Machine learning-based predictive models for equivalent damping ratio of RC shear walls

Energy-based seismic design is being rapidly developed and suggests that the seismic demands are met by the energy dissipation capacity of the structural members. Equivalent damping ratio is a measure of energy dissipation in structural members that accounts for the post-elastic behavior of the member and provides insight regarding the dynamic response reduction during a seismic event. The present study implements a machine learning algorithm to estimate the equivalent damping ratio in reinforced concrete shear walls at displacements corresponding to a 1.0% lateral drift ratio. Five different machine learning models, namely, Robust Linear Regression, K-Nearest Neighbor Regression, Kernel Ridge Regression, Support Vector Regression, and Gaussian process regression were evaluated in order to choose the model with the highest accuracy. Among all models, Gaussian process regression, a machine learning method with successful implementation experiences in civil/structural engineering related problems, is selected to identify the equivalent damping ratio. The developed GPR-based algorithm uses a database of 161 rectangular shear walls subjected to quasi-static reversed cyclic loading with geometry and mechanical properties commonly found in building stocks of many earthquake-prone countries. The proposed algorithm estimates the equivalent damping ratio for each specimen by predicting the cyclic dissipated energy and lateral force values as two dependent variables. The model validation results show a mean coefficient of determination (R2) of about 0.89; a relative root mean square error of about 0.14 and a mean absolute percentage error of 10.44%, which is considered a substantially accurate prediction for such a complex problem. An open-source model and the entire database are provided which can be used by researchers and also design engineers. The proposed predictive model enables comparing the damping capacity of shear walls and the outcomes of this study are believed to contribute to the energy-based design or performance evaluation procedures in terms of predicting the energy capacity of shear walls.


Introduction
Reinforced concrete shear walls are the most common lateral load resisting structural members used to withstand seismic loads and deflections in earthquake-prone countries in addition to resisting gravity loads. As the main lateral load-bearing member of the structural system in many buildings, shear walls might experience severe loads and extreme deformations during their life span.
Nowadays, evaluating behavior and deformations of multi-degree of freedom (MDOF) nonlinear systems subjected to transient loads is possible using high-fidelity finite element-based structural analysis software. However, despite all advances in sophisticated methods, more straightforward approaches are preferred for computational efficiency and saving time. Particularly, two approximate approaches are broadly used to determine the maximum responses of nonlinear systems under earthquake loading. The first approach uses inelastic response spectra or modifies the maximum response of an elastic single degree of freedom (SDOF) system with a modification factor according to the damping and lateral stiffness properties of the nonlinear system. The second approach defines an equivalent linear SDOF model with proper stiffness and damping ratio (smaller stiffness and larger damping) to represent the original nonlinear system behavior (Miranda and Ruiz-Garcia 2002;Dwairi and Kowalsky 2004;Priestley et al. 2007). In the latter approach, realistic estimation of the equivalent linear SDOF system properties such as equivalent damping ratio is crucial for accurate judgment of performance and behavior of the MDOF nonlinear structure. This is particularly important if analyses are carried out using a displacement-based approach as this methodology utilizes equivalent structural properties to evaluate target displacements based on drift limitations (Dwairi 2004). Notably, there is a growing interest in displacement-based design philosophy as an alternative to the force-based design approach (Priestley 2003;Priestley et al. 2007;Zaharia and Taucer 2008;Sullivan 2018). Furthermore, an accurate approximation of the equivalent damping ratio is important as it controls the estimation of structural dissipated energy in energy-based design methods. Introduced by Housner (1956) and followed by Trifunac and Brady (1975) and Akiyama (1985), the energy-based design approach aims to address the shortcomings of force and displacement-based design methods by considering the balance between the input energy to a structure and structural energy dissipation capacity as well as the duration of a ground motion (Benavent-Climent et. al. 2010;Erberik et. al. 2012;Dindar et. al. 2015).
