Multivariate joint probability distributions for seismic design parameters across multiple building codes

In order to contribute to the lack of statistical anomaly detection and sensitivity visualization of the response parameters from international seismic codes, this study presents a comparative analysis between multiple Chilean codes for the seismic design of reinforced concrete buildings. The seismic response and its similar implications to the US codes are studied from a model of a reinforced concrete building in the city of Viña del Mar, Chile. The first study case is the NCh433 Of.96 Mod.2009, the second case is the NCh433 Of.96 Mod.2009 + D.S. 117 and 118, and the third case is the NCh433 Of.96 Mod.2009 + D.S. 60 and 61. The results show that the seismic response does not present great differences, but they show noticeable dependencies between multiple parameters within the three study cases. The third case presents the most oversized values of the response parameters, which require a design against a higher base shear. An analysis of the bio-seismic profile is included, which is a procedure commonly used for buildings in Chile, and its indexes are analyzed with statistical models that provide an easy overview of the data structure. Additionally, a pier wall with an opening, and a pier with a T-shaped wall are selected, which present an increase in the amount of reinforcing steel from the first code case to the third, and then to the second code case.


Introduction
The rapid expansion of large cities has led to urban development, which presents significant challenges in architectural and structural design with increasingly complex calculations and modeling procedures.However, the optimization of design outcomes may differ based on regional site conditions, code requirements, and design preferences.To account for these differences, it is necessary to modify the mathematical formulation and structural analysis to adapt to specific local conditions in accordance with the code of practice and relevant design codes in a specific region (Gan et al. 2019).Designing Extended author information available on the last page of the article and constructing structures with the ability to withstand natural disasters in countries prone to seismic activity, represents a significant obstacle in the field of engineering.Seismic countries, such as Chile, present an excellent opportunity to investigate the frequency and attributes of sizable earthquakes, with the acquired knowledge being applicable to other seismic-prone areas worldwide.The pronounced seismic activity in the region can be attributed to the subduction of the Nazca plate beneath the South American plate in the area (Salazar et al. 2022).Since 1900, 13 events with M w ≥ 8.0 have affected the country.
The establishment of a national seismic code to direct optimal practices for building design in Chile resulted from significant national and international seismic events that brought about numerous changes in the country's engineering field and worldwide.Notable examples include the 1906 San Francisco earthquake, the 1906 Valparaiso earthquake, and the 1923 Tokyo earthquake.The first version of the national seismic code for building design dates back to 1972, known as NCh433 Of.72, which adopted the philosophy from the German code DIN 1045 and incorporated the experience of the 1960 Valdivia earthquake ( M w = 9.5 ), the most powerful earthquake ever recorded.The Chilean seismic code NCh433 had major changes between 1993 and 1996, where lessons learned after the 1985 Algarrobo earthquake ( M w = 8.0 ) were incorporated.Service-level design lateral forces and analysis procedures for buildings were defined (NCh433 Of.96).Currently, Chile has several earthquake-related loading and design codes, differentiated by their functionality or structural system.The loading codes are: NCh2369 for industrial facilities (INN 2003a), NCh2745 for base isolated buildings (INN 2003b), and NCh433 for residential and office buildings (INN 1996).
The seismic analysis procedures established in NCh433 Of.96, for Modal Response Spectra Analysis, are essentially the same as in UBC-94 (ICBO 1994), except that forces from the code represent allowable stress level and must be amplified for 1.4 for ultimate load level.Additionally, design requirements for reinforced concrete (RC) buildings have historically followed ACI 318-95 (ACI 1995) with few exceptions.The most notable exception is the exclusion of the requirement for transverse reinforcement in boundary elements in walls.In addition, it is similar to ASCE 7-10 (ASCE 2010), whose design spectra are defined based on proximity to the fault zone, soil conditions, and structural system behavior.A new version of the seismic design code was presented in 2009(NCh433 Of.96 Modified in 2009(INN 2009)).Nonetheless, it exhibited a markedly limited duration of practical applicability due to the Maule earthquake ( M w = 8.8 ), the sixth strongest earth- quake instrumentally recorded in history (Vigny et al. 2011).In light of the compressed timeframe for response after the seismic event, two Supreme Decrees were presented: DS 117 and DS 118 (INN 2010), which complemented NCh433 Of.96 Mod.2009.The main novelty in these Supreme Decrees was that a more detailed methodology was incorporated to obtain the input parameters related to the soil.As detailed in the work by Vicencio and Alexander (2018), slight variations in the input parameters related to the interaction of the soil and the structure, can present significant differences in the response.
The 2010 Maule earthquake presented a new scenario for the analysis and review of the Chilean seismic codes.Statistics regarding the observed damage due to this earthquake reported by the Real State Committee of Chile (CChC 2012) indicate that only four collapses (between 4 and 18 stories) occurred, and nearly 40 buildings were severely damaged and had to be demolished, which represents less than 1% of the total number of new residential buildings built between the 1985 and 2010 earthquakes.This can be considered a successful resilient performance from a statistical point of view and for the interpretation of the seismic design code in terms of its effectiveness.Nevertheless, there was a interruption in construction activity lasting a minimum of 2 years, during which real estate firms refrained from undertaking new projects owing to the prevailing uncertainty (CChC 2012).
