Probabilistic Seismic Performance Assessment of a Representative Soft-First-Story Building in an Earthquake-Prone Region

Abstract

The structural performance of mid-rise buildings with a soft first story is a critical issue in earthquake-prone regions. This paper presents a detailed assessment of both the seismic performance and the structural reliability of a confined masonry mid-rise building with a soft reinforced-concrete first-story irregularity located in Mexico. This structure was designed according to outdated building codes to reflect construction practices that remain common in some parts of the country. Nonlinear dynamic analyses were conducted using ETABS v21. To simulate various seismic scenarios, ground motion records associated with return periods of 72, 475, and 975 years, respectively, were implemented. The results demonstrated that maximum inter-story drift is predominantly concentrated at the first story, exceeding the performance thresholds for immediate occupancy, life safety, and collapse prevention. Furthermore, a probabilistic performance assessment was developed considering the randomness of inter-story drift responses. Then, reliability index (β) was calculated for each seismic scenario. In all cases, β values remained consistently below the minimum recommended limit. These findings confirm the formation of a soft-story mechanism at the first level and are relevant for buildings designed under construction provisions like those used in the present case study.

1. Introduction

Structural irregularities are a key consideration in the seismic assessment of buildings, as they can significantly affect the distribution of lateral forces and the dynamic response under earthquake loads. Both in-plane and vertical irregularities may lead to localized stress concentrations and amplified inter-story drifts. Extensive research has shown that conventional lineal analysis methods often fail to fully capture these effects, while nonlinear static and dynamic approaches provide a more accurate representation of structural performance under seismic demands [1,2,3].

Soft- or weak-first-story buildings are a common vertical irregularity in urban areas, typically featuring a ground floor with markedly reduced stiffness and strength due to large openings, while upper stories include confined-masonry walls that significantly increase lateral rigidity. This discontinuity concentrates seismic demands on first-story columns and has repeatedly led to severe damage or collapse during strong earthquakes [4,5,6,7,8,9]. These conditions highlight the need for quantitative assessments of the seismic performance and reliability of mixed/combined of hybrid RC-masonry systems, especially those designed under earlier seismic codes and still widely used. Furthermore, how seismic codes define and address these irregularities is crucial for their proper identification and evaluation, emphasizing the importance of revisiting design provisions for existing structures built under outdated regulations [10,11].

Seismic events have repeatedly caused severe damage or collapse in buildings with soft first stories, highlighting their vulnerability. Notable examples include the 1971 San Fernando earthquake (Mw = 6.6), where Olive View Hospital suffered extensive damage due to weak first stories and irregular configurations [12]; the 1985 Michoacan earthquake in Mexico City, which caused about 10,000 fatalities, with 8% of damaged RC building exhibiting vertical irregularities [4]; the 1986 El Salvador earthquake, affecting nearly 60,000 buildings with first-story failures due to floor stiffness differences [13,14]; the 2010 Maule earthquake in Chile, where diaphragm-type shear walls did not prevent weak-story failures [15]; the 2016 Manabí, Ecuador earthquake, where taller first stories increased flexibility beyond structural capacity, causing severe damage [16]; and the 2017 Mexico earthquake (Mw = 7.1), which resulted in 369 fatalities, over 300 severely damaged buildings, and 46 collapses, with around 50% of affected structures exhibiting weak stories, including some that had previously been retrofitted [9,17]. These cases demonstrate that significant stiffness and strength discontinuities at the first story amplify ductility demands, promote stress concentrations, and reduce structural reliability, underscoring the need for quantitative assessments of seismic performance in such buildings.

In addition to the seismic damage documented in past earthquakes, several retrofit strategies such as base isolation systems with curved surface sliders and lead rubber bearings, as well as buckling restrained based braces, have been shown to effectively reduce soft-story demands and improve the seismic safety of existing structures [18,19]. Although these mitigation measures lie beyond the scope of the present study, acknowledging their relevance reinforces the need for reliable assessment methodologies that can support future retrofit decisions.

Previous investigations documented failures of soft-first-story buildings and led to improvements in nonlinear modeling and reliability-based seismic assessment. Pseudo-dynamic tests on RC buildings with masonry infills assessed their impact on seismic behavior [20]. Comparisons between buildings with uniform infills and those lacking first-floor infills showed that the latter promoted a weak-story mechanism, increasing inter-story drifts and collapse risk [21]. This research analyzed force–deformation curves and proposed a method to estimate the minimum safety factor in order to prevent collapse, while [22] developed an innovative model using frame and link elements to simulate lateral collapse of soft-first-story buildings. This approach allowed for efficient dynamic analysis by reducing the computational demand and avoiding unnecessary growth in the stiffness matrix as link elements increased. Although the results were reasonable, the absence of experimental data limited full validation, highlighting the need for further studies to confirm the model’s accuracy. Additionally, ref. [23] showed that preliminary seismic risk assessments are crucial for identifying soft-first-story buildings, where concentrated seismic demand increases collapse probability, as evidenced by their pushover analyses. Similarly, ref. [24] analyzed soft-first-story buildings affected by the 2017 Mexico City earthquake. Four-, six-, and eight-story structures designed under 1970s and 1980s codes were evaluated using nearby recorded accelerograms, showing that first-story drift and rotation demands matched observed damage; over 50% of the collapsed buildings had this typology on soft soil. A limitation was assuming elastic masonry walls. Additional numerical studies on four- and eight-story reinforced-concrete frames designed to modern codes, using nonlinear static and incremental dynamic analyses, showed increased drift capacity and reduced collapse probability. Model calibration enhanced result accuracy, supporting more realistic assessments of soft-first-story behavior [25]. A comparison of buildings designed under the 2017 and 1976 Mexico City codes [26,27], using records from the 2017 Puebla-Morelos earthquake, was presented in [28]. The 2017 code improved drift distribution, prevented concentration at the first story, and delayed soft-story mechanisms. In contrast, buildings designed with older provisions showed greater vulnerability, consistent with post-earthquake observations. Calibrated numerical models improved the accuracy of the seismic behavior assessment and confirmed the increased resilience provided by the updated code. Ref. [29] evaluated reliability indices of soft-ground-floor buildings using models of structures heavily damaged in the 2017 earthquake. Three-, five-, and eight-story buildings were analyzed with 45 recorded accelerograms. The original structures showed large displacements and low reliability indices, especially the five-story buildings, consistent with observed damage. The study underscores the need for seismic retrofitting and notes that the findings apply specifically to the examined buildings, as results may vary in regions with different seismic conditions and construction practices. In [30], soft stories, defined as having about 70% less stiffness than upper levels, were identified as highly vulnerable, potentially leading to severe damage or collapse when columns are weak. The results also indicate that soft stories located at higher levels show greater stiffness and significantly improve the building’s seismic performance.

