Numerous engineering cases (Guan, Sci China Ser e: Technol Sci 52:820–834, 2009; Liu et al., J Perform Constr Facil 29:04,014,129, 2015), dynamic numerical analysis (Zou et al., Comput Geotech 49:111–122, 2013; Uddin, Comput Struct 72:409–421, 1999) and dynamic physical model tests (Zhu et al., Journal of Harbin Institute of Technology 18:132–138, 2011; Liu et al., Soil Dyn Earthq Eng 82:11–23, 2016) demonstrated that the instability of dam slopes is a major engineering concern for high CFRDs under earthquake excitation. It is explicitly mandated that seismic stability calculations for earth-rock dams should be included in seismic analysis in China's Hydraulic Seismic Design Code (NB 35047–2015). The comprehensive evaluation of slope stability is further specified to consider the factors such as the position, depth, and extent of the slip surface, as well as the duration and magnitude of stability index exceeding limits.
7.1 Introduction
Numerous engineering cases (Guan 2009; Liu et al. 2015), dynamic numerical analysis (Zou et al. 2013; Uddin 1999) and dynamic physical model tests (Zhu et al. 2011; Liu et al. 2016) demonstrated that the instability of dam slopes is a major engineering concern for high CFRDs under earthquake excitation. It is explicitly mandated that seismic stability calculations for earth-rock dams should be included in seismic analysis in China's Hydraulic Seismic Design Code (NB 35047–2015). The comprehensive evaluation of slope stability is further specified to consider the factors such as the position, depth, and extent of the slip surface, as well as the duration and magnitude of stability index exceeding limits. Therefore, in addition to the traditional safety factor, the cumulative time of \({\text{F}}_{\text{S}}{<1.0}\) and cumulative slippage were adopted by many scholars to evaluate the stability of dam slopes (Zhang and Li 2014). This approach, along with associated performance indices, was commonly utilized in numerous engineering practices. The seismic response of dam slope is characterized by a multitude of uncertainties, primarily including the stochastic seismic excitation and the material parameter randomness. However, deterministic dynamic time history analysis based on one or several seismic waves is more commonly employed in the current methods of dam slopes seismic performance assessment. The current approach fails to quantitatively analyze the impact of uncertainty factors on the seismic stability of dam slopes and inadequately evaluates the seismic safety of dam slopes from a performance perspective. Therefore, studying dam slope seismic stability based on stochastic dynamic and probabilistic analysis can effectively supplement deterministic analysis methods by fully considering uncertain factors in seismic response processes and introducing reasonable performance indices for dam slope stability evaluation. This approach is of significant importance and can progressively enhance performance-based seismic stability assessments of high CFRD slopes.
In addition, some studies have shown that instability of dam slopes tends to be shallow layer sliding at the top of the dam body, with slip surfaces often occurring under low confining pressures. The softening effects of rockfill materials are gradually manifested under low confining pressures subjected to earthquakes (Zou et al. 2016), especially strong ones and this will exacerbate damage to the dam slopes (Skempton 1985). Hence, it is of great significance to analyze the seismic performance of the high CFRD slopes considering the softening effects and some researchers have conducted a lot of studies. For instance, Chen et al. (1992) conducted finite element analysis by using strain softening model to determine the degree of weakening along potential slip surfaces. Potts et al. (1990) conducted numerical simulations using a combination of finite element method and strain softening model to investigate the delayed failure and progressive cumulative failure of an excavated stiff clay slope. Zhang et al. (2007) proposed a methodology for analyzing the progressive failure of strain softening slopes based on the strength reduction method and strain softening model. Liu and Ling (2012) investigated the effects of strain softening of backfill on the deformation and reinforcement load of geosynthetic-reinforced soil structures (GRS) walls. Wang et al. (2017) considered the strain-softening characteristics of soils and simulated the mechanical behavior of slope failure using an improved finite element method. These studies indicate that if the strain softening characteristics of the slope are neglected, the slope safety might be overestimated, particularly under earthquakes. However, few studies have been conducted to consider the impact of rockfill softening characteristics on the seismic safety of dam slopes. Zhou et al. (2016) performed a deterministic comparison of the influence on the safety of dam slopes by considering and without considering softening characteristics of rockfill materials, and preliminary findings suggested that the potential for dam slope instability increased when considering the softening effects. However, deterministic analyses fall short in providing a comprehensive reflection of the stochastic ground motions and the effects of different seismic intensities on the stability of dam slopes. Hence, conducting thorough research on the impact of rockfill material softening effects on the seismic safety of dam slopes from a stochastic and probabilistic perspective is imperative. Additionally, the conventional dam slope stability evaluation methods relying on the pseudo-static approach have limitations in accurately portraying the input characteristics of ground motions and their dynamic responses, which significantly influence dam slope stability. Therefore, employing finite element dynamic time-history analysis methods is necessary for a comprehensive assessment of dam slope seismic stability. Furthermore, selecting appropriate performance indices to establish corresponding performance-based seismic safety evaluation criteria is crucial.
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In this chapter, the randomness of material parameters, stochastic seismic excitation, and their coupling are comprehensively taken into account. The dynamic finite element time-history method for dam slope stability analysis was applied, incorporating the coupled calculation method for softening strength parameters variation of the rockfill materials. A methodology for generating stochastic ground motions and random parameters was established and the fragility analysis as well as the GPDEM were introduced. The impact of rockfill material softening effects on the seismic stability of dam slopes is evaluated from a stochastic and probabilistic perspective based on three performance indices: safety factor, cumulative time of \({\text{F}}_{\text{S}}{<1.0}\) and cumulative slippage, and the stochastic dynamic response pattern of dam slopes stability is revealed. Furthermore, a probabilistic analysis method for dam slope stability considering the softening effects is established, along with a performance-based seismic safety evaluation framework of high CFRD slopes.
