Reliability of steel structures with Chevron bracing systems considering the performance-based seismic design philosophy

In this paper, the seismic performance and reliability of steel buildings with Chevron-Braced frames are studied integrating a novel probabilistic approach and the performance-based seismic design concept. The seismic response of models is extracted using response history analyses with the help the commercial software SAP2000. In this sense, three variables associated with the seismic response of the structure are studied: overall lateral displacement, rotation of connections, and inter-story drift. Those responses are evaluated by exciting the structure with eleven characteristic ground motions of the zone with respect to three performance levels: immediate occupancy, life safety, and collapse prevention. Once the seismic response is extracted for every performance level, the reliability of the models is calculated with respect to inter-story drift as described next. First, considering the seismic response in terms of inter-story drift for every ground motion, the associated histogram is constructed. Then, using 13 Probability Density Functions (PDFs), a Chi-square test is performed to identify the best-fitted PDF associated to the histogram of inter-story drift. Afterwards, with the best-fitted PDF of inter-story drift, the probability of failure and reliability index are extracted considering serviceability limits for every performance level. This represents a unique approach to extract the risk of structures subjected to ground motions associated to different performance levels. In addition to the structural reliability, a study about the cost of the structures with and without Chevron braces is developed, and then, it is documented the best option. Finally, based on the results reported in this paper, it is demonstrated that steel buildings with Chevron-braced frames present a better seismic performance than steel moment resisting frames without any bracing system. In summary, overall lateral drifts are reduced between 40 and 60% when Chevron braces are implemented in comparison to steel moment resisting frames without braces. On the other hand, if Chevron bracing systems are not used, i.e., in steel moment resisting frames, the inter-story drifts are about 300% higher than those of steel structures with Chevron braces. Hence, structural damages can be considerably reduced if Chevron-braced frames are implemented in steel structures that may be excited by characteristic ground motions of the zone where they are located.


