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Seismic fragility analysis of a High Arch Dam-Foundation System based on seismic instability failure mode
Soil Dynamics and Earthquake Engineering 130 (2020) 105981
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Seismic fragility analysis of a High Arch Dam-Foundation System based on seismic instability failure mode Hui Liang a, b, Jin Tu a, b, Shengshan Guo a, b, *, Jianxin Liao c, Deyu Li a, b, Shiqi Peng c a
Earthquake Engineering Research Centre, China Institute of Water Resources and Hydropower Research, Beijing, China State Key Laboratory of Simulation and Regulation of Water Cycle in River Basin, Beijing, 100048, China c China Three Gorges Projects Development Co., Ltd, Beijing, 100038, China b
A R T I C L E I N F O
A B S T R A C T
Keywords: High arch dam-foundation system Seismic instability failure mode Dynamic contact model Latin hypercube sampling Seismic fragility analysis
Seismic stability of concrete arch dams has always been a critical issue in seismic safety evaluation. In this paper, a high concrete arch dam is selected as a case study for the seismic fragility analysis based on instability failure mode. A comprehensive approach considering the opening and closing of contraction joint, probable sliding rock mass of dam abutments, the interface between the dam and foundation as well as the effect of foundation ra diation damping is presented. The random and uncertain parameters containing the friction coefficients and cohesions are generated with Latin hypercube sampling method (LHS). The approximate IDA is performed, and the slippage and sliding area ratio are chosen as the engineer demand parameters (EDP). Different damage levels are identified by the slippage-based rule and sliding area ratio-based rule respectively according to their cor responding overall mean IDA curves. In addition, seismic fragility curves are developed for the defined damage levels. The results reveal that the damage levels defined based on different EDP-based rules could be different, while the ultimate capacity of the concrete arch dam reflected by selected EDPs are basically the same. Sliding area ratio-based rule can better reveal the whole process of sliding development of the arch dam abutment and the influence of residual cohesions. The existence of residual cohesion delays the sliding development and en hances the seismic stability of arch dam-foundation systems. Furthermore, a comparative analysis is performed based on the extended limit states. Seismic fragility analysis is of great significance for quantitatively evaluating the seismic performance of concrete dams from the probability point of view.
1. Introduction Seismic safety of high dams is an inevitable problem during the construction of momentous water conservancy projects. Existing prac tical examples of high dams subjected to strong earthquakes show that dam abutment is the weak part. The overall instability of the damfoundation system caused by its damage is one of the most important concerns in the seismic design and safety assessment of high dams, which faces with complicated geological conditions and various loads. Moreover, there are various uncertainties, such as epistemic (modeling and material parameters) and aleatory (earthquake ground motion, flood etc.) uncertainty during the design, construction and operation of dams [1–3], which significantly affect the seismic perfor mance of concrete dams. Fragility analysis, which plays an essential role in seismic probabilistic risk assessment within the framework of performance-based seismic design, is gradually applied to water
conservancy and hydropower projects, such as gravity dams and arch dams. Hariri-Ardebili M A et al. [4] provide a comprehensive state-of-the-art review of existing application on fragility analysis of concrete dams. The detailed procedures and different methods are summarized and a comparative review of major researches and publi cations is presented. A consistent procedure is particularly presented to perform fragility analysis for the sliding failure mode of concrete gravity dams in order to identify and track natural and epistemic uncertainty separately [8]. However, among those described researches, few attempts have been partly made on the effect of various uncertainties on seismic perfor mance of concrete arch dams. Amirpour and Mirzabozorg [9] adopt nine actual ground motion records divided into three levels as seismic input for quantitative and qualitative analysis of the ultimate seismic failure state of arch dams using IDA method [10]. Zhonghong et al. [11] perform seismic risk analysis of a concrete arch dam considering the
* Corresponding author. Earthquake Engineering Research Centre, China Institute of Water Resources and Hydropower Research, Beijing, China. E-mail addresses: [email protected], [email protected] (S. Guo). https://doi.org/10.1016/j.soildyn.2019.105981 Received 22 July 2019; Received in revised form 6 November 2019; Accepted 17 November 2019 Available online 22 November 2019 0267-7261/© 2019 Elsevier Ltd. All rights reserved.
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non-homogeneity of concrete material, uncertainty of earthquake ground motions and uncertainty of concrete material parameters. IDA approach is adopted to analyze a thin high arc dam by taking parameters of Sa, PGA and PGV as intensity measure (IM) and the overstressed area (OSA) as engineering demand parameter, and based on the defined limit states, the seismic fragility curves are obtained [12]. In view that the quantitative study of progressive failure of concrete dams is an impor tant part of risk analysis, Hariri-Ardebili, Furgani and Meghella [13] compare and analyze more than 20 indexes using Endurance Time Analysis method (ETA) based on the concept of damage index and performance index, which provides an effective method for the selection of engineering demand parameters in seismic performance evaluation of dams. The seismic fragility analysis of a high arch dam including dam concrete cracking and contraction joint opening and closing is per formed considering both epistemic and aleatory uncertainties, and seismic fragility curves under different damage levels are drawn [14]. In general, seismic fragility analysis of high arch dams is still in its infancy. The fragility analysis based on the instability failure mode of arch dam-foundation systems is basically absent. Thus, this paper fo cuses on the seismic fragility analysis of a high concrete arch damfoundation system based on the seismic instability failure mode. The arch dam abutment and the dam are coupled as a system. A high con crete arch dam in China is taken as a case study. The friction coefficients and cohesions of the probable sliding block affecting the seismic stability of the arch dam are taken as random parameters. Both two models i.e. model I without residual cohesions and model II with residual cohesions (30% of the corresponding peak cohesions) are considered for a comparative analysis of the effect of residual cohesion on the seismic fragility of the arch dam. The PGA is used as intensity measure (IM), and the slippage and sliding area ratio of the bottom sliding surface are chosen as engineer demand parameters (EDP) respectively. IDA curves are obtained by conducting the IDA with Latin Hypercube Sampling (LHS). Different damage levels are defined through slippage-based rules and sliding area ratio-based rules. Furthermore, the corresponding fragility curves are extracted for the seismic stability evaluation of concrete arch dams.
Fig. 1. Aerial view from downstream of Baihetan arch dam.
intensity measures (IM, such as PGA, PGV). N and M represents the number of random samples and the number of ground motions. NLS is the number of the dam reaching or exceeding a certain LS at each earthquake intensity level among all the N � M dynamic analyses. (2) Fragility curves can be fitted with different probability distribu tion functions based on the data obtained from the above results (Equation (1)). At present, the widely used analytic function of fragility curve is lognormal distribution function [6,18,19], shown as following Equations (2)–(4). Thus, seismic fragility analysis of concrete dams can be performed and fragility curves can be developed according to the above-mentioned procedures. � � ln im ηim FðimÞ ¼ PðLSjIM ¼ imÞ ¼ Φ (2) βim
ηim ¼ βim ¼
2. Computational methodology
Fragility is expressed as the probability of engineering structure reaching a certain limit state or performance level under different earthquake intensity levels in seismic engineering [16,17]. A large amount of data is needed in order to obtain fragility curves of structures with reasonable and accurate consideration of the effects of un certainties on structural performance. Analytical fragility analysis is adopted, of which the data base of is obtained from the structural dy namic response under a series of earthquakes with increasing seismic intensity levels. Procedures for analytical fragility analysis include structural dynamic response analysis, definition of limit states and for mation of fragility curve. Herein, IDA method is used to develop seismic fragility curves of the concrete arch dam-foundation system. A series nonlinear dynamic analyses are performed considering epistemic un certainty. The seismic fragility curves of concrete dams can be obtained according to the following procedures.
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi XN�M 1 ðlnðimÞ ηim Þ2 N � M 1 i¼1
(4)
3. Nonlinear modeling of a high arch dam-foundation system 3.1. Finite element model The Baihetan double-curvature arch dam is selected as a case study (Fig. 1), located in strong earthquake area in Southwest China. The height of the dam is 289 m, and the crest is at El. 834 m. The thickness of the dam crest is 14.0 m, and the maximum thickness of dam bottom is 83.91 m. The valley of the dam site is relatively symmetrical ‘V’ type. The left bank slope is gentle and the right bank slope is steep. Moreover, there are many gentle dip structural planes, faults and cracks in the dam abutments. The combination of all these unfavorable factors constitutes a geological background resulting in the probable seismic instability of dam abutments under earthquakes. According to the topographic and geological characteristics of the dam, a three-dimensional finite element mesh of arch dam-foundation system is established, shown in Fig. 2(a). The whole system is dis cretized by three-dimensional solid elements. Six layers of threedimensional solid elements are arranged along the thickness direction of the dam body. The whole model contains 130,000 nodes and 120,000 solid elements. Fig. 2(b) describes the potential sliding block of the arch dam abutment. The sliding block is composed of bottom sliding surface,
(1) The probability of a concrete dam reaching a certain limit state (LS) under a given earthquake intensity level can be obtained by the following formula. NLS N�M
(3)
where FðimÞ is the probability that the dam reaches or exceeds a certain LS at an earthquake intensity level. Φð⋅Þ is standard normal distribution function. ηim and βim are the logarithmic mean and logarithmic standard deviation of im.
