Based on the multiple stripes analysis method, an investigation of the estimation of aleatory randomness by S_{a}(T_{1})-based intensity measures (IMs) in the fragility analysis is carried out for two typical low- and medium-rise reinforced concrete (RC) frame structures with 4 and 8 stories, respectively. The sensitivity of the aleatory randomness estimated in fragility curves to various S_{a}(T_{1})-based IMs is analyzed at three damage limit states, i.e., immediate occupancy, life safety, and collapse prevention. In addition, the effect of characterization methods of bidirectional ground motion intensity on the record-to-record variability is investigated. It is found that the damage limit state of the structure has an important influence on the applicability of the ground motion IM. The S_{a}(T_{1})-based IMs, considering the effect of softened period, can maintain lower record-to-record variability in the three limit states, and the S_{a}(T_{1})-based IMs, considering the effect of higher modes, do not show their advantage over S_{a}(T_{1}). Furthermore, the optimal multiplier C and exponent α in the dual-parameter ground motion IM are proposed to obtain a lower record-to-record variability in the fragility analysis of different damage limit state. Finally, the improved dual-parameter ground motion IM is applied in the risk assessment of the 8-story frame structure.
Reinforced concrete (RC) frame structures, as a structural system that enables flexible space separation, are widely used in Chinese cities. Previous earthquake damage surveys have shown that reinforced concrete frame structures are often severely damaged or even collapse during earthquakes [1–4]. Therefore, accurate and solid seismic fragility analysis of RC frame structures is important and necessary. Seismic fragility analysis is an effective method for evaluating the seismic performance of a structure from a probabilistic perspective [5–10].
The first and important step in the fragility analysis is to select a ground motion intensity measure (IM) to measure the aleatory randomness caused by record-to-record variability. The seismic IM is used as an intermediate variable to connect the seismic hazard and engineering performance demand measure (DM). To date, a variety of ground motion IMs have been proposed, and the selection of IMs depends not only on the structural system concerned but also on the engineering performance DM of interest [11–15]. Existing studies are mainly focused on the macro analysis of the relationship between ground motion IMs and engineering performance DMs. At the same time, the applicability of different IMs for seismic fragility analysis at various damage limit states has not yet been analyzed in depth.
The seismic fragility curve illustrates a structure's probability of exceeding a particular damage state under the action of earthquakes with different intensities. However, the establishment of fragility curves requires a significant amount of computations. Therefore, it is critical to quickly and easily estimate fragility curves. Commonly used methods for establishing fragility curves are incremental dynamic analysis [16–20], truncated incremental dynamic analysis [21], multiple stripes analysis [22,23], and static pushover analysis methods [24–26]. Compared with incremental dynamic analysis, the multiple stripes analysis method does not need to scale the amplitude of all the ground motions to a level that causes the damage limit state of interest, but only needs to analyze the structure at specific ground motion intensity levels [21]. The multiple stripes analysis method is equivalent to a special case of the IDA method. Even compared with the truncated incremental dynamic analysis, the calculation amount of the multiple stripes analysis is still smaller [21]. The multiple stripes analysis method can consider the influence of high-modes characteristics of high-rise structures and the hysteretic characteristics of structural members compared with static pushover analysis methods. Therefore, the multiple stripes analysis method can easily present the fragility curves while ensuring accuracy compared to other methods.
Spectral acceleration at fundamental period S_{a}(T_{1}), a convenient and efficient intensity measure for first mode dominated structures [27], has been widely adopted in seismic design codes and research in many countries. In recent years, various forms of IMs have been proposed based on S_{a}(T_{1}). The purpose of this paper is to investigate the estimation of aleatory randomness by using different S_{a}(T_{1})-based ground motion IMs in the fragility analysis in terms of different damage limit states for the RC frame structures. With thirty pairs of far-field records, multiple stripes analysis is carried out for two RC frame structures with 4 and 8 stories to explore the effect of S_{a}(T_{1})-based IMs on the record-to-record variability estimated in fragility curves of different damage limit states. The influence of the bidirectional ground motion intensity characterization methods on the seismic fragility assessment is also explored. Furthermore, the optimal parameters in the dual-parameter ground motion IM for the fragility analysis at different damage states are proposed through parameter analysis. Finally, the improved dual-parameter ground motion IM is applied to the risk assessment of exceeding each limit state for the 8-story frame structure according to the Chinses seismic code.
There are many types of ground motion IMs, and the application scopes of different ground motion IMs are different [11,12]. Due to the structural characteristics of RC frame structures, frame buildings in seismic high-risk areas are often structurally not high, and then their dynamic characteristics are generally dominated by the first vibration mode. The spectral acceleration at the first vibration period S_{a}(T_{1}) is a convenient and efficient ground motion IM in the seismic analysis of structures dominated by the first mode [27]. However, RC frame structures are prone to be elasto-plastic under the action of strong earthquakes, and a single S_{a}(T_{1}) cannot reflect this characteristic. Cordova et al. [28] considered the effect of structural stiffness degradation and proposed the dual-parameter ground motion IM S* (see Table 1), where the softened period T_{f} = 2T_{1} and the combination coefficient α = 0.5. Mehanny [29] proposed the improved IM-CR and IM-SR based on S*. By introducing the self-adaptive parameter R_{IM}, both IM-CR and IM-SR can be applied to situations with different nonlinear levels. R_{IM} was recommended to be 2. Bojórquez et al. [30] proposed an IM exploring the geometric mean of spectral acceleration at multiple periods, in which the parameter N_{p} is used to capture the spectral shape, and T_{N} = 2T_{1} and α = 0.4 are recommended.
