Optimum buckling-restrained braces application to enhance seismic performance of RC frame with curtailed walls

. Reinforced concrete (RC) frames are commonly built together with shear walls. In high seismicity regions, constructing shear walls along the frame’s height is proven ineffective. Thus, in numerous studies, using shear walls at a certain height is beneficial, and this kind of structure is popularly known as curtailed shear walls. However, the area above the curtailed walls could suffer significant deformation under high seismic load and upgrading the seismic performance in those upper parts is needed. In this study, the seismic performance of 2-dimension RC frame building with curtailed walls is improved by installing buckling-restrained braces (BRBs). The seismic response is performed through non-linear dynamic analysis using an open-licensed software, STERA_3D. To determine the ideal number and location of the BRBs above the curtailed walls, a classical genetic algorithm is exercised using Python language programming. The parameters involved in optimizations are inter-story drifts, the number of BRBs, and damage indices surround the frames. The result shows that the configuration of BRBs resulted from the optimization could reduce the excessive amount of displacement along the building height. Moreover, the genetic algorithm could give the fittest number and location of BRBs installation to upgrade the seismic response of RC frames with partial shear walls.


Introduction
In prone earthquake area countries with relatively dense populations, it is often to use reinforced concrete (RC) as a structure for the buildings.As the needs of vertical residential and office buildings increase, more RC buildings could be vulnerable to the earthquake.Thus, to support the lateral loadings due to seismic activities, additional structures are added, for example, shear walls.In some cases, shear walls are installed full height along the RC frames [1][2][3][4].However, there were some proofs that installing RC shear wall with the full height of frame is ineffective, because the seismic responses such as story drift [5] of the structure at the upper stories, and the flexural and shear responses [6] at the upper stories developed significantly.This phenomenon is also shown by another study which perform analysis for RC with shear wall structures [7][8].
To overcome this situation, installing shear walls at a certain level to reduce the bending deflection at the upper part could be beneficial.In some studies, this installation of shear walls at partial levels is referred as curtailed wall.Removing certain upper part of shear walls and only using the lower parts could diminish the referred large responses at the top part of the structure.The optimization height for applying shear walls in the RC frames had been studied by several researchers, [8][9][10].However, these studies only discussed about the frame with partial shear walls subjected to static linear lateral loads, while the nonlinearity of dynamic analysis *Corresponding author: taufiq.im@ft.umy.ac.id is disregarded.There had been research about the effect under nonlinear dynamic analyses [11], but this study was only highlighting the enormous upper structure's deflection without the solution to reduce it.
In this study, the seismic performance of RC frame with curtailed walls, especially at the structure's upper part, is enhanced with additional lateral resisting structures, namely buckling-restrained braces, and popularly known as BRB.This study is extensive research from the previous proposal by the author [12].
In the world, buckling-restrained braces have been implemented in many infrastructures due to the benefit of this structural element.It diminishes the disadvantage of using conventional steel braces, which could buckle if the axial compressive load in the bracing element excessive.By covering the steel element with encasing mortar, the BRBs could prevent from buckle.Thus, the element will directly enter to the yielding stage and at this stage, the overall structural ductility could significantly increase and could sustain large seismic events.The example of BRBs application in Japanese building is presented in Fig. 1, while the configuration of BRB elements are illustrated in Fig. 2.
The originality of the study is that the structure will be subjected under Indonesian seismicity by adopting the latest national earthquake standard and the recent response spectrum, as a representative of the high seismicity region.The BRBs are installed at the upper part to fix the large deformation due to the difference stiffness between upper parts and lower parts, in the E3S Web of Conferences 429, 05029 (2023) https://doi.org/10.1051/e3sconf/202342905029ICCIM 2023 condition of sustaining lateral dynamic load from earthquake.To use the resource of BRBs intelligently, the determination of number and location of BRBs is made by optimizing the seismic responses to fit a certain criterion.To perform the optimization, a simple yet powerful Genetic Algorithm is adopted in this study.The optimization procedures and results are discussed in the following passages.

