It is widely recognized that high CFRDs, especially those with thinner slabs, exhibit significant three-dimensional effects. Three-dimensional finite element analysis offers a more realistic depiction of the stress distribution of the dam. However, there is a lack of studies that utilize stochastic dynamics and probabilistic analyses to investigate the seismic safety of three-dimensional CFRDs and establish corresponding safety evaluation criteria.
6.1 Introduction
It is widely recognized that high CFRDs, especially those with thinner slabs, exhibit significant three-dimensional effects. Three-dimensional finite element analysis offers a more realistic depiction of the stress distribution of the dam. However, there is a lack of studies that utilize stochastic dynamics and probabilistic analyses to investigate the seismic safety of three-dimensional CFRDs and establish corresponding safety evaluation criteria. Building upon the analyses conducted in the preceding chapters, it is evident that the randomness of ground motions is the primary factor influencing the stochastic dynamic response of CFRDs. In this chapter, drawing from the performance-based seismic safety evaluation studies of two-dimensional high CFRDs presented in Chaps. 3 and 5, we delve into the distribution and variations of acceleration, deformation, and slab stress of three-dimensional high CFRDs, considering the stochastic dynamics and probability perspectives. Additionally, we elucidate the interconnection with two-dimensional analyses. Ultimately, we identify dam crest subsidence and the safety of face-slabs impermeable body as key indicators for defining the limit states of CFRDs. Moreover, we enhance the performance-based framework for seismic safety evaluation to assess the reliability of CFRDs.
6.2 Basic Information of High CFRDs
The 250 m three-dimensional CFRD finite element mesh model is shown in Fig. 6.1. The upstream and downstream slopes have a ratio of 1:1.4 and 1:1.6. The width of dam crest is 18 m, the thickness of concrete face-slabs is 1.175 m. The horizontal width of the cushion and transition areas setting under the concrete slab is 4 and 6 m, respectively. The slab was poured in three phases, 75 m, 150 m and 250 m respectively. The dam was filled in 50 layers, and the water storage was divided into 48 steps up to 240 m. The element simulating the dam comprises hexahedral iso-parametric elements and a minor proportion of degenerate tetrahedral elements. The Goodman contact surface unit without thickness is set up between face-slabs and cushion Goodman (1968). Moreover, vertical and peripheral joints are simulated using 8-node spatial jointing element. Seismic input using fluctuation input method based on viscoelastic artificial boundary setting. To simplify the calculation, bedrock is not considered in this study. The additional mass method in Chap. 3 is used to analyze the hydrodynamic pressure on slabs. Both static and dynamic processes are simulated using a uniform generalized plastic model for rockfill, transition, and cushion materials. The contact surface between face-slabs and cushion can be simulated using the generalized plastic contact surface model. Material parameters using values from Sect. 3.4.1. The face-slabs simulated by a linear elastic model for C30 concrete. The following parameters were used: E = 3.1 × 104 MPa, ρ = 2.40 g/cm3, ν = 0.167, fc = 27.6 MPa.
A 3-dimensional triangular mesh with a width of 250 meters, a height of 18 meters, and a thickness of 6 meters. The upstream and downstream slopes are 1 is to 1.4 and 1 is to 1.6, respectively. The surface is of 4 layers of face slabs, cushion, transition, and rockfill material.
Fig. 6.1
Finite element mesh model of the high CFRD
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6.3 Stochastic Dynamic Response and Probability Analysis of High CFRDs
Non-stationary random ground motions were simulated using the intensity-frequency fully non-stationary stochastic ground motion generation methods. The ground motions were generated at PGA intervals of 0.1 g, ranging from 0.1 g to 1.0 g. For each PGA, 144 ground motions were created, resulting in a total of 1440. In addition, the vertical acceleration is two thirds that of the horizontal acceleration. The stochastic dynamic response and the associated probability information under different intensity ground motions is obtained by performing finite element calculations and probabilistic density evolution analysis on the model.
6.3.1 Dam Acceleration
Figure 6.2 shows the distribution of the maximum acceleration of the dam based on a single sample of ground motion at PGA of 0.5 g. Figure 6.3 shows the distribution of the mean values of the 144 acceleration responses. Regardless of the individual response or the mean value, the amplification of acceleration response is most obvious in the dam crest region. In addition, the downstream dam slope is also the concentration area of larger acceleration response. This may be due to the confinement of the upstream slab and the reflection of the fluctuation of the downstream dam slope, essentially the same as the two-dimensional analysis. From the mean value analysis, the dam acceleration amplification is circularly or elliptically distributed in the top region of the dam and roughly symmetrically distributed along the dam axis with respect to the center line of the river valley. The amplification effect is most obvious at the dam crest location at the river valley. It also indicates that the stochastic ground excitation has a great influence on the acceleration response of the dam body.