2.1. Generation of seismic fragility curve
PðLSjIM ¼ imÞ ¼
1 XN�M lnðimi Þ N � M i¼1
(1)
where PðLSjIM ¼ imÞ is the conditional probability that the concrete dam reaches or exceeds a certain LS under an earthquake intensity level of 2
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Soil Dynamics and Earthquake Engineering 130 (2020) 105981
Fig. 2. Finite element mesh of the arch dam-foundation system.
Fig. 3. Selected observation point (number 6272).
pronounced than that of elastic modulus of concrete and bedrock. Thus, shear strength parameters on each potential sliding surface are consid ered as random variables. They are taken as μboottom, cboottom and μcrack, ccrack and μside, cside respectively for the bottom sliding surface, cracking surface and side sliding surface. In view that the overall seismic insta bility of the arch dam-foundation system reflected by the sudden and rapid increase of the slippage is a process of gradual development and accumulation of the total deformation including local cracking and sliding of each part, the residual slippage of the characteristic point (node number 6272, Fig. 3) at the bottom sliding surface is selected as one of the indexes to quantitatively evaluate the seismic sliding stability of the arch dam abutment. Moreover, the sliding area ratio (ratio of the sliding area to the bottom sliding surface) is chosen as another damage index for evaluating seismic stability of the arch dam-foundation.
side sliding surface, cracking surface and free surface. Fig. 3 shows the location of the selected observation point (node 6272), of which the slippage can be used to represent the overall seismic stability of the damfoundation system. Since the opening and closing of contraction joints of arch dams are important factors affecting their seismic performance [20–22], the dam is separated into 29 adjacent monoliths by 28 vertical contraction joints. Generally, dam-foundation interface is the weak part of the dam-foundation system. Therefore, a joint interface is considered to simulate the failure of the dam-foundation interface. Moreover, for the purpose of research on seismic stability of the dam abutment, each po tential sliding surface of the probable sliding block that affects the overall stability of the dam is also considered to be a joint interface Fig. 2 (b). Thus, a comprehensive approach composed of contraction joint opening and closing, failure along dam-foundation interface and sliding of the probable sliding rock mass is developed for seismic stability analysis of the arch dam. The viscous-spring artificial boundary is used to simulate the radia tion damping effect caused by the dynamic interaction between the dam and foundation. The widely used Westergaard added mass model [23] is adopted to consider the hydrodynamic pressure. The joint interfaces which are discretized into contact node pairs are modeled using a dy namic contact model based on Lagrange multiplier method, proposed by Liu [15]. The sliding resistance force at sliding surfaces are supposed to be controlled exclusively by friction coefficients and cohesions in the dynamic contact model. Since this research focuses on the seismic sta bility of arch dam abutments, the influence of shear strength parameters on contact surfaces of the probable sliding rock mass is more
3.2. Material parameters The material parameters of dam concrete are as follows: mass den sity ¼ 2400.0 kg/m3 and Poisson’s ratio ¼ 0.167. The static elastic modulus is 24.0 GPa and dynamic elastic modulus is 1.5 times of the static elastic modulus. The linear expansion coefficient is 6.5 � 10 6/� C. The actual stratification of bedrock materials is considered, of which the average static comprehensive deformation modulus of each layer is about 11.0 GPa. The density and Poisson’s ratio are 2700.0 kg/m3 and 0.23 respectively. The initial tensile strength of dam foundation inter face is 3.42 MPa, and the friction coefficient and cohesion are 1.15 and 1.10 MPa, respectively. The parameters of the contraction joint are taken as: friction coefficient μjoint ¼ 0.8, cohesion cjoint ¼ 0. 3
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3.3. Static and dynamic loads
Table 1 The material properties and probability distributions for the random parameters. Material parameters
Mean/Basecase model
COV
Minimum
Maximum
type
Friction coefficient
0.432
0.200
0.2592
0.6048
Normal
0.193
0.400
0.08293
0.3872
1.100
0.200
0.6600
1.5400
Lognormal Normal
1.200
0.400
0.5156
2.4076
0.420
0.200
0.2520
0.5880
0.090
0.400
0.03867
0.1806
μbottom
Cohesion cbottom (MPa) Friction coefficient
μcrack
Cohesion ccrack (MPa) Friction coefficient
μside
Cohesion cside (MPa)
The static loads include self-gravity of the dam body, water pressure in the upstream and downstream under the normal water level, up stream sediment pressure and temperature load. Upstream and down stream normal water levels are 825.00 m and 604.00 m respectively. The upstream sediment elevation is 710 m. The bulk density and internal friction angle are 5 kN/m3 and 0� . The design temperature load of the arch ring is shown in Table 2. In view that the record-to-record uncertainty is beyond consideration in this paper, three uncorrelated artificial ground motions in three di rections are generated based on the site-specific design response spec trum corresponding to the exceedance probability of 2% in 100 years. The PGA in vertical direction is two-third of that in stream direction in the analysis. The representative value of horizontal design seismic ac celeration of Baihetan arch dam is 0.406 g, which is equivalent to 2% of the exceeding probability in 100 years, and the vertical value is 2/3 of the horizontal value, which is 0.271 g. All the components of the arti ficial ground motion are normalized and shown in Fig. 4.
Lognormal Normal Lognormal
Uncertainty is an inevitable objective phenomenon in the process of dam design, construction and operation. They are important factors to be considered in seismic fragility analysis of concrete dams. Considering various uncertainties in the framework of probability statistics can provide a basis for fully understanding the expected range of structural dynamic response. Herein, epistemic uncertainty due to the lack of in formation of the engineering geological condition is considered for the seismic fragility analysis. As above-mentioned, six parameters μboottom, cboottom and μcrack, ccrack and μside, cside are treated as random variables in the dynamic contact model. Normal distribution and lognormal distri bution are considered for friction coefficients and cohesions respectively [24]. A mean value and coefficient of variation (COV) for each param eter is assigned according to engineering design data. The distribution of each parameter is truncated with a reasonable minimum and maximum to meet with physical limits. The hard limits set at 2 standard deviations away from the central value are considered to cover most of the samples. Thus, the parameters used in the dynamic contact model are defined using the properties listed in Table 1. The widely used Latin hypercube sampling method (LHS) [25] is adopted to handle the current impact of parameter uncertainties in view of its accuracy and efficiency for the probabilistic analysis.
4. Probabilistic analysis Generally, it is assumed that the cohesion on the contact surface becomes zero and ineffective when the sliding happens. In fact, when the relative sliding of the contact surface happens, the effect of cohesion on the contact surface is not necessarily completely ineffective, and its ef fect may be attenuated to a certain extent. Herein, the initial cohesion parameters between the contact surfaces can be defined as peak cohe sion. When sliding of the contact surface happens, the peak cohesions change to the residual cohesions. In order to study the effect of residual cohesions on the seismic stability of arch dam-foundation systems, two models, i.e. model I without residual cohesions and model II with re sidual cohesions (30% of the corresponding peak cohesions is selected as a case study), are considered for the probabilistic analysis with the advanced dynamic contact model. The friction coefficients and co hesions, which are the main parameters affecting the seismic sliding stability of arc dam-foundation systems, are sampled using Latin hy percube sampling method. There is no specific standard for the size N of random variables. The size N of samples is related to the number of random parameters and the response of structures affected by them. Generally, a relative high sample size N will provide reasonable and
Table 2 Temperature load. Elevation Normal water level þ temperature drop
uniformity linear
834.0
800.0
760.0
720.0
680.0
640.0
610.0
580.0
565.0
545.0
5.99 0.00
2.84 7.27
3.52 10.98
3.23 12.06
4.27 12.28
4.28 12.38
5.29 12.4
1.77 5.89
1.39 5.28
1.15 4.91
Fig. 4. The normalized artificial acceleration time histories. 4
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Table 3 Slippage-based limit states. EDP
Limit states
Slippage-based
Local sliding damage
Overall sliding instability
0.00 m
0.40 m
Table 4 Slippage-based limit states in Reference [13]. EDP
Slippagebased
Limit states Initiation
Joint opeining/ sliding
Concrete cracking
Global yielding
Dam failure
0.0003 m
0.0015 m
0.0053 m
0.0365 m
0.0393 m
Fig. 5. Slippage-based limit states definition.
accurate assessment for a structure. LHS method is used to obtain a total number of 20 samples for each uncertain parameter in the seismic fragility analysis of concrete gravity dams [7]. The sample sizes N of 30 and 200 are obtained by LHS separately [5,26]. Herein, the sample size N is taken as 50 for probabilistic analysis. In general, seismic sliding stability of arch dam abutment is not only affected by the arch thrust, but also by inertia force of the abutment sliders. Considering that this paper focuses on the seismic stability of an arch dam-foundation system, PGA is selected as IM, scaled to multiple levels from zero to 1.0 g with steps of 0.1 g in the IDA of arch dams. The slippage of the selected characteristic point and the sliding area ratio are chosen as EDPs to assess seismic stability of the arch dam-foundation system. Thus, the total number 10 � N ¼ 500 of nonlinear dynamic analyses are performed for both model I and II, and IDA curves are generated from discrete points with spline interpolation. Since the IDA curves can be used for the definition of limit states [27], the overall mean IDA curves with their 50% fractile are obtained. Consequently, based on the seismic fragility theory, seismic fragility curves are ob tained to quantitatively describe the seismic stability of arch dam-foundations.
