2. Probability Analysis Method of Seismic Response for Earth-Rockfill Dams

A considerable number of actual seismic records have indicated the significant uncertainty in seismic motion characteristics. Therefore, a very limited number of computational samples can hardly capture the influence of various factors related to seismic motion stochastic characteristics on the response of earth-rock dam structures, such as the frequency content, amplitude variations, peak values, duration, and arrangement order of different-amplitude pulses within the seismic motion. As early as the 1990s, scholars began exploring the stochastic dynamic response and reliability of earth-rock dams under stochastic seismic excitation. For example, Yu et al. (1993) generated 100 artificial earthquake waves through random simulation, introduced the concept of a state point, employed dynamic programming to locate the most probable sliding surface, and calculated the probability of permanent displacement of earth-rock dams. Chen et al. (1995) used a power function to describe the variation of average shear modulus with dam height and proposed a simplified method for analyzing the stochastic seismic response of heterogeneous earth dams based on a one-dimensional shear beam model. Liu (1996) and Liu et al. (1996) simulated the seismic process as a stationary Gaussian filtered white noise process, utilized the equivalent nodal force method, conducted random seismic response analysis, established an analysis method for the average permanent deformation failure probability of earth-rock dams, and developed a nonlinear random response and dynamic reliability analysis method based on the theory of random vibration and the virtual excitation method. They validated the method's rationality through numerical examples related to earth-rock dams. Shao et al. (1999) simulated the seismic action process as a zero-mean stationary Gaussian process, conducted random seismic response analysis to calculate the stochastic dynamic response of earth-rock dams, and employed the Hook-Jeeves search method to locate the most dangerous sliding plane and the minimum safety factor in the mean sense. This was compared with shake table model tests to validate the effectiveness and rationality of the approach. Wang et al. (2006) proposed a simple seismic motion model that simulates stationary random processes based on stochastic process theory. They investigated the seismic response characteristics of an actual homogeneous earth dam under random load excitation.

Earth-rock dams are constructed using natural materials, and their properties vary naturally, making them complex with diverse physical and mechanical characteristics. Additionally, their gradation ranges widely, and on-site construction is primarily controlled by the void ratio. It's challenging to precisely control gradation, leading to significant variability and uncertainty in deformation and strength parameters of the dam materials (Wichtmann and Triantafyllidis 2013). However, both deformation and strength parameters have a substantial impact on the numerical analysis results of seismic dynamic responses for high-faced rockfill dams. Currently, in the seismic probabilistic analysis of earth-rock dams, most efforts have been directed towards the uncertainty of seismic motion, with fewer studies investigating the influence of uncertainty in dam construction materials on dam dynamic responses. Zhang and Liu (1994) collected physical and mechanical test data for construction materials from 95 dam engineering projects in China. They mainly established a statistical database based on c and φ parameters and developed a comprehensive probability statistical analysis program, resulting in many valuable outcomes. Wu et al. (1991) considered the uncertainty of rockfill mass, dynamic modulus, and damping. They combined frequency domain analysis and perturbation method in structural dynamic response analysis to study the variation of dam slope displacement. Sanchez et al. (2014) utilized the Karhunen–Loeve expansion method to simulate the stochastic fields of rockfill and dam foundation materials, and solved for the random slope displacement of an earth-rock dam. Kartal et al. (2010) using the example of the Torul rockfill dam, applied an improved response surface method to consider the uncertainty of material and geometric properties of panels and rockfill. They explored the crack resistance and compression reliability of panels with different thicknesses under seismic actions. Wang et al. (2013) combined the mechanical parameter samples of dam construction materials using orthogonal design. Using dam crest settlement as an evaluation indicator, they studied the vulnerability of earth-rock dams. Yang and Zhu (2016) proposed a combined technique of stochastic field simulation and finite element method in the stochastic finite element method. They discussed the effects of spatial uncertainty in dry density, void ratio, coefficient of non-uniformity, average particle size, Duncan-Chang E-B model modulus coefficient, and initial friction angle on the random seismic response and permanent deformation of earth-rock dams.

Generally speaking, the factors influencing the structural performance requirements can be represented by two random comprehensive variables, namely the comprehensive effect S of the structure and the comprehensive resistance R of the structure. Therefore, the functional expression of a structure reaching a certain limit resistance can be quantified as:

Therefore, Z is also a random variable. When Z > 0, the structural response is within the safe region, indicating that the functional safety requirements are satisfied, and this quantified indicator is represented by P_f. When Z < 0, the structural response is outside the safe region, signifying structural functional failure, and this quantified indicator is represented by P_f. Since the factors influencing S and R are a series of more fundamental random variables (such as the strength parameters of earth-rock dams, seismic loads, displacement responses, etc.), let these fundamental random variables be X₁, X₂, …, X_n. Then, the general form of the functional expression can be represented as:

The limit state equation is:

The basic principle of the First-order Second-moment Method (FOSM) (Hasofer and Lind 1974) is to expand the limit state function at a specific point using a Taylor series, considering the uncertainty of random variable distributions. The linearized first-order term is selected, and the mean and standard deviation of the random variables are used to calculate reliability indices. Expanding the limit state function at the point \(X_{0i} {(}i = 1,2, \, \ldots , \, n{) }\) using a Taylor series, we have:

In order to derive the linear limit state equation, only the first-order term is approximated:

\(\left( {\frac{\partial g}{{\partial X_{i} }}} \right)_{{X_{0} }}\) represents the value of that derivative at point \(X_{0i} (i = 1,2, \, ..., \, n) \, \). Equation (2.5) is a commonly used formula for linearizing the functional expression in reliability analysis.

