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Seismic fragility for high CFRDs based on deformation and damage index through incremental dynamic analysis
Soil Dynamics and Earthquake Engineering 104 (2018) 432–436
Contents lists available at ScienceDirect
Soil Dynamics and Earthquake Engineering journal homepage: www.elsevier.com/locate/soildyn
Seismic fragility for high CFRDs based on deformation and damage index through incremental dynamic analysis
T
⁎
Rui Pangb, Bin Xua,b, , Xianjing Konga,b, Degao Zoua,b a b
State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, China School of Hydraulic Engineering, Faculty of Infrastructure Engineering, Dalian University of Technology, Dalian 116024, China
A R T I C L E I N F O
A B S T R A C T
Keywords: High CFRDs Deformation Damage index IDA Fragility Seismic performance
In this paper, a seismic fragility analysis method based on incremental dynamic analysis (IDA) is extended to evaluate the seismic performance of high concrete face rockfill dams (CFRDs). Permanent deformation and faceslab damage index using a modified generalized plasticity model for rockfills and a plastic-damage model for face-slabs are considered to be dam damage measures (DMs) after defining a new face-slab damage index. The verification to damage index through the Zipingpu CFRD and previous research indicates that the grading standards are reasonable. Fragility curves and the probabilities are determined for each DM under different earthquake intensities. The results of fragility analysis demonstrate that this method can provide a strong scientific basis for predicting the earthquake destruction and loss of high CFRDs.
1. Introduction
2. Fragility analysis method based on IDA
In China, many high concrete face rockfill dams (CFRDs) have been built or designed. These dams are commonly distributed in areas experiencing strong ground motions, therefore, seismic performance assessments must be performed for these dams. Seismic fragility analysis is one of the most effective methods to evaluate seismic performance. This method can predict when these structures will reach or exceed a certain limit state's probabilities under different strengths of seismic action and employs fragility curves or matrices to describe the probability distributions of all limit states. IDA is a parametric analysis method based on nonlinear dynamic time history analysis and has been widely used in structural fragility analysis [1]. However, due to the complexity of high CFRDs, there have been few related reports on their fragility analysis based on IDA applied in this engineering field. In this paper, a seismic performance assessment for high CFRDs is performed based on a fragility analysis using IDA. Deformation and face-slab damage are the two major forms of destruction of CFRDs; CFRDs that have exhibited such destruction as the Zipingpu CFRD in 2008 (156 m) [2]. Considering the nonlinear characteristics of concrete, Lee and Fenves [3] proposed a plastic damage model to independently determine the damage in pull and pressure modes and the stiffness recovery phenomenon in the reverse loading of concrete, and this model has been successfully applied to concrete face-slabs [4]. Thus, it is reasonable and feasible to regard the relative settlement of dam crests and damage index as evaluation indices of seismic performance.
IDA method: Several scholars have attempted to introduce IDA into the preliminary safety assessment of dams. For example, Kong and Pang et al. [5] first introduced IDA to seismic safety assessment of high CFRDs based on three aspects permanent deformation, stability of dam slope, safety of face-slabs for the first time using an equivalent linear constitutive model, and gained the fragility curves and probabilities. Hariri-Ardebili and Saouma [6] applied IDA to obtain the collapse fragility curves of concrete dams. Mohammad Alembagheri and Mohsen Ghaemian [7] performed a damage assessment of a typical arch dam through IDA subjected to a set of 12 earthquakes, and damage propagation was investigated and various IDA curves were created. These studies achieved a preliminary assessment of dam safety and demonstrated that the IDA method is suitable for large water and hydropower engineering. Fragility analysis method: The seismic fragility curves provide the conditional probabilities of the structural response reaching or exceeding certain limit states corresponding to the seismic capacity under different earthquake intensities. After calculating the response of the structures under different ground motion intensities with the IDA, the relationships between the limit states and DMs are quantified by combination with the definition of the limit states, and then, the seismic fragility is determined. According to the previous research [8], in general, the seismic fragility is assumed to follow a lognormal cumulative distribution function of double parameters, defined as follows:
⁎
Corresponding author at: State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, China. E-mail address: [email protected] (B. Xu).
https://doi.org/10.1016/j.soildyn.2017.11.017 Received 8 September 2017; Received in revised form 8 November 2017; Accepted 9 November 2017 0267-7261/ © 2017 Elsevier Ltd. All rights reserved.
