge at the face of the column causes large strain on the flanges of the column and on the beam-column connections causing damage, FEMA 267 [47]. As a result, recent standards such as ANSI/AISC 358-16 [48] and ANSI/AISC 341-16 [36] specify that the position of the plastic hinges should be further from the face of the column by a distance of at least half the depth of the beam. Therefore, strengthening this type of buildings is essential to ensure that life-safety is protected.

Description of the Pre-Northridge 9-Storey Building
The building was designed as an office building located in Seattle, USA and rested on stiff soil (type S2 according to UBC-94 [34]. It consists of 9 floors and a basement with one level while its structural system relies on four steel moment resisting frames (SMRFs) located on the perimeter and gravity columns support the interior beams as shown in Fig. 9a. The roof and floors are composite slabs, each has 140 mm total thickness. The gravity loads for floors, roof and penthouse are illustrated in Fig. 9a. The four SMRFs are approximately similar; therefore the North-South (N-S) frame is examined as in SAC project [33]. The frame dimensions are illustrated in Fig. 9b. Column bases have simple connections and the Lateral displacement at the basement floor is restricted. Compact sections were used for the design with sufficient lateral bracing to eliminate the effects of local and lateral torsional buckling. Steel ASTM A572 Gr. 50 was used for columns and girders, which has nominal yield strength of 345 MPa. The cross sections of the columns and girders as well as the positions of splices designed in the SAC project [33] are shown in Table 3.

BRBs Design to Rehabilitate the Pre-Northridge SMRF
The adequate design of the BRBs is important to obtain a stable inelastic response in addition, it grants that failure occurs  in the core plate and not in projections, connections or in the restraining system. In this study, inverted V-shaped BRBs were added in two bays in the perimeter frames to resist the lateral load on each floor by more than one brace resulting in smaller cross sections of the core plates thus reducing the additional forces on the surrounding columns and girders as shown in Fig. 9c. The material properties of BRBs and the length of yielding segment were taken as in the validation study. The seismic load and load combinations were calculated based on ASCE/SEI 7-16 [35] taking into account that the soil type is C, which is close to the soil S2 in UBC-94 [34]. The design spectral acceleration for the building site at a short period (S DS ) and at a period of 1.0 s (S D1 ) is equal to 1.4 g and 0.488 g, respectively, and the seismic design category is D. By calculating the maximum axial force in each brace, the inner core section can be designed according the following equation, P ysc F ysc A sc (9) where P ysc is the design axial strength of the brace, P ysc is the axial yield strength of steel core , equals 0.9 according to ANSI/AISC 341-16 [36], F ysc is the yield stress and A sc is the area of yielding segment. The designed cross section areas for core plates on each floor are shown in Table 3.The restraining system is designed to have a critical buckling load greater than twice the yield strength of the inner core to prevent the global buckling of this system as recommended in [7,17]. The flexural, shear and axial strength of the girders and columns were checked considering the adjusted compressive and tensile strength of the BRBs which are calculated according to ANSI/AISC 341-16 [36] based on the following equations The adjusted brace strength in compression βωR y F ysc A sc (10) The adjusted brace strength in tension ω R y F ysc A sc (11) where the modified factor ω represents the effect of strain hardening while β represents the effect of plate expansion and frictional forces in case of compressive forces. These values should be determined based on a qualification test under deformations equal to twice the design value of the storey drift. Kersting et al. [49] identified typical values of ω, which range from 1.3 to 1.5 and β values range from 1.05 to 1.15. In this study ω and β are taken equal to 1.4 and 1.1, respectively, while R y , which represents the ratio between actual and nominal yield stress, is taken equal to 1.0 because F ysc is specified based on the coupon test performed by Tremblay et al. [31]. The design results showed that the columns and most of the girders are able to bear the additional force from the BRBs, with the exception of the girders in the last two floors which had a cross section of W610 × 92 due to the axial compressive force that affected them. This section was strengthened with plates 10 mm thick. Simple connections were established between the BRBs and the surrounding girders and columns where the proposed FEM doesn't take into account the details of the connections since they are designed to behave elastically during analysis. In addition, such details require a detailed model of the whole building, which gives unacceptable time for analysis and consumes a lot of storage space, especially when NTHA is performed for multi-storey building this is according to the capability of ordinary computers available these days.