However, the abovementioned ground motion IMs only capture the effect of period elongation and do not reflect the effect of higher vibration modes for long-period structures. Some studies have pointed out that ground motion IMs considering the higher vibration mode are suitable for high-rise buildings. The linear combination-type IMs Sa12∗andSa123∗ proposed by Shome et al. [27] considers the spectral accelerations at the first two and three periods, respectively, and the combination coefficients are the weighted average of the modal mass participation coefficients of a 20-story building. Vamvatsikos et al. [31] proposed IMs in flexible combination forms, namely, IM_{12} and IM_{123}. For IM_{12}, the power exponent β is set as 1/2. The parameters τ_{a} and τ_{b} are suggested to be T_{1} and T_{f}, respectively, when IM_{12} is used in low-rise buildings, that is, IM_{12} is equivalent to S* in such cases. The parameters τ_{a} and τ_{b} are suggested to be T_{1} and T_{2}, respectively, when IM_{12} is used in medium-rise buildings. For IM_{123}, the power exponents β and γ are recommended to be 1/3, and the first three periods T_{1}, T_{2} and T_{3} are included in terms of high-rise buildings. Similarly, Lin et al. [32] proposed IM S_{N1} considering the softened period and IM S_{N2} considering the first two vibration periods T_{1} and T_{2}, and suggested C = 1.5, α = 0.5 and β = 0.75. Lu et al. [33] investigated the relationship between the optimal number of combined modes n and the first vibration period T_{1} and proposed ground motion IM S¯a. Kazantzi et al. [34] proposed ground motion IM S_{a,gm}(T_{i}) based on the geometric mean form considering spectral acceleration at multiple periods for building classes. It is pointed out that the IM including five spectral values, (T_{i})_{5}, can be well applied to low and high building classes. To assess the collapse capacity of generic moment frames, Adam et al. [35] proposed an optimal IM_{opt} that includes an unfixed lower bound period T_{0.95M}. T_{0.95M} can be estimated as T0.95M≈T1/[1+3(m0.95M−1)/2], in which m0.95M=ceilN is the mode associated with the exceedance of 95% of the total mass and N is the number of stories in the building.
Basic InformationStructural Model
Two typical 4- and 8-story RC frame structures with seismic design [36,37] according to the current Chinese Code for Seismic Design of Building [38] are selected as the analysis cases, as shown in Fig. 1. The total heights of the building are 16.2 and 31.8 m, respectively, of which the height of the first story is 4.5 m and the other layers are 3.9 m. The typical floor plan of those buildings is shown in Fig. 2. The frames belong to the seismic precautionary building of Category-C with a design precautionary intensity of 8. Both frames belong to the first design earthquake group, and have type II site classification. The roof dead load and live load are set as 7 and 0.7 kN/m^{2}, respectively. The floor dead and live load are taken as 5 and 2 kN/m^{2}. The load of the infill wall is evenly distributed on the beam, 6 kN/m for the outer wall and 3 kN/m for the inner wall. The frame beams and columns are made of C40 concrete, and the longitudinal reinforcement is HRB335. The detailed dimensions and reinforcement information of frame beams and columns can be found in Wang [36] and He et al. [37].
The two RC frames are modeled in the program OpenSEES developed by the Pacific Earthquake Engineering Research (PEER) center [39]. Concrete02 is used to describe the behavior of concrete, and Steel02 is used to represent the behavior of steel. Concrete02 can consider the tensile mechanical properties of concrete and the degradation of unloading stiffness [40]. Steel02 can reflect the isotropic hardening effect and the Bauschinger effect. It is very efficient in calculation because it uses the explicit function expression of strain, and at the same time it maintains good consistency with the results of the cyclical loading tests of steel bars [41]. Columns and beams are modeled by fiber dispBeamColumn elements, by which the nonlinear characteristics of components can be precisely simulated with a small calculation cost compared to standard beam finite elements in general-purpose engineering simulation software. From the aspect of material constitutive, one-dimensional material constitutive is enough for fiber dispBeamColumn elements. From the aspect of simulating deformation, fiber dispBeamColumn elements can simulate bending deformation and axial deformation well, and steel and concrete can be considered separately from the aspect of modeling. The P-delta effect is considered in the models, and Rayleigh damping is used herein with a damping ratio of 5% assumed. Table 2 summarizes some transverse and longitudinal vibration periods of the two RC frames. More detailed modeling information can be obtained referring to Wang [36] and He et al. [37].