Soil Dynamics and Earthquake Engineering 104 (2018) 432–436
R. Pang et al.
PR (C|IM = X ) = Φ ⎡ ⎣
ln(Χ / θ) ⎤ σ ⎦
For concrete dams, for example, Wang and Zhang et al. [13] considered the influence of the damage location on the overall structures of gravity dams and proposed the following dividing values: a damage index of 0.05 for intact and minor failure, a damage index of 0.15 for minor and moderate failure, and a damage index of 0.45 for moderate and severe failure; the upper limit of severe failure was 0.75. Mohammad Alembagheri and Mohsen Ghaemian [7] carried out damage assessment for a typical arch dam through nonlinear IDA. In this paper, referring to the structural damage index described above and considering the importance of the location of the damage on the CFRDs, we believe that the higher the damage location is, however the smaller the impact will be on the structure; finally, we define the damage index. In accordance with the research results of Xu et al. [14], the face-slab damage mainly occurs in the range of 0.4H-0.9 H (H: dam height) under the action of an earthquake. The damage index of face slabs is proposed as:
(1)
3. Seismic records and performance parameters 3.1. Selection of seismic records In this paper, the ground motion inputs adopt actual and artificial seismic records respectively. The response spectra based on the site conditions of a high earth-rockfill dam in the southwest of China are chosen to be the target response spectra, as described in Formula (2), β(T) is the magnification response spectrum of ground motion acceleration, and T1 = 0.12 s, T2 = 0.34 s, βmax = 2.5, and γ = 1.0. According to the proposal of Vamvatsikos et al. [9], 10–20 earthquake records meet the requirements of IDA analysis. First, 10 actual seismic records which are well agreeable with the target response spectra based on site conditions are selected in PEER [10]. Then one seismic wave is artificially generated based on the target response spectrum and is used for a comparison purpose. The acceleration response spectra of 11 earthquake waves are shown in Fig. 1.
⎧1 ⎪1 + (βmax − 1) T − 0.04 ⎪ T1 − 0.04 β (T ) = ⎨ βmax ⎪ T ⎪ βmax ( T2 )γ ⎩
n
DI = α⋅
n ∑i = 1
(
0.9H − hi 0.25H
0.9H − h Si⋅ 0.25H i
)
)
(3)
where di is the damage factor of the ith element of the face slab, n is the number of face slab elements in 0.4–0.9 H, Si is the area of the ith element of the face slab, hi is the center height to the face-slab bottom of the ith element of the face slab, DI is the damage index of the face slabs; and α is the influence coefficient, which is generally 2.0. As shown in Fig. 2. In this paper, referring to the related literatures of the failure grading of gravity dam, we regard DI = 0.03, 0.15, and 0.45 as the dividing values of minor, moderate and severe failure, respectively.
T ≤ 0.04s 0.04s < T ≤ T1 T1 < T ≤ T2 T2 < T ≤ 6s
(
∑i = 1 d i⋅Si⋅
(2)
3.2. Proposal of dam failure grades 4. Finite element analysis
Relative settlement ratio of the dam crest: Kong and Pang et al. [5] considered relative settlement ratios of 0.4%, 0.7%, and 1% of the dam crest (crest settlement values/height of the dam) as the assessment limitation and analyzed the fragility when this dam exhibited minor, moderate and severe failure. Swaisgood et al. [11] surveyed 69 dams, regarded the relative settlement ratio of the dam crest as an index, and divided the degree of earthquake damage into four failure grades: healthy (< 0.1%), minor (0.012–0.5%), moderate (0.1–1.0%) and severe (> 0.5%). In this paper, we refer to the related safety assessment and grading standards described above and consider that the simulated maximum post-earthquake deformation results based on the generalized plastic model are smaller than measured values [12], e.g., the maximum settlement deformation simulated with a generalized plasticity model is about 0.77 m, whereas the actual measured value was 1 m for the Zipingpu CFRD. Finally, we establish three limit failure states with relative settlement ratios of 0.2%, 0.4%, and 0.6% of the dam crest corresponding to three failure grades (minor, moderate and severe). Damage index of face-slabs: In the earthquake damage prediction and post-earthquake evaluation of concrete structures, many scholars use the damage index to quantitatively describe the degree of failure.