3 triangular contour plots. A, high acceleration in the horizontal direction with a color scale ranging from 5 to 12. B, high acceleration in the vertical direction with a color scale ranging from 2.5 to 10. C, high acceleration in the dam axial direction with a color scale ranging from 5 to 12.5.
Fig. 6.2
Dam acceleration distribution based on the single sample. a Horizontal direction b Vertical direction c Dam axial direction
3 triangular contour plots. A, high acceleration in the horizontal direction with a color scale ranging from 5 to 12. B, high acceleration in the vertical direction with a color scale ranging from 2.5 to 10. C, high acceleration in the dam axial direction with a color scale ranging from 5 to 12.5.
Fig. 6.3
Dam acceleration mean distribution based on the 144 samples. a Horizontal direction b Vertical direction c Dam axial direction
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Figure 6.4 illustrates distribution of the maximum acceleration and amplification in downstream direction along dam height under different seismic intensities, respectively. Different ground motions cause different response curves, but the curves have the same trend. A sudden alteration happens at around 0.8H, the amplification effect at the top of the dam is particularly pronounced, showing a strong “whip-sheath” effect. The acceleration amplification under 0.5 g PGA in Fig. 6.4c also shows a large dispersion. These features are similar to the 2-D acceleration response. Meanwhile, Fig. 6.4d shows that the amplification of the mean acceleration tends to decrease as the intensity of ground motion increases. Therefore, it is necessary to investigate the acceleration response of CFRDs based on stochastic ground motions.
4 line graphs. A and B, height versus acceleration plots increasing lines for mean, numbers 16, 28, and 68 earthquakes. C and D, height versus amplification of acceleration. D plots increasing lines for P G A equals 0.1, 0.2, 0.4, 0.6, and 1 gram.
Fig. 6.4
Distribution of maximum horizontal acceleration along dam height. a PGA = 0.4 g b PGA = 1.0 g. c PGA = 0.5 g d Mean of different PGA
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The horizontal coordinates of Fig. 6.5 are the normalized dam crest axis and the vertical coordinates are the down-river acceleration at the dam crest. Figure 6.5d shows the amplification of the mean acceleration corresponding to earthquakes of different intensities. The acceleration amplification effect is not obvious near banks, but increases suddenly near the middle of the valley, showing obvious three-dimensional effect, and the response laws of different ground motions are quite different. For the same ground motion, there are certain differences in the response patterns at different intensities. In addition, as the intensity of ground motion increases, the amplification of acceleration along the axial direction of the dam tends to decrease.
4 line graphs. A and B, acceleration versus l over L plots lines for mean, numbers 16, 28, and 68 earthquakes. C, acceleration amplification versus l over L. D, amplification of mean acceleration versus l over L plots lines for P G A equals 0.2, 0.6, and 1 gram.
Fig. 6.5
Distribution of maximum horizontal acceleration along dam axial direction. a PGA = 0.2 g b PGA = 1.0 g c PGA = 0.5 g d Mean of different PGA
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Figure 6.6 shows the discrete point of the maximum horizontal acceleration at the top under different seismic intensities, which can provide a reference for the seismic safety design and control criteria for CFRDs. The maximum acceleration distribution is relatively discrete, with 95 and 5% exceedance probabilities. The difference between the maximum and minimum values is large, with reaching 2–2.5 times, which suggests that the acceleration response of the dam body under ground motions generated in this research is statistically significant. The maximum acceleration response under different seismic intensities is basically linearly distributed, similar to the 2-D trend.
A line graph of maximum acceleration versus P G A indicates. Increasing lines for mean and exceedance probability of 5 percent, 50 percent, and 95 percent are plotted.
Fig. 6.6
Discrete point distribution of maximum horizontal acceleration
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6.3.2 Dam Deformation
Figure 6.7 shows the horizontal and vertical residual deformation responses obtained based on a single sample of ground motion under PGA = 0.5 g, respectively, and Fig. 6.8 shows the mean of 144 responses. The response occurs mainly in the center of the river valley at the top of the dam, with an approximately elliptical distribution. The result shows a similar pattern to the acceleration responses, indicating a correlation between them.
