With the improvement in the efficiency and accuracy of finite element numerical calculations, the nonlinear time history analysis method has gradually become the mainstream seismic safety assessment approach in the field of earth-rock dam engineering.
3.1 Introduction
With the improvement in the efficiency and accuracy of finite element numerical calculations, the nonlinear time history analysis method has gradually become the mainstream seismic safety assessment approach in the field of earth-rock dam engineering. However, seismic loads exhibit significant uncertainties, leading to a certain level of randomness in the seismic response of high CFRD. The results obtained from seismic time history analysis vary significantly with different ground motions or intensity levels. Therefore, relying on only a few seismic records makes it challenging to comprehensively understand the seismic performance of high CFRD. On the other hand, as the theory of performance-based seismic safety assessment continues to evolve, there is a need to understand the performance levels and seismic safety of complex and critically important engineering structures like high CFRD under different seismic intensities. Traditional deterministic analysis methods are insufficient for meeting such requirements. Thus, there is a need to delve into the performance levels of dams under future seismic actions from the perspective of fragility. The purpose of probability analysis is to predict the probability of high CFRD reaching various performance levels under different ground motions or intensity levels. It is a crucial component of performance-based seismic safety assessment, allowing for the consideration of uncertainties in structural seismic responses. Currently, there is limited research on the seismic safety assessment of rockfill dams, especially high CFRD, from a probabilistic perspective.
However, it is well known that many 200 m or even 300 m high CFRDs are under construction or planned in the western regions of China, and they are located in high-intensity seismic zones. Therefore, it is necessary to gradually improve their seismic safety assessment system and establish a performance-based seismic safety assessment method. However, the dynamic response and failure modes of high CFRDs under seismic action are complex. There are few reported instances of seismic damage to high CFRDs, especially from different seismic intensities, making the study of seismic damage modes and their quantitative description, analysis, and classification still in the preliminary research stage. Furthermore, the rockfill material of high CFRDs exhibits strong nonlinear characteristics, and seismic, especially strong seismic, actions have a significant impact on its response. Therefore, advanced numerical models need to be selected to simulate its dynamic response. Lastly, because probabilistic analysis requires many finite element dynamic time history analysis considering various uncertainties, it is challenging to analyze large-volume, strongly nonlinear high CFRDs or the analysis takes a long time. Therefore, precise and rapid probabilistic analysis methods need to be adopted.
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In this chapter, based on the above problems, the stochastic ground motion is fully considered, and the spectral expression-random function method is used to obtain the sample timescales of ground motion acceleration with rich probabilistic features in the same set system and the assigned probability corresponding to each ground motion. Through the high-performance geotechnical engineering nonlinear dynamic analysis program, GEODYNA, developed by the research team, a series of finite element elastoplastic dynamic analyses are conducted on high CFRDs. Combined with the generalized probability density evolution method, it obtains information on random dynamic responses and probability distributions. This forms the preliminary framework for a performance-based seismic safety assessment under random seismic actions. Firstly, it examines the random dynamic and probabilistic response patterns of several commonly used response indicators for high CFRDs based on elastoplastic analysis. This includes dam body acceleration, deformation, and panel stress. From the perspectives of random dynamics and probability, numerical distribution ranges for these response indicators under different seismic intensities are suggested, providing reference values for seismic design and ultimate seismic capacity analysis of high CFRDs. Subsequently, based on performance indicators, such as dam top settlement deformation and the panel demand-to-capacity ratio, combined with the super-stress duration, preliminary recommendations for corresponding performance levels are provided. The study establishes a seismic safety assessment method based on a multi-seismic intensity, multi-performance target, and exceedance probability performance relationship, along with fragility probability curves.
3.2 Computational Constitutive Models
This chapter primarily applies an improved generalized plasticity constitutive model for rockfill, based on the generalized plasticity theory, boundary surface theory, and critical state theory. This model can better reflect the actual conditions of soil, including cyclic hardening–softening, shear dilation-shrinkage, and particle crushing. It can analyze the entire process of seismic dynamic response and permanent deformation, making it theoretically more reasonable (Feng et al. 2010). On the other hand, to address the contact issues between the rockfill and the panel under dynamic loading, some contact constitutive models corresponding to the generalized plasticity model have also been proposed and developed, such as the generalized plasticity contact surface model (Liu et al. 2012).
3.2.1 Generalized Plastic Static and Dynamic Unified Model for Rockfill
To address the requirements of significant variations in the average principal stress, Dalian University of Technology has implemented a series of enhancements to the original generalized plasticity model (Xu et al. 2012; Zou et al. 2013; Liu et al. 2015) These improvements take into account the stress correlation between the elastic modulus and loading/unloading modulus of dam construction materials, enabling effective consideration of phenomena such as shear expansion, shear contraction, and cyclic cumulative deformation under dynamic conditions. The framework is clear and utilizes a set of parameters to facilitate static and dynamic simulation and analysis of rock pile materials.