Despite extensive studies on soft-first-story buildings, their structural reliability remains insufficiently explored, particularly for mid-rise structures with a soft reinforced-concrete first story and confined-masonry walls above, a configuration rarely analyzed. This work addressed that gap by evaluating the seismic reliability of a representative building using a nonlinear dynamic model in which masonry walls are represented through a wide-column approach calibrated with experimental data to capture shear-dominated nonlinear behavior. This study implements an extended probabilistic framework that estimates the reliability index $\beta$ across several performance limits and three seismic hazard levels, using multiple probability density functions and goodness of fit criteria. This provides a practical methodology applicable to buildings commonly constructed under outdated seismic codes, offering a straightforward yet robust means to determinate whether such structures meet acceptable safety levels. By quantifying inter-story drift demands and reliability indices under region-specific ground motions, this study raises a central question: to what extent do past seismic design provisions affect the current safety of these buildings, and how can a simplified probabilistic approach guide retrofit decisions and risk mitigation in earthquake-prone regions?

2. Case Study: Confined Masonry Building with a Soft Story in the First Level

The selected case of study in this paper is a five-story structure with a regular floor plan configuration. However, the structure presents a vertical irregularity due to a sudden change in stiffness and strength between the first and upper stories. Such an irregularity in the structural system consists of reinforced-concrete moment frames at the first story and confined-masonry walls in upper floors. The ground level is designated for vehicle parking, while the upper stories are intended for residential use. This structural configuration is representative of numerous buildings in seismic-prone regions, where soft first stories are commonly employed for functional and architectural reasons. To better illustrate these characteristics, Figure 1 and Figure 2 present the building’s floor plan and elevation view, respectively, offering a detailed representation of its geometric and structural features.

Figure 1. Structural plan distribution of the building at the first story and upper stories (Dimensions in meters, m).

Figure 2. Front and side elevations of the case-study building (Dimensions in meters, m).

The selected case study has been chosen as a representative benchmark because it exemplifies a structural typology commonly found in seismic-prone regions, where open ground floors are combined with confined masonry upper stories. Many mid-rise buildings in these areas exhibit similar soft-first-story conditions and were designed under older seismic codes, making them potentially vulnerable to earthquake-induced damage. As such, this building provides a realistic scenario to assess the seismic performance and reliability of soft-story structures, allowing the results to be applicable to a winder class of comparable buildings.

The mid-rise building under consideration is in Acapulco, Guerrero, in southwestern Mexico, a region of high seismic activity due to its proximity to the Pacific Ring of Fire. The seismicity in this area is primarily driven by the subduction of the Cocos Plate beneath the North American Plate, a process that has historically generated large-magnitude earthquakes in Mexico [31]. According to Mexican seismic regulations, Acapulco is classified within seismic zone D, the highest seismic hazard category in the country [32]. The structural analysis was conducted using a three-dimensional model developed in the commercial software ETABS v21 [33], assuming firm soil conditions. The reinforced-concrete beams and columns were modeled as frame elements with linear elastic behavior, consistent with the modal response spectrum analysis carried out in this stage of this study. The linear behavior of the confined-masonry walls was modeled using the wide-column method, following the approach presented in [34]. This method was chosen because it allows for the representation of shear-dominated behavior in the walls, incorporates the contribution of adjacent boundary columns, and ensures proper connectivity with other structural elements through the use of rigid horizontal links. The wide-column element is placed at the centroid of the wall, with the transformed areas of the boundary columns included to accurately capture the section properties. Figure 3 shows the structural model, where the panels correspond to wide-column elements visualized in extruded view for clarity. Additionally, the floor system was modeled as a rigid diaphragm to ensure the appropriate distribution of lateral forces. A detailed description of the nonlinear components, needed only for the subsequent time-history analyses, is provided later in Section 4, where the plastic hinges and other inelastic features are introduced.

Figure 3. Three-dimensional structural model of the building.

The structure was originally designed in accordance with the Manual of Civil Structures Design, Earthquake Design of Federal Electricity Commission (MDOC-CFE in Spanish) [35], as well as the Mexico City Building Code and its Complementary Technical Standards for the Desing of Masonry Structures and Concrete Structures [36]. The structural design did consider a seismic behavior factor of Q = 2, with an additional reduction factor applied due to vertical irregularity. In accordance with [35], structures that do not meet regularity conditions must employ a reduced seismic behavior factor Q’, obtained as Q times 0.8, to account for ductility effects in the seismic forces. Consequently, the seismic design forces increase due to the penalization of structural irregularity. However, the displacements obtained with Q’ are multiplied by Q to obtain the actual displacements. Figure 4 illustrates the spectrum adopted for the design. Following these design criteria, the cross-sectional dimensions of the structural elements were determined and are summarized in Table 1.

Figure 4. Seismic design spectrum for the building.

Table 1. Dimensions of structural elements.

