PSI - Issue 78

Parvane Rezaei Ranjbar et al. / Procedia Structural Integrity 78 (2026) 615–622

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simultaneously to the building during time-history analyses. These analyses produced suites of IDA curves (Fig. 4) for both the existing and retrofitted buildings, in the X- and Y-directions, respectively. The fundamental periods ( T 1 ) used to calculate the first-mode 5%-damped spectral acceleration, S a ( T 1 , 5%), are T 1x =0.743 s and T 1y =0.450 s for the existing building, and T 1x =0.325 s and T 1y =0.290 s for the retrofitted building. Each graph illustrates the structural demands imposed by individual ground motion records at varying intensity levels, revealing both similarities and differences depending on the earthquake input. In each case, drift demands are compared against specific drift ratio thresholds - Slight, Moderate, Extensive, and Complete - corresponding to the four damage levels defined by Hazus (2024) (Table 2). To construct the fragility curves, linear regression was applied to the IDA curve data (Fig. 5). This process estimates how the structural response (e.g., drift or acceleration) evolves with increasing seismic intensity. Based on the method described in Section 2.2, and using the regression coefficients, two critical parameters of the fragility function are derived for each damage mode: the median, representing the seismic intensity at which there is a 50% probability of reaching a given damage state, and the standard deviation, representing the associated uncertainty. The high values of the coefficient of determination in Fig. 5 ( R² =0.9367 and 0.9563, respectively, in the X- and Y-directions for the existing structure; R² =0.8927 and 0.8637 for the retrofitted structure) confirm the strength of the linear model fit to the data. Fig. 6 shows the resulting fragility curves for both the existing and retrofitted buildings in the X- and Y-directions. In the X-direction, the fragility curves indicate greater structural vulnerability compared to the Y-direction. This finding is consistent with the structural capacity analysis, which revealed that the building’s lateral force -resisting system in the X-direction is composed of only a few moment frames. These observations underscore the importance of three-dimensional structural modeling and the application of bi-directional ground motions in achieving accurate fragility curve calibration.

Collapse Prevention Limit State N. ID Earthquake Name Date

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Spectral acceleration [g] Period [sec] 212xa (SF=6.65) 212ya (SF= 6.72) 343xa (SF=2.35) 343ya (SF=3.03) 388xa (SF=3.19) 388ya (SF=3.38) 390xa (SF=9.18) 390ya (SF=9.31) 391xa (SF=5.19) 391ya (SF=4.49) 459xa (SF=1.67) 459ya (SF=1.93) 461xa (SF=0.73) 461ya (SF=0.72) Avarege S. Target

M w Ep.Dist. [km]

a g (X) [m/s 2 ]

a g (Y) [m/s 2 ]

E1 212 Duzce 2 E2 343 Christchurch

06/06/2000 6.0 15.23 0.629 0.623

21 /0 2 / 2011 6.2 16.52 1 . 779 1 . 382 E3 388 Christchurch 13/06/2011 6.0 13.44 1.311 1.240 E4 390 Christchurch 13/06/2011 6.0 29.44 0.456 0.450 E5 391 Christchurch 13/06/2011 6.0 14.86 0.807 0.913 E6 459 Northridge 17/01/1994 6.7 20.19 2.507 2.173 E7 461 Northridge 17/01/1994 6.7 20.25 5.717 5.783

(a)

(a) Fig. 3. (a) Acceleration spectra for the Collapse Prevention limit state; (b) Parameters of earthquake records.

0.00 0.25 0.50 0.75 1.00 1.25 1.50 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50

0.00 0.25 0.50 0.75 1.00 1.25 1.50 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50

212 343 388 390 391 459 461

212 343 388 390 391 459 461

damage

Complete

Slight damage

Slight damage

Extensive damage

Extensive damage

Moderate damage

Moderate damage

Spectral Acceleration (g)

Spectral Acceleration (g)

damage

Complete

0.000

0.005

0.010

0.015

0.020

0.000

0.005

0.010

0.015

0.020

Max Drift Ratio -Y

Max Drift Ratio -X

(a)

(b)

212 343 388 461 390 391 459

212 343 388 461 390 391 459

Moderate damage Extensive damage

Moderate damage Extensive damage

Slight damage

Slight damage

Spectral Acceleration (g)

Spectral Acceleration (g)

damage

damage

0.020 Complete

0.020 Complete

0.000

0.005

0.010

0.015

0.000

0.005

0.010

0.015

(d) Fig. 4. IDA Curves: (a) Existing Structure - X-Direction; (b) Existing Structure - Y-Direction; (c) Retrofitted Structure - X-Direction; (b)Retrofitted Structure - Y-Direction. Max Drift Ratio -X Max Drift Ratio -Y

(c)