Energy-based seismic design is being rapidly developed today and is expected to be used in parallel to displacement-based design methods in the near future for more precise analysis. Equivalent damping ratio is a measure of energy dissipation in structural members that accounts for the post-elastic behavior of the member and provides insight regarding the dynamic response reduction during a seismic event. For the design and assessment of reinforced concrete members and structures, it is important to accurately estimate the energy dissipated by the structural model and structural properties that influence the equivalent damping ratio. The concept of equivalent damping ratio was pioneered by Jacobsen (1930) and has been investigated thoroughly by several researchers (e.g., Jacobsen 1960;Rosenblueth and Herrera 1964;Hudson 1965;Iwan 1980;Fardis and Panagiotakos 1996;Miranda and Ruiz-Garcia 2002;Blandon and Priestley 2005). Some studies have also proposed relations for estimating equivalent damping ratio based on hysteresis models for various structural components; e.g. the equation by Gulkan and Sozen (1974) for one bay one story RC frames, Kowalsky et al. (1995) relation for isolated bridge columns, and Priestley (2003) equations for several structural components (e.g. concrete walls and columns, steel members, precast walls and frames).
Damping characteristics of reinforced concrete walls, masonry walls, infilled frames, and other isolated structural components using experimental methods have been the subject of numerous studies. Buttmann (1983) tested two RC and masonry specimens under horizontal sinusoidal excitation. Evaluating the derived resonance curves showed that 11% and 24% damping values could be respectively attributed to RC and masonry walls under excitation amplitudes corresponding to safe shut down earthquake whereas 4% damping is a realistic estimate for both masonry and RC walls subjected to operating basis earthquake. Farrar and Baker (1995) stipulated the role of material properties (aggregate size and concrete strength) on damping properties of low-aspect-ratio shear walls and noted that the effective damping value could become as high as 22% for sufficiently cracked walls. The damping characteristics of bare, infilled and carbon fiber reinforced polymer (CFRP) retrofitted infilled RC frames were studied by Ozkaynak (2010) and Ozkaynak et al (2014) conducting quasi-static and pseudo-dynamic tests on 16 one-bay, one-story 1/3-scaled specimens. Evaluating test results, equivalent damping ratio values of 5%, 12%, and 14% were determined for bare, infilled and CFRP retrofitted frames and a new method based on the energy balance of the system was proposed for quantifying the damping properties of the specimens. Implementing shaking table test results, Yang and Park (2021) concluded that a value of 3.5% can be considered as the mean damping ratio for squat shear walls with limited damage subjected to earthquake loadings with peak ground acceleration of 0.1 g.
The well-known fact that large deformations induced by intense earthquakes frequently cause serious damage to structural and non-structural members has driven building code developers to impose certain provisions on lateral displacements. In seismic codes and building standards of the early 2000s, the maximum lateral drift ratio is recommended to be limited to 1.0-2.0% for the ultimate limit state (Park 2003). According to Eurocode 8 Part 1 (2004), the 1.0% lateral deflection restriction for the ultimate limit state applies to buildings with non-structural elements not interfering with structural deformations or buildings without non-structural elements. ASCE/SEI 7-16 requires that for a great variety of structures in risk category IV (buildings designed as essential facilities) maximum lateral drift ratio should not exceed 1.0% (ASCE 2016). FEMA 356 (2000 standard also imposes a 1.0% transient drift ratio limit for concrete wall structures at Life Safety (LS) performance level whereas the limit drift for Collapse Prevention (CP) for the same structure category is 2.0%. The National Building Code of Canada 2015 (NBC 2015) enforces that a maximum permissible lateral deflection of 0.01 story height must be considered for post-disaster buildings (i.e. buildings necessary to give service after disasters). The Colombian NSR-10 has similar requirements to Eurocode 8 Part 1 such that the lateral drift ratio has been limited to 1.0% by the standard (NSR-10 2010; Arroyo et al. 2018). Finally, according to the Turkish Building Seismic Code-2018 (TBSC-2018), even though the inter-story drift ratio is not directly implemented to evaluate the structural seismic performance of buildings, the drifts are limited to prevent infill wall failures. In the Turkish code, the drift ratio limits vary from approximately 1.0% to 2.0% depending on the infill wall-to-structural member connection type and the seismicity of the region. The literature revealed that although the necessity to restrict lateral deflections is recognized in seismic design codes globally, there is no consensus on the maximum allowable deformation limit. In this study, 1.0% lateral drift ratio is considered as a commonly accepted criterion for restricting damages caused by lateral deformations and therefore has been used as the reference displacement level.