Elements	Dimensions (m)	Reinforcement *	Stories
Confined-masonry walls	0.12 × 1.30 0.12 × 3.00 0.12 × 3.10	Confined with tie columns (0.18 × 0.18 m) and bond beams (0.15 × 0.30 m).	2nd–5th
Beams	0.30 × 0.55	Longitudinal reinforcement: 3ø15.9 (top) + 3ø19.1 (bottom); Stirrups: ø9.5 @ 0.20 m (center) and ø9.5 @ 0.10 m (ends).	1st
Beams	0.30 × 0.55	Longitudinal reinforcement: 2ø19.1 + 1ø15.9 (top) + 3ø19.1 (bottom); Stirrups: ø9.5 @ 0.20 m (center) and ø9.5 @ 0.10 m (ends).	1st
Beams	0.30 × 0.55	Longitudinal reinforcement: 3ø15.9 (top) + 2ø19.1 + 1ø15.9 (bottom); Stirrups: ø9.5 @ 0.20 m (center) and ø9.5 @ 0.10 m (ends).	1st
Columns	0.35 × 0.35	Longitudinal reinforcement: 4ø19.1 per side (total 12 bars); Ties: ø9.5 @ 0.15 m (center) and ø9.5 @ 0.10 m (ends).	1st
Columns	0.35 × 0.35	Longitudinal reinforcement: 4ø19.1 + 2ø15.9 (face 1) and 2ø19.1 + 2ø15.9 (face 2). (total 16 bars); Ties: ø9.5 @ 0.15 m (center) and ø9.5 @ 0.10 m (ends).	1st
Columns	0.35 × 0.35	Longitudinal reinforcement: 6ø15.9 (face 1) and 2ø19.1 + 2ø15.9 (face 2). (total 16 bars); Ties: ø9.5 @ 0.15 m (center) and ø9.5 @ 0.10 m (ends).	1st

* Reinforcement: Bar diameters are indicated by “ø” followed by the nominal diameter in millimeters. Longitudinal reinforcement is reported as quantity and bar diameter (e.g., “3ø15.9 (top)”) and transverse reinforcement as bar diameter @ spacing in meters (e.g., “ø9.5 @ 0.10 m”).

3. Seismic Ground Motions

For the nonlinear dynamic analyses of the building, a total of 66 pairs of ground motion records were selected, comprising 22 pairs for each of the two principal horizontal directions of the structural model. These records are grouped into three intensity levels, corresponding to seismic hazard levels with return periods of 72, 475, and 975 years. Each return period is probabilistically associated with a specific performance level: Immediate Occupancy (IO), Life Safety (LS), and Collapse Prevention (CP), respectively. The relationship between return periods and performance levels is summarized in Table 2.

Table 2. Return periods associated with structural performance objectives [37].

Return Period	Annual Exceedance Probability	Performance Level
72 years	1.39%	IO
475 years	0.21%	LS
975 years	0.10%	CP

The selection process for ground motion records began with the definition of probabilistic target spectra corresponding to the three considered return periods, which are specifically constructed for the location where the structure is situated (Acapulco, Mexico). A database containing approximately 20,000 historical earthquake records from Mexico was used, most of which were recorded in the southwestern region by the Institute of Engineering of the National Autonomous University of Mexico and the Center for Scientific Research and Higher Education of Ensenada (CICESE, in Spanish). A key step involved converting the original acceleration time histories into response spectra, considering the simultaneous action of the two orthogonal horizontal components of the seismic input. Equation (1) was used to combine the spectral ordinates of both components and generate a single representative spectrum, in accordance with three-dimensional modeling approach employed in the nonlinear dynamic analyses, following the current Complementary Technical Standards for the Seismic Design of Mexico City [38]. As part of the procedure, the selected records were processed using Equation (2) by applying the geometric mean spectral acceleration ratio (Sa_avg) [39] and were scaled using the geometric mean scaling factor (see Equation (3)), within the period range 0.2 T₁ to 2.0 T₁, where T₁ is the fundamental period of the structure. For the analyzed building, T₁ was 0.47 s, calculated as the average of the two principal horizontal directions. Table A1, Table A2 and Table A3 present the characteristics of the 66 selected pairs of ground motion records for return periods of 72, 475, and 975 years, respectively (See Appendix A). In addition, the duration of the selected accelerograms was shortened using the Arias Intensity method, aiming to obtain a 5% to 95% interval, representing the period of greatest intensity of the ground motion records [40].

$$S{a}_{es}(T)=\sqrt{{\displaystyle \frac{S{a}_{EW}^2(T)+{Sa}_{NS}^2(T)}{2}}}$$

(1)

where $a_{es}$ is the ordinate of the spectrum compared with the probabilistic spectrum, and ${Sa}_{EW}$ and ${Sa}_{NS}$ are the spectral ordinates corresponding to the pseudo-acceleration spectra of the two individual horizontal components of the candidate seismic ground motions, expressed as a fraction of gravity.

$${Sa}_{avg}({0.2T}_1,\dots ,{2.0T}_1)={\left(\prod _{i=1}^{n}Sa\left(T_i\right)\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$n$}\right.}}$$

(2)

where $n$ is the number of vibration periods considered.

$$SF({0.2T}_1,\dots ,{2.0T}_1)={\left(\prod _{i=1}^n{\displaystyle \frac{Sa(T_i)}{S_i(T_i)}}\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$n$}\right.}}$$

(3)

where $Sa(T_i$) denotes the spectral acceleration of the target spectrum at period $T_i$, and $S_i(T_i)$ corresponds to the spectral acceleration of the selected spectrum at same period.

4. Nonlinear Modeling

Nonlinear dynamic analyses were conducted using ETABS v21 software [33], based on a three-dimensional structural model. The wide-column approach, previously adopted for the linear analysis, was used to represent the confined-masonry walls in the upper stories. This method allows for the incorporation of shear hinges and concentrated plasticity at the center of the wall, reproducing the inelastic behavior observed experimentally. The approach follows established guidelines defining the transformed cross-sectional area, shear areas in parallel and transverse directions, moments of inertia, and torsional rigidity, thereby accounting for the contribution of adjacent reinforced-concrete tie columns. Although a macro-element model could capture local wall response with greater detail, its implementation in ETABS v21 is considerably more complex and requires specialized software for full calibration. Furthermore, only a limited number of studies have adopted the wide-column methodology for mid-rise buildings with reinforced-concrete moment frames in the first story and confined-masonry walls above, yet those applications have demonstrated that this approach provides a reliable representation of the global structural response under seismic loading.