The generalized elastoplastic stress increment of the generalized plasticity model can be expressed as:
The strain increment consists of both elastic strain increment \({\text{d}}{\upvarepsilon}_{\text{e}}\) and plastic strain increment \({\text{d}}{\upvarepsilon}_{\text{p}}\):
where \({\text{D}}^{\text{ep}}\) represents the elastoplastic matrix, which is influenced by factors such as the current stress state, stress level, stress history, loading and unloading directions, and changes in the microstructure of particles. L and U represent loading and unloading, respectively. \({\text{D}}^{\text{e}}\) represents the elastic matrix. \({\text{n}}_{{\text{g}}{\text{L}}}\) and \({\text{n}}_{{\text{g}}{\text{U}}}\) represent the plastic flow directions during loading and unloading, respectively, and they signify the direction of plastic strain increments. \({\text{n}}\) is the loading direction vector, representing the direction of the yield surface normal. \({\text{H}}_{\text{L}}\) and \({\text{H}}_{\text{U}}\) are the plastic moduli during loading and unloading, respectively.
The generalized plasticity P-Z model was primarily proposed to address the liquefaction problem in sandy soils. In the analysis of soil liquefaction, the variation in confining pressure is relatively small. However, with high earth-rock dams, there is a significant variation in the average principal stress within the dam body. When considering pressure dependency in the P-Z model, its parameters are greatly influenced by the average principal stress. Therefore, there are some limitations in the application of this model for static and dynamic analysis of high earth and rock dam. Dalian University of Technology has made improvements to the P-Z model, and the modified values of the plastic modulus includes loading and unloading modulus and elastic modulus are as follows:
$$ K = K_{0} p_{\text{a}} (p/p_{\text{a}} )^{{m_{\text{v}} }} $$
(3.5)
$$ G = G_{0} p_{\text{a}} (p/p_{\text{a}} )^{{m_{\text{s}} }} $$
(3.6)
In order to be able to better consider the hysteresis properties of the rock pile, the stress history function \({\text{H}}_{\text{DM}}\) is also modified for reloading as:
Meanwhile the cyclic densification effect has been considered, where \({\text{H}}_{\text{den}} = {\text{e}}^{{\upgamma}_{\text{d}}{\upvarepsilon}_{\text{v}}}\) represents the densification coefficient to consider the cyclic hardening characteristics of rockfill. Determine each parameter of the above model by particle swarm optimization algorithm, mainly including: G0, ms, K0, mv, αg, αf, Mg, Mf, H0, m1, β0, β1, HU0, mu, γd, γDM, γU seventeen parameters.
3.2.2 Generalized Plastic Interface Model
Liu et al. (2008) proposed a two-dimensional contact surface model based on the critical state and the generalized plasticity framework. Based on this model and the boundary surface theory, academician Kong et al. (2014) developed a static and dynamic unified two- and three-dimensional generalized plastic contact surface model for panel dams on the base of the above-mentioned generalized plasticity model of rockfill, which can effectively reflect the characteristics of the contact surface, such as shear expansion and contraction, hardening and softening, residual deformation, and particle crushing. As shown in Fig. 3.1, two boundary surfaces are defined on the normalized shear surface \(\frac{{\uptau}_{\text{x}}}{{\upsigma}_{\text{n}}} - \frac{{\uptau}_{\text{y}}}{{\upsigma}_{\text{n}}}\), which is approximated as a circle. These surfaces include the peak stress boundary surface and the maximum stress history boundary surface. Additionally, the maximum stress history boundary surface is also defined in \({\uptau}-{\upsigma}_{\text{n}}\) space:
Three schematics. A. A rectangular boundary surface is drawn on the 3-D plane of sigma n, versus tau x, versus tau y. B. Two concentric circular boundaries are drawn on a tau y upon sigma n versus tau x upon sigma n plane. A graph of tau versus sigma n plots the parabolic trend of the stress surface.
Fig. 3.1
Definitions of some basic concepts on the boundary surface, a stress–strain, b definition of boundary surface, c maximum stress surface in τ-σn space
then, the relationship between the stress increment and displacement increment at the contact surface under three-dimensional conditions can be obtained and expressed as:
where @@\({\text{d}}{\upsigma}={\left({\text{d}}{\uptau}_{\text{x}}\text{,d}{\uptau}_{\text{y}}\text{,d}{\upsigma}_{\text{n}}\right)}^{\text{T}}\) is the stress increment, \({\text{d}}{\upsigma}={\left({\text{d}}{\text{u}}_{\text{x}}\text{,d}{\text{u}}_{\text{y}}\text{,d}{\text{v}}_{\text{n}}\right)}^{\text{T}}\text{/}{\text{t}}\) is the strain increment, t is the thickness of the contact surface, generally equal to 5–10 times the average particle size.
\( \varvec{n}{:} \, {\text{d}}{\upsigma}^{\text{e}}={0}\) represents the neutral variable load, where \({\text{d}}{\upsigma}^{\text{e}} \, = \, {\text{D}}^{\text{e}}\text{d}\upvarepsilon\).
However, unlike the conventional generalized plasticity model, the stress state in the loading direction \({\text{n}}\) is determined when there is a backbend point by replacing the absolute stress state \(\stackrel{\text{-}}{\upsigma}\) by the stress state \({\upsigma}\) at the mapped point on the maximum stress boundary plane.
The model parameters include the elasticity parameter \({\text{D}}_{\text{n0}}\), \({\text{D}}_{\text{s0}}\); critical state parameter \({\text{e}}_{\uptau}\), \( \uplambda \), \({\text{M}}_{\text{c}}\); Plastic flow direction \( \upalpha \), \({\upgamma}_{\text{d}}\),\({\text{k}}_{\text{m}}\); Load Direction Parameters \({\text{M}}_{\text{f}}\); Plastic modulus parameters H0, k, fh; Particle Breaking Parameters a, b, c, c0 = 0.0001, Most of these parameters can be determined directly from test results.