In this paper, the distribution characteristics and change rules of every DM under different seismic records with different earthquake intensities are analyzed in detail by performing two-dimensional (2-D) nonlinear finite element numerical calculations for a typical CFRD with a height of 250 m (Fig. 2) based on GEODYNA [15]. The static and dynamic calculations of the rockfill, transition and cushion are all simulated by the modified generalized plasticity constitutive model [16], and the model parameters are shown in Table 1. The contact between the face slab and cushion is simulated by the generalized plasticity interface model [17], whose parameters are based on the literature [18]. The plastic-damage behavior of concrete face-slabs simulated with the plastic-damage model [4,14], and the parameters are obtained from the literature [4]. Ground motion inputs are added in the form of a viscoelastic boundary combined with an equivalent load at the boundary of the finite element model to simulate the interaction of finite fields with infinite domains [19]. 5. Results of IDA and fragility analysis 5.1. IDA results This study selects the PGA of the earthquake as the earthquake IM and modulates the amplitude at equal intervals (the range is 0.1 g) until the failure of different DMs to severity. The IDA curves corresponding to different ground motion records of the PGA-relative settlement ratio of the dam crest and PGA-damage index are obtained after a large number of nonlinear finite element calculations, as shown in Fig. 3. It may be seen from Fig. 3(a) that with the increase of the IM value, the changes of the relative settlement ratio of the dam crest become slower, which indicates that the rockfills are denser under a stronger earthquake. However, Fig. 3(b) shows that the damage index changes slowly under a weak earthquake and quickly under a strong earthquake, which demonstrates that the face-slab safety is high under a weak earthquake, but the risk will increase under a strong earthquake.
Fig. 1. The curves of earthquake acceleration response spectra.
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Fig. 2. Finite element mesh of the CFRD.
strength) and the cumulative overstress duration (COD, the cumulative time that the calculated tensile stress exceeds the tensile strength, i.e. DCR > 1) based on stress with a linear elasticity analysis of face-slabs, is used by Kong and Pang et al. [5] to assess the seismic performance in two grades, minor and moderate. Ghanaat [20] introduced this assessment method using an elastic time-history analysis of concrete dams. As shown in Fig. 5 (typical performance curves under Wave No. 1), the recommended failure grades are as follows: (1) Minor or no failure: DCR ≤ 1; (2) Minor-moderate failure: the DCR and COD are in the shaded part of Fig. 5; (3) Severe failure: DCR > 2 or the COD is beyond the shaded part; a nonlinear time-history analysis may be required in this situation. Fig. 6 shows a comparison of the failure probabilities based on stress with the damage index using PGA. Clearly, the difference between them is very small, which indicates that the dividing standards of the failure grades based on the damage index are reasonable and feasible.
Table 1 Rockfill parameters for modified generalized plasticity model. ρ/(kg/m3)
G0
K0
Mg
Mf
αf
αg
H0
HU0
2180
1387
1664
1.68
1.20
0.2
0.45
800
1600
ms 0.58
mv 0.58
ml 0.44
mu 0.44
rd 140
γDM 20
γu 7
β0 20
β1 0.026
5.2. Fragility analysis results The large dams should be designed based on two levels to ensure their seismic safety, the operating basis earthquake (OBE) and the maximum credible earthquake (MCE). The ground motion levels of the dam with OBE and MCE are set to be 0.40 g and 0.65 g based on a CFRD currently being built in China, respectively, to investigate its fragility under different earthquake levels. Fig. 4 illustrates the seismic fragility curves of every performance index, and the probabilities of exceeding each limit state with different DMs are acquired. The seismic fragility matrices in each limit state under the OBE and MCE levels are shown in Table 2.
6. Conclusion This study introduced a seismic fragility analysis based on IDA to the seismic safety assessment field of high CFRDs and developed a method for assessing the seismic performance of a CFRD based on a deformation and damage index. The fragility method considered the randomness of the ground motion inputs, comprehensively assessed the seismic performance of the dam, and provided a reference for the dam seismic performance under a strong earthquake.
5.3. Verification of failure grades based on damage index Verification of severe failure through Zipingpu CFRD: The Wenchuan earthquake resulted in a significant deformation and destruction of the impervious system to the Zipingpu CFRD. Generally, the horizontal PGA of the dam foundation in the Wenchuan earthquake was 0.55 g [16]. The damage index defined in Section 3.2 was 0.412 from a 2-D nonlinear finite element calculation. According to the dividing standards above, the Zipingpu CFRD is in the state of moderate-severe failure, which is consistent with the actual situation. Verification of minor and moderate failure based on stress: A faceslab destroying index, that simultaneously considers the stress amplitude of exceeding the tensile strength (called the demand capacity ratio, DCR = σt/fs, where σt is the calculated tensile stress and fs is the tensile
(1) In this paper, DMs with a relative settlement ratio of the dam crest and damage index were established, and the damage index of the face slab was defined. The basis for the selection and definition was also discussed and provided suitable failure DMs for performing the fragility analysis. (2) Standards for dividing the dam failure grades based on the relative settlement ratio of the dam crest and the damage index were suggested, and three failure levels of the seismic performance were
Fig. 3. IDA curves for every DM: (a) relative settlement ratio of dam crest, (b) damage index of face-slabs.