In the ground story, where soft-story mechanisms are critical, plastic hinges of type P-M2-M3 with fiber-section models were assigned to all columns to accurately capture the nonlinear interaction between axial load and biaxial bending [41]. Beam elements were assigned M3 hinges to represent flexural plasticity [41]. Concrete nonlinear behavior was defined using the Mander stress–strain curve with kinematic hysteresis to represent its cyclic response. In the upper stories, the confined-masonry walls, already represented as wide columns, were assigned shear-dominated plasticity using hinges calibrated from experimental data. The nonlinear behavior followed the approach proposed by [42], using geometric and material properties obtained from a reserved cyclic loading test on a concrete block masonry wall [43]. To ensure a shear-governed failure mechanism and avoid unrealistic flexural deformation, a rigid beam with infinite stiffness was introduced at the top of each wide-column element. Table 3 summarizes the mechanical properties used in the model, and Figure 5 illustrates the idealized wall configuration.

Table 3. Mechanical properties of the materials (Mpa units) [43].

Calibrated Masonry Wall
Confining Elements		Mortar	Masonry
Compressive strength	Modulus of elasticity	Compressive strength	Compressive strength	Diagonal compressive strength
21.38	21,226	23.01	6.65	0.54

Figure 5. Idealized model of the masonry wall (Dimensions in meters, m).

The masonry wall tested under reversed cyclic loading provided displacement data used to simulate its nonlinear behavior analytically. A nonlinear dynamic analysis was performed using the displacement history obtained from the experimental test, allowing for accurate reproduction of the observed hysteresis cycles. The hysteretic response was modeled using the Pivot model, whose calibration parameters (α₁, α₂, β₁, β₂ y n) allowed for a realistic representation of stiffness degradation and energy dissipation, which are key aspects of masonry behavior under seismic demands. Table 4 presents the shear force–deformation hinge parameters and the final Pivot model coefficients obtained through sensitivity analysis. Figure 6 illustrates the applied displacement history and the comparison between the experimental and numerical hysteresis loops. This calibrated behavior was incorporated into the masonry walls of the building model. However, a limitation of this study is that the actual structure includes walls with different lengths, and no experimental data are currently available for those specific geometries. Although this limitation is acknowledged, the implemented calibration provides a suitable approximation of the nonlinear response of the masonry walls, as it was based on the formula for the maximum shear strength (V_mR) established in the Complementary Technical Standards for Masonry Construction [44]. This approach allowed for an appropriate adjustment of the shear capacity of each wall according to its specific length within the structure. It also acknowledged that more elaborate calibration schemes exist, such as the framework presented in [45], which offers a highly detailed procedure for characterizing masonry nonlinear behavior. Although such approaches provide increased resolution at the material and component level, they could also support a more refined calibration of the confined-masonry walls represented as wide-column elements in the upper stories of the reinforced-concrete frame system. Nevertheless, the level of detail required for that type of calibration falls beyond the scope of the present paper.

Table 4. Shear force–deformation hinge and Pivot hysteresis model parameters.

Shear Force–Deformation Hinge Model				Pivot Hysteresis Model Parameters
Force SF = 126.5 kN		Disp SF = 0.0024 m		Pivot-type hysteresis
Point	Force/SF		Disp/SF	α1	4
E (−)	−0.3		−4	α2	4
D (−)	−0.85		−3.8	β1	0.8
C (−)	−0.875		−1	β2	0.8
B (−)	−0.38		0	n	0.5
A	0		0
B	0.38		0
C	0.875		1
D	0.85		3.8
E	0.3		4

Figure 6. Displacements history and calibrated hysteresis response of masonry wall.

Force SF and Disp SF are reference shear force and displacement, respectively, used to normalize the force–deformation behavior of the hinge. The parameters defining the hysteresis response (points −E to E) are expressed as dimensionless ratios with respect to these reference values.

5. Structural Performance

5.1. PBSD Methodology

The Performance-Based Seismic Design (PBSD) methodology is an iterative process that begins with the definition of performance objectives associated with a specific seismic hazard level. This is followed by estimating the forces and deformations generated by an earthquake in the structure and its components through a detailed analysis, preferably non-linear. Finally, the performance of the structure is evaluated both globally and locally to ensure that the established objectives have been met [46]. The PBSD approach aims to predict and assess building performance in a realistic and reliable manner [47]. The application of PBSD surpasses traditional methods based on prescriptive specifications, as it requires the structure to perform as expected under different seismic intensities in a probabilistic manner. Additionally, this approach seeks to provide a clear understanding of the risk of structural damage and repair costs, which contrasts with conventional codes that only aim to ensure the necessary strength to prevent collapse. However, the implementation of PBSD remains limited due to the complexity in design. For this reason, it is mainly used in high-priority buildings or in structural systems with uncertainty in their performance [48].

PBSD is not only used in design processes but is also effective as a seismic assessment method. This evaluation approach allows for the analysis and determination of whether a building meets the established seismic performance levels under different seismic intensities based on its actual behavior in response to seismic demand [49]. The engineering process of evaluation using the PBSD philosophy is a crucial step towards resilience. Evaluating the structural behavior of buildings is essential, particularly for those located in high-seismic-risk areas or those presenting structural uncertainty, such as buildings with open ground floor plans. Figure 7 illustrates the PBSD process with an emphasis on evaluation.

Figure 7. Methodology for structural evaluation based on PBSD.