Three-dimensional model of the two RC frames (a) 4-story (b) 8-storyTypical floor plan of the two RC frame structuresDynamic properties of the two RC frames
Frame
Period
T_{1}(s)
T_{2}(s)
T_{3}(s)
4-story
Transverse
0.55
0.17
0.08
Longitudinal
0.55
0.16
0.08
8-story
Transverse
1.11
0.34
0.18
Longitudinal
1.09
0.34
0.18
Selection of Ground Motions
Thirty pairs of far-field ground motions (see Table 3) are selected here from the PEER strong motion database [42–44] according to the selection criteria recommended in FEMA P695 [45]. The source-to-distance of the far-field ground motions is greater than 10 km, and enough records from large-magnitude earthquake events are chosen to ensure record-to-record variability. The moment magnitudes of earthquakes are larger than 6.2 with a mean value of 7.0. PGA_{1} (PGA of Component 1) and PGA_{2} (PGA of Component 2) of the records are listed in Table 3. The average value of the maximum peak ground acceleration PGA_{max} of the two components is 0.46 g. The acceleration response spectra of the ground motions with a 5% damping ratio are illustrated in Fig. 3.
Selected far-field ground motion records
ID No.
Earthquake
Component 1
Component 2
Station
Year
PGA_{1}(g)
PGA_{2}(g)
M
R(km)
1
San Fernando
SFERN/PEL090
SFERN/PEL180
LA-Hollywood Stor FF
1971
0.22
0.19
6.61
22.77
2
Imperial Valley-06
IMPVALL.H/H-DLT262
IMPVALL.H/H-DLT352
Delta
1979
0.24
0.35
6.53
22.03
3
Imperial Valley-06
IMPVALL.H/H-E11140
IMPVALL.H/H-E11230
El Centro Array #11
1979
0.37
0.38
6.53
12.56
4
Morgan Hill
MORGAN/G03090
MORGAN/G03000
Gilroy Array #3
1984
0.20
0.19
6.19
13.02
5
Superstition Hills-02
SUPER.B/B-ICC000
SUPER.B/B-ICC090
El Centro Imp. Co. Cent
1987
0.36
0.26
6.54
18.2
6
Superstition Hills-02
SUPER.B/B-IVW360
SUPER.B/B-IVW090
Imperial Valley Wildlife Liquefaction Array
1987
0.21
0.18
6.54
23.85
7
Loma Prieta
LOMAP/A02043
LOMAP/A02133
APEEL 2-Redwood City
1989
0.27
0.22
6.93
43.23
8
Loma Prieta
LOMAP/AND250
LOMAP/AND340
Anderson Dam (Downstream)
1989
0.25
0.24
6.93
20.26
9
Loma Prieta
LOMAP/OHW000
LOMAP/OHW270
Oakland-Outer Harbor Wharf
1989
0.29
0.27
6.93
74.26
10
Loma Prieta
LOMAP/SFO000
LOMAP/SFO090
SF Intern. Airport
1989
0.24
0.33
6.93
58.65
11
Landers
LANDERS/CLW-LN
LANDERS/CLW-TR
Coolwater
1992
0.28
0.28
7.28
19.74
12
Landers
LANDERS/YER270
LANDERS/YER360
Yermo Fire Station
1992
0.24
0.15
7.28
23.62
13
Kobe/Japan
KOBE/ABN090
KOBE/ABN000
Abeno
1995
0.23
0.22
6.9
24.85
14
Kobe/Japan
KOBE/FKS090
KOBE/FKS000
Fukushima
1995
0.22
0.18
6.9
17.85
15
Kocaeli/Turkey
KOCAELI/ARE000
KOCAELI/ARE090
Arcelik
1999
0.21
0.13
7.51
13.49
16
Kocaeli/Turkey
KOCAELI/DZC180
KOCAELI/DZC270
Duzce
1999
0.31
0.36
7.51
15.37
17
Chi-Chi/Taiwan
CHICHI/CHY101-E
CHICHI/CHY101-N
CHY101
1999
0.34
0.40
7.62
9.94
18
Chi-Chi/Taiwan
CHICHI/TCU045-E
CHICHI/TCU045-N
TCU045
1999
0.47
0.51
7.62
26
19
Duzce/Turkey
DUZCE/BOL000
DUZCE/BOL090
Bolu
1999
0.74
0.81
7.14
12.04
20
Hector Mine
HECTOR/HEC000
HECTOR/HEC090
Hector
1999
0.27
0.33
7.13
11.66
21
Loma Prieta
LOMAP/WAH000
LOMAP/WAH090
WAHO
1989
0.37
0.65
6.93
17.47
22
Northridge-01
NORTHR/TAR360
NORTHR/TAR090
Tarzana-Cedar Hill A
1994
0.99
1.78
6.69
15.6
23
Chi-Chi/Taiwan
CHICHI/TCU088-E
CHICHI/TCU088-N
TCU088
1999
0.52
0.53
7.62
18.16
24
Chi-Chi/Taiwan
CHICHI/TCU095-E
CHICHI/TCU095-N
TCU095
1999
0.37
0.70
7.62
45.18
25
Niigata/Japan
NIIGATA/NIG023EW
NIIGATA/NIG023NS
NIG023
2004
0.28
0.40
6.63
25.82
26
Chuetsu-oki/Japan
CHUETSU/65005EW
CHUETSU/65005NS
JoetsuUragawaraku Kamabucchi
2007
0.56
0.26
6.8
22.74
27
Chuetsu-oki/Japan
CHUETSU/65025EW
CHUETSU/65025NS
Kashiwazaki City Center
2007
0.65
0.48
6.8
11.09
28
Chuetsu-oki/Japan
CHUETSU/65056EW
CHUETSU/65056NS
Kashiwazaki City Takayanagicho
2007
0.36
0.73
6.8
20.03
29
Chuetsu-oki/Japan
CHUETSU/65057EW
CHUETSU/65057NS
OguniNagaoka
2007
0.63
0.51
6.8
20
30
Chuetsu-oki/Japan
CHUETSU/6CB51EW
CHUETSU/6CB51NS
TamatiYoneIzumozaki
2007
0.50
0.63
6.8
11.48
Acceleration response spectra of the selected records with 5% damping ratioMultiple Stripes Analysis and Criteria for Each Damage Limit State
