# Collapse Assessment of Mid-Rise RC Dual Wall-Frame Buildings Subjected to Subduction Earthquakes

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## Abstract

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## 1. Introduction

_{w}8.0 1985 Valparaiso, M

_{w}8.8 2010 Maule, M

_{w}8.2 2014 Iquique, M

_{w}8.3 2015 Illapel, and M

_{w}7.6 2016 Chiloe [1]), and the seismic behavior of these buildings was deemed satisfactory. During the 2010 earthquake, only 2.0% of RC buildings with nine or more stories and only 0.4% of buildings with three or more stories suffered severe damage due to strong shaking [2]. In particular, this event caused only a few partial collapses (e.g., the O’Higgins Tower) and one total collapse (i.e., the Alto Rio building) [3], as shown in Figure 1.

_{c}, was found to be 2.61 × 10

^{−5}and the collapse probability in 50 years, P

_{c}(50), was found to be 0.13%. Cando et al. [15] analyzed the seismic performance of a suite of four code-conforming 20-story shear-wall buildings located in Santiago. The λ

_{c}values were found to range from 2.7 × 10

^{−4}to 6.9 × 10

^{−4}and the values of P

_{c}(50) were found to range from 1.3% to 3.4%. The latter probabilities were not only higher than those calculated by Araya-Letelier et al. [14] but also exceeded the 1% in 50 years target collapse probability indicated in ASCE 7-22 [10]. According to Cando et al. [15], these relatively high P

_{c}(50) values may be related to the assumed soil type (soil type B in [14], soil type C in [15], as defined by the Chilean seismic design code [9]) and small differences in the seismic hazard model. It must be noted that in each of these studies, only a single building archetype located at a particular location on a specific soil type was analyzed. In other words, the potential influence of different seismicity levels and soil types on the collapse probability when buildings are subjected to subduction earthquakes was not accounted for. Moreover, since these studies focused on the performance of tall buildings, the quantitative collapse performance of mid-rise dual wall-frame RC office buildings in Chile remains unclear.

_{c}, P

_{c}(50)) were calculated. Finally, all relevant findings are discussed in Section 6.

## 2. Collapse Assessment Methodology

_{a}(T

_{1}). Based on recent studies, S

_{a}(T

_{1}) is still an adequate predictor of collapse for stiff RC structures whose seismic response is dominated by the fundamental vibration mode [28,29] (e.g., mid-rise RC dual wall-frame buildings). Consequently, for each archetype building model and GM record, nonlinear response history (NLRH) analyses at increasing S

_{a}(T

_{1}) levels were performed until collapse. Further details of this methodology are provided in the following subsections.

#### 2.1. Estimation of S_{a}(T_{1}) Intensities That Trigger Collapse

_{a}(T

_{1}) values at 0.05 g steps until collapse. The sets of subduction GM records represented the aleatory uncertainty of the seismic hazard (called RTR variability), which is the dominant source of uncertainty compared to epistemic uncertainty (i.e., modeling uncertainty) [13]. Still, important epistemic uncertainty, such as damage and material decay due to, for example, climate factors, should be incorporated in future studies. To reduce the significant computational costs, the IDAs were performed only along the shorter horizontal plan direction, i.e., the direction along which the collapse fragility was greater. The lowest S

_{a}(T

_{1}) value at which either the global or local structural collapse criteria were met was recorded for each structural model and each GM. Since earthquake-induced collapse should be evaluated in a probabilistic way, the dispersion of the S

_{a}(T

_{1}) collapse intensity values was calculated to develop the respective building-specific collapse fragility functions.

#### 2.2. Estimation of Collapse Fragility Functions

_{a}(T

_{1}), which is represented by P(C|S

_{a}(T

_{1}), and assume a lognormal probability function (PF) [30,31,32]. These PFs are defined by two parameters: (1) the median (i.e., median collapse capacity, $\widehat{\mathsf{\theta}}$); and (2) the logarithmic standard deviation (i.e., dispersion, $\widehat{\mathsf{\beta}}$). Both parameters were estimated using the Maximum Likelihood Method (MLM) [33]. Then, Kolmogorov–Smirnov (K–S) and Lilliefors goodness-of-fit tests, both at a 5% significance level, were used to evaluate the quality of the lognormal PFs. The K–S test is a distribution-free (non-parametric) test used for continuous distributions [34], whereas the Lilliefors test, recommended by FEMA P-58 [13], is a more severe (compared to the K–S test) test and is recommended when the parameters are not specified but estimated from the sample [35].

#### 2.3. Collapse Performance Metrics: P(C|S_{a}(T_{1})_{MCE}), CMR, λ_{c}, and P_{c}(50)

_{a}(T

_{1})

_{MCE}), is obtained from the collapse fragility curves. This collapse performance metric is used to assess safety considerations, for instance, US code-conforming buildings must not have values greater than 10% [10]. Another collapse performance metric is the Collapse Margin Ratio (CMR), which is defined by FEMA P695 [16] as the median collapse capacity $\widehat{\mathsf{\theta}}$ divided by S

_{a}(T

_{1})

_{MCE}, as shown in Equation (1).

_{c}, which represents the average number of earthquake-induced structural collapses of a given structure per year. This λ

_{c}value is a combination of the product of the collapse fragility and the seismic hazard (Equation (2), where dλ

_{Sa}/dS

_{a}is the derivative of the seismic hazard curve).

