Structural Reliability Assessment of Dual RC Buildings for Different Shear Wall Configuration

Abstract

Shear walls, integrated into conventional reinforced concrete (RC) moment-resisting frame systems (RC frame–shear wall building), have proven to be effective in improving the structural performance and reliability of buildings; however, the seismic behavior of the building depends directly on the location of these elements. For this reason, this paper evaluates the structural reliability of ten medium-rise RC buildings designed based on the Mexico City Building Code, considering different shear wall configurations. With the aim to estimate and compare the seismic reliability, the buildings are modeled as complex 3D structures via the OpenSees 3.5 software, which are subjected to different ground motion records representative of the soft soil of Mexico City scaled at different intensity values in order to compute incremental dynamic analysis (IDA). Furthermore, the parameter used to estimate the reliability is the maximum interstory drift (MID), which is obtained from the incremental dynamic analysis in order to assess the structural fragility curves. Finally, the structural reliability estimation is computed via probabilistic models by combining the fragility and seismic hazard curves. It is concluded from the results that the structural reliability is maximized when shear walls are symmetrically distributed. On the other hand, the configuration with walls concentrated in the center of the building tends to oversize the frames to reach a reliability level comparable to that of symmetrical arrangements.

1. Introduction

Earthquake design of buildings that can develop good structural performance is a major concern, especially in regions of high seismicity, as in the case of Mexico City. In this context, dual RC systems, which combine frames and shear walls, are an effective solution due to their capacity to resist both gravity loads and earthquake-induced lateral forces [1]. Several studies have demonstrated the importance of shear walls in improving seismic performance, even in comparison with other protection systems like bracings [2,3], proving particularly effective in medium- and high-rise buildings [4,5].

The inclusion of these elements not only provides additional stiffness and strength but also helps to significantly reduce lateral displacements compared to a solution that relies solely on frames [6]; this consequently results in smaller mechanical elements in the framing system [7], thus obtaining smaller column and beam sections. The above-described are important factors in the mitigation of structural and non-structural damages that help to increase structural reliability and reduce repair costs after an earthquake [8]. Despite these advantages, the effectiveness of shear walls does not depend solely on their presence but is strongly influenced by factors such as their location and configuration within the building plan [9,10,11], the detailing of sections to avoid fragile failures and achieve adequate ductility [12], attention to confining zones at the ends of the walls called “boundary elements” [13], as well as the correct modeling of these elements in structural analysis software [14].

Another important aspect arises from the prescriptive methods for the design of shear walls, which, although they have contributed to the development of seismic regulations, do not always capture multiple sources of uncertainty—such as the nonlinear response of materials, analysis models, and the characteristics of seismic action—that influence the actual performance of structures [15,16]. In this sense, the adoption of seismic design methods based on reliability and probabilistic evaluation of the structural response [17] are becoming increasingly relevant within the building analysis and design procedure, since it allows a more accurate estimation of the expected structural behavior in the face of seismic events. Among some relevant methods for the estimation of the seismic performance of structural systems considering explicitly random and epistemic uncertainties such as the one proposed by Jafari et al. [18], where a framework representing the structural demand in a three-dimensional space consisting of the reliability index, the intensity measure and the allowable engineering demand parameter (β-IM-EDP_all) is proposed [18]; on the other hand Rahgozar et al. [19], reinforces the aspect of employing probabilistic tools to estimate the structural reliability. Motivated by the need to evaluate the structural reliability of complex dual systems. In this work, structural reliability is evaluated using demand hazard curves that are calculated using Equation (1) proposed by Esteva [20], as discussed in the methodology below. For this aim, based on the above mentioned, this paper evaluates the influence of the shear wall configuration on the structural reliability of buildings with dual systems, by comparing five eight-story and five twelve-story RC buildings designed under the Mexican City building code [21,22], maintaining the same geometric configuration, both in plan and height, but varying the location of the shear walls; and subsequently subjected to IDA executed by means of three-dimensional models via OpenSees [23] where constitutive laws for confined and unconfined concrete were applied to adequately represent the boundary elements and the wall web; in addition, SFI-MVLEM-3D elements that allow to consider an axial-flexural-shear interaction in the nonlinear analysis subject to multidirectional loads is used.