Fig. 6. Sliding area ratio-based limit states definition.
5.1.1. Slippage-based rule Fig. 5 describes different slippage-based limit states defined for the concrete arch dam-foundation system through the overall mean IDA curve and 50% fractile IDA curve. It shows that the curve is divided into three stages by two turning points. In the first stage, with the increase of ground motion, only slight sliding occurs on the abutment slider, and the residual slippage of the characteristic points is very small and almost zero, which indicates that the arch dam-foundation system is of good stability and it is in a stable region. In the second stage, the residual slippage increases slowly with the increase of PGA levels. Although the local sliding happens, the maximum residual displacement is less than 0.04 m, and the overall stability of the dam is still in a controllable state. In the third stage, the residual sliding displacement of characteristic points increases rapidly, the stability of arch dam foundation system is in an uncontrollable state, and the overall sliding failure has occurred. Therefore, according to the appearance of turning points in each stage, the two limit states and the corresponding residual slippage threshold of arch dam foundation system based on sliding instability failure mode can be defined as (Table 3): (1) Local sliding damage, corresponding to the residual slippage of 0.0 m of the characteristic point at the first turning point, i.e. the initial sliding of the characteristic point; (2) Overall sliding failure, corresponding to the residual slippage of 0.04 m (40.0 mm) at the second turning point. According to HaririArdebili M A [13], the maximum joint sliding ΩðδJn Þ corresponding to the initial dam sliding and the ultimate dam failure is 0.3 mm and 39.3 mm, which is basically consistent with this paper of 0.00 and 40.0 mm,
5. Results and discussion 5.1. Limit state definition The definition of limit states for the arch dam is the premise of seismic fragility analysis. As for concrete dams, their limit states are related to different failure modes. Existing examples of dams subjected to earthquake hazards and researches show that main potential failure modes of arch dams include dam cracking caused by excessive tensile cantilever stress and sliding failure of abutment slider. Up to now, there is no completely unified and recognized scheme to define the limit states of concrete dams. Shen huaizhi [28] proposed preliminary suggestions on performance-based safety evaluation index according to the yield ratio of the gravity dam-foundation interface and the performance of drainage system. Alembagheri et al. [27] reasonably analyzed the bearing capacity and ultimate state of concrete gravity dams using IDA curves. Kadkhodayan et al. [12] defined three performance levels for a concrete arch dam based on IDA curves. The overstress area is taken as EDP and Sa, PGA and PGV are taken as IMs. In this paper, IDA curves are used to define the limit states of the arch dam-foundation. The overall mean slippage-based and sliding area ratio-based IDA curves with their 50% fractile are obtained from probabilistic analysis. The different limit states of seismic stability of the arch dam-foundation system under earthquakes are defined for the seismic fragility analysis. 5
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with the increase of PGA levels, and the arch dam can still maintain a relatively good stability. (3) Moderate sliding stage. The sliding area ratio increases to more than 50%, which indicates that more than half of the joints on the bottom sliding surface have slid. At present, the mod erate sliding damage of the arch dam has happened, which affects the normal operation of the reservoir power station; (4) Overall sliding instability stage. The arch dam-foundation system is in overall insta bility state, which is difficult to repair. In summary, based on the sliding area ratio-based rule, the limit states of the arch dam foundation system can be divided into (Table 5): (1) Local/Slight sliding damage, corresponding to the sliding area ratio of 10%; (2) Moderate sliding damage, corresponding to the sliding area ratio of 50%; (3) Overall sliding failure, corresponding to the sliding area ratio of 90%.
Table 5 Sliding area ratio-based limit states. EDP
Limit states
Sliding area ratio-based
Local sliding damage
Moderate sliding damage
Overall sliding instability
10%
50%
90%
shown in Table 4. Moreover, a comparative analysis is performed to present the difference of the seismic fragility curves based on these defined limit states. 5.1.2. Sliding area-based rule Fig. 6 depicts the sliding area ratio-based limit states of the concrete arch dam-foundation system. Different regions are divided by the red dotted line for each limit state, detailly shown in Fig. 6. According to Fig. 6, based on the sliding area ratio versus PGA curve, the process of seismic sliding stability of arch dam foundation system can be divided into four stages: (1) Stability stage. The sliding area ratio is small, less than 10% of the whole sliding surface area, and at this stage, it increases slowly with the increase of PGA levels, and the arch dam is of good stability; (2) Slight sliding stage. The sliding area ratio increases rapidly
5.2. Seismic fragility curves 5.2.1. Slippage-based rule The seismic fragility curves corresponding to two slippage-based limit states of the concrete arch dam-foundation system are given in Fig. 7 (a) and (b). Fig. 8 (a) and (b) presents the comparison of fragility curves of model I and II for different limit states. Seismic fragility pa rameters are listed in Table 6.
Fig. 7. Seismic fragility curves for different limit states: (a) model I; (b) model II.
Fig. 8. Comparison of fragility curves of model I and II for different limit states: (a) Local sliding damage; (c) Overall sliding instability. 6
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zero when PGA reaches 0.481 g. It can be considered that the arch dam can maintain stability under check earthquake. Moreover, from Fig. 8 (a) and (b), slight sliding damage of the arch dam-foundation for model I may occur until PGA reaches about 0.1 g, while the corresponding PGA for model II is about 0.15 g. The overall sliding failure of the arch dam is likely to occur when PGA comes to 0.5 g for model I and 0.55 g for model II. For model I, the probability of occurrence of overall sliding failure is almost 100% when PGA is 0.7 g, which is 0.75 g for model II. It can be drawn that the existence of re sidual cohesion delays the sliding development and enhances the seismic stability of arch dam-foundation systems. Furthermore, a comparative analysis is performed based on the extended limit states defined in Reference [13]. The fragility curves as well as those obtained by this paper are shown in Fig. 9. From the discrete data obtained by statistical method, the probability of the dam reaching the local sliding damage/initiation and joint sliding are of slight difference. Besides, an additional limit state i.e. concrete cracking is defined in Reference [13], and the corresponding fragility curve is extracted. The fragility curves corresponding to the limit states of global yielding and dam failure/overall sliding instability defined by joint sliding in Reference [13] are almost the same. Based on the comparative analysis, it is preferable to add one limit state of concrete cracking into this research for a more comprehensive evaluation of seismic fragility of the concrete arch dam based on instability failure mode.
Table 6 Fragility parameters for model I and model II. EDP
Limit states Local sliding
ηPGA Slippage-based
1.2338 1.1425
Overall sliding instability βPGA 0.4124 0.3220
ηPGA 0.4263 0.3857
βPGA 0.0819 0.0721
5.2.2. Sliding area ratio-based rule The seismic fragility curves corresponding to three sliding area ratiobased limit states of the concrete arch dam-foundation system are given in Fig. 10 (a) and (b). Fig. 11(a), (b) and (c) presents the comparison of fragility curves of model I and II for different limit states. Seismic fragility parameters are listed in Table 7. The difference of the proba bility of the arch dam reaching a certain limit state under different PGA levels for model I and II are shown in Table 8. According to Fig. 10 (a) and (b), for model I, when PGA reaches 0.406 g, the probability of occurrence of slight sliding damage is about 90%, and that of moderate sliding damage is less than 55%. The prob ability of occurrence of overall sliding failure is about 0.5%. When PGA reaches 0.481 g, the probability of overall sliding failure of the arch dam foundation system is about 20%. As for model II under design earth quake, the corresponding probability of occurrence of slight sliding damage is about 70% and it’s less than 30% for moderate sliding dam age. Meanwhile, the probability of occurrence of overall sliding failure is
Fig. 9. Comparison of fragility curves based on the extended LSs.
Clearly, as shown in Fig. 7 (a) and (b), for model I, the probability of occurrence of local sliding damage is about 80%, and it is zero for overall sliding failure under design earthquake (2% probability of exceedance in 100 years, PGA ¼ 0.406 g). Even if PGA reaches check earthquake (1% probability of exceedance in 100 years, PGA ¼ 0.481 g), the probability of occurrence of overall sliding failure of the arch dam-foundation sys tem is still zero. As for model II, the probability of occurrence local sliding of the arch dam-foundation system decreases slightly, which is similarly approximate 80% under design earthquake i.e. PGA of 0.406 g. The probability of occurrence of local sliding of arch dam foundation system is about 90%, and the probability of overall sliding failure is still
Fig. 10. Seismic fragility curves for different limit states: (a) model I; (b) model II. 7
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Fig. 11. Comparison of fragility curves of model I and II for different limit states: (a) Local sliding damage; (b) Moderate sliding damage; (c) Overall sliding instability. Table 8 Differencein probability of model I and model II.