At the end of November 2011, DS 60 and DS 61 were published (INN 2011), which replaced the aforementioned DS 117 and DS 118, imposing new provisions in seismic design.As detailed by Massone (2013), one of the first modifications involves that the compressive concrete strain cannot exceed 0.008, indirectly limiting the axial load, which is one of the potential precursors of the damage.Restraining buckling is also provided by requiring transversal reinforcement with a maximum spacing of 6d b , where d b is the longitudinal rebar diameter.Other limitations were also imposed to improve the concrete placement and effectiveness of confinement, as well as the prevention of global buckling.Further details can be found in Lagos et al. (2012Lagos et al. ( , 2017)), and Lagos et al. (2020), where the authors elucidate several key insights derived from the design of a large number of buildings that have exhibited effective seismic performance during earthquakes of magnitude 8.0 or greater.
In general, the most typical structural systems used in mid-to high-rise residential buildings in Chile to withstand the lateral forces produced by seismic events are RC shear wall systems.RC shear walls provide the required stiffness and strength to resist the loads produced by strong ground motions.These structural elements prevent lateral sway improving stability in that plane accompanied by small lateral drifts.However, the NCh433 code in its multiple versions allows the use of a linear analysis model that does not match some terms of the same code, such as achieving structures without damage to seismic events of moderate intensity, limiting damage to non-structural elements with seismic events of medium intensity, or avoiding collapse during seismic events of exceptionally severe intensity even if they present damage.The formal study of the seismic behavior of a structure would also require the use of nonlinear theoretical models with high computational costs, which in some consulting firms is not profitable.From this, design procedures are generated, such as the bio-seismic profile methodology developed by Guendelman et al. (1997), which generates a seismic qualification of Chilean RC buildings through the evaluation of 13 indexes.The indexes present in the bio-seismic profile are grouped according to stiffness indexes, coupling indexes, and indexes of structural redundancy and ductility requirement.Henoch (2007) determined the bio-seismic qualification for 4 high-rise buildings in the world (buildings with more than 50 floors or higher than 200 ms).It is concluded that the H/T parameter achieves to successfully qualify the buildings.It is also observed that the coupling indexes present good performance, with the exception of the Jin Mao Tower in Shanghai due to it presents torsion in the first mode.Therefore, it implies that it does not achieve the minimum value for some indexes.Teranishi and García (2017) determined the bio-seismic qualification for two RC building located in Mexico City.The results show that the first building presents certain level of translational-torsion coupling, however, the building is a torsionally rigid structure.In contrast, the second building presents a different behavior.Although this structure is torsionally flexible structure in terms of bio-seismic indexes, the symmetry it has favors a good seismic behavior.This does not necessarily imply that torsion is less important in the structural response.It is concluded that the two buildings were designed and built before the onset of the current regulations.Their structural conditions are satisfactory indicating that they have experienced a proper seismic behavior before intense earthquake as the one in Mexico City in 1985 ( M w = 8.1 ), which is consistent with the result of the application of the methodology proposed by the bioseismic profile.Additionally, Music and Soto (2021) determined the bio-seismic qualification for eight RC buildings located in Antofagasta, Chile.Among the conclusions, six out of eight buildings presented the density of walls in compression index out of the recommended range by the bio-seismic profile.This implies that the walls of the first floor of these buildings might be subjected to high compression stresses.Moreover, the structural response index presented a good relationship with the index for performance-based seismic design.For this case, an elastic overall seismic response is expected for the building supported on bedrock.However, in terms of individual elements, an inelastic behavior may occur.With this wide background generated by literature and experience from a seismic country, there are still questions from multiple practitioners regarding if there is greater or lesser variability in the building response when using a more updated seismic code, or if there is a higher or lower sensitivity in the response parameters if the seismic code requires more specific inputs.In order to address these doubts, this article presents a statistical analysis of the response parameters obtained from an RC building model using various seismic design provisions.The building under study is a Chilean RC structure analyzed according to three versions of the NCh433 code: NCh433 Of.96 Mod.2009 (Case 1), NCh433 Of.96 Mod.2009 + DS 117 andDS 118 (Case 2), andNCh433 Of.96 Mod.2009 + DS 60 andDS 61 (Case 3).The objective of analyzing the response parameters in the first part of this work is to observe the variability between the values obtained through contour plots with multivariate normal distributions, which provide the dependence between two variables.Then, an analysis of the bio-seismic profile methodology is included and the variability between the indexes is analyzed using multivariate kernel density estimations.This type of statistical model allows to visualize if there is dependency between the two directions of the seismic movement with the bio-seismic indexes.Finally, two types of pier walls are selected to be studied (pier wall with an opening, and a pier with a T-shaped wall section).An analysis of its flexural and shear capacity is carried out, together with the impact on the amount of reinforcing steel at each level using Gaussian mixture models.The main objective of this article is to contribute to the lack of methodologies and statistical analysis for the parameters of international seismic codes in the literature, taking as example 3 Chilean codes.This type of analysis can serve as a reference to quantify the differences between multiple codes based on the variability of the response parameters.As previously described, the Chilean code is strongly related to the evolution of the US code, so the sensitivity of the response parameters can also be extrapolated to what has been learned from international experience.

