8. Performance Seismic Safety Evaluation

Liu et al. (2012) conducted a statistical analysis of the seismic settlement rates of 123 earth-rock dams with heights exceeding 15 m worldwide. Combining publicly available seismic damage data and numerical calculation results, the numerical analysis mainly referred to high earth-rock dam projects under construction or planned in China, including 11 dams such as Nuozhadu, Lianghekou, Shuangjiangkou, Longpan, Houziyan, Liangfengtai, Wenquan, Jishi Gorge, and Longshou. It concluded that the majority of dam crest relative settlement rates are below 1%, with those exceeding 1% being primarily earth or hydraulic fill dams. It is preliminarily indicated that earth-rock dams constructed using modern heavy compaction techniques experience a dam crest relative settlement rate of approximately 1% under seismic peak ground acceleration less than 0.6 g. The Nanjing Hydraulic Research Institute (1998) initially suggested allowing a settlement rate of 2% for earth-rock dams below 100 m, 1.5% for those above 100 m, and 1.0% for those above 200 m. Zhao et al. (2015) proposed setting the dam crest relative settlement rate between 0.6% and 0.8% as the ultimate control standard for high panel rockfill dams. Tian et al. (2013) recommended setting the evaluation limits for dam heights of 100 m, 150 m, 200 m, 250 m, and 300 m at 2%, 1.5%, 1%, 0.85%, and 0.75% of the dam crest relative settlement rate, respectively. Chen et al. (2013) proposed using a dam crest settlement rate below 1% as the seismic deformation control criterion for core wall rockfill dams and below 0.6% as the ultimate control standard for high panel rockfill dams. Swaisgood et al. (2003) studied a total of 69 earth-rock dams (including panel rockfill dams, core wall rockfill dams, hydraulic fill dams, and earth dams) both domestically and internationally, using dam crest relative settlement rate as an indicator. They categorized the damage situation into intact (below 0.1%), slight damage (0.012%-0.5%), moderate damage (0.1%-1.0%), and severe damage (above 0.5%), but primarily focused on lower earth-rock dams. Figure 8.1 shows the curves for the relationship between the average PGA-dam top relative seismic subsidence rate and the 5% exceedance probability. It can be seen that the relative subsidence at the top of the dam increases with the increase in PGA, but the trend change gradually becomes slower. Combined with the literature in above, 0.3%, 0.7%, and 1.0% of the relative subsidence are suggested as the criticality of the performance level, which corresponds to the critical states of mild, moderate, and severe damages. In the Wenchuan earthquake, the 156 m-high Zipingpu faced rockfill dam suffered PGA = 0.55 g earthquake, and the subsidence was 0.81 m, which was about 0.519% of the height of the dam (Zhou 2012). According to the average fitting formula, the relative subsidence rate of the dam top was 0.521%, which was almost no different from the monitoring value, which proved the accuracy of the stochastic dynamic analysis, and it was in the range between mild and moderate damage according to the damage classification standard, consistent with the idea of ‘medium damage can be repaired. According to the damage classification criteria, the dam is in the range of mild-moderate damage, which is consistent with the idea of ‘medium earthquake can be repaired’, and the Zipingpu dam has been repaired and is in normal use. In addition, we suggest that the derivative of the curve obtained from the 5% beyond probability fit is 0, which corresponds to the no-breakage critical state, at which time, PGA = 1.207 g, corresponding to a relative seismic subsidence of the dam top of 1.151%, which is taken as 1.1% for safety.

A line graph of relative settlement of dam crest in percentage versus P G A in gravity. 2 lines follow increasing trends with equations, where the r-squared value is 1 and 0.9999. The monitored value is 0.0519%, and the predicted value is 0.521%.