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(a)
(b)
Fig. 4. Seismic fragility curves for every DM: (a) permanent deformation, (b) damage index.
Table 2 Seismic fragility matrices under various limit states. Performance Indices
Permanent deformation
Damage index
Limit states
No-failure
Minor-moderate
Moderate-severe
Severe
No-failure
Minor-moderate
Moderate-severe
Severe
OBE MCE
1.4% 0
72.7% 29.5%
25.2% 58.5%
0.7% 12.0%
11.4% 0.7%
52.6% 18.8%
28.9% 44.6%
7.1% 36.0%
ground motion levels (OBE and MCE) were obtained, Table 2 showed the various probabilities. Acknowledgements This work was supported by the National Key Research and Development Program of China (Grant No. 2017YFC0404900) and the National Natural Science Foundation of China (Grant Nos. 51678113, 51679029, 51508071 and 51779034). This financial support is gratefully acknowledged. References Fig. 5. Performance curves for the face-slabs based on stress using linear elastic analysis. [1] Goda Katsuichiro, Yoshikawa Hiromichi. Incremental dynamic analysis of woodframe houses in Canada: effects of dominant earthquake scenarios on seismic fragility. Soil Dyn Earthq Eng 2013;48:1–14. [2] Kong Xianjing, Zhou Yang, Zou Degao, et al. Numerical analysis of dislocations of the face slabs of the Zipingpu concrete faced rockfill dam during the Wenchuan earthquake. Earthq Eng Eng Vib 2011;10(4):581–9. [3] Lee J, Fenves LG. Plastic-damage model for cyclic loading of concrete structures. J Eng Mech 1998;124(3):892–900. [4] Xu Bin, Zou Degao, Kong Xianjing, et al. Dynamic damage evaluation on the slabs of the concrete faced rockfill dam with the plastic-damage model. Comput Geotech 2015;65:258–65. [5] Xian-jing Kong, Rui Pang, De-gao Zou, Bin Xu, Yang Zhou. Seismic performance evaluation of high CFRD based on incremental dynamic analysis. Chin J Geotech Eng 2017;1–6. [2017-06-20] [[in Chinese]. [6] Hariri-Ardebili MA, Saouma VE. Collapse fragility curves for concrete dams: comprehensive study. J Struct Eng 2016;142(10):04016075. [7] Alembagheri Mohammad, Ghaemian Mohsen. Damage assessment of a concrete arch dam through nonlinear incremental dynamic analysis. Soil Dyn Earthq Eng 2013;44:127–37. [8] Baker JW. Efficient analytical fragility function fitting using dynamic structural analysis. Earthq Spectra 2013;31(1):579–99. [9] Vamvatsikos D. Incremental dynamic analysis. Earthq Eng Struct Dyn 2002;31(3):491–514. [10] 〈http://ngawest2.berkeley.edu/〉 [24 May]. [11] SWAISGOOD JR. Embankment dam deformations caused by earthquakes, In: Proceedings of the Pacific Conference on Earthquake Engineering. Christchurch; 2003. [12] Zou Degao, Xu Bin, Kong Xianjing, Liu Huabei, Zhou Yang. Numerical simulation of the seismic response of the Zipingpu concrete face rockfill dam during the Wenchuan earthquake based on a generalized plasticity model. Comput Geotech 2013;49:111–22.
Fig. 6. Comparison of failure probabilities based on stress with damage index.
determined: minor, moderate and severe failure. The verification results by the Zipingpu CFRD and literature [5] indicate that the dividing standards of the failure grades based on the damage index are reasonable and feasible. (3) The fragility curves and failure probabilities of each DM corresponding to the different failure grades were obtained through a fragility analysis of each DM for a typical CFRD, and then the probabilities of the different performance states under different 435
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[17] Liu Jingmao, Zou Degao, Kong Xianjing. A three-dimensional state-dependent model of soil–structure interface for monotonic and cyclic loadings. Comput Geotech 2014;61:166–77. [18] Xianjing Kong, Jingmao Liu, Degao Zou. Numerical simulation of the separation between concrete face slabs and cushion layer of Zipingpu dam during the Wenchuan earthquake. Sci China-Technol Sci 2016;59(4):531–9. [19] Jingbo Liu, Yandong Lu. A direct method for analysis of dynamic soil-structure interaction based on interface idea. China Civil Eng J 1998;83(3):261–76. [20] Ghanaat Y. Failure modes approach to safety evaluation of dams, In: Proceedings of the 13th World Conference on Earthquake Engineering. Vancouver, Canada; 2004.
