Impact of Probabilistic Modeling Alternatives on the Seismic Fragility Analysis of Reinforced Concrete Dual Wall–Frame Buildings towards Resilient Designs
Abstract
:1. Introduction
2. Alternative PDFs to Generate Fragility Functions
2.1. Lognormal Distribution
2.2. Weibull Distribution
2.3. Gumbel Distribution
2.4. Gamma Distribution
2.5. Cauchy Distribution
2.6. Inverse Weibull Distribution
2.7. Inverse Gamma Distribution
3. Development of Fragility Functions of RC Buildings
3.1. Fitting of Fragility Functions Procedure
3.2. Assessment of the Quality of the Fragility Functions
3.2.1. GoodnessofFit Tests
3.2.2. Identification of Data Outliers
3.2.3. Comparison of Estimated Parameters
3.2.4. Comparison of Alternative PDFs
4. Impact of Alternative Fragility Functions on Annualized PerformanceBased Metrics
Metrics Variability
5. Conclusions
 Even though the Lognormal PDF is, by far, the most widely used PDF within PerformanceBased Earthquake Engineering (PBEE), there are other PDFs, such as Weibull or Gamma, that are technically valid (e.g., positiveskewed distribution) to represent fragility functions, that pass goodnessoffit tests, but that provide significantly different adjustments in some part of the fragility functions (e.g., the Lognormal PDF usually adjusts better to the data in the lower part of the fragility function, whereas the Weibull PDF often adjusts better to the data in the upper part of the fragility function. Actually, a Lognormal PDF is not always the best representation of the data).
 Even though the simulated data used in this study passed all alternative goodnessoffit tests, the Lilliefors test always resulted in the stricter test for the same confidence level when compared with the Kolmogorov–Smirnov (K–S) and Anderson–Darling (A–D) tests. It is worth noting that several studies have addressed PBEE use of the K–S, which is a more relaxed test that might lead to fragility functions that the Lilliefors test could have rejected.
 Nevertheless, the use of outlier detection methods to remove unusual data and the subsequent estimation of new parameters (when the difference between previous and posterior parameters differs by more than 20%) is recommended by FEMA P58; several studies within PBEE do not even mention the application of this process. In this study, several PDFs presented differences larger than 20% (never the Lognormal PDF, but still showed a −19.21% difference in one case), highlighting the need to implement these outlier detection methods. When comparing past and new estimated parameters after the elimination of data outliers, there is no correlation between which PDF is the one that exhibits the lower or higher variability. This variability is also not explained by a correlation between the leverage or Peirce’s criteria.
 The leverage criterion tends to be more sensitive to the suspicious data of the lower tails of the fragility functions. In contrast, Peirce’s criterion tends to be more sensitive to the upper tail of the fragility functions. The latter effects might generate significant differences, since lower values (located at the lower tail of the fragility functions) tend to affect more annualized metrics associated with intensity measures that occur more often. The latter explains why applying the leverage criterion exhibits the most variability in the assessment measures.
 The use of information criteria allows the quantitative evaluation of alternative PDF based on their adjustment to data, and there is no correlation between bestranked fitted functions for the eight buildings used in this study. In particular, the Cauchy PDF is always the worst function. At the same time, the Inverse Gamma and Inverse Weibull usually fit better after eliminating the data outlier for the case of collapse LS. However, the Mixture PDF proposed in this study (a combination of a Lognormal PDF and Weibull PDF from 0 to 50% and from 50% to 100%, respectively) tends to fit better for midrise RC buildings for ServiceLevel LS.
 Finally, it is essential to highlight that even though there have been several efforts to advance to more accurate experimental and numerical methods within PBEE to better predict the performance of our built environment (in terms of casualties, environmental impacts, economic losses, and downtime) fewer efforts have been placed on assessing alternative probabilistic methods, whose impacts can be even more significant than some differences generated by some, for instance, structural modeling methods. Consequently, this study recommends the use of multicomparative analyses for probabilistic model selection (including the alternative outlier detection methods, the PDFs, and the information criteria analyses proposed in this study, except the Cauchy PDF) on a casetocase basis for extreme LS, such as collapse. These extreme LSs are characterized by large variability (high nonlinear behavior), and significant differences are detected when applying alternative probabilistic methods. Thus, the implementation of these computationally very lowcost intensive probabilistic methods, when compared with other computational methods used in PBEE (e.g., finite element models or IDAs), is recommended to provide the best fitting to each particular buildingspecific data set to obtain more accurate PBEE estimations. In the case of the SL LS, since the differences obtained from applying these alternative probabilistic methods have significantly lower impacts, the step of multicomparative analysis for modelselection assessment could be skipped compared with the collapse LS.