In line with several scientific disciplines, ML methods have also found their place in solving problems related to civil engineering. Starting from simpler applications such as the conceptual design of linear steel members under bending or conceptual design of wind bracings in tall buildings (Arciszewski et. al. 1987;Arciszewski 1994;Reich 1996), ML methods are being employed nowadays to deal with a vast variety of subjects in the construction industry; e.g. bridge engineering, geotechnical engineering, structural health monitoring, design optimization, etc. (Lu et al. 2017;Zhang et al. 2016;Tseranidis et al. 2016;Pan et al. 2018;Huang et al. 2019). In terms of evaluating structural properties of RC or masonry members, ML models are particularly very efficient in addressing problems in which the relationships between multiple parameters are not comprehensively known (Siam et al. 2019;Song et al. 2020). For instance, estimation of shear strength of RC beamcolumn joints, shear strength of RC beams, and the maximum shear strength of reinforced concrete shear walls were studied using ML models by Jeon et al (2014), Chou et al (2020), and Song et al (2020) respectively. Luo and Paal (2018) used integration of multioutput least-squares support vector machine regression procedure and grid search algorithm for predicting backbone curves of rectangular RC columns subjected to reversed cyclic loading. Vu and Hoang (2016) studied the punching shear capacity of concrete slabs reinforced by Fiber-Reinforced Polymer (FRP) bars using a hybrid ML approach (LS-SVM). Siam et al. (2019) addressed predicting the ultimate drift ratio and failure modes of reinforced masonry shear walls using Principal Component Analysis and Projection to Latent Structures algorithms, respectively. The research by Mangalathu et al. (2020) is an example of implementing several ML models to classify the failure modes of reinforced concrete shear walls, providing a model with the highest classification performance.
In literature, to the best of the authors' knowledge, equivalent damping has not been estimated for RC shear walls using ML-based models. To fill this gap, an ML algorithm was developed in this study to identify the equivalent damping ratio of rectangular RC shear walls tested under quasi-static reversed cyclic loading at displacements corresponding to 1.0% lateral drift ratio. The developed ML algorithm utilizes a set of shear wall experimental data to estimate the equivalent damping ratio of each specimen. The employed ML method is Gaussian Process Regression (GPR) which is a non-parametric supervised technique with successful application records in structural and earthquake engineering problems. Some examples are: Hoang et al. (2016) employed GPR for constructing a model to predict the compressive strength of high-performance concrete, Momeni et al. (2020) used a GPR-based solution for studying the bearing capacity of piles, Sheibani and Ou (2020) developed a GPR-based model for evaluating the intensity of damage to structures in an earthquake region, whereas Deger and Taskin (2022a, b) proposed GPRbased models for predicting backbone curves of RC shear walls. Such studies stipulated the capabilities of the GPR method by showing high accuracy prediction performance using a limited number of samples.

Definition of the equivalent viscous damping and estimation procedure by using load-displacement curves
The equivalent damping ratio of each specimen from the given dataset is calculated by Eq. 1, which is based on the equivalent damping expression of the resonant steady-state proposed by Jacobsen (1930) and(1960).
where eq is the equivalent damping ratio, and E D is the cyclic dissipated energy. The E so is the elastic strain energy calculated as E so = 1 2 ku 2 0 where k refers to the equivalent stiffness corresponding to the loop maximum lateral displacement u 0 . Figure 1a presents these parameters for a hypothetical hysteresis loop, where f 0 represents the lateral force corresponding to the maximum lateral displacement u 0 .