Table 8.1 lists the exceedance probabilities corresponding to different levels of dam crest relative settlement under various seismic intensities. This establishes a relationship table between multiple seismic intensities, dam crest relative settlement, and exceedance probability, providing reference for performance-based seismic safety design. The significance lies in determining control standards for dam crest relative settlement corresponding to different exceedance probabilities, based on the owner's or practical requirements. According to Table 8.1, the exceedance probability for the Zipingpu CFRD faced rockfill dam falls between 50 and 60%, indicating a relatively high probability of seismic damage occurrence, which aligns with the actual situation. Furthermore, Table 8.1 holds significant reference value for studying the ultimate seismic resistance capacity of high CFRDs. The following recommendations are proposed: For PGA values ranging from 0.1 to 0.3 g, an exceedance probability of approximately 25% is suggested. At PGA = 0.1 g, the relative settlement is 0.15%; at PGA = 0.2 g, it is 0.25%; and at PGA = 0.3 g, it is 0.35%. For PGA values between 0.4 and 0.7 g, an exceedance probability of around 50% is recommended. Specifically, at PGA = 0.4 g, the relative settlement is 0.4%; at PGA = 0.5 g, it is 0.5%; at PGA = 0.6 g, it is 0.55%; and at PGA = 0.7 g, it is 0.6%. For PGA values between 0.8 and 1.0 g, an exceedance probability of approximately 75% is suggested. At PGA = 0.8 g, the relative settlement is 0.55%; at PGA = 0.9 g, it is 0.6%; and at PGA = 1.0 g, it is 0.65%. Figure 8.2 depicts the damage probability corresponding to different peak accelerations for various damage levels, i.e., fragility curves, obtained through B-Spline interpolation fitting. Different performance levels of damage probability can be defined based on these curves.

A multiline graph of the probability of destruction in percentage versus P G A in gravity. 2 lines follow an increasing trend. Another line moves horizontally on the x-axis, and then increases.

Furthermore, in line with the design philosophy of “minor earthquakes do not cause damage, moderate earthquakes are reparable, and major earthquakes do not cause collapse,” the following recommendations are proposed: For minor earthquakes (0–0.2 g), the dam remains essentially intact, with only a very small percentage (below 5%) experiencing “mild damage.“ Instances of mild damage or above are rare, and the relative settlement rate at the dam crest is 0.3% when subjected to seismic action of 0.2 g, corresponding to a damage probability of around 7.4%, approximately 5%. For moderate earthquakes (0.2–0.5 g), the dam is in a state of “mild to moderate” damage, with only a very small percentage (below 5%) reaching moderate damage. Instances of moderate damage or above are rare, and the relative settlement rate at the dam crest is 0.7% when subjected to seismic action of 0.5 g, corresponding to a damage probability of around 6.2%, approximately 5%. For major earthquakes (0.5–0.8 g), the dam is in a state of “moderate to severe” damage, with only a very small percentage (below 5%) reaching severe damage. Instances of dam collapse are rare, and the relative settlement rate at the dam crest is 1.0% when subjected to seismic action of 0.8 g, corresponding to a damage probability of around 5.4%, approximately 5%. For earthquakes exceeding 1.0 g, a very small percentage (below 5%) of dam collapses may occur. The exceedance probability of a dam crest settlement rate of 1.1% under seismic action of 1.0 g is around 5.4%, approximately 5%, and earthquakes exceeding 1.0 g are generally rare. Therefore, the above analysis from a probabilistic perspective essentially demonstrates the rationality of the performance level classification criteria mentioned above.

In the evaluation of seismic safety of panel impermeable body, the current related research mainly focuses on panel stress and joint deformation as the control criteria, however, the short time panel overstress may not cause damage, so the effect of overstress holding time should be considered. Ghanaat (2004) used the results of elastic time-range analysis to propose the damage classification standard of concrete dams based on the ratio of tensile stress to concrete tensile strength obtained by calculation, i.e., the demand capacity ratio (DCR) and the cumulative overstress duration (COD) that the dam tensile stress is greater than the concrete tensile strength in the seismic process, i.e., the accumulated time of overstress holding time, and the damage classification standard of concrete dams was proposed, and the COD of 0.4 s was determined when the DCR = 1 and 0 s for the DCR = 2, and the damage limit of slight and moderate damage was determined. When DCR = 1, the COD is 0.4 s, and when DCR = 2, the COD is 0 s, which determines the boundaries of slight and moderate damage, and this concept was introduced in the seismic performance evaluation of concrete gravity dams by Shen et al (2007).