[13] Chao Wang, Sherong Zhang, Man Li, et al. Classification of earthquake damage to gravity dams based on a damage index model. Earthq Eng Eng Dyn 2014;34(6):218–26. [in Chinese]. [14] Xu Bin, Zou Degao, Kong Xianjing, et al. Concrete slab dynamic damage analysis of CFRD based on concrete nonuniformity. Int J Geomech 2017;17(9):04017055. [15] Degao Zou, Xianjing Kong, Bin Xu. User manual for geotechnical dynamic nonlinear analysis. Dalian: Institute of Earthquake Engineering, Dalian University of Technology; 2005. [16] Xu Bin, Zou Degao, Liu Huabei. Three-dimensional simulation of the construction process of the Zipingpu concrete face rockfill dam based on a generalized plasticity model. Comput Geotech 2012;43:143–54.
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Contents lists available at ScienceDirect
Soil Dynamics and Earthquake Engineering journal homepage: www.elsevier.com/locate/soildyn
Seismic fragility for high CFRDs based on deformation and damage index through incremental dynamic analysis
T
⁎
Rui Pangb, Bin Xua,b, , Xianjing Konga,b, Degao Zoua,b a b
State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, China School of Hydraulic Engineering, Faculty of Infrastructure Engineering, Dalian University of Technology, Dalian 116024, China
A R T I C L E I N F O
A B S T R A C T
Keywords: High CFRDs Deformation Damage index IDA Fragility Seismic performance
In this paper, a seismic fragility analysis method based on incremental dynamic analysis (IDA) is extended to evaluate the seismic performance of high concrete face rockfill dams (CFRDs). Permanent deformation and faceslab damage index using a modified generalized plasticity model for rockfills and a plastic-damage model for face-slabs are considered to be dam damage measures (DMs) after defining a new face-slab damage index. The verification to damage index through the Zipingpu CFRD and previous research indicates that the grading standards are reasonable. Fragility curves and the probabilities are determined for each DM under different earthquake intensities. The results of fragility analysis demonstrate that this method can provide a strong scientific basis for predicting the earthquake destruction and loss of high CFRDs.
1. Introduction
2. Fragility analysis method based on IDA
In China, many high concrete face rockfill dams (CFRDs) have been built or designed. These dams are commonly distributed in areas experiencing strong ground motions, therefore, seismic performance assessments must be performed for these dams. Seismic fragility analysis is one of the most effective methods to evaluate seismic performance. This method can predict when these structures will reach or exceed a certain limit state's probabilities under different strengths of seismic action and employs fragility curves or matrices to describe the probability distributions of all limit states. IDA is a parametric analysis method based on nonlinear dynamic time history analysis and has been widely used in structural fragility analysis [1]. However, due to the complexity of high CFRDs, there have been few related reports on their fragility analysis based on IDA applied in this engineering field. In this paper, a seismic performance assessment for high CFRDs is performed based on a fragility analysis using IDA. Deformation and face-slab damage are the two major forms of destruction of CFRDs; CFRDs that have exhibited such destruction as the Zipingpu CFRD in 2008 (156 m) [2]. Considering the nonlinear characteristics of concrete, Lee and Fenves [3] proposed a plastic damage model to independently determine the damage in pull and pressure modes and the stiffness recovery phenomenon in the reverse loading of concrete, and this model has been successfully applied to concrete face-slabs [4]. Thus, it is reasonable and feasible to regard the relative settlement of dam crests and damage index as evaluation indices of seismic performance.
IDA method: Several scholars have attempted to introduce IDA into the preliminary safety assessment of dams. For example, Kong and Pang et al. [5] first introduced IDA to seismic safety assessment of high CFRDs based on three aspects permanent deformation, stability of dam slope, safety of face-slabs for the first time using an equivalent linear constitutive model, and gained the fragility curves and probabilities. Hariri-Ardebili and Saouma [6] applied IDA to obtain the collapse fragility curves of concrete dams. Mohammad Alembagheri and Mohsen Ghaemian [7] performed a damage assessment of a typical arch dam through IDA subjected to a set of 12 earthquakes, and damage propagation was investigated and various IDA curves were created. These studies achieved a preliminary assessment of dam safety and demonstrated that the IDA method is suitable for large water and hydropower engineering. Fragility analysis method: The seismic fragility curves provide the conditional probabilities of the structural response reaching or exceeding certain limit states corresponding to the seismic capacity under different earthquake intensities. After calculating the response of the structures under different ground motion intensities with the IDA, the relationships between the limit states and DMs are quantified by combination with the definition of the limit states, and then, the seismic fragility is determined. According to the previous research [8], in general, the seismic fragility is assumed to follow a lognormal cumulative distribution function of double parameters, defined as follows:
⁎
Corresponding author at: State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, China. E-mail address: [email protected] (B. Xu).
https://doi.org/10.1016/j.soildyn.2017.11.017 Received 8 September 2017; Received in revised form 8 November 2017; Accepted 9 November 2017 0267-7261/ © 2017 Elsevier Ltd. All rights reserved.