The multiple stripes analysis method is selected here to obtain the fragility curves at different damage limit states, as it does not need to scale all the ground motions to the IM levels that cause the damage limit state of interest. Beacuse multiple stripes analysis can only produce the fractions of the damage limit state at some IM levels, the maximum likelihood estimation method, an appropriate fitting approach for the multiple stripes analysis method, is utilized [5,21]. The seismic response from each ground motion is assumed to be independent of the results from other ground motions. The maxima of fragility curve parameters can be obtained by Eq. (1). A detailed introduction to the application of the fitting approach in multiple stripes analysis can be found in Baker [21].
{η^,β^RTR}=argmaxη,βRTR∑j=1m{ln(njzj)+zjlnΦ(ln(IMj/η)βRTR)+(nj−zj)ln[1−Φ(ln(IMj/η)βRTR)]}
where η is the median capacity at a specific limit state; β_{RTR} is the lognormal standard deviation, which represents the record-to-record variability; m is the number of IM levels; Φ() is the standard normal cumulative distribution function; z_{j} is the number of observations of the limit state out of n_{j} ground motions in the case of intensity level IM_{j}.
The bidirectional ground motions in Table 3 are applied to the two RC frames to obtain the fragility curves at different damage limit states by using multiple stripes analyses. Component 1 of the ground motions is applied to the transverse direction of the frames, and Component 2 is applied to the longitudinal direction. The amplitude scaling is based on Component 1, and the same scale factor is used for Component 2 to preserve the relative amplitude of the two components of the record. In the fragility analysis, the criteria of each damage limit state are determined according to FEMA 356 [46] as shown in Table 4. For the immediate occupancy (IO) limit state, the maximum inter-story drift ratio θ_{max} of RC frames is set as 1%; for the life safety (LS) limit state, the ratio θ_{max} is suggested as 2%; and for the collapse prevention (CP) limit state, the ratio θ_{max} is set as 4%.
It should be noted that the number of IM levels m (see Eq. (1)) is a key parameter for the fitted results in the multiple stripes analysis. If the number of IM levels is larger, the computational cost increases, while if the number of IM levels is smaller, the accuracy of the results cannot be guaranteed. Eads et al. [47] found that the analysis result is mainly affected by the IM levels at the lower half of the fragility curve. Additionally, Baker [21] pointed out that the fragility function can be effectively estimated by the IM levels with a low probability. Héctor Dávalos et al. [48] recommended that it is better to conduct multiple stripes analyses at only two intensity levels, namely, enhanced two-stripe analysis (E2SA). Zhang et al. [44] suggested using three IM levels with two levels lower than the estimated median limit capacity in the multiple stripes analysis. In general, an accurate result can be obtained by the IM levels focused on the lower half of the fragility curve. Therefore, six IM levels with PGA_{1} (i.e., PGA of Component 1of the records) equal to 0.25, 0.35, 0.45, 0.85, 1.25 and 1.65 g are used in the multiple stripes analysis of the three fragility limit states in this paper. This strategy ensures that there are no less than two fraction points in the lower half of the fragility curves of the three limit states, as shown in Figs. 4 to 5.
Criteria for each damge limit state of RC frames
Damage limit state
Maximum inter-story drift ratio θ_{max}
Immediate Occupancy (IO)
1%
Life Safety (LS)
2%
Collapse Prevention (CP)
4%
Fragility curves characterized by PGA<sub>1</sub> for the 4-story building (a) maximum inter-story drift ratio θ<sub>max</sub> under different PGA<sub>1</sub> levels; (b) fragility curve estimated for IO limit state; (c) fragility curve estimated for LS limit state; (d) fragility curve estimated for CP limit state
In Fig. 4, the median capacity at IO limit state η_{IO} for the 4-story building is 0.42, the median capacity at LS limit state η_{LS} is 0.75 g and the median capacity at CP limit state η_{CP} is 1.22 g. In Fig. 5, the median capacity at IO limit state η_{IO} for the 8-story building is 0.36 g, the median capacity at LS limit state η_{LS} is 0.72 g and the median capacity at CP limit state η_{CP} is 1.55 g. From the IO to CP limit states, the lognormal standard deviations β_{RTR} increase from 0.41 to 0.44 and then to 0.43 in the case of the 4-story building. However, in the case of 8-story building, β_{RTR} is 0.75, 0.57 and 0.61, which do not show a certain regularity. This may be caused by the characteristics of the building itself.