The DCR of a concrete panel can be defined as the ratio of the tensile stress of the panel to the tensile strength of the concrete:where σ_t is the tensile stress for the linear elastic analysis of the panel; ft is the static tensile strength of the concrete of the panel, which can be obtained from the formula suggested by Raphael (1984):f_t is the compressive strength of plain concrete. Since the dynamic tensile strength of concrete material is increased more than the static tensile strength due to the effect of seismic deformation rate and the assumption of linear elasticity finite element factor, Raphael suggests that the nominal tensile strength of concrete under seismic dynamic loading is:

Therefore, the maximum allowable DCR value for the panel concrete in the linear elasticity analysis is divided by the nominal tensile strength under seismic loading (DCR = 2), and the corresponding stress is two times the static tensile strength, as shown in Fig. 8.3.

A line graph of stress versus strain. Line 1 linearly ascends with the peak point labeled B nominal tensile strength. Line 2 ascends and descends. The horizontal distance between the start point of line 2 and the endpoint of line 1 is labeled the damage control area.

Cumulative Overstress Duration (COD) is defined as the total duration of tensile stress when DCR ≥ 1. The longer the cumulative duration, the greater the possibility of panel damage. The permissible COD is determined as follows: the COD of five simple harmonic stresses when the stress harmonic amplitude with a period of T = 0.2 s reaches two times the static tensile strength (DCR = 1) is calculated by Eq. (8.4), as shown in Fig. 8.4. For DCR = 1, the COD is 0.6 s (recommended value for panel dams); when DCR = 2, the COD is 0 s, and Fig. 8.5 shows the time history of panel stress change under the action of two different intensities of ground shaking.

A line graph of D C R versus time. A solid line T = 0.24 second is plotted from the y-axis at (0.0, 1), and another sinusoidal line is over the solid line. A dashed horizontal line is plotted from the y-axis at (0.0, 0).

2 line graphs of stress in megapascal versus time in seconds. 1 and 2, a solid and a dashed line move horizontally in each graph. Another line fluctuates throughout.

Therefore, according to the dynamic change of panel stress, the preliminary evaluation of panel safety and damage classification, the proposed damage levels are as follows: no damage or mild damage, the panel does not exceed the tensile strength, \(\text{DCR} \le 1\), the panel is in the linear elasticity range; mild-moderate damage, the panel is cracked, but at an acceptable level, the \({\text{DCR}}\) and the COD are in the shaded part of Fig. 8.6. Moderate-heavy damage or severe damage with \(\text{DCR} \ge 2\) or COD outside the shaded portion of Fig. 8.6 should be analyzed using nonlinear elastic–plastic time-course analysis to further assess the degree of damage and seismic safety of the panels.

A graph of C O R versus D C R. A linear line decreases and plots triangular regions with the following labels. The shaded portion on the lower left is labeled mild-moderate damage linear elastic analysis. The unshaded upper right is labeled severe damage nonlinear analysis.

Figure 8.7 shows the exceeding probability of COD under different DCR, e.g., \(\text{DCR } = \, \text{1.5}\) indicates that the COD is greater than 1.5; when PGA = 0.1 g, the COD = 0 at \(\text{DCR } = \, {1}\), which indicates that the stresses are all within the line elasticity range, and the panel is not damaged. And the exceeding probability that the panel damage indexes are in different ranges under each seismic intensity can be obtained, which provides the basis for performance-based seismic safety evaluation of CFRD from the perspective of panel damage.

From the above exceedance probability to get the fragility curves as shown in Fig. 8.8, when PGA = 0.1 g, the panel is completely in the linear elastic range, no damage occurs; when PGA = 0.2 g, the panel occurs mild damage probability of 28.6%, the probability of moderate damage 3.4%; when PGA = 0.3 g, the panel occurs mild damage probability of 77.5%, the probability of moderate damage 37.1%; when PGA = 0.4 g, the probability of mild destruction of the panel is 92.5% and the probability of moderate destruction is 69.9%; when PGA = 0.5 g, the probability of mild destruction of the panel is 97.8% and the probability of moderate destruction is 87.1%; when PGA = 0.6 g, the probability of mild destruction of the panel is 100% and the probability of moderate destruction is 94.0%; when PGA = 0.7 g, the probability of mild destruction of the panel is 100% and the probability of moderate destruction is 94.0%. However, the damage of the panel based on the two-dimensional elastic–plastic analysis is not very accurate, and we can only understand its general distribution law and analyze its damage state qualitatively and quantitatively from the perspective of probability.