Soil Dynamics and Earthquake Engineering 104 (2018) 432–436
R. Pang et al.
PR (C|IM = X ) = Φ ⎡ ⎣
ln(Χ / θ) ⎤ σ ⎦
For concrete dams, for example, Wang and Zhang et al. [13] considered the influence of the damage location on the overall structures of gravity dams and proposed the following dividing values: a damage index of 0.05 for intact and minor failure, a damage index of 0.15 for minor and moderate failure, and a damage index of 0.45 for moderate and severe failure; the upper limit of severe failure was 0.75. Mohammad Alembagheri and Mohsen Ghaemian [7] carried out damage assessment for a typical arch dam through nonlinear IDA. In this paper, referring to the structural damage index described above and considering the importance of the location of the damage on the CFRDs, we believe that the higher the damage location is, however the smaller the impact will be on the structure; finally, we define the damage index. In accordance with the research results of Xu et al. [14], the face-slab damage mainly occurs in the range of 0.4H-0.9 H (H: dam height) under the action of an earthquake. The damage index of face slabs is proposed as:
(1)
3. Seismic records and performance parameters 3.1. Selection of seismic records In this paper, the ground motion inputs adopt actual and artificial seismic records respectively. The response spectra based on the site conditions of a high earth-rockfill dam in the southwest of China are chosen to be the target response spectra, as described in Formula (2), β(T) is the magnification response spectrum of ground motion acceleration, and T1 = 0.12 s, T2 = 0.34 s, βmax = 2.5, and γ = 1.0. According to the proposal of Vamvatsikos et al. [9], 10–20 earthquake records meet the requirements of IDA analysis. First, 10 actual seismic records which are well agreeable with the target response spectra based on site conditions are selected in PEER [10]. Then one seismic wave is artificially generated based on the target response spectrum and is used for a comparison purpose. The acceleration response spectra of 11 earthquake waves are shown in Fig. 1.
⎧1 ⎪1 + (βmax − 1) T − 0.04 ⎪ T1 − 0.04 β (T ) = ⎨ βmax ⎪ T ⎪ βmax ( T2 )γ ⎩
n
DI = α⋅
n ∑i = 1
(
0.9H − hi 0.25H
0.9H − h Si⋅ 0.25H i
)
)
(3)
where di is the damage factor of the ith element of the face slab, n is the number of face slab elements in 0.4–0.9 H, Si is the area of the ith element of the face slab, hi is the center height to the face-slab bottom of the ith element of the face slab, DI is the damage index of the face slabs; and α is the influence coefficient, which is generally 2.0. As shown in Fig. 2. In this paper, referring to the related literatures of the failure grading of gravity dam, we regard DI = 0.03, 0.15, and 0.45 as the dividing values of minor, moderate and severe failure, respectively.
T ≤ 0.04s 0.04s < T ≤ T1 T1 < T ≤ T2 T2 < T ≤ 6s
(
∑i = 1 d i⋅Si⋅
(2)
3.2. Proposal of dam failure grades 4. Finite element analysis
Relative settlement ratio of the dam crest: Kong and Pang et al. [5] considered relative settlement ratios of 0.4%, 0.7%, and 1% of the dam crest (crest settlement values/height of the dam) as the assessment limitation and analyzed the fragility when this dam exhibited minor, moderate and severe failure. Swaisgood et al. [11] surveyed 69 dams, regarded the relative settlement ratio of the dam crest as an index, and divided the degree of earthquake damage into four failure grades: healthy (< 0.1%), minor (0.012–0.5%), moderate (0.1–1.0%) and severe (> 0.5%). In this paper, we refer to the related safety assessment and grading standards described above and consider that the simulated maximum post-earthquake deformation results based on the generalized plastic model are smaller than measured values [12], e.g., the maximum settlement deformation simulated with a generalized plasticity model is about 0.77 m, whereas the actual measured value was 1 m for the Zipingpu CFRD. Finally, we establish three limit failure states with relative settlement ratios of 0.2%, 0.4%, and 0.6% of the dam crest corresponding to three failure grades (minor, moderate and severe). Damage index of face-slabs: In the earthquake damage prediction and post-earthquake evaluation of concrete structures, many scholars use the damage index to quantitatively describe the degree of failure.