5.2. Performance Levels

Performance levels with a structural evaluation focus are used to determine damage in a real or simulated structure subjected to a seismic event. The main objective of this approach is to verify whether the building meets the previously established performance levels, identifying whether it can remain operational in a safe manner with potential damage that does not compromise occupant safety or whether structural intervention is required. When applying a performance-based evaluation, the selected performance levels are linked to specific seismic intensities (see Table 2) and are defined using quantifiable parameters that allow for the assessment not only of damage magnitude but also of structural functionality, occupant safety, and a realistic estimate of repair costs [50]. In this paper, the global inter-story drift of the building is evaluated using the performance levels IO, LS, and CP, with the aim of identifying the presence of the soft-story mechanism under different seismic intensities. Table 5 presents the damage state associated with each performance level, as well as the acceptable deformation limits considered, which correspond to the reinforced-concrete frames since they represent the most vulnerable structural zone of the building.

Table 5. Seismic performance levels and allowable drift limits [37].

Performance Level	Description	Allowable Drift Ratio
IO	The structure functions normally after the seismic event, exhibiting minor damage to both structural and nonstructural components.	0.002–0.005
LS	Life safety for occupants is substantially ensured; however, structural components have sustained moderate to extensive damage, necessitating intervention though an appropriate rehabilitation method.	0.005–0.015
CP	Severe damage compromises occupant life safety. The primary goal at this performance level is to prevent partial or complete structural collapse.	0.015–0.025

5.3. Structural Response at IO Level Under 72-Year Return Period Earthquakes

The nonlinear dynamic response of the building, in terms of inter-story drift, under seismic records corresponding to a 72-year return period revealed significant flexibility at the ground floor, as drifts increased markedly in that zone. Figure 8 and Figure 9 show the inter-story drift profiles along the building height, where a notable concentration of deformations is observed in the reinforced-concrete frames located at the first level. This behavior is consistent with observations from various seismic events worldwide, where the upper stories of this type of structure typically experience minor to moderate damage, while the ground floor concentrates the highest structural demand.

Figure 8. Maximum inter-story drifts under 72-year return period (X direction).

Figure 9. Maximum inter-story drifts under 72-year return period (Y direction).

According to the previous section, the performance limit considered for earthquake records with a 72-year return period is 0.005. This value is established for concrete structures, as most of the building’s deformation tends to concentrate in this type of structural system. Based on the maximum inter-story drifts obtained from each seismic record, the results indicate that, despite the low intensity of these events, the structure did not remain below this threshold. This suggests the potential for damage that could compromise immediate post-earthquake occupancy. Figure 10 and Figure 11 graphically illustrate the performance level reached by the structure under this demand scenario.

Figure 10. Seismic demand for 72-year return period records (X direction).

Figure 11. Seismic demand for 72-year return period records (Y direction).

5.4. Structural Response at LS Level Under 475-Year Return Period Earthquakes

The accelerograms associated with a 475-year return period significantly increased the global deformation of the structure in both directions of analysis (see Figure 12 and Figure 13). Despite the increased seismic demand, the lateral deformation pattern remained like that observed under frequent earthquakes, with deformation concentrated at the ground floor. The plots of maximum inter-story drift profiles clearly reveal the soft-story mechanism, primarily caused by abrupt changes in stiffness and strength between the reinforced-concrete frames at the ground level and the confined-masonry walls above. Although the vulnerability of this configuration is well known, the construction of such buildings is still not explicitly prohibited in most cities worldwide. In Mexico, recent code updates have addressed this issue by recognizing the limited ductility of ground-floor elements and requiring more robust sections; however, most buildings of this type were built decades ago under less-stringent regulations that did not effectively control this structural irregularity, as is the case with the building analyzed in this paper. These results demonstrate that, under rare seismic events, such buildings are already highly susceptible to severe damage.

Figure 12. Maximum inter-story drifts under 475-year return period (X direction).

Figure 13. Maximum inter-story drifts under 475-year return period (Y direction).

The seismic performance assessment for this intensity level showed that the accelerograms pushed the structure beyond the CP threshold, with an average peak inter-story drift of 0.030 in the X-direction and 0.027 in the Y-direction. These results indicate that the demand significantly exceeded the LS performance level, doubling its expected limits and entering a range associated with potential collapse. Figure 14 and Figure 15 clearly illustrate this high demand, reflected in the deterioration of the building’s overall seismic performance.

Figure 14. Seismic demand for 475-year return period records (X direction).

Figure 15. Seismic demand for 475-year return period records (Y direction).

5.5. Structural Response at CP Level Under 975-Year Return Period Earthquakes

The highest-intensity earthquakes, associated with a 975-year return period, caused nearly all inter-story drift responses to exceed the maximum allowable threshold of 0.025 by a considerable margin, as shown in Figure 16 and Figure 17. Similar to the previously analyzed events, the global deformation was concentrated at the ground floor, further accentuating the vulnerability of the reinforced-concrete frames at this level. Although such seismic events are of low probability, their occurrence cannot be ruled out in regions with high seismic activity, where buildings with combined structural systems like the one analyzed may lose their load-bearing capacity under extreme seismic demand. These conditions can lead to severe and often irreparable damage, especially in structures designed under outdated or less-stringent building codes or those with significant structural irregularities.

Figure 16. Maximum inter-story drifts under 975-year return period (X direction).

Figure 17. Maximum inter-story drifts under 975-year return period (Y direction).

The seismic performance evaluation of the building under earthquakes associated with the CP level did not align with the expected behavior. As shown in Figure 18 and Figure 19, the seismic demand significantly weakened the global capacity of the structure. The maximum inter-story drift values in both directions were found to be near or within the collapse range, with average values of 0.0045 in the X-direction and 0.038 in the Y-direction. According to Table 5, this performance level is expected to result in severe structural damage without total collapse. However, the results suggest that the structural elements at the ground floor are being put under excessive demand, potentially leading to a localized collapse in that area. This condition has been observed in past seismic events, where ground-floor column instability led to the crushing of that story under the weight of the upper floors. Furthermore, the results indicate that for all three intensity levels analyzed, the upper stories tend to behave as a rigid block, producing similar inter-story drift values across the confined-masonry wall levels, which is consistent with behavior observed in real buildings of this typology.

Figure 18. Seismic demand for 975-year return period records (X direction).

Figure 19. Seismic demand for 975-year return period records (Y direction).