2. Methodology

The methodology for this study is focused on evaluating the optimal configuration of shear walls in RC shear wall-frame buildings by comparing the structural reliability across five different shear wall distributions for each building. For this purpose, the mean annual exceedance rate, νD(d), proposed by Esteva [20]. This measure quantifies the expected annual number of times that the seismic demands, d, exceed a threshold MID, evaluated for a specific seismic intensity, Sa (in this case the spectral acceleration of vibration of the structure Sa(T₁) or Sa). In other words, this approach combines the probability of exceeding a certain demand level (acceleration, velocity, etc.), which is related to the seismic hazard of the region where the building will be located [24], with the structure’s propensity to incur damage in its various elements and components when subjected to seismic forces at different intensity levels in order to compute structural reliability [25]. The mathematical equation that describes this relationship is as follows:

$${\nu }_D\left(d\right)=\int \left|{\displaystyle \frac{d\nu \left(s_a\right)}{d\left(s_a\right)}}\right|P\left(MID>d\left|s_a\right.\right)d\left(s_a\right)$$

(1)

where ${\displaystyle \frac{d\nu \left(s_a\right)}{d\left(s_a\right)}}$ is the derivative of the seismic hazard curve for the site, representing the variation in the exceedance rate with respect to the Sa, and $P\left(MID>d\left|s_a\right.\right)$ is the conditional probability that the structural demand, MID, exceeds a specific value d, given a level of spectral acceleration Sa. Notice that in the present work spectral acceleration at first mode of vibration of the structure was selected as intensity measures since the ground motion records have similar spectral shape (similar N_p or N_pg values) [26,27].

Seismic hazard curves collect data on spectral acceleration and its annual exceedance rate for the soft soil of Mexico City, which, for this study, were obtained from the INGEN-UNAM database. The development of fragility curves was based on the collection of MID data through incremental dynamic analyses performed for different values of Sa. For each Sa value, a lognormal function was fitted to the recorded MID data, as suggested by Bojórquez et al. [28,29,30]. The mean of the natural logarithm of MID, ${\widehat{\mu }}_{\left.{ln}_D\right|Sa}$, and its standard deviation, ${\sigma }_{\left.{ln}_D\right|Sa}$, were calculated. Using these parameters and applying Equation (2), the conditional probability of MID exceeding specific drift levels (d) was estimated for each seismic intensity level.

$$P\left(MID>d\left|s_a\right.\right)=1-\mathsf{\varnothing }\left({\displaystyle \frac{\mathrm{l}\mathrm{n}(d)-{\widehat{D}}_{\left.{ln}_D\right|Sa}}{{\sigma }_{\left.{ln}_D\right|Sa}}}\right)$$

(2)

The research methodology is illustrated according to the flowchart of Figure 1.

3. Structural Models

3.1. Geometric Characteristics of the Buildings

The proposed methodology is exemplified by two groups of buildings: five eight-story buildings and five twelve-story buildings. Figure 2 illustrates the general architectural disposition of the building, where we have a regular building maintaining the same floor plan distribution and mezzanine height on all levels.

Five proposed floor plan locations were made for each group of buildings, the main criterion being to maintain the same number of walls in all cases, except for one model where, for reasons of space limitations and architectural functionality, two walls were omitted (see Figure 3e and Figure 4e), so that the differences in seismic response could be directly attributed to the distribution of these walls and not to variations in the total mass or in the overall stiffness and strength between buildings. The most typical distributions observed in architecture were considered. The five proposed configurations are shown in Figure 3 for the eight- and twelve-story building, respectively, which were defined as M-12_(i) and M-8_(i), where “i” represents the configuration number. From this point on, they will be referred to as follows.

3.2. Building Models and Structural Design

As for the design actions that are expected to affect the building during its useful life, three categories were considered: permanent, variable, and accidental. The dead loads were estimated from the dimensions of the structural elements and the corresponding volumetric weight of the materials, while the live loads were obtained from the statistical recommendations included in the Mexican City building code for an office-use destination [31]. As for the design seismic actions (accidental loads), they were obtained from the spectra associated with earthquakes with a recurrence period equal to or greater than 250 years corresponding to a soft soil scenario characterized by a dominant period of Ts = 2.2 s through the Seismic Hazard Information System of Mexico City (SASID) [32]. Figure 5a shows a visual representation of the building density in the study area, with buildings from approximately 6 to 15 floors; this was taken into account when defining the number of floors for the present study. The spectra obtained for the corresponding evaluations are shown in Figure 5b. In this figure, UHS means uniform hazard spectrum.