Fragility curves characterized by PGA<sub>1</sub> for the 8-story building (a) maximum inter-story drift ratio θ<sub>max</sub> under different PGA<sub>1</sub> levels; (b) fragility curve estimated for IO limit state; (c) fragility curve estimated for LS limit state; (d) fragility curve estimated for CP limit stateEstimation of Aleatory Randomness
In addition to the fourteen IMs listed in Table 1, three simple amplitude-type IMs PGA, peak ground velocity (PGV) and peak ground displacement (PGD) are also evaluated here. For S*, α is set as 0.5 and T_{f} is set to be 2T_{1}. For IM-CR and IM-SR, α is set as 0.5 and R_{IM} is taken as 2. In I_{Np}, T_{N} is taken as 2T_{1} and α is set as 0.4. The original parameters in Table 1 are used in Sa12∗ and Sa123∗. The first two vibration periods and the first three vibration periods are applied to IM_{12} and IM_{123}, respectively. For S_{N1}, C is taken as 1.5 and α is taken as 0.5, and for S_{N2}, β is set to be 0.75. In S¯a, n is taken as 2. Five spectral values, (T_{i})_{5} = {T_{2m}, min[(T_{2m} + T_{1m})/2,1.5 T_{2m}], T_{1m}, 1.5T_{1m}, 2T_{1m}}, is used in S_{a,gm}(T_{i}), in which the class-average vibrations periods, T_{1m} and T_{2m}, are taken as the first two periods. In IM_{opt}, the un-fixed lower bound period T_{0.95M} is estimated according to the aforementioned formula about the story number of the buildings.
Lucchini et al. [49] and Kostinakis et al. [50] pointed out that the characterization methods of bidirectional ground motion intensity using a single value influence the correlation between the ground motion IMs and engineering performance DM. Therefore, the influence of different characterization methods on the record-to-record variability in the fragility analysis is also investigated herein. As shown in Table 5, six characterization methods for bidirectional ground motion intensity are used here [50].
Characterization methods for bidirectional ground motion intensity
Expression
Definition
IM_{1}
The IM value of Component 1
IM_{2}
The IM value of Component 2
IMAMV=IM1+IM22
Arithmetic mean value (AMV)
IMGMV=IM1⋅IM2
Geometric mean value (GMV)
IMSRSS=IM12+IM22
Square root of the sum of squares (SRSS)
IM_{max} = max(IM_{1}, IM_{2})
Maximum value
It should be noted that each ground motion has a different IM level when using different characterization methods or IMs other than PGA_{1}. In such a situation, to estimate the fragility curves, n_{j}in Eq. (1) is set to 1, and the probability of a particular limit state is set to 1 if θ_{max} is larger than the corresponding criterion. Taking S_{a}(T_{1}) as an example, the analysis processes of the 4-story frame using the two bidirectional characterization methods, i.e., IM_{1} and IM_{SRSS}, are shown in Figs. 6 and 7, respectively. Figs. 6 and 7 show that β_{RTR} estimated by S_{a}(T_{1}) increases roughly with the shift of the limit state from IO to CP in both the bidirectional ground motion intensity characterization methods IM_{1} and IM_{SRSS}.
Fragility curves characterized by <italic>S<sub>a</sub></italic>(<italic>T<sub>1</sub></italic>) in the case of IM1 for the 4-story frame (a) maximum inter-story drift ratio θ<sub>max</sub> under different IM<sub>1</sub>-<italic>S<sub>a</sub></italic>(<italic>T<sub>1</sub></italic>)) levels; (b) fragility curve estimated for IO limit state; (c) fragility curve estimated for LS limit state; (d) fragility curve estimated for CP limit state Fragility curves characterized by <italic>S<sub>a</sub></italic>(<italic>T<sub>1</sub></italic>) in the case of IMSRSS for the 4-story frame (a) maximum inter-story drift ratio θ<sub>max</sub> under different IM<sub>SRSS</sub>-<italic>S<sub>a</sub></italic>(<italic>T<sub>1</sub></italic>) levels; (b) fragility curve estimated for IO limit state; (c) fragility curve estimated for LS limit state; (d) fragility curve estimated for CP limit state
Figs. 8 to 9 illustrate the uncertainties of the ground motion β_{RTR} estimated in three limit states of the two buildings using different bidirectional characterization methods. It can be seen from the figures that the limit state of the structure has an important influence on the applicability of the ground motion IM. With the development of the damage state, that is, from the IO limit state to the CP limit state, the uncertainties of the ground motion β_{RTR} corresponding to each IM generally show an increasing trend.