From the above fundamental principles, it can be seen that the first-order second-moment method, including its various improved forms, is straightforward and computationally convenient. However, for complex nonlinear problems, its accuracy is difficult to ensure. Convergence of iterations is often problematic, and it is no longer applicable to engineering structures that frequently involve implicit functions or complex state functions. In conclusion, it is challenging to apply this method to probabilistic analysis of large and complex nonlinear structures.

The Monte Carlo method is a numerical computation approach that involves generating random samples of variables based on different probability distribution characteristics. These random variable samples are then used as inputs to obtain samples of a functional function. By counting the number of samples that result in failure or destruction, the method estimates the probability of failure. This approach has a clear conceptual framework, is easy to use, and finds wide application in the field of probability analysis. As the number of simulations increases, the accuracy of the Monte Carlo method improves, but the computational complexity also increases significantly. This makes it challenging to apply to practical engineering scenarios, especially for large-scale and strongly nonlinear projects such as high earth-rock dams. Nonetheless, it is often used to validate the accuracy of other new probability analysis methods.

To enhance the efficiency of numerical simulation methods, various techniques have been developed based on the Monte Carlo foundation, such as importance sampling, subset simulation, line sampling, directional sampling, and Latin hypercube sampling. While these methods improve computational efficiency, they require the determination of design points for limit state functions to obtain important density functions. However, obtaining design points for limit states can be difficult in practical engineering, especially for complex projects, presenting significant challenges to widespread adoption. The estimated probability of failure obtained using the Monte Carlo method can be summarized briefly as follows:in which, n_f represents the number of samples in the failure region, and n is the total number of samples.

For large and complex structures, the limit state function is often implicit. The response surface methodology (RSM) involves constructing a simple explicit function that progressively approximates the actual implicit (or explicit) limit state function. This simplification aids in the calculation of probabilistic reliability and has gained increasing attention in recent years. In 1951, Box and Wilson (1951) first introduced the response surface methodology, focusing on how to use statistical methods to obtain an explicit function for approximating a complex implicit function. Wong (1985) was the first to apply the response surface methodology to analyze the reliability of slope stability. Subsequently, numerous scholars both domestically and internationally have further expanded the research related to this methodology. Based on random variables, the form is as follows:

Determining the constant a and constant matrices B and C requires an adequate number of sample points.

Regarding the reliability analysis of structures, the associated random variables are often numerous. To determine the unknown coefficients in Eq. (2.7), a substantial number of sample points need to be analyzed and computed. This can impact the computational efficiency of the response surface methodology. In order to achieve both sufficient computational accuracy and improved efficiency, in 1990, Bucher and Bourgund (1990) introduced the quadratic response surface function, which has been widely utilized. Its form is as follows:

In the equation: coefficients a, b_i, and c_i need to be determined by obtaining a sufficient number of equations using 2n + 1 sample points. However, since the interaction terms do not appear in the response surface function, the number of sample points required for determining the response surface function decreases. The specific calculation method is as follows:

The first step involves considering the mean as the central point and selecting sample points within the interval \({(}m_{\text{X}} - f_{{\sigma_{\text{X}} }} , \, m_{\text{X}} + f_{{\sigma_{\text{X}} }} {)}\). Literature suggests that a suitable value for f is in the range of 1–3. Utilizing the chosen sample points yields 2n + 1 values of the function g(X). Subsequently, the unknown coefficients in the response surface function can be calculated. Once the response surface function is obtained, approximate values for the design check points X_D on the limit state surface can be calculated.

The second step involves selecting a new center point, X_M, which can be chosen from the line connecting the mean point m_X and X_D, while ensuring the validity of the limit equation, \(g{(}X{)} = 0\), namely:

This selection of a new center point aims to ensure that the original limit state surface's information is encompassed by the chosen sample points as much as possible.

The third step involves considering X_M as the center point to select a new set of sample points and then repeating the process from the first step. This will yield numerical values for the design check points on the limit state surface and related reliability indicators. The entire procedure requires solving for the values of 4n + 3 functions g(X). Figure 2.1 illustrates the schematic diagram of the entire approximation process.

Two mathematical diagrams. Both depict a curve labeled g X = 0 and a dashed curve representing g dash X = 0. The diagrams include points marked on the curves, along with horizontal and vertical lines extending from the axes to these points.