The proposed ML algorithm requires two intermediate output parameters ( E D and f 0 ). These parameters are extracted from experimental load-displacement curves of database specimens. An example of the extraction procedure has been demonstrated in Fig. 1b for specimen MSW3 (Salonikios et al. (1999)) which is cantilever wall with 1800 mm height (Fig. 2).
As observed in Fig. 1b, from the three cycles in the vicinity of 1.0% drift ratio, the first cycle has been selected to avoid the additional effect of cyclic degradation in calculations. The equivalent damping ratio values at various displacements are evaluated from  (Salonikios et al. 1999) Eq. 1 knowing the two aforementioned parameters and u 0 . For instance, u 0 for the loop corresponding to the 1.0% drift ratio during the experiment is 0.0105 × wall height= 0.0105 × 1800 = 18.9 mm , f 0 is 134.79 kN, E D is 3513 kN mm, and the equivalent damping ratio is calculated as 21.9%.
In Fig. 3, maximum hysteresis loop drifts and the corresponding equivalent damping ratio values calculated by the described procedure have been demonstrated for specimen MSW3.
After several attempts of directly predicting the equivalent damping ratio, eq with different machine learning methods, the output accuracy was found to be insufficient. Alternatively, two individual GPR based models were developed for predicting the cyclic dissipated energy corresponding to the loop at 1.0% drift ratio ( E D ) and the lateral force (f 0 ) as the two intermediate outputs. Individually predicted f 0 and E D values were used for estimating equivalent damping ratio, eq as per Eq. 1. A concise introduction to theoretical principles of GPR method has been provided in Appendix I.

Reinforced concrete shear wall database
Databases/datasets constitute the backbone of data-driven modeling procedures based on machine learning methods (Fayyad et al. 1996;Montáns et al. 2019). In the past few years, several databases summarizing information about RC shear walls have been developed or collected by various researchers. A certain number of data from existing literature has been used to derive empirical relations for predicting or modeling various RC wall properties such as shear strength, deformation capacity, hysteresis behavior (Sengupta and Li 2014;Looi and Su 2017;Deger and Basdogan 2019;Song et al. 2020), whereas others were used to evaluate provisions by certain building standards (Orakcal et al. 2009;Shegay et al. 2015). Some of the available databases, on the other hand, were developed for very specific reasons such as the one by Nie et al. (2020) which is a database of 15 specimens to study shear-critical RC walls under combined tension-bending-shear loads. In addition, Mangalathu et al. (2020) assembled a database of 393 samples to develop ML models for predicting failure modes of RC shear walls.
The database used in this research is an expansion of the one assembled by Deger and Basdogan (2019) which includes RC shear wall specimens subjected to incremental reversed cyclic loading. The specimens with the following properties were not included in the database: (1) specimens subjected to monotonic loading; (2) walls with unconventional reinforcing including those with diagonal web reinforcement, composite material strengthening, mechanical couplers or reinforced with steel profiles; (3) retrofitted, repaired, or re-tested walls; (4) specimens with highly confined boundary elements (representing bridge piers, etc.); and (5) walls with openings.
After adding new data, the database was consisting of 447 specimens tested under quasi-static reversed cyclic loading where 276 walls had rectangular cross-sections. Considering the higher number of rectangular cross-sectioned walls compared to the rest of the specimens with barbell or other cross-sectional shapes, the study was focused on the rectangular category. It is noteworthy that rectangular shear walls have been increasingly used since the late 1980s replacing barbell-shaped walls in many countries (Wallace 2012). To assemble a database representing walls commonly used in residential/office type buildings, samples with the following characteristics were also excluded: (1) walls taller than 3.25 m; (2) walls without concentrated longitudinal rebars at extreme ends; (3) specimens undergone cyclic loading but lacking a reported and precise load-displacement curve (some references do not present hysteresis curves of all tested specimens); (4) specimens subjected to varying axial load, out of plane loading or very specific loading protocols (e.g. protocols with loads decreasing over time); and (5) specimens failed at ultimate drift ratios less than 1.0%.