_{c}(50), is obtained using Equation (3) and assumes a Poisson process. This performance metric is the collapse potential in 50 years and is also used to analyze safety considerations. For instance, US code-conforming buildings must not have P

_{c}(50) values greater than 1% [10].

#### 2.4. Deaggregation of λc

_{a}(T

_{1}) value to λ

_{c}is obtained through the deaggregation of λ

_{c}. Deaggregated λ

_{c}values based on S

_{a}(T

_{1}) are obtained using Equation (4), whose terms have been previously defined.

## 3. Methodology to Define Code-Conforming Archetype Buildings

#### 3.1. Statistical Evaluation of Chilean RC Buildings

^{2}.

#### 3.2. Structural Layout and Code-Conforming Design Procedure of the Archetype Buildings

^{2}), whereas the beam span lengths are 8.0 m. The underground levels are surrounded by 20 cm thick perimeter walls. Figure 3a shows a typical floor plan view and Figure 3b shows a typical underground floor plan view. Likewise, Figure 3c depicts the lateral transverse view and a 3D view is provided in Figure 3d.

_{c}) of 35 MPa and nominal yield strength (f

_{y}) of 420 MPa, respectively, were assumed. A dead load of 2 kPa and a live load of 3 kPa were used for the underground levels, whereas 5 kPa and 2 kPa were considered for the typical stories and the roof, respectively.

^{eff}). The initial estimate of the response modification factor (R*) depends exclusively on the building’s fundamental period, the structural system, and the soil type. If the base-shear demand, V

_{b}, divided by the seismic weight of the building, W, (normalized base-shear demand, C) satisfies both the upper (C

_{max}) and lower (C

_{min}) limits (i.e., C

_{min}≤ C ≤ C

_{max}), then R

^{eff}= R*. Or else, R* is modified so that C

_{min}≤ C ≤ C

_{max}, and in this case, R

^{eff}is set equal to the modified value of R*. The relevant design factors are summarized in Table 1.

_{min}), which depends on the importance factor (equal to 1.0 due to occupation category II), seismic zone, and soil type. As explained later, the design of the remaining archetypes in the transverse direction (i.e., Bld-07-z2-sD and Bld-07-z3-sD) was controlled by the axial deformation demand on the RC core walls. Figure 4 shows the elastic and reduced code-conforming design spectra (NCh 433) for each archetype building. The vertical dashed lines indicate the fundamental period of vibration (T

_{1}) in the transverse direction (i.e., the direction of analysis). The reduced values of S

_{a}(T

_{1}) used in the design of the structural members are indicated in each figure.

_{a}(T

_{1}) values are indicated in each figure. The UHS and elastic NCh 433 spectra of the archetype buildings on soil type B (Figure 5a,c) have similar spectral shapes but differences in the values of the spectral ordinates are evident. On the other hand, the UHS and elastic code spectra of the archetype buildings on soil type D (Figure 5b,d) have dissimilar spectral shapes but the corresponding spectral ordinates are very close to each other at the fundamental period. It is worth mentioning that the elastic spectra specified in the current Chilean seismic code [8,9] are not probabilistic in nature but calibrated for the structural demands observed in recent earthquakes (1985 and 2010 earthquakes [12]). Therefore, differences between the DBE-UHS and elastic code spectra were expected. The process used to obtain the UHS spectra is explained in Section 4.

_{1}values of 0.92 s, 0.84 s, and 0.91 s, respectively). On the other hand, archetype Bld-07-z3-sD had larger member sections (walls, beams, and columns) and was a significantly stiffer structure (T

_{1}= 0.67 s).

#### 3.3. Modeling Details

_{c}

^{eff}A = 0.1 G

_{c}A, where G

_{c}= 0.4 E

_{c}, A is the cross-section, and E

_{c}is the elastic modulus) [45]. This procedure is an indirect way to consider shear cracking because the shear-wall element captures neither nonlinear shear deformations nor coupling with nonlinear flexural deformations under cyclic loading. Figure 7 presents examples of the constitutive functions of the concrete and steel fibers of the core walls of archetype Bld-07-z3-sD.

_{p}at both ends and a linear-elastic region in between. YULRX backbone functions were also adopted to represent the uniaxial stress–strain relationships between steel and concrete materials. Length L

_{p}was determined following recommendations found in the literature [59] (L

_{p}= 0.5 h, where h is the depth of the element). Geometric nonlinearity was also implemented to consider the P-delta effects in the columns and walls. As indicated in previous studies [60,61], long-span post-tensioned slabs in core-wall buildings were considered so as not to substantially affect the building response. Thus, these elements were not explicitly modeled. In its place, rigid diaphragms were implemented at all floor levels and the self-weight and mass of the slabs were incorporated into the models. Lastly, energy dissipation was mostly modeled directly by the hysteretic force–deformation response of the structural components. Modal damping was set to 2.4% for all modes and the additional Rayleigh damping was set to 0.1% at 0.2 T

_{1}and 1.5 T

_{1}[43,44].

#### 3.4. Seismic Collapse Criteria

_{a}(T

_{1})), were not detected.

#### 3.5. Pushover Analyses

_{b}was evaluated at the ground level (see Figure 3) and normalized by the seismic weight of the stories above. The capacity curves in the positive direction were not equal to those in the negative direction due to the asymmetric plan layout of the core walls (see Figure 3a). Moreover, Figure 9 depicts some points of interest: (1) the first fiber at any wall that reached steel yielding, concrete crushing, shear strength, and steel fracture; and (2) the SDR reaching 5% at any story.