7.2 Dynamic Finite Element Time History Analysis Method Considering Softening Behaviors of Rockfill for Dam Slope Stability
7.2.1 Dynamic Finite Element Time History Analysis Method Considering Softening Behaviors of Rockfill for Dam Slope Stability
Although the pseudo-static method is simple and has rich practical experience, it fails to adequately consider the stress–strain relationship within the soil mass. The calculation results only represent the average values of assumed potential sliding surfaces and cannot provide information on the magnitude of soil deformation. Therefore, methods based on seismic response analysis have gradually gained attention and development. A method that calculates the factor of safety against sliding for each moment by considering the instantaneous stress changes during seismic events is referred to as the finite element time-history method.
in which, \({\text{c}}_{\text{i}}\) and \({\varphi}_{\text{i}}\) are the cohesion and internal friction angle of element i, respectively, and \({\text{l}}_{\text{i}}\) is the length of the element i in the slip circle. \({\sigma }_{\text{i}}\) and \({\tau }_{\text{i}}\) are the normal stress and tangential stress of element i, respectively, and as expressed follows:
where, \({\sigma}_{\text{x}}{=(}{\sigma}_{\text{x}}^{\text{s}}{+}{\sigma}_{\text{x}}^{\text{d}}\text{)}\), \({\sigma}_{\text{y}}{=(}{\sigma}_{\text{y}}^{\text{s}}{+}{\sigma}_{\text{y}}^{\text{d}}{)}\); \({\tau}_{\text{xy}}{=(}{\tau}_{\text{xy}}^{\text{s}}\text{+}{\tau}_{\text{xy}}^{\text{d}}\text{)}\); \({\sigma}_{\text{x}}^{\text{s}}\) is the horizontal static stress; \({\sigma}_{\text{x}}^{\text{d}}\) is the horizontal dynamic stress; \({\sigma}_{\text{y}}^{\text{s}}\) is the vertical static stress; \({\sigma}_{\text{y}}^{\text{d}}\) is the vertical dynamic stress; \({\tau }_{\text{xy}}^{\text{s}}\) is the static shear stress; and \({\tau }_{\text{xy}}^{\text{d}}\) is the dynamic shear stress; α denotes the inclination of the base of the slice with respect to the horizontal.
For any slip surface, the sliding angular acceleration of the slider can be expressed as:
$$\user2{\ddot{\theta }}\left( t \right) = \frac{{\varvec{M}}}{{\varvec{I}}}$$
The schematic diagram of the safety factor calculation is show in Fig. 7.1.
A diagram of an inclined plane with point O at the top, a line labeled R extending from O with a marked angle labeled theta. A square on the left depicts stress components sigma x, sigma y, and tau x y.
Fig. 7.1
Schematic diagram of the safety factor calculation
×
7.2.2 Rockfill Softening
In the currently commonly used block slip method, the decrease in shear strength of the slip surface is not considered when \({\text{F}}_{\text{S}}\text{<1.0}\). However, the peak strength of the rockfill materials gradually decreases along with the sliding. In this section, the software FEMSTABLE 2.0 developed by Dalian University of Technology was adopted to perform stochastic dynamic response analysis of the dam slope and the softening characteristic of the rockfill materials was considered. The effect of stochastic ground motions on the minimum safety factor, cumulative time of \({\text{F}}_{\text{S}}\text{<1.0}\) and cumulative slippage was thoroughly investigated. The specific steps are described as below:
1.
Five sets of ground motions were generated using the stochastic ground motion model. The PGA was adjusted from 0.1 g to 0.5 g in intervals of 0. 1 g and each set had 233 stochastic ground motions.
2.
For each ground motion, the static and dynamic calculations of the dam are conducted, and the finite element time-history analysis of the dam slope is performed based on the accumulated static and dynamic stress results, then the safety factor of the dam slope is obtained.
3.
The cumulative slippage of the slip surface was calculated using the block slip method if the safety factor was less than 1.0. The shear strain on the slip surface can be expressed as:
$$\gamma_{\text{s}} = \frac{{D^{\text{k}} }}{d}$$
(7.9)
where \({\text{D}}^{\text{k}}\) is the cumulative slippage of the slip block, d is the shear band width and \({\gamma}_{\text{S}}\) is the average shear strain.
The relationship between shear strain and post-peak strength is determined using the stress–strain curve of the rockfill materials. If the minimum safety factor is less than 1.0, the subsequent finite element dynamic stability analysis is conducted using the post-peak strength, conversely, using the peak strength.
4.
Check if the earthquake had ended. If the earthquake was ongoing, continue the calculation. If the earthquake had ended, output the time history of safety factor and cumulative slippage. Repeat this process for each ground motion and a series of dynamic stability information under different earthquake intensities were obtained.
5.
The GPDEM were introduced and the probabilistic information of safety factor, cumulative time of \({\text{F}}_{\text{S}}{<1.0}\) and cumulative slippage under different seismic intensities were obtained. The effects of considering or without considering the softening effects on the stability of the dam slopes were analyzed from a probabilistic perspective.
7.3 Effect of Softening Characteristics of Rockfill Materials Based on Stochastic Dynamic and Probabilistic Analysis
A 250-m CFRD was used to perform finite element time history analysis of dam slope stability in order to compare the effects on the slope stability of high CFRDs considering and without considering softening. First, five sets of stochastic ground motions were generated from PGA = 0. 1 g to 0. 5 g with 0. 1 g intervals and each set had 233 samples. Then, for each ground motion, the 2-D nonlinear finite element numerical analysis of the CFRD was performed including static, dynamic and stability calculations using the software GEODYNA and FEMSTABLE 2.0. Finally, the influence of softening effects of rockfill materials was evaluated based on the minimum safety factor, cumulative time of \({\text{F}}_{\text{S}}{<1.0}\) and cumulative slippage from the perspective of stochastic dynamics and probability, so as to the basis and reference of performance-based seismic safety evaluation for high CFRD slopes.
7.3.1 Calculating Basic Information
The calculation models, load conditions and boundary conditions remain consistent with those described in Sect. 2.7.4, and the static and dynamic parameters are also adopted from the numerical values presented in Sect. 2.7.4. An improved Newmark method was adopted to perform the stability calculations, considering and without considering softening. Ten groups of consolidated drained triaxial tests results of rockfill materials from existing or planned high earth-rock dams were statistically analyzed, and the relationship between post-peak shear strain of rockfill materials and dimensionless values (ratio of normalized post-peak strength to peak strength) was obtained, also a normalized strength parameter curve was fitted. Figures 7.2 and 7.3 illustrate the relationships between \(\frac{{\varphi _{0} }}{{\varphi _{{\max }} }}\) and \(\frac{{\Delta \varphi }}{{\Delta \varphi _{{\max }} }}\) with post-peak shear strain, respectively. As can be seen, when the post-peak shear strain increases, the softening effects of the rockfill materials become evident. The relationship equation obtained from the fitting curves are Eqs. (7.10) and (7.11). Therefore, the relationship between post-peak shear strain and post-peak strength parameters used for the CFRD can be obtained, as shown in Table 7.1. The softening characteristic was considered only in rockfill A and B in this paper.