2 triangular contour plots. A, the deformation is high at the top and low at the bottom with a color scale ranging from 0 to 1. B, the deformation is high at the center and low at the bottom with a color scale ranging from 0 to 1.2.
Fig. 6.7
Residual deformation based on the single sample. a Horizontal direction b Vertical direction
2 triangular contour plots. A, the deformation is highest in the center and decreases towards the right with a color scale ranging from 0 to 1. B, the deformation is highest in the center and decreases towards the edges with a color scale ranging from 0 to 1.2.
Fig. 6.8
Residual deformation mean based on the 144 samples. a Horizontal direction (b) Vertical direction
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The exceedance probability curves of horizontal and vertical residual deformation are shown in Fig. 6.9, which serves as a useful point of reference for subsequent evaluations of seismic safety based on deformation. Figure 6.10 shows the distribution of discrete points of residual deformation in the horizontal and vertical direction for different seismic intensities, which has a certain degree of dispersion and is similar to the pattern of the two-dimensional. Table 6.1 lists the horizontal and vertical residual deformations corresponding to different exceedance probabilities under different PGA. Based on the 95% and 5% exceedance probabilities, the maximum value in the horizontal direction is about 3–4 times of the minimum value, and the maximum value in the vertical direction is approximately twice that of the minimum., which can provide reference for CFRDs ultimate seismic capacity analysis under different PGA.
2 line graphs plot decreasing lines for P G A equals 0.1, 0.2, 0.30, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1 gram. A, exceedance probability versus horizontal residual deformation. B, exceedance probability versus vertical residual deformation.
Fig. 6.9
Exceedance probability of horizontal and vertical residual deformation. a Horizontal direction. b Vertical direction
2 line graphs. A, horizontal residual deformation versus seismic intensity. B, vertical residual deformation versus seismic intensity. Lines for mean and exceedance probability 96, 5, and 50 percent are plotted.
Fig. 6.10
Horizontal and vertical residual deformation under different PGA. a Horizontal direction. b vertical direction
Table 6.1
The horizontal and vertical residual deformation of dam crest based on different exceedance probability under different PGA
Exceedance probability
PGA (g)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Horizontal residual deformation (m)
Mean
0.096
0.261
0.477
0.721
0.979
1.241
1.508
1.776
2.052
2.326
5%
0.160
0.442
0.809
1.187
1.585
1.974
2.379
2.760
3.157
3.578
50%
0.091
0.249
0.456
0.695
0.948
1.208
1.468
1.728
1.997
2.261
95%
0.044
0.106
0.196
0.335
0.480
0.644
0.786
0.967
1.144
1.291
Vertical residual deformation (m)
Mean
0.334
0.610
0.840
1.031
1.199
1.347
1.477
1.591
1.688
1.776
5%
0.477
0.849
1.152
1.392
1.606
1.820
2.005
2.188
2.341
2.495
50%
0.327
0.600
0.829
1.015
1.179
1.323
1.447
1.556
1.647
1.731
95%
0209
0.395
0.558
0.715
0.853
0.952
1.059
1.126
1.201
1.218
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6.3.3 Overstress Volume Ratio and Overstress Cumulative Time of Faced-Slab
(1) Dynamic stresses of faced-slab
Figure 6.11 shows the distributions of stress at 6 s moment, post-seismic stress and maximum stress (positive for tensile stresses and negative for compressive stresses) obtained based on a single sample for PGA = 0.4 g. Figure 6.12 shows the corresponding distribution of mean values. In terms of individual samples, the bottom of the faced-slab is mainly subjected to compressive stresses, and tensile stresses are mainly concentrated in the upper part of the faced-slab. However, from the mean value, the whole faced-slab is mainly subjected to compressive stress at the bottom, indicating that stresses are varying over time. Therefore, it is necessary to explore the damage standard based on the time accumulation effect.
3 contour plots with the color scales. A has the stress at the bottom with a color scale ranging from negative 18 to 6. B has the stress at the center with a color scale ranging form negative 16 to 1.5. C has the stress at the bottom with a color scale ranging from negative 7.5 to 15.