Table 7 Fragility parameters for model I and model II. EDP
Sliding area ratio-based
Limit states Local/slight sliding damage
Moderate sliding damage
Overall sliding instability
ηPGA
ηPGA
ηPGA
1.2343 1.0751
βPGA 0.3062 0.3167
0.8733 0.7584
βPGA 0.1691 0.1737
0.6367 0.5559
βPGA 0.1385 0.1428
about zero. There is a 5% decrease for the probability of occurrence of overall sliding failure under check earthquake with respect to model I. From Fig. 11 (a), (b) and (c), it is clearly that only when PGA is 8
Prob I-Prob II
Limit state Local sliidng damage
moderate sliidng damamge
Overall sliding instability
PGA
0 0.02 0.18 0.32 0.06 0 0 0 0 0
0 0 0.06 0.12 0.3 0.02 0 0 0 0
0 0 0 0.04 0.08 0.3 0 0 0 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
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slightly greater than 0.1 g, slight sliding damage may occur for model I. While the corresponding PGA increase to 0.15 g for model II. When PGA reaches 0.7 g and 0.75 g for model I and model II respectively, the probability of occurrence of overall sliding instability reaches 100%, which is consistent with the conclusion obtained based on slippagebased fragility curves. It can be obtained that the residual cohesion plays an important role in improving the seismic stability of concrete dams. It presents to be relatively conservative for the seismic design of concrete dams without considering the residual cohesion, but to some extent, the real seismic performance of concrete dams is underestimated. Moreover, compared with slippage-based fragility curves, sliding area ratio-based curves can reflect the difference between model I and model II more obviously. It can better reveal the whole process of sliding development of the arch dam abutment and the influence of residual cohesions. The comparison of slippage-based and sliding area rationbased fragility curves of the arch dam-foundation system with or without residual cohesion shows that the probability of occurrence of overall failure of dam foundation system given by sliding area ratiobased rule is higher than that given by slippage-based rule under check earthquake. It is safer to use sliding area ratio-based rule to evaluate the seismic stability of dam-foundation system.
the PGA corresponding to the 100% probability of ultimate sliding instability are basically the same with the value of 0.7 g and 0.75 g respectively for model I and model II. Moreover, sliding area ratio-based curves can reflect the difference between model I and model II more obviously. It can better reveal the whole process of sliding development of the arch dam abutment and the influence of residual cohesion. The comparison of slippage-based and sliding area ration-based fragility curves of the arch dam-foundation system with or without residual cohesion shows that the probability of occurrence of overall failure of dam foundation system given by sliding area ratio-based rule is higher than that given by slippage-based rule under check earthquake. It is safer to use sliding area ratio-based rule to evaluate the seismic stability of dam-foundation system. It is notable that seismic performance of arch dams can be quantitatively evaluated by seismic fragility analysis which is of great significance for managing uncertainties in the seismic design and providing reinforcement and rehabilitation strategies of concrete dams. According to the comparative analysis based on the extended LSs, it is preferable to add one limit state of concrete cracking into this research for a more comprehensive evaluation of seismic fragility of the concrete arch dam based on instability failure mode. The current study has the limitations in view that only an example of a concrete arch dam is presented without considering all possible sources of nonlinearity and uncertainty. However, it’s no doubt that the research is of great significance for the seismic safety evaluation of concrete dams and can serve as a reference for the following study. The further study can be extended from the above perspectives: (1) the un certainty caused by record-to-record uncertainty; (2) material nonline arity of the dam concrete; (3) the correlation between the random parameters.
6. Conclusions This paper presents the development of seismic fragility curves of a high concrete arch dam based on seismic instability failure mode. The epistemic uncertainty is considered by taking the friction coefficients and cohesions as random parameters affecting the seismic stability performance of arch dams. Both the cohesion and residual cohesion are considered for a comparative analysis of the effect of residual cohesion on the seismic fragility of the arch dam. The PGA is taken as intensity measure, and the slippage and the sliding area ratio of the bottom sliding surface are used as engineering demand parameters (EDP). The IDA method is implemented to quantitatively define the damage levels of the high curvature arch dam through slippage-based rule and sliding area ratio-based rules respectively. Consequently, the seismic fragility curves are obtained for the defined damage levels based on the different EDPbased rules. Based on the obtained IDA curves, the damage levels are defined first according to slippage-based rule. The developing process of seismic stability of the arch dam can be divided into three stages based on the overall mean slippage-based IDA curves: (1) Stability region; (2) Local sliding damage region; (3) Overall sliding instability region. Meanwhile, the two damage levels can be defined from the corresponding slippage of the two turning points. The slippage with zero value can be a good choice for the limit state of local sliding damage initiation. The slippage of 0.04 m (40 mm) can be a critical value for the limit state of ultimate instability initiation. Afterwards, the comprehensive developing process of seismic stability of the arch dam can be obtained from the overall mean sliding area ratio-based IDA curves: (1) Stability region; (2) Local/ slight sliding damage region; (3) Moderate sliding damage region; (4) Overall sliding instability region. Thus, the corresponding three damage levels can be identified based on the sliding area ratio-based rule. The sliding area ratio with 10%value can be a good choice for the limit state of local sliding damage initiation. The sliding area ratio of 50% can be a critical value for the limit state of moderate sliding damage initiation. The overall sliding damage initiation can be defined at the sliding area ratio of 90%. Moreover, seismic fragility curves are extracted for the seismic sta bility of concrete arch dams under earthquake hazards for both two models, i.e. model I without residual cohesions and model II with re sidual cohesions (30% of the corresponding peak cohesion). From the obtained fragility curves, it can be said that the arch dam is able to maintain good stability under MCE. Compared slippage-based fragility curves and sliding area ratio-based curves, though there is a difference in the probability of occurrence of ultimate sliding instability under MCE,
Declaration of competing interest The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the paper (entitled with ‘Seismic Fragility Analysis of a High Arch DamFoundation System based on Seismic Instability Failure Mode’, Paper No.: SOILDYN_2019_790_R1) submitted to Soil Dynamics and Earth quake Engineering. Acknowledgements This study is supported by the National Key Research and Develop ment Program [grant number 2017YFC0404903], National Natural Science Foundation of China (grant number 51709283) and State Key Laboratory of Water Cycle Simulation and Regulation and Natural Sci ence for Youth Foundation [grant number 2016CG03]. The authors are grateful for these supports. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.soildyn.2019.105981. References [1] Hoffman FO, Hammonds JS. Propagation of uncertainty in risk assessments: the need to distinguish between uncertainty due to lack of knowledge and uncertainty due to variability. Risk Anal 1994;14:707–12. [2] Ferson S, Ginzburg LR. Different methods are needed to propagate ignorance and variability. Reliab Eng Syst Saf 1996;54:133–44. [3] Kiureghian AD, Ditlevsen O. Aleatory or epistemic? does it matter? Struct Saf 2009; 31:105–12. [4] Hariri-Ardebili MA, Saouma VE. Seismic fragility analysis of concrete dams: a stateof-the-art review. Eng Struct 2016;128:374–99. [5] Alembagheri M, Seyedkazemi M. Seismic performance sensitivity and uncertainty analysis of gravity dams. Earthq Eng Struct Dyn 2015;44(1):41–58. [6] Tekie PB, Ellingwood BR. Seismic fragility assessment of concrete gravity dams. Earthq Eng Struct Dyn 2003;32(14):2221–40.
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[7] Bernier C, Padgett JE, Proulx J, et al. Seismic fragility of concrete gravity dams with spatial variation of angle of friction: case study. J Struct Eng 2015;142(5). 05015002-1: 05015002-05015011. [8] Morales-Torres A, Escuder-Bueno I, Altarejos-García L, et al. Building fragility curves of sliding failure of concrete gravity dams integrating natural and epistemic uncertainties. Eng Struct 2016;125:227–35. [9] Amirpour A, Mirzabozorg H. Quantifying the qualitative limit-states using IDA approach in concrete arch dams. Arabian J Sci Eng 2014;39(11):7729–40. [10] Vamvatsikos D, Cornell CA. Incremental dynamic analysis. Earthq Eng Struct Dyn 2002;31(3):491–514. [11] Zhong H, Li HJ, Bao YL. Seismic risk analysis of an arch dam. Appl Mech Mater 2013;353–356:2020–3. [12] Kadkhodayan V, Aghajanzadeh SM, Mirzabozorg H. Seismic assessment of arch dams using fragility curves. Civ Eng J 2016;1(2):14–20. [13] Hariri-Ardebili MA, Furgani L, Meghella M, et al. A new class of seismic damage and performance indices for arch dams via ETA method. Eng Struct 2016;110: 145–60. [14] Jin Ting Wang, Ming Xu Zhang, Ai Yun Jin, Chu Han Zhang. Seismic fragility of arch dams based on damage analysis. Soil Dyn Earthq Eng 2018;109:58–68. [15] Liu JB, Wang D, Yao L. A Contact force model in the dynamic analysis of contactable cracks. Acta Mech Solida Sin 1993;6(4):439–52. [16] Varley A. The exceptional and the everyday: vulnerability analysis in the international decade for natural disaster reduction. In: Varley A, editor. Disasters, development and environment. Chinchesterl: John Wiley & Sons Ltd; 1994. [17] Barbat AH, Moya, Ye0 pez Fabricio, Canas Jose0 A. Damage scenarios simulation for seismic risk assessment in urban zones. Earthq Spectra 1996;12(3):371–94. [18] Goodman J. Structural fragility and principle of maximum entropy. Struct Saf 1985;3(1):37–46.