A scatter plot of phi 0 over phi max versus gamma S in percentage. It plots a decreasing linear trend represented by a fitting curve. The highest point is 1.00 at 0%, while the lowest point is approximately near 0.87 at around 21%.
Fig. 7.2
Normalized relationships between γs and φ0/φmax
A scatter plot of delta phi over delta phi max versus gamma S in percentage. It plots a downward linear trend indicated by the fitting curve. The plot starts at 1.0 for 0% gamma S and decreases to around 0.5 as gamma approaches 20%. Values are approximated.
The stochastic dynamic results of three indices (the minimum safety factor, cumulative time of \({\text{F}}_{\text{S}}{<1.0}\) and cumulative slippage) for dam slope stability under different PGA (0. 1 g, 0.2 g, 0.3 g, 0.4 g, 0. 5 g) considering and without considering softening were obtained. The probabilistic information of these three indices was also obtained based on the GPDEM and equivalent extreme event theory.
(1)
Safety factor
Figure 7.4 illustrates the second-order statistical time histories (mean and standard deviation) of the safety factor with PGA adjusting from 0. 1 g to 0. 5 g. The individual safety factor time history analysis suggests that the impact of considering softening on the safety factor is not obvious. However, the mean and standard deviation of the safety factor indicate that considering softening effects leads a decrease of the safety factor. There is little difference in the safety factor time history between considering and without considering the softening effects under weak earthquake (e.g., PGA = 0.1 g), which is because the rockfill material has not reached the peak strength to show the softening effects. However, as the seismic intensity increases (e.g., PGA = 0.5 g), the difference becomes more obvious. Furthermore, the difference in the safety factor between the two cases gradually increases with the increase of seismic intensity, indicating a progressive trend. The standard deviation increases with time, indicating that the change of safety factor increases with the development of nonlinear behavior of rockfill materials which is further illustrated by the sharp fluctuation of 3–9 s; the standard deviation also decreases over time, part of the reason is that the change of safety factor decreases as the history of ground motion time decreases with time. This indicates that the softening of the rockfill materials is a progressive process. Furthermore, the stability of the mean and standard deviation of the safety factor time history demonstrates the statistical significance of the generated stochastic ground motions. It also indicates that the randomness of the ground motions has a significant impact on the safety factor. Based on the above analysis, it is evident that the dynamic response of dam slope stability is sensitive to the stochastic ground motions and different ground motions have a significant impact on the safety factor of the dam slope. The softening of the rockfill materials under earthquake is a progressive process. Therefore, it is necessary and meaningful to consider the softening of the rockfill materials and introduce the stochastic ground motions for the analysis of dam slope stability based on safety factor.
A series of fifteen multi-line graphs, for safety factor versus time, with different conditions unsoftening and softening represented by fluctuating lines. The graph in the first column has multiple peaks, the second column graphs have nearly stable trends and the last column plots have a bell-shaped curve with slight fluctuations.
Fig. 7.4
Safety factor time history and their mean and standard deviation. a Safety factor. b Mean c. Standard deviation d. Safety factor e. Mean f Standard deviation g. Safety factor h. Mean i. Standard deviation j. Safety factor k. Mean l. Standard deviation m. Safety factor n. Mean o. Standard deviation
×
Figures 7.5 and 7.6 illustrate the PDFs and CDFs of the minimum safety factor with PGA = 0. 2 g and 0. 5 g based on the equivalent extreme event theory. It is evident that there is negligible difference in the PDFs and CDFs of the minimum safety factor with PGA = 0.2 g between considering and without considering softening. However, the difference increases significantly with PGA = 0. 5 g and it is consistent with the results of mean and standard deviation. The results show that as the PGA increases, the safety factor time history and the minimum safety factor of the dam slope show an increasing difference between considering and without considering softening during the earthquake process. This difference is attributed to the progressive softening effects of the rockfill materials with the increasing seismic intensity. The dam slope reliabilities with the minimum \({\text{F}}_{\text{S}}{=1.0}\) are listed in Table 7.2. It can be observed that there is almost no difference in probability between considering and without considering softening corresponding to different earthquake intensities. This result also indicates that it is unreasonable to evaluate the dam slope stability only based on the minimum safety factor.
2 multi-line graphs of the probability density function and cumulative distribution function of the minimum safety factor, compare the unsoftening and softening conditions. Both graphs illustrate close overlap between conditions, depicting an S-shaped curve in the second plot and a peak in the first.
Fig. 7.5
PDFs and CDFs of minimum safety factor under 0.2 g (a) PDF (b) CDF
2 multi-line graphs of P D F and C D F of the minimum safety factor, with distinct differences between Unsoftening and softening conditions. a. It peaks around (0.2, 2.5) for Unsoftening and (0.2, 3) for softening, b. It plots a steeper curve for Unsoftening and softening.