Introduction
In the aftermath of earthquake occurrence, it has been documented the devastation in terms of human losses as well as economic costs because of damages in buildings and infrastructure [1,2].For example, the 1985 Mexico City, 1989 Loma Prieta, 1994 Northridge, and 1995 Kobe earthquakes caused damages of around 4, 6, 30, and 150 billon dollars, respectively [1][2][3].These are some of the reasons why earthquake-resistant building codes are being updated frequently as structural engineers acquire new knowledge about the seismic performance of structures.In a general sense, it is supposed that if structural engineers follow recommendations and/or codes during the design process of buildings, the probability of collapse is reduced to very low levels.In other words, the main objective of prescriptive building codes is to prevent the collapse of the structure under the most probable loadings that the structure may experience during its life period.However, this is not always the case since several collapses have been reported in structures around the world because of the shaking provoked by ground motions.Furthermore, very few is recommended in building codes and/or guidelines about how to control and reduce the damage potential in buildings subjected to earthquakes [4].The above and some other issues related to the resilient design of buildings motivated the introduction of a new philosophy to design structures called Performance-Based Seismic Design (PBSD) [5].
One of the first documents about the PBSD philosophy was published by the Structural Engineers Association of California (SEAOC) in two volumes titled Performance Based Seismic Engineering of Buildings [6].These guidelines reported a set of recommendations for engineering procedures to design buildings of predictable and defined seismic performance.In 1996, another technical advancement was documented by the Applied Technology Council (ATC) in a report titled ATC-40 Seismic Evaluation and Retrofit of Concrete Buildings [7].One year later, in 1997, the Federal Emergency Management Agency (FEMA) documented one of the most important guidelines about the seismic rehabilitation of buildings [8].These guidelines were expected to be a tool for design engineers, a reference for building regulatory officials, and the base for the generation and implementation of building code standards.To improve the prescriptive design guidelines, in 2000, a major study funded by FEMA was conducted by the joint venture SAC [SEAOC, ATC, and California Universities for Research in Earthquake Engineering (CUREE)].The main purpose of such a study was to generate recommendations for more robust construction and seismic design of steel structures.The major findings of the research developed by SAC were documented in a series of reports [9][10][11][12][13][14][15].In certain way, the above-documented reports and guidelines can be credited as the first technical documents for the PBSD of structures.Few years later, because of raising studies about PBSD, many comprehensive recommendations and/or guidelines were documented in the literature about the PBSD philosophy.Some of them are listed as follows: Quantification of Building Seismic Performance Factors [16], Guidelines for Performance-Based Seismic Design of Tall Buildings [17], Seismic Provisions for Structural Steel Buildings [18], An Alternative Procedure for Seismic Analysis and Design of Tall Buildings Located in the Los Angeles Region [19], Seismic Performance Assessment of Buildings [20], NEHRP Recommended Seismic Provisions: Design Examples [21], and Seismic Evaluation and Retrofit of Existing Buildings [22].More recently, Heshmati & Aghakouchak [23] evaluated the response modification, overstrength, and deflection amplification factors, respectively, for steel diagrid structural systems.In this research, the FEMA P-695 report [16] was implemented to quantify these factors.As part of a wide-range studies of PBSD philosophy, Mohsenian et al., [24] developed a comprehensive investigation about the seismic reliability of 16-, 24-, and 32-story diagrid systems considering 12 compatible response spectra.Return periods of 475-and 2475-years, respectively, were implemented to evaluate the seismic performance of the selected structures.The main result of this research was that multi-level response modification factors proposed by Mohsenian et al. [24] can be implemented in the PBSD of diagrid structures to achieve diverse performance targets corresponding to different seismic hazards.In 2021, another research about the probabilistic seismic demand of steel diagrid systems was documented [25].Four diagrid structural systems with 4-, 8-, 16-, and 24-story were subjected to incremental dynamic analyses, and it was demonstrated that the most important intensity measures for steel diagrid systems are the peak ground velocity (PGV) and the ratio of the PGV and peak ground acceleration (PGA), in other words, PGV/PGA.One year later, in 2022, buckling restrained braces were implemented in diagrid structures to form a hybrid structural configuration [26].In total, 32 diagrid steel models with different configurations were designed to be studied in such research.Pushover (nonlinear static) and incremental dynamic analysis were implemented in the study to extract the seismic response of the steel diagrid models.It was documented that the findings of the above research can help structural engineers to decide how to use bracing systems in the height of diagrid structures.Very recently, a novel model to extract the seismic response of mid-rise steel modular building systems (MBS) under near-field ground motions was documented [27].In this investigation, the seismic vulnerability of mid-rise steel modular building systems was studied with the help of fragility and seismic demand hazards curves, respectively.The main finding was the introduction of seismic demand models for PBSD methods.
Based on the previous discussion, it is clear that the PBSD of structures was implemented for steel and reinforced concrete buildings.In this paper, the debate is focused on the PBSD implementation of structures with Chevron braces.Thus, a review of some of the most important investigation is documented as follows.Back in 1988, Khatib et al. [28] published a technical report titled "Seismic Behavior of Concentrically Braced Steel Frames".In such research, Khatib et al. [28] reported that Chevron-braced frames have an inelastic cyclic behavior that is mainly characterized by a deterioration of strength, quick redistribution of internal forces, tendency to form soft stories, and fracture due to excessive structural deformation.The main contribution was the introduction of a novel structural system including vertical linkage elements in structures with Chevron braces which was demonstrated to be economical and practical [28].More than ten years later, to identify improved design procedures and code provisions, Sabelli et al. [29] detected ground motion and structural characteristics that control the seismic response of concentrically braced steel frames.In such a study, response history analyses were implemented to study the effect on the most important