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
A–D  Anderson–Darling 
AIC  Akaike Information Criteria 
BIC  Bayesian Information Criteria 
B072B  7story building located in seismic zone 2 on soil type B 
B072D  7story building located in seismic zone 2 on soil type D 
B073B  7story building located in seismic zone 3 on soil type B 
B073D  7story building located in seismic zone 3 on soil type D 
B162B  16story building located in seismic zone 2 on soil type B 
B162D  16story building located in seismic zone 2 on soil type D 
B163B  16story building located in seismic zone 3 on soil type B 
B163D  16story building located in seismic zone 3 on soil type D 
CDF  Cumulative Distribution Function 
DS  Damage State 
EDP  Engineering Demand Parameter 
IDA  Incremental Dynamic Analysis 
IM  Intensity Measure 
K–S  Kolmogorov–Smirnov goodnessoffit test 
LevC.  Leverage criterion for data outlier 
Lill.  Lilliefors goodnessoffit test 
LS  Limit State 
MLM  Maximum Likelihood Method 
SL  Service Level 
PBEE  PerformanceBased Earthquake Engineering 
PrcC.  Peirce’s criterion for data outlier 
Probability Distribution Function  
PFA  Peak Floor Acceleration 
PSDR  Peak Story Drift Ratio 
RC  Reinforced Concrete 
RDR  Roof Drift Ratio 
Notation list  
${\mathrm{S}}_{\mathrm{a}}\left({\mathrm{T}}_{1}\right)$  Spectral acceleration ordinate at the fundamental period of the structure 
$\mathrm{P}\left(\mathrm{LS}{\mathrm{S}}_{\mathrm{a}}\left({\mathrm{T}}_{1}\right)\right)$  Probability of a specific limit state conditioned on the spectral pseudo acceleration 
${\mathsf{\lambda}}_{\mathrm{LS}}$  Mean annual frequency of a specific limit state 
${\mathrm{P}}_{\mathrm{LS}}\left(50\right)$  Probability of exceeding a specific limit state in a time of 50 years 
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Name  1st Parameter  2nd Parameter  Equation  

Lognormal ✓  ${\mathrm{P}(\mathrm{D}\mathrm{S}\ge \mathrm{d}\mathrm{s}}_{\mathrm{i}}\left{\mathrm{S}}_{\mathrm{a}}\left({\mathrm{T}}_{1}\right)=\mathrm{x}\right)=\mathsf{\Phi}\left(\frac{\mathrm{l}\mathrm{n}(\mathrm{x}/{\mathsf{\theta}}_{\mathrm{i}})}{{\mathsf{\beta}}_{\mathrm{i}}}\right)$  ${\mathsf{\theta}}_{\mathrm{i}}=$ median  ${\mathsf{\beta}}_{\mathrm{i}}=$ logarithmic standard deviation  (1) 
Weibull ✓  ${\mathrm{P}(\mathrm{D}\mathrm{S}\ge \mathrm{d}\mathrm{s}}_{\mathrm{i}}\left{\mathrm{S}}_{\mathrm{a}}\left({\mathrm{T}}_{1}\right)=\mathrm{x}\right)=1{\mathrm{e}}^{{\left(\mathrm{x}/{\mathrm{b}}_{\mathrm{i}}\right)}^{{\mathrm{a}}_{\mathrm{i}}}}$  ${\mathrm{a}}_{\mathrm{i}}=$ shape  ${\mathrm{b}}_{\mathrm{i}}=$ scale  (2) 