A multiline graph of the probability of destruction percentage versus P G A in units of gravity. 2 plotted lines increase and move horizontally.

Table 8.3 lists the relationship between multiple earthquake intensities-relative settlement of dam crest-exceedance probability based on 3-D stochastic dynamic response analysis. Figure 8.10 shows the fragility curves for the performance-based seismic safety evaluation of CFRDs. There are some differences between the subsidence values calculated in 2-D and 3-D. The damage probability in 3-D can be obtained by complementing the damage probability in 2-D for different seismic intensities.

A multiline graph of the probability of damage percentage versus P G A in units of gravity. It plots 3 curves of mild damage, moderate damage, and severe damage in an increasing trend. 3 dashed lines follow the corresponding curves.

In this paper, the overstress volume ratio of more than 55% or the accumulation time of more than 8 s is taken as the critical state, and faced-slab seismic safety evaluation index and damage grade standard based on the overstress volume ratio and the accumulation time of overstress is established, as shown in Fig. 8.11. Furthermore, Table 8.4 lists the relationship between multiple earthquake intensities-cumulative time-exceedance probability. Figure 8.12 shows state lines under different critical ground motion intensities. The fragility curves of different critical state based on faced-slab damage is shown in Fig. 8.13, while the fragility curve based on relative settlement rate of dam crest is added to refine the evaluation framework as the dotted line. Combining the above research and engineering practice, this paper suggests that the seismic intensity be divided into [0,0.2 g], [0.2–0.5 g], [0.5–0.8 g], [0.8–1.0 g], corresponding to four states of the CFRD being in the “basically intact”, “mildly-moderately” damaged, “ moderately-severely” damaged, and “rarely failed” states, respectively. It can be seen that the slab damage of Zipingpu CFRD is in mild-moderate condition, which is consistent with the actual situation. Under mild earthquakes (0–0.2 g), the dam is basically intact, with only a very small number of dams (less than 5 percent) experiencing “mild damage”, and basically no damage occurring above the threshold. At the limit of mild damage, the probability of exceeding the PGA of 0.2 g is about 1.5%. Under moderate earthquakes (0.2–0.5 g), the dam is in the state of “mild-moderate” damage, only a very small number (less than 5%) reaches moderate damage. The exceedance probability of moderate dam damage ranges from 1.3% to 3.8% under PGA = 0.5 g, with only a few cases. For severe earthquakes (0.5–0.8 g), the dam is in a state of “moderate-to-severe” damage, with very few (less than 5%) reaching severe damage, and basically no dam failures. The probability of exceeding the threshold of severe damage corresponding to a 0.8 g PGA is 3.7%-6.6%. For larger earthquakes (PGA = 1.0 g), very few (less than 5%) dam failures are allowed, with an exceedance probability of 3.7% for 1.0 g. Earthquakes above 1.0 g are generally less frequently encountered. The result shows that the dam deformation is more likely to reach a mild damage compared to faced-slab damage. Moderate and severe damage is then more likely to be caused by faced-slab damage. The above analyses basically show the reasonableness of the performance standard division criterion in this paper from the perspective of probability. It shows a certain correspondence with the deformation-based division criterion.

A multiline graph of the accumulation time in seconds versus the overstressed volume ratio in percentage. 5 linear lines decrease at different elevations, and another line of critical state is plotted linearly in a decreasing manner 4 times between other lines.

9 multiline graphs of accumulation time in seconds versus overstressed volume ratio in percentage. A to I, 4 lines decrease linearly. All 9 graphs consist of dashed lines moving downwards with slight variations, and there is an increase in the number of dashed lines from one graph to the next.