3.3 Ground Motion Input Method
With the increasing height of the dam, the limitations of the traditional consistency input method, which cannot consider the radiative damping effect of infinite foundations and the influence of traveling wave effects, become more and more prominent. Therefore, scholars at home and abroad have carried out in-depth research on artificial boundaries (Lysmer et al. 1969; Deeks et al. 1994; Liu et al. 1998, 2005, 2006) among which viscoelastic boundaries have been widely used in simulating dam-foundation interactions due to their ability to simultaneously simulate the scattering effect of waves and the elastic recovery of semi-infinite foundations and their ability to overcome the low-frequency drift caused by viscous boundaries. In this chapter, ground motion inputs are implemented using viscoelastic artificial boundaries and equivalent nodal loads, as shown in Fig. 3.2.
A schematic diagram of a viscoelastic artificial boundary element. It shows a rectangular box with two springs and two dampers. The springs are labeled k n and k t. The dampers are labeled c n and c t.
Fig. 3.2
Viscoelastic artificial boundary element
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The boundary cell spring and damping coefficients are:
where \({\text{v}}_{\text{p}}\) is the P-wave speed,\({\text{v}}_{\text{p}}=\sqrt{ {2(1-}{ \upmu}{)} \, {(1-2}{\upmu}{)}{\text{v}}_{\text{s}}}\), \({\text{v}}_{\text{s}}\) is S-wave speed, \({\text{v}}_{\text{s}}=\sqrt{\text{G}/\uprho}\).
The equivalent nodal load calculation expression is:
where \({\dot{\varvec{u}}}_{\text{b}}^{\text{ef}}\),\({ \, {\varvec{u}}}_{\text{b}}^{\text{ef}}\), \({\varvec{R}}_{\text{b}}^{\text{ef}}\) are the velocity, displacement and force vectors induced by the free wave field at the boundary nodes, respectively; \({\varvec{K}}_{\text{b}}\), \({\varvec{C}}_{\text{b}}\) is the additional stiffness matrix and the additional damping matrix, respectively; and \({\varvec{F}}_{\text{b}}\) is the equivalent nodal load applied on the boundary.
3.4 Stochastic Dynamic Response and Probabilistic Analysis for High CFRDs
In this section, considering the randomness of ground motion, joint random ground motion generation method, generalized probability density evolution method, reliability probabilistic analysis method, susceptibility analysis method, and generalized plasticity model of rock pile material and contact surface, etc., stochastic dynamic and probabilistic analyses are carried out on a 250 m panel rockfill dam, to reveal the stochastic dynamic response law and damage probability under the action of different ground motion and different seismic intensities, to provide references for the seismic safety evaluation of CFRD based on their performance.
3.4.1 Finite Element Model and Material Parameter Information
The finite element grid of the concrete panel rockfill dam is shown in Fig. 3.3, with a dam height of 250 m, upstream dam slope gradient of 1:1.4, and downstream dam slope gradient of 1:1.6. The width of the top of the dam is 20 m, with a bedding zone and a transition zone below the panels, in which the width of the bedding zone is 3 m, the width of the transition zone is 4 m. According to “Code for Design of Concrete Face Rockfill Dams” (SL228-2013), the panel thickness is obtained as 0.30 + 0.0035 H) m, and H is the dam height. The panels are poured in three phases, 75, 150, and 250 m respectively. The dam body is filled in 50 layers, and the water is stored in 48 steps to 240 m. The unit is simulated by a quadrilateral unit, and the contact surface between the panel and the cushion layer is a Goodman element without thickness. The thickness and width of the dam foundation shall be 1/2 of the length of the dam bottom. After filling and impounding, perform dynamic calculations under seismic loads, in which static calculations provide initial stress, strain and displacement fields for dynamic calculations. The seismic input adopts the wave input method based on viscoelastic artificial boundary setting, and the hydrodynamic pressure on the panel is simulated by the additional mass method.
An F E mesh model of a C F R D. The dam is 250 meters high and 770 meters wide at the base. The dam is divided into a grid of squares. The diagram also presents the viscoelastic boundaries of the dam.
Fig. 3.3
Finite element mesh of the CFRD
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The generalized plastic model of rockfill materials is used to simulate the static and dynamic processes for rockfill materials, cushion materials, and transition materials, and the parameters are the numerical values from literature (Xu et al. 2012). Refer to Table 3.1. In the static and dynamic analysis, the contact surface between the panel and the cushion is simulated by the generalized plastic contact surface model, and the parameters are the values in the literature (Liu et al. 2014), refer to Table 3.2. Both are the dynamic parameters of the Zipingpu CFRD. The bedrock is simulated by a linear elastic model, and the parameters are density ρ = 2600 kg/m3, elastic modulus E = 2.0 × 104 MPa, and Poisson's ratio ν = 0.25. The panel is simulated by a linear elastic model, which is C30 concrete with a density of ρ = 2.40 g/cm3, an elastic modulus of E = 3.1 × 104 MPa, a Poisson’s ratio of ν = 0.167, and a concrete compressive strength fc = 27.6 MPa. The tensile strength under static and seismic dynamic loads is calculated using the formula suggested by Raphael (1984).