Structural description of the building
Mid-to high-rise buildings in Chile can be classified according to their use in two main categories: residential and office buildings.The former must have partitions for occupant privacy, while the latter requires large open spaces.For this study, the structure is a residential building located in Viña del Mar (Chile), which is a city located in a hazard zone of maximum effective acceleration due to the proximity with the subduction between tectonic plates.The structure incorporates RC shear walls.It has 17 floors above ground level and two floors in the basement, resulting in a total height of 51.4 m from the foundation level.The total area below ground level is 2500 m 2 , and the total area above ground level is 8900 m 2 .It is founded on dense gravel soil that, according to the Chilean codes, is clas- sified as Type II (Case 1 and 2) or Type C (Case 3), which corresponds to an average shear wave velocity for the top 30 m of soil, Vs 30 ≥ 350 m/s.The floor plans of the building have the classical characteristics of typical Chilean residential buildings, i.e., high density of RC walls distributed throughout the plan to separate apartments and rooms, with a variety of cross section shapes, such as I, C, L, T, etc.The thickness of the walls ranges between 20 cm and 30 cm.The slabs are 14 cm thick for typical floors and between 20 cm and 22 cm in the basement.There are two elevator openings of 4.10 m by 2.35 m on each floor and stairs opening of around 3.32 m by 4.10 m for the typical floor and around 1.52 m by 4.10 m in the basement.The height between floors is 2.52 m for the typical floor, 3.23 m in the first basement, and 2.60 m in the second basement.
The structural design of the building is performed using a elastic and lineal model based on the procedures of 3 described versions of the Chilean code NCh433, and to reduce the computational cost of building modeling.According to the design criterion, the building is based on a concrete compressive strength of 25 MPa, between levels 4 and 17 and 35 MPa for the remaining levels.The yield strength of the reinforcing steel used is 420 MPa.As mentioned in the previous section, the NCh433 has similar criterion to the ASCE 7-10 code for the conformation of the acceleration design spectra based on soil conditions, proximity to the fault zone, and type of structural system, among others.As will be detailed in the following sections, the later versions of the NCh433 include new input parameters for the seismic design spectra.

Building model response
Computational models are often used for response prediction and performance assessment of large and complex civil structures.Herein, detailed three-dimensional elastic and linear finite element (FE) models are developed using the software ETABS v9.7.4 (Fig. 1).It should be mentioned that a rigid diaphragm is considered, without considering cracking.With this model, the input parameters are entered according to the case, and then the response parameters are obtained.Additionally, the inter-story relative displacements from centers of mass and the relative displacements from extreme points are obtained in order to identify important differences in the magnitudes of the displacements among each case of study.

Building response for case 1
This case study is based on the procedure of the NCh433 Of.96 Modified in 2009 code.This code presents similar criterion to the US codes for seismic design, such as proximity to the fault zone, the importance of the structure related to the building category, soil conditions, type of structural system, fundamental periods of the structure, etc. Table 1 presents the inputs for design spectra of Case 1, where A o is the zone maximum effective acceleration, I is the importance factor of the structure, T o , p and s are soil parameters, R o is the structural system parameter, T * x and T * y are the periods of the mode with largest translational mass in the direction of analysis, and R * x and R * y are the reduction factors.The previous input parameters lead to the design spectra illustrated in Fig. 2. It should be mentioned that, compared to the US codes, the drifts in the Chilean standard incorporate reduced spectra.The result of the analysis show that the inter-story displacements measured from the centers of mass in Fig. 3a.do not exceed 0.11%, and the inter-story relative displacements in Fig. 4a do not exceed 0.12%.

Building response for case 2
This case study is based on the methodology of the NCh433 Of.96 Modified in 2009 code, including DS 117 and DS 118 caused as a result of the M w = 8.8 earthquake in Chile on February 27, 2010 (Rojas et al. 2010(Rojas et al. , 2011)).The input parameters presented in Table 2 and Table 3 are mainly associated with the characteristics of the soil, and in contrast to Case 1, Case 2 presents a greater number of input parameters.It can also be seen in Fig. 2 that the design spectra curve presents a larger design space for periods and accelerations than the curve of the previous code in Fig. 2. Given the input parameters presented, the result of the analysis show that the inter-story displacements in Fig. 3b.do not exceed 0.12%, and the interstory relative displacements in Fig. 4b do not exceed 0.14%.

Building response for case 3
This case study is based on the procedure of the NCh433 Of.96 Modified in 2009 code, including DS 60 and DS 61 published in 2011, which repeal the aforementioned DS 117 and 118 that redefined the design criterion for seismic events.Given the conditions of the soil where the building is located and under the design criterion of the case study, the parameters are determined for the conformation of the design spectra presented in Fig. 2 with the data grouped in Table 4.In contrast to the previous codes, the design spectra from Fig. 2 presents a curve similar to Case 1; however, it presents a higher peak acceleration.Additionally, the input parameters presented in Table 4 are similar to Case 1.Given the input parameters presented, the result of the analysis show that the inter-story displacements measured from the centers of mass in Fig. 3c.do not exceed 0.12%, and the inter-story relative displacements in Fig. 4c do not exceed 0.15%.