[19] Baker JW. Efficient Analytical fragility function fitting using dynamic structural analysis. Earthq Spectra 2015;31(1):579–99. [20] Clough RW. Non-linear mechanisms in the seismic response of arch dams. In: Proceedings of the international research conference earthquake Engineering. Yugoslavia: Skopje; 1980. p. 669–84. [21] Fenves GL, Mojtahedi S, Reimer RB. Effect of contraction joints on earthquake response of an arch dam. J Struct Eng ASCE 1992;118:1039–55. [22] Zhang CH, Xu YJ, Wang GL, Jin F. Non-linear seismic response of arch dams with contraction joint opening and joint reinforcements. Earthq Eng Struct Dyn 2000; 29:1547–66. [23] Westergaard HM. Water pressure on dams during earthquakes. Trans Am Soc Civ Eng 1933;98:418–72. [24] Xu Jianqiang, Huang Wei, Gao Ya Fen. International code for design and specification of concrete gravity dam: sub coefficient determination and experimental design. In: International Conference on dam technology and longterm performance; 2011. p. 30–5 [in Chinese)]. [25] Iman RL, Conover WJ. A distribution-free approach to inducing rank correlation among input variables. Commun Stat Simulat Comput 1982;11:311–34. [26] Vamvatsikos D, Fragiadakis M. Incremental dynamic analysis for estimating seismic performance sensitivity and uncertainty. Earthq Eng Struct Dyn 2010;39 (2):141–63. [27] Alembagheri M, Ghaemian M. Seismic assessment of concrete gravity dams using capacity estimation and damage indexes. Earthq Eng Struct Dyn 2013;42(1): 123–44. [28] Shen Huaizhi. Performance-based seismic damage analysis and risk assessment of concrete dam-foundation system. Tsinghua University; 2007.
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Seismic fragility analysis of a High Arch Dam-Foundation System based on seismic instability failure mode Hui Liang a, b, Jin Tu a, b, Shengshan Guo a, b, *, Jianxin Liao c, Deyu Li a, b, Shiqi Peng c a
Earthquake Engineering Research Centre, China Institute of Water Resources and Hydropower Research, Beijing, China State Key Laboratory of Simulation and Regulation of Water Cycle in River Basin, Beijing, 100048, China c China Three Gorges Projects Development Co., Ltd, Beijing, 100038, China b
A R T I C L E I N F O
A B S T R A C T
Keywords: High arch dam-foundation system Seismic instability failure mode Dynamic contact model Latin hypercube sampling Seismic fragility analysis
Seismic stability of concrete arch dams has always been a critical issue in seismic safety evaluation. In this paper, a high concrete arch dam is selected as a case study for the seismic fragility analysis based on instability failure mode. A comprehensive approach considering the opening and closing of contraction joint, probable sliding rock mass of dam abutments, the interface between the dam and foundation as well as the effect of foundation ra diation damping is presented. The random and uncertain parameters containing the friction coefficients and cohesions are generated with Latin hypercube sampling method (LHS). The approximate IDA is performed, and the slippage and sliding area ratio are chosen as the engineer demand parameters (EDP). Different damage levels are identified by the slippage-based rule and sliding area ratio-based rule respectively according to their cor responding overall mean IDA curves. In addition, seismic fragility curves are developed for the defined damage levels. The results reveal that the damage levels defined based on different EDP-based rules could be different, while the ultimate capacity of the concrete arch dam reflected by selected EDPs are basically the same. Sliding area ratio-based rule can better reveal the whole process of sliding development of the arch dam abutment and the influence of residual cohesions. The existence of residual cohesion delays the sliding development and en hances the seismic stability of arch dam-foundation systems. Furthermore, a comparative analysis is performed based on the extended limit states. Seismic fragility analysis is of great significance for quantitatively evaluating the seismic performance of concrete dams from the probability point of view.
1. Introduction Seismic safety of high dams is an inevitable problem during the construction of momentous water conservancy projects. Existing prac tical examples of high dams subjected to strong earthquakes show that dam abutment is the weak part. The overall instability of the damfoundation system caused by its damage is one of the most important concerns in the seismic design and safety assessment of high dams, which faces with complicated geological conditions and various loads. Moreover, there are various uncertainties, such as epistemic (modeling and material parameters) and aleatory (earthquake ground motion, flood etc.) uncertainty during the design, construction and operation of dams [1–3], which significantly affect the seismic perfor mance of concrete dams. Fragility analysis, which plays an essential role in seismic probabilistic risk assessment within the framework of performance-based seismic design, is gradually applied to water
conservancy and hydropower projects, such as gravity dams and arch dams. Hariri-Ardebili M A et al. [4] provide a comprehensive state-of-the-art review of existing application on fragility analysis of concrete dams. The detailed procedures and different methods are summarized and a comparative review of major researches and publi cations is presented. A consistent procedure is particularly presented to perform fragility analysis for the sliding failure mode of concrete gravity dams in order to identify and track natural and epistemic uncertainty separately [8]. However, among those described researches, few attempts have been partly made on the effect of various uncertainties on seismic perfor mance of concrete arch dams. Amirpour and Mirzabozorg [9] adopt nine actual ground motion records divided into three levels as seismic input for quantitative and qualitative analysis of the ultimate seismic failure state of arch dams using IDA method [10]. Zhonghong et al. [11] perform seismic risk analysis of a concrete arch dam considering the
* Corresponding author. Earthquake Engineering Research Centre, China Institute of Water Resources and Hydropower Research, Beijing, China. E-mail addresses: [email protected], [email protected] (S. Guo). https://doi.org/10.1016/j.soildyn.2019.105981 Received 22 July 2019; Received in revised form 6 November 2019; Accepted 17 November 2019 Available online 22 November 2019 0267-7261/© 2019 Elsevier Ltd. All rights reserved.
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non-homogeneity of concrete material, uncertainty of earthquake ground motions and uncertainty of concrete material parameters. IDA approach is adopted to analyze a thin high arc dam by taking parameters of Sa, PGA and PGV as intensity measure (IM) and the overstressed area (OSA) as engineering demand parameter, and based on the defined limit states, the seismic fragility curves are obtained [12]. In view that the quantitative study of progressive failure of concrete dams is an impor tant part of risk analysis, Hariri-Ardebili, Furgani and Meghella [13] compare and analyze more than 20 indexes using Endurance Time Analysis method (ETA) based on the concept of damage index and performance index, which provides an effective method for the selection of engineering demand parameters in seismic performance evaluation of dams. The seismic fragility analysis of a high arch dam including dam concrete cracking and contraction joint opening and closing is per formed considering both epistemic and aleatory uncertainties, and seismic fragility curves under different damage levels are drawn [14]. In general, seismic fragility analysis of high arch dams is still in its infancy. The fragility analysis based on the instability failure mode of arch dam-foundation systems is basically absent. Thus, this paper fo cuses on the seismic fragility analysis of a high concrete arch damfoundation system based on the seismic instability failure mode. The arch dam abutment and the dam are coupled as a system. A high con crete arch dam in China is taken as a case study. The friction coefficients and cohesions of the probable sliding block affecting the seismic stability of the arch dam are taken as random parameters. Both two models i.e. model I without residual cohesions and model II with residual cohesions (30% of the corresponding peak cohesions) are considered for a comparative analysis of the effect of residual cohesion on the seismic fragility of the arch dam. The PGA is used as intensity measure (IM), and the slippage and sliding area ratio of the bottom sliding surface are chosen as engineer demand parameters (EDP) respectively. IDA curves are obtained by conducting the IDA with Latin Hypercube Sampling (LHS). Different damage levels are defined through slippage-based rules and sliding area ratio-based rules. Furthermore, the corresponding fragility curves are extracted for the seismic stability evaluation of concrete arch dams.
Fig. 1. Aerial view from downstream of Baihetan arch dam.
intensity measures (IM, such as PGA, PGV). N and M represents the number of random samples and the number of ground motions. NLS is the number of the dam reaching or exceeding a certain LS at each earthquake intensity level among all the N � M dynamic analyses. (2) Fragility curves can be fitted with different probability distribu tion functions based on the data obtained from the above results (Equation (1)). At present, the widely used analytic function of fragility curve is lognormal distribution function [6,18,19], shown as following Equations (2)–(4). Thus, seismic fragility analysis of concrete dams can be performed and fragility curves can be developed according to the above-mentioned procedures. � � ln im ηim FðimÞ ¼ PðLSjIM ¼ imÞ ¼ Φ (2) βim
ηim ¼ βim ¼
2. Computational methodology
Fragility is expressed as the probability of engineering structure reaching a certain limit state or performance level under different earthquake intensity levels in seismic engineering [16,17]. A large amount of data is needed in order to obtain fragility curves of structures with reasonable and accurate consideration of the effects of un certainties on structural performance. Analytical fragility analysis is adopted, of which the data base of is obtained from the structural dy namic response under a series of earthquakes with increasing seismic intensity levels. Procedures for analytical fragility analysis include structural dynamic response analysis, definition of limit states and for mation of fragility curve. Herein, IDA method is used to develop seismic fragility curves of the concrete arch dam-foundation system. A series nonlinear dynamic analyses are performed considering epistemic un certainty. The seismic fragility curves of concrete dams can be obtained according to the following procedures.