Fig. 7.6
PDFs and CDFs of minimum safety factor under 0.5 g. a. PDF b CDF
Table 7.2
Seismic reliability under different earthquake levels
Softening effects
Cumulative time (s)
PGA
0.1 g
0.2 g
0.3 g
0.4 g
0.5 g
Unsoftening
0
0.9738
0.0931
0.0043
0
0
1
1
0.9828
0.6314
0.1226
0.0199
2
1
1
0.9977
0.9132
0.5685
Softening
0
0.9738
0.0931
0.0043
0
0
1
1
0.9355
0.3314
0.0274
0
2
1
1
0.8661
0.3132
0.0593
Note 0 represents the strain before peak value
×
×
(2) Cumulative time of \({\text{F}}_{\text{S}}{<1.0}\)
The cumulative time of \({\text{F}}_{\text{S}}{<1.0}\) is a new indicator for the dam slope stability evaluation, which has been gradually adopted by researchers. Some scholars and engineers suggest that the dam slope will lose stability if the cumulative time of \({\text{F}}_{\text{S}}{<1.0}\) exceeds 1–2 s. Figures 7.7 and 7.8 illustrate the PDFs and CDFs of the cumulative time of \({\text{F}}_{\text{S}}{<1.0}\) with PGA = 0. 2 g and 0. 5 g based on the equivalent extreme event theory, respectively. It can be observed that under weak earthquake, there is little difference between considering and without considering softening. However, as the seismic intensity increases, the difference becomes more obvious, indicating that the softening effects of the rockfill materials gradually become evident under strong earthquake actions. Furthermore, the difference in CDFs between considering and without considering softening gradually increases after the cumulative time of 0.5 s with PGA = 0. 2 g. This is because once the rockfill materials became softening, the post-peak strength decreases, leading to lower calculated safety factors and an increase in the cumulative time of \({\text{F}}_{\text{S}}{<1.0}\). The reliability based on the cumulative time of \({\text{F}}_{\text{S}}{<1.0}\) corresponding to different earthquake intensities are listed in Table 7.2. The results show that as the seismic intensity increases, the difference in probabilities between considering and without considering softening becomes more obvious and this also suggests that evaluating the stability of the dam slope based on the cumulative time of \({\text{F}}_{\text{S}}{<1.0}\) is reasonable.
2 graphs of P D F and C D F of cumulative time under Unsoftening and softening conditions. a. It plots a steep initial peak for Unsoftening while softening has a more gradual slope. The C D F graph reflects the rapid accumulation of time under Unsoftening compared to softening.
Fig. 7.7
Probability information of cumulative time of \({\text{F}}_{\text{S}}{<1.0}\) under 0.2 g (a) PDF (b) CDF
2 multi-line graphs of P D F and C D F of cumulative time under Unsoftening and softening conditions, highlighting distinct trends. a. It plots separate peaks for both conditions, with Unsoftening peaking earlier. The C D F indicates an S-shaped curve for both the legends.
Fig. 7.8
Probability information of cumulative time of \({\text{F}}_{\text{S}}{<1.0}\) under 0.5 g
×
×
(3) Slip surfaces and cumulative slippage.
Figure 7.9 illustrates the most dangerous slip surfaces of dam slope with PGA = 0. 5 g considering and without considering softening. The results show that the positions of the most dangerous slip surfaces are basically not affected by considering and without considering softening corresponding to different earthquakes, but the positions of some slip surfaces are also different, which proves that different earthquake motions have certain influence on the position of the most dangerous slip surface, and shows the necessity of studying the influence of softening effects on dam slope slip from the perspective of stochastic dynamics and probability. Figure 7.10 illustrates the cumulative slippage with PGA = 0. 5 g considering and without considering softening. It is evident that the cumulative slippage considering softening is significantly greater than that without considering softening under strong earthquake. Figures 7.11 and 7.12 illustrate the probability information with PGA = 0. 2 g and 0. 5 g considering and without considering softening. Table 7.3 list the reliability of the cumulative slippage of 5, 50, and 100 cm corresponding to different PGA, both considering and without considering softening. The probability information demonstrates that under weak earthquake, the dam slope exhibits almost no slippage, while significant slippage occurs and the softening effect has a substantial impact on the slippage under strong earthquake. This is because with the earthquake intensity increases, the softening effects becomes more obvious and the slippage and slip shear strain also increase.
Two triangle diagrams, comparing Unsoftening and softening conditions on slopes. The left diagram illustrates a more significant deformation zone for Unsoftening, while the right depicts a less pronounced area for softening,
Fig. 7.9
The slide surface corresponding to the minimum safety factor (0. 5 g)
2 multi-line graphs of cumulative slippage versus time for softening and softening conditions. The left graph and the right graph have softening with multiple lines indicating progressive slippage over time, with Unsoftening reaching over 600 centimeters and softening just below 900 centimeters.
Fig. 7.10
The cumulative slippage of dam slope corresponding to the minimum safety factor (0. 5 g) (a) Unsoftening (b) Softening
2 multi-line graphs of P D F and C D F of cumulative slippage for Unsoftening and softening conditions. The P D F graph plots a rapid decline from 100 to near zero within 40 centimeters. The C D F graph sharply rises to 1.0 at around 100 centimeters, indicating complete slippage.
Fig. 7.11
Probability information of cumulative slippage under 0.2 g (a) PDF (b) CDF
2 multi-line graphs of P D F and C D F of cumulative slippage for Unsoftening and softening conditions. The P D F graph has peaks around 150 centimeters for Unsoftening, and 300 centimeters for softening, while the C D F graph demonstrates faster accumulation for Unsoftening, with both reaching 1.0.
Fig. 7.12
Probability information of cumulative slippage under 0.5 g. a PDF. b CDF
Table 7.3
Seismic reliability under different earthquake levels
Softening effects
Cumulative slippage (cm)
PGA
0.1 g
0.2 g
0.3 g
0.4 g
0.5 g
Unsoftening
5
1
0.7121
0.1340
0.0218
0
50
1
0.9976
0.8299
0.3773
0.1266
100
1
1
0.9734
0.7297
0.3522
Softening
5
1
0.6869
0.1067
0.0102
0
50
1
0.9732
0.5378
0.1252
0.0240
100
1
0.9969
0.8170
0.3305
0.0979
×
×
×
×
7.3.3 Conclusion
The rockfills of earth-rockfill dams gradually show softening effects subjected to earthquakes, especially strong ones, which will significantly affect the safety of dam slopes. In order to evaluate the softening effects on the stability of dam slopes, a probability analysis method of dam slope safety based on the equivalent extreme event theory and GPDEM was introduced considering the stochastic earthquake excitation. A 250-m CFRD was used to perform stochastic dynamic response analysis and probabilistic reliability analysis based on three physical parameters of dam slope stability, the minimum safety factor, cumulative time of \({\text{F}}_{\text{S}}{<1.0}\) and cumulative slippage. Three main conclusions are as follows：
1.