A multiline graph of the probability of damage in percentage versus P G A in units of gravity. It plots 3 curves of mild damage, moderate damage, and severe damage in an increasing trend. 3 dashed lines follow the increasing trends.

In seismic design of earth-rock dams in China, the pseudo-static method occupies an important position and is widely used with consensus on judgment criteria. When the safety factor Fs is less than 1.0, slope instability is indicated. However, this method is no longer suitable for high-intensity areas with high earth-rock dams. Currently, the finite element dynamic time-history method is commonly used, but there are significant differences in evaluation criteria. Li et al. (2010) suggested that when the cumulative time of Fs < 1.0 exceeds 2 s, slope instability occurs. Zhao et al. (2015) argue that using the dynamic time-history method, when Fs < 1.0, slope instability is observed. When using the dynamic equivalent value method, Fs < 1.1 indicates slope instability. Chen et al. (2013) proposed that when the cumulative time of Fs < 1.0 exceeds 2 s, slope instability occurs. Tian et al. (2014) recommend that when conducting numerical analysis of slope stability, the maximum thickness of the sliding block should not be less than 5 m, and when the cumulative time of Fs < 1.0 exceeds 1 s, slope instability occurs. Some domestic and international studies and standards also use cumulative slip displacement as an indicator to evaluate slope stability. Ozkan (1998) set a seismic slip deformation limit of 1 m. Switzerland adopts a two-level seismic design criterion (Darbre 2004), allowing deformations of 20 cm for shallow sliding and 50 cm for deep sliding. Tian et al. (2013) propose that when the cumulative slip displacement exceeds 1 m or exceeds 1% of the length of the sliding body, slope instability may occur. Shao et al. (2011) indicate that when the slip displacement exceeds 2% of the length of the sliding body, slope instability occurs.

Figure 8.14 shows the relationship between the average cumulative time between PGA. As the seismic intensity increases to a certain threshold, the dam slope sliding suddenly occurs and there is a significant increase in cumulative time at a specific PGA, followed by a stable increase. This indicates that when the seismic intensity reaches a certain level, the dam slope becomes completely instability. Based on the relationship between the average cumulative time and PGA, the performance boundaries were determined combined with the discussion in Sect. 1.5 using a turning point method. The preliminary classification criteria for failure levels are as follows: when the safety factor is less than 1.0, indicating the onset of cumulative time which is corresponding to “sliding critical” in Table 8.5; mild to moderate damage (0–0.4 s), with 0.5 s as the boundary for moderate damage; cumulative time of 1.5 s as the boundary for severe damage; and 2 s is suggested to be the boundary of partial dam break. Therefore, it is preliminarily recommended to use the cumulative time thresholds of 0 s, 0.5 s, and 1.5 s as performance levels for the classification of high CRFD slopes damage, corresponding to mild, moderate, and severe damage levels, respectively. Furthermore, a multiple earthquake intensities-cumulative time-exceedance probability relationship table was established in Table 8.5, and the corresponding fragility curve is shown in Fig. 8.15, providing valuable references for performance-based dam slope stability and safety design.

A line graph of the P G A in units of gravity versus cumulative time in seconds. Values are approximated. 4 dashed vertical lines are plotted with a gap in between. A line increases from (0, 0), through (0.12, 0.4) to (2.34, 1). Another dashed line increases linearly from (1.49, 0.74) to (2.34, 1).

A multiline graph of the probability of damage in percentage versus P G A in units of gravity. It plots 3 curves of mild damage, moderate damage, and severe damage in an increasing trend.