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi XN�M 1 ðlnðimÞ ηim Þ2 N � M 1 i¼1
(4)
3. Nonlinear modeling of a high arch dam-foundation system 3.1. Finite element model The Baihetan double-curvature arch dam is selected as a case study (Fig. 1), located in strong earthquake area in Southwest China. The height of the dam is 289 m, and the crest is at El. 834 m. The thickness of the dam crest is 14.0 m, and the maximum thickness of dam bottom is 83.91 m. The valley of the dam site is relatively symmetrical ‘V’ type. The left bank slope is gentle and the right bank slope is steep. Moreover, there are many gentle dip structural planes, faults and cracks in the dam abutments. The combination of all these unfavorable factors constitutes a geological background resulting in the probable seismic instability of dam abutments under earthquakes. According to the topographic and geological characteristics of the dam, a three-dimensional finite element mesh of arch dam-foundation system is established, shown in Fig. 2(a). The whole system is dis cretized by three-dimensional solid elements. Six layers of threedimensional solid elements are arranged along the thickness direction of the dam body. The whole model contains 130,000 nodes and 120,000 solid elements. Fig. 2(b) describes the potential sliding block of the arch dam abutment. The sliding block is composed of bottom sliding surface,
(1) The probability of a concrete dam reaching a certain limit state (LS) under a given earthquake intensity level can be obtained by the following formula. NLS N�M
(3)
where FðimÞ is the probability that the dam reaches or exceeds a certain LS at an earthquake intensity level. Φð⋅Þ is standard normal distribution function. ηim and βim are the logarithmic mean and logarithmic standard deviation of im.
2.1. Generation of seismic fragility curve
PðLSjIM ¼ imÞ ¼
1 XN�M lnðimi Þ N � M i¼1
(1)
where PðLSjIM ¼ imÞ is the conditional probability that the concrete dam reaches or exceeds a certain LS under an earthquake intensity level of 2
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Fig. 2. Finite element mesh of the arch dam-foundation system.
Fig. 3. Selected observation point (number 6272).
pronounced than that of elastic modulus of concrete and bedrock. Thus, shear strength parameters on each potential sliding surface are consid ered as random variables. They are taken as μboottom, cboottom and μcrack, ccrack and μside, cside respectively for the bottom sliding surface, cracking surface and side sliding surface. In view that the overall seismic insta bility of the arch dam-foundation system reflected by the sudden and rapid increase of the slippage is a process of gradual development and accumulation of the total deformation including local cracking and sliding of each part, the residual slippage of the characteristic point (node number 6272, Fig. 3) at the bottom sliding surface is selected as one of the indexes to quantitatively evaluate the seismic sliding stability of the arch dam abutment. Moreover, the sliding area ratio (ratio of the sliding area to the bottom sliding surface) is chosen as another damage index for evaluating seismic stability of the arch dam-foundation.
side sliding surface, cracking surface and free surface. Fig. 3 shows the location of the selected observation point (node 6272), of which the slippage can be used to represent the overall seismic stability of the damfoundation system. Since the opening and closing of contraction joints of arch dams are important factors affecting their seismic performance [20–22], the dam is separated into 29 adjacent monoliths by 28 vertical contraction joints. Generally, dam-foundation interface is the weak part of the dam-foundation system. Therefore, a joint interface is considered to simulate the failure of the dam-foundation interface. Moreover, for the purpose of research on seismic stability of the dam abutment, each po tential sliding surface of the probable sliding block that affects the overall stability of the dam is also considered to be a joint interface Fig. 2 (b). Thus, a comprehensive approach composed of contraction joint opening and closing, failure along dam-foundation interface and sliding of the probable sliding rock mass is developed for seismic stability analysis of the arch dam. The viscous-spring artificial boundary is used to simulate the radia tion damping effect caused by the dynamic interaction between the dam and foundation. The widely used Westergaard added mass model [23] is adopted to consider the hydrodynamic pressure. The joint interfaces which are discretized into contact node pairs are modeled using a dy namic contact model based on Lagrange multiplier method, proposed by Liu [15]. The sliding resistance force at sliding surfaces are supposed to be controlled exclusively by friction coefficients and cohesions in the dynamic contact model. Since this research focuses on the seismic sta bility of arch dam abutments, the influence of shear strength parameters on contact surfaces of the probable sliding rock mass is more
3.2. Material parameters The material parameters of dam concrete are as follows: mass den sity ¼ 2400.0 kg/m3 and Poisson’s ratio ¼ 0.167. The static elastic modulus is 24.0 GPa and dynamic elastic modulus is 1.5 times of the static elastic modulus. The linear expansion coefficient is 6.5 � 10 6/� C. The actual stratification of bedrock materials is considered, of which the average static comprehensive deformation modulus of each layer is about 11.0 GPa. The density and Poisson’s ratio are 2700.0 kg/m3 and 0.23 respectively. The initial tensile strength of dam foundation inter face is 3.42 MPa, and the friction coefficient and cohesion are 1.15 and 1.10 MPa, respectively. The parameters of the contraction joint are taken as: friction coefficient μjoint ¼ 0.8, cohesion cjoint ¼ 0. 3
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3.3. Static and dynamic loads
Table 1 The material properties and probability distributions for the random parameters. Material parameters
Mean/Basecase model
COV
Minimum
Maximum
type
Friction coefficient
0.432
0.200
0.2592
0.6048
Normal
0.193
0.400
0.08293
0.3872
1.100
0.200
0.6600
1.5400
Lognormal Normal
1.200
0.400
0.5156
2.4076
0.420
0.200
0.2520
0.5880
0.090
0.400
0.03867
0.1806
μbottom
Cohesion cbottom (MPa) Friction coefficient
μcrack
Cohesion ccrack (MPa) Friction coefficient
μside
Cohesion cside (MPa)
The static loads include self-gravity of the dam body, water pressure in the upstream and downstream under the normal water level, up stream sediment pressure and temperature load. Upstream and down stream normal water levels are 825.00 m and 604.00 m respectively. The upstream sediment elevation is 710 m. The bulk density and internal friction angle are 5 kN/m3 and 0� . The design temperature load of the arch ring is shown in Table 2. In view that the record-to-record uncertainty is beyond consideration in this paper, three uncorrelated artificial ground motions in three di rections are generated based on the site-specific design response spec trum corresponding to the exceedance probability of 2% in 100 years. The PGA in vertical direction is two-third of that in stream direction in the analysis. The representative value of horizontal design seismic ac celeration of Baihetan arch dam is 0.406 g, which is equivalent to 2% of the exceeding probability in 100 years, and the vertical value is 2/3 of the horizontal value, which is 0.271 g. All the components of the arti ficial ground motion are normalized and shown in Fig. 4.
Lognormal Normal Lognormal
Uncertainty is an inevitable objective phenomenon in the process of dam design, construction and operation. They are important factors to be considered in seismic fragility analysis of concrete dams. Considering various uncertainties in the framework of probability statistics can provide a basis for fully understanding the expected range of structural dynamic response. Herein, epistemic uncertainty due to the lack of in formation of the engineering geological condition is considered for the seismic fragility analysis. As above-mentioned, six parameters μboottom, cboottom and μcrack, ccrack and μside, cside are treated as random variables in the dynamic contact model. Normal distribution and lognormal distri bution are considered for friction coefficients and cohesions respectively [24]. A mean value and coefficient of variation (COV) for each param eter is assigned according to engineering design data. The distribution of each parameter is truncated with a reasonable minimum and maximum to meet with physical limits. The hard limits set at 2 standard deviations away from the central value are considered to cover most of the samples. Thus, the parameters used in the dynamic contact model are defined using the properties listed in Table 1. The widely used Latin hypercube sampling method (LHS) [25] is adopted to handle the current impact of parameter uncertainties in view of its accuracy and efficiency for the probabilistic analysis.
