4.1. Methodology and Process for Developing Seismic Fragility Curves
The seismic fragility curve is a mathematical function that illustrates the relation between a given intensity measure (IM) for a seismic event and the probability that the response of the component exceeds the limit state where the IM occurs. In the present study, the seismic fragility curves of the mold transformer component were established using truncated incremental dynamic analysis (TIDA) [
28]. This approach involves a set of accelerograms scaled to certain levels of seismic intensities until the component collapses. Then, time history analyses were performed for each intensity level, and the maximum response of each component was recorded. The main advantage of this method is that it reduces the number of structural analyses and computational efforts. The uncertainties in the modeling components are considered by using a set of corresponding parameters after Zentner et al. [
29].
Figure 11 shows a flowchart of the process of developing the seismic fragility curves for the mold transformer.
The fragility curves used for seismic vulnerable evaluation are natural logarithmic functions providing the probability of exceeding an identified response of a component for different intensity levels (IL) of the earthquake corresponding to a specified performance level. Accordingly, log-normal distribution is widely used to establish the fragility curves, not only for structural elements, according to the study of Cornell et al. [
30], but also for nonstructural elements [
16,
17]. For a specified damage state (DS), the relation between the probability of exceedance and the specific IL values can be presented in terms of a log-normal distribution function as follows [
28,
31,
32,
33]
where
P is the probability for the cases that the component response will exceed a specified performance level at a given ground motion with
IL =
xi,
is the standard normal cumulative distribution function, and
θ and
β are the median and the standard deviation of the fragility function, respectively.
In order to fit the fragility curves from the observed analytical data,
θ and
β could be evaluated by using the maximum likelihood estimator (LE) method for the entire data set [
28,
34] as follows
where
m is the number of PGA levels,
pi is the probability that the component response will exceed a specified performance level at a given ground motion with
IL = xi based on analytical results,
zi is the number of exceedances out of
ni ground motion, and
denotes a product over all PGA levels. Finally, parameters
θ and
β were determined by maximizing the likelihood.
4.2. Characteristics of Input Ground Acceleration Time Histories Used for Dynamic Analyses
In order to assess the seismic vulnerability of the mold transformer using fragility curves, the selection of input ground motion for time history analyses is a key parameter. Vamvatsikos and Cornell [
35] proposed the use of 10–20 input acceleration time histories when performing TIDA, while ATC-58 [
36] recommended 20 time histories. Thus, in this study, 20 ordinary records of natural ground motion with both horizontal and vertical components were selected from the PEER database [
37].
The time history ground motions were derived from historical recordings so that their mean response spectrum matched the target design acceleration spectrum evaluated according to the Korean Building Code (KBC 2016) [
38]:
Sds = 0.497 and
Sd1 = 0.287, and their shear wave velocity (
Vs,30) in a range of 180 m/s to 760 m/s, which was complied with the soil type C and D [
38]. The SRSS-based pseudo-acceleration response spectrum of selected ground motion was demonstrated in
Figure 12. The selected earthquake ground motions were presented in detail in
Table 5. These earthquake records cover the ranges of magnitude from 5.5 to 7.5, of shear wave velocity (
Vs,30) from 198.77 to 634.33 m/sec, and of closet ruptured distance (
Rrup) from 4.06 to 199.84 km. Scatter diagrams of magnitude versus closet ruptured distance (
Rrup) and magnitude versus peak ground acceleration of the selected ground motions are illustrated in
Figure 13 and
Figure 14, respectively.
4.3. Intensity Measure, Intensity Levels, and Uncertainty in Modeling Parameters Used for Dynamic Analyses
In order to construct the fragility curves using the TIDA approach, a series of time history analyses have been performed for a set of acceleration records, which were scaled to varying and increasing levels of seismic intensity. The parameter that informs the scaling of the acceleration records is typically known as the intensity measure (IM), while the degree of seismic intensity is referred to as the intensity level (IL). A common IM that has been widely used in seismic fragility analysis, peak ground acceleration (PGA) [
16,
39,
40,
41], was chosen in the present study. In particular, various levels of PGA with a range of 0.05 to 2 g were used as IL. This range of PGA was considered to ensure that the mold transformer could exhibit different performance levels to exceed the limit states.
Moreover, in reality, the mold transformers can be manufactured with different configurations and specifications, which can lead to variations in performance when subjected to the same earthquake. In order to consider such uncertainty in the modeling parameters of the mold transformer when estimating the fragility curves, various dynamic parameters of the mold transformer were investigated in terms of the effective stiffness and the coil mass. Specifically, three cases of effective translational stiffness (0.8 Kt, 1.0 Kt, and 1.2 Kt), three cases of effective rotational stiffness (0.8 Kθ, 1.0 Kθ, and 1.2 Kθ), and three cases of coil mass (0.6 Mc, 1.0 Mc, and 1.2 Mc) were investigated for each time history analysis; based on the test results, 1.0 Kt, 1.0 Kθ, and 1.0 Mc were the values of the modeling parameters chosen for the prototype analytical model.