Gumbel ✓  ${\mathrm{P}(\mathrm{D}\mathrm{S}\ge \mathrm{d}\mathrm{s}}_{\mathrm{i}}\left{\mathrm{S}}_{\mathrm{a}}\left({\mathrm{T}}_{1}\right)=\mathrm{x}\right)=\mathrm{e}\mathrm{x}\mathrm{p}\left({\mathrm{e}}^{{(\mathrm{x}{\mathsf{\mu}}_{\mathrm{i}})}^{/{\mathrm{b}}_{\mathrm{i}}}}\right)$  ${\mathsf{\mu}}_{\mathrm{i}}=$ location  ${\mathrm{b}}_{\mathrm{i}}=$ scale  (3) 
Gamma ✓  ${\mathrm{P}(\mathrm{D}\mathrm{S}\ge \mathrm{d}\mathrm{s}}_{\mathrm{i}}\left{\mathrm{S}}_{\mathrm{a}}\left({\mathrm{T}}_{1}\right)=\mathrm{x}\right)=\frac{\mathsf{\gamma}({\mathrm{a}}_{\mathrm{i}},{\mathrm{b}}_{\mathrm{i}}\mathrm{x})}{\mathsf{\Gamma}\left({\mathrm{a}}_{\mathrm{i}}\right)}$  ${\mathrm{a}}_{\mathrm{i}}=$ shape  ${\mathrm{b}}_{\mathrm{i}}=$ rate  (4) 
Cauchy ✓✓  ${\mathrm{P}(\mathrm{D}\mathrm{S}\ge \mathrm{d}\mathrm{s}}_{\mathrm{i}}\left{\mathrm{S}}_{\mathrm{a}}\left({\mathrm{T}}_{1}\right)=\mathrm{x}\right)=\frac{1}{2}+\frac{1}{\mathsf{\pi}}\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\left(\frac{\mathrm{x}{\mathsf{\mu}}_{\mathrm{i}}}{{\mathrm{b}}_{\mathrm{i}}}\right)$  ${\mathsf{\mu}}_{\mathrm{i}}=$ location  ${\mathrm{b}}_{\mathrm{i}}=$ scale  (5) 
Inverse Weibull ✓✓  ${\mathrm{P}(\mathrm{D}\mathrm{S}\ge \mathrm{d}\mathrm{s}}_{\mathrm{i}}\left{\mathrm{S}}_{\mathrm{a}}\left({\mathrm{T}}_{1}\right)=\mathrm{x}\right)={\mathrm{e}}^{{\left(\mathrm{x}{\mathrm{b}}_{\mathrm{i}}\right)}^{{\mathrm{a}}_{\mathrm{i}}}}$  ${\mathrm{a}}_{\mathrm{i}}=$ shape  ${\mathrm{b}}_{\mathrm{i}}=$ scale  (6) 
Inverse Gamma ✓✓  ${\mathrm{P}(\mathrm{D}\mathrm{S}\ge \mathrm{d}\mathrm{s}}_{\mathrm{i}}\left{\mathrm{S}}_{\mathrm{a}}\left({\mathrm{T}}_{1}\right)=\mathrm{x}\right)=\frac{\mathsf{\Gamma}({\mathrm{a}}_{\mathrm{i}},1/{\mathrm{b}}_{\mathrm{i}}\mathrm{x})}{\mathsf{\Gamma}\left({\mathrm{a}}_{\mathrm{i}}\right)}$  ${\mathrm{a}}_{\mathrm{i}}=$ shape  ${\mathrm{b}}_{\mathrm{i}}=$ scale  (7) 
Requirement  FEMA P581 (Lognormal PDF only)  This Study (All PDFs) 

Peer reviewed  Yes  Yes 
Number of specimens  ≥5  44 
Goodnessoffitness  Lilliefors test (α = 5%) ^{1}  Lill. test (α = 2.5%, α = 5%) K–S test (α = 2.5%, α = 5%) A–D (α = 2.5%, α = 5%) 
Elimination of outliers  Peirce’s criterion  Peirce’s criterion Leverage criterion 
Reestimation of PDF parameters after removal of outliers 


Outliers Criteria  B072B  B072D  B073B  B073D  

1st P  2nd P  1st P  2nd P  1st P  2nd P  1st P  2nd P  
Peirce  Lognormal  −4.51%  −13.01%  0.00%  0.00%  −4.03%  −9.48%  −1.58%  −2.94% 
Weibull  26.23%  −7.01%  0.00%  0.00%  23.36%  −6.42%  6.02%  −2.22%  
Mixt (LN + Wbl)  −4.51%  −13.01%  0.00%  0.00%  −4.03%  −9.48%  −1.58%  −2.94%  
Gumbel  −12.08%  −3.64%  0.00%  0.00%  −10.74%  −3.50%  −3.77%  −1.40%  
Gamma  37.02%  45.44%  0.00%  0.00%  26.40%  34.00%  6.80%  8.97%  
Cauchy  −1.58%  −10.24%  0.00%  0.00%  −0.91%  −6.40%  −0.97%  −3.70%  
Inverse Weibull  5.90%  −3.05%  0.00%  0.00%  5.39%  −2.63%  1.89%  −1.13%  