^{max}) was reached at roughly the same value of the RDR (≈1.5~2.0%). Finally, there was always a clear drop in strength after the first concrete crushing when the seismic loads were applied in the positive direction. It is interesting to note that this first concrete crushing occurred at the SBEs of the walls located at the first story, where the shear and moment demands were expected to be more significant, and at RDR values higher than 2.5% (i.e., excessive drift demands).

^{max}to the design base shear V

^{des}, was calculated. Table 3 summarizes the pushover analysis results, where the ratios of V

^{des}(also above ground level) and V

^{max}to the seismic weight are also shown. Along the positive transverse direction, the Ω

^{+}values ranged from 2.4 to 4.4, with an average value of 3.3. Along the negative transverse direction, on the other hand, the Ω

^{−}values ranged from 2.5 to 3.7, with an average value of 3.0. Again, the high values of V

^{max}for archetype Bld-07-z3-sD (2918 tonf for V

^{max+}, and 3149 tonf for V

^{max−}) were consistent with the larger cross-section dimensions of the structural members, resulting in an archetype that had more strength.

## 4. Seismic Hazard Analyses and Selection of Subduction Ground Motions

#### 4.1. Seismic Hazard Analyses

_{a}(T

_{1}) versus the mean annual frequency of exceedance, λ

_{Sa(T1)}, and Figure 11b depicts the S

_{a}(T

_{1}) versus the mean return period, Tr. Horizontal lines are also drawn to represent the traditional λ

_{Sa}hazard levels related to return periods of 2475, 475, and 72 years, i.e., the maximum considered earthquake (MCE), design-basis earthquake (DBE), and service-level earthquake (SLE), respectively.

_{a}(T

_{1}) values for the different hazard levels are presented in Table 4, along with the values of both the Chilean code elastic design spectra, S

_{a}(T

_{1})

^{des,e}, and the reduced design spectra, S

_{a}(T

_{1})

^{des,red}, (see Figure 4). It is noted that there were relatively small differences in terms of the S

_{a}(T

_{1})

_{SLE}, S

_{a}(T

_{1})

_{DBE}, and S

_{a}(T

_{1})

_{MCE}values between the archetypes on soil type B (i.e., Bld-07-z2-sB and Bld-07-z3-sB), which had very similar values of T

_{1}. On the other hand, the S

_{a}(T

_{1}) values for the archetype Bld-07-z3-sD were almost 1.5 times higher than those for archetype Bld-07-z2-sD, indicating a considerable increase in seismic demand. It is interesting to note that the values of S

_{a}(T

_{1})

^{des,e}were different from those of S

_{a}(T

_{1})

_{DBE}. These differences were expected because the elastic design spectra of the Chilean seismic design code NCh 433 [8,9] are not uniform-hazard spectra. In addition, the Chilean code does not explicitly define the SLE and MCE hazard levels. These levels were defined as indicated in ASCE 7-22 [10].

_{a}(T

_{1})

^{des,e}(i.e., spectral ordinates of the Chilean code elastic design spectra). For the archetypes on soil type D, the probabilities of exceedance in 50 years were nearly 10% (Tr ~475 years), which is a worldwide standard for the DBE hazard level. For the archetypes on soil type B, on the other hand, the probabilities of exceedance in 50 years were higher than 22% (Tr ~200 years). In other words, the hazard level associated with the elastic design NCh 433 spectra for soil type B was smaller than that associated with the DBE. These observations are consistent with the previous results shown in Figure 5.

#### 4.2. Selection of Ground Motions

_{a}(T) distribution (with conditional mean and conditional standard deviation at each period within the range of interest). The CS calculations considered the target S

_{a}(T

_{1}) for the 2%—50-year hazard level and the mean causal magnitude M, distance R, and epsilon ε obtained from the PSHA deaggregation, where each GMM was considered separately. The mean and standard deviation of logarithmic S

_{a}(T) also considered the correlation model proposed by Candia et al. [69] for the Chilean subduction zone, where the correlations are generally higher than those for other subduction zones such as Japan. Finally, single GMM calculations were combined using assumed logic-tree weights, following method 2, which was suggested by Lin et al. [70], to obtain the composite CS of each case. Although logic-tree weights are not rigorously correct, it was considered a convenient approximation for this investigation.

_{a}(T), at a period between 0.2 T

_{1}and 2.0 T

_{1}, following the procedure proposed by Baker and Lee [72]. Since NLRH analyses of the 3D archetypes were performed only in the transverse direction, each ground motion was a horizontal component. Following current selection guidelines [16,73], these components met the selection criteria and the objectives of consistency, representativeness, and statistical sufficiency to permit the statistical evaluation of the RTR variability in the structural response. To avoid bias in the probability of collapse when S

_{a}(T

_{1}) was used as the IM, the amplitude scaling factor that modifies the ground motions to achieve the desired intensity level was limited to a maximum value of 5.0, as suggested by recent studies [74,75] where the spectral shape was appropriately accounted for in the selection process.

_{a}(T), was higher for archetypes on soil type D (Figure 12b,d) than for those on soil type B (Figure 12a,c). This variability was inherited from the Chilean GMMs used in the PSHA and may impact the variability of the structural response.