Table 3.1
Parameters of the generalized plasticity model of rockfill material
G0
K0
Mg
Mf
αf
αg
H0
HU0
ms
1000
1400
1.8
1.38
0.45
0.4
1800
3000
0.5
ms
mv
ml
mu
rd
γDM
γu
β0
β1
0.5
0.5
0.2
0.2
180
50
4
35
0.022
Table 3.2
Parameters of the generalized plasticity model of contact surface model
Ds0/kPa
Dn0/kPa
Mc
er
λ
a/kPa0.5
b
c
1000
1500
0.88
0.4
0.091
224
0.06
3
a
rd
km
Mf
k
H0/kPa
fh
t/m
0.65
0.2
0.6
0.65
0.5
8500
2
0.1
Based on the above non-stationary stochastic ground motion generation method, 144 ground motions were generated. By comparing with the target value, the error of several control values is still within 10%, which meets the needs of stochastic dynamics and probability analysis of CFRD. The vertical seismic acceleration is 2/3 of the horizontal seismic acceleration, and the horizontal bedrock peak acceleration is adjusted to 0.1–1.0 g, with an interval of 0.1 g. Input the ground motion and perform a series of finite element calculations. A total of 1440 working conditions need to be calculated for 10 seismic peak accelerations to obtain random dynamic information under the action of different intensities of ground motions. Then, based on the above numerical method, the probability density evolution equation is solved to obtain the probability information of seismic response of CFRD at each time. In the following, the influence of stochastic ground motion will be analyzed from the random dynamic response of dam body acceleration, dam body deformation, and face plate downslope stress, as well as the stochastic dynamic probability of these physical quantities.
3.4.2 Dam Acceleration
Figure 3.4 shows a cloud of the maximum horizontally oriented acceleration response based on a single sample and a cloud of the mean value of the response of 144 samples at 0.5 g seismic intensity. It can be seen that no matter based on the response obtained from a single sample or the average value of 144 sample responses, the acceleration response amplification in the crest area is the most obvious. In addition, the downstream dam slope is also a concentration of larger acceleration responses. However, the average distribution of the acceleration response is relatively regular and gentle, indicating that the stochastic seismic excitation has a huge impact on the maximum acceleration response of the dam body, and the stochastic seismic excitation makes the maximum acceleration response of each region tend to a constant value.
Two contour plots present the maximum horizontal acceleration response of a dam. Left. Response from a single sample. Right. Mean response of 144 samples. Both plots exhibit the highest response at the top and near an oblique side.
Fig. 3.4
Maximum horizontal acceleration response, a Single sample acceleration response, b 144 sample acceleration response mean
×
In order to further study the distribution law and influencing factors of the acceleration response of the dam body, Fig. 3.5 lists the distribution law of the maximum horizontal acceleration response with the dam height under the action of 0.2, 0.6 and 1.0 g ground motion intensity. We can see that the maximum horizontal acceleration responses caused by different ground motions are distributed differently along the dam height, but the overall trend is the same. A large turning point begins to appear at the dam height of 200 m, which is 0.8 H, and the magnification effect of the dam crest is abnormally obvious. It shows a strong “whipping effect” effect, and these characteristics are uniformly reflected in the distribution of the mean value along the dam height. From the diagram of acceleration magnification (Fig. 3.5d), we can see that as the intensity of ground motion increases, the magnification decreases, and the acceleration response changes more dramatically with the increase of seismic intensity. Under the seismic intensity of 0.1 g, the average acceleration at the dam crest is approximately three times that at the dam base. At PGA (Peak Ground Acceleration) of 0.2, 0.6, and 1.0 g, the average acceleration at the dam crest is about 2.9, 2.5, and 2.4 times that at the dam base, respectively. The maximum horizontal acceleration along the dam height and amplification factors under other seismic intensities are listed in Table 3.3 (values in parentheses indicate amplification relative to the dam base). Interestingly, as the seismic intensity increases, the amplification effect decreases.
4 multi-line graphs. A, B, and C. The height of the dam versus acceleration graph compares rising curves of mean, 16, 28, and 68 earthquakes. D. The height of the dam versus magnification graph compares rising trends of P G A equals 0.1, 0.2, 0.4, 0.6, and 1 g.