Inverse Gamma  26.43%  22.03%  0.00%  0.00%  17.45%  14.13%  5.30%  4.02%  
Leverage  Lognormal  1.99%  −19.21%  0.00%  0.00%  −2.17%  −13.73%  5.94%  −15.46% 
Weibull  16.38%  −0.54%  0.00%  0.00%  27.64%  −5.30%  11.12%  3.57%  
Mixt (LN + Wbl)  1.99%  −19.21%  0.00%  0.00%  −2.17%  −13.73%  5.94%  −15.46%  
Gumbel  −21.74%  3.29%  0.00%  0.00%  −13.25%  −1.39%  −11.76%  6.81%  
Gamma  46.35%  45.77%  0.00%  0.00%  37.79%  44.07%  33.56%  28.18%  
Cauchy  0.83%  −8.25%  0.00%  0.00%  0.21%  −9.10%  3.20%  −5.17%  
Inverse Weibull  52.59%  5.90%  0.00%  0.00%  11.24%  0.22%  30.42%  9.66%  
Inverse Gamma  60.66%  67.05%  0.00%  0.00%  29.85%  29.51%  44.94%  57.04% 
Outliers Criteria  B162B  B162D  B163B  B163D  

1st P  2nd P  1st P  2nd P  1st P  2nd P  1st P  2nd P  
Peirce  Lognormal  0.00%  0.00%  −4.79%  −11.78%  −1.17%  −5.00%  −1.80%  −6.50% 
Weibull  0.00%  0.00%  20.29%  −7.00%  8.56%  −1.84%  13.25%  −3.01%  
Mixt (LN + Wbl)  0.00%  0.00%  −4.79%  −11.78%  −1.17%  −5.00%  −1.80%  −6.50%  
Gumbel  0.00%  0.00%  −14.12%  −4.01%  −4.54%  −0.87%  −6.78%  −1.38%  
Gamma  0.00%  0.00%  30.65%  39.21%  11.64%  13.27%  16.31%  19.29%  
Cauchy  0.00%  0.00%  −2.25%  −11.15%  −0.62%  −4.01%  −0.46%  −4.68%  
Inverse Weibull  0.00%  0.00%  8.85%  −3.13%  2.94%  −0.73%  4.00%  −1.04%  
Inverse Gamma  0.00%  0.00%  25.15%  20.59%  9.84%  8.79%  12.32%  10.87%  
Leverage  Lognormal  1.35%  −4.97%  0.03%  −7.91%  −0.01%  −10.15%  −1.80%  −6.50% 
Weibull  3.10%  0.85%  7.67%  −1.27%  11.81%  −1.11%  13.25%  −3.01%  
Mixt (LN + Wbl)  1.35%  −4.97%  0.03%  −7.91%  −0.01%  −10.15%  −1.80%  −6.50%  
Gumbel  −4.92%  1.69%  −8.45%  0.53%  −11.96%  0.67%  −6.78%  −1.38%  
Gamma  9.54%  8.39%  16.90%  17.90%  23.25%  23.86%  16.31%  19.29%  
Cauchy  0.61%  −2.66%  0.20%  −5.52%  −0.04%  −5.64%  −0.46%  −4.68%  
Inverse Weibull  9.12%  2.16%  13.25%  1.50%  18.48%  1.22%  4.00%  −1.04%  
Inverse Gamma  11.79%  13.70%  18.31%  19.44%  24.30%  24.91%  12.32%  10.87% 
Outliers Criteria  B072B  B072D  B073B  B073D  

1st P  2nd P  1st P  2nd P  1st P  2nd P  1st P  2nd P  
Peirce  Lognormal  0.00%  0.00%  −1.24%  −4.44%  0.00%  0.00%  0.00%  0.00% 
Weibull  0.00%  0.00%  6.85%  −1.88%  0.00%  0.00%  0.00%  0.00%  
Mixt (LN + Wbl)  0.00%  0.00%  −1.24%  −4.44%  0.00%  0.00%  0.00%  0.00%  
Gumbel  0.00%  0.00%  −4.45%  −0.95%  0.00%  0.00%  0.00%  0.00%  
Gamma  0.00%  0.00%  10.14%  11.86%  0.00%  0.00%  0.00%  0.00%  
Cauchy  0.00%  0.00%  −0.37%  −3.52%  0.00%  0.00%  0.00%  0.00%  
Inverse Weibull  0.00%  0.00%  2.90%  −0.79%  0.00%  0.00%  0.00%  0.00%  
Inverse Gamma  0.00%  0.00%  8.65%  7.56%  0.00%  0.00%  0.00%  0.00%  
Leverage  Lognormal  0.00%  0.00%  −1.24%  −4.44%  0.00%  0.00%  6.21%  −30.94% 
Weibull  0.00%  0.00%  6.85%  −1.88%  0.00%  0.00%  17.60%  3.08%  