Modeling and Analysis Specifications
Two FEM were carried out for the N-S frame, one representing the SMRF and the other representing the same frame after retrofitting using 18 inverted V-shaped BRBs distributed in two bays in all floors except for the basement as shown in Fig. 9b, c. The same elements and methodologies described previously were used to implement the models. The aspect ratio of the solid elements used to simulate inner core plates and restraining systems of BRBs was kept in the recommended range (1:5) to obtain accurate results. All inverted V-shaped BRBs contain two members of the proposed FEM. Local coordinates were created for each member to correctly coupling the solid-beam transitional segments and accordingly, the forces properly transfer to the core plate. The performance of pre-Northridge SMRF and retrofitted BRBF models was assessed by exposing them to different earthquake excitations scaled to simulate both the DBE and the MCE of the building site. Nonlinear time history analyses were performed using ANSYS [38] taking into account large deformation effects as well as material nonlinearity. In these analyses, the nonlinear structural dynamics equations were solved numerically using Newmark Method in association with the Newton-Raphson iterative algorithm. To obtain accurate results, some attempts were initially made to choose the appropriate time step in which the results are not affected by reducing it. The perimeter frame on which the seismic analyses were performed resist 50% of the seismic load, so the masses involved in the horizontal direction of the earthquake were set to equal half the seismic weight of the building. As for the masses participating in the vertical direction, they were equal to the proportion of the loads borne by the perimeter frame on each floor. The seismic mass of the building was calculated on the basis of dead load and half live load and a Rayleigh damping of 3% was adopted as in the SAC project [33]. Eight historical earthquakes were used in the analysis. The seismological properties and identification data of these earthquake are summarized in Table 4. The amplitude scaling specified by the ASCE/SEI 7-16 [35] was applied to each record. In this method, the maximum direction of each earthquake considered in the analysis is scaled so that the average spectrum of all excitations matches, or overrides but is not less than 90% of the target spectrum over a period of 20% to 200% of the fundamental period of the structures. Figure 10a, b shows a comparison of the average spectral accelerations of the scaled earthquakes and the target to be achieved in the case of DBE and MCE, respectively.

Evaluation of Seismic Performance
The effect of retrofitting appeared once the modal analysis was performed, which is the first step in performing the dynamic analysis. As this analysis showed that the fundamental time period of the pre-Northridge building is 2.462 s, which exceeds the upper limit of the time period (T ) of the SMRF that equals to C u × 0.0724h 0.8 n in accordance with ASCE/SEI 7-16 [35]. C u is a coefficient that varies between 1.4 and 1.7 according to the design spectral acceleration at a period of 1 s (S D1 ). In Seattle, S D1 is greater than Table 4 Seismological properties and identification data of the selected historical earthquakes   The inter-storey drift ratio under the DBE for a pre-Northridge SMRF, b BRBF 0.3 g, so C u is equal to 1.4. h n is the building height measured from the base to the highest point of the lateral load resisting system. On the other hand, the fundamental time period changed after retrofitting using BRBs to be 1.589 s due to the additional lateral stiffness from BRBs only as the model analysis based mainly on the calculation of masses and stiffness. This improvement in lateral stiffness makes the building fall within the code limits for T which is equal to C u × 0.0731h 0.75 n in the case of BRBF. By performing nonlinear time history analyses on models using the DBEs and MCEs, the BRBF model achieves even greater improvements. Figure 11 shows the inter-storey drift ratios for all floors under all earthquakes, and average and standard deviation SD, at the time of the maximal value under the influence of DBEs, of the pre-Northridge SMRF and BRBF models. The SMRF gave a wide variance in the behavior and values of inter-storey drift ratios for different earthquakes, in addition to that three from the eight earthquakes and the average exceeded the code limit which equals the allowable drift a divided by the redundancy factor ρ in the case of the SMRF located in seismic categories D-F. The allowable drift a for both SMRF and BRBF is equal to 0.02 h sx (h sx is the height of the storey under level x) while ρ is equal to 1.3 for seismic category D in accordance with ASCE/SEI 7-16 [35]. On   Fig. 12 Comparison between lateral top displacements of the SMRF and BRBF under the effect of a 1995 Dinar earthquake, Turkey, and b 1994 Northridge earthquake, USA the other side, adding BRBs resulted in a remarkable convergence in the profiles and values of the inter-storey drift ratios under the influence of various earthquakes. Moreover, for all earthquakes, the storey drift ratio is much less than the allowable drift. The percentage of improvement reaches 71.7% compared to the average results of pre-Northridge SMRF. This is attributed to the energy dissipated during the occurrence of plastic deformations in the core plates of the BRBs as well as the improvement in the lateral stiffness of the building. These results reflect the efficiency of the BRBs in serving as seismic structural fuses. This efficiency is clearly demonstrated under the influence of MCE, as it protected the building from collapse or major permanent damage, as happened in the case of SMRF model. Where Fig. 12a, b shows the top lateral displacement of the two models under two earthquakes scaled to represent the MCE for the building site, Dinar, Turkey earthquake and Northridge, USA earthquake, respectively. Under the influence of the Dinar earthquake, a large permanent deformation occurred in the SMRF model where the residual drift reached 8.3 cm as shown in Fig. 12a. In this model, the yield started at the ends of the lower beams, and then the stress increased in these sections forming plastic hinges that spread to most of the beams over time as shown in Fig. 13a, b. These hinges were subjected to large rotations, which weakened the lateral stiffness of the SMRF resulting in large drift and permanent plastic deformations captured in the top lateral displacement and in the momentrotation hysteresis loops at the ends of the beams, Figs. 12a and 13b. On the contrary, no plastic hinges were formed in the BRBF model under this earthquake and most of the plastic deformations were confined to the core plates of the BRBs as shown in Fig. 13a, b. The plastic deformations in the core plates of BRBs caused a small residual drift reaching 0.65 cm in the top floor, which is negligible compared to the corresponding value in the SMRF model. The improvement in top lateral displacement and in residual drift by using BRBs reaches 78.06% and 92.17%, respectively, compared to the pre-Northridge SMRF. When the models were affected by the Northridge earthquake, the SMRF collapsed due to the stresses in the region of the beam-column connections and at the flanges of the columns reached the ultimate limit as shown in Fig. 14a. This failure mode is the same as that detected by FEMA 267 [47] in pre-Northridge buildings with moment resistant frames where fracture occurred in the beam-column connections because the position of the plastic hinges was designed to occur at the faces of the columns. On the other hand, the performance of the BRBF model is the same for all earthquakes scaled to simulate the MCE of the building site, as all the significant plastic deformations were concentrated in the inner cores of the BRBs as shown in Fig. 15. This figure also shows the stress-strain hysteresis loops for the core plate of one of the BRBs on the first floor and on the last two floors, showing that most the BRBs had plastic deformations as designed but that the most affected were on the first floors. This plastic behavior is preferred, as it is easy to change the inner core plates after the earthquake has ended to restore the building's resistance to lateral loads again making the BRBs a good choice as a lateral load resisting system or for the rehabilitation of existing buildings. This study also demonstrates the ability of the proposed FEM to represent multi-storey BRBFs and reach accurate results under complex and timeconsuming analysis such as the NTHA, making it reliable for studying and designing this type of structure.