Fig. 3.5
Distribution of maximum horizontal acceleration along dam height, a PGA = 0.2 g, b PGA = 0.6 g, c PGA = 1.0 g, d different seismic intensities
Table 3.3
The mean of the maximum horizontal acceleration along the dam height and the magnification
Mean
PGA
0.1 g
0.2 g
0.3 g
0.4 g
0.5 g
0.6 g
0.7 g
0.8 g
0.9 g
1.0 g
Height of dam
0 m
0.56
1.13
1.69
2.26
2.81
3.38
3.98
4.54
5.13
5.69
50 m
0.58
1.15
1.72
2.27
2.80
3.33
3.86
4.37
4.88
5.37
100 m
0.65
1.28
1.89
2.48
3.06
3.63
4.18
4.70
5.23
5.72
150 m
0.79
1.55
2.26
2.91
3.51
4.08
4.65
5.17
5.71
6.24
200 m
0.78
1.52
2.20
2.85
3.44
4.03
4.63
5.19
5.87
6.54
250 m
1.70 (3.04)
3.28 (2.90)
4.72 (2.79)
6.03 (2.67)
7.23 (2.57)
8.44 (2.50)
9.75 (2.45)
11.04 (2.43)
12.32 (2.40)
13.68 (2.40)
Note 0 m dam height indicates the bottom of the dam, and magnification is indicated in parentheses
×
Figure 3.6 shows the time history of the horizontal acceleration response of the dam crest based on a single sample and the mean and standard deviation time history of 144 acceleration responses under the action of PGA = 0.5 g seismic intensity. There is a certain correlation between the time course of acceleration response and the time course of ground motion, and the mean value of acceleration of multiple ground motion tends to 0, indicating that the acceleration response has a strong variability, and the time course of the acceleration response varies a lot with different ground motion, so we should analyze the acceleration response of the CFRD based on the stochasticity of the ground motion. The time history of the standard deviation curve rises first, indicating that with the development of the nonlinear characteristics of the rockfill material, the variability of the acceleration response increases, and then decreases, partly because the variability of the acceleration response decreases with the decrease in the fluctuation amplitude of the time history of the ground motion. These changes prove that the acceleration response of CFRD is very sensitive to different ground motions, and shows that ground motion is a random process, so it is necessary to analyze the acceleration response from the perspective of random dynamics. Figure 3.7 shows the probabilistic evolution information of the dam top acceleration with time evolution under the action of PGA = 0.5 g seismic intensity, which can be obtained by the finite difference method in TVD format, including the probability density function at typical moments, the probability density evolution surface, and the probability density contour, which shows that the probability distribution of the acceleration response is irregular, and the probability density function surface is like the rolling peaks, which indicates that the acceleration response Rise and fall and evolve with time, also shows that the probability flows in the space state, which is better reflected in the probability density contour plots, and also shows that the acceleration response evolves with time with great variability.
Three-line graphs. A. An acceleration versus time graph depicts a trend fluctuating between negative 10 and 10. B. The mean versus time graph depicts a trend, almost stable at 0. C. The standard deviation versus time graph depicts a first rising then decreasing trend.
Fig. 3.6
Response, mean and standard deviation time-history of horizontal acceleration, a typical sample, b mean, c standard deviation
A. A probability density function versus acceleration depicts three first rising then decreasing trends for 12, 4, and 8 seconds. B. A 3-D surface plot presents the P D F of acceleration as a function of time and acceleration. C. A contour plot of acceleration versus time.
Fig. 3.7
Probability evolution information of horizontal acceleration in the dam crest, a PDF, b PDF evolution surfaces, c PDF contour
×
×
Figure 3.8 shows the maximum horizontal acceleration exceeding probability at different dam heights under the 0.6 g seismic intensity, and the distribution of acceleration along the dam height under several typical exceeding probabilities, which can be obtained by constructing a virtual probability density evolution process combined with the SUPG format finite element method. It can be clearly seen that under different exceeding probabilities, the "whipping effect" effect is present above the dam height of 0.8 H, and the exceeding probability of the acceleration at the dam crest is much higher than that at other elevations, and the amplification effect is obvious. The 50% exceedance probability acceleration distribution is basically the same as the mean value, indicating that the generated stochastic ground motion has strong statistical laws and probabilistic significance.
Two multi-line graphs. A. Exceedance probability versus acceleration depicts 6 falling trends for dam heights of 0, 50, 100, 150, 200, and 250 meters. B. Height of dam versus acceleration graph depicts 3 rising trends for 5, 25, and 50 exceedance probabilities.
Fig. 3.8
The exceedance probability of the horizontal maximum acceleration and the distribution of acceleration along the dam height, a acceleration exceedance probability at different dam heights, b distribution of acceleration along the dam height for typical overtopping probability
×
Figure 3.9a shows the discrete point plots of the maximum horizontal acceleration at the top of the dam under the seismic intensity of 0.6 g. The point distributions have a certain degree of discretization, and the maximum value reaches about three times the minimum value, which indicates that the generated acceleration response under ground motion is statistically more significant. From the distributions of the mean maximum acceleration, 95% exceeding probability and 5% exceeding probability under different seismic intensities (Fig. 3.9b), it can be seen that the maximum horizontal acceleration basically has a linear distribution, and the trend of acceleration can be predicted more accurately by linear fitting; on the other hand, the maximum acceleration of the dam roof should be between 95 and 5% exceeding probability, when the PGA is 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 and 1.0 g, respectively. 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0 g, the maximum horizontal acceleration response ranges from 0.9–2.61 m/s2, 1.88–4.83 m/s2, 2.99–6.70 m/s2, 3.63–8.62 m/s2, and 4.36–10.32 m/s2 respectively, 4.93–12.24 m/s2, 5.82–14.16 m/s2, 6.67–16.15 m/s2, 7.25–18.18 m/s2, and 8.12–20.09 m/s2, which can provide a reference for the seismic safety design of CFRD.
A. A scatter plot of maximum acceleration versus number, with a mean of 8.439 and a standard deviation of 1.422. B. A line graph of max acceleration versus P G A, with three rising lines representing mean, 95 percent exceedance probability, and 5 percent exceedance probability.
Fig. 3.9
Discrete point distribution of maximum acceleration response under different PGA. a Discrete point distribution of maximum acceleration (PGA = 0.6 g), b maximum acceleration at different seismic intensities
×
3.4.3 Dam Deformation
Figure 3.10 shows the horizontal residual deformation obtained based on a single sample at 0.5 g seismic intensity and the mean value of the response of 144 samples, while Fig. 3.11 shows the vertical residual deformation. It can be seen that both horizontal and vertical residual deformations occur at the top of the dam after the earthquake, whether based on the response of a single sample or the mean value of the response of 144 samples, and the distribution patterns of the response and the mean value of the response of a single sample are almost completely similar, which indicates that the effects of the ground motion on the deformation of the dam body of the various stripes of the law is basically similar.