All the buildings were designed considering the bidirectional effects of the earthquake, as indicated in the design standards, using commercial software by means of three-dimensional finite element modeling (see Figure 3 and Figure 4), using “Frame” type elements for the definition of beams and columns, “Slab” type elements for slab design, and “Wall” type elements for the modeling of shear walls; following the guidelines established in [21,22] and incorporating important provisions of the American Concrete Institute (ACI 318-19) [33] to solve some limitations of the Mexico City codes regarding the design of structures with limited ductility (Q = 2) in seismic regions [34]. In addition, the effect of cracking in the structural elements was considered, and verifications of the strength and serviceability limits were carried out in accordance with the regulations established in the standards.

The main characteristics of each system are compared using different engineering indicators. The fundamental vibration periods obtained for each of the study models are summarized in

Table 1. Fundamental vibration periods of each of the models.

Models	M-12_1/M-8_1	M-12_2/M-8_2	M-12_3/M-8_3	M-12_4/M-8_4	M-12_5/M-8_5
Twelve-story buildings	1.08 s	1.08 s	0.94 s	0.94 s	1.20 s
Eight-story buildings	0.75 s	0.75 s	0.69 s	0.69 s	0.91 s

. All models presented their first vibration mode translationally in the “Y” direction, with the exception of models M-8_5 and M-12_5, which presented a rotational behavior in their first mode; additionally, they are the most flexible buildings since they have a longer period than the rest of the models in each of their corresponding groups, which, for this particular case, having structural periods of approximately 2 s represents a disadvantage since we would be going up in the ascending branch of the design spectrum and approaching the period of the soil in the study area. Figure 6 and Figure 7 illustrate the vibration modes for each of the models of both study groups.

Figure 8 and Figure 9 show the comparisons in terms of maximum floor-to-floor drifts for the most unfavorable direction of analysis for the group of twelve- and eight-story buildings, respectively; where a dashed line represents the MID for the limit state associated with a life safety performance level (γ_sv = 0.01) established by the Mexican City building code for structures with a low ductility RC frame–shear wall. In the group of eight-story buildings, the M-8_2 model is the one with the highest interstory drifts, but still within the allowable limit, while for the group of twelve-story buildings, it is the M-12_1 model. Notice that, although the M-8_5 and M-12_5 models present on average the lowest interstory drift, as will be seen below, when evaluating their dynamic behavior where aspects such as higher modes and the degradation of stiffness and resistance in the evaluation of the structural behavior response, these models present important deficiencies in the structural performance.

Figure 10 and Figure 11 illustrate the final design sections for the twelve- and eight-story models, respectively, together with the layout of the reinforcing steel in the columns and beams and the distribution of the levels where these sections were applied. As previously mentioned, the design of the structural elements was standardized for the five models that make up each corresponding study group, taking as a base reference the M-8_1 and M-12_1 models, with the purpose of making a fairer comparison between them where the only variable is the layout of walls in plan, and thus having better control in the comparison of the performance of each of the elements, as in their demand/capacity (D/C) ratio, for example.

Figure 12 and Figure 13 present the designs of the shear walls located in the 6 and 7 m spans, corresponding to the eight- and twelve-story buildings, respectively. To standardize the evaluation of seismic behavior and simplify the nonlinear analysis by reducing processing times, the same reinforcement distribution was adopted along the entire height, from level 1 to the roof. The thickness of the web of the walls was kept constant over the entire height at 300 mm for the 8-story buildings and 350 mm for the 12-story buildings. The sections shown in this subchapter were used to perform all the revisions required by the standards for the spectral modal analysis, as well as for the application of the IDA and pushover analysis.

Although the Mexican City Building Code does not specify special detailing of structural elements for buildings designed with limited ductility (Q = 2), the design recommendations for medium ductility were followed to ensure that elements working primarily in flexure do not fail in shear before plastic hinges can form at their ends. An example of detailing considerations for beam elements is shown in Figure 14.

4. Pushover and Nonlinear Modelling

4.1. Pushover Analysis

To develop the pushover analyses, initial gravity loads were applied considering 100% of the dead load and 100% of the instantaneous live load. Subsequently, a pattern of lateral loads associated with height was applied at the centers of mass at each level, based on a pattern of equivalent forces obtained as a function of the percentage of participation in the total basal shear and the total weight of the structure. Figure 15 shows the pushover curves for the most critical direction of analysis, which corresponded to the X direction.