However, in traditional quadratic response surface methods, when computing nonlinear functional functions, there are often issues with convergence failure and significant errors. Many scholars have proposed improved response surface methods, as well as intelligent response surface methods like Kriging (Luo et al. 2012), Artificial Neural Networks (ANN) (Cho 2009), Radial Basis Functions (RBF) (Deng et al. 2005), and Support Vector Machines (SVM) (Zhao 2008), to enhance the accuracy of structural probability reliability analysis. However, this also increases the complexity of the solution process. Additionally, for highly nonlinear structures such as high earth-rock dams, the effectiveness of sample training for these intelligent methods might not be ideal, and the improvement in accuracy for failure probability estimation might not be significant. It is important to note that first-order second-moment methods and response surface methods are generally used for the probability analysis of static systems. They are less frequently used in dynamic systems or are limited to simple structural probability analysis. Moreover, they cannot adequately capture the stochastic dynamic response process.

In conclusion, the first-order second-moment method, although conceptually simple and computationally convenient, often lacks accuracy and struggles with convergence for complex problems. It may not guarantee precision and is unsuitable for solving engineering structural probability with implicit functions or complex state functions. The Monte Carlo method is versatile, but it's computationally inefficient and challenging to apply to complex real-world engineering problems. However, it can be used to validate the accuracy of other probability analysis methods. The response surface method is primarily employed for solving probability analysis problems with implicit state functions, but it has drawbacks such as convergence failures and significant errors. Some of its improvement methods often suffer from suboptimal sample training results and increased complexity in the solution process.

It is important to emphasize that the first-order second-moment method and the response surface method are generally used for the probability analysis of static systems. They are less frequently used in dynamic systems or are limited to simple structural probability analysis, and they cannot capture the stochastic dynamic response process. In recent years, the probability density evolution method based on stochastic vibration theory has made significant contributions to the field of structural probability analysis. It decouples the solution from complex state functions by solving physical equations and probability density evolution equations. It can incorporate all probability information about structural response and has been successfully applied to structural probability analysis. This approach has played a prominent role in the development and application of structural reliability probability theory.

As is well known, external excitations, system parameters, and initial conditions of structural dynamic responses are all characterized by randomness. Structural dynamic responses can be regarded as stochastic processes, and their statistical probability characteristics are entirely determined by the aforementioned sources of randomness. Therefore, based on the principle of probability conservation, by establishing the probability density evolution rules from the source random factors to the target random responses, it is possible to solve the stochastic dynamic processes and reliability probabilities of complex nonlinear structures. This allows for a comprehensive understanding of their seismic performance under different levels of earthquake excitation.

Based on the knowledge of structural dynamics, the equation of motion for an n-degree-of-freedom system under dynamic loading excitation can be represented as follows:where, \(\overline{M}\) C, and K represent the effective mass, damping, and stiffness matrices of the structure, with their fundamental parameters possibly exhibiting randomness; \({\mathbf{\ddot{X}}}({\text{t}} )\), \({\dot{\mathbf{X}}}({\text{t}} )\) and \({\mathbf{X}}({\text{t}} )\) are the acceleration, velocity, and displacement vectors of the structural response, respectively; \({\mathbf{\ddot{X}}}_{\text{g}} ({\varvec{\Theta} ,}\;{\text{t}} )\) is the stochastic dynamic excitation process; Θ is the random vector within the entire system, and the solution of Eq. (3.​1) uniquely and continuously depends on Θ. For convenience, the solution of Eq. (2.10) can be written as:where, H = (H₁, H₂, …, H_n)^T. The velocity time history can be expressed as:where, h = (h₁, h₂, …, h_n)^T.

Therefore, more generally, any physical quantity of response in the structural system, such as deformation, displacement, velocity, acceleration, and stress, uniquely and continuously depend on Θ. Denote the interested physical quantity as Z = (Z₁, Z₂, …, Z_m)^T, then:where, h = (h_{z, 1}, h_{z, 2}, …, h_Z,m)^T.

Clearly, Eq. (2.13) can also be regarded as a stochastic dynamic system, where the source random factors are entirely described by Θ. Considering the augmented system formed by (Z, Θ), since all the probabilistic factors are encompassed, it represents a conservative probabilistic system. The joint probability density function (PDF) of (Z, Θ) can be denoted as p_ZΘ(z, θ, t). According to the principle of probability conservation (Chen and Li 2009), the following equation can be obtained:

After undergoing certain mathematical manipulations (El Hami and Radi 2016), the above equation can be transformed into:

Combining Eq. (2.14), a more explicit conclusion can be drawn:

Therefore, the joint probability density function of Z(t), denoted as p_Z(z, t), is:

When considering only a specific response physical quantity, Eq. (2.16) degenerates into a one-dimensional partial differential equation, namely:

The dimensionality m of Eq. (2.16) is independent of the original physical system's degree of freedom n. Regardless of whether the source random factors originate from initial conditions, structural parameters, or external excitations, the governing equation takes the form of Eq. (2.16). Therefore, Eq. (2.16) is referred to as the Generalized Probability Density Evolution Equation, sometimes simply referred to as the Generalized Density Evolution Equation (GDEE). From the above derivation, it is evident that the physical law revealed by the GDEE indicates that, during the evolution process of a general dynamic system, the rate of change of the joint probability density function distribution of the generalized displacement (which can represent actual displacement, stress, deformation, etc.) with respect to time is proportional to the rate of change of the generalized displacement. The proportionality coefficient is determined by the instantaneous generalized velocity, implying that the process of probability density evolution follows strict physical laws. The initial and boundary conditions for Eq. (2.16) are as follows:where z₀ represents the deterministic initial value. Hence, solving the GDEE is a combination of solving physical equations and solving the Probability Density Evolution Equation. For some simpler problems, analytical solutions can be obtained using methods like characteristic lines. However, for the majority of engineering practical problems, numerical solutions are needed. From Eq. (2.17), it's apparent that this is a linear partial differential equation. To obtain a numerical solution for this partial differential equation, the coefficients of the equation need to be obtained, which are the derivatives of the physical quantity under consideration when {Θ = θ}. These derivatives can be acquired from the solutions of Eqs. (2.10) and (2.14). Therefore, the numerical implementation of the probability density evolution theory can follow the steps below, as shown in Fig. 2.2. The process of solving the GDEE can be divided into the following four steps:

A flowchart. It starts with selecting discrete representative points within a space, then increments q by 1. Next, random samples are inserted into a dynamic system. The G D E E is solved based on initial and boundary conditions. The results are summed up using a formula.

As seen, the solution of the Generalized Probability Density Evolution process is essentially based on the principle of probability conservation. It transforms the stochastic dynamic system into a series of deterministic physical equations that possess inherent probabilistic connections. Through the GDEE, it acquires the probabilistic information about the system's physical state. This process combines the solutions of a series of deterministic dynamic systems and the solution of the GDEE. This exactly embodies the fundamental idea that the evolution of probability density functions depends on the evolution mechanism of the physical system's state. Moreover, the solution of the dynamic system and the solution of the GDEE are decoupled, avoiding the need for repetitive iterations, or solving implicit functions as in traditional reliability probability methods. This approach demonstrates good applicability for complex nonlinear dynamic systems.

The selection of discrete representative points in probabilistic space is one of the crucial techniques in the application of the probability density evolution method. Assuming a multidimensional probabilistic space Ω_Θ contains a set of discrete representative points θ_q = (θ_{1, q}, θ_{2, q}, …, θ_{s, q}), where q = 1, 2, …, n_sel. For each representative point, if we take the Voronoi region as the representative volume (as shown in Fig. 2.3), the probability within this representative volume corresponds to the assigned probability of that representative point:

2 illustrations. The left diagram depicts overlapping circles with dots inside and a shaded intersection area. The right diagram illustrates a Voronoi diagram with polygons surrounding points labeled. Both diagrams represent spatial relationships and partitioning within a plane.

where r_cv is the covering radius of the point set \({\mathcal{P}}_{{{\text{sel}}}} = \left\{ {{\varvec{\theta}}_{\text{q}} = (\theta_{{1,\text{q}}} ,\theta_{{2,\text{q}}} , \cdot \cdot \cdot ,\theta_{{\text{s},\text{q}}} );\;q = 1,2, \cdot \cdot \cdot ,n_{{\text{sel}}} } \right\}\). It can be easily deduced that:

This is the compatibility condition of assigned probabilities. In the probabilistic space, the selection of discrete fundamental point sets should have minimal deviations; based on the probability distribution type, suitable transformations are applied to the fundamental point set to minimize the F-deviation. The formula for calculating the F-deviation is:where \({\mathcal{F}}(x)\) is the joint distribution function and \({\mathcal{F}}_{n} (x)\) is the empirical distribution function of the representative point set. \(I\left\{ \cdot \right\}\) is the indicator function. Considering the assigned probabilities of the representative point set, the F-deviation in Eq. (2.29) can be further modified as:

Typically, discrete representative point selection can be achieved using methods such as the sphere-cutting method, lattice point method, number-theoretic methods, and the GF-deviation-based optimization method, significantly reducing the number of selected points. Here, we will briefly introduce the number-theoretic point selection method and the GF-deviation-based point set optimization method used in this paper.

In the expression, \(F_{i}^{ - 1} ( \cdot )\) is the inverse cumulative distribution function of the i^th random variable. Step 2: Transform each random variable so that the assigned probabilities of the sets of n points become mutually close. The assigned probabilities \(x_{q}^{ * } = (x_{q,\;1}^{ * } ,\;x_{q,\;2}^{ * } ,\; \ldots \;,\;x_{{q, i}}^{ * } )\) points are estimated:

To reduce the GF bias, the following transformation is applied:

Finally, \(x_{q}^{ * * } = (x_{q,\;1}^{ * * } ,x_{q,\;2}^{ * * } , \ldots ,x_{q,\;i}^{ * * } )\) represents the representative set of points used.