Multivariate normal distributions for design parameters
The building designs obtained from input parameters detailed in the previous sections exhibit similar fundamental periods among the three cases.Consequently, the stiffness of the buildings do not vary significantly.The design parameters obtained are summarized in Table 5, which include the elastic base shear (Q), the minimum base shear ( Q min ), the effective shear ( Q ef ), the effective force modification factor ( R ef ), and the seismic weight (W), defined as the sum of total dead load and a percentage of the total live load (25% for each case).The variation of the seismic weight in Case 2 presented in Table 5 is due to the variation of the minimum thicknesses for the RC walls that is established in the respective code.This leads to the fact that there are some walls in Case 2 that are larger than in the other cases, which implies an increase in the total weight.In addition, the shear force design corresponds to around 7% of the seismic weight in all cases.
According to Westenenk et al. (2013), there is some positive correlation between damage and building orientation, soil quality, plan and vertical building irregularities, slenderness and plan aspect ratio, high levels of compression stresses, lack of boundary element confinement, and small wall thicknesses.Given the lack of statistical analysis available with respect to some of the parameters in each model class under study, it seems desirable to explore the parameter variability and analyze the influence that each of the governing parameters has from the model responses.Thus, the influence of the previously described seismic design codes is investigated.The analytical results provide valuable insight into the mean and covariance of the model parameters, their dependence, and variability in the model parameter estimates arising from the data-set, as well as the uncertainty of the model parameters obtained from each data-set.
With the intention of exploring an appropriate domain of the parameters covering all observed values, the multivariate normal (MVN) distributions, and the coefficient of variation for each probability density function (i.e., the ratio between standard deviation, , and the mean of a given probability density function, ), are selected so that the analysis is wider than 3 study cases with lateral forces in 2 directions for each case.The joint probability density function (PDF) can be utilized in different probabilistic-based analysis and parameter-dependent analyses, such as bayesian modeling, rock mechanics, ground motion intensity measures, among others (Birrell et al. 2021;Contreras et al. 2018;Fayaz et al. 2023;Jia et al. 2022).In this section, MVN distribution fits are generated over different  where X and Y represent the mean of random variable X and Y, respectively, X and Y represent the standard deviation of random variable X and Y, respectively, and represents the linear correlation between random variables X and Y.The bivariate density function takes the shape of a bell-curve, generating an ellipse which is the projection of a three-dimensional solid into a two-dimensional plane.The ellipse contours indicate the strength and direction of the linear correlation.If the linear correlation is closer to zero, (1)  then the contours will appear circular (Paolino 2020).From Eq. 1, the elliptical width in the x-dimension and y-dimension is given by X and Y , respectively.The results presented in this section are analyzed by a bivariate normal distribution, which is a particular case of an MVN distribution for two design parameters, including pairwise correlations.Figure 5 depicts a matrix designed to illustrate the interdependence among the parameters, and serves as a visual representation of the obtained results.The sub-figures in the diagonal show the marginal PDF of each parameter, While the remaining figures display contour plots for each pair of parameters.The multiple contour plots present the correlation of dependency between two variables through the width of the ellipse.For example, a low correlation between the seismic weight W and to the rest of the parameters can be observed in Fig. 5, since this parameter depends on the fundamental periods of the building and in their design methodology for each case, which implies an important variation in the shear responses.In contrast, it can be observed that the scale factor R ef presents a high correla- tion to the elastic base shear Q, and a similar correlation between Q ef and Q min , where in some cases, the effective base shear is equal to the minimum base shear.It is emphasized that this analysis only has visual validity, since the number of samples is very low for 3 study cases, which might mean that the standard errors can be very large (Table 6), and also the samples are not independent, since the response parameters come from one building sample with different spectra design.
Regarding the uncertainty of the model parameters presented in Table 6, the variability of scale factors and the base shear present an important difference with the variability of the seismic weight, minimum shear, and the effective shear, respectively.Table 6 presents that the model response is more sensitive to the seismic weight, minimum shear, and effective shear ( c.o.v.< 3 %) than the scale factor and base shear ( c.o.v.≈ 20%).The uncertainty in such estimates is inversely proportional to the square root of the number of data-sets, and thus is expected to be higher for a smaller number of data-sets used.This is reasonable since the model response is more sensitive to the seismic weight, the minimum shear, and the effective shear than the scale factor and the base shear, where the values of the three most sensitive parameters are repeated in more than one case.Both the values of the model response, as well as the visualization of the contour plots, show that the use of various versions of the NCh433 seismic code turns out to have a low variation in the response of the studied building.