Figure 15 shows that models M-8_2, M-8_4, and M-8_5 have a higher initial stiffness (506.49, 504.17, and 316.41 ton/cm, respectively) than models M-8_1 and M-8_3 (181.25 and 177.74 ton/cm, respectively) but have a lower deformation capacity, so that when evaluating the performance points of each structure (PP), the latter two models are much higher than the first three. A similar behavior was observed in the group of twelve levels. Finally, in models M-8_5 and M-12_5, it was observed that, although they have a good resistance capacity, there is little margin between the displacement associated with creep (where plastic hinges begin to be generated in the elements) and the maximum displacement, which can be interpreted as a possible global brittle failure of the structure once the linear design range is exceeded.

4.2. Nonlinear Modeling to Incremental Dinamic Analysis

To perform the incremental nonlinear dynamic analyses, three-dimensional nonlinear models of the buildings were developed using Open-Sees 3.5 software [23]. Each model was subjected to seismic recordings that were progressively scaled to different intensity levels to identify the critical points of yielding and collapse. To better capture the inelastic behavior, the beams and columns were represented by a fiber-type approach, using the distributed plasticity method [35], where each subdivided fiber is associated with a “UniaxialMaterial” command that is implemented in the software to characterize the uniaxial stress-strain relationships for both confined and unconfined concrete.

Confined concrete was modeled following the approach proposed by Mander et al. [36], while unconfined concrete was modeled using the parabolic model proposed by Todeschini, both implemented as “Concrete02” [37]. For the reinforcing steel, the uniaxial formulation “Steel02” [38], capable of reproducing the effects of strain hardening and cyclic degradation, was used. Regarding the discretization of the elements, for beams and columns they followed the recommendations of Kostic and Filippou [39], adopting an 8×8 fiber mesh in the confined core and a 5 × 5 mesh in the cladding of the elements. To model the cor-tant walls, the recommendations of Kolozvari et al. [40] were followed using SFI-MVLEM-3D elements, which allow capturing in detail the flexural–shear interaction in a three-dimensional context. Figure 16 presents a conceptual diagram of this formulation, ideal for representing the hysteretic behavior of the walls under seismic loads. The concrete of the walls was defined using ConcreteCM, a uniaxial hysteretic constitutive model, and the reinforcing steel was defined using SteelMPF, as proposed by Kolozvari et al. [41]. A concrete compressive strength of f’c = 250 kg/cm² and a yield strength of the reinforcing steel of Fy = 4200 kg/cm² (for horizontal and vertical reinforcement) were considered in the design of all structural elements.

5. Seismic Hazard Characterization

The structures are analyzed by nonlinear dynamic analysis with a set of seismic records specifically selected to represent the seismic hazard of the study area. The selection criteria were based on the fact that the selected records should follow a spectral form similar to that of the uniform hazard spectrum of the site, which takes into account two direct hazards: intermediate-depth earthquakes and subduction earthquakes, both of which are particularly characteristic of soft ground. Additionally, accelerograms of earthquakes occurring at different times were considered, since the soft soil of Mexico City tends to rigidize over time due to water extraction, resulting in a reduction in the maximum peaks of the spectral acceleration and a shortening of the period of the soil of the maximum peaks in the spectral acceleration and a shortening of the soil period. Twelve seismic records were selected from a complete family of accelerograms, with the additional condition of having moment magnitudes (M_w) equal to or greater than 6 and epicentral distances close to 300 km from the study site, in addition to those mentioned in the previous paragraph. The main characteristics of these seismic events are summarized in

Table 2. Characteristics of seismic events.

Seismic	Date	Epicenter Coordinates	Magnitude	Station
S1	11 January 1997	17.910 N; 103.04 W	6.9	Valle Gómez
S2	14 September 1995	16.31 N; 98.88 W	7.4	Tlatelolco
S3	9 October 1995	18.74 N; 104.67 W	7.3	Garibaldi
S4	25 April 1989	16.603 N; 99.4 W	6.9	Alameda
S5	9 October 1995	18.74 N; 104.67 W	7.3	Liverpool
S6	11 January 1997	17.9 N; 103 W	6.9	Cordoba
S7	25 April 1989	16.603 N; 99.4 W	6.9	C.U. Juarez
S8	14 September 1995	16.31 N; 98.88 W	7.2	Cujp
S9	19 September 1985	18.08 N; 102.942 W	8.1	SCT B-1
S10	9 October 1995	18.74 N; 104.67 W	7.3	sector popular
S11	19 September 2017	19.394 N; 99.148 W	7.1	SCT B-2
S12	19 September 2017	19.449 N; 99.137 W	7.1	Tlatelolco

.