Methodology and objects 2.1 Evaluation of seismic performance through nonlinear dynamic analyses
Selecting a proper tool to perform nonlinear dynamic analyses is an important step on this study.The analysed building consists of four main elements: RC beam as a part of RC frame, RC column as a part of RC frame, RC walls acting as structural shear walls, and bucklingrestrained braces or BRB as lateral braces.These four elements are modelled using specific idealizations according to the software adaptation.In this study, an open licensed software developed by Professor Taiki Saito from Toyohashi University of Technology, Japan [13], is employed.The Graphical User Interface of the software is depicted in Fig. 3.This open licensed software had also been utilized in other similar studies [14][15][16].
Both of column and beam elements are idealized as single line element with two nonlinear bending springs at both ends and single nonlinear shear spring at the half element, as illustrated in Fig. 4. Since the column also supports axial load, the multi spring model initially presented by Lai, Will and Otani in 1984 [17] is adapted, as illustrated in Fig. 5.The nonlinearity of all structural members is accommodated via nonlinear shear spring and nonlinear flexural spring as idealized in Fig. 6 and Fig. 7.The buckling-restrained braces (BRBs) is modelled as a lateral damper, which is modelled as bilinear model shear spring as shown in Fig. 8.The detail of all idealizations is elaborated in the program manual [13].

Applied earthquake ground motions
The input seismic excitations in this study are chosen based on the specific location, which is one private campus in Indonesia: Universitas Muhammadiyah Yogyakarta, Indonesia.In 2006, Special Province of Yogyakarta experienced great earthquake and the location is still an earthquake prone area with high seismicity, thus there are still many rooms for development and improvement regarding to the building technologies for seismic resistance.The specific latitude and longitude are -7.80956,110.32173, as shown in Fig. 9. Based on this location, the response spectrum is then generated from reading the earthquake hazard map, provided in Indonesian standard SNI 1726:2019 and the chosen seismic waves are matched with this response spectrum.The selection and matching process follow Indonesian Standard SNI 8899:2020.The selected ground motions in this study are elaborated in Table 1.The matched response spectrum of all ground motions is shown in Fig. 10, and the time history forms can be found in Fig. 11, which are recorded by PEER [18].10.Response spectrum of indonesian earthquake hazard map and selected ground motions.

Building objects
To simplify the analysis, the building is subjected to only one direction of seismic excitation.Therefore, the 2D model is adequate for this study.The building has 5 bays and 10 story levels, with the bay's span is 6.4 m with inter-story height is 3 m, as depicted in Fig. 12.The shear walls stop at level 4, which represent the wall coverage of 40% frame height.The detail of structural element dimension and rebars is presented in Table 2-4.
The determination of the concrete and rebar strength and member dimensions refer from the previous study [12], as this study is an extension of previous research.The building has natural period of 0.467 second, and the damping modelled as stiffness proportional damping with coefficient h1 is 0.03 and h2 is 0.02.For the building with academic purposes, the limit inter-story drift of the building based on SNI 1726:2019 is 1% of height.Thus, to show the performance of BRB and the optimization results, the earthquake is scaled so that the drift is 1.33%.It is expected that after the installation of BRBs, the drift could be reduced according to the standard.

Buckling-restrained braces
In this study, the strength of BRBs follow Fig. 8 and the adopted parameter component is elaborated in Table 5, which refers to the previous study [12], where the additional maximum shear capacity from BRBs is about 10 percent of the frame shear capacity.

Adoption of simple genetic algorithm
Genetic algorithm (GA) has been a popular optimization method, influenced by the natural selection procedures.The concept is to use the survival of the fittest approach https://doi.org/10.1051/e3sconf/202342905029ICCIM 2023 [20].In this research, the flow of simple GA process is elaborated as many steps.Table 5. Adopted parameters of BRBs in this study [12].Step 1: Idealize the location of BRBs using the binary and arrange it as a chromosome, as shown in Fig. 13.Fig. 13.Idealization of chromosome using binaries.
Step 2: Build a set of population consist of 6 chromosomes, as an example in Table 6.Table 6.Production of population. No.
Initial Arrangement Step 3: Evaluate and rank the population regarding to the value of fitness function, as an example in Table 7. Step 4: Match the first half of first chromosomes with the last half of second chromosomes.This step is referred as crossover, as an example in Fig. 14.Step 5: Change one binary number randomly.This step is referred as mutation, as an example in Fig. 15.Step 6: Re-evaluate and rank the new population after crossover and mutation, and pick the 6 top ranks for iterations, by evaluating the fitness function.
Step 7: Iterate the process until the objective is achieved: the chromosome has the fittest value of fitness function.

Fitness function
Fitness function is a function to determine whether the arrangement of BRBs in the RC frame with curtailed wall is effective and it has enough element to reduce the seismic response of the building.The determination of fitness function is based on three parameters: the number of BRBs, the inter-story drift response of the structure after installing BRBs, and the damage indices appear after installation of BRBs.The number of BRBs already represent the cost of BRBs, since the larger number of BRBs needed, the higher cost could be expected.The formula of fitness function is shown in Equation 1. ( where: : weighting factor, taken from previous trial and error, with =0.3, = 2, and = 0.2 : the representative of the number of BRBs : the representative of inter-story deflection : the representative of damage indices of reinforced concrete member, surrounding BRBs member.