The imposed five conditions resulted in the elimination of 115 samples from the initial set and the research was conducted on the remaining 161 wall specimens. It is worth also noting that no additional data cleaning to remove outliers (i.e. data standing away or deviating from other data) was performed to maintain the generality of the ML model, as advised by some researchers (e.g. Momeni et al. 2020).
Several researchers have reported significant changes of damping in correlation with the level of damage in reinforced concrete shear walls and other structural members (e.g. Aristizabal-Ochoa 1983; Consuegra and Irfanoglu 2008;Curadelli et al. 2008;Chopra 2020). The damage level at target drift level of 1% was investigated within the dataset used in this study. For this purpose, the damage progression reports of the specimens were scanned revealing that apart from 15 walls, most damage descriptions, when available, lack direct data about damage status at 1.0% drift ratio. For such specimens, the damage status at 1.0% drift was estimated using damage descriptions at smaller and larger drift ratios and/or their hysteresis loops considering the key lateral load-displacement points (i.e. coordinates of the yield point, the maximum lateral load, observable pinching shape, etc.) The described review showed that more than 95% of the specimens (154 out of 161 walls) exceeded their yield displacement at 1.0% drift ratio and can be classified as "irreparably damaged" according to Ghobarah (2004).
A more refined damage classification could be carried out using the Homogenised Reinforced Concrete damage scale (HRC scale) developed by Rossetto and Elnashai (2003). Using the mentioned reference, it is estimated that at 1.0% drift ratio, Homogenised Reinforced Concrete Damage Index, DI HRC of "70 to 80" can be attributed to the majority of database specimens (99 out of 161 walls) indicating that they are in post-yield status and reaching their ultimate load capacity. For 32 specimens, DI HRC , at 1.0% drift ratio, is estimated as "50 to 60" corresponding to moderate damage (preyield or yielding status) and for the remaining 30 specimens as "90 or larger" which corresponds to immediately before partial collapse or partial collapse. According to Rossetto and Elnashai (2003), HRC-damage index value ranges of 50 to 60, 70 to 80 and 90 or larger respectively correspond to moderate, heavy and major damage scales given by ATC-13, 1985.
For completeness of the discussion on damage, it is also beneficial to mention the failure mechanism of database specimens. It is expected to observe diagonal tension cracks, web crushing and continuous horizontal cracks along the base in shear-controlled walls; rebar buckling and concrete spalling at toes of flexure-controlled walls and a combination of diagonal cracks and boundary element longitudinal steel buckling in shear-flexure controlled walls (Deger et al. 2022;Benavent-Climent et al. 2012;Faraone et al. 2020). This study was carried out consulting descriptions of the specimens' failure mechanisms, available photographs of specimens' final status and lateral load-displacement hysteresis curves in the corresponding references. It was revealed that the database consists of 54 shear-controlled, 67 flexure-controlled and 40 shear-flexure controlled walls, respectively corresponding to about 33.5%, 41.5% and 25% of database specimens which may be considered as a relatively even distribution.
Machine learning approaches use measurable properties of databases (features) as input factors for making predictions. Consequently, the feature set for an ML model needs to be selected carefully to yield optimal results. Figures 4, 5, and Table 1 demonstrate the shear wall features used by the proposed algorithm to identify f 0 and E D (and subsequently, eq ) dependent variables. In addition to the essential shear wall features, Fig. 4 a Schematic view of a database wall; b Some of the model features u 0 is also included as a feature that corresponds to the actual measured maximum displacement value that the specimen has experienced at the loading cycles. As it can be seen from Figs. 1b and 3, the maximum lateral displacement that has been realized  during the tests might slightly differ from the target value of 1.0% drift due to the test setup accuracy. This effect has been reflected in the study by the addition of realized displacement u 0 in the feature set. The feature selection process for the two discussed GPR-based models evaluating f 0 and E D benefitted from the experimental equations for the peak shear strength and energy dissipation capacity of RC structural walls. Although f 0 and E D (lateral force and dissipated energy at 1.0% drift ratio) are apparently very different parameters from the peak shear strength and energy dissipation capacity factors, the discussed study proved to be helpful in determining the final feature sets.