Record-to-record variability <italic>β</italic><sub>RTR</sub> estimated in fragility analysis of the three limit states for the 4-story building by using six bidirectional characterization methods (a) IM<sub>1</sub>, IM<sub>2</sub> and IM<sub>AMV</sub> (b) IM<sub>GMV</sub>, IM<sub>SRSS</sub> and IM<sub>max</sub>
For both 4- and 8-story buildings, the IMs considering the effect of higher modes, such as Sa12∗, Sa123∗, IM_{12}, IM_{123}, S_{N2} and S¯a, are not advantageous with larger β_{RTR} compared to S_{a}(T_{1}) in the three limit states for most bidirectional characterization methods. This may be because the spectral acceleration corresponding to the higher modes is not highly correlated with the structural damage in low and medium buildings [43]. However, the IMs considering the effect of the softened period, such as S*, IM-CR, IM-SR, I_{NP} and S_{N1}, even the IM S_{a,gm}(T_{i}) which considers both the effect of the softened period and higher modes, have lower β_{RTR} relative to S_{a}(T_{1}) and those IMs considering higher modes in most cases. Moreover, as damage develops, the advantages of IMs considering the effect of the softened period become increasingly obvious, especially for S*. This may be because the effect of the softened period becomes increasingly significant with the deepening of nonlinearity and damage [51], and the correlation between the structural response and the spectral acceleration corresponding to the softened period is strengthening.
With respect to most bidirectional characterization methods, both dual-parameter IMs S* and S_{N1} can maintain lower β_{RTR} in the three limit states for the two buildings. Although IM-CR, IM-SR and I_{NP} are also combination-type ground motion IMs that consider the effect of period elongation, their differences in the selected softened period or the power exponents from S* and S_{N1} caused significant differences in the evaluation of ground motion uncertainty. It is worth noting that in the IO state, the β_{RTR} of S_{N1}, IM-CR and IM-SR is smaller than S*. The selected softened periods in S_{N1}, IM-CR and IM-SR are smaller than that of S*. This shows that the selected softened period should not be a fixed value in different damage states. As analyzed, when the appropriate softened period is selected, that is, the appropriate parameter C in T_{f} = CT_{1}, the smallest β_{RTR} can be obtained. For the three amplitude-type IMs, PGV has apparent advantages in the three limit states of the buildings. Although IM_{opt} considered the impact of the number of floors in the structure, its performance is not satisfactory, which may be due to the failure to consider the fundamental period of the structure.
Record-to-record variability <italic>β</italic><sub>RTR</sub> estimated in fragility analysis of the three limit states for the 8-story building by using six bidirectional characterization methods (a) IM<sub>1</sub>, IM<sub>2</sub> and IM<sub>AMV</sub> (b) IM<sub>GMV</sub>, IM<sub>SRSS</sub> and IM<sub>max</sub>
The uncertainties estimated by a specific IM using different bidirectional ground motion intensity characterization methods are different, as shown in Figs. 8 and 9. It can be seen from the above analysis that S* and S_{N1} have apparent advantages in estimating of the uncertainty of ground motion. Taking S* and S_{N1} as examples, the β_{RTR} estimated in the six bidirectional characterization methods are illustrated in Figs. 10 and 11, respectively. Since only one direction of ground motion intensity is considered, there is a large difference between IM_{1} and IM_{2}. Structural damage may be related to the intensity of ground motion in both directions. When considering the combination of the intensity in the two directions, IM_{AMV}, IM_{GMV} and IM_{SRSS} can predict relatively stable and similar β_{RTR} in the two structures for each limit state. The β_{RTR} obtained by the two methods is close to the minimum in most cases.
<italic>β</italic><sub>RTR</sub> estimated by <italic>S*</italic> with the six bidirectional characterization methods<italic>β</italic><sub>RTR</sub> estimated by <italic>S</italic><sub>N1</sub> with the six bidirectional characterization methodsImproved IM for Fragility Analysis at Different Limit StatesCorrelation Analysis
From the analysis in the previous section, it can be concluded that the selected softened period in the dual-parameter IM (see Eq. (2)) should not be a fixed value in different damage states. As analyzed, when the appropriate softening period is selected, that is, parameter C in T_{f} = CT_{1} as listed in Eq. (2), a smaller β_{RTR} can be obtained. Shown in Fig. 12 are the β_{RTR} estimated by the spectral acceleration at any vibration period of interest. It should be noted that the geometric mean value IM_{GMV} is used here to consider the bidirectional ground motion intensities. Obviously, in the fragility analysis at the three limit states, all the minimum β_{RTR} values are achieved at the softened period in the three structures. As the damage intensifies, the point where the minimum β_{RTR} is obtained moves to the right. As shown in the figure, the point of obtaining the minimum β_{RTR} moves from position A to position B and finally moves to position C from IO to LS and then to CP. That is, as the damage intensifies, the spectral acceleration at a longer softened period has a better correlation with structural damage, namely, a smaller dispersion is achieved. The softened period T_{f} in Eq. (2) should be adjusted according to the damage limit state of the structure in the fragility analysis.
S∗=(Sa(T1))1−α(Sa(Tf))α
<italic>β</italic><sub>RTR</sub> estimated by the spectral acceleration at any vibration period for the three limit states
In addition, Fig. 12 shows that the spectral acceleration corresponding to the higher modes often achieves a higher β_{RTR}, which is why the IMs considering the effect of higher modes do not perform well, as analyzed in the previous section. Studies have shown that the spectral acceleration corresponding to the higher modes can achieve a higher correlation with structural damage in super high-rise buildings [33,43].