To evaluate the structures under different levels of seismic intensity, the amplitudes of the recorded ground motions are modified using a scaling factor (SF). This scaling factor is obtained from the ratio between the spectral acceleration at the target intensity and the spectral acceleration corresponding to the vibration period of the structure under study. In the dynamic analyses, both components of the records are considered to act simultaneously. The intensities of these components are combined using the quadratic mean, as represented by the following equation:

$${Sa}_{\left(g\right)}=\sqrt{{\displaystyle \frac{{{Sa}^2}_{EW}\left(T\right)+{{Sa}^2}_{NS}(T)}{2}}}$$

(3)

where Sa_{EW (T)} is the spectral acceleration of the east–west component of the record for the structure’s period, and Sa_NS(T) is the corresponding value for the north–south component. Figure 17 and Figure 18 show the pseudo-acceleration response spectra for the group of twelve- and eight-story buildings, respectively, considering both horizontal components and scaled according to the fundamental period of vibration of each structural model. The dashed lines indicate the fundamental period associated with each building, highlighting that models M-12_1 and M-12_2 share the same period, as do M-12_3 and M-12_4, giving rise to three sets of response spectra for the twelve-story case. Similarly, for the eight-story building, models M-8_1 and M-8_2 had the same period, as did M-8_3 and M-8_4, also generating three sets of response spectra. To apply the ADIs, the seismic records were scaled for Sa(g) values from 0.1 g until 1.5 g, or until the structure reached collapse.

6. Incremental Dynamic Analysis and Structural Fragility

6.1. Results of Nonlinear Dynamic Analysis

The IDA methodology was implemented using a step-by-step nonlinear approach, scaling acceleration records to reach different levels of seismic intensity, in accordance with the proposal by Vamvatsikos and Cornell [42]. This approach provides a more detailed representation of the structural behavior under increasing demands, allowing for a more precise estimation of the safety and effectiveness of seismic design. IDA enables the plotting of curves that relate performance indices with intensity parameters. The key point is that the damage to structural components explicitly depends on these structural performance parameters [43]. In this way, capacity curves are obtained that cover the entire range of response, from elastic behavior to collapse.

In this study, MID was adopted as an engineering demand parameter to estimate the structural demand at different seismic intensity levels. Figure 19 shows the capacity curves for the twelve-story models, where a low dispersion below 600 gals was identified, characterized by an almost linear trend that indicates a yield limit. Up to this intensity, the buildings remain essentially elastic in accordance with the suggestion in [42], with drifts less than 0.5%, a value associated with immediate occupancy performance [22]. A similar behavior was observed in the eight-story models, as can be seen in Figure 20. On the other hand, the collapse exhibited greater dispersion due to the nonlinear nature in the prediction of the structural response and the uncertainties in the analysis assumptions. However, following the SAC/FEMA approach [43], the imminent collapse was defined as the point at which the slope of the IDA curve drops to 20% of the initial elastic slope, which occurred for intensities above 1300 gals, associated with MIDs greater than 1% (life safety performance [22]). This performance is consistent with the site’s maximum probable acceleration, close to 1400 gals for T ≈ 2 s. However, in models M-12_5 and M-8_5, an abrupt failure was detected immediately after the yield limit (750 gals and 1250 gals, respectively), as reflected by a sudden change in the slope of all the IDA curves. In such a way that the results suggest brittle behavior and a rapid degradation of structural capacity once the Sa (T₁) limit is exceeded.