Mixt (LN + Wbl)  0.00%  0.00%  −1.24%  −4.44%  0.00%  0.00%  6.21%  −30.94%  
Gumbel  0.00%  0.00%  −4.45%  −0.95%  0.00%  0.00%  −31.74%  8.30%  
Gamma  0.00%  0.00%  10.14%  11.86%  0.00%  0.00%  86.63%  79.06%  
Cauchy  0.00%  0.00%  −0.37%  −3.52%  0.00%  0.00%  1.13%  −7.92%  
Inverse Weibull  0.00%  0.00%  2.90%  −0.79%  0.00%  0.00%  89.46%  13.04%  
Inverse Gamma  0.00%  0.00%  8.65%  7.56%  0.00%  0.00%  135.59%  157.71% 
Outliers Criteria  B162B  B162D  B163B  B163D  

1st P  2nd P  1st P  2nd P  1st P  2nd P  1st P  2nd P  
Peirce  Lognormal  −0.93%  −4.04%  −1.79%  −11.37%  0.00%  0.00%  −0.99%  −6.99% 
Weibull  10.54%  −1.46%  27.82%  −2.98%  0.00%  0.00%  22.27%  −1.82%  
Mixt (LN + Wbl)  −0.93%  −4.04%  −1.79%  −11.37%  0.00%  0.00%  −0.99%  −6.99%  
Gumbel  −3.00%  −0.72%  −8.69%  −1.26%  0.00%  0.00%  −4.07%  −0.68%  
Gamma  9.52%  10.73%  29.67%  32.45%  0.00%  0.00%  17.64%  19.07%  
Cauchy  −0.38%  −2.80%  −0.71%  −6.64%  0.00%  0.00%  −0.27%  −2.82%  
Inverse Weibull  1.92%  −0.64%  6.95%  −1.09%  0.00%  0.00%  2.66%  −0.60%  
Inverse Gamma  7.65%  6.80%  24.93%  23.03%  0.00%  0.00%  13.73%  12.79%  
Leverage  Lognormal  2.29%  −23.09%  −1.79%  −11.37%  1.19%  −16.08%  0.04%  −15.28% 
Weibull  24.52%  0.34%  27.82%  −2.98%  23.59%  −0.23%  27.67%  −1.26%  
Mixt (LN + Wbl)  2.29%  −23.09%  −1.79%  −11.37%  1.19%  −16.08%  0.04%  −15.28%  
Gumbel  −24.20%  3.83%  −8.69%  −1.26%  −15.07%  2.16%  −16.57%  0.87%  
Gamma  64.95%  62.41%  29.67%  32.45%  40.80%  39.84%  39.33%  39.83%  
Cauchy  1.16%  −11.44%  −0.71%  −6.64%  0.42%  −8.81%  0.18%  −5.78%  
Inverse Weibull  44.26%  4.93%  6.95%  −1.09%  22.72%  2.82%  26.26%  1.36%  
Inverse Gamma  72.77%  78.17%  24.93%  23.03%  42.80%  45.26%  39.42%  40.04% 
Case  Total Data  PrcC. Outl.  LevC. Outl.  

AIC  BIC  AIC  BIC  AIC  BIC  
B072B  Gumbel  95.67  99.23  Gamma  72.93  76.36  Gumbel  72.38  75.81 
Lognormal  95.85  99.42  Lognormal  74.38  77.81  Inverse Gamma  72.54  75.96  
Gamma  96.28  99.85  Mixt (LN + Wbl)  74.64  78.07  Inverse Weibull  72.60  76.02  
Inverse Gamma  97.23  100.80  Weibull  74.78  78.20  Lognormal  73.72  77.15  
Mixt (LN + Wbl)  99.95  103.52  Gumbel  75.54  78.97  Gamma  75.57  79.00  
Weibull  102.29  105.86  Inverse Gamma  77.18  80.61  Mixt (LN + Wbl)  80.10  83.53  
Inverse Weibull  106.07  109.63  Cauchy  87.46  90.89  Weibull  83.85  87.28  
Cauchy  108.74  112.31  Inverse Weibull  88.97  92.40  Cauchy  91.33  94.76  
B072D  Gamma  153.02  156.59  Gamma  153.02  156.59  Gamma  153.02  156.59 
Lognormal  153.27  156.84  Lognormal  153.27  156.84  Lognormal  153.27  156.84  
Gumbel  153.78  157.35  Gumbel  153.78  157.35  Gumbel  153.78  157.35  
Mixt (LN + Wbl)  154.37  157.94  Mixt (LN + Wbl)  154.37  157.94  Mixt (LN + Wbl)  154.37  157.94  