Conclusions
Buckling restrained braces (BRBs) are highly nonlinear members. To simulate their behavior, several types of models have been made in previous studies, some of which are detailed and give accurate results, but they consume a lot of time and large storage space. Other types of models are simplified and give acceptable results but they cannot simulate the high-mode buckling of the core plate of BRBs under compressive forces, while such large deformations raise energy dissipation and lead to better performance during earthquakes. Therefore, in this research, a mixed FEM is proposed to study the actual behavior of the BRBFs in multi-storey buildings without consuming expensive computational time or requiring large storage space for results file especially when implementing (NTHA). This is done by using different elements and material models to simulate each part as needed to represent its nonlinear behavior. Hence, the need for a detailed FEM from solid and contact elements is reduced and confined to the yielding segment and the surrounding restraining system. The remaining parts of the BRBF such as the projection parts of BRBs, girders and columns are simulated using beam elements with taking into account material nonlinearity and large deformation effects. Thus the proposed mixed FEM combined the advantages of the detailed models and the simplified models that were mentioned in the introduction. This model was validated by simulating the experimental test of Tremblay et al. [31], as it has been shown to be effective in representing the actual behavior of BRBF. In the second part of this research a seismic rehabilitation for a pre-Northridge 9-storey SMRF that was part of the SAC project [33] is conducted using the proposed FEM. The most important conclusions achieved by this model are summarized in the following points: • Using a detailed model for the inner part of BRB helps capture the different high-mode buckling of the core plate and plastic deformations that form under repeated compressive and tensile forces. Such deformations cause significant energy dissipation and this is what distinguishes the BRBs and makes them act as seismic structural fuses. This was very clear in the results of rehabilitation study, whereas under most earthquakes the main plastic deformations occurred in the inner plates of the BRBs and no plastic hinges formed in the girders contrary to the case of pre-Northridge SMRF without retrofitting. • Simulation of the frictional and contact forces between the yielding segment and the restraining system depending on the penalty algorithm is sufficient to predict their effect on the increase in compressive strength since the difference between the maximum compressive forces resulted from the experimental test and from the proposed FEM does not exceed 0.5%.
• The proposed beam-solid transfer mechanism that transfers forces between the beam elements representing the non-yielding segments of the BRBs and the solid elements representing the inner part of BRBs proved successful when the model was validated, whereas the maximum difference between the computed values of the initial stiffness, maximum tensile and compressive forces using the proposed mixed FEM and the corresponding measured values from the experimental test doesn't exceed 5.5%. • The use of beam elements to represent members of steel frames and non-yielding segments of the BRBs simplifies the model and reduces the number of elements needed for analysis. This resulted in the proposed model being able to save 91.3% and 88.5% of analysis time and storage space, respectively, compared to detailed FEM, which enabled the model to simulate large frames with different numbers of BRBs. Characterizing the nonlinear properties and taking the effect of large deformations of these elements contributes to the representation of the plastic hinge formation, plastic rotations, residual displacements, positions of maximum stress and in predicting failure modes. All these benefits make the proposed FEM reliable for use in the study and design of BRBFs.
Funding Open access funding provided by The Science, Technology & Innovation Funding Authority (STDF) in cooperation with The Egyptian Knowledge Bank (EKB). The author did not receive support from any organization for the submitted work.

Conflict of interest
The author has no conflict of interest to report.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecomm ons.org/licenses/by/4.0/.