6.2. Structural Fragility and Seismic Hazard Curves

As previously mentioned, structural demand hazard curves represent the average number of times per year that a certain maximum inter-story drift value, d, is exceeded. These curves allow for the evaluation of the reliability of structures using the proposed methodology. To determine a structural demand hazard curve, it is essential to know the seismic hazard curve, which shows the average number of times per year that a certain seismic intensity, Sa, is exceeded; these are associated with each of the fundamental vibration periods of the structures. For the present study, they were obtained from the INNGEN-UNAM database (see Figure 21). Additionally, fragility curves for each building were estimated by applying Equation (2) for different DME limit states according to ASCE/SEI 41, which establishes the following damage limit states; these are consistent with those established in [22], adding only one extra value (collapse state) to evaluate the imminent collapse point that occurs in the nonlinear range of buildings:

• Operational State (O): Generally, an inter-story drift of less than 0.2% of the story height is allowed.
• Moderate Damage State (DM): Allows an inter-story drift in the range of 0.5%.
• Extensive Damage State (DE): Allows an inter-story drift in the range of 1%.
• Collapse State (C): Allows an inter-story drift greater than 2%.

Figure 22 shows the set of fragility curves for each of the damage states compared among the five models of the twelve-story building, for a range of drifts from 0.002, representing the MID at the service or operational limit state, to 0.02, which would represent a collapse limit state, considering various levels of seismic intensity. Similarly, Figure 23 shows the fragility curves for the 5 models of the eight-story building for the same damage limit states.

7. Numerical Results

7.1. Incremental Dynamic Nonlinear Analysis

A comparative analysis was performed for the two groups of buildings (twelve-story and eight-story) between the capacity levels and deformations, as determined from the medians calculated from the capacity curves obtained through the incremental dynamic analyses applied to each model. These comparisons are shown in Figure 24.

Eight-Story Building (M-8_1 to M-8_5):

For low values (100–400 gals), the MID values obtained from the analyses are below 0.003, with means and medians quite close and low variances, suggesting an essentially elastic behavior and very little variation in all configurations. On the other hand, when considering spectral demands from 600–700 gals, the demand of M-8_5 begins to be significantly higher with respect to the rest of the models; in addition, larger deformations at low intensity values tend to increase, which could translate into a potential global brittle failure. The highest demands in all buildings result in values close to 1200–1300 gals, although models M-8_1 to M-8_4 do not show a probable point of incipient collapse as marked as the one obtained for M-8_5.

Twelve-Story Building (M-12_1 to M-12_5):

When examining the interval from 0 to 600 gals, all models show MID values that remain below 0.007, which means that they behave similarly and are close to the elastic range. However, once we exceed this intensity, model M-12_5 starts to lose its slope significantly, which causes a noticeable decrease in the overall structural stiffness. Between 700 and 1000 gals, M-12_5 almost doubles the drifts compared to M-12_1, M-12_2, M-12_3, and M-12_4, and this difference becomes even more pronounced around 1200–1300 gals. On the other hand, M-12_3 shows the lowest MIDs, especially in the collapse zone, indicating that it has a more effective shear wall distribution to manage lateral displacements. At 1500 gals, the peak drifts: M-12_5 reaches around 0.064, which is significantly higher than M-12_3 (at around 0.019). All models maintain, in general, a good structural capacity up to medium intensity levels. M-12_5 shows the least efficient structural behavior, while M-12_3 stands out as the best performer by effectively limiting deformations during high seismic intensities.

7.2. Structural Fragility

Figure 25 shows the structural fragility curves obtained for the two groups of buildings studied. These curves were evaluated for MID of 0.01, which corresponds to an extensive damage state.

Eight-Story Building (M-8_1 to M-8_5):

Looking at the structural fragility linked to an extensive damage state (MID = 0.01) for the eight-story building models (M-8_1 to M-8_5), we find that the probability of exceeding this damage level is below 10% over a wide initial range of spectral accelerations, approximately up to 930–940 cm/s². When higher spectral accelerations are considered, between 1000 and 1200 cm/s², the differences between the models are more noticeable: M-8_5 and M-8_4 show a faster increase in the probability of exceeding MID = 0.01, indicating that they are more vulnerable to extensive damage. For high seismic demands, approximately at 1400–1500 cm/s², M-8_5 almost reaches the probability of 1, while M-8_4 goes beyond 0.65 and M-8_1 is around 0.50. In contrast, M-8_2 and M-8_3 remain below 0.42 and 0.37, respectively, showing that they have lower structural fragility at these intensity levels.