For estimating peak shear strength value of RC shear walls, equations by ACI 318-14 (2014) Table 1, which represent geometrical and mechanical specifications of specimens, were selected as model features. On the other hand, some parameters such as those indicating confinement properties of boundary elements (e.g. spacing and yield strength of lateral boundary reinforcement) or ductility were observed as inefficient in improving the model prediction performance and were disregarded consequently.
A comparable procedure was carried out for determining the feature set for E D considering studies on the effect of various parameters influencing energy dissipation capacity of RC shear walls. Some of these parameters are compressive strength of concrete (Yan et al. 2008), amount of flexural rebars (Park and Eom 2004), mechanical properties of reinforcement (Belmouden and Lestuzzi 2007), axial load ratio (Su and Wong 2007;Yan et al. 2008) and confinement properties of boundary elements (Oh et al. 2002;Song et al. 2019). Benefitting from such studies, for the model predicting E D , the set of input factors were specified as features No.1 to No.9, feature No.11 ( f ybl ) and feature No.12 (u 0 ). Once again, it was observed that introducing confinement properties of boundary elements did not contribute to prediction quality. This may be attributed to the fact that boundary element confinement is mostly influential on energy dissipation properties when larger deformations occur (Oh et al. 2002;Song et al. 2019).
Statistical properties of the three dependent variables (f 0 , E D and eq ) can be found in Table 2 and Fig. 6. In terms of code compliancy; the database consists of specimens collected from experiments carried out around the world between years 1982 to 2019. Although various construction codes may prescribe different conditions for seismic load resisting shear walls, database is examined according to ACI 318-11 (2011) to determine the conformity degree of the specimens to modern standards. The study showed that only 10 out of 161 walls fully comply with the requirements of ACI 318-11 considering provisions on (1) structural concrete strength (Sect. 1.1); (2) minimum lateral and longitudinal rebar ratio and their maximum distances (Sect. 11.9.9); (3) boundary elements of special structural walls (Sect. 21.9.6); and (4) ties of compression members (Sect. 7.10.5). For instance, although all specimens have longitudinal rebar concentration at their extreme ends, 62 out of 161 specimens lack any lateral reinforcement/stirrups for their boundary elements and from the remaining 99 specimens, 11 fail to satisfy 3 or 4 of the aforementioned requirements all together. Therefore, it can be concluded that the majority of walls in the database can be categorized as sub-standard with regard to current regulations.

The proposed ML approach
Theoretical principles of the employed ML technique (GPR) has been introduced in Appendix I. This section deals with introducing the proposed algorithm used for predicting equivalent damping ratio, eq values. The success rate of the algorithm has been evaluated considering the correlation between the predicted and observed values. The coefficient of determination (R 2 ), as defined by Eq. 2, has been selected as the statistic indicator for As it is obvious from Eq. 1, equivalent damping ratio is defined as a function of two dependent variables ( E D and f 0 ). Considering that eq values only at displacements corresponding to 1.0% lateral drift ratio are of interest u 0 can be assumed as a known parameter. To identify the two intermediate parameters ( E D , f 0 ) and estimating eq subsequently, an algorithm consisting of three stages was developed (Fig. 7): The developed algorithm for predicting equivalent damping ratio (ξ eq ) (1) In the first stage, for n times, data gets shuffled and divided into training and test sets. Two GPR models are trained for each set and f 0 and E D values are predicted. (2) In the second stage, the two most successful models in predicting f 0 and E D values are selected using the results of the first stage analyses. Subsequently, f 0 and E D values in the n test sets are predicted using the two selected models. It was observed through trial and error that the greatest correlation between predicted and observed eq values are obtained if results from the model with the maximum R 2 for f 0 and the model with the minimum RELRMSE for E D are used as the two most successful models. (3) As the third stage, the predicted dependent variables (f 0 and E D ) are employed for calculating eq values using Eq. 1. Predicted eq values by the described process will be compared with test eq data to control the quality of prediction and for model validation.