## 5. Response and Collapse Assessment Results

#### 5.1. IDAs, Collapse Fragility Functions, P(C|S_{a}(T_{1})MCE), and CMR

_{a}(T

_{1}) level. As expected, the median PSDRs increased with an increasing return period (i.e., increasing hazard level). For comparison purposes, the response for a ~50,000-year return period is also presented. Although this is an extremely large return period, the corresponding S

_{a}(T

_{1}) values were close to the median S

_{a}(T

_{1}) collapse values (units of gravity, g) that are presented later.

_{a}(T

_{1}) values that triggered a collapse and the collapse mode (i.e., either simulated or non-simulated) were identified and recorded for each ground motion. The collapse modes were found to depend on the archetypes. For instance, more than 60% of collapses of archetypes Bld-07-z2-sB, Bld-07-z3-sB, and Bld-07-z2-sD were due to local collapse criteria. Concrete crushing and, at the same time, steel buckling at the wall boundary elements when the buildings were loaded in the positive transverse direction (i.e., boundary elements in compression) were mostly observed. On average, 40% of collapses occurred in the negative transverse direction (i.e., boundary elements in tension) due to the global collapse criteria (SRD ≥ 5%). On the other hand, 70% of the collapses of archetype Bld-07-z3-sD were due to the global collapse criteria (in either the negative or positive transverse direction).

_{a}(T

_{1}) values. The figures also present the 50th collapse percentile (median), as well as the 16th and 84th collapse percentiles (equal to one logarithmic standard deviation below and above the mean when a lognormal distribution is assumed). Moreover, Figure 14e–h shows the estimated lognormal collapse fragility curves and the values of the S

_{a}(T

_{1}) collapse intensity.

_{a}(T

_{1}) collapse intensity data. In more detail, seismic zone 2 Bld-07-z2-sB had S

_{a}(T

_{1}) collapse values that ranged from 1.40 g to 3.95 g, with $\widehat{\mathsf{\theta}}$ = 2.21 g, whereas Bld-07-z2-sD had S

_{a}(T

_{1}) collapse intensity values ranging from 1.75 g to 6.85 g, with a greater $\widehat{\mathsf{\theta}}$ of 3.49 g. On the other hand, seismic zone 3 Bld-07-z3-sB exhibited S

_{a}(T

_{1}) collapse values that ranged from 1.00 g to 4.00 g, with $\widehat{\mathsf{\theta}}$ = 2.10 g, whereas Bld-07-z3-sD exhibited S

_{a}(T

_{1}) collapse intensities ranging from 2.90 g to 9.70 g, with a significantly greater $\widehat{\mathsf{\theta}}$ of 5.72 g. In particular, the $\widehat{\mathsf{\theta}}$ results were consistent with the seismic design code strength requirement (i.e., $\widehat{\mathsf{\theta}}$ = 2.21 g for Bld-07-z2-sB (V

_{b}

^{des}= 0.05 W) was smaller than $\widehat{\mathsf{\theta}}$ = 5.72 g for Bld-07-z3-sD (V

_{b}

^{des}= 0.10 W)).

_{a}(T

_{1}) collapse values seemed to be rather high, the estimated values of $\widehat{\mathsf{\beta}}$ were either equal to or smaller than those recommended by FEMA P-58 [13].

_{a}(T

_{1})) are presented in Figure 15a and it can be observed that the fragility curves of archetypes Bld-07-z2-sB and Bld-07-z3-sB are quite similar to each other, whereas those of archetypes Bld-07-z2-sD and Bld-07-z3-sD are situated far to the right. At first glance, this observation could suggest that the latter archetype buildings have a superior seismic collapse performance, but it is important to note that for a specified S

_{a}(T

_{1}) value, the corresponding λ

_{Sa(T1)}value may differ significantly for different archetype buildings depending on the fundamental period, seismic zone, and soil type. Thus, a direct comparison of the fragility functions expressed in terms of S

_{a}(T

_{1}) can be misleading [38]. Consequently, Figure 15b shows the collapse fragility curves but plotted as a function of the corresponding λ

_{Sa(T1)}values, where vertical lines at different hazard levels (i.e., probabilities of exceedance equal to 2%, 10%, and 50% in 50 years, or, in other words, return periods of 72, 475 and 2475 years, respectively) are also shown.

_{a}(T

_{1}) was misleading. For instance, the S

_{a}(T

_{1}) collapse fragility curve of archetype Bld-07-z2-sB was located to the left of the remaining curves in Figure 15a (which may have been misunderstood as being inferior performance), whereas the corresponding λ

_{Sa(T1)}collapse fragility curve was located to the right of the other curves, which indicates a superior collapse performance (relative to that of the archetypes on soil type D). Although fragility functions plotted as a function of λ

_{Sa(T1)}may not provide sufficient information to quantitatively rank the earthquake-induced collapse performance of different buildings (as opposed to the information provided by λ

_{c}, for example), this analysis provides a useful tool to compare different buildings to each other.