6. Structural Reliability

The structural reliability of the open-ground-story building analyzed in this paper is evaluated using the reliability index ($\beta$), determined through a probabilistic approach that employs probability density functions (PDFs). The procedure begins with the collection of structural response data in terms of inter-story drift, obtained from nonlinear dynamic analyses conducted for each selected ground motion record. These data are used to construct histograms that represent the random behavior of the variable, which are then normalized to obtain an empirical probability distribution indicating the likelihood of different drift levels. A smooth curve corresponding to a theorical distribution is then fitted over this empirical distribution [51]. In this paper, 12 theorical distributions were considered for the fitting procedure (see Table 6). The fitting was performed using Equation (4), which corresponds to the Chi-squared goodness-of-fit test applied to the cumulative distribution function (CDFs) of these theorical distributions [52].

$$x^2=\sum _{i=1}^k{\displaystyle \frac{{(O_i-E_i)}^2}{E_i}}$$

(4)

where $x^2$ is a calculated value that follows a Chi-square distribution, $O_i$ is the number of times an inter-story drift was observed in interval $i$, $E_i$ is the expected number of observations in that same interval, and $k$ is the total number of intervals into which the drift data was divided. To select the best-fit probability function, the one with the smallest $x^2$ value is chosen.

Table 6. Theorical distributions used for the fitting procedure [53].

Distribution Number	Distribution Name	Distribution Number	Distribution Name
1	Normal	7	Logistic
2	Lognormal	8	Kernel
3	Birnbaum-Saunders	9	Loglogistic
4	Extreme value	10	Stable
5	Gamma	11	t Location scale
6	Generalized extreme value	12	Weibull

To compute the reliability index ($\beta$), it is necessary to define its relationship with the probability of unacceptable performance ($p_{up}$) and the probability of acceptable performance ($p_{ap}$), which characterize the unsafe and safe structural conditions, respectively. In this study, these parameters are evaluated based on the inter-story drift limits established by the PBSD framework.

The $p_{ap}$ is computed by integrating the PDF fitted to the drift data within the interval defined by the thresholds $a$ and $b$, as shown in Equation (5):

$$p_{ap}= {\int }_a^b{f}_x(x)dx$$

(5)

where $x$ represents the inter-story drift, $f_x(x)$ is the PDF of the fitted distribution, and $a$ and $b$ correspond to the drift limits associated with the target performance level.

Subsequently, the $p_{up}$ is obtained as the complement of the $p_{ap}$:

$$p_{up}=1-p_{ap}$$

(6)

Finally, the $\beta$ is expressed as a numerical value: the higher the value, the greater the reliability of the structure. Equation (7) presents the expression used for its calculation. A reference threshold of $\beta$= 3.5 is adopted to assess the structural reliability of the building, a value consistent with international code recommendations and previous papers on buildings with similar structural typologies [20].

$$\beta ={\Phi }^{-1}(1-p_{up})$$

(7)

where ${\Phi }^{-1}$ is the inverse of the CDF, evaluated for the best-fit distribution selected using the Chi-squared goodness-of-fit test.

The structural reliability assessment of the building focused on the inter-story drift response at the first story, as this level concentrates the highest inelastic demand during seismic events due to the vertical irregularity inherent in the structural configuration. This concentration of the deformation at the ground level, consistently observed across all analyzed seismic loading scenarios, represents the critical performance zone and thus serves as the most representative reference point for estimating structural reliability indices.

6.1. Structural Reliability at IO Performance Level

The structural response of the building to seismic records associated with a 72-year return period resulted in reliability values significantly below the minimum threshold of $\beta$= 3.5. As shown in Table 7, the IO-level reliability analysis yielded mean $\beta$ values of 1.46 for the EW component and 1.30 for the NS component, with their corresponding standards deviations and coefficients of variation indicating considerable dispersion among records. These results show that the structure reaches only about half of the minimum required reliability for immediate occupancy, implying a high probability of damage under moderate seismic events, which is consistent with the deficient structural response observed in Figure 10 and Figure 11. One record even produced a negative $\beta$ value, meaning that the probability of unacceptable performance exceeded 50% as inter-story drift surpassed the IO threshold for more than half the record duration. For each seismic input, the probability distribution was selected based on the lowest $x^2$ value and p-values ≥ 0.005, ensuring a statistically acceptable fit of the drift data before computing the reliability index.

Table 7. Structural reliability indices for IO performance level.

EW Component Seismic Records					NS Component Seismic Records
Record	PDF	${\mathit{x}}^2$	p-Value	$\mathit{\beta }$	Record	PDF	${\mathit{x}}^2$	p-Value	$\mathit{\beta }$
E1-IO	Kernel	5.69	0.68	1.80	E1-IO	Kernel	3.75	0.92	2.18
E2-IO	Kernel	3.07	0.96	1.68	E2-IO	Kernel	4.28	0.89	1.91
E3-IO	Kernel	9.08	0.43	1.53	E3-IO	Kernel	4.07	0.91	1.45
E4-IO	Kernel	1.66	0.99	1.87	E4-IO	Kernel	6.50	0.69	0.89
E5-IO	Kernel	1.38	0.99	2.19	E5-IO	Kernel	1.63	0.99	1.20
E6-IO	Kernel	3.63	0.93	1.93	E6-IO	Kernel	2.76	0.97	1.33
E7-IO	Kernel	2.42	0.98	1.50	E7-IO	Kernel	21.34	0.08	1.27
E8-IO	Kernel	2.56	0.98	1.44	E8-IO	Kernel	8.58	0.48	0.58
E9-IO	Kernel	3.11	0.80	2.57	E9-IO	Kernel	2.20	0.98	1.18
E10-IO	Kernel	3.72	0.93	1.23	E10-IO	Kernel	2.71	0.97	1.18
E11-IO	Kernel	8.07	0.53	0.77	E11-IO	Kernel	6.28	0.71	0.66
E12-IO	Kernel	2.78	0.97	1.29	E12-IO	Kernel	3.12	0.96	1.16
E13-IO	Kernel	3.74	0.93	1.10	E13-IO	Kernel	3.24	0.95	1.37
E14-IO	Kernel	1.68	0.99	1.89	E14-IO	Kernel	1.37	0.99	1.81
E15-IO	Kernel	5.40	0.80	0.92	E15-IO	Kernel	4.45	0.88	0.83
E16-IO	Kernel	14.99	0.09	0.39	E16-IO	Kernel	1.87	0.99	1.08
E17-IO	Kernel	3.45	0.94	1.22	E17-IO	Kernel	1.97	0.99	1.99
E18-IO	Kernel	2.80	0.97	2.11	E18-IO	Kernel	1.44	0.99	1.53
E19-IO	Kernel	0.66	0.99	1.60	E19-IO	Kernel	5.78	0.76	1.78
E20-IO	Kernel	1.40	0.99	1.67	E20-IO	Kernel	3.33	0.95	1.74
E21-IO	Kernel	6.40	0.70	−0.25	E21-IO	Kernel	9.90	0.36	0.26
E22-IO	Kernel	1.19	0.06	1.61	E22-IO	Kernel	7.16	0.62	1.28
$\mathrm{Mean} \beta$				1.46	$\mathrm{Mean} \beta$				1.30
$\mathrm{Standard} \mathrm{deviation} \beta$				0.56	$\mathrm{Standard} \mathrm{deviation} \beta$				0.47
$\mathrm{Coefficient} \mathrm{of} \mathrm{variation} \beta$				0.38	$\mathrm{Coefficient} \mathrm{of} \mathrm{variation} \beta$				0.36