Inverse Gamma  154.59  158.16  Inverse Gamma  154.59  158.16  Inverse Gamma  154.59  158.16  
Weibull  154.96  158.53  Weibull  154.96  158.53  Weibull  154.96  158.53  
Inverse Weibull  158.87  162.43  Inverse Weibull  158.87  162.43  Inverse Weibull  158.87  162.43  
Cauchy  176.94  180.51  Cauchy  176.94  180.51  Cauchy  176.94  180.51  
B073B  Lognormal  117.74  121.30  Gamma  99.43  102.91  Gamma  95.07  98.49 
Gumbel  118.06  121.63  Weibull  100.01  103.49  Lognormal  96.07  99.50  
Gamma  118.39  121.96  Mixt (LN + Wbl)  100.27  103.74  Mixt (LN + Wbl)  96.10  99.53  
Inverse Gamma  119.06  122.63  Lognormal  100.74  104.22  Weibull  96.12  99.55  
Mixt (LN + Wbl)  121.14  124.71  Gumbel  100.86  104.33  Gumbel  96.35  99.77  
Weibull  123.09  126.66  Inverse Gamma  103.29  106.76  Inverse Gamma  98.21  101.63  
Inverse Weibull  124.09  127.66  Inverse Weibull  109.85  113.33  Inverse Weibull  104.55  107.98  
Cauchy  134.85  138.42  Cauchy  118.81  122.29  Cauchy  113.88  117.31  
B073D  Gamma  191.70  195.27  Weibull  182.70  186.22  Lognormal  171.10  174.52 
Weibull  192.37  195.94  Gamma  183.13  186.65  Gamma  171.31  174.74  
Mixt (LN + Wbl)  192.61  196.18  Mixt (LN + Wbl)  183.47  186.99  Gumbel  171.48  174.91  
Lognormal  193.02  196.59  Lognormal  184.79  188.31  Inverse Gamma  171.59  175.02  
Gumbel  193.04  196.61  Gumbel  184.88  188.40  Mixt (LN + Wbl)  172.99  176.42  
Inverse Gamma  195.74  199.31  Inverse Gamma  187.70  191.22  Weibull  174.23  177.65  
Inverse Weibull  202.91  206.48  Inverse Weibull  195.19  198.71  Inverse Weibull  175.10  178.52  
Cauchy  213.66  217.23  Cauchy  205.01  208.53  Cauchy  193.84  197.26 
Case  Total Data  PrcC. Outl.  LevC. Outl.  

AIC  BIC  AIC  BIC  AIC  BIC  
B162B  Gamma  −0.93  2.64  Gamma  −0.93  2.64  Lognormal  −3.98  −0.45 
Lognormal  −0.86  2.71  Lognormal  −0.86  2.71  Inverse Gamma  −3.74  −0.22  
Inverse Gamma  −0.10  3.47  Inverse Gamma  −0.10  3.47  Gamma  −3.66  −0.14  
Gumbel  −0.09  3.47  Gumbel  −0.09  3.47  Gumbel  −3.60  −0.08  
Mixt (LN + Wbl)  1.38  4.95  Mixt (LN + Wbl)  1.38  4.95  Mixt (LN + Wbl)  −0.88  2.64  
Weibull  2.33  5.90  Weibull  2.33  5.90  Weibull  0.45  3.98  
Inverse Weibull  5.81  9.37  Inverse Weibull  5.81  9.37  Inverse Weibull  0.50  4.02  
Cauchy  18.78  22.34  Cauchy  18.78  22.34  Cauchy  16.16  19.68  
B162D  Inverse Gamma  17.52  21.09  Lognormal  2.60  6.03  Inverse Gamma  9.63  13.10 
Lognormal  17.85  21.42  Gamma  2.83  6.25  Inverse Weibull  10.30  13.77  
Gumbel  18.13  21.70  Gumbel  2.94  6.37  Gumbel  10.32  13.79  
Gamma  19.23  22.80  Inverse Gamma  3.07  6.49  Lognormal  10.33  13.80  
Inverse Weibull  19.83  23.40  Mixt (LN + Wbl)  4.84  8.26  Gamma  11.77  15.24  