Figure 8.16 shows the relationship between the cumulative slippage and PGA. As the seismic intensity increases to a certain threshold, the dam slope sliding suddenly occurs, and there is a significant increase in cumulative slippage at a specific PGA, followed by a stable increase. This indicates that when the seismic intensity reaches a certain level, the dam slope becomes completely instability. Based on the relationship between the average cumulative slippage and PGA, the performance boundaries were determined combined with the discussion in Sect. 1.5 using a turning point method. The preliminary classification standards for damage levels are as follows: mild damage occurs when the safety factor is less than 1.0, corresponding to “minimal sliding” in Table 8.6; mild to moderate damage (0–15 cm), with 20 cm as the moderate damage limit; 100 cm as the severe damage limit, which is also the classification criteria commonly used in the analysis of Sect. 1.5; and it is suggested to set 150 cm as the criterion for partial dam break. Therefore, it is preliminarily recommended to use cumulative slippage of 0, 20, and 100 cm as performance thresholds for high CRFD failure, corresponding to mild, moderate, and severe damage levels, respectively. It can be observed that this classification method is somewhat similar to that established based on the cumulative time. A multiple earthquake intensities-cumulative slippage-exceedance probability relationship table was established in Table 8.6, providing valuable references for performance-based dam slope stability and safety design. Additionally, it can be seen that both of the two methods based on the cumulative time and the cumulative slippage exhibit a certain degree of correlation from Fig. 8.17.

A line graph of the P G A in units of gravity versus cumulative slippage in centimeters. Values are approximated. 4 dashed vertical lines are plotted with a gap in between. A line increases from (0, 0), through (13, 0.5) to (280, 1).

A multiline graph of the probability of damage in percentage versus P G A in units of gravity. It plots 3 curves of mild damage, moderate damage, and severe damage in an increasing trend. 3 dashed lines follow the curves with slight variations.

In this section, the coupling randomness of seismic motion and shear strength parameters are considered and 144 sets of non-stationary acceleration time histories as well as the random shear strength parameters \({\varphi}_{0}\) and \(\Delta\varphi\) are generated. 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. 1440 acceleration time histories were generated, with PGA = 0.1 to 1.0 g with a 0.1 g interval, and each level had 144 stochastic ground motions. Subsequently, the GPDEM, reliability probability analysis, and fragility analysis are applied to study the impact of coupling randomness on the stability of high CFRD from a perspective of stochastic dynamics and probability. This further improves the performance-based seismic safety evaluation system for dam slope stability.

Figure 8.18 illustrates the time history of mean and standard deviation of the safety factor with PGA = 0.5 g, which shows that the influence of the coupling randomness of seismic motion and material parameters on the safety factor is essentially consistent with the influence of seismic motion randomness. However, it differs significantly from the influence of material parameter randomness. The discrete distribution and exceedance probability of the minimum safety factor with PGA = 0.5 g shows that the response caused by the coupling randomness has minor differences compared to the response induced by seismic motion randomness, but substantial differences compared to the response affected by material parameter randomness, seen from Fig. 8.19. These results demonstrate that seismic motion randomness plays a dominant role in the safety factor response.

2 multiline graphs of the mean and standard deviation versus time. A, 2 lines move horizontally. Another line fluctuates throughout. B, 3 lines increase, fluctuate, and decrease.

2 graphs. A, a scatterplot of the minimum safety factor versus numbers. 3 types of plots fluctuate along a horizontal solid line. B, a line graph of the exceedance probability versus minimum safety factor plots 3 decreasing lines.

Figure 8.20 illustrates the discrete distribution and exceedance probability of the cumulative time of \({\text{F}}_{\text{S}}{<1.0}\) with PGA = 0.5 g and it is evident that the coupling randomness has a significant impact on the cumulative time of \({\text{F}}_{\text{S}}{<1.0}\), showing notable differences compared to the influence of seismic motion randomness alone. This indicates that the shear strength parameters have a substantial effect on the cumulative time of \({\text{F}}_{\text{S}}{<1.0}\). Therefore, it is crucial to fully consider the coupling randomness for the performance-based seismic safety evaluation of dam slopes.

2 graphs. A, a scatterplot of the cumulative time versus numbers. 3 types of plots are clustered and mostly spread across near the x-axis. B, exceedance probability percentage versus cumulative time in seconds plots 3 decreasing curves.