Two triangular contour plots present the horizontal residual deformation. A. For a single sample. B. For 144 samples. The deformation values range from 0.0 to 1.0. The deformation is highest at the top of the triangle and decreases towards the base.
Fig. 3.10
Horizontal residual deformation, a single sample, b 144 samples
Two triangular contour plots present the vertical residual deformation. A. For a single sample. B. For 144 samples. The deformation values range from 0.0 to 1.2. The deformation is highest at the top of the triangle and decreases towards the base.
Fig. 3.11
Vertical residual deformation, a single sample, b 144 samples
×
×
Figure 3.12 shows the distribution pattern of horizontal residual deformation along the dam height for three seismic intensities of 0.1, 0.5 and 1.0 g. It can be seen that the horizontal residual deformation caused by different ground motions has different distribution patterns along the dam height, but the trend is basically the same, it gradually increases along the dam height and reaches the maximum value at the top of the dam, which is consistent with the conclusion of the above mentioned distribution pattern of cloud diagrams, and the maximum horizontal residual deformation occurs at the top of the dam, but under the effect of strong earthquakes such as the 1.0 g earthquake the horizontal residual deformation does not appear completely at the top of the dam. Residual deformations caused by different ground motion are more discrete, and different seismic intensities also affect their distribution patterns, indicating that horizontal displacements are more sensitive to the effects of ground motion. Figure 3.13 shows the distribution of vertical residual deformation along the dam height under three seismic intensities of 0.1, 0.5 and 1.0 g, which is basically similar to that of horizontal residual deformation. The above analysis shows that the deformation of the dam body is more sensitive to the effect of ground motion, and the deformation dynamic response of CFRD should be analyzed from a stochastic point of view.
4 multi-line graphs. A, B, and C. The height of the dam versus horizontal residual deformation graph compares rising curves of mean, 16, 28, and 68 earthquakes. D. The height of the dam versus horizontal residual deformation graph compares rising trends for 10 P G A values.
Fig. 3.12
Distribution of horizontal residual deformation along dam height, a PGA = 0.1 g, b PGA = 0.5 g, c PGA = 1.0 g, d different seismic intensities
4 multi-line graphs. A, B, and C. The height of the dam versus vertical residual deformation graph compares rising curves of mean, 16, 28, and 68 earthquakes. D. The height of the dam versus vertical residual deformation graph compares rising trends for 10 P G A values.
Fig. 3.13
Distribution of vertical residual deformation along dam height, a PGA = 0.1 g, b PGA = 0.5 g, c PGA = 1.0 g, d different seismic intensities
×
×
Figure 3.14 shows the horizontal displacement of the dam roof based on a single sample and the mean and standard deviation of 144 samples under PGA = 0.5 g seismic intensity, and Fig. 3.15 shows the vertical displacement. As seismic ground motion changes, dam crest deformation continuously increases and eventually stabilizes, exhibiting cumulative effects due to the plastic properties of the dam materials. However, it also shows slight fluctuations, indicating that the dam materials possess certain elastic properties and reloading process. The final stability of the mean value of the deformation is better, indicating that the generated stochastic ground motion is statistically better; the standard deviation generally increases, indicating that the variability of the deformation of the dam crest gradually increases with the development of the nonlinear behavior of the rockfill material; from the mean and standard deviation, the deformation is more sensitive to different ground motions. Figure 3.16 shows typical momentary probability density curves, probability density evolution surfaces, and probability density evolution contours for the horizontal displacement of the dam roof at 0.5 g seismic intensity, and Fig. 3.17 shows the probability information of vertical displacement. The probability distribution of deformation is not normal or lognormal as often assumed, but an irregular probability curve, the probability density surface shows the characteristics of peaks “high and low”, and the contour lines flow like “water”, which are the results of the irregular flow of probability in space. This is the result of the irregular flow in space, and the deformation evolves with time, which also reveals the transmission process of probability statistical information, and shows the sensitivity of deformation to different ground motion, and it is necessary to analyze the seismic deformation response of CFRD from the viewpoint of stochastic dynamics.
Three-line graphs. A. A horizontal displacement versus time graph depicts a trend fluctuating between negative 0.5 and 1.75. B. The mean versus time graph depicts a rising trend from 0.0 to 1.0. The standard deviation versus time graph depicts a first rising then stable trend.
Fig. 3.14
Mean and standard deviation time history of horizontal displacement, a typical sample, b mean, c standard deviation
Three-line graphs. A. A vertical displacement versus time graph depicts a trend fluctuating between negative 0.5 and 1.75. B. The mean versus time graph depicts a rising trend from 0.0 to 1.0. C. The standard deviation versus time graph depicts a first rising then stable trend.
Fig. 3.15
Mean and standard deviation time history of vertical displacement, a typical sample, b mean, c standard deviation
A. A probability density function versus horizontal displacement graph depicts three first rising then decreasing trends for 12, 4, and 8 seconds. B. A 3-D surface plot presents the P D F of acceleration as a function of time and displacement. C. A contour plot of displacement versus time.