In this paper, the distribution characteristics and change rules of every DM under different seismic records with different earthquake intensities are analyzed in detail by performing two-dimensional (2-D) nonlinear finite element numerical calculations for a typical CFRD with a height of 250 m (Fig. 2) based on GEODYNA [15]. The static and dynamic calculations of the rockfill, transition and cushion are all simulated by the modified generalized plasticity constitutive model [16], and the model parameters are shown in Table 1. The contact between the face slab and cushion is simulated by the generalized plasticity interface model [17], whose parameters are based on the literature [18]. The plastic-damage behavior of concrete face-slabs simulated with the plastic-damage model [4,14], and the parameters are obtained from the literature [4]. Ground motion inputs are added in the form of a viscoelastic boundary combined with an equivalent load at the boundary of the finite element model to simulate the interaction of finite fields with infinite domains [19]. 5. Results of IDA and fragility analysis 5.1. IDA results This study selects the PGA of the earthquake as the earthquake IM and modulates the amplitude at equal intervals (the range is 0.1 g) until the failure of different DMs to severity. The IDA curves corresponding to different ground motion records of the PGA-relative settlement ratio of the dam crest and PGA-damage index are obtained after a large number of nonlinear finite element calculations, as shown in Fig. 3. It may be seen from Fig. 3(a) that with the increase of the IM value, the changes of the relative settlement ratio of the dam crest become slower, which indicates that the rockfills are denser under a stronger earthquake. However, Fig. 3(b) shows that the damage index changes slowly under a weak earthquake and quickly under a strong earthquake, which demonstrates that the face-slab safety is high under a weak earthquake, but the risk will increase under a strong earthquake.
Fig. 1. The curves of earthquake acceleration response spectra.
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Fig. 2. Finite element mesh of the CFRD.
strength) and the cumulative overstress duration (COD, the cumulative time that the calculated tensile stress exceeds the tensile strength, i.e. DCR > 1) based on stress with a linear elasticity analysis of face-slabs, is used by Kong and Pang et al. [5] to assess the seismic performance in two grades, minor and moderate. Ghanaat [20] introduced this assessment method using an elastic time-history analysis of concrete dams. As shown in Fig. 5 (typical performance curves under Wave No. 1), the recommended failure grades are as follows: (1) Minor or no failure: DCR ≤ 1; (2) Minor-moderate failure: the DCR and COD are in the shaded part of Fig. 5; (3) Severe failure: DCR > 2 or the COD is beyond the shaded part; a nonlinear time-history analysis may be required in this situation. Fig. 6 shows a comparison of the failure probabilities based on stress with the damage index using PGA. Clearly, the difference between them is very small, which indicates that the dividing standards of the failure grades based on the damage index are reasonable and feasible.
Table 1 Rockfill parameters for modified generalized plasticity model. ρ/(kg/m3)
G0
K0
Mg
Mf
αf
αg
H0
HU0
2180
1387
1664
1.68
1.20
0.2
0.45
800
1600
ms 0.58
mv 0.58
ml 0.44
mu 0.44
rd 140
γDM 20
γu 7
β0 20
β1 0.026
5.2. Fragility analysis results The large dams should be designed based on two levels to ensure their seismic safety, the operating basis earthquake (OBE) and the maximum credible earthquake (MCE). The ground motion levels of the dam with OBE and MCE are set to be 0.40 g and 0.65 g based on a CFRD currently being built in China, respectively, to investigate its fragility under different earthquake levels. Fig. 4 illustrates the seismic fragility curves of every performance index, and the probabilities of exceeding each limit state with different DMs are acquired. The seismic fragility matrices in each limit state under the OBE and MCE levels are shown in Table 2.
6. Conclusion This study introduced a seismic fragility analysis based on IDA to the seismic safety assessment field of high CFRDs and developed a method for assessing the seismic performance of a CFRD based on a deformation and damage index. The fragility method considered the randomness of the ground motion inputs, comprehensively assessed the seismic performance of the dam, and provided a reference for the dam seismic performance under a strong earthquake.