structural response parameters of diverse structural configurations.In 2005, Kim & Choi [30] studied the seismic performance of ordinary concentric braced frames and special concentric braced frames, respectively, implementing Chevron bracing systems.Several pushover analyses were used to extract the seismic behavior of steel structures with different span lengths and stories.The main conclusion of the study was that response modification factors, extracted with the help of pushover analyses, were in most of the cases smaller than the values recommended in building codes.Few years later, Tapia-Hernández and Tena-Colunga [31] investigated the structural behavior of 26 regular steel moment resisting frames with concentrically Chevron bracing systems using pushover analyses.They designed the 26 steel structures to be in seismic zones in Mexico.As a result, they reported that there is a relationship between collapse mechanisms with respect to the height of the structure, this is something that is not considered in construction guidelines nor building codes.In addition, as part of the main results, Tapia-Hernández and Tena-Colunga [31] developed a new equation to calculate the minimum strength ratio between Chevron bracing system and columns of moment resisting frames to achieve the following collapse mechanism: strong column-weak beam-weaker bracing.Additionally to the above-mentioned studies, the seismic performance of several reinforced concrete structures with Chevron braces were studied by Godinez-Dominguez et al. [32].Reinforced concrete buildings were designed considering the capacity design method for three different soil conditions in Mexico.Nonlinear response history analyses were implemented using simulated ground motions corresponding to a target design spectrum representing the maximum considered earthquake of the zone.Based on the results of such a research, Godinez-Dominguez et al. [32] documented that proper ductility capacities and overstrength demands can be obtained if capacity design parameters recommended in building codes are used.In general, they reported a satisfactory structural performance of the structures with Chevron braces.As a complement of the last reported research, in 2015, a very interesting study about the seismic performance of reinforced concrete buildings with Chevron bracing systems was documented in the literature [33].In this study, Huang et al. [33] reported that guidelines for the use of Chevron bracing systems in reinforced concrete spatial frames still require improvement.Cyclic test of three specimens were conducted on laboratory to evaluate the seismic behavior of reinforced concrete spatial frames with Chevron braces.As a result of this experimental study, it was validated that structures with Chevron braces presented better performance than structures without bracings.These results were documented in terms of hysteresis loop, energy dissipation, strength, and stiffness degradation.Furthermore, when Chevron braces are implemented, an overall mechanism of failure is developed instead of the normal inter-story mechanism of failure.Also, it was found that after the failure of Chevron braces, the structure continues transferring and supporting loads.In general, Huang et al. [33] demonstrated that braced frames develop a better seismic response than systems without them.Another important research is the one documented by Asada et al. [34] where a study was developed about the seismic performance of Chevron-configured special concentrically braced frames with yielding beams.In brief, two structure prototypes were studied three-and nine-story frames braced with Chevron systems including yielding beams.Nonlinear dynamic analyses were utilized to study the seismic behavior of the structures considering two simulated seismic hazard levels.Brace fracture, beam vertical deformation, and collapse were studied.At the end of the research, Asada et al. [34] stated that the seismic performance of steel frames with Chevron braces with limited beam yielding performs similar or even better than steel structures with X bracing systems.In 2022, Seki et al. [35] presented research about the structural behavior of steel structures with Chevron braces subjected to seismic loading based on Japanese practice.In such study, six braced frames and nine isolated braced structures were tested in the lab to analyze their seismic performance.The structures were designed and constructed following the Japanese practice.Round HSS or I steel sections were implemented for the braces and the frames were subjected to cyclic loading like the protocols specified for steel moment resisting frames in the US.It was validated that Chevron braced frames with compact sections can safely develop large drifts.Very recently, several Chevron braced frames were studied to determine their seismic performance [36].Arc-shaped, polyline, and lateral rod Chevron bracing systems were proposed and analyzed in the paper.For every Chevron bracing system, the bearing capacity, monotonic loading, vertical displacement at the brace point intersection, failure mode, hysteresis performance, energy dissipation, and fracture trend were studied and compared with other structural systems.The results validated that the bearing capacity of normal Chevron braces underwent a rapid change during the loading process.On the other hand, the studied Chevron bracing systems did not experience such a behavior.
Based on the above information, the main research gap detected in this paper is the lack of computational tools to extract the risk of steel structures with bracing systems.Thus, it is necessary to extend the study of bracing systems from deterministic to probabilistic approaches.In this sense, in this paper, an alternative reliability technique is implemented to extract the risk of steel structures with and without Chevron braces in terms of probability of failure and reliability index.Within this context, the PBSD concept is implemented in this paper considering three performance levels: Immediate Occupancy (IO), Life Safety (LS), and Collapse Prevention (CP), respectively.The significance and importance of the applied methodology is mainly based on the intelligent integration of a unique reliability technique to extract risk and the PBSD philosophy concept.This is something quite unique in the literature since in more of the cases the seismic performance of structures is considered only in deterministic analyses.Thus, compared with previous research studies, the implementation of this methodology may contribute to the earthquake-resistant design of resilient structures.Hence, the paper is organized as follows.First, the PBSD concept is introduced to clarify all the technical aspects behind this novel design philosophy.Then, the methodology to select ground motions is explained in detail, this part is very important since the selected ground motions will be used for the reliability calculation of the structures under study.In Sect.4, the probabilistic approach to extract risk is comprehensively documented.In Sect.5, the numerical examples are introduced.Finally, in Sects.6 and 7 results and conclusions are reported, respectively.