6.2. Structural Reliability at LS Performance Level

The structural reliability of the building under seismic events associated with a 475-year return period resulted in $\beta$ values well below the minimum acceptable threshold of $\beta$ = 3.5. As shown in Table 8, the EW and NS components exhibit average $\beta$ values of 1.62 and 1.65, respectively. To quantify the dispersion of the results, the standard deviations were found to be 1.46 for EW and 1.49 for NS, yielding coefficients of variation close to 0.90 in both directions. These high coefficients of variation indicate substantial record-to-record variability, meaning that the structural response is highly sensitive to the characteristics of each seismic record rather than showing a stable or uniform performance. This level of variability implies that, although some records result in moderate $\beta$ values, many others fall significantly below the average and approach critical performance conditions. Consequently, the structure shows a high probability of exceeding the inter-story drift limit of 0.015 under rare seismic events, reflecting insufficient reliability for the LS performance level. This interpretation is consistent with the performance trends shown in Figure 14 and Figure 15, where several analyses approach collapse-prevention conditions. Additionally, the Kernel distribution again provided the best statistical fit to the drift data, confirming a consistent non-parametric behavior across performance levels and reinforcing the robustness of the probabilistic modeling approach.

Table 8. Structural reliability indices for LS performance level.

EW Component Seismic Records					NS Component Seismic Records
Record	PDF	${\mathit{x}}^2$	p-Value	$\mathit{\beta }$	Record	PDF	${\mathit{x}}^2$	p-Value	$\mathit{\beta }$
E1-LS	Kernel	2.71	0.97	2.73	E1-LS	Kernel	3.61	0.94	2.01
E2-LS	Kernel	4.03	0.91	0.98	E2-LS	Kernel	14.55	0.10	1.22
E3-LS	Kernel	1.18	0.99	1.94	E3-LS	Kernel	2.79	0.97	1.48
E4-LS	Kernel	8.43	0.49	1.99	E4-LS	Kernel	5.02	0.83	1.45
E5-LS	Kernel	22.13	0.08	1.04	E5-LS	Kernel	2.28	0.98	1.80
E6-LS	Kernel	7.05	0.63	0.95	E6-LS	Kernel	1.81	0.99	1.85
E7-LS	Kernel	9.88	0.36	1.07	E7-LS	Kernel	7.82	0.55	1.94
E8-LS	Kernel	1.88	0.99	1.62	E8-LS	Kernel	6.60	0.68	1.38
E9-LS	Kernel	2.43	0.98	1.47	E9-LS	Kernel	12.91	0.17	0.50
E10-LS	Kernel	14.56	0.10	0.48	E10-LS	Kernel	13.89	0.13	0.83
E11-LS	Kernel	0.95	0.99	1.11	E11-LS	Kernel	0.13	0.99	7.73
E12-LS	Kernel	12.42	0.19	1.36	E12-LS	Kernel	1.66	0.99	1.80
E13-LS	Kernel	3.35	0.94	1.89	E13-LS	Kernel	6.24	0.72	2.39
E14-LS	Kernel	0.98	0.99	0.74	E14-LS	Kernel	14.19	0.12	0.31
E15-LS	Kernel	1.77	0.99	1.38	E15-LS	Kernel	13.04	0.16	0.71
E16-LS	Kernel	1.08	0.99	2.26	E16-LS	Kernel	4.03	0.91	1.75
E17-LS	Kernel	2.90	0.97	1.92	E17-LS	Kernel	3.61	0.74	2.01
E18-LS	Kernel	2.10	0.98	7.61	E18-LS	Kernel	3.44	0.74	1.95
E19-LS	Kernel	1.24	0.99	0.60	E19-LS	Kernel	2.12	0.98	0.38
E20-LS	Kernel	3.49	0.96	0.71	E20-LS	Kernel	8.09	0.53	0.99
E21-LS	Kernel	5.76	0.76	1.20	E21-LS	Kernel	6.60	0.68	1.34
E22-LS	Kernel	2.72	0.97	0.63	E22-LS	Kernel	7.37	0.60	0.55
$\mathrm{Mean} \beta$				1.62	$\mathrm{Mean} \beta$				1.65
$\mathrm{Standard} \mathrm{deviation} \beta$				1.46	$\mathrm{Standard} \mathrm{deviation} \beta$				1.49
$\mathrm{Coefficient} \mathrm{of} \mathrm{variation} \beta$				0.90	$\mathrm{Coefficient} \mathrm{of} \mathrm{variation} \beta$				0.90