Optimal C and α in Dual-Parameters IM
In this section, the optimal multiplier C and exponent α suitable for fragility analysis at different damage limits are suggested through parameter analysis. Figs. 13 to 14, presented in the form of contour maps, illustrate the influence of different combinations of C and α on standard deviations, β_{RTR}. The blue part depicts a smaller value of β_{RTR}, and the red part depicts a larger value of β_{RTR}. Note that the geometric mean value IM_{GMV} is used here to reflect the bidirectional ground motion intensity.
Under different combinations of C and α, the lognormal standard deviations β_{RTR} are different and reflect apparent regularity. The optimal combination of the parameters, C and α, is determined when β_{RTR} reaches its minimum. It can be seen from the figure that the position of the smaller dispersion β_{RTR} (see the blue part) is shifted to the right part with a larger C as the damage increases in each structure. That is, structural damage has a better correlation with a longer softening period from the IO limit state to the CP limit state. In addition, the position to achieve a smaller dispersion β_{RTR} moves downward, especially from the IO limit state to the LS limit state. That is, the weight of the softened period, i.e., the combination coefficient α in the IM, is decreasing.
Relation between <italic>β</italic><sub>RTR</sub> and <italic>C, α</italic> for the 4-story buildingRelation between <italic>β</italic><sub>RTR</sub> and <italic>C, α</italic> for the 8-story building
Table 6 lists the ranges of C and α by which the two structures can achieve relatively smaller dispersion β_{RTR} under the three limit states. As seen from the table, the value of C increases from IO to CP. In addition, α is getting smaller, especially from IO to LS. Thus, to achieve analytical simplicity, for the IO state, it is recommended to set α to 0.9 and C to 1.3; for the LS state, it is recommended to set α to 0.7 and C to 1.8; for the CP state, it is recommended to set α to 0.6 and C to 2.4, as shown in the improved IM SIM∗ in Eq. (3). The values suggested here do not allow the two structures to achieve the minimum β_{RTR}, but it is guaranteed that the β_{RTR} at the proximity of its minimum can be achieved.
SIM∗=(Sa(T1))1−α(Sa(CT1))α,{α=0.9,C=1.3forIOlimitstateα=0.7,C=1.8forLSlimitstateα=0.6,C=2.4forCPlimitstate
The ranges of <italic>C</italic> and <italic>α</italic> to obtain relatively smaller <italic>β</italic><sub>RTR</sub>
Building
C
α
IO
LS
CP
IO
LS
CP
4-story
1.2∼1.4
1.7∼2.0
2.4∼2.7
0.8∼1.0
0.4∼0.8
0.5∼0.8
8-story
1.2∼1.4
1.6∼1.9
2.3∼2.8
0.8∼1.0
0.6∼0.9
0.5∼0.8
Fig. 15 shows the dispersion β_{RTR} estimated with the improved IM by adopting the parameters suggested in Eq. (3). Compared with S* and S_{N1} (see Figs. 10 and 11), the improved IM SIM∗ can achieve lower dispersion in all the bidirectional characterization methods. Similar to S* and S_{N1}, the improved IM SIM∗ achieves relatively higher dispersion in the methods of IM_{2} and IM_{1}.
<italic>β</italic><sub>RTR</sub> estimated by <inline-formula id="ieqn-22"><mml:math id="mml-ieqn-22"><mml:msubsup><mml:mi>S</mml:mi><mml:mrow><mml:mi>I</mml:mi><mml:mi>M</mml:mi></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:math></inline-formula> with the six bidirectional characterization methodsRisk Assessment by the Improved IM
In this section, the improved IM is further applied to the risk assessment at the three limit states for the 8-story frame structure according to Chinese codes. Generally, the fragility curve corresponding to one limit state is assumed to be a lognormal cumulative distribution function (see Eq. (4)) as used above in Eq. (1).
P(Limitstate∣IM)=Φ(ln(IM/η)βRTR)
where P(Limitstate∣IM) is the probability of exceeding one specific limit state caused by a ground motion at the intensity level IM, and η and β_{RTR} are the median capacity and lognormal standard deviation, respectively. The mean hazard curve, λ¯IM, as shown in Eq. (5), can be estimated by fitting multiple hazard data.
λ¯IM(IM)=k0IM−k
where k_{0} and k are two empirical factors. These S_{a}(T_{1}) and S_{a}(T_{f}) corresponding to the frequent earthquake (63.2% probability of exceedance in 50 years), the design basis earthquake (10% probability of exceedance in 50 years) and the maximum considered earthquake (2% probability of exceedance in 50 years) are obtained referring to the Code for Seismic Design of Buildings [38] for the regions with seismic precautionary intensity of 8. Then, the SIM∗ corresponding to the three limit states can be obtained by adopting the suggested parameters in Eq. (3). The annual probability of exceedance\; λ¯IM for each SIM∗ can be obtained by assuming that the ground motion intensity level is Poisson distributed. Subsequently, the seismic hazard curves with respect to the three limit states of the 8-story frame can be obtained by fitting Eq. (5) with three discrete points (SIM∗, λ¯IM) at each limit state, as illustrated in Fig. 16. Due to the different IMs used in each limit state, the hazard risk curve in each limit state is slightly different.