Twelve-Story Building (M-12_1 to M-12_5):

Similarly, for the group of twelve-level structures, it is observed that the probability of passing is very low or zero for intensities below 500 cm/s². In the 1000–1200 cm/s² range, there are high chances for M-12_1 and M-12_4 to pass a small damage limit (0.01), with chances of approximately 0.40. M-12_5; nevertheless, they have a much higher chance, almost doubling to 0.80, to reach this damage limit. Around 1400–1500 cm/s², M-12_5 reaches a probability almost at 2; similarly, M-12_1 and M-12_2, while M-12_3 and M-12_4 close to that probability but remain below the other models.

7.3. Structural Reliability

Structural reliability is assessed using structural demand hazard curves. Figure 26 shows the curves obtained for each of the groups under study. Based on the mean annual exceedance rate, the variation in the probability of exceeding different levels of MID for each of the eight-story models (M-8_1 to M-8_5) is observed. The analysis focuses on the interval 0.002 ≤ MID ≤ 0.02, which clearly distinguishes the structural behavior at low, intermediate, and high seismic demand levels.

Eight-Story Building (M-8_1 to M-8_5):

Model M-8_4 exhibits the highest exceedance rates (a MID of 0.002 is expected to be exceeded every 20 years), while the other models maintain a return period of 66 years for exceeding this limit. When the MID increases to 0.004, M-8_4 remains the most vulnerable (this limit is expected to be exceeded every 127 years), followed by M-8_5 (277 years). Models M-8_1, M-8_2, and M-8_3 show significantly lower values. As the MID increases, M-8_5 demonstrates a trend toward greater structural vulnerability at several points. For instance, when evaluating exceedance probabilities for MID > 0.008, it emerges as the model with the highest exceedance probability.

Twelve-Story Building (M-12_1 to M-12_5):

In the case of the twelve-story building group, considering the exceedance rate for a MID = 0.002, M-12_5 reaches a return period of 19 years for exceeding this limit, significantly shorter than the 45–52 years observed for the other models. This indicates that, even at initial deformation levels, M-12_5 has a high probability of surpassing this limit. For MID values close to 0.006 and 0.008, M-12_5 maintains a lower return period than the other models, with a period of 75 years to reach this damage level.

8. Conclusions

In this research, the structural design of ten buildings was carried out considering RC shear wall frames, divided into two groups: five models with eight stories and five models with twelve stories and different shear wall configurations. Structural reliability was used to verify the design from a probabilistic point of view and in order to observe the influence of shear wall location. The study was conducted on buildings with a 3D regular configuration in both plan and elevation, which are supposed to be located on the soft soil of Mexico City. From the results, the following conclusions can be drawn:

Importance of shear wall location

Throughout the study, it was verified that the location of shear walls has a very important influence on the seismic response of buildings. When they are evenly distributed in the plan—combining exterior walls with symmetrically arranged interior walls (as in models M-8_3 and M-12_3)—deformations are reduced and the probability of damage is decreased, even under higher intensity earthquake motions.

Models with greater damage propensity

Buildings M-8_5 and M-12_5 stood out by exhibiting the highest exceedance values for deformations and damage probabilities, even at seismic demands lower than those presented in the UHS for the fundamental structural period of these models. In these cases, the shear walls were concentrated in the central area of the building, which is a common practice to accommodate elevators and/or installations; however, positioning these elements in the center is not the most favorable option from a structural standpoint, since, to achieve the same level of structural reliability compared to placing the walls symmetrically and on the exterior, the frame system sections must be significantly increased.

Comparison between eight- and twelve-story buildings

Although both groups of models exhibited generally similar behavior, the twelve-story models showed significantly greater differences in their structural response at high seismic intensity levels. This highlights the need to apply more rigorous design criteria as building height increases, as the location and distribution of shear walls become critical factors in determining the overall structural performance.

Design recommendations and future research

• Balanced Distribution: Placing the shear walls in a perimeter or symmetrical arrangement helps to better control deformations; therefore, it reduces the probability of severe damage.
• Probabilistic Approach: Incorporating fragility and seismic hazard evaluations from the outset of the design process enables an estimation of safety throughout the building’s service life.
• New Research Directions: The results presented in this study are based on the evaluation of the structural behavior of ten regular buildings both in plan and elevation. Therefore, future work is encouraged to expand the number of buildings studied, varying the architecture and number of levels to gain a better understanding of the behavior of these types of structures. Additionally, future studies could focus on analyzing the economic impact and architectural feasibility of the different configurations, as well as further exploring the relationship between costs and structural reliability, especially by using advanced ground motion intensity measures.