Fig. 3.16
Probability evolution information of horizontal displacement in the dam crest a, PDF, b PDF evolution surface, c PDF contours
A. A probability density function versus vertical displacement depicts three first rising then decreasing trends for 12, 4, and 8 seconds. B. A 3-D surface plot presents the P D F of acceleration as a function of time and displacement. C. A contour plot of displacement versus time.
Fig. 3.17
Probability evolution information of vertical displacement in the dam crest, a PDF, b PDF evolution surface, c PDF contours
×
×
×
×
The exceeding probability of horizontal and vertical residual deformation of the dam roof is obtained by solving the constructed virtual stochastic process and generalized probability density evolution equations by the finite element method in SUPG format, as shown in Fig. 3.18 From the exceedance probability curves of horizontal and vertical residual deformations under different seismic intensities, the exceedance probability of each deformation value can be obtained to provide a basis for performance-based seismic design and safety evaluation of CFRD.
2 multi-line graphs of exceedance probability versus vertical residual deformation of the dam roof. Both graphs present the decreasing trends for P G A = 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, and 1.0.
Fig. 3.18
Exceedance probability of horizontal and vertical residual deformation under different PGA. a horizontal residual deformation exceeding probability, b vertical residual deformation beyond probability
×
Figure 3.19a shows the discrete point distribution of the horizontal residual deformation of the dam roof under 0.5 g seismic intensity, the difference between the maximum and minimum values is large, the maximum value reaches about 5 times of the minimum value, but it is more concentrated near the mean value, which indicates that the generated horizontal displacement of the dam body under the action of the ground motion is more statistically significant. From the discrete point distribution of the horizontal residual deformation of the dam roof under different seismic intensities and the distribution curves of the mean value, 50% exceeding probability, 95% exceeding probability, and 5% exceeding probability (Fig. 3.19b), the horizontal residual deformation basically shows a linear distribution, and the trend of the horizontal residual deformation can be predicted more accurately by the fitting of the formula; and it is obvious that the curves of the mean value and 50% exceeding probability almost coincide, indicating that the generated random ground motion has a stronger statistical regularity. almost coincide, indicating that the generated random ground motion has a strong statistical regularity. On the other hand, the post-earthquake horizontal deformation of the dam roof should be distributed between 95 and 5% beyond probability, with the variation ranging from 0.09–0.29 m, 0.18–0.58 m, 0.27–0.89 m when the PGA is 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9 and 1.0 g, respectively, 0.37–1.21 m, 0.44–1.58 m, 0.56–1.92 m, 0.65–2.25 m, 0.68–2.58 m, 0.84–2.87 m, and 0.91–3.20 m, respectively, which can provide a reference for the seismic safety design of CFRD. Figure 3.20a shows the distribution point plot of the vertical residual deformation of the dam roof under 0.5 g seismic intensity, and the maximum value is about four times of the minimum value; from the discrete point distribution of the vertical residual deformation under different seismic intensities and the distribution curves of the mean, 50% exceeding probability, 95% exceeding probability, and 5% exceeding probability (Fig. 3.20b), with the increase of the seismic intensity, as the stacked rock material becomes more and more dense resulting in the change of distribution curve gradually becomes slower; when the PGA varies from 0.1 to 1.0 g, the range of post seismic vertical deformation of the dam roof should be 0.14–0.41 m, 0.28–0.79 m, 0.40–1.17 m, 0.53–1.51 m, 0.67–1.79 m, 0.79–2.07 m, 0.92–2.28 m, 0.94–2.52 m, 1.07–2.66 m, and 1.16–2.77 m, which provide references for the dam seismic design and ultimate seismic capacity analysis. Table 3.4 lists the horizontal and vertical residual deformations of the dam roof at 50, 95 and 5% exceeding probability.
A. A scatter plot of horizontal residual deformation versus number, with a mean of 0.957 and a standard deviation of 0.281. B. A line graph of horizontal residual deformation versus P G A, with 4 rising lines representing mean, 5, 95, and 50 percent exceedance probability.
Fig. 3.19
Horizontal residual deformation of dam crest under different PGA, a discrete point distribution of horizontal residual deformation (pga = 0.5 g), b horizontal residual deformation at different seismic intensities
A. A scatter plot of vertical residual deformation versus number, with a mean of 0.203 and a standard deviation of 0.256. B. A line graph of vertical residual deformation versus P G A, with 4 rising lines representing mean, 5, 95, and 50 percent exceedance probability.