PBSD concept
The PBSD concept is a very recent design philosophy in the structural engineering community.In this part of the paper, it will be documented.The main objective of PBSD is to design structures that can achieve established performance levels for different ground motion intensities, controlling, in certain way, the possible damages that they may experience under the action of earthquakes.In other words, the PBSD represents an alternative for structural engineers when deterministic approaches, as those reported in building codes, are not efficient enough for the complexity of the project.In general terms, the process of the PBSD concept is illustrated in Fig. 1.The first step is to define the performance levels to be studied in the structure, these levels are generally selected by the structural engineer in consensus with the owner of the building and the contractor.Then, a preliminary design is proposed based on deterministic code approaches, which will be obviously updated as the process advances.The third step is to evaluate the performance capability of the structure considering the selected performance levels.Thus, if the performance of the structure satisfies the selected performance levels, the design process is done, if not, the structural design must be updated.
It is important to mention that one of the most important steps in the PBSD concept is the definition of performance levels.As previously mentioned, in this paper, three performance levels are considered in this research: IO, LS, and CP.They are described in Table 1.

Ground motion selection technique
The ground motion selection strategy must be documented in this part of the paper.It is well-known that if nonlinear response history analyses will be implemented in the calculation of the seismic response of structures, a suite of representative earthquake records must be selected.Within the following context, the ground motion selection technique is implemented.It is important to state that the ground motion selection is developed in terms of response spectra.Thus, the first step is the generation and/or construction of the corresponding target response spectrum considering the seismic hazard of the zone.Then, a database of more than 20,000 ground motion records occurred in Mexico is used to select the eleven most representative earthquakes of the zone following the next criterion.The raw acceleration of the 20,000 records is converted to spectral acceleration in the form of response spectra, then, scale factors are applied to every response spectrum to match as much as possible the form of the target response spectrum.In addition, scale factors must help to the spectral acceleration of the 11 selected candidates to match the spectral acceleration of the target response spectrum corresponding to the first period of vibration of the structure [37].Figure 2 illustrates a flowchart of the ground motion selection strategy.

IO
After the impact of ground motions related to IO performance level, the structure presents very few structural and non-structural damages.The structure can be reoccupied right after a quick inspection of a structural engineer LS Considerable damages have occurred in the structure after the impact of earthquakes related to LS performance level.However, the chance of the structure to collapse is very low.Structural repair and/or retrofit will be necessary before the building can be reoccupied CP The building is vaguely standing after the shaking of ground motions related to CP performance level.The damage of the structure is considerable high.The structure is about to collapse.To repair and/or retrofit may be very expensive Based on the above paragraph, it is evident that the selection of the ground motions must be developed in terms of a seismic hazard.In this paper, the seismic hazard of the zone is considered with respect to probability of exceedances and return period related to the three performance levels under consideration as summarized in Table 2.

Probabilistic approach to extract risk
The probabilistic approach to extract the structural reliability of steel structures with Chevron braces is presented in this section of the manuscript.The safety and/or security of structures is generally represented in terms of probability of failure ( p f ) and reliability index ( ) [38].Within this context, when structures are subjected to seismic loading their responses can be studied in terms of overall and/or inter-story drifts, respectively, as well as rotation of connections.In nature, those structural responses are random variables, and their stochastic behavior can be represented by their corresponding Probability Density Functions (PDFs).In this paper, the probabilistic approach to extract risk is used considering the inter-story drift of the structures under study.To clarify the concept behind the novel probabilistic approach, please suppose that Fig. 3 is illustrating the inter-story drift response of one structure under the action of a specific ground motion.For illustration purposes only, a Normal PDF is considered in Fig. 3.
In Fig. 3, the vertical and horizontal axes represent the PDF value and inter-story drift, respectively.It can be observed the limits a and b as well representing boundaries between the safe and failure region respectively, their values will be documented later in the paper.In addition, in Fig. 3, the p f value is represented by the blue shaded area which can be computed as [38]: where x represents the inter-story drift provoked for a particular ground motion, and P(a ≤ x ≤ b) is the probability of inter-story drift taking a value between the a and b limits, which can be extracted as [38]: where f x (x) is the best-fitted PDF of inter-story drift used for the calculation of the p f of the structure under study.It is important to mention that a Chi-Squared test is performed by the reliability approach to select the best-fitted PDF between the following eleven distribution functions: (1) Normal, (2) Lognormal, (3) BirnbaunSaunders, (4) Extreme Value (EV), (5) Gamma, (6) Generalized Extreme Value (GEV), (7) Logistic, (8) Log-logistic, (9) Stable, (10) t Location Scale (tLS), and (11) Weibull [39].
(1)  Once p f is extracted, it must be utilized to obtain the corresponding value of which is a safety indicator that repre- sents the reliability of the structure under the action of the corresponding ground motion.The reliability indicator can be calculated as [38]: where Φ −1 is the inverse value of the Cumulative Density Function (CDF) of the best-fitted PDF selected in the previous steps of the methodology.
Within this frame of reference, it must be stated that large values of represents a high reliability of the structure.In other words, when p f is small, is high, and vice versa.In general terms, to be considered as safe structures, buildings must present values higher than 1.285 which represents p f of about 10%.This will be discussed more in detail in Sect.6 where the results are documented.