In this section, the fundamental idea of GPDEM was employed to construct a stochastic process with a "virtual time parameter" regarding the extreme dynamic responses of the dam slope. The GPDEM equation was derived and the extreme value distribution probability and reliability of stochastic structural dynamic reaction are obtained. This method exhibits wide applicability in the probabilistic analysis of complex engineering structures, enabling a more accurate assessment of the reliability of earth-rockfill dam slopes.
2.
The results show that the difference between considering and without considering softening gradually increases with the increase of earthquake intensity based on the minimum safety factor, cumulative time of \({\text{F}}_{\text{S}}{<1.0}\) and cumulative slippage which was because the softening effects of rockfills were gradually revealed during the earthquake. Meanwhile, the softening was a gradual process and it is of great significance to analyze the seismic performance of the high CFRDs considering the softening effects.
3.
The results of reliability analysis show that it is unreasonable to study the stability of earth-rockfill dams only based on the minimum safety factor, and it is necessary to combine the cumulative time of \({\text{F}}_{\text{S}}{<1.0}\) and cumulative slippage to fully evaluate the safety of dam slope. The proposed stochastic probabilistic analysis method can give a more accurate evaluation of the reliability of high earth-rockfill dam slopes and also provide theoretical support for seismic design and risk assessment of earth and rock dams.
7.4 Statistical Analysis of Shear Strength Parameters of High CFRDs
Because the rockfill materials of earth-rockfill dams are sourced from different quarries or are located in different areas, there exists a certain level of discreteness and variability. In order establish a universal, unified and referential safety evaluation system for seismic performance of high CFRD slopes, the shear strength parameters of 40 CFRDs with a height of more than 100 m are statistically collected to obtain exact parameter statistical characteristics, as shown in Table 7.4.
Table 7.4
The shear strength parameter statistics of high CFRDs over 100 m in China
No
Engineering
Dam height /m
φ0 /°
Δφ /°
No
Engineering
Dam height /m
φ0 /°
Δφ /°
1
Shuibuya
233
51.2
9.1
13
Longma
135
51.7
11.0
52.0
8.5
14
Shanxi
132
56.1
11.6
50.0
8.4
54.4
9.8
52.0
8.5
15
Yinzidu
130
50.6
10.9
2
Houziyan
223
49.6
7.5
16
Jiemian
126
52.5
9.5
49.8
7.2
53.0
9.0
48.0
7.5
52.0
10.0
50.0
8.2
17
Eping
125
47.0
6.8
3
Jiangpinghe
210
47.7
6.3
18
Heiquan
124
47.0
6.8
53.0
6.5
48.0
7.0
48.6
5.5
46.0
6.5
48.3
5.2
19
Baixi
124
47.3
6.3
4
Sanbanxi
186
51.6
8.5
47.0
7.0
56.2
12.5
47.9
5.0
55.7
12.4
20
Baiyun
120
53.2
6.4
52.7
10.6
21
Qinshan
120
46.3
8.6
55.7
12.4
50.0
6.1
46.3
7.4
48.0
10.0
5
Hongjiadu
180
53.0
9.0
22
Gudongkou
118
48.4
7.2
52.0
10.0
23
Sujiahekou
117
51.0
10.5
52.3
7.3
24
Bajiaohe
115
49.9
10.0
51.3
6.9
49.5
10.0
57.0
13.1
25
Sinanjiang
115
42.8
8.3
52.8
9.6
26
Gaotang
112
45.6
5.9
6
Tianshengqiao
178
52.9
11.5
27
Chahanwusu
110
52.8
10.0
58.0
14.0
53.2
10.0
51.6
16.7
51.4
9.3
58.9
13.1
54.4
10.7
54.0
13.0
53.2
10.4
54.0
13.5
51.4
9.3
48.0
10.0
54.4
10.7
46.9
4.5
28
Nalan
109
49.4
7.3
48.7
5.7
56.0
14.0
53.5
8.7
47.0
6.4
57.0
13.0
29
Yutiao
106
44.6
5.1
50.5
10.0
42.6
5.7
7
Tankeng
161
54.7
11.4
50.0
8.4
55.8
12.0
42.0
5.5
55.4
11.9
30
Qiezishan
106
48.6
8.3
54.4
11.5
31
Liyutang
105
45.5
8.5
56.0
12.1
47.0
8.0
55.8
12.1
44.0
9.0
54.7
11.5
32
Dongba
105
52.1
9.6
55.4
11.9
33
Panshitou
103
54.2
12.4
56.3
12.4
43.9
10.0
57.1
12.4
44.9
11.5
8
Jilintai
157
55.9
12.6
34
Sianjiang
103
46.7
6.6
49.8
9.2
35
Chaishitan
103
47.4
10.2
51.2
13.5
36
Baishuikeng
101
52.6
14.3
57.0
14.0
58.4
13.4
53.0
8.1
37
Kajiwa
171
51.4
9.0
9
Zipingpu
156
53.6
11.2
50.6
8.7
55.4
10.6
52.4
9.5
10
Malutang
154
51.9
10.6
52.6
9.5
55.0
15.0
38
Cihaxia
254
54.2
8.1
11
Gongboxia
139
49.8
9.4
52.7
8.2
50.0
9.0
39
Gushui
242
55.5
11.3
46.7
8.1
53.0
11.0
47.2
7.1
53.5
10.7
45.2
13.0
54.4
10.6
46.8
4.4
55.0
12.2
52.0
9.8
52.0
9.9
51.3
14.8
40
Jiudianxia
137
50.9
8.5
12
Wuluwati
138
43.9
3.3
φ0: Mean of 51.1; Std.D of 3.8;
Δφ: Mean of 9.5; Std.D of 2.7
43.5
3.0
44.2
3.6
Figures 7.13 and 7.14 illustrate the frequency histograms and statistical features of \({\varphi}_{0}\) and \({\Delta\varphi}\), respectively. It is evident that \({\varphi}_{0}\) and \({\Delta\varphi}\) basically obey a normal distribution. The mean and standard deviation are listed in Table 7.4. The relationship between post-peak strength and post-peak strain based on the mean shear strength can be obtained according to Eqs. (7.10) and (7.11). The corresponding values are presented in Table 7.5.