The analyses were conducted 10 times and each attempt included 1000 repeats (n = 1000) to ascertain model efficiency. In the following sections, regression analysis results for the models predicting f 0 , E D and eq and the role of data transformation in enhancing results will be discussed.

Data transformation
Data pre-processing has been proved to be an efficient technique for improving the performance of machine learning algorithms by transforming data to a more informative format (Jiang et al. 2008). Log transformation of independent and dependent variables is the data pre-processing method used in this study. Figure 8 demonstrates how log-transformation converts the distribution of f 0 and E D variables to better fit the normal distribution. This could be observed also in Fig. 9 where normal probability plots of the two output parameters before and after log transformation are presented with uniform order statistics medians approximated according to the approach by Filliben (1975) where m i values are normal order statistic medians and n is the number of the transformed data. It is observed that the transformed data points in Fig. 9b and d show a closer distribution to the theoretical normal distribution compared to the original data ( Fig. 9a and c).

Model validation AND regression analysis results
In this section, regression analysis results in the testing set using the GPR method will be investigated. In each repeat, the database was randomly split into training and testing sets with 137 specimens for the training set (85% of the database) and 24 specimens for the testing set (15% of the database). Table 3 demonstrates a summary of outputs from the two GPR models predicting the two dependent parameters (f 0 and E D ) and results obtained from the developed algorithm for estimating eq . From the minimum R 2 values in Table 3, it is clear that in some cases the developed models fail to establish a strong correlation between the predicted and observed data. To study this issue further, histograms of R 2 for f 0 , E D and eq in the 1 st attempt (Table 3) where poor predictions are obtained for f 0 and E D have been presented in Fig. 10. The three histograms demonstrate that such poor estimations could be recognized as exceptions. Finally, Fig. 11 shows analysis results in the testing set for one of the models developed for predicting eq which is the model with the highest coefficient of determination (R 2 = 0.991) among models built in the analysis set of Table 3 (RELRMSE of this model is 0.078). As it is observed from the summary statistics in Fig. 11(b), the mean predicted to observed eq ratios by the model is 1.078.
The developed algorithm was tested using other machine learning methods to examine the efficiency of the GPR technique. These methods include Robust Linear Regression (RLR), Support Vector Regression (SVR), K-Nearest Neighbor Regression (KNNR) and Kernel Ridge Regression (KRR). As summarized in Table 4, the best results among alternative techniques were obtained by the KRR method with a mean R 2 value of 0.823 and a mean REL-RMSE value of 0.193. This shows GPR method yields the results with the highest accuracy for the given problem.

Conclusions
Reinforced concrete shear walls are lateral load resisting members commonly used in seismic-prone areas. Due to the increasing implementations of energy-based design methods and the broad utilization of RC shear walls, having a good understanding of their behavior in displacements beyond the elastic limit is of great importance. This paper aims to investigate the equivalent damping ratio ( eq ) of RC shear walls at displacements corresponding to 1.0% drift ratio by using machine learning methods. Results of the research might be particularly useful for simplifying complicated structural systems and avoiding sophisticated analyses.