#### 5.2. Values of λ_{c} and P_{c}(50)

_{c}and P

_{c}(50) are presented in Table 6 and it can be observed that these values ranged from 2.17 × 10

^{−5}to 7.24 × 10

^{−5}and from 0.11% to 0.36%, respectively. The estimated values of P

_{c}(50) were small and consistent with the seismic response of modern Chilean RC buildings empirically observed in recent earthquakes. In addition, the target maximum probability of collapse of 1% in 50 years indicated by ASCE 7-22 [10] was achieved by each archetype building and, therefore, post-2010 Chilean RC mid-rise dual wall-frame buildings are expected to reach the collapse prevention limit state (at least at the locations and soil types considered in this study).

_{c}(50) values. For soil type B, the P

_{c}(50) value for seismic zone 3 was higher than that for seismic zone 2. Specifically, the P

_{c}(50) values for archetypes Bld-07-z3-sB and Bld-07-z2-sB were 0.31% and 0.11%, respectively. In contrast, for soil type D, the P

_{c}(50) value for seismic zone 3 was smaller than that for seismic zone 2. Specifically, values of P

_{c}(50) for archetypes Bld-07-z3-sD and Bld-07-z2-sD were 0.18% and 0.36%, respectively. Regarding the influence of the soil type, the P

_{c}(50) values for archetypes Bld-07-z2-sB and Bld-07-z2-sD were 0.11% and 0.36%, respectively, whereas those for archetypes Bld-07-z3-sB and Bld-07-z3-sD P

_{c}(50) were 0.31% and 0.18%, respectively. Although the effect of the seismic zone and soil type on the P

_{c}(50) values may appear counter-intuitive (even chaotic), it is important to highlight that Chilean seismic design codes are mostly prescriptive and do not include explicit PBEE design targets. Given that normalized design base shears, V

^{des}/W, for the archetype buildings on soil type D were higher than those for the archetype buildings on soil type B (see Table 3), higher P

_{c}(50) values for the soil type D archetype buildings seemed to indicate that the difference between the actual demand and the design demand was greater on soil type D than on soil type B. Since current Chilean design codes are prescriptive and lack PBEE design targets (such as uniform collapse risk on different soil types), the substantial differences observed in the P

_{c}(50) values may be expected (which does not make them less unacceptable).

#### 5.3. Deaggregation Values of λ_{c}

_{c}based on the S

_{a}(T

_{1}) intensity values are shown in Figure 16a, and it can be observed that the areas below the curves for archetypes Bld-07-z2-sD and Bld-07-z3-sB are significantly greater than those for archetypes Bld-07-z2-sB and Bld-07-z3-sD, which was expected because these areas represent the values of λ

_{c}that were summarized in Table 6. Moreover, the deaggregation curve of archetype Bld-07-z3-sB is located more to the left, which indicates that the contribution of small S

_{a}(T

_{1}) values to λ

_{c}was greater for this archetype building than for the remaining buildings. Consequently, archetype Bld-07-z3-sB was more susceptible (in terms of collapse during its lifetime) to small/medium S

_{a}(T

_{1}) intensities. In contrast, the deaggregation curve of archetype Bld-07-z3-sD is located more to the right, which indicates that the contribution of high S

_{a}(T

_{1}) values to λ

_{c}was greater for this archetype building than for the remaining buildings. Consequently, archetype Bld-07-z3-sD was more susceptible (in terms of collapse during its lifetime) to high S

_{a}(T

_{1}) intensities. Figure 16a also shows three sets of lines that indicate the values of $\widehat{\mathsf{\theta}}$, values of S

_{a}(T

_{1}) at 50% of λ

_{c}, and values of S

_{a}(T

_{1}) at 75% of λ

_{c}.

_{a}(T

_{1}) intensity values at 50% of λ

_{c}but were always smaller than the S

_{a}(T

_{1}) intensity values at 75% of λ

_{c}. For instance, for archetype Bld-07-z2-sD, $\widehat{\mathsf{\theta}}$ = 3.49 g was 28% higher than the S

_{a}(T

_{1}) value at 50% of λ

_{c}(= 2.73 g), whereas for archetype Bld-07-z3-sB, $\widehat{\mathsf{\theta}}$ = 2.10 g was 17% higher than the S

_{a}(T

_{1}) value at 50% of λ

_{c}(= 1.80 g). Although the precise characterization of the collapse fragility functions is necessary for the entire range of the S

_{a}(T

_{1}) intensity values, these observations indicate that this characterization is needed more at S

_{a}(T

_{1}) intensities smaller than $\widehat{\mathsf{\theta}}$ because these S

_{a}(T

_{1}) values contribute the most to λ

_{c}(and, as a result, also to P

_{c}(50)), which is in agreement with previous studies [22,38].

_{c}deaggregation curves for different archetype buildings. The deaggregation curves of λ

_{c}as a function of the corresponding λ

_{Sa(T1)}values are shown in Figure 16b, where vertical lines indicate the previously defined λ

_{Sa(T1)}hazard levels. At the SLE level, each archetype building presents small (almost negligible) values of deaggregated λ

_{c}and this is in agreement with the observed seismic performance of Chilean RC dual wall-frame buildings in recent earthquakes. Regarding the DBE level, apart from archetype Bld-07-z2-sD, the remaining archetype buildings exhibited negligible values of deaggregated λc, whereas at the MCE level, only the archetype Bld-07-z2-sB showed negligible values of deaggregated λ

_{c}. As seen in Figure 16b, the deaggregation curves of archetypes Bld-07-z2-sD and Bld-07-z3-sB are located more to the left, which shows that the influence of high λ