In some simpler cases, analytical solutions for the probability density evolution equation can be obtained. However, for complex multi-degree-of-freedom structural solutions, numerical methods are typically employed, such as various forms of finite difference methods and finite element methods. Here, we briefly introduce the Total Variation Diminishing (TVD) finite difference method and the Streamline Upwind Petrov–Galerkin (SUPG) finite element method used in this paper.

To solve the GDEE, the Total Variation Diminishing (TVD) finite difference method is often employed. Equation (2.21) can be discretized as follows:where:

Introducing the expression for flux limiter:

In the equation, \(u( \cdot )\) represents the Heaviside function, and \(\psi_{0} (r) = \max (0,\min (2r,1),\min (r,2))\). Research indicates that the difference scheme (Eq. 2.38) possesses the TVD (Total Variation Diminishing) property:where the total variation is:For the case of discrete curves:

The SUPG finite element method possesses better convergence properties and can obtain the probability density function at boundary locations more effectively. Based on similarity, Eq. (2.21) can be expressed as follows:where w = (1, α(x)) represents the velocity field, α(x) is the flux. and \(\Omega =\Omega_{\text{u}} \times \Omega_{\text{x}} \subset R^{2}\) denotes the computational domain of the finite element partition, where \( e = 1,2, \ldots ,N_{{\text{el}}}\). As the domain is two-dimensional, we can discretize it using quadrilateral elements.

According to reference (Elman et al. 2014), we can define the trial solution space as, where \({\mathcal{V}} = \left\{ {\left. p \right|p \in {\mathcal{H}}^{1} \left( \Omega \right),p = g\;\;on \, \partial \Omega } \right\}\) is the Sobolev space. We also define the space: \({\mathcal{V}}_{0} = \left\{ {\left. \Psi \right|\Psi \in {\mathcal{H}}^{1} \left( \Omega \right),\Psi = 0\;\;on \, \partial \Omega } \right\}\).The fundamental idea of Petrov–Galerkin approximation is to specify a weak formulation where the space of test (weighting) functions is different from that of the trial solution. More specifically, the test space is spanned by functions of the form given in Eq. (2.44):where \(\Psi \in {\mathcal{V}}_{0}\) is the Galerkin-type weighting function, τ is a coefficient, and element e is given by the following expression:

In the above expression, e represents the characteristic length of element \( \tau_{\text{e}} = \min \frac{{\lambda_{\text{x}} }}{\cos \vartheta }, \frac{{\lambda_{\text{u}} }}{\sin \vartheta }\), λ_x and λ_u denote the lengths of the rectangle in the x and u directions, respectively, where \(\vartheta =\arctan \left( {\left| {\frac{{\omega_{\text{u}} }}{{\omega_{\text{x}} }}} \right|} \right)\). The following equations will be denoted by subscript h to indicate the discrete finite element problem. The weak form of Eq. (2.43) is:

To further enhance the accuracy of the method, we incorporate the approach proposed in (Hughes et al. 1986) into Eq. (2.46), resulting in:Here, w_|| represents the projection of w onto the values on \(\nabla p^{\text{h}}\):

Large-scale structures, especially earth-rock dams, are quite intricate. Under seismic actions, they exhibit nonlinear and even strongly nonlinear effects. Nonlinear dynamic analysis is necessary. The traditional response spectrum method is no longer applicable, and seismic acceleration time history analysis must be employed to comprehensively understand the seismic response process. Conventional artificial synthesis methods for stochastic seismic acceleration time histories seldom consider the influence of non-stationary characteristics of seismic motion, which is unreasonable. Therefore, it is appropriate to use frequency-dependent non-stationary power spectral models to synthesize seismic acceleration time histories. In recent years, some researchers have introduced frequency-dependent non-stationary power spectral models into the field of hydraulic structures to study their effects on dam bodies. However, this area has been less explored in the context of earth-rock dams, especially high-panel block dams. Nonetheless, based on the characteristics of high-panel block dams, the study of non-stationary seismic stochastic dynamic excitation is one of the effective paths for the performance-based seismic safety assessment of high-panel block dams. In 1996, Deodatis proposed an evolving power spectral model for fully non-stationary seismic acceleration time histories based on the stationary Clough-Penzien power spectral model. In 2011, Cacciola and Deodatis improved the model. The bilateral evolving power spectral density function can be constructed using the following equation:

In the equation: A(t) is the intensity modulation function, recommended to be taken as (Cacciola and Deodatis 2011):

In the equation, c represents the time at which the peak ground acceleration (PGA) of the seismic motion occurs, and in this paper, it is set to 4 s; d is a parameter controlling the shape of A(t), and in this paper, it is set to 2. In the modulation function of the evolving power spectral density, the following parameters reflect its frequency-dependent non-stationary characteristics:

In the above equation, ω₀ and ξ₀ are the initial circular frequency and initial damping ratio, which can be determined by site characteristics. In this paper, they are taken as 25 and 0.45 rad/s, respectively. a and b are parameters determined based on site characteristics and seismic category. Their values are 3.5 and 0.3 rad/s, respectively. T represents the duration of the seismic acceleration time history, and its value varies depending on the site. Typically, for Site Classes I₀, I₁, II, III and IV, the values are taken as 12s, 15s, 20s, 25s, and 30s respectively. The spectral parameters that reflect spectral intensity can be expressed as:where \(\overline{a}_{\max }\) is the mean of the peak ground acceleration (PGA) of the seismic motion, and γ is the equivalent peak factor, taken as Eq. (2.7).