Numerical examples
In this part of the paper, the numerical examples are presented which will be used to implement the novel reliability approach introduced in Sect. 4. Within this frame of reference, following the building code of Mexico [40], two 7-story steel structures were designed considering that they would be constructed in Culiacan, Mexico, respectively.The plan view of both buildings is similar, it is showed in Fig. 4.
First, a steel building without Chevron braces was designed, it is illustrated in Fig. 5.It mainly represents a steel moment resisting frame.On the other hand, another steel structure was designed implementing Chevron braces to reduce lateral drifts that the building may experience.The steel structure with Chevron braces can be seen in Fig. 6.It must be stated that loadings used for the final design of both structures, with and without Chevron bracing systems, were calculated following the building code of Mexico [40].For the case of vertical loading, dead and live loads were determined considering the use of the structure (office building).On the other hand, for lateral loadings, the demand of wind was calculated considering the location of the structure and the external form of the buildings as well as the regional velocity of the wind for the zone under consideration.For the case of earthquake loading, based on the type of structure, location, and earthquake hazard of the zone, a design response spectrum was constructed.In other words, for seismic loading, the following steps were implemented in the design of the steel structures.The first step is to calculate the weight of the structure per floor.Then, the structure must be classified depending on its importance, size, and type, respectively.Next, based on the seismic hazard of the zone, the maximum acceleration in rock is calculated and converted to the corresponding soil type.Afterwards, four factors are calculated: seismic performance factor, ductility reduction factor, over-resistance factor, and redundancy factor.Finally, with all the factor previously calculated the lateral seismic loading can be calculated for the structure under consideration.For the bracing elements, HSS frame elements were used in the structure as illustrated in Fig. 6.Once the structures were properly designed, the ground motion selection strategy introduced earlier in Sect. 3 was implemented to select 11 ground motions for IO, LS, and CP performance levels, respectively.In other words, 33 ground motions were selected for every structure under consideration.It is important to mention that the selected ground motions are real and recorded in Mexico, and they are part of a data base containing more than 20,000 candidates.Figure 7 illustrates the target response spectrum for every performance level which is used for the proper selection of ground motions.Tables 3 and 4 summarize the ground motions corresponding to every performance level for the steel structures without and with Chevron braces, respectively.Information in Tables 3 and 4 are organized in terms of performance level, earthquake name, scale factor, magnitude, focal depth, peak ground acceleration (PGA), location, and date.Every ground motion was used to perform nonlinear response history analyses of the structures and determine their respective structural response.The frames analyzed in this paper (Figs. 5 and 6) are part of a 3D building.They are in the perimeter of the steel structures, thus, the frames under consideration in this research are exterior frames.One more reason to only select exterior frames for the dynamic analyses is because Chevron braces are placed in the perimeter of the 3D buildings.With respect to the nonlinear dynamic analyses of the 2D frames, it is important to mention that they were performed by considering their corresponding masses and they were excited with the selected ground motions.Thus, seismic loading was applied in 2D, and the structural responses were extracted to be used in the calculation of the risk in terms of probability of failure and reliability index, respectively.The next section summarizes the results of the research.