Fig. 6.11
Stress distribution in the slope direction of single sample at different time. a Stress at 6 s moment. b post-seismic stress. c maximum stress
3 contour plots with the color scales. A has the stress at the bottom with a color scale ranging from negative 18 to 0. B has the stress at the center with a color scale ranging from negative 16 to 0. C has a stress at the bottom with a color scale ranging from negative 7.5 to 15.
Fig. 6.12
Mean stress distribution in the slope direction at different time. a Stress at 6 s moment b Post-seismic stress. c Maximum stress
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Figure 6.13 shows the stress distribution of slab in the slope direction along dam height under PGA = 0.4 g (positive for tensile stresses and negative for compressive stresses). The distribution pattern of stress response along dam height caused by different ground motions is different, in which the tensile stress is mainly concentrated at the top of the dam. However, from the mean value, the whole faced-slab is mainly subjected to compressive stress. Figure 6.13b shows the distribution of maximum stress along the dam height, which is basically similar to 2-D analysis. The distribution law of the response caused by different ground motions is basically same, but there is a big difference in the values, reflecting the necessity of analyzing based on the stochasticity of ground motions.
3 line graphs. A, height versus stress in slope direction plots lines for mean, numbers 16, 28, and 68 earthquakes. B, height versus maximum stress plots lines for mean of 16, 28, and 68 earthquakes. C, a graph of height versus mean stress per megapascal plots lines for 3, 6, 9, and 15 seconds.
Fig. 6.13
Stress distribution in the slope direction along dam height. a Stress at 6 s moment b Maximum stress c Mean stress at typical time
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The four characteristic elements shown in Fig. 6.14 are selected to explore the stochastic dynamic response of the faced-slab during the seismic process in detail. Figure 6.15 shows the distribution pattern of the faced-slab in the slope direction along dam axial direction at different moments under PGA of 0.4 g (height of point C). The distribution pattern of downslope stresses under different ground motions varies greatly. Overall, the faced-slab stress tends to zero value at all moments, and the maximum stress is distributed at the center of the valley. Figure 6.16 shows the stress variation time course of the four feature elements, the tensile stresses are mainly distributed in the upper region of the faced-slab and the lower part of the faced-slab is mainly subjected to compressive stresses. From the mean value, the stress time course tends to be constant. From the standard deviation, the patterns at the top and bottom of the dam are closer, demonstrating a certain symmetry in the values. In summary, it is necessary to study the stress response law of faced-slab from the perspective of ground motion stochasticity.
A graph of Y versus Z has an inverted triangular grid between (0, 250) and (600, 250), (250. 250) and (300, 600). Four points A, B, C, and D are plotted at (285, 20), (285, 140), (285, 190), and (285, 230). Values are approximated.
Fig. 6.14
Sketch of feature points locations
4 line graphs. A and B plot stress in slope direction versus l over L. Lines for mean, numbers 16, 28, and 68 earthquakes are plotted. C, maximum stress versus l over L. D, mean stress versus l over L plots lines for maximum stress, 3 seconds, 6 seconds, 9seconds, and 15 seconds.
Fig. 6.15
Stress distribution in the slope direction along dam axial direction. a Stress at 6 s moment b Post-seismic stress c Maximum stress d Mean stress of typical time
3 line graphs plots stress in megapascal versus time in seconds. Lines for A, B, C, and D are plotted. A, 4 fluctuating curves are observed. B, 4 irregular horizontal lines are observed. C, 4 Increasing fluctuating curves are observed.
Fig. 6.16
Stress time history in the slope direction. a Typical sample b Mean c Standard deviation
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(2) Overstress volume ratio
To represent the damage of the concrete slab more intuitively, this paper defines the ratio of the volume of tensile stress exceeding the tensile strength to the whole slab at different seismic strengths as the overstressed volume ratio (VolR). Figure 6.17 shows the VolR time history under different PGA, for example. Figure 6.18 shows the mean and the standard deviation time history. The VolR varies significantly with time evolution and seismic intensity, with an increasing trend for increasing seismic intensity. Therefore, considering the effect of time accumulation, the damage standard of faced-slab based on the VolR and overstress accumulation time can be initially explored.
10 graphs of overstressed volume ratio in percentage versus time in seconds plots lines for mean, numbers 16, 28, and 68 earthquakes. A has sparse fluctuations. B to J have dense fluctuations with mean, numbers 16, 28, and 68 earthquakes.