A combined line and a histogram of the frequency versus phi 0 with a fitted normal distribution curve and a C D F plot with 4 increasing curves for empirical, normal distribution, and confidence limits. The histogram peaks near (53, 0.20), Values are approximated.
Fig. 7.13
Frequency histograms and statistical features of \({\varphi}_{0}\)a Frequency histogram b Comparison of CDFs
A combined line and a histogram of the frequency versus phi 0 with a fitted normal distribution curve and a C D F plot with 4 increasing curves for empirical, normal distribution, and confidence limits. The histogram peaks near approximately (10. 0.24),
Fig. 7.14
Frequency histograms and statistical features of \({\Delta\varphi}.\)a Frequency histogram. b Comparison of CDFs
Table 7.5
The post-peak shear strain and strength based on the mean of shear strength
Post-peak shear strain
φ0（°）
∆φ（°）
0
51.1
9.5
0.03
50.1
8.8
0.06
49.2
8.1
0.09
48.2
7.5
0.12
47.2
6.9
0.15
46.3
6.2
0.18
45.3
5.6
0.21
44.3
4.9
×
×
7.5 Dam Slope Stability Performance Evaluation Considering Randomness of Ground Motion
In this section, the randomness of the ground motion is considered. The GPDEM, reliability probability analysis method and fragility analysis method are employed to assess the seismic safety of a 250-m CFRD slope considering the softening effects of the rockfill materials. This evaluation reveals the evolution patterns of its stochastic dynamic response and changes in failure probability under different levels of seismic intensity, providing valuable references for the safety evaluation of dam slope stability.
7.5.1 Basic Information
The calculation models, load conditions remain consistent with those described in Sect. 3.4.1. A Duncan-Chang E-B constitutive model was employed for the static calculations and then the initial stress conditions were provided for the subsequent dynamic calculations. The Hardin-Drnevich constitutive model was adopted to simulate the non-linear behaviour of the rockfill materials during the dynamic calculations. The rockfill material parameters of the two constitutive models are provided in Tables 7.6 and 7.7, respectively. The parameters of bedrock and faced-slabs are the same as in Sect. 3.4.1; The peak and post-peak strengths are listed in Table 7.5 in order to consider the softening effects. The PGA was adjusted from 0. 1 to 1.0 g with 0. 1 g intervals, resulting in a total of 1440 acceleration time histories.
Table 7.6
Parameters for duncan E-B model
ρ /(kg/m3)
K
n
Rf
Kb
m
φ0 /(°)
Δφ /(°)
2160
1350
0.28
0.80
780
0.18
51.1
9.5
Table 7.7
Parameters for Hardin-Drnevich model
K
n
ν
2660
0.444
0.33
7.5.2 Stochastic Dynamic Response for Slope Stability of High CFRDs
(1) safety factor
Figure 7.15 illustrates the safety factor time histories of three typical samples and the mean and standard deviation time histories with PGA = 0. 1, 0. 5 and 1. 0 g, respectively. As can be seen, when the seismic intensity increases, the safety factor exhibits more drastic fluctuations over time, and the standard deviation also increases. The mean time history of safety factor tends to be stable, indicating that the random ground motion samples generated have good statistical characteristics. Figure 7.16 illustrates the probability information for safety factor derived by using GPDEM with PGA = 0. 5 g. The PDFs at three typical times are entirely distinct and display two or even multiple peaks, not fitting the regular distributions like normal or log-normal distributions. The PDF evolution surface vividly illustrates the evolving process of the PDF of safety factor over time which has significant fluctuations and transformations, as shown in Fig. 7.16b. The evolution of the PDF also indicates larger variability among different seismic samples, and the fluctuations and evolutions of this variability is more evident in the PDF contour map, as shown in Fig. 7.16c.
Three multi-line graphs of safety factor, mean, and standard deviation versus time under various P G A conditions. Each plot has three fluctuating curves highlighting variability in safety factors over time.
Fig. 7.15
Response, mean and standard deviation time history of safety factor a Safety factor b Mean c Standard deviation
3 graphs of safety factor P D Fs over time. A P D F line plot, a 3-D plot, and a contour plot for various F s values. a. It plots distinct peaks at different times, the 3-D plot visualizes P D F changes over time, and the contour plot outlines variations in safety factors.
Fig. 7.16
Probability evolution information of safety factor.a PDFs at three typical times. b PDF evolution surface. c PDF contour map
×
×
Figure 7.17 presents the discrete point distribution of minimum safety factor with PGA = 0. 5 g and 1. 0 g. The point distribution has a large dispersion, and the difference between the maximum and minimum values is significant. The maximum value with PGA = 0. 5 g is 1.17 and the minimum value is 0.42, while the maximum value with PGA = 1. 0 g is 0.66 and the minimum value is 0, which indicates that the safety factor has high statistical significance subjected to stochastic earthquake, emphasizing the importance of analyzing from the perspective of stochastic dynamics based on the minimum safety factor. The exceedance probability increases at a faster rate with higher seismic intensity, seen from Fig. 7.18. This may be attributed to the softening effects of rockfill materials under strong earthquake. And the minimum safety factors under different seismic intensities are primarily distributed between the 95% and 5% exceedance probabilities. The numerical range of safety factor is as follows: 1.31–1.87, 1.05–1.64, 0.84–1.45, 0.67–1.30, 0.50–1.17, 0.35–1.04, 0.19–0.92, 0.08–0.80, 0–0.71, and 0–0.61. These findings serve as a reference for the seismic stability design of high CRFD under earthquake.
2 scatter plots of minimum safety factor versus sample number. a. It plots scatter dots with a mean of 0.837 and a standard deviation of 0.157. b. It plots scatter dots with a mean of 0.277 and a standard deviation of 0.177, indicating lower safety factors.
Fig. 7.17
Discrete point distribution of minimum safety factor under 0. 5 g and 1. 0 g. a. PGA = 0. 5 g .b PGA = 1. 0 g
2, multi-line graphs of mean and standard deviation of V o R versus time under different P G A conditions. Both plots have 10 fluctuating curves. a. It plots a rise in V o R mean with time, peaking around 6 seconds. b. It indicates variability in V o R, with standard deviations reaching up to 15%.