4.4. Identifications of Damage States and Limit States
From the shaking table test results, three weak points of the mold transformer were adopted as critical damage states, as shown in
Figure 15. The first damage states (DS1) correspond to the failure of the spacers, the second damage states (DS2) correspond to the excessive coil movement in Y-direction, and the third damage states (DS3) correspond to the loosening of linked bolts between the bottom and bed beams. According to the recommendation specified in ASCE 41-17 Standard [
42], the target performance levels used to evaluate the seismic vulnerability of the nonstructural elements for buildings can be selected among Operational (OP), Position Retention (PR), and Life Safety (LS). Specifically, the OP performance level involves the functional nonstructural equipment required for normal use regardless of minor damage; the PR performance level involves nonstructural equipment, which is secured in place and might be able to function if necessary service is available; and the LS performance level involves nonstructural equipment that can sustain significant damage but not become dislodged and fall in a manner that could cause death or serious injury to occupants or people. In order to evaluate the seismic performance of the mold transformer, the target performance levels were selected based on the correlation with critical damage states, as shown in
Figure 16. Specifically, damage in spacers (DS1) can lead to slippage of the HV and LV coils away from the original positions and the interaction between them, which involve the OP and PR performance levels; and excessive movement in the Y-direction (DS2) can cause interaction with the adjacent equipment and breaking of the electrical wires, which involve the OP and PR performance levels; and the degree of loosening at the bottom bolts (DS3) can affect the functional operation or lead to the collapse of the entire transformer and threaten human life, which involve the OP, PR, and LS performance levels.
When correlating the performance levels and damage states, the values of the limit states were defined for each target performance level corresponding to the damage states. The parameters by which the limit states were evaluated were partly based on the observation from shaking table test results [
27] and partly based on judgment. Specifically, for the DS1, the peak response acceleration at the coils (PRA) in the Y-direction is strongly related to the spacer damage due to the fact that the lowest effective stiffness is in this direction. Accordingly, for the DS1, the values of the limit states corresponding to the OP and PR performance levels were respectively determined to be 0.22 g, which corresponds to the time when the cracks were initially formed, and 0.69 g, which corresponds to the time when the spacers were failed, as observed during shaking table tests. For the DS2, the maximum displacement in the Y-direction at the top beams is strongly related to the excessive movement of the transformer. Accordingly, the values of the limit states corresponding to the OP and PR performance levels were determined to be 50 mm and 75 mm, respectively, based on the provisions as well as the recommendations of the Korea Research Institute [
43]. For the DS3, the maximum displacement in the Z-direction at the bottom beam is strongly related to the degree of loosening of the link bolted between the bottom beam and bed beam; and the values of the limit states corresponding to the OP, PR, and LS performance levels were determined to be 2.2 mm, 6.6 mm, and 12.45 mm, respectively, based on the experiment test results [
27]. The summary of the value of the limit state corresponding to each damage state is presented in
Table 6.
4.5. Fragility Curves and Discussions
The peak response results obtained from the analytical model of the mold transformer are illustrated in
Figure 17 in terms of the different parameters represented for damage states: the peak response acceleration at the coils in the Y-direction (
Figure 17a), maximum displacement at the top beam in the Y-direction (
Figure 17b), and maximum displacement at the bottom beam in the Z-direction (
Figure 17c). As the peak ground acceleration increases, the maximum response from the analytical model increases as well, indicating increased physical damage to the mold transformer. Moreover, the limit values corresponding to the damage states for each performance level are presented in
Figure 17. In order to establish the fragility curves, these analytical results provide the fraction of the analyses that lead to the exceedance of limit states corresponding to specific performance levels. In total, 1120 cases were analyzed with consideration of the variety of earthquake ground motion characteristics and uncertainty in modeling parameters to establish the fragility curves of the hybrid mold transformers.
To further elucidate the effects of uncertain modeling parameters on the performance of the mold transformer in an earthquake,
Figure 18 presents the characteristics of the Friuli_Italy-01 Earthquake used for dynamic analysis in terms of the acceleration time histories in the Y and Z-directions (
Figure 18a,c) and corresponding FT results in the frequency domain (
Figure 18b,d). In both directions, at least four resonance frequencies could be clearly observed in the frequency domain, from 1.67 Hz to 19.3 Hz in the Y-direction.