Bio-seismic profile of buildings
The bio-seismic profile is a methodology that entails the computation of multiple indexes in order to generate a seismic evaluation of reinforced concrete structures.These indexes are denominated and defined as follows.
• 1 is the ratio between total building height above ground level (H), and the first lateral period (T).• 2 is the ratio between the moment given the P − Δ effect ( M P−Δ ), and the basal moment in the analysis direction ( M P−Δ ).• 3 is the ratio between the top level displacement ( ), and the total building height (H) multiplied by 1000.• 4 is the ratio between the maximum inter-story displacement measured at the centers of mass ( cg ), and the total building height (H) multiplied by 1000.
• 5 is the ratio between the maximum inter-story displacement at extreme points ( ep ), and the total height of the building (H) multiplied by 1000.• 6 is the ratio between rotational period ( T r ), and lateral (translational) period ( T t ).
• 7 is the ratio between coupled rotational equivalent mass ( M cr ), and direct lateral equivalent mass ( M dt ).• 8 is the ratio between the base torsional moment over the base shear force to the base ( M t ∕Q b ), and the radius of rotation ( r b ).• 9 is the ratio between coupled lateral equivalent mass ( M cr ), and direct lateral equivalent mass ( M dt ).• 10 is the ratio between coupled basal shear ( Q cb ), and direct basal shear ( Q db ).
• 11 is the ratio between basal coupled overturning moment ( M co ), and direct overturning moment in the analysis direction ( M do ).• 12 is the number of relevant elements in seismic resistance ( N r ).

Q ef (tonf)
Fig. 5 Contour plots using multivariate normal distribution • 13 is the effective spectral reduction factor R * * calculated as the ratio between the spec- tral acceleration reduction factor R * , and 1.4f max f min , where f min is the minimum basal shear amplification factor, and f max is the maximum basal shear amplification factor.
The stiffness indexes are 1 to 5 ; the coupling indexes are 6 to 11 ; and the indexes of structural redundancy and ductility requirement are 12 and 13 , respectively.Recommended ranges for each index were suggested by Guendelman et al. (1997) based on 585 buildings in Chile, and the qualification in colors (similar to the bio-chemical profile) was presented by Guendelman et al. (2010).The foundation of the bio-seismic profile is rooted in the need to provide a seismic assessment for buildings that can satisfy the following requirements: (1) compliance with the NCh433 code (in its various versions) using a linear analysis model that may not guarantee adherence to some of the code's provisions, such as withstanding earthquakes of moderate intensity without sustaining damage, limiting damage to non-structural elements during earthquakes of moderate intensity, or preventing collapse during exceptionally severe earthquakes, even in the presence of damage; (2) the formal investigation of a seismic behavior of the structure would need the application of nonlinear theoretical models; and (3) accumulated experience and normative analysis can be leveraged to identify the factors that have contributed to satisfactory performance of Chilean buildings in past earthquakes.This procedure facilitates the identification of structural deficiencies and provides insights into potential corrections and analytical considerations for the design of the analyzed structure.This methodology is presented as a tool for comprehending the structural behavior and serves as a complement to the structural calculation of a building.The computed index values offer information on the anticipated structural response of the building during a significant seismic event, enabling a seismic qualification to be established.

Bio-seismic profile results
Figure 6 summarized the results obtained using the bio-seismic profile methodology.Stable ranges are in the green box, slightly stable ranges are in the yellow box, and out-ofrange values are in the red box.X and Y denote the seismic direction.The results present that the index numbers with stable range is greater when applying the provisions of Case 1, then slightly by Case 3 and later by Case 2. The stiffness indexes present acceptable values and in accordance with the seismic strains required in the Chilean codes regarding relative displacements at certain points, which have not undergone variation since the provisions of Case 1.According to a previously described, buildings are classified as flexible if H∕T < 40 m/s, and structures with normal stiffness if 40 < H∕T < 70 m/s.These indexes are important because the assumed stiffness of the structural elements show a significant influence on the prediction of the seismic axial load requirements on the walls of a building.This also agrees with the previous section, where it is concluded that having similar fundamental periods, stiffness of the building do not vary much.In addition, the interstory displacements measured from the centers of mass do not exceed 0.12% on average for the three study cases, being located within the permissible ranges of deformation according to the Case 1.

Multivariate kernel density estimations for bio-seismic indexes
Multivariate kernel density estimations (KDE) are generated over different sets of measurements to account for the variability in seismic indexes across datasets.In statistics, KDE is the use of a non-parametric method to estimate the PDF of a random variable based on smoothing coefficients as weights called kernels.The most frequently used bivariate kernel function is symmetric: where, x i , y i are random sample vectors with a density function f, with i = 1, 2, ..., n , b x and b y are smoothing coefficients called bandwidths, and K is the kernel function.Many types of kernel functions can be found in the literature (Wu 2017;Węglarczyk 2018), which determines how the influence of each observation is distributed.Therefore, the kernel can have a notable impact on the estimation of the resulting density function.In this article, the estimate is based on a symmetrical and normal kernel function described in Eq. 3. It is considered that the histogram biases the data of their individual location replacing their locations with a bin location.This causes the histogram shape to become discontinuous and flat in each interval.KDE has not these drawbacks, due to it produces a smooth empirical PDF based on individual locations of all sample data. (2) Fig. 6 Bio-seismic profile results The aim of KDE is to make inferences about the underlying PDF everywhere, even in areas where no data have been recorded, based on a finite sample.With KDE, the contribution of each data point is spread out and smoothed into a region of space surrounding it.Combining the smoothed contributions generates a summary of the composition of the data and its corresponding PDF. Figure 7 displays the contour plots for the KDE of pairs of seismic indexes, separated by the seismic directions S x and S y , show that there is no distinct indexes pairs by direction for 1 , 3 , 6 , 7 , 8 , 9 , and 12 .The circular plots and the scatter of both directions explain why the correlations are close to zero.This also coincides with those indexes that depend in greater percentage on the inertia in their respective direction of displacement.The results also show that there are more related seismic indexes between both directions than completely separate seismic indexes, which follows that the displacement in the X-axis is not an independent variable of displacement in the Y-axis.Based on the intermediate results presented in Figs. 5 and 7, it appears plausible to hypothesize that KDE can discriminate observations sampled from PDFs other than the MVN distribution.Specifically, if feature space regions that contain normal data have a relatively high density, it logically follows that other regions would have a relatively low density, as the integral of the PDF must be normalized to one.