Figure 8.21 illustrates the exceedance probabilities of cumulative time of \({\text{F}}_{\text{S}}{<1.0}\) with different seismic intensities, mainly distributed between 95 and 5% exceedance probabilities. Under the influence of the coupling randomness, the variability becomes more obvious. The numerical ranges are as follows: 0–0.34 s (0.3 g), 0–1.40 s (0.4 g), 0–2.18 s (0.5 g), 0–2.68 s (0.6 g), 0.06–3.15 s (0.7 g), 0.23–3.58 s (0.8 g), 0.48–3.87 s (0.9 g), and 0.68–4.16 s (1.0 g). These results provide valuable references for the performance-based seismic safety evaluation of dam slopes. Table 8.7 lists the exceedance probabilities for cumulative time under different seismic intensities, and the fragility curve based on the cumulative time of \({\text{F}}_{\text{S}}{<1.0}\) is derived, as shown in Fig. 8.22. Notably, there are discernible differences compared to the exceedance probabilities obtained only considering seismic motion randomness.

2 multiline graphs. A, the exceedance probability in percentage versus cumulative time in seconds plots 8 decreasing lines with varying elevations. B, the cumulative time versus P G A plots 4 increasing lines. Different shades of dot plots are stacked and increase from 0.0 to 1.0 on the x-axis.

A multiline graph of the exceedance probability in percentage versus P G A in gravity. It plots 3 curves of mild damage, moderate damage, and severe damage in an increasing trend. 3 dashed lines follow the 3 curves with slight variations.

Figure 8.23 illustrates the discrete point distribution and exceedance probabilities of cumulative slippage with PGA = 0.5 g, which indicates that the coupling randomness of ground motions and material parameters significantly influences the cumulative slippage, showing notable differences compared to the influences of seismic motion randomness and material parameter randomness alone. This indicates that the shear strength parameters have a large effect on the cumulative slippage. Therefore, it is crucial to fully consider the coupling randomness for the performance-based seismic safety evaluation of dam slopes.

2 graphs. A, a scatterplot of the cumulative slippage versus numbers. 3 types of plots are mostly clustered near the x-axis and scattered above. B, the line graph of the exceedance probability in percentage versus cumulative slippage plots 3 decreasing lines.

Figure 8.24 shows the exceedance probabilities of cumulative slippage and Fig. 8.25 illustrates the cumulative slippage under different seismic intensities, which are mainly distributed between 95 and 5% exceedance probabilities and they exhibit greater variability when considering the coupling randomness. The numerical ranges are as follows: 0–3 cm (0.3 g), 0–45 cm (0.4 g), 0–129 cm (0.5 g), 0–231 cm (0.6 g), 0–335 cm (0.7 g), 0–462 cm (0.8 g), 0–628 cm (0.9 g), and 14–777 cm (1.0 g). These results serve as a reference for performance-based seismic safety evaluations of dam slopes. Table 8.8 lists the exceedance probabilities for cumulative slippage under different seismic intensities, and the fragility curve based on the cumulative time of FS is derived, as shown in Fig. 8.26. Notably, there are discernible differences compared to the exceedance probabilities obtained solely considering the ground motion randomness.

8 multiline graphs of the exceedance probability in percentage versus cumulative slippage. A to H, 2 lines of the ground motion and coupling randomness follow a decreasing trend.

A graph of cumulative slippage in centimeters versus P G A in gravity. 4 lines follow an increasing trend. Different shades of stacked scatter plots for 8 vertical lines with a gap in between are plotted.

A multiline graph of the probability of damage in percentage versus P G A in units of gravity. It plots 3 curves of mild damage, moderate damage, and severe damage in an increasing trend. 3 dashed lines follow the curves with slight variations.

The correlation analysis of cumulative time and cumulative slippage shown in Fig. 8.27 reveals that considering the coupling randomness of ground motion and material parameters results in a stronger correlation between the two factors and a broader range of numerical variability. This indicates the necessity of incorporating the coupling randomness of ground motion and material parameters in the analysis of dam slope stability and seismic safety.

2 scatterplots of the cumulative slippage in centimeters versus cumulative time in seconds. A, 2 types of plots are clustered in the lower left and follow 2 dashed increasing lines. B, 2 types of plots follow 2 dashed increasing lines and are clustered in the center.