A data set of 161 rectangular cross-sectioned reinforced concrete shear walls subjected to quasi-static reversed cyclic loading was assembled for the purposes of the present research. Equivalent damping ratio was estimated for each specimen using parameters derived from experimental load-displacement curves. The following conclusions were obtained as a result: • An algorithm was developed based on selecting the two most successful models, one predicting lateral force corresponding to the maximum lateral displacement, f 0 and the other one predicting cyclic dissipated energy, E D from a certain number of repeats in the training sets. The implemented ML technique used for predicting the two dependent variables is Gaussian process regression (GPR). The two selected models were used subsequently for predicting the dependent parameters (E D and f 0 ) in testing sets and finally, equivalent damping ratio eq values were calculated from predicted f 0 and E D values. Validation of the results was performed by comparing model outcomes with eq data in the testing set. • The GPR based ML method for predicting the equivalent damping ratio, eq of RC shear walls at their presumed displacements yielded good results considering statistical indicators of R 2 , RELRMSE and MAPE with mean R 2 being 0.89, mean REL-   Fig. 11 a Results of the model predicting ξ eq in the testing set with R 2 = 0.991; b Distribution of the predicted to observed values and the corresponding statistical information nearly homogeneous distribution of specimens. It is also estimated that, at 1.0% drift ratio, 62% of walls had experienced heavy damage while about 20% and 18% of specimens could be classified as respectively moderately and majorly damaged according to ATC-13. Considering that various failure mechanisms tend to show different damage progression patterns, it is concluded that the developed algorithm works well for specimens of different failure modes and varying scales of damage. • Equivalent damping ratio values at displacements corresponding to 1.0% lateral drift ratio for the 161 database specimens range from 3.02% to 26.6% and have a mean value of 11.1%. These figures have been derived for rectangular prototypes with concentrated rebars at extreme ends and heights equal to or less than 3.25 m which represent walls with extensive use in residential/office buildings around the world. • It should be noted that for different drift levels e.g. 2% or larger, different features will be influential on the damping capacity. Nevertheless, similarly high accuracy is expected in different drift levels by ML methods proposed here.
Reinforced concrete shear walls are effective in reducing structural lateral displacement, while resisting high seismic forces and dissipating substantial energy. Energy serves as an indicator of structural damage associated with plastic deformations under cyclic loading, and is a promising index for future earthquake engineering when the energy-based approach become widely employed. The proposed predictive model developed by Machine Learning enables comparing the damping capacity of shear walls at a commonly accepted deformation level. The outcomes of this study are believed to contribute to the energybased design/performance evaluation procedures in terms of predicting the energy capacity of shear walls.
zero. The covariance function, the so-called kernel, specifies the similarities between two points in a function. Similar to many other applications, kernel has been selected as the squared exponential covariance function in this study: where 2 f is the maximum value that the covariance function can take and is named the signal variance whereas is the length-scale or the length parameter. As it is clear from Eq. 9, kernel results in greater values when points i and j are close to each other indicating a stronger correlation between f i and f j .
If * represents predictions at testing data points * , the joint distribution of and * function is represented as Eq. 10.
The observed data is generally noisy because of reasons such as incompleteness in measurements. Considering the effect of noise, the training data could be collected as D = { , } where vector represents the noisy training outputs.
It is frequently assumed for noise to have a Gaussian distribution with zero mean and variance of 2 or ( ∼ N 0, 2 ) . The joint distribution of training and test data has a multivariate Gaussian distribution by definition: where is an identity matrix. Function values of the test set, * , could be derived from the posterior distribution of p(f |D) using Bayes' theorem. The conditional joint Gaussian distribution, p( * | , y, * ) , has the mean and covariance matrices as of Eqs. 13 and 14 respectively.
Another aspect is optimizing hyperparameters of the kernel function ( , f and ) which has been performed by maximizing a log-posterior probability function, called the log marginal likelihood. To be more exact, by specifying the covariance matrix of noisy observations as y = K( , ) + 2 , hyper-parameter values ( = ( , 2 f , 2 ) ) are determined by maximizing logp( | , ):  The signal variance Author contribution The authors have no relevant financial or non-financial interests to disclose. All authors contributed to the study conception and design. All authors read and approved the final manuscript.

Funding
The study has been supported by funds from the Scientific and Technological Research Council of Turkey (TUBITAK) under Project No: 218M535 and ITU-BAP Project No: MGA-2022-43379. Opinions, findings, and conclusions in this paper are those of the authors and do not necessarily represent those of the funding agencies.

Data availability
The entire database used in this study and the MATLAB code for the machine learning model are provided in GitHub for researchers and engineers in the field (https:// github. com/ Siama kTY/ MLfor-Equiv alent-Dampi ng-Ratio. git).