Parameter
To accommodate the number of BRBs in consideration of fitness function, a simple function is developed as presented in Equation 2. ( where: : Number of BRBs in story i. N : Total number of levels.k : The level where the wall stops.

Parameter
To integrate the inter-story drift of each level in the consideration of fitness function, a simple function is adopted as shown in Equations 3-4. ( where: : maximum story drift for each story : limit of story drift, which is 1% from 3 meter of level's height.Thus, this value equals to 3 cm.

Parameter
To adopt the damage indices of structural element after BRBs' installation, in the consideration of fitness function, a simple function is adapted as written in Equation 5. (5) where: j : bays number : damage indices of beams on left node : damage indices of beams on right node

Damage indices
To calculate , the value of beam's damage indices (DI) is needed.The damage indices are calculated by using the popular Park-Ang damage index [21][22] for reinforced concrete elements.The DI formula is presented in Equation 6.The specimen in Fig. 10 is applied with the earthquake motions in Fig. 9 and it is expected to have large value of inter-story drift, larger than the limit from SNI 1726:2019 (1% of height for academic building) so that the BRBs could reduce the seismic response.To make the deformation uniform before optimization, the time history records were scaled so that the maximum response of inter-story drift is 1.33%.To simplify the discussion, only 4 results from 4 different earthquake are discussed in this paper.The scale of the earthquake is presented in Table 8.As depicted in Fig. 16, the dashed line represents the RC frame with curtailed wall's responses under each seismic excitation.The GA was then performed and the result is the solid line in the same figure, where the solid line inter-story drift result is always less than the allowable drift, which is 1% of inter-story height.

Configuration of BRBs after optimization
The configuration of BRBs is based on the fitness written in Equation 1, where the lowest fitness function means the best solution of the BRBs implementation.All four optimization results are shown in Table 9, and Fig. 17-20 show BRBs location and the BRBs ductility responses under respective seismic motions.The ductility value must be larger than 1, as the value represent the ratio between maximum displacement to yielding displacement.All values of BRB's ductility are guaranteed more than 1, thus all BRBs develop yielding stage and the benefit of BRBs are utilized optimally.This study limit only to show how the GA could perform and give the result of optimized number and location of BRBs for each earthquake.Therefore, the optimized number of BRBs has different value for different earthquake, and should be studied further to have a universal solution of optimize location and number of BRBs for every input earthquake.In this study, most of the results showed that the BRBs are needed at the level 6, 7, and 8, where the inter-story drift is large before applying BRBs.It is correct that the GA chose most of the BRBs location at these story levels.

BRBs responses during the earthquakes
Fig. 21 shows one response of the BRBs element, which modelled as a damper in program.The BRBs is idealized as bilinear model, and under the seismic excitation of scaled 1940 El Centro NS earthquake, the BRB at the position of dash circle shows the nonlinear response, which develop the nonlinearity and increase the ductility.This action shows that the BRBs works and the benefit of buckling resistance from the element could be practised.

Conclusions
Structure comprises of reinforced concrete frame and partial shear walls can be used to sustain lateral seismic loadings.However, under dynamic nonlinear analyses, the significant drift of upper part without walls could suffer large deformation.To tackle the situation, BRBs are employed and the number and location is determined by using Genetic Algorithm to install the BRBs optimally.The study shows a good enhancement of inter-story drift responses under each seismic excitation after installing BRBs.The number of BRBs could be different for each earthquake.Thus, further study should be developed to determine a universal solution of number and location of BRBs for every earthquake.

:
negative parameter based on repeated loading influence : Yielding force : highest lateral drift responses due to seismic ℎ Hysteretic energy dissipated by beam's element : largest drift capacity during monotonic loading

Fig. 17 .
Fig. 17.Optimize location of BRBs under Scaled 1940 El Centro NS earthquake and the value of maximum BRB's ductility.

Fig. 18 .
Fig. 18.Optimize location of BRBs under Scaled 1952 Taft EW earthquake and the value of maximum BRB's ductility.

Fig. 21 .
Fig. 21.Optimize location of BRBs under Scaled 1940 El Centro NS earthquake and the BRB's shear-deformation response at selected location.

Table 1 .
Selected ground motions for this study.

Table 3 .
Details of beam elements.

Table 7 .
Ranking of fitness functions.

Table 8 .
Earthquake scale to reach inter-story drift of 1.33% inter-story height.

Table 9 .
Optimization results from genetic algorithm.