4. Probabilistic analysis Generally, it is assumed that the cohesion on the contact surface becomes zero and ineffective when the sliding happens. In fact, when the relative sliding of the contact surface happens, the effect of cohesion on the contact surface is not necessarily completely ineffective, and its ef fect may be attenuated to a certain extent. Herein, the initial cohesion parameters between the contact surfaces can be defined as peak cohe sion. When sliding of the contact surface happens, the peak cohesions change to the residual cohesions. In order to study the effect of residual cohesions on the seismic stability of arch dam-foundation systems, two models, i.e. model I without residual cohesions and model II with re sidual cohesions (30% of the corresponding peak cohesions is selected as a case study), are considered for the probabilistic analysis with the advanced dynamic contact model. The friction coefficients and co hesions, which are the main parameters affecting the seismic sliding stability of arc dam-foundation systems, are sampled using Latin hy percube sampling method. There is no specific standard for the size N of random variables. The size N of samples is related to the number of random parameters and the response of structures affected by them. Generally, a relative high sample size N will provide reasonable and
Table 2 Temperature load. Elevation Normal water level þ temperature drop
uniformity linear
834.0
800.0
760.0
720.0
680.0
640.0
610.0
580.0
565.0
545.0
5.99 0.00
2.84 7.27
3.52 10.98
3.23 12.06
4.27 12.28
4.28 12.38
5.29 12.4
1.77 5.89
1.39 5.28
1.15 4.91
Fig. 4. The normalized artificial acceleration time histories. 4
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Table 3 Slippage-based limit states. EDP
Limit states
Slippage-based
Local sliding damage
Overall sliding instability
0.00 m
0.40 m
Table 4 Slippage-based limit states in Reference [13]. EDP
Slippagebased
Limit states Initiation
Joint opeining/ sliding
Concrete cracking
Global yielding
Dam failure
0.0003 m
0.0015 m
0.0053 m
0.0365 m
0.0393 m
Fig. 5. Slippage-based limit states definition.
accurate assessment for a structure. LHS method is used to obtain a total number of 20 samples for each uncertain parameter in the seismic fragility analysis of concrete gravity dams [7]. The sample sizes N of 30 and 200 are obtained by LHS separately [5,26]. Herein, the sample size N is taken as 50 for probabilistic analysis. In general, seismic sliding stability of arch dam abutment is not only affected by the arch thrust, but also by inertia force of the abutment sliders. Considering that this paper focuses on the seismic stability of an arch dam-foundation system, PGA is selected as IM, scaled to multiple levels from zero to 1.0 g with steps of 0.1 g in the IDA of arch dams. The slippage of the selected characteristic point and the sliding area ratio are chosen as EDPs to assess seismic stability of the arch dam-foundation system. Thus, the total number 10 � N ¼ 500 of nonlinear dynamic analyses are performed for both model I and II, and IDA curves are generated from discrete points with spline interpolation. Since the IDA curves can be used for the definition of limit states [27], the overall mean IDA curves with their 50% fractile are obtained. Consequently, based on the seismic fragility theory, seismic fragility curves are ob tained to quantitatively describe the seismic stability of arch dam-foundations.
Fig. 6. Sliding area ratio-based limit states definition.
5.1.1. Slippage-based rule Fig. 5 describes different slippage-based limit states defined for the concrete arch dam-foundation system through the overall mean IDA curve and 50% fractile IDA curve. It shows that the curve is divided into three stages by two turning points. In the first stage, with the increase of ground motion, only slight sliding occurs on the abutment slider, and the residual slippage of the characteristic points is very small and almost zero, which indicates that the arch dam-foundation system is of good stability and it is in a stable region. In the second stage, the residual slippage increases slowly with the increase of PGA levels. Although the local sliding happens, the maximum residual displacement is less than 0.04 m, and the overall stability of the dam is still in a controllable state. In the third stage, the residual sliding displacement of characteristic points increases rapidly, the stability of arch dam foundation system is in an uncontrollable state, and the overall sliding failure has occurred. Therefore, according to the appearance of turning points in each stage, the two limit states and the corresponding residual slippage threshold of arch dam foundation system based on sliding instability failure mode can be defined as (Table 3): (1) Local sliding damage, corresponding to the residual slippage of 0.0 m of the characteristic point at the first turning point, i.e. the initial sliding of the characteristic point; (2) Overall sliding failure, corresponding to the residual slippage of 0.04 m (40.0 mm) at the second turning point. According to HaririArdebili M A [13], the maximum joint sliding ΩðδJn Þ corresponding to the initial dam sliding and the ultimate dam failure is 0.3 mm and 39.3 mm, which is basically consistent with this paper of 0.00 and 40.0 mm,
5. Results and discussion 5.1. Limit state definition The definition of limit states for the arch dam is the premise of seismic fragility analysis. As for concrete dams, their limit states are related to different failure modes. Existing examples of dams subjected to earthquake hazards and researches show that main potential failure modes of arch dams include dam cracking caused by excessive tensile cantilever stress and sliding failure of abutment slider. Up to now, there is no completely unified and recognized scheme to define the limit states of concrete dams. Shen huaizhi [28] proposed preliminary suggestions on performance-based safety evaluation index according to the yield ratio of the gravity dam-foundation interface and the performance of drainage system. Alembagheri et al. [27] reasonably analyzed the bearing capacity and ultimate state of concrete gravity dams using IDA curves. Kadkhodayan et al. [12] defined three performance levels for a concrete arch dam based on IDA curves. The overstress area is taken as EDP and Sa, PGA and PGV are taken as IMs. In this paper, IDA curves are used to define the limit states of the arch dam-foundation. The overall mean slippage-based and sliding area ratio-based IDA curves with their 50% fractile are obtained from probabilistic analysis. The different limit states of seismic stability of the arch dam-foundation system under earthquakes are defined for the seismic fragility analysis. 5
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with the increase of PGA levels, and the arch dam can still maintain a relatively good stability. (3) Moderate sliding stage. The sliding area ratio increases to more than 50%, which indicates that more than half of the joints on the bottom sliding surface have slid. At present, the mod erate sliding damage of the arch dam has happened, which affects the normal operation of the reservoir power station; (4) Overall sliding instability stage. The arch dam-foundation system is in overall insta bility state, which is difficult to repair. In summary, based on the sliding area ratio-based rule, the limit states of the arch dam foundation system can be divided into (Table 5): (1) Local/Slight sliding damage, corresponding to the sliding area ratio of 10%; (2) Moderate sliding damage, corresponding to the sliding area ratio of 50%; (3) Overall sliding failure, corresponding to the sliding area ratio of 90%.
Table 5 Sliding area ratio-based limit states. EDP
Limit states
Sliding area ratio-based
Local sliding damage
Moderate sliding damage
Overall sliding instability
10%
50%
90%
shown in Table 4. Moreover, a comparative analysis is performed to present the difference of the seismic fragility curves based on these defined limit states. 5.1.2. Sliding area-based rule Fig. 6 depicts the sliding area ratio-based limit states of the concrete arch dam-foundation system. Different regions are divided by the red dotted line for each limit state, detailly shown in Fig. 6. According to Fig. 6, based on the sliding area ratio versus PGA curve, the process of seismic sliding stability of arch dam foundation system can be divided into four stages: (1) Stability stage. The sliding area ratio is small, less than 10% of the whole sliding surface area, and at this stage, it increases slowly with the increase of PGA levels, and the arch dam is of good stability; (2) Slight sliding stage. The sliding area ratio increases rapidly
5.2. Seismic fragility curves 5.2.1. Slippage-based rule The seismic fragility curves corresponding to two slippage-based limit states of the concrete arch dam-foundation system are given in Fig. 7 (a) and (b). Fig. 8 (a) and (b) presents the comparison of fragility curves of model I and II for different limit states. Seismic fragility pa rameters are listed in Table 6.
Fig. 7. Seismic fragility curves for different limit states: (a) model I; (b) model II.
Fig. 8. Comparison of fragility curves of model I and II for different limit states: (a) Local sliding damage; (c) Overall sliding instability. 6
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zero when PGA reaches 0.481 g. It can be considered that the arch dam can maintain stability under check earthquake. Moreover, from Fig. 8 (a) and (b), slight sliding damage of the arch dam-foundation for model I may occur until PGA reaches about 0.1 g, while the corresponding PGA for model II is about 0.15 g. The overall sliding failure of the arch dam is likely to occur when PGA comes to 0.5 g for model I and 0.55 g for model II. For model I, the probability of occurrence of overall sliding failure is almost 100% when PGA is 0.7 g, which is 0.75 g for model II. It can be drawn that the existence of re sidual cohesion delays the sliding development and enhances the seismic stability of arch dam-foundation systems. Furthermore, a comparative analysis is performed based on the extended limit states defined in Reference [13]. The fragility curves as well as those obtained by this paper are shown in Fig. 9. From the discrete data obtained by statistical method, the probability of the dam reaching the local sliding damage/initiation and joint sliding are of slight difference. Besides, an additional limit state i.e. concrete cracking is defined in Reference [13], and the corresponding fragility curve is extracted. The fragility curves corresponding to the limit states of global yielding and dam failure/overall sliding instability defined by joint sliding in Reference [13] are almost the same. Based on the comparative analysis, it is preferable to add one limit state of concrete cracking into this research for a more comprehensive evaluation of seismic fragility of the concrete arch dam based on instability failure mode.
Table 6 Fragility parameters for model I and model II. EDP
Limit states Local sliding
ηPGA Slippage-based
1.2338 1.1425
Overall sliding instability βPGA 0.4124 0.3220
ηPGA 0.4263 0.3857
βPGA 0.0819 0.0721
5.2.2. Sliding area ratio-based rule The seismic fragility curves corresponding to three sliding area ratiobased limit states of the concrete arch dam-foundation system are given in Fig. 10 (a) and (b). Fig. 11(a), (b) and (c) presents the comparison of fragility curves of model I and II for different limit states. Seismic fragility parameters are listed in Table 7. The difference of the proba bility of the arch dam reaching a certain limit state under different PGA levels for model I and II are shown in Table 8. According to Fig. 10 (a) and (b), for model I, when PGA reaches 0.406 g, the probability of occurrence of slight sliding damage is about 90%, and that of moderate sliding damage is less than 55%. The prob ability of occurrence of overall sliding failure is about 0.5%. When PGA reaches 0.481 g, the probability of overall sliding failure of the arch dam foundation system is about 20%. As for model II under design earth quake, the corresponding probability of occurrence of slight sliding damage is about 70% and it’s less than 30% for moderate sliding dam age. Meanwhile, the probability of occurrence of overall sliding failure is
Fig. 9. Comparison of fragility curves based on the extended LSs.