Figure 19 shows the effect of the mass coil, M
c, on the main responses of the transformer: peak response acceleration at the coil in the Y-direction, maximum displacement at the top beam in the Y-direction, and maximum displacement at the bottom beam in the Z-direction. In general, the peak response acceleration at the coil and the maximum displacement at the top beam in the Y-direction showed a nonlinear increase when the PGA was greater than 1.0 g, while the maximum displacement at the bottom beam in the Z-direction showed a linear increasing trend. Moreover, when the coil mass varied in the range of 0.6 to 1.2 M
c, at the PGA of 2.0 g, the peak acceleration response in the Y-direction varied in the range of 17 %, the maximum displacement at the top beam in the Y-direction varied in the range of 15%, and the maximum displacement at the bottom beam in the Z-direction varied in the range of 30%. In
Figure 20, the effect of the translational stiffness,
Kt, on the main parameters was also presented. As shown in
Figure 20a, the effective translational stiffness has a great influence on the peak acceleration response. For example, at the PGA of 2.0, the peak acceleration response in the Y-direction in the case of 1.0
Kt was about 50% lower than that of 0.8
Kt and about 30% greater than that of 1.2
Kt. Meanwhile, the effective translational stiffness had a negligible influence on the displacement in the Y-direction and the Z-direction (
Figure 20b,c). Moreover, in all cases, the effect of the rotational stiffness in the range of 0.8 to 1.2
Kθ on parameters was also negligible.
Figure 21,
Figure 22 and
Figure 23 present the effects of uncertainty in the modeling parameters on the fragility curves for the coil mass, effective translational stiffness, and rotational stiffness of the mold transformer. As shown in
Figure 21, the variability of coil mass ultimately has little effect on the fragility curves corresponding to DS1; meanwhile, the fragility curves corresponding to DS2 and DS3 become flatter as the mass coil decreases. This is mainly attributed to the fact that the decrease of the mass coil leads to an increase in the natural time period, which leads to the decrease of deformation and finally the probability of exceeding limit states.
Figure 22 shows that the variation of translational stiffness in the range of 0.8 to 1.2
Kt has almost no effect on the fragility curves corresponding to DS1 and DS2, but does have a significant effect on DS3. The increased translational stiffness significantly reduced the deformation in the Z-direction with the increase of PGA, leading to the decreased probability of exceeding limit states and the fragility curves becoming flatter. Similarly,
Figure 23 indicates that the rotational stiffness has no effect on response acceleration and deformation in the Y-direction, leading to no change in the fragility curves corresponding to DS1 and DS2. Nonetheless, the change in rotational stiffness leads to slight variation of vertical deformation of the mold transformer, triggering the variation in the fragility curve with respect to DS3; however, this variation is not significant overall.
In order to estimate the complete fragility curves of the mold transformers, all analytical cases were combined to consider both variability in PGA and uncertain modeling parameters. Accordingly, the values of the median (
) and the standard deviation (
) were evaluated to establish fragility curves for different performance levels corresponding to damage states, and these are summarized in
Table 7. By using Equations (6) and (7), the fragility curves of the mold transformer were derived for various damage states according to different performance levels, and these were presented in
Figure 24. In
Figure 24, the data points represented the fraction of exceedance of limit states for specific performance levels calculated from dynamic analyses. In addition,
Figure 25 illustrates the differences in the probability of exceedance between analytical data and the fitted model according to PGA of different damage states.
The critical PGA values corresponding to a specific probability of exceedance, which may play an important role in the seismic vulnerable evaluation; seismic design purposes were also determined from the fragility curves. Previous studies by Kildashti et al. [
44], Parool et al. [
45], and Talaat et al. [
46] considered 50% probability of exceedance as the acceptance criteria. In this study, in order to consider the safety and the abundance of electrical transformer components in buildings, the value of 40% probability of exceedance was proposed as the acceptance criteria.
Figure 24 also presents the critical PGA values corresponding to 40% probability of exceedance of the mold transformer. In detail, for the DS1 (
Figure 24a), the 40% probability of exceedance corresponding to the OP and PR performance levels were predicted at PGA levels of 0.15 g and 0.4 g, respectively. For the DS2 (
Figure 24b), the 40% probability of exceedance corresponding to the OP and PR performance levels were predicted at PGA levels of 0.3 g and 0.45 g, respectively. For the DS3 (
Figure 24c), the 40% probability of exceedance corresponding to the OP, PR, and LS performance levels were predicted at PGA levels of 0.2 g, 0.6 g, and 1.2 g, respectively. For another value of the probability of exceedance, the critical PGA for each performance level corresponding to specific damage states could also be determined by using the fragility curves derived from the values of the median and the standard deviation presented in
Table 7.