Results
Results are presented in this section in terms of the seismic response of every structure and their corresponding structural reliability.In addition, the cost of the steel structures with and without Chevron braces is discussed.Hence, using the ground motions summarized in Tables 3 and 4, nonlinear response history analyses were performed using SAP2000 finite element software [41].It is important to mention as well that direct integration method was used for the time history analyses, the motion type was transient, proportional damping was implemented, and the method of Newmark was used for time integration.Next, the results are discussed.1 3

Seismic response of buildings with and without Chevron braces
Table 5 summarizes the mean values of maximum inter-story drifts considering the IO, LS, and CP performance levels for the fourth floor of the structures.For every performance level, 11 ground motions were selected for the steel structures with and without Chevron braces, respectively.In this sense, the maximum inter-story drifts were extracted under the action of earthquake loading with the help of nonlinear dynamic analyses performed via SAP2000 software [41].The main observation with respect to Table 5 is that, for every performance level under study, the mean value of the maximum inter-story drift was larger for steel structures without Chevon braces.This demonstrates one of the benefits of the implementation of bracing systems to reduce lateral drifts.Furthermore, based on what is summarized in Table 5, it is demonstrated for both steel structures that mean values of maximum inter-story drifts are increasing from IO to LS and from LS to CP performance levels, respectively.This is quite logical since the magnitude of ground motions related to IO are smaller than those related to LS performance level, and ground motions related to CP are the largest in terms of magnitude.Table 6 presents the mean values of maximum overall lateral drifts.This structural response represents the maximum lateral displacement that the structure presented during the earthquake excitation.In summary, it is clearly observed in Table 6 that steel structures with Chevron bracing systems are presenting smaller displacements than the structure without braces.In addition, larger lateral displacements are detected for the case of ground motions 1 3 related to CP performance level in comparison with LS and IO, respectively.Thus, in general terms, the steel structure with Chevron braces is improving the structural response for both buildings under consideration.Another structural response that was studied for steel structures with and without Chevron braces was the rotation of the beam-to-column connection of the node located in the corner of the top roof level of the structure.Table 7 summarizes this response in terms of the mean values of the maximum rotation presented in such a node for every ground motion under study.It is documented in Table 7 that the steel structure with Chevron braces is reducing the rotation of the connection, hence, if damage is related to rotation of beam-to-column connections [42], Chevron braces may contribute to reduce it.Additionally, like the other two structural responses, rotations are increasing from IO to LS and from LS to CP performance level, respectively.Finally, based on the deterministic results summarized in Tables 5, 6, 7, the implementation of Chevron braces in steel structures may improve the structural response of them under the action of earthquakes.In the next section, it is presented the structural reliability of both structures considering the inter-story drift of them.

Structural reliability of buildings with and without Chevron braces
It was demonstrated in the previous subsection that the implementation of Chevron braces in steel structures improves the structural behavior when such structures are excited by ground motions of different intensities.In this part of the paper, the novel probabilistic approach presented earlier in Sect. 4 is implemented to extract the risk of the structures in terms of reliability index ( ).Because of the sake of space and the limitation in terms structural performance limits, the reliability analysis is limited to extract the risk considering the inter-story drift of the structures under the action of the selected ground motions.Thus, the structural performance limits reported in Table 8 are used to extract the reliability of the structures under consideration.In Table 8, the value of H represents the height of the floor under study.In the case of this paper, based on what is observed in Figs. 4 and 5, for the fourth story, the height is 3.5 m.
The structural reliability considering the IO performance level for the structures with and without Chevron braces is summarized in Table 9.For the case of the steel structure without the implementation of Chevron braces, it is observed that the values of are considerably smaller with respect to the structure with Chevron bracing systems.In addition, for some cases, negative values of are presented, these negative values are related to probability of failure superior to 50%.On the other hand, the meaning of values of = ∞ is that none of the inter-story drifts were higher than the performance limits presented in Table 8.Thus, in general, based on the results reported in Table 9, for IO performance level, steel structures with Chevron braces are more reliable than structures with no bracings at all.Table 10 summarizes the structural reliability of steel structures with and without Chevron bracing systems for the LS performance level.It can be detected in the results presented in Table 10 that the values for the buildings with Chevron braces are higher than for the case of steel structures with no braces.This is justified because Chevron braces are reducing the inter-story drifts of the structure under consideration.In other words, it is observed once more that structures with Chevron are more reliable.One more aspect to consider in the results presented in Table 10 is that the PDF used for the calculation of structural reliability are very different, this can be happening because of the high uncertainty of the seismic loading.
The structural reliability for the case of strong ground motions corresponding to the CP performance level is summarized in Table 11.In general, somehow the same is happening than for the other levels.Higher reliability indexes are observed for the steel structures with Chevron bracing systems.On the other hand, for some cases, negative values of can be seen in Table 11.In summary, once more again, structures with bracing systems are more reliable.

