Currently, in probabilistic fragility analysis, the randomness of ground motion and material parameters is typically addressed by randomly combining several ground motions selected from seismic databases and sampled material parameters (Xu et al., Journal of Hydroelectric Engineering 37:31–38, 2018). However, this approach lacks coupling effects, suffers from insufficient sample sizes or involves extensive computational efforts, thereby limiting the acquisition of comprehensive probability information. There are few studies on the coupling randomness of ground motion and material parameters on the dynamic response of structures, and even less on the seismic response of high CFRDs.
5.1 Introduction
Currently, in probabilistic fragility analysis, the randomness of ground motion and material parameters is typically addressed by randomly combining several ground motions selected from seismic databases and sampled material parameters (Xu et al. 2018). However, this approach lacks coupling effects, suffers from insufficient sample sizes or involves extensive computational efforts, thereby limiting the acquisition of comprehensive probability information. There are few studies on the coupling randomness of ground motion and material parameters on the dynamic response of structures, and even less on the seismic response of high CFRDs.
This chapter considers the coupled randomness of ground motion and material parameters by simultaneously generating stochastic ground motion acceleration time histories and random material parameters. Combined with methods such as the generalized probability density evolution method (GPDEM), reliability probability analysis, and fragility analysis, it reveals the impact of coupled randomness on the seismic dynamic response of high CFRDs from the perspectives of stochastic dynamics and probability. Subsequently, it enhances the performance-based seismic safety evaluation framework.
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5.2 Basic Information
Considering the coupled randomness of ground motion and material parameters, there are 10 random parameters, including 2 uniformly distributed random variables Θ1 and Θ2 in the ground motion spectrum expression in Sect. 2.5 (interval [0, 2π] uniformly distributed and independent of each other) and 8 random variables of material parameters in Sect. 4.3.1 (normally distributed and independent of each other, coefficient of variation 0.1). Based on GF-discrepancy method, 144 groups of stochastic ground motion and random material parameter samples are simultaneously generated. The vertical seismic acceleration is taken as 2/3 of the horizontal seismic acceleration, and the peak acceleration of bedrock along the river is adjusted to 0.1 g-1.0 g, respectively, with an interval of 0.1 g. A series of finite element dynamic calculations were carried out with the input of ground motion. A total of 1440 working conditions were calculated for 10 peak accelerations of ground motion to obtain the stochastic dynamic information under the action of different seismic intensities. Then the probability density evolution equation was solved based on the above numerical method to obtain the probability information of seismic response of high CFRDs at each time. In the following, the effects of the coupling randomness of ground motion and material parameters are analyzed in detail from the aspects of dam acceleration, deformation and random dynamic response of the slope stress of the panel, as well as the probability of these physical indicators, and the corresponding performance-based seismic safety evaluation framework is established. The finite element model, loading conditions, and other material parameter information in this chapter are completely consistent with Sect. 3.4.1.
5.3 Stochastic Dynamic and Probabilistic Analysis of High CFRD
5.3.1 Dam Acceleration
Figure 5.1 shows the maximum horizontal acceleration response based on a single sample and the mean distribution of 144 groups of sample responses at PGA = 0.5 g, respectively. It can be seen that the distribution law is similar to that considering only the randomness of ground motion, but it is different from that considering only the randomness of parameters, indicating that the randomness of ground motion plays a more important role in acceleration response.
Two contour plots labeled a and b, a triangular shape with varying shade gradients. A color gradient scale with values ranging from 3.0 to 6.0 is presented.
Fig. 5.1
Maximum horizontal acceleration response. a A single sample. b Mean value
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In order to further study the influence of the coupled randomness of ground motion and material parameters on the acceleration response of the dam, Fig. 5.2 illustrates the distribution curve and average value of the maximum horizontal acceleration response with the dam height under the seismic intensity of 0.5 g. It can be seen that under the action of coupled randomness, the distribution of acceleration is similar to that of ground motion, which is more obvious in the mean value. From the mean value, the distribution of acceleration response along dam height caused by coupled randomness basically coincides with the response caused by ground motion randomness. The means and standard deviation time history in Fig. 5.3 again proves the above conclusion, but it is quite different from the response considering only the random material parameters.
2 graphs. A and B plot the height of dam in meters versus acceleration. Graph A features multiple light shade curves with a dark shade line that plots mean deformation. Graph B plots increasing lines for stochastic ground motion, random material parameters, and coupled randomness.
Fig. 5.2
Distribution of maximum horizontal acceleration along dam height. a Response to coupled randomness. b Comparison of responses under different randomization factors
Two line graphs. Graph A plots mean verus time. Lines for stochastic ground motion, random material parameters, and coupled randomness are plotted. Graph B plots the standard deviation versus time, with similar comparisons across the three conditions.