Clearly, as shown in Fig. 7 (a) and (b), for model I, the probability of occurrence of local sliding damage is about 80%, and it is zero for overall sliding failure under design earthquake (2% probability of exceedance in 100 years, PGA ¼ 0.406 g). Even if PGA reaches check earthquake (1% probability of exceedance in 100 years, PGA ¼ 0.481 g), the probability of occurrence of overall sliding failure of the arch dam-foundation sys tem is still zero. As for model II, the probability of occurrence local sliding of the arch dam-foundation system decreases slightly, which is similarly approximate 80% under design earthquake i.e. PGA of 0.406 g. The probability of occurrence of local sliding of arch dam foundation system is about 90%, and the probability of overall sliding failure is still
Fig. 10. Seismic fragility curves for different limit states: (a) model I; (b) model II. 7
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Fig. 11. Comparison of fragility curves of model I and II for different limit states: (a) Local sliding damage; (b) Moderate sliding damage; (c) Overall sliding instability. Table 8 Differencein probability of model I and model II.
Table 7 Fragility parameters for model I and model II. EDP
Sliding area ratio-based
Limit states Local/slight sliding damage
Moderate sliding damage
Overall sliding instability
ηPGA
ηPGA
ηPGA
1.2343 1.0751
βPGA 0.3062 0.3167
0.8733 0.7584
βPGA 0.1691 0.1737
0.6367 0.5559
βPGA 0.1385 0.1428
about zero. There is a 5% decrease for the probability of occurrence of overall sliding failure under check earthquake with respect to model I. From Fig. 11 (a), (b) and (c), it is clearly that only when PGA is 8
Prob I-Prob II
Limit state Local sliidng damage
moderate sliidng damamge
Overall sliding instability
PGA
0 0.02 0.18 0.32 0.06 0 0 0 0 0
0 0 0.06 0.12 0.3 0.02 0 0 0 0
0 0 0 0.04 0.08 0.3 0 0 0 0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
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slightly greater than 0.1 g, slight sliding damage may occur for model I. While the corresponding PGA increase to 0.15 g for model II. When PGA reaches 0.7 g and 0.75 g for model I and model II respectively, the probability of occurrence of overall sliding instability reaches 100%, which is consistent with the conclusion obtained based on slippagebased fragility curves. It can be obtained that the residual cohesion plays an important role in improving the seismic stability of concrete dams. It presents to be relatively conservative for the seismic design of concrete dams without considering the residual cohesion, but to some extent, the real seismic performance of concrete dams is underestimated. Moreover, compared with slippage-based fragility curves, sliding area ratio-based curves can reflect the difference between model I and model II more obviously. It can better reveal the whole process of sliding development of the arch dam abutment and the influence of residual cohesions. The comparison of slippage-based and sliding area rationbased fragility curves of the arch dam-foundation system with or without residual cohesion shows that the probability of occurrence of overall failure of dam foundation system given by sliding area ratiobased rule is higher than that given by slippage-based rule under check earthquake. It is safer to use sliding area ratio-based rule to evaluate the seismic stability of dam-foundation system.
the PGA corresponding to the 100% probability of ultimate sliding instability are basically the same with the value of 0.7 g and 0.75 g respectively for model I and model II. Moreover, sliding area ratio-based curves can reflect the difference between model I and model II more obviously. It can better reveal the whole process of sliding development of the arch dam abutment and the influence of residual cohesion. The comparison of slippage-based and sliding area ration-based fragility curves of the arch dam-foundation system with or without residual cohesion shows that the probability of occurrence of overall failure of dam foundation system given by sliding area ratio-based rule is higher than that given by slippage-based rule under check earthquake. It is safer to use sliding area ratio-based rule to evaluate the seismic stability of dam-foundation system. It is notable that seismic performance of arch dams can be quantitatively evaluated by seismic fragility analysis which is of great significance for managing uncertainties in the seismic design and providing reinforcement and rehabilitation strategies of concrete dams. According to the comparative analysis based on the extended LSs, it is preferable to add one limit state of concrete cracking into this research for a more comprehensive evaluation of seismic fragility of the concrete arch dam based on instability failure mode. The current study has the limitations in view that only an example of a concrete arch dam is presented without considering all possible sources of nonlinearity and uncertainty. However, it’s no doubt that the research is of great significance for the seismic safety evaluation of concrete dams and can serve as a reference for the following study. The further study can be extended from the above perspectives: (1) the un certainty caused by record-to-record uncertainty; (2) material nonline arity of the dam concrete; (3) the correlation between the random parameters.
6. Conclusions This paper presents the development of seismic fragility curves of a high concrete arch dam based on seismic instability failure mode. The epistemic uncertainty is considered by taking the friction coefficients and cohesions as random parameters affecting the seismic stability performance of arch dams. Both the cohesion and residual cohesion are considered for a comparative analysis of the effect of residual cohesion on the seismic fragility of the arch dam. The PGA is taken as intensity measure, and the slippage and the sliding area ratio of the bottom sliding surface are used as engineering demand parameters (EDP). The IDA method is implemented to quantitatively define the damage levels of the high curvature arch dam through slippage-based rule and sliding area ratio-based rules respectively. Consequently, the seismic fragility curves are obtained for the defined damage levels based on the different EDPbased rules. Based on the obtained IDA curves, the damage levels are defined first according to slippage-based rule. The developing process of seismic stability of the arch dam can be divided into three stages based on the overall mean slippage-based IDA curves: (1) Stability region; (2) Local sliding damage region; (3) Overall sliding instability region. Meanwhile, the two damage levels can be defined from the corresponding slippage of the two turning points. The slippage with zero value can be a good choice for the limit state of local sliding damage initiation. The slippage of 0.04 m (40 mm) can be a critical value for the limit state of ultimate instability initiation. Afterwards, the comprehensive developing process of seismic stability of the arch dam can be obtained from the overall mean sliding area ratio-based IDA curves: (1) Stability region; (2) Local/ slight sliding damage region; (3) Moderate sliding damage region; (4) Overall sliding instability region. Thus, the corresponding three damage levels can be identified based on the sliding area ratio-based rule. The sliding area ratio with 10%value can be a good choice for the limit state of local sliding damage initiation. The sliding area ratio of 50% can be a critical value for the limit state of moderate sliding damage initiation. The overall sliding damage initiation can be defined at the sliding area ratio of 90%. Moreover, seismic fragility curves are extracted for the seismic sta bility of concrete arch dams under earthquake hazards for both two models, i.e. model I without residual cohesions and model II with re sidual cohesions (30% of the corresponding peak cohesion). From the obtained fragility curves, it can be said that the arch dam is able to maintain good stability under MCE. Compared slippage-based fragility curves and sliding area ratio-based curves, though there is a difference in the probability of occurrence of ultimate sliding instability under MCE,
Declaration of competing interest The authors declared that they have no conflicts of interest to this work. We declare that we do not have any commercial or associative interest that represents a conflict of interest in connection with the paper (entitled with ‘Seismic Fragility Analysis of a High Arch DamFoundation System based on Seismic Instability Failure Mode’, Paper No.: SOILDYN_2019_790_R1) submitted to Soil Dynamics and Earth quake Engineering. Acknowledgements This study is supported by the National Key Research and Develop ment Program [grant number 2017YFC0404903], National Natural Science Foundation of China (grant number 51709283) and State Key Laboratory of Water Cycle Simulation and Regulation and Natural Sci ence for Youth Foundation [grant number 2016CG03]. The authors are grateful for these supports. Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.soildyn.2019.105981. References [1] Hoffman FO, Hammonds JS. Propagation of uncertainty in risk assessments: the need to distinguish between uncertainty due to lack of knowledge and uncertainty due to variability. Risk Anal 1994;14:707–12. [2] Ferson S, Ginzburg LR. Different methods are needed to propagate ignorance and variability. Reliab Eng Syst Saf 1996;54:133–44. [3] Kiureghian AD, Ditlevsen O. Aleatory or epistemic? does it matter? Struct Saf 2009; 31:105–12. [4] Hariri-Ardebili MA, Saouma VE. Seismic fragility analysis of concrete dams: a stateof-the-art review. Eng Struct 2016;128:374–99. [5] Alembagheri M, Seyedkazemi M. Seismic performance sensitivity and uncertainty analysis of gravity dams. Earthq Eng Struct Dyn 2015;44(1):41–58. [6] Tekie PB, Ellingwood BR. Seismic fragility assessment of concrete gravity dams. Earthq Eng Struct Dyn 2003;32(14):2221–40.
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