Cost of buildings with and without Chevron braces
So far in this paper, the deterministic and probabilistic structural response of the steel structures with and without Chevron braces has been studied.In general, it has been demonstrated that when bracing systems are implemented in steel structures, their inter-story drift deformation can be reduced, and their corresponding structural reliability increased.However, another important aspect to be considered in the decision about the structural system to use is the respective cost.In this context, Table 12 summarizes the cost of the steel structures with and without Chevron braces.The reported costs were calculated based on the prices of the year 2020, when this research was performed.The costs included connections, price of steel, fabrication, and construction of the structures.It is observed in Table 12 that is more expensive to construct a steel structure using Chevron bracing systems, but the structural response can be considerably reduced and so the probable damages on the buildings.Hence, to take the decision about the structural system to implement in a building is not easy and several factors must be integrated in the discussion before taking the decision.Nevertheless, based on the results presented in this paper, if possible damages of structures need to be reduced, the main option would be to implement Chevron bracing systems.

Conclusions
Based on the results presented in this paper, the Authors have reached the following conclusions: • The PBSD philosophy represents a very promising alternative for the design of earthquake-resistant structures since damages can be reduced and/or controlled, it was demonstrated its implementation to steel structures with Chevron braces in this paper.
• For every performance level under consideration, overall lateral drifts for the steel structure with Chevron braces were smaller.
Thus, if overall lateral drifts need to be controlled, it seems that Chevron braces are a good option.In addition, if one compares the overall lateral drifts of the structure with and without Chevron braces, respectively, it can be observed a reduction of about 40% of overall lateral drifts for the IO and LS, and 60% for the CP performance level.• The structural response related to inter-story drifts was larger for the steel structures without Chevron bracing systems.Thus, if inter-story drifts in a structure excited by an earthquake causes damage on it, the use of a structural system with Chevron braces could be the best option to prevent such damage.Within this context, it was detected that inter-story drifts are about 300% larger for the steel structure without Chevron braces for the IO performance level.For the LS and CP performance levels the inter-story drifts were approximately 266 and 263% larger, respectively, for the steel structure without Chevron braces.• Rotations of beam-to-column connections were studied as well.It was detected, for most of the performance levels, that rotations of connections are larger for the steel structure without Chevron braces.• Reliability of the structure with Chevron braces was always superior in comparison with the steel moment resisting frame.
For some cases, the structure without Chevron braces presented negative values of , representing a probability of failure superior to 50%.On the other hand, some values of = ∞ were detected for the structure with Chevron braces, indicating the highest reliability which can be obtained with the novel probabilistic approach.In summary, in terms of , the use of Chevron braces always incremented the reliability of the building.• In terms of construction cost, it is demonstrated that the steel structure with Chevron braces is more expensive than the steel moment resisting frame, however, such an increment in costs can be justified with the reduction of the structural response that will be achieved if Chevron braces are implemented.It is important to mention that reducing lateral and/ or inter-story drifts may result in a decrease of the possible damages of the structures.• It is important to mention that some of the main limitations of the study presented in this paper are the following.The structural response of the structures under consideration were extracted considering only 2-D analyses, thus, it is recommended to use 3-D analyses for future investigations.Only steel structures were used in this paper to implement the novel probabilistic approach, then, it maybe feasible to validate the reliability technique for reinforced concrete buildings as well.The selected ground motions are coming from a data base of only 20,000 candidates, it is recommended to expand such a data base.• Finally, due to post-buckling compressive strength deterioration and full yielding of braces, unbalanced force of Chevron bracing systems could be an effective factor that may reduce the structural reliability of the system.This must be investigated more in detail in future research.

Fig. 4
Fig. 4 Plan view of the steel structure (units in meters)

Fig. 5
Fig. 5 Steel structure designed without Chevron braces

Table 3
Ground motions selected for steel structure without Chevron braces

Table 4
Ground motions selected for steel structure with Chevron braces