Fig. 5.3
Mean and standard deviation time-history of horizontal acceleration of dam crest based on different randomness. a Mean. b Standard deviation
3 graphs. Graph A plots maximum acceleration versus sample number. Data points for stochastic ground motion mean 7.23, random material parameters mean 8.23, and coupled randomness mean 7.41 are plotted. Graph B plots exceedance probability versus maximum acceleration.
Fig. 5.4
Discrete point distribution, exceedance probability of maximum acceleration. a Scatter distribution of maximum acceleration. b Exceedance probability of maximum acceleration. c Distribution of maximum acceleration along the dam height under typical exceedance probabilities
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5.3.2 Dam Deformation
Figure 5.5 depicts the residual horizontal deformation responses based on single-group samples and the averaged responses from 144 sample groups under PGA = 0.5 g. Figure 5.6 illustrates the vertical residual deformation. It is evident that regardless of the response from single-group samples or the averaged response from multiple sample groups, both horizontal and vertical residual deformations occur at the crest of the dam. Furthermore, these deformations closely align with the distribution pattern of responses based on seismic randomness.
Two contour plots labeled a and b, a triangular shape with varying shade gradients. A color gradient scale with values ranging from 0.0 to 0.8.
Fig. 5.5
Horizontal residual deformation. a Horizontal residual deformation of a single group of samples. b Mean horizontal residual deformation
Two contour plots labeled a and b, a triangular shape with varying shade gradients. A color gradient scale with values ranging from 0.0 to 1.0.
Fig. 5.6
Vertical residual deformation. a Vertical residual deformation of one group of samples. b Mean vertical residual deformation
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Figure 5.7 displays the distribution pattern of horizontal displacements along the dam height at a seismic intensity of 0.5 g at the final moment. It can be observed that the distribution pattern based on the coupling of seismic and material parameters randomness is similar to that considering seismic randomness. This similarity is particularly evident in the mean distribution along the dam height, as the two lines nearly coincide. However, there is a notable difference when compared to the distribution pattern considering only material parameter randomness. Figure 5.8 illustrates the distribution pattern of vertical displacements along the dam height at a seismic intensity of 0.5 g at the final moment. Similarly, it can be concluded that seismic randomness seems to exert a controlling influence on the vertical deformation, but there is a slight difference in the numerical values at the crest. The vertical deformation based on coupled randomness is 1.16 m, while the vertical deformation based on seismic randomness is 1.20 m, indicating a difference of approximately 3.4%.
2 graphs. A and B plot the height of dam in meters versus horizontal residual deformation. Graph A features multiple light shade curves with a dark shade line that plots mean deformation. Graph B plots increasing lines for stochastic ground motion, random material parameters, and coupled randomness.
Fig. 5.7
Distribution of horizontal residual deformation along the dam height. a Coupled randomness. b Comparison of responses under different random factors
2 graphs. A and B plot the height of dam in meters versus vertical residual deformation. Graph A features multiple light shade curves with a dark shade line that plots mean deformation. Graph B plots increasing lines for stochastic ground motion, random material parameters, and coupled randomness.
Fig. 5.8
Distribution of vertical residual deformation along the dam height. a Coupled randomness. b Three random factors
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From the horizontal displacement in Fig. 5.9 and the vertical displacement in Fig. 5.10, both mean and standard deviation time histories are presented (PGA = 0.5 g). It can be observed that during the seismic process, the horizontal displacement exhibits almost no difference compared to the seismic response induced by seismic randomness. The temporal pattern of vertical displacement is largely consistent, differing mainly in numerical values. However, both of these responses significantly deviate from the randomness attributed to material parameter randomness. Upon observing the standard deviation time history, it becomes apparent that the standard deviation time history of seismic responses due to the coupling of seismic input and material parameter randomness is notably similar to that induced by seismic input randomness. Nevertheless, there exists a certain discrepancy in terms of numerical values.
2 graphs. Graph A plots mean deformation versus time, comparing stochastic ground motion, random material parameters, and coupled randomness. Graph B plots the standard deviation of deformation versus time, with similar comparisons.
Fig. 5.9
Mean and standard deviation time history of horizontal displacement. a Mean time history. b Standard deviation time history
2 graphs. Graph A plots mean deformation versus time, comparing stochastic ground motion, random material parameters, and coupled randomness. Graph B plots the standard deviation of deformation over time.
Fig. 5.10
Mean and standard deviation time history of vertical displacement. a Mean time history. b Standard deviation time history
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The scatter plots in Fig. 5.11, illustrating the residual horizontal and vertical deformations at the crest (PGA = 0.5 g), further corroborate the aforementioned observations. However, it is notable that the deformation responses prompted by seismic randomness and coupled randomness exhibit greater variability compared to the deformations induced by material parameter randomness. This suggests that seismic randomness exerts a more significant influence on dam deformation responses, indicating a certain controlling effect.