Finite element analysis of pier walls
An analytical study of two critical pier walls of the building indicated in Fig. 8a and b are described in this section.The study aims to compare the roof displacement capacity and seismic behavior in these pier walls in order to check if the design presents important variations between each case.Based on the simplified compatibility model, according to the ACI-318 (ACI 2008), and illustrated in Fig. 9, the capacities of the maximum axial load acting on the cross sections are evaluated.Then, the curvature requirements are verified, establishing the parameters set forth in the results (Table 8).The compatibility models presented in Fig. 9 assume a cantilever wall, in which the wall length is l w , and lateral loads either distributed or concentrated over the wall height h w .The traditional curvature distri- bution model, presented in Fig. 9a, assumes an elastic component until the yield curvature y is achieved, and an inelastic component accounts for the additional curvature once the ultimate curvature u has been achieved.Although the inelastic curvature tends to increase linearly, simplified models such as the one presented in Fig. 9.b have described the inelastic component as an equivalent rectangular distribution of curvature of length l p named plastic hinge length.Thus, the wall top lateral displacement u can be determined using Eq. 4, if a triangular distribution of the forces is applied along h w , or the simplified model adopted by the ACI 318-08 code in Eq. 5, which only one component is considered (Wallace and Orakcal 2002).
To obtain the displacement capacity at top of the pier wall (roof displacement), the strength values from the model are first obtained, and then the axial force-moment relationships are estimated using a model of the pier wall cross section.The roof displacement capacities of the pier walls are analytically estimated using the same ACI 318-08 procedure through the Eq.( 6).
where c lim is the greatest depth of the neutral axis calculated for the factored axial force and nominal moment resistance consistent with the design displacement u .The wall aspect ratio is defined as , where M is the bending moment at the base of the wall, and V is the shear force.The conventional ACI 318-08 method for predicting the roof displacement capacity is insufficient due to the 3D interaction between the wall and other parts of the structure, which is not adequately accounted for Alarcon et al. (2015).The choice of pier walls is based on the fact that these elements present the highest values of shear stresses, which implies that they are related to greater tributary areas, and that they should not present torsional effects either.This criteria is important because lateral forces are more relevant in seismic countries.Secondly, it should be emphasized that the continuity of the structural elements is essential.Walls with setback discontinuities and walls with openings are common in Chile (Massone et al. 2019(Massone et al. , 2017)).The most critical direction in this building is that with the highest pier wall density due to the discontinuity in the floor diaphragm, which coincides with what is obtained in the Tables 5 and 6, and with the pier walls densities presented in Fig. 8a.With this, the pier walls with the greatest design difficulties are selected.The selected structural sections are a pier wall elevation with an opening named Pier 1, and a pier elevation with a T-shaped wall named Pier 2. Regarding the T-shaped cross section, different authors and design codes have proposed expressions for the determination of the effective width b eff as a function of the wall height, flange thicknesses, and other parameters (Rojas et al. 2021).One of the most used estimates is the one obtained by Wallace (1996), who through experimental tests, suggests the estimation presented in Eq. 7.
(6) c lim = l w 600 u ∕h w where s c is the distance between adjacent walls.This simple expression is the one implemented for the analysis of Pier 2. Although all model results presented herein consistently show that the selected pier wall behaviors are critical, it is not possible to completely discard that the strain concentrations on the pier walls could be caused by a completely different mechanism with lighter axial loads and cyclic behaviors as the results presented by Jünemann et al. (2016).In addition, the importance of having walls with openings and T-shaped walls will depend on the stiffness matrix and the global seismic behavior of the structure.It should be mentioned that floors with rectangular elements and without major singularities are used in this project, for which the results are referential and cannot be extrapolated in terms of structural response.
The structural components under consideration are the pier walls created using SAP2000 v.14, as illustrated in Figs.8a and b.It is defined as two piers, in which the left length (Pier 1) is 7.50 m with 1.50 m 2 and a thickness of 0.20 m.The right length (Pier 2) is 8.20 m with 1.64 m 2 and a thickness of 0.25 m.Both pier walls have heights of 48.72 m.These selections are made on the element that take a large contribution with respect to the basal shear of the building.For Case 1 is 17.12%, for Case 2 is 25.08%, and for Case 3 is 21.74%.The interaction diagrams are made between various pairs of axial and moment stresses, according to the geometry and construction materials of the element under study before the interaction of the required loads.The geometry of the requested sections are considered in its entirety, and the b eff criterion described in Eq. 7 for the Pier 2 is considered.Additionally, the criterion of axial force-moment interaction diagram and confinement in the sections are evaluated in the design by strains.
In Case 1, the critical section of the elements are defined on the first floor to establish two parameters of this evaluation.Then, the need for confinement of the sections is verified.Finally, the results are shown in Fig. 10a.Table 8 shows that the aspect ratios are distant, M∕Vl w ≤ 1.0 in Pier 1, and M∕Vl w ≥ 1.0 in Pier 2 for all the cases, which suggests that the behavior of the pier walls should be controlled by shear on the Pier 1, and flexure on the Pier 2. From the results presented Tables 7, 8, and the interaction diagram illustrated in Fig. 10a, it can be observed that the pier walls in Case 1 satisfy the general design requirements since amplified loads are smaller than the reduced capacity.According to the ACI 318-08 code regarding the necessary condition to establish confinement, this is not met in the section since the magnitudes of the distance to the neutral axis c does not exceed the established limit magnitude c lim , so it is not considered confinement in the section.In contrast, the strain demand satisfies the general design requirement.Finally, the total amount of reinforcing steel in both pier walls is 181770 cm 2 .
For Case 2, it is considered that in any cross section of the walls, the unit elongation of the longitudinal reinforcement must exceed 0.004 when the concrete in the opposite extreme fiber reaches a unit shortening equal to 0.003.Additionally, the maximum axial load acting on the cross sections of the walls must be less than or equal to 0.3 f ′ c A g , where f ′ c corresponds to the compressive strength of concrete, and A g the wall cross section area.The foregoing is considered a general philosophy, however, the provisions of the reinforcement and the yield stress of the reinforcing steel used, among other factors, must be considered, which can lead to different values of 0.3 f ′ c A g .Tables 7, 8, and Fig. 10b show that the pier walls satisfy the gen- eral design requirements.According to the ACI 318-08 code and D.S. 118, the sections do not require confinement.Regarding the bending capacity, this complies with the codes currently (7) b eff = min 0.25h w , s c 2 considered, the latter not being an explicit requirement of D.S. 118.Finally, the total amount of reinforcing steel is 245778 cm 2 .For Case 3, the maximum axial load acting on the transversal section of the wall must be less than or equal to 0.35 f ′ c A g .Tables 7, 8, and the interaction diagram of the selected pier walls presented in Fig. 10c shows that the walls satisfies the general design requirements.According to the ACI 318-08 code and D.S. 60, the pier wall sections do not require confinement.Regarding the curvature requirements, it is fulfilled satisfactorily given what is stated in