Building upon the generalized Clough-Penzien power spectral model, this paper employs the concept of stochastic processes to achieve the generation of non-stationary seismic motion using a spectral representation based on hydraulic seismic design codes. By combining this approach with the generalized probability density evolution method, the paper conducts refined stochastic dynamic response and seismic reliability analysis for high-panel block dams. Typically, non-stationary seismic acceleration random processes with zero mean can be generated using the following formula (Ou and Wang 1998):

In the equation, ω_k = kΔω, and \({\text{S}}_{{\ddot{X}_{{\text{g}}} }}\) is the bilateral evolving power spectral density function. At the frequency ω = 0, it should satisfy:

In Eq. (2.54), \(\{ X_{{\text{k}}} ,Y_{{\text{k}}} \}\) (k = 1, 2, …, N) are standard orthogonal random variables that satisfy the following fundamental conditions:

In the equation, E [⋅] represents the mathematical expectation, and δ_jk is the Kronecker delta. The relative error of the mean for simulating non-stationary seismic acceleration processes can be expressed as:

In the equation: ω_u = NΔω represents the truncation frequency, T is the duration of the non-stationary seismic acceleration process. Typically, the relative error of the mean ε(N) ≤ 1.0 is less than or equal to 1.0. In this paper, Δω is set to 0.15 rad/s, and the truncation term N is taken as 1600.

Then, based on the concept of stochastic processes, the random function expression of the standard orthogonal random vector \(\{ X_{{\text{k}}} ,Y_{{\text{k}}} \}\) can be constructed. Assuming that any two sets of standard orthogonal random vectors \(\overline{X}_{n}\) and \(\overline{{Y_{n} }}\) (n = 1, 2, …, N) are functions of two independent random variables Θ₁ and Θ₂, the random function can be denoted as:

In the equation, \(cas({\text{x}}) = \cos ({\text{x}}) + \sin ({\text{x}})\) is the Hartley orthogonal basis function. The fundamental random variables Θ₁ and Θ₂ are uniformly distributed and mutually independent in the interval [0, 2π], which can usually be obtained using number-theoretical methods. After a certain deterministic mapping, \(\{ X_{{\text{k}}} ,Y_{{\text{k}}} \}\) becomes the standard orthogonal basis random variable needed for Eq. (2.54) and is uniquely determined.

Taking Site I₁ as an example, an acceleration time history with a duration of 15s is generated. The discretized representative points Θ_1,i and Θ_2,i are obtained using number-theoretical methods. To reduce fitting errors, the evolving power spectral density can often be corrected using the following equation:

\(S_{{{\ddot{\text{X}}}_{{\text{g}}} }} (t,\omega )|_{{{\text{m + }}1}}\) and \(S_{{{\ddot{\text{X}}}_{{\text{g}}} }} (t,\omega )|_{{\text{m}}}\) are the evolving power spectra obtained after the m + 1 and m iterations, respectively. \(RSA^{{\text{S}}} (\omega ,\zeta )^{2} |_{{\text{m}}}\) is the average response spectrum obtained in the m iteration. \(RSA^{{\text{T}}} (\omega ,\zeta )^{2}\) is the target spectrum, often based on seismic design codes. Here, ω = 2π/T₀, T₀ is the natural period of vibration. ζ is the damping ratio, typically taken as 0.05 for earth-rock dams. ω_c is the cutoff frequency, which can be set as 1.57 rad/s. By substituting Eq. (2.60) into Eq. (2.54), a set of representative acceleration time histories can be generated. Generally, after a few iterations, the required accuracy can be achieved. To provide a detailed description of the differences between the generated acceleration time histories and the target values, the following relative error controls are commonly employed:

In the equation, ε_m represents the average relative error of the mean acceleration time history; ε_s is the average relative error of the standard deviation of the acceleration time history; ε_r is the average relative error of the response spectrum mean. X(t_k) and Y(t_k) are the mean and standard deviation of the target values at the k^th control point. \(\overline{X} (t_{k} )\) is the sample mean of the k^th control point, and \(\overline{Y} (t_{k} )\) is the sample standard deviation. N_m = 1500 is the number of time history discrete points. R(T₀, k) is the response spectrum obtained based on seismic design codes, such as hydraulic seismic design codes. \(\overline{R} (T_{{{0, }k}} )\) is the response spectrum of the generated samples. N_r = 400 is the number of discrete points for natural period. When generating 233 sets of acceleration time histories, the resulting errors are 4.8% for ε_m, 4.2% for ε_s, and 3.4% for ε_r. Figure 2.4 provides a comparison between the generated seismic motion sample mean, standard deviation, and response spectrum, and the target values (with \(\overline{a}_{\max } = 2\)). It can be observed that the fit is very good.