2 scatter plots. Graph A plots horizontal residual deformation in meters against the sample number. Graph B plots vertical residual deformation against the same sample number. Data points representing stochastic ground motion, random material parameters, and coupled randomness are plotted.
Fig. 5.11
Discrete point distribution of horizontal and vertical deformation. a Horizontal residual deformation. b Vertical residual deformation
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Figure 5.12 presents the exceedance probability curves of residual horizontal and vertical deformations for various seismic intensities (PGA = 0.1–1.0 g, in increments of 0.1 g). It is evident that the exceedance probability curves for residual deformations induced by the coupling of seismic input and material parameters randomness closely align with those resulting from seismic randomness alone. However, as seismic intensity increases, the difference in exceedance probability based on horizontal displacement slightly grows, while the difference based on vertical displacement diminishes slightly.
2 line graphs. A, exceedance probability versus horizontal deformation. It plots 5 decreasing curves for P G A equals 0.1, 0.2, 0.3, 0.4, and 0.5 g. B, exceedance probability versus vertical deformation. It plots 5 decreasing curves for P G A equals 0.1, 0.2, 0.3, 0.4, and 0.5 g.
Fig. 5.12
Exceedance probability of horizontal and vertical residual deformation Faced-slab Stress. a Horizontal residual deformation. b Vertical residual deformation
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From the distribution pattern of maximum downslope stress along the dam height under a seismic intensity of 0.5 g in Fig. 5.13 and the variation of downslope stress mean over time in Fig. 5.14, it can be observed that there is minimal difference between the stress responses induced by the coupling randomness and those caused by seismic randomness. This indicates that seismic input plays a primary role in faced-slab stress response. Figure 5.15, which illustrates the exceedance probability of cumulative overstress duration (COD) for various Demand-Capacity Ratios (DCRs) and the resultant performance states, further validates this point. Hence, the analysis of faced-slab failure probability can focus solely on the impact of seismic randomness.
A line graph plots the height of dam versus the mean maximum stress. Lines for stochastic round motion, random material parameter, and coupled randomness are plotted.
Fig. 5.13
The maximum downslope stress distribution along the dam height
A line graph plots mean stress versus time. Two increasing lines for stochastic proud motion and coupled randomness and a fluctuating line for random material parameter are plotted.
Fig. 5.14
The mean downslope stress time history
2 line graphs. A plots exceedance probability versus C O D. Decreasing lines for D C R equals 1, 1.2, 1.5, 1.75, and 2 are plotted. B plots C O D versus D C R. Decreasing lines for 77.8, 80, 82.8, 84.9, 85.1, 87.1, 89.2, 90.5, 93.2, 93.3, 96.5, 94.5, and 9.8 percent are plotted.
Fig. 5.15
Exceedance probability of faced-slab damage index
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5.4 Conclusion
This chapter investigates in detail the effects of coupled randomness of ground motion and material parameters on the dynamic response and seismic safety of high CFRDs from the perspectives of stochastic dynamics and probability. Firstly, stochastic ground motion time histories and random material parameter samples are generated simultaneously by combining spectral representation-random function and material parameter random variables. Then, the stochastic dynamic response and probability characteristics of high CFRDs under the influence of coupled randomness of ground motion and material parameters are analyzed. By comparing the stochastic dynamic and probability results of dam body acceleration, deformation, and panel stress caused by seismic randomness, material parameter uncertainty, and coupled randomness, it is demonstrated that the seismic randomness has a predominant control effect on seismic response, and the influence of material parameter randomness can be neglected to some extent when conducting comprehensive random dynamic response analysis of dams. Finally, based on the above performance indicators and performance levels, a seismic safety evaluation framework is established for different seismic intensity levels, multiple seismic intensity-multiple performance objective-surpassing probability scenarios. Vulnerability curves for different damage levels are obtained. The results show that there is little difference in performance safety evaluation probability obtained from seismic randomness and coupled randomness, within 10%. Therefore, when performing performance-based seismic safety evaluation of high CFRDs, considering seismic randomness alone is sufficient to meet the requirements.
The above chapter mainly conducts seismic safety analysis of two-dimensional high CFRDs from the perspectives of stochastic dynamics and probability. However, three-dimensional effects have a certain influence on the seismic response of high CFRDs, especially for relatively thin panels. Two-dimensional analysis often struggles to accurately describe and define their failure states. Nevertheless, the above research can qualitatively and quantitatively study and compare panel failure states to some extent from the perspectives of stochastic dynamics and probability. Additionally, three-dimensional analysis is commonly used in engineering applications to study the deformation of dam bodies, especially the failure patterns of panels. Therefore, the following research will explore the stochastic dynamic response and probability variation laws of high CFRDs based on three-dimensional numerical calculations, revealing the relationship with the aforementioned two-dimensional stochastic dynamic analysis, and thereby establishing corresponding methods and frameworks for performance safety evaluation.
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