Fig. 6.17
Time history of overstress volume ratio under different PGA. a PGA = 0.1 g b PGA = 0.2 g c PGA = 0.3 g d PGA = 0.4 g e PGA = 0.5 g f PGA = 0.6 g g PGA = 0.7 g h PGA = 0.8 g i PGA = 0.9 g j PGA = 1.0 g
2 line graphs plot lines for P G A equals 0.1, 0.2, 0.30, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1 gram. A plots mean of Vol R versus time. B plots the standard of Vol R versus time.
Fig. 6.18
Mean and standard deviation time history of overstress volume ratio under different PGA. a Mean b Standard deviation
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Figure 6.19 illustrated the probability density evolution information of the overstressed volume ratio, including PDF curves of typical times, the PDF surfaces and contours of the overstressed volume ratio over the time from 8 to 12 s. The great variability and multi-peak shape shown in Fig. 6.19a greatly affects the dynamic reliability of faced-slab. Figure 6.19b, c show the irregular surface of PDF and the transfer process of probability information, proving the overstress volume ratio has great variability with the evolution of time.
3 graphs. A, a line graph of P D F versus volume R has 3 narrow bell curves of t equals 4, 8, and 12 seconds. B, a 3-dimensional graph of P D F versus volume R versus t over s. C, a line graph of volume R versus t over s has irregular shaded fluctuations with a color scale ranging from 0 to 0.3.
Fig. 6.19
Probability evolution information of overstress volume ratio. a PDF curves of typical times b PDF surfaces c Contour plot of PDF surface
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Figure 6.20 shows the discrete point distribution and exceedance probability curves of different overstress volume ratios corresponding to cumulative time under PGA = 0.5 g, which can define the corresponding damage standard of faced-slab.
A, a scatterplot of cumulative time versus number. Plots for volume R equals 5 and 10 percent and lines for mean equals 2.29 and 1.25 are plotted. B, a graph of exceedance probability versus cumulative time plots decreasing lines for vol R equals 5, 10, 15, 20, 25, 30, 35, 40, 45, and 50 percent.
Fig. 6.20
Cumulative time and exceedance probability of different overstress volume ratios. a Discrete point distribution. b Exceedance probability curves
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6.4 Conclusion
In this chapter, the stochastic dynamic response and probabilistic evolution of acceleration, deformation and faced-slab stress of a three-dimensional high CFRD are analyzed with full consideration of the randomness of ground motions, and the range of distributional variations of the physical quantities is suggested from the stochastic dynamic and probabilistic perspectives by combining the stochastic ground motion generation method, the generalized probability density evolution method, the fragility analysis method, and the elasto-plasticity analysis method of CFRDs. Finally, the performance safety evaluation standards based on the relative seismic subsidence rate of the dam crest are compared, and the faced-slab seismic safety evaluation indicators based on the overstress volume ratio combined with the overstress accumulation time is preliminarily explored. The main work and conclusions are as follows:
(1) From the perspective of stochastic dynamics, it is revealed that the acceleration and deformation of the dam body are mainly distributed in the central part of the valley at the top of the dam, and the top area of the dam shows strong “whip sheath” effect and three-dimensional effect. From the perspective of stochastic dynamics and probability, the acceleration, deformation and stress response have large variability, both in distribution and value. It is necessary to analyze the non-linear dynamic response of high CFRDs under earthquake action from the stochastic dynamics point of view. The result has certain reference significance for the seismic safety evaluation and ultimate seismic capacity analysis of high CFRDs.
(2) From the perspective of probability and performance evaluation, the damage probability relationship between 2-D deformation and 3-D deformation is revealed; the seismic safety evaluation framework based on the overstressed volume ratio of faced-slabs combined with the cumulative time is preliminarily explored and suggested, as described in Sect. 8.1, which basically indicates the reasonableness of the division of the performance level through the probability, and basically corresponds to the deformation-based performance safety evaluation standard, and further improves the performance-based seismic safety evaluation framework for CFRDs.
It should be noted that for panels, it is most reasonable to establish the corresponding seismic performance safety evaluation standards based on elastic–plastic cumulative damage, but the current numerical calculations based on the 3D concrete elastic–plastic damage model are particularly large in scale and are not well suited for stochastic dynamic and probabilistic analyses. Nevertheless, the performance indicators established in this chapter can basically quantify the damage degree of faced-slabs, which provides a reference for performance-based seismic safety evaluation of CFRDs.
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