Fig. 3.20
Vertical residual deformation of dam crest under different PGA, a discrete point distribution of vertical residual deformation (PGA = 0.5 g), b vertical residual deformation under different seismic intensities
Table 3.4
The horizontal and vertical residual deformation of dam crest based on different exceedance probability under different PGA
Exceedance probability situation
PGA
0.1 g
0.2 g
0.3 g
0.4 g
0.5 g
0.6 g
0.7 g
0.8 g
0.9 g
1.0 g
Horizontal displacement (m)
Mean
0.192
0.367
0.553
0.749
0.957
1.174
1.388
1.579
1.790
1.979
5%
0.292
0.584
0.886
1.205
1.580
1.920
2.246
2.577
2.872
3.20
50%
0.178
0.355
0.534
0.725
0.920
1.131
1.346
1.539
1.741
1.921
95%
0.089
0.179
0.273
0.370
0.440
0.560
0.653
0.683
0.838
0.910
Vertical displacement (m)
Mean
0.268
0.518
0.762
0.990
1.203
1.401
1.574
1.707
1.845
1.948
5%
0.408
0.788
1.171
1.505
1.791
2.071
2.284
2.518
2.662
2.774
50%
0.262
0.506
0.744
0.965
1.178
1.362
1.551
1.686
1.823
1.927
95%
0.142
0.278
0.396
0.525
0.674
0.786
0.917
0.935
1.073
1.160
×
×
3.4.4 Panel Stress
Figure 3.21 shows the distribution pattern of panel downslope stresses along the dam height of 0.4 g seismic intensity (where tensile stresses are positive and compressive stresses are negative). The distribution pattern of stress response along the dam height caused by different ground motions is different, but the trend is basically the same; the average value of the downslope tensile stress is mainly concentrated in the range of 100–250 m of the dam height, and reaches the maximum value around 0.75 H of the dam height. From the distribution of stress averages along the dam height at 3, 9 and 15 s moments, the panels are mainly subjected to compressive stresses and are more compressive at the bottom of the panels, which are generally in an extruded state. The above analysis shows that, from a random perspective, the values and distribution patterns of panel stress response caused by different ground motions are quite different, and the stress response analysis of individual ground motions cannot effectively evaluate the change rule, reflecting the necessity of analysis based on the randomness of ground motions.
2 multiline graphs. A. A graph of the height of the dam versus downslope stress plots 5 falling curves for mean, 16, 28, and 68 earthquakes. b. A graph of dam height versus downslope stress depicts 3 falling trends for 3, 9, and 15 seconds.
Fig. 3.21
The maximum stress distribution along the dam height, a maximum downhill stress, b mean value of downslope stress at different moments
×
As a representative, Fig. 3.22 gives the stress time course based on a single sample and the mean and standard deviation time course of the stress of 144 samples under 0.4 g seismic intensity. The ups and downs of the stresses show the open and closed state of the concrete; the characteristics of the mean stresses with time show that the panels are mainly in the compression state from the stochastic dynamic analysis considerations; the different ground motions have a great influence on the stress changes of the panels; the standard deviation of the stresses increases firstly and then decreases with time, which demonstrates the opening and closing of the concrete's linear properties, which is caused by the variation of the strength of the ground motions. Figure 3.23 illustrates the state of evolution of the stress probability density with time, which shows the sensitivity of the panel stresses to different ground motions.
Three line graphs. A. A downhill stress versus time graph depicts a trend fluctuating between negative 30 and 10. B. The mean versus time graph depicts a rising trend. C. The standard deviation versus time graph depicts a first rising then decreasing trend.
Fig. 3.22
Stress response, mean and standard deviation time history, a typical sample, b mean, c standard deviation
A. A probability density function versus vertical displacement depicts three first rising then decreasing trends for 12, 4, and 8 seconds. B. A 3-D surface plot presents the P D F of acceleration as a function of time and displacement. C. A contour plot of displacement versus time.
Fig. 3.23
Probability density evolution information of stress, a typical PDF curves, b PDF evolution surface, c PDF contours
×
×
3.5 Conclusion
In this chapter, the uncertainty of ground motion input is fully considered, and the random ground motion generation method, generalized probability density evolution method, probabilistic analysis of susceptibility, and elastic–plastic analysis of panel rockfill dams are jointly applied to reveal the stochastic dynamic response and probabilistic rule of change of the seismic process and after earthquake of CFRD based on the physical quantities of acceleration, deformation, and panel stresses of the dam, and the distribution change ranges of the physical quantities are proposed from the stochastic dynamics and probability points of view. Finally, considering the two evaluation aspects of dam body deformation and panel impermeable body safety, appropriate performance indexes are selected, different performance level classification standards are initially proposed, and the performance safety evaluation framework of multi-seismic intensity-multi-performance target-exceeding probability is established. The main work and conclusions are as follows:
(1)
From the perspective of stochastic dynamics, it is revealed that the maximum horizontal acceleration distribution of the dam body is mainly concentrated in the top of the dam and the downstream slope area, and above 0.8 H (H is the height of the dam) shows a strong “whip-sheath” effect, and the amplification multiples of the top of the dam are different under the action of different ground motion intensities; through the stochastic dynamics analysis and probabilistic analysis of strong statistical significance, it is shown that the maximum value is three times or even five times of the minimum value. Probability analysis shows that the randomness of ground motion has a large impact on the acceleration, deformation and panel stress response of the dam body, and the maximum value is three times or even five times of the minimum value, so it is necessary to comprehensively evaluate the seismic capacity of CFRD from the random point of view of ground motion; reveals the characteristics of the probability of the response of various physical quantities of the spatial irregularities in the flow of the characteristics of the probability of the distribution of the response of the response of the response of the response of the response of each physical quantity in the space is not the regular probability of distribution such as the characteristics of the normal distribution; based on the 5, 50% and Based on 5, 50 and 95% exceeding probability, the range of calculated values of different physical quantities is suggested, which provides a reference for the numerical calculation results and the analysis of ultimate seismic capacity of CFRD.
Uncertainty in rockfill material parameters also has an impact on the establishment of a unified performance safety evaluation standard for CFRD, so the stochastic dynamic response and probabilistic analysis results of CFRD should be further investigated on the basis of full consideration of ground motion stochasticity, material parameter uncertainty including ground motion-material parameter coupled stochasticity, and then the performance-based seismic safety evaluation framework should be improved.
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