4 line graphs. a. A fluctuating line of acceleration versus time. b. 2 overlapping flat lines labeled target and 233 samples for mean versus time. c. 2 overlapping lines labeled target and 233 samples form a peak for standard deviation versus time. d. 2 overlapping lines labeled target and 233 samples form a concave up decreasing trend for beta T versus period.

This section verifies the accuracy and efficiency of GPDEM based on an undamped single-degree-of-freedom (SDOF) system. The equation of free vibration is as follows:where the initial conditions are as follows:

The natural frequency ω is a random parameter uniformly distributed in the interval \(\left[ {\frac{5\pi }{4},\;\frac{7\pi }{4}} \right]\), with initial values of \(x_{0} = 1.0\text{m}\) and \(\dot{x}_{0} = 0.0\text{m}\), respectively. When \(\dot{x}_{0} = 0.0\text{m}\), the displacement solution for the dynamic system Eq. (2.70) is:

Clearly, X(t) is a stochastic process. Ultimately, the probability density function (PDF) of the displacement and the second-order statistical time history (mean and standard deviation) can be obtained, as shown in Figs. 2.5 and 2.6. The excellent fit between the analytical solution and GPDEM demonstrates the high accuracy of this method.

A multiline graph of P D F versus displacement in meters. It plots 2 M-shaped overlapping curves for an analytical solution and G P D E M.

2 multiline graphs. 1. Mean versus time. 2 overlapping sinusoids with diminishing amplitude labeled G P D E M and analytical solution. 2. Standard deviation versus time. 2 overlapping sinusoids on a rising trend labeled G P D E M and analytical solution.

To verify the accuracy and efficiency of employing the GPDEM method in solving the stochastic processes and reliability of nonlinear dynamic systems under random seismic effects, further investigation is conducted based on the typical nonlinear vibration system, the Duffing oscillator (Sekar and Narayanan 1994). The response of the Duffing oscillator under random seismic effects can be expressed as follows:Here, where a₀ = 2.0 rad/s, ζ₀ = 0.05 and μ = 200 m represent the natural frequency, damping ratio, and nonlinearity coefficient, respectively. Finally, through comparison with the second-order statistical values obtained using the Monte Carlo Method (MCM) (Fig. 2.7) and the cumulative distribution functions (CDFs) at different time instances (Fig. 2.8), the efficiency and accuracy of GPDEM in dealing with stochastic processes and reliability of nonlinear structural responses under stochastic seismic excitations are verified.

2 multiline graphs. 1. Mean versus time. 2 overlapping flat trends for 10000 M C S and 1000 G P D E M. 2. Standard deviation versus time. 2 overlapping trends for 10000 M C M and 1000 G P D E M forms a peak.

A multiline graph of the C D F in % from 0 to 100 versus displacement from negative 0.010 to 0.010 in meters. It plots 2 sets of 2 overlapping lines crisscrossing each other. The first set 6 s has M C S and G P D E M. The second set 12 s has M C S and G P D E M.

High-dimensional random parameter samples were generated using the GF-bias optimization-based point selection method. By combining this approach with the GPDEM, a probabilistic analysis of multi-layered soil-rock engineering slopes considering the randomness of material parameters was performed. By comparing the results with those obtained using traditional Monte Carlo simulation methods, this approach proves to be effective in generating samples and conducting structural reliability analysis involving dozens or even scores of random variables. It demonstrates high efficiency and reliability. The probabilistic analysis employed deterministic seismic motion, with acceleration time histories shown in Fig. 2.9, where the peak ground acceleration PGA = 0.2g.

A line graph of acceleration from negative 2 to 2 meters per second square versus time from 0 to 15 seconds. It has a fluctuating waveform with several peaks.

The first example involves a multi-layered embankment, as described in reference (Reale et al. 2016). Figure 2.10 shows its finite element mesh, consisting of two layers of fill material, topped with a 10 m-deep layer of hard silty soil. The parameter details are provided in Table 2.1. The mesh comprises 3985 elements and 4107 nodes. All random parameters are assumed to follow independent normal distributions, totaling six random variables. Monte Carlo simulation was performed 10,000 times, while GPDEM simulation was carried out 600 times.

A schematic of a multi-layered embankment. It has 3 layers. Layer 1 has a height of 10 meters and an angle of 40 degrees. Layer 2 has a height of 9 meters and a width of 55 meters. Layer 3 has a height of 10 meters and a width of 70 meters.

Figure 2.11 presents the stochastic process of safety factors, including a comparison of means and standard deviations. Figure 2.12 showcases probability density functions and cumulative distribution functions at several representative time instances. Figure 2.13 illustrates the probability density function (PDF) and cumulative distribution function (CDF) of the minimum safety factor. The reliability based on the Monte Carlo Method (MCM) and GPDEM is 0.8159 and 0.8173, respectively. It can be observed that the two results are in good agreement, yet the GPDEM demonstrates higher efficiency.