Mixt (LN + Wbl)  22.62  26.19  Weibull  5.62  9.05  Mixt (LN + Wbl)  15.29  18.76  
Weibull  24.41  27.98  Inverse Weibull  6.78  10.21  Weibull  17.15  20.62  
Cauchy  40.35  43.92  Cauchy  24.84  28.26  Cauchy  32.81  36.29  
B163B  Gumbel  −3.38  0.19  Inverse Gamma  −8.26  −4.74  Gumbel  −12.76  −9.28 
Inverse Gamma  −3.32  0.25  Lognormal  −8.19  −4.67  Inverse Gamma  −12.26  −8.79  
Lognormal  −2.92  0.65  Gumbel  −8.06  −4.54  Inverse Weibull  −11.88  −8.40  
Gamma  −1.97  1.59  Gamma  −7.66  −4.14  Lognormal  −11.60  −8.13  
Inverse Weibull  −0.19  3.38  Mixt (LN + Wbl)  −4.27  −0.75  Gamma  −10.63  −7.15  
Mixt (LN + Wbl)  2.23  5.80  Inverse Weibull  −4.18  −0.65  Mixt (LN + Wbl)  −6.65  −3.18  
Weibull  4.78  8.34  Weibull  −2.36  1.16  Weibull  −4.20  −0.73  
Cauchy  16.99  20.56  Cauchy  12.07  15.59  Cauchy  9.74  13.22  
B163D  Inverse Gamma  65.28  68.85  Inverse Gamma  57.20  60.72  Inverse Gamma  57.20  60.72 
Inverse Weibull  65.40  68.97  Gumbel  57.48  61.00  Gumbel  57.48  61.00  
Gumbel  66.09  69.65  Lognormal  57.89  61.42  Lognormal  57.89  61.42  
Lognormal  66.66  70.22  Inverse Weibull  58.42  61.94  Inverse Weibull  58.42  61.94  
Gamma  68.99  72.56  Gamma  59.26  62.78  Gamma  59.26  62.78  
Mixt (LN + Wbl)  72.79  76.36  Mixt (LN + Wbl)  62.14  65.66  Mixt (LN + Wbl)  62.14  65.66  
Weibull  76.80  80.37  Weibull  64.83  68.35  Weibull  64.83  68.35  
Cauchy  85.55  89.11  Cauchy  77.76  81.28  Cauchy  77.76  81.28 
Case  Total Data  Case  Total Data  LevC. Outl.  

AIC  BIC  AIC  BIC  AIC  BIC  
B072B  Mixt (LN + Wbl)  −112.84  −111.06  B073B  Mixt (LN + Wbl)  −112.67  −110.88  
Lognormal  −112.37  −108.81  Lognormal  −111.97  −108.40  
Gamma  −112.07  −108.50  Gamma  −111.71  −108.14  
Gumbel  −112.04  −108.48  Inverse Gamma  −111.69  −108.12  
Inverse Gamma  −112.02  −108.45  Gumbel  −111.56  −107.99  
Weibull  −109.06  −105.49  Weibull  −109.16  −105.59  
Inverse Weibull  −108.53  −104.96  Inverse Weibull  −108.99  −105.42  
Cauchy  −89.23  −85.66  Cauchy  −85.35  −81.78  
B072D  Inverse Gamma  −108.32  −104.75  B073D  Weibull  −67.33  −63.76  Mixt (LN + Wbl)  −81.93  −80.22 
Gumbel  −108.30  −104.74  Gamma  −62.77  −59.20  Lognormal  −79.69  −76.26  
Lognormal  −108.06  −104.49  Mixt (LN + Wbl)  −60.64  −58.86  Inverse Gamma  −79.62  −76.19  
Mixt (LN + Wbl)  −107.82  −106.04  Lognormal  −58.55  −54.98  Gamma  −79.37  −75.94  
Gamma  −107.16  −103.59  Gumbel  −57.76  −54.19  Gumbel  −79.24  −75.81  
Inverse Weibull  −105.26  −101.69  Cauchy  −53.22  −49.65  Inverse Weibull  −75.42  −71.99  
Weibull  −101.12  −97.55  Inverse Gamma  −52.49  −48.92  Weibull  −74.90  −71.48  
Cauchy  −89.94  −86.37  Inverse Weibull  −37.33  −33.76  Cauchy  −59.70  −56.27 
Case  Total Data  PrcC. Outl.  LevC. Outl.  