6.3. Structural Reliability at CP Performance Level

For the ground motion records associated with a 975-year return period, the structural reliability of the building yielded values far below the minimum threshold of $\beta$ = 3.5. As shown in Table 9, the EW component presented a mean $\beta$ of 1.39, while the NS component reached a mean $\beta$ of 2.05. The corresponding standards deviations (1.10 for EW and 1.66 for NS) and coefficients of variation (0.79 and 0.81, respectively) reveal a markedly high dispersion in the $\beta$ values, indicating not only a low average reliability but also a highly unstable and record-dependent structural response. In other words, the probability of unacceptable performance varies substantially among the seismic inputs, reflecting significant sensitivity of the structure to the characteristics of individual ground motions. These results demonstrate that, at the CP performance level, the structure consistently exceeds the inter-story drift threshold of 0.025 under very rare seismic events, implying a significant likelihood of collapse initiation. The interpretation aligns with Figure 18 and Figure 19, where the maximum drift demands surpass the prescribed limit and fall within the collapse range. Additionally, the p-values obtained for the best-fitting probability distributions (Kernel for all records) confirm statistically acceptable fits, supporting the reliability of the probabilistic representation of drift demands across the evaluated intensity levels.

Table 9. Structural reliability indices for CP performance level.

EW Component Seismic Records					NS Component Seismic Records
Record	PDF	${\mathit{x}}^2$	p-Value	$\mathit{\beta }$	Record	PDF	${\mathit{x}}^2$	p-Value	$\mathit{\beta }$
E1-CP	Kernel	22.38	0.07	0.46	E1-CP	Kernel	2.45	0.98	2.24
E2-CP	Kernel	12.70	0.18	1.43	E2-CP	Kernel	8.76	0.46	0.97
E3-CP	Kernel	1.23	0.99	3.01	E3-CP	Kernel	22.07	0.007	1.85
E4-CP	Kernel	1.22	0.99	2.46	E4-CP	Kernel	9.88	0.36	2.01
E5-CP	Kernel	23.00	0.06	1.10	E5-CP	Kernel	11.39	0.25	2.21
E6-CP	Kernel	11.96	0.22	1.89	E6-CP	Kernel	5.59	0.78	1.19
E7-CP	Kernel	1.59	0.99	2.03	E7-CP	Kernel	1.52	0.99	2.57
E8-CP	Kernel	2.65	0.97	4.82	E8-CP	Kernel	1.80	0.99	1.38
E9-CP	Kernel	4.91	0.84	1.51	E9-CP	Kernel	15.07	0.09	1.41
E10-CP	Kernel	15.48	0.08	0.95	E10-CP	Kernel	9.05	0.43	0.95
E11-CP	Kernel	29.19	0.03	0.88	E11-CP	Kernel	0.57	0.99	1.62
E12-CP	Kernel	7.33	0.60	1.44	E12-CP	Kernel	2.81	0.97	1.51
E13-CP	Kernel	1.73	0.99	0.84	E13-CP	Kernel	0.18	0.99	7.72
E14-CP	Kernel	20.29	0.08	0.61	E14-CP	Kernel	16.73	0.09	0.15
E15-CP	Kernel	6.67	0.67	0.58	E15-CP	Kernel	8.45	0.49	0.38
E16-CP	Kernel	6.04	0.74	0.80	E16-CP	Kernel	11.26	0.25	2.01
E17-CP	Kernel	6.98	0.64	1.13	E17-CP	Kernel	20.41	0.07	1.85
E18-CP	Kernel	1.02	0.99	0.61	E18-CP	Kernel	1.57	0.99	4.86
E19-CP	Kernel	21.73	0.07	2.59	E19-CP	Kernel	2.40	0.98	2.75
E20-CP	Kernel	10.32	0.32	0.34	E20-CP	Kernel	7.06	0.63	0.95
E21-CP	Kernel	1.61	0.99	1.33	E21-CP	Kernel	22.51	0.07	0.65
E22-CP	Kernel	19.27	0.08	−0.28	E22-CP	Kernel	2.94	0.97	3.81
$\mathrm{Mean} \beta$				1.39	$\mathrm{Mean} \beta$				2.05
$\mathrm{Standard} \mathrm{deviation} \beta$				1.10	$\mathrm{Standard} \mathrm{deviation} \beta$				1.66
$\mathrm{Coefficient} \mathrm{of} \mathrm{variation} \beta$				0.79	$\mathrm{Coefficient} \mathrm{of} \mathrm{variation} \beta$				0.81

7. Conclusions

This study provides a reliability-based seismic assessment of a mid-rise building with a soft reinforced-concrete first story and confined-masonry walls in the upper levels, a configuration that remains scarcely explored. The calibrated wide-column approach for masonry walls, combined with an extended probabilistic framework to estimate $\beta$ indices at different hazard levels, allows for a comprehensive understanding of the seismic performance of this type of structure.

The results highlight that vertical irregularities, particularly at the ground floor, significantly amplify seismic demands, with structural vulnerability closely linked to the design provisions of outdated codes. The nonlinear time-history analyses reveal that even moderate seismic events can produce inter-story drifts exceeding current performance thresholds, emphasizing the need for careful evaluation of similar existing structures.

The probabilistic assessment shows that the Kernel distribution provides the best fit for inter-story drift data, supporting robust estimates of reliability indices across different hazard levels. These findings demonstrate the applicability of a simplified probabilistic approach as a step in the right direction to guide structural engineers in detecting zones to retrofit in buildings affected by earthquake loading.

Finally, this study is limited to mid-rise buildings and focuses on inter-story drift, with model calibration based on available experimental data. Sensitivity evaluations of the model are recommended to improve the validity of results. Future research could extend the methodology to different building heights, additional structural response parameters, updated design codes, retrofit interventions to improve structural reliability, and the incorporation of uncertainties of material properties, damping ratios, or modeling assumptions. In addition, it is recommended to implement the methodology to other types of structural systems located in different seismic regions with high seismic activity.