Gaussian mixture models for amount of reinforcement
Gaussian mixture model (GMM) is a probabilistic model in which the observations are considered to follow a probabilistic distribution formed by the combination of multiple normal distributions (k Gaussians distributions) with independent means, variances, and weights.
Once the probability density of each observation has been calculated, it can be used as a criterion to identify anomalies and value trends.Observations are anomalous when they have a very low predicted probability.The GMM for an empirical distribution of feature values can be formally defined as: where i and Σ i , are the mean and covariance matrix of the ith Gaussian, respectively, and w i is the probability that x belongs to the ith Gaussian, or the Gaussian weight.Except for the number of Gaussians, the model parameters are defined iteratively in a data-driven manner.It estimates 3 parameters (mean, covariance and Gaussian weight) using an expectation-maximization (EM) algorithm that searches for the maximum a posteriori probability (Souza et al. 2019).In this article, a probabilistic fit based on the calibration of the Ground Motion Model (GMM) using the reinforcing steel quantities in the finite element pier walls has been employed to identify the steel concentration and design value anomalies (independent of the story).It is possible to visually identify from histograms in Fig. 11 three data concentrations.Therefore, for the GMMs used in this analysis, we consider k=3 Gaussian components.Note how the mixture of Gaussians enables the different reinforcement distributions to be described and how the lowest-weighted distributions occupy a different space compared with the higher-weighted ones for rebar contribution A s in Fig. 11.The results presented in Fig. 11a, b, and c illustrates the mixed distribution for each case with similarities in its means, standard deviations, weights, and normalized histograms.As previously detailed, Case 2 presents the largest amount of reinforcing steel with 24.58 m 2 , while Case 1 presents the least amount with 18.18 m 2 .Regarding the analysis of each graph, the highest concentration of steel reinforcement on average presents μ = 0.65 m 2 , σ = 0.03 , and w = 0.57 , which coincides with what is obtained on stories 6 to 17.In contrast, the lowest concentration of reinforcing steel bars on average presents μ = 2.95 m 2 , σ = 0.22 and w = 0.11 , which coincides with what is obtained on the 1st floor and on the first basement floor.This last value coincides with the greatest stresses and the critical zone for determining the curvature capacity of the analyzed pier walls, which means a higher requirement for steel reinforcement.These concentrations, on average, present similar values to the results obtained in Fig. 11d when considering all the case studies.Additionally, it is observed in Fig. 11d that the relative frequency is lower than those obtained in Fig. 11a, b, and c, since although there are higher frequencies for steel concentrations, the number of samples is also greater.
Although it may not hold that individual steel contribution will follow a normal distribution, using a mixture of Gaussian distributions should enable approximation of the statistical distributions of the steel amount values.It is therefore hypothesized that the GMM ( 8) is useful for the detection of anomalies and for assessing trend accuracy.The impact of the reinforcing steel in a RC wall during an earthquake is very important because linear and non-linear building models from the literature demonstrate that providing special boundary detailing to the walls considerably improves performance (Massone et al. 2021;Ni and Birely 2018), while border reinforcement detailing experimentally influences fatigue performance (Egger et al. 2021;Massone et al. 2023a, b).It should also be mentioned that, despite not presenting significant differences between design codes, the variations in the amount of steel are still important in terms of costs.A reduction in the amount of steel can be very important economically for a project under construction.