AIC  BIC  AIC  BIC  AIC  BIC  
B162B  Mixt (LN + Wbl)  −201.74  −199.96  Mixt (LN + Wbl)  −203.69  −201.93  Gamma  −200.60  −197.22 
Gamma  −200.83  −197.26  Weibull  −202.68  −199.16  Mixt (LN + Wbl)  −200.49  −198.80  
Weibull  −199.75  −196.18  Gamma  −200.93  −197.41  Lognormal  −200.39  −197.01  
Lognormal  −199.72  −196.15  Lognormal  −199.44  −195.91  Inverse Gamma  −199.98  −196.61  
Inverse Gamma  −198.08  −194.51  Inverse Gamma  −197.50  −193.97  Gumbel  −197.93  −194.55  
Gumbel  −195.88  −192.31  Gumbel  −194.70  −191.18  Weibull  −197.61  −194.23  
Inverse Weibull  −187.18  −183.61  Inverse Weibull  −185.70  −182.18  Inverse Weibull  −194.34  −190.96  
Cauchy  −184.88  −181.31  Cauchy  −184.31  −180.79  Cauchy  −180.83  −177.45  
B162D  Inverse Gamma  −225.81  −222.24  Mixt (LN + Wbl)  −227.67  −225.93  Mixt (LN + Wbl)  −227.67  −225.93 
Lognormal  −225.50  −221.93  Gamma  −226.98  −223.51  Gamma  −226.98  −223.51  
Gumbel  −225.49  −221.92  Lognormal  −226.72  −223.25  Lognormal  −226.72  −223.25  
Gamma  −224.87  −221.31  Inverse Gamma  −226.30  −222.82  Inverse Gamma  −226.30  −222.82  
Inverse Weibull  −223.24  −219.67  Weibull  −225.29  −221.81  Weibull  −225.29  −221.81  
Mixt (LN + Wbl)  −221.35  −219.57  Gumbel  −224.32  −220.84  Gumbel  −224.32  −220.84  
Weibull  −217.08  −213.52  Inverse Weibull  −221.12  −217.64  Inverse Weibull  −221.12  −217.64  
Cauchy  −206.69  −203.13  Cauchy  −205.56  −202.09  Cauchy  −205.56  −202.09  
B163B  Mixt (LN + Wbl)  −208.47  −206.68  Mixt (LN + Wbl)  −208.47  −206.68  Mixt (LN + Wbl)  −208.37  −206.66 
Weibull  −206.95  −203.39  Weibull  −206.95  −203.39  Weibull  −207.20  −203.77  
Gamma  −206.51  −202.94  Gamma  −206.51  −202.94  Gamma  −205.36  −201.93  
Lognormal  −205.34  −201.77  Lognormal  −205.34  −201.77  Lognormal  −204.48  −201.05  
Inverse Gamma  −203.77  −200.20  Inverse Gamma  −203.77  −200.20  Inverse Gamma  −203.40  −199.97  
Gumbel  −201.31  −197.74  Gumbel  −201.31  −197.74  Gumbel  −200.29  −196.86  
Inverse Weibull  −193.68  −190.11  Inverse Weibull  −193.68  −190.11  Inverse Weibull  −195.03  −191.60  
Cauchy  −189.40  −185.83  Cauchy  −189.40  −185.83  Cauchy  −186.18  −182.75  
B163D  Gamma  −155.15  −151.58  Mixt (LN + Wbl)  −160.75  −158.99  Mixt (LN + Wbl)  −162.64  −160.91 
Lognormal  −155.06  −151.50  Gamma  −159.42  −155.90  Gamma  −161.93  −158.45  
Inverse Gamma  −154.51  −150.94  Weibull  −159.06  −155.54  Lognormal  −161.72  −158.25  
Mixt (LN + Wbl)  −154.44  −152.66  Lognormal  −158.54  −155.02  Inverse Gamma  −161.34  −157.86  
Gumbel  −152.19  −148.62  Inverse Gamma  −157.34  −153.82  Gumbel  −159.09  −155.62  
Weibull  −148.56  −144.99  Gumbel  −153.60  −150.07  Weibull  −159.05  −155.57  
Inverse Weibull  −145.58  −142.01  Inverse Weibull  −146.19  −142.66  Inverse Weibull  −155.56  −152.08  
Cauchy  −139.60  −136.04  Cauchy  −141.04  −137.52  Cauchy  −141.29  −137.81 
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Martinez, I.; Gallegos, M.F.; ArayaLetelier, G.; LopezGarcia, D. Impact of Probabilistic Modeling Alternatives on the Seismic Fragility Analysis of Reinforced Concrete Dual Wall–Frame Buildings towards Resilient Designs. Sustainability 2024, 16, 1668. https://doi.org/10.3390/su16041668
Martinez I, Gallegos MF, ArayaLetelier G, LopezGarcia D. Impact of Probabilistic Modeling Alternatives on the Seismic Fragility Analysis of Reinforced Concrete Dual Wall–Frame Buildings towards Resilient Designs. Sustainability. 2024; 16(4):1668. https://doi.org/10.3390/su16041668
Chicago/Turabian StyleMartinez, Ivanna, Marco F. Gallegos, Gerardo ArayaLetelier, and Diego LopezGarcia. 2024. "Impact of Probabilistic Modeling Alternatives on the Seismic Fragility Analysis of Reinforced Concrete Dual Wall–Frame Buildings towards Resilient Designs" Sustainability 16, no. 4: 1668. https://doi.org/10.3390/su16041668