Mean hazard curves expressed by <inline-formula id="ieqn-23"><mml:math id="mml-ieqn-23"><mml:msubsup><mml:mi>S</mml:mi><mml:mrow><mml:mi>I</mml:mi><mml:mi>M</mml:mi></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:math></inline-formula> in the cases of three limit states for the 8-story building
The annual probability of exceeding one specific limit state, as listed in Eq. (6), can be obtained through the total probability theory by combining the fragility curve (Eq. (4)) with the seismic hazard curve (Eq. (5)).
λlimitstate=∫P(Limitstate∣IM)⋅|dλ¯IM(IM)|
Furthermore, Eq. (7) can be obtained by substituting and simplifying Eq. (6) [52]. It should be noted that there is no consideration of epistemic uncertainty in Eq. (7).
λlimitstate=λ¯IM(η)exp(12k2βRTR2)=k0η−kexp(12k2βRTR2)
Table 7 shows the annual probability of exceeding each limit state for the 8-story frame structure. Note that the median capacity η and lognormal standard deviation β_{RTR} are estimated by SIM∗ using the bidirectional intensity characterization method of IM_{GMV}, as shown in Fig. 17. For the 8-story frame structure, the annual probability of exceeding the CP limit state is 7.40 × 10^{−5}, which is lower than the acceptable annual collapse probability of 1 × 10^{−4} for Category-C buildings determined in Zhang et al. [53].
Fragility curves characterized by <inline-formula id="ieqn-24"><mml:math id="mml-ieqn-24"><mml:msubsup><mml:mi>S</mml:mi><mml:mrow><mml:mi>I</mml:mi><mml:mi>M</mml:mi></mml:mrow><mml:mo>∗</mml:mo></mml:msubsup></mml:math></inline-formula> in the case of IM<sub>GMV</sub> for the 8-story frameAnnual probability of exceeding each limit state for the 8-story frame structure
Building
Annual probability of exceeding each limit state
IO
LS
CP
8-story
4.5 × 10^{−4}
1.76 × 10^{−4}
7.40 × 10^{−5}
Conclusions
Based on two low- and medium-rise RC frame structures, an investigation is first provided in this study regarding the estimation of record-to-record variability β_{RTR} by using S_{a}(T_{1})-based IMs in the fragility curves of three limit states (i.e., IO, LS and CP). Subsequently, the optimal multiplier C and exponent α in the dual-parameter IM for different limit states are suggested through parameter analysis. Furthermore, the improved dual-parameter IM is applied to the risk assessment at the three limit states. Several observations can be reached from the case study on the RC frame structure, as follows:
The limit state of the structure has an important influence on the applicability of S_{a}(T_{1})-based IMs. From the IO limit state to the CP limit state, the uncertainties of the ground motion β_{RTR} corresponding to most IMs generally show an increasing trend. For low- and medium–rise RC frame structures, the S_{a}(T_{1})-based IMs considering the effect of a softened period, i.e., S* and S_{N1}, can maintain a lower β_{RTR} in the three limit states compared to the S_{a}(T_{1})-based IMs considering the effect of higher modes.
With increasing structural damage, spectral acceleration and structural damage have a better correlation at longer softening period. The selected softened period in the dual-parameter IM should not be a fixed value in different damage states. In addition, as the damage develops, the combination index α in dual-parameter IM should be reduced to obtain a better correlation. Taking into account the different damage states, a dual variable-parameter is adopted in the improved IM.
The record-to-record variability estimated using different bidirectional ground motion intensity characterization methods with respect to a specific IM is continually varying. The characterization method that only considers the intensity of ground motion in a single direction, IM_{1} or IM_{2}, lacks stability. When considering the combination of the intensity in the two directions, IM_{AMV}, IM_{GMV} and IM_{SRSS} can predict relatively stable and similar β_{RTR}. By adopting IM_{GMV}, the annual probabilities of exceeding each limit state for the 8-story frame structure were analyzed.
Only two typical low- and medium-rise frame buildings with 4 and 8 stories are employed here. In addition, only 30 pairs of far-field records are selected in this paper. More structure cases and ground motion records with different spectral characteristics, such as near-field records, are needed to verify the suggested parameters in the improved IM.
Funding Statement: This research was financially supported by the Jiangsu Youth Fund Projects (SBK2021044269), the National Natural Science Foundation of China Youth Fund (52108457, 52108133), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (20KJB560014), Fundamental Research Funds for the Central Universities (B210201019), High-level Talent Research Fund of Nanjing Forestry University (163050115), Nanjing Forestry University Undergraduate Innovation Training Program (2021NFUSPITP0221, 2020NFUSPITP0352 and 2020NFUSPITP0373), and Jiangsu Undergraduate Innovation Training Program (202110298079Y).
Conflicts of Interest: The authors declare that they have no conflicts of interest to report regarding the present study.
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