_{Sa(T1)}values on λ

_{c}was greater for these archetype buildings than for the remaining ones, indicating that the former archetype buildings were more susceptible (in terms of collapse during their lifetime) to more frequent ground motions (small and medium intensities). It can also be observed that, again, the influence of the seismic zone and soil type on the deaggregated λ

_{c}curves is unclear. For example, at λ

_{Sa(T1)}= 10

^{−4}, the area under the deaggregation curve is 42.1% of λ

_{c}for archetype Bld-z2-sD but only 25.7% of λ

_{c}for archetype Bld-z3-sD (see Table 6). On the other hand, at λ

_{Sa(T1)}= 10

^{−4}, the area under the deaggregation curve is 5.4% of λ

_{c}for archetype Bld-z2-sB but 32.6% of λ

_{c}for archetype Bld-z3-sB (see Table 6). Therefore, it cannot be concluded that archetype buildings located in a high seismicity zone are less susceptible to collapse during their lifetime (e.g., 50 years) to more recurrent ground motions than those in a moderate seismicity zone. As a summary, Figure 17 shows a bar plot, where the λ

_{c}and P

_{c}(50) values are plotted for the four archetype buildings.

## 6. Summary and Closing Remarks

_{w}8.8 2010 Chilean earthquake. This group of archetype buildings is characterized by two site locations (i.e., a high seismic zone and a moderate seismic zone, which are denoted as seismic zones 3 and 2, respectively), two soil types (B and D, where the latter is less stiff than the former), and one building height (i.e., 7 stories). The archetype buildings use the naming convention of Bld-07-zX-sY, where X is the seismic zone (i.e., 2 or 3) and Y is the soil type (i.e., B or D). The assessment of the collapse performance was obtained by implementing the latest advances in PBEE proposed by the PEER Center, following the well-established FEMA P-58 methodology. The seismic collapse assessment was evaluated through 3D nonlinear finite-element models subjected to IDAs using 44 carefully chosen and scaled Chilean subduction ground-motion records. The collapse performance of the archetype buildings was characterized by the estimation of (1) the collapse fragility functions; (2) the probability of collapse at the Maximum Considered Earthquake (MCE) intensity, P(C|S

_{a}(T

_{1})

_{MCE}); (3) the collapse margin ratio, CMR; (4) the mean annual frequency of collapse, λ

_{c}; (5) the probability of collapse in 50 years, P

_{c}(50); and (6) the deaggregation of λ

_{c}. In summary, the following conclusions can be drawn:

- The lognormal distribution is an adequate representation of the collapse fragility function of the archetype buildings since each collapse fragility function passed the Kolmogorov–Smirnov and Lilliefors goodness-of-fit tests, both at a 5% significance level. Other distribution functions can be tested in further studies.
- The estimated median collapse ($\widehat{\mathsf{\theta}}$) and logarithmic dispersion ($\widehat{\mathsf{\beta}}$) values were 2.21 g, 3.49 g, 2.10 g, and 5.72 g, and 0.28, 0.40, 0.35, and 0.36, for archetype buildings Bld-07-z2-sB, Bld-07-z2-sD, Bld-07-z3-sB, and Bld-07-z3-sD, respectively.
- The analysis and comparison of the collapse fragilities based on λ
_{Sa(T1)}(as an alternative to S_{a}(T_{1})) generated meaningful information since adequate direct comparisons of the conditional collapse probability can be obtained at different hazard levels such as the SLE, 50%—50 years; DBE, 10%—50 years; and MCE, 2%—50 years. Specifically, all archetype buildings had a negligible collapse probability at the SLE and DBE levels. At the MCE level, the archetype buildings Bld-07-z2-sD and Bld-07-z3-sB had non-negligible collapse probabilities (2.78% and 1.58%, respectively). However, all these MCE collapse probabilities were smaller than the 10% target defined in ASCE 7-22. Consequently, current Chilean seismic design standards seem to provide adequate levels of collapse prevention, which is in agreement with the observed performance of modern RC buildings in recent earthquakes. - The calculated values of the CMR (i.e., $\widehat{\mathsf{\theta}}$ divided by the MCE intensity) ranged from 2.1 to 2.7, with a mean of 2.3. This mean value was higher (i.e., lower collapse risk) than the values stated in previous studies on RC frame buildings designed using US seismic codes and subjected to crustal ground motions. No clear influence of the seismic zone and soil type on the CMR was identified.
- The estimated values of λ
_{c}and P_{c}(50) were 2.17 × 10^{−5}, 7.24 × 10^{−5}, 6.31 × 10^{−5}, and 3.56 × 10^{−5}, and 0.11%, 0.36%, 0.31%, and 0.18% for archetypes Bld-07-z2-sB, Bld-07-z2-sD, Bld-07-z3-sB, and Bld-07-z3-sD, respectively. The values of P_{c}(50) largely fulfilled the maximum 1% target collapse probability in 50 years stated by ASCE 7-22, indicating, once more, the adequate level of collapse prevention provided by current Chilean seismic design standards. - Non-negligible differences were found between the values of P
_{c}(50) for the different archetype buildings but no clear influence of the seismic zone and soil type on P_{c}(50) was observed. Although Chilean seismic design codes are mostly prescriptive and do not include explicit PBEE design targets, the collapse risk should nevertheless be more uniform, suggesting that current Chilean seismic design codes (particularly the design spectra) might require a revision. - Lastly, the deaggregation of λ
_{c}(as a function of λ_{Sa(T1)}, which allows adequate direct comparisons between the deaggregation functions) showed that the values of $\widehat{\mathsf{\theta}}$ were always higher than the S_{a}(T_{1}) values at 50% of λ_{c}. In other words, in terms of the collapse risk, the contribution of the S_{a}(T_{1}) values at the S_{a}(T_{1}) < $\widehat{\mathsf{\theta}}$ range was greater than that of the S_{a}(T_{1}) values at the S_{a}(T_{1}) > $\widehat{\mathsf{\theta}}$ range. This observation means that, contrary to intuition, the accurate characterization of collapse fragilities is more important at the S_{a}(T_{1}) < $\widehat{\mathsf{\theta}}$ range than at the S_{a}(T_{1}) > $\widehat{\mathsf{\theta}}$ range.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Buildings damaged by the M