Fig. 7.18
Exceedance probability of minimum safety factor under different PGA. a Exceedance probability. b The minimum safety factor
×
×
(2) Cumulative time of \({\text{F}}_{\text{S}}{<1.0}\)
Figure 7.19 presents the discrete point distribution of cumulative time of \({\text{F}}_{\text{S}}< \text{1} {.0}\) with PGA = 0.6 g and 1. 0 g. The point distribution has a large dispersion, and the difference between the maximum and minimum values is significant. The maximum value with PGA = 0.6 g is 2.31 s and the minimum value is 0 s, while the maximum value with PGA = 1. 0 g is 4.02 s and the minimum value is 0.65 s. This indicates the importance of analyzing from the perspective of stochastic dynamics based on the cumulative time with \({\text{F}}_{\text{S}}{<1.0}\). Figure 7.20 shows the exceedance probability of cumulative time with \({\text{F}}_{\text{S}}{<1.0}\) under different seismic intensities. The probabilities primarily range between 5 and 95% and the numerical ranges for each seismic intensity are as follows: 0–0.11 s (0.3 g), 0–0.48 s (0.4 g), 0–1.27 s (0. 5 g), 0.02–1.85 s (0.6 g), 0.21–2.30 s (0.7 g), 0.51–2.71 s (0.8 g), 0.77–3.12 s (0.9 g), and 1.18–3.51 s (1. 0 g). These findings provide references for performance-based seismic stability evaluation of high CRFD.
2 scatter plots of cumulative time versus sample number. a. It Plots scatter dots with a mean of 0.764 and a standard deviation of 0.545. b. It plots scatter dots with a mean of 2.332 and a standard deviation of 0.638.
Fig. 7.19
Distribution of cumulative time with \({\text{F}}_{\text{S}}{<1.0}\) under 0.6 g and 1. 0 g. a PGA = 0.6 g. b PGA = 1. 0 g
2 multiple-line plots of exceedance probability and cumulative time versus P G A. a. It plots 8 declining exceedance probabilities as cumulative time increases for different P G A levels. b. It plots 4 increasing lines for cumulative time rising with increasing P G A with different exceedance probabilities.
Fig. 7.20
Exceedance probability of cumulative time with \({\text{F}}_{\text{S}}{<1.0}\) under different PGA. a Exceedance probability. b The cumulative time with \({\text{F}}_{\text{S}}{<1.0}\)
×
×
(3) Cumulative slippage
Figure 7.21 illustrates the slip surfaces corresponding to the minimum safety factor with PGA = 0.4 g and 0.6 g. It is evident that the position and size of the slip surfaces vary with different seismic intensities. The slip surface is smaller but more + prone to shallow sliding under strong earthquakes. Figure 7.22 illustrates the discrete point distribution of the cumulative slippage with PGA = 0.6 g and 1. 0 g. The point distribution has a large dispersion, and the difference between the maximum and minimum values is very significant. The maximum value with PGA = 0.6 g is 331 cm and the minimum value is 0 cm, while the maximum value with PGA = 1. 0 g is 848 cm and the minimum value is 13 cm and this indicates the importance of analyzing from the perspective of stochastic dynamics based on the cumulative slippage. Figures 7.23 and 7.24 shows the exceedance probability and scatter plot of cumulative slippage under different seismic intensities. The probabilities primarily range between 5 and 95%. The numerical ranges for each seismic intensity are as follows: 0–0.3 cm (0.3 g), 0–11 cm (0.4 g), 0–67 cm (0. 5 g), 0–148 cm (0.6 g), 0–261 cm (0.7 g), 5–350 cm (0.8 g), 15–498 cm (0.9 g), 46–624 cm (1.0 g). These findings provide references for the performance-based stability assessment of high CRFD.
Two triangle diagrams comparing slope deformation under different P G A values. The left triangle with P G A = 0.4 gram has significant deformation, while the right triangle with P G A = 0.6 gram exhibits less deformation, indicating varying material responses.
Fig. 7.21
The slide surface corresponding to the minimum safety factor
Two scatter plots of cumulative slippage versus sample number. a. It plots scatter dots with a mean slippage of 40.297 and a standard deviation of 55.641. b. It plots scatter dots with a higher mean slippage of 279.183 and a standard deviation of 175.640.
Fig. 7.22
Distribution of cumulative slippage under 0.6 g and 1. 0 g. a PGA = 0.6 g. b PGA = 1. 0 g
Six line graphs of exceedance probability versus cumulative slippage under various conditions. Graphs a to f illustrate decreasing probabilities as slippage increases, with the lowest probabilities occurring at the highest slippage values, showing distinct trends for different conditions.
Fig. 7.23
Exceedance probability of cumulative slippage under different PGA. a PGA = 0.3 g. b PGA = 0.4 g. c PGA = 0.5 g. d PGA = 0.6 g. e PGA = 0.7 g. f PGA = 0.8 g. g PGA = 0.9 g. h PGA = 1.0 g
A multi-line graph of cumulative slippage versus P G A with multiple curves representing average values and exceedance probabilities of 5%, 50%, and 95%. Slippage increases with P G A, reaching up to 900 cm at 1.0 g, highlighting the correlation between P G A and slippage.
Fig. 7.24
Cumulative slippage under different PGA
×
×
×
×
(4) Discussion on the relationship between cumulative time of \({\text{F}}_{\text{S}}{<1.0}\) and cumulative slippage
The cumulative time of \({\text{F}}_{\text{S}}{<1.0}\) and the cumulative slippage both exhibit cumulative effects, suggesting a certain degree of correlation. Therefore, this section investigates the preliminary relationship between the cumulative time of \({\text{F}}_{\text{S}}{<1.0}\) and the cumulative slippage under different seismic intensities through extensive sample analysis. It can be observed that there is a certain correlation between the cumulative time of \({\text{F}}_{\text{S}}{<1.0}\) and the cumulative slippage, seen from Fig. 7.25. However, the correlation decreases as the seismic intensity increases. Based on the distribution pattern, it is inferred that a complex correlation exists between the two indices and it is influenced by seismic intensity.