5.3. Verification of failure grades based on damage index Verification of severe failure through Zipingpu CFRD: The Wenchuan earthquake resulted in a significant deformation and destruction of the impervious system to the Zipingpu CFRD. Generally, the horizontal PGA of the dam foundation in the Wenchuan earthquake was 0.55 g [16]. The damage index defined in Section 3.2 was 0.412 from a 2-D nonlinear finite element calculation. According to the dividing standards above, the Zipingpu CFRD is in the state of moderate-severe failure, which is consistent with the actual situation. Verification of minor and moderate failure based on stress: A faceslab destroying index, that simultaneously considers the stress amplitude of exceeding the tensile strength (called the demand capacity ratio, DCR = σt/fs, where σt is the calculated tensile stress and fs is the tensile
(1) In this paper, DMs with a relative settlement ratio of the dam crest and damage index were established, and the damage index of the face slab was defined. The basis for the selection and definition was also discussed and provided suitable failure DMs for performing the fragility analysis. (2) Standards for dividing the dam failure grades based on the relative settlement ratio of the dam crest and the damage index were suggested, and three failure levels of the seismic performance were
Fig. 3. IDA curves for every DM: (a) relative settlement ratio of dam crest, (b) damage index of face-slabs.
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(a)
(b)
Fig. 4. Seismic fragility curves for every DM: (a) permanent deformation, (b) damage index.
Table 2 Seismic fragility matrices under various limit states. Performance Indices
Permanent deformation
Damage index
Limit states
No-failure
Minor-moderate
Moderate-severe
Severe
No-failure
Minor-moderate
Moderate-severe
Severe
OBE MCE
1.4% 0
72.7% 29.5%
25.2% 58.5%
0.7% 12.0%
11.4% 0.7%
52.6% 18.8%
28.9% 44.6%
7.1% 36.0%
ground motion levels (OBE and MCE) were obtained, Table 2 showed the various probabilities. Acknowledgements This work was supported by the National Key Research and Development Program of China (Grant No. 2017YFC0404900) and the National Natural Science Foundation of China (Grant Nos. 51678113, 51679029, 51508071 and 51779034). This financial support is gratefully acknowledged. References Fig. 5. Performance curves for the face-slabs based on stress using linear elastic analysis. [1] Goda Katsuichiro, Yoshikawa Hiromichi. Incremental dynamic analysis of woodframe houses in Canada: effects of dominant earthquake scenarios on seismic fragility. Soil Dyn Earthq Eng 2013;48:1–14. [2] Kong Xianjing, Zhou Yang, Zou Degao, et al. Numerical analysis of dislocations of the face slabs of the Zipingpu concrete faced rockfill dam during the Wenchuan earthquake. Earthq Eng Eng Vib 2011;10(4):581–9. [3] Lee J, Fenves LG. Plastic-damage model for cyclic loading of concrete structures. J Eng Mech 1998;124(3):892–900. [4] Xu Bin, Zou Degao, Kong Xianjing, et al. Dynamic damage evaluation on the slabs of the concrete faced rockfill dam with the plastic-damage model. Comput Geotech 2015;65:258–65. [5] Xian-jing Kong, Rui Pang, De-gao Zou, Bin Xu, Yang Zhou. Seismic performance evaluation of high CFRD based on incremental dynamic analysis. Chin J Geotech Eng 2017;1–6. [2017-06-20] [[in Chinese]. [6] Hariri-Ardebili MA, Saouma VE. Collapse fragility curves for concrete dams: comprehensive study. J Struct Eng 2016;142(10):04016075. [7] Alembagheri Mohammad, Ghaemian Mohsen. Damage assessment of a concrete arch dam through nonlinear incremental dynamic analysis. Soil Dyn Earthq Eng 2013;44:127–37. [8] Baker JW. Efficient analytical fragility function fitting using dynamic structural analysis. Earthq Spectra 2013;31(1):579–99. [9] Vamvatsikos D. Incremental dynamic analysis. Earthq Eng Struct Dyn 2002;31(3):491–514. [10] 〈http://ngawest2.berkeley.edu/〉 [24 May]. [11] SWAISGOOD JR. Embankment dam deformations caused by earthquakes, In: Proceedings of the Pacific Conference on Earthquake Engineering. Christchurch; 2003. [12] Zou Degao, Xu Bin, Kong Xianjing, Liu Huabei, Zhou Yang. Numerical simulation of the seismic response of the Zipingpu concrete face rockfill dam during the Wenchuan earthquake based on a generalized plasticity model. Comput Geotech 2013;49:111–22.
Fig. 6. Comparison of failure probabilities based on stress with damage index.
determined: minor, moderate and severe failure. The verification results by the Zipingpu CFRD and literature [5] indicate that the dividing standards of the failure grades based on the damage index are reasonable and feasible. (3) The fragility curves and failure probabilities of each DM corresponding to the different failure grades were obtained through a fragility analysis of each DM for a typical CFRD, and then the probabilities of the different performance states under different 435
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