_{w}8.8 2010 Chilean earthquake: (

**a**) O’Higgins office building; (

**b**) Alto Rio residential building.

**Figure 2.**(

**a**) Chilean RC buildings 2002–2020: number of office buildings and number of stories. (

**b**) Definition of high-rise building from IBC-2021 [37]. MR-Region = Metropolitan Region; V-Region = Valparaiso Region.

**Figure 3.**Archetype buildings. (

**a**) Typical floor plan view. (

**b**) Underground floor plan view. (

**c**) Lateral transverse view. (

**d**) 3D view.

**Figure 4.**Comparison of elastic NCh 433 and reduced design spectra (transverse direction). (

**a**) Bld-07-z2-sB; (

**b**) Bld-07-z2-sD; (

**c**) Bld-07-z3-sB; (

**d**) Bld-07-z3-sD. Note: in all subfigures, dashed lines and continuous lines represent the elastic NCh433 spectrum and reduced spectrum, respectively.

**Figure 5.**Comparison of DBE-UHS and elastic NCh 433 spectra. (

**a**) Bld-07-z2-sB; (

**b**) Bld-07-z2-sD; (

**c**) Bld-07-z3-sB; (

**d**) Bld-07-z3-sD. Note: in all subfigures, dot lines and dashed lines represent the DBE-UHS spectrum and the elastic NCh433 spectrum, respectively.

**Figure 7.**Archetype Bld-07-z3-sD: material models for the shear walls. (

**a**) Confined concrete. (

**b**) Unconfined concrete. (

**c**) Steel. (

**d**) Wall shear. SBE = special boundary element; OBE = ordinary boundary element; h = story height; e = number of vertical shear-wall elements per story.

**Figure 9.**Capacity curves (pushover). (

**a**,

**b**) Base shear vs. roof displacement. (

**c**,

**d**) Base-shear coefficient versus roof-drift ratio.

**Figure 10.**Strains at the boundary elements of archetype Bld-07-z2-sB (pushover along the positive transverse direction) (

**a**) at V

_{b}

^{max}(RDR ≈ 1.5%), and (

**b**) at the first concrete crushing point R (RDR = 3.0%). (

**c**) Schematic YULRX curve: color scale.

**Figure 11.**Seismic hazard curves plotted as a function of (

**a**) the annual probability of exceedance, λ

_{Sa(T1)}, and (

**b**) the return period, Tr.

**Figure 12.**Ground-motion selection. (

**a**) Bld-07-z2-sB; (

**b**) Bld-07-z2-sD; (

**c**) Bld-07-z3-sB; (

**d**) Bld-07-z3-sD. Note: the 3rd, 4th, 5th and 6th series in the legend of subfigure (

**a**) also apply for the other subfigures.

**Figure 13.**Median values of Peak Story Drift Ratios (PSDR). (

**a**) Bld-07-z2-sB; (

**b**) Bld-07-z2-sD; (

**c**) Bld-07-z3-sB; (

**d**) Bld-07-z3-sD.

**Figure 14.**IDA results: (

**a**) Bld-07-z2-sB; (

**b**) Bld-07-z2-sD; (

**c**) Bld-07-z3-sB; (

**d**) Bld-07-z3-sD. Collapse simulation results and estimated collapse fragility functions: (

**e**) Bld-07-z2-sB; (

**f**) Bld-07-z2-sD; (

**g**) Bld-07-z3-sB; (

**h**) Bld-07-z3-sD. Note: the 1st, 2nd and 4th series in the legend of subfigure (

**a**) also apply for subfigures (

**b**–

**d**); and the 2nd, 3rd and 4th series in the legend of subfigure (

**e**) also apply for subfigures (

**f**–

**h**).

Archetype | Transverse Direction | Longitudinal Direction | C_{min} | ||||||
---|---|---|---|---|---|---|---|---|---|

T (s) | R* | R^{eff} | C (%) | T (s) | R* | R^{eff} | C (%) | (%) | |

Bld-07-z2-sB | 0.923 | 8.6 | 6.5 | 5.0 | 0.465 | 6.2 | 6.2 | 6.0 | 5.0 |

Bld-07-z2-sD | 0.835 | 6.1 | 6.1 | 7.0 | 0.426 | 3.7 | 3.7 | 11.2 | 6.0 |

Bld-07-z3-sB | 0.913 | 8.6 | 6.5 | 6.7 | 0.464 | 6.2 | 6.2 | 8.1 | 6.7 |

Bld-07-z3-sD | 0.673 | 5.9 | 5.9 | 9.8 | 0.357 | 3.6 | 3.6 | 15.7 | 8.0 |

^{eff}= effective response modification factor; C = normalized base-shear demand.