Conclusions
This study presents an analytical comparison of three different versions of Chilean seismic codes for RC buildings, which includes a design procedure known as the bio-seismic profile.Subsequently, two pier wall sections are analyzed.This type of analysis can serve as an example to quantify the variations between different codes through their sensitivity to seismic parameters, which contributes to the deficit of statistical anomaly identification and visualization of parameter sensitivity in the literature.The assessment yields the following conclusions: • The inter-story displacements measured from the centers of mass do not exceed 0.12%, and the inter-story relative displacements do not exceed 0.15% for the 3 cases, which are within the allowable ranges of displacements according to the different seismic codes without important variations.The fundamental periods are quite similar between the cases, which suggests that the buildings stiffness do not change much.The most updated code (Case 3) presents the most oversized values of the response parameters of the studied building, while the oldest version of the code (Case 1) presents lower values.In practice, this means that by having a higher basal shear, such as the one the obtained from the most updated code, the structural elements must be designed to have a higher seismic capacity.
The MVN density functions plotted as contour plots show that the model response is more sensitive to the seismic weight, minimum shear, and effective shear than the scale factor and base shear.Both the values of the model response, as well as the visualization of the contour plots, show that the use of various versions of the NCh433 seismic code turns out to have a low variation in the response of the studied building.• The results related to the bio-seismic profile show that the number of indexes into the stable range is greater when applying the provisions of Case 1, then followed slightly by Case 3, and finally by Case 2. The contour plots of KDE of seismic indexes separated by the seismic directions show 7 of 13 indexes with a correlation close to zero.This is related to the data scatter and those seismic indexes that depend at a greater rate on the inertia in their respective direction of displacement.The implementation of multivariate KDE for each seismic direction generates inferences about the underlying PDF, including where no data points are observed, providing an easy overview of the data structure and better sampling fit.This is why KDE has an advantage in discriminating observations sampled from PDFs other than the MVN in terms of the intermediate results obtained.• The behavior of the analyzed FE pier walls for each case is controlled by shear on the Pier 1 and flexure on the Pier 2, satisfying the design requirements from the interaction diagram.In relation to the structural design, the highest requirement for steel is presented for Case 2 with a 7.8% and 35.2% increase compared to Case 1 and Case 3, respectively.The concentration of steel is analyzed with a GMM for each case separately and together, which gives a result that the highest trend the of amount of reinforcing steel is 0.7 m 2 on average for most floors, while the lowest trend is 2.9 m 2 on average corresponding to the floors with the highest stress concentrations (1st basement and 1st floor).Although each Gaussian in the mixture is fitted independently, GMMs offer a good solution to capture different amounts of reinforcement steel for the analyzed pier walls, assigning them to Gaussians with different means, standard deviations and weights.Despite not presenting significant differences between design codes, the reduction in the amount of steel can be economically important for a building under construction.

Fig. 1
Fig. 1 3D linear FE model view of the 17-story building

Fig. 2 Fig. 3
Fig. 2 Elastic design spectra for each case study

Fig. 4
Fig. 4 Peak relative displacements from extreme points

Fig. 9
Fig. 9 Compatibility models according to the ACI 318-08 code: a elastic and inelastic component, and b simplified plastic hinge model

Fig. 11
Fig. 11 Normalized histograms and Gaussian mixture models (GMM) of amounts of reinforcing steel in the pier walls

Table 1
Inputs for design spectra of Case 1A o I T o (sec) p s R o T * x (sec) T * y (sec) R *

Table 2
Input parameters according to soil type of Case 2

Table 5
Design parameters of Case 1, Case 2, and Case 3 30 modes are considered for each case to achieve a total mass participation of 99.99% in X direction, and 99.99% in Y direction

Table 6
Summary of mean, variance, and coefficient of variation

Table 7
Moment-curvature parameters from pier walls

Table 8
Ultimate stresses for interaction diagram and maximum axial loads for each case on the 1st floor .S. 60 and the general provisions of the case.Finally, the total amount of reinforcing steel is 227194 cm 2 .
Fig. 10 Interaction diagram for each case study D