8 scatter plots of cumulative slippage versus cumulative time under various conditions. Each plot features a concave up increasing curve as the best fit, with increasing slippage over time. The highest R-squared value is 0.955.
Fig. 7.25
The relationship between cumulative time and cumulative slippage. a PGA = 0.3 g. b PGA = 0.4 g. c PGA = 0. 5 g. d PGA = 0.6 g. e PGA = 0.7 g. f PGA = 0.8 g. g PGA = 0.9 g. h PGA = 1. 0 g
×
7.6 Dam Slope Stability Stochastic Dynamic Analysis Considering Randomness of Shear Strength Parameters
7.6.1 Basic Information
The working conditions and load conditions of the high CFRD are the same as those in Sect. 3.4.1. The static and dynamic parameters are adopted those in Sect. 7.5.1 The mean and standard deviation of \({\varphi}_{0}\) and \({\Delta\varphi}\) as well as the type of distribution are the values obtained in Sect. 7.4, and 144 sets of shear strength parameters were obtained based on the GF-discrepancy method. Considering the softening effects, the post-peak strengths of \({\varphi}_{0}\) and \({\Delta\varphi}\) is obtained by using Eqs. (7.10) and (7.11), respectively. A series of finite element dynamic stability calculations were performed with PGA = 0. 5 g, and the stochastic dynamic and probabilistic information of the safety factor, the cumulative time of \({\text{F}}_{\text{S}}{<1.0}\) and the cumulative slippage were obtained by combining with the GPDEM, so as to provide a reference for safety evaluation of dam slope stability performance based on parameter randomness.
7.6.2 Safety Factor
Figure 7.26 illustrates the safety factor time history with deterministic mean material parameters and the mean and standard deviation time history of safety factor. It can be observed that the safety factor exhibits different patterns compared to seismic randomness, indicating the significance of considering the material parameters uncertainties. The significant fluctuations in the standard deviation further emphasize the strong influence of material parameters uncertainties on the safety factor under seismic excitation.
3 line graphs of safety factor, safety factor mean, and standard deviation versus time. Each plots a fluctuating curve. a. Peaks around 2 and 7. b. Mean safety factors with peaks around 2 and 7. c. Standard deviation with peaks around 2 and 7. Values are approximated.
Fig. 7.26
Response, mean and standard deviation time history of safety factor. a Safety factor. b Mean. c Standard deviation
×
Figure 7.27 shows that different material parameters lead to varying minimum safety factors, with a range between 1.19 and 0.46. The value at 95% exceedance probability is 0.48, while it is 1.10 at 5% exceedance probability, with a difference of approximately 2.3 times. There are noticeable distinctions between the safety factor exceedance probabilities considering the seismic randomness and material parameter randomness. Hence, it is essential to consider the effects of material parameters randomness on safety factor.
2 scatter plots of cumulative time versus sample number and exceedance probability versus cumulative time. a. It plots scatter data for ground motion and strength parameters with means of 0.384 and 0.548. b. It plots two concave up-decreasing curves for ground motion and strength parameters.
Fig. 7.27
Distribution and exceedance probability of minimum safety factor. a Distribution. b Exceedance probability
×
7.6.3 Cumulative Time of\({\text{F}}_{\text{S}}{<1.0}\)
Figure 7.28 shows that different material parameters lead to varying cumulative time of \({ \, {\text{F}}}_{\text{S}}{<1.0}\), with a range between 3.34 and 0 s. The value at 95% exceedance probability is 0 s, while it is 1.77 s at 5% exceedance probability. There are noticeable distinctions between the safety factor exceedance probabilities considering the seismic randomness and material parameter randomness. Hence, it is essential to consider the impacts of the material parameters randomness on the cumulative time of \({\text{F}}_{\text{S}}{<1.0}\). According to the damage level division criteria in Sect. 8, the corresponding probabilities for mild damage (0 s), moderate damage (0.5 s), and severe damage (1.5 s) are 92.4%, 35.3%, and 7.6%, respectively.
2 scatter plots of cumulative time versus sample number and exceedance probability versus cumulative time. a. It plots scatter data for ground motion and strength parameters with means of 0.384 and 0.548. b. It plots two concave up-decreasing curves for ground motion and strength parameters.
Fig. 7.28
Distribution and exceedance probability of cumulative time. a Distribution. b Exceedance probability
×
7.6.4 Cumulative Slippage
Figure 7.29 shows slip surfaces corresponding the minimum safety factor obtained considering seismic randomness (red line) and shear strength parameters randomness (blue line) with PGA = 0. 5 g. It is evident that different randomness significantly affects the position and size of the slip surfaces. Figure 7.30 shows that different material parameters result in varying cumulative slippage, with a significant difference between the maximum value of 363 cm and the minimum value of 0 cm. The value at 95% exceedance probability is 0 cm, while it is 141 cm at 5% exceedance probability, indicating a substantial variation. There are noticeable distinctions between the cumulative slippage considering the seismic randomness and material parameter randomness. Hence, it is crucial to consider the effects of the material parameters randomness on the cumulative slippage. According to the damage level division criteria in Sect. 8, the corresponding probabilities for mild damage (0 cm), moderate damage (20 cm), and severe damage (100 cm) are 80.1%, 28.9%, and 7.1%, respectively. The correlation analysis of cumulative time and cumulative slippage shown in Fig. 7.31 reveals that considering material parameter randomness results in better correlation between the two factors.
A triangle illustrates the side surface deformation under combined conditions. The overlapping areas at the top right indicate varying degrees of deformation due to different factors.
Fig. 7.29
The slide surface corresponding to the minimum safety factor
2 scatter plots of cumulative slippage versus sample number and exceedance probability versus cumulative slippage. a. It plots scatter data for ground motion and strength parameters. b. It plots two concave up-decreasing curves for ground motion and strength parameters.
Fig. 7.30
Distribution and exceedance probability of cumulative slippage. a Distribution. b Exceedance probability
A scatter plot of cumulative slippage versus cumulative time with data points for ground motion and strength parameters. Ground motion data increases steadily with R squared = 0.90 and reaches around 200 at 2 seconds. Strength parameters follow a more rapid increase, reaching 400 by 3 seconds.
Fig. 7.31
The relationship between cumulative time and cumulative slippage
×
×
×
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