Archetype | Core Walls | Beams | Columns | ||||
---|---|---|---|---|---|---|---|

Flanges | Webs | (b × h) | (b × h) | ||||

l | t | l | l | us3–s2° | s3°–s7° | ||

Bld-07-z2-sB | 4.2 | 0.35 | 16.0 | 0.25 | 0.6 × 0.5 | 0.7 × 0.7 | 0.6 × 0.6 |

Bld-07-z2-sD | 4.2 | 0.45 | 16.0 | 0.30 | 0.6 × 0.5 | 0.7 × 0.7 | 0.6 × 0.6 |

Bld-07-z3-sB | 4.2 | 0.35 | 16.0 | 0.25 | 0.6 × 0.5 | 0.7 × 0.7 | 0.6 × 0.6 |

Bld-07-z3-sD | 4.2 | 0.55 | 16.0 | 0.40 | 0.7 × 0.6 | 1.0 × 1.0 | 0.8 × 0.8 |

Archetype | V^{des} | V^{des}/W | V^{max+} | V^{max+}/W | Ω^{+} | V^{max−} | V^{max−}/W | Ω^{−} |
---|---|---|---|---|---|---|---|---|

(tonf) | (%) | (tonf) | (%) | (tonf) | (%) | |||

Bld-07-z2-sB | 348 | 4.7 | 1524 | 20.4 | 4.4 | 1283 | 17.2 | 3.7 |

Bld-07-z2-sD | 826 | 10.7 | 2036 | 26.3 | 2.5 | 2038 | 26.3 | 2.5 |

Bld-07-z3-sB | 464 | 6.2 | 1827 | 24.4 | 3.9 | 1495 | 20.0 | 3.2 |

Bld-07-z3-sD | 1231 | 15.1 | 2918 | 35.7 | 2.4 | 3149 | 38.5 | 2.6 |

Archetype | S_{a}(T_{1})_{SLE} (g) | S_{a}(T_{1})_{DBE} (g) | S_{a}(T_{1})_{MCE} (g) | S_{a}(T_{1})^{des,red} (g) | S_{a}(T_{1})^{des,e} (g) | P(%) in 50 y | Tr (y) |
---|---|---|---|---|---|---|---|

Bld-07-z2-sB | 0.16 | 0.41 | 0.81 | 0.040 | 0.261 | 24 | 180 |

Bld-07-z2-sD | 0.34 | 0.86 | 1.63 | 0.155 | 0.943 | 8 | 600 |

Bld-07-z3-sB | 0.19 | 0.52 | 0.99 | 0.054 | 0.348 | 22 | 200 |

Bld-07-z3-sD | 0.55 | 1.31 | 2.36 | 0.233 | 1.377 | 9 | 530 |

Archetype | $\widehat{\mathsf{\theta}}$ | $\widehat{\mathsf{\beta}}$ | K–S | Lilliefors | P(C|S_{a}(T_{1})_{MCE}) | CMR |
---|---|---|---|---|---|---|

(g) | Test? | Test? | (%) | |||

Bld-07-z2-sB | 2.21 | 0.28 | Pass | Pass | 0.02 | 2.7 |

Bld-07-z2-sD | 3.49 | 0.40 | Pass | Pass | 2.78 | 2.1 |

Bld-07-z3-sB | 2.10 | 0.35 | Pass | Pass | 1.58 | 2.1 |

Bld-07-z3-sD | 5.72 | 0.36 | Pass | Pass | 0.66 | 2.4 |

Archetype | λ_{c} (1/Year) | P_{c}(50) (%) | λ_{c}(λ_{Sa(T1)} = 10^{−4})/λ_{c} (%) |
---|---|---|---|

Bld-07-z2-sB | 2.17 × 10^{−5} | 0.11 | 5.4 |

Bld-07-z2-sD | 7.24 × 10^{−5} | 0.36 | 42.1 |

Bld-07-z3-sB | 6.31 × 10^{−5} | 0.31 | 32.6 |

Bld-07-z3-sD | 3.56 × 10^{−5} | 0.18 | 25.7 |

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## Share and Cite

**MDPI and ACS Style**

Gallegos, M.F.; Araya-Letelier, G.; Lopez-Garcia, D.; Parra, P.F.
Collapse Assessment of Mid-Rise RC Dual Wall-Frame Buildings Subjected to Subduction Earthquakes. *Buildings* **2023**, *13*, 880.
https://doi.org/10.3390/buildings13040880

**AMA Style**

Gallegos MF, Araya-Letelier G, Lopez-Garcia D, Parra PF.
Collapse Assessment of Mid-Rise RC Dual Wall-Frame Buildings Subjected to Subduction Earthquakes. *Buildings*. 2023; 13(4):880.
https://doi.org/10.3390/buildings13040880

**Chicago/Turabian Style**

Gallegos, Marco F., Gerardo Araya-Letelier, Diego Lopez-Garcia, and Pablo F. Parra.
2023. "Collapse Assessment of Mid-Rise RC Dual Wall-Frame Buildings Subjected to Subduction Earthquakes" *Buildings* 13, no. 4: 880.
https://doi.org/10.3390/buildings13040880