3 line graphs. 1. A fluctuating line graph of safety factor versus time. 2. A line graph of mean versus time plots 2 overlapping fluctuating curves for M C M and G P D E M. The curves overlap each other. 3. A line graph of standard deviation versus time plots 2 overlapping fluctuating curves for M C M and G P D E M.

2 multiline graphs. 1. P D F versus safety factor. 3 pairs of overlapping lines for M C M and G P D E M form 3 adjacent peaks with overlap for 3 s, 8 s, and 12 s. 2. C D F versus safety factor. 3 pairs of overlapping lines for M C M and G P D E M form 3 adjacent sigmoid curves for 3 s, 8 s, and 12 s.

2 multiline graphs. A. P D F versus minimum safety factor. 2 overlapping lines for 10000 M C M and 600 G P D E M form a peak. B. C D F versus minimum safety factor. 2 overlapping lines for 10000 M C M and 600 G P D E M form a sigmoid curve.

The second example involves a complex four-layer soil profile, detailed in reference (Zolfaghari et al. 2005). Figure 2.14 depicts its finite element mesh. There are a total of eight random parameters, each following an independent lognormal distribution, as specified in Table 2.2. Monte Carlo simulation was conducted 10,000 times, while GPDEM simulation was performed 800 times.

A finite element mesh for case study 2 is displayed. The mesh is divided into four layers, each with different shades. The mesh is labeled with coordinates at the corners of each layer. The coordinates of the bottom layer are (0, 0) and (25, 0).

Figure 2.15 shows the stochastic process of safety factors, including a comparison of means and standard deviations. Figure 2.16 presents the probability density functions and cumulative distribution functions at several representative time instances. Figure 2.17 illustrates the probability density function and cumulative distribution function of the minimum safety factor. The reliabilities based on the MCM and GPDEM are 0.6701 and 0.6675, respectively.

3 multiline graphs. A. Safety factor versus time. B. Mean versus time. C. Standard deviations versus time. Each graph features a fluctuating trend for 10000 M C M and a flat trend for 800 G P D E M.

2 multiline graphs. A. P D F versus safety factor. It features 3 pairs of almost overlapping peaks. Each peak features 2 trends for M C M and G P D E M, namely 3 s, 8 s, and 15 s. B. C D F versus safety factor. It features 3 pairs of almost overlapping sigmoid curves. Each curve features 2 trends for M C M and G P D E M, namely 3 s, 8 s, and 15 s.

2 multiline graphs. A. P D F versus minimum safety factor. It features a pair of overlapping peaks comprising 10000 M C M and 800 G P D E M. B. C D F versus minimum safety factor. It features a pair of overlapping sigmoid curves comprising 10000 M C M and 800 G P D E M.

To further verify the effectiveness and efficiency of the GPDEM combined with the non-stationary seismic generation method for random dynamic simulation and probabilistic analysis in highly nonlinear structural panel-stacked rockfill dams, this section contrasts the results with those obtained using traditional MCM. It examines their influence on the dam crest acceleration, panel stress stochastic dynamic response, and associated probabilities.

The finite element mesh of the concrete panel-stacked rockfill dam is shown in Fig. 2.18. In the calculations, the elements use quadrilateral 4-node isoparametric elements, with a total of 3777 elements and 3795 nodes. The computations were conducted on a PC machine with an Intel i7 8-core CPU and 32GB of RAM. The dynamic calculation for a single sample takes about 3 min, while the stable calculation takes around 30 s. The dynamic water pressure is applied using the Westergaard additional mass method (Westergaard 1933). The considered panel-stacked rockfill dam has a crest width of 25 m, a height of 245 m, an upstream dam slope of 1:1.5, an upper downstream ramp slope of 1:1.7, and a lower downstream ramp slope of 1:1.4. The dam body consists of panels and five stacking areas: stacking area A, stacking area B, stacking area C, transition zone, and cushion zone. The reservoir water level is 225 m.

A meshed triangular shape is divided into three sections. The top section is labeled Rockfill A. The middle section is labeled Rockfill B. The bottom section is labeled Rockfill C.

The static analysis of the rockfill material uses the Duncan E-B nonlinear elastic model (Duncan and Chang 1970), and Table 2.3 provides the parameters for the static model. The dynamic analysis employs the Hardin-Drnevich equivalent linear viscoelastic model (Hardin and Drnevich 1972), with parameters as presented in Table 2.4. The peak ground acceleration of the seismic motion is adjusted to PGA = 0.4g, and a total of 377 and 5000 seismic records are generated using the number theoretic method and the Monte Carlo method, denoted as 377 GPDEM and 5000 MCM, respectively. Through a series of finite element dynamic time history analyses and solutions of probability density evolution equations, the dam crest acceleration and panel stress stochastic dynamic response second-order statistical values, as well as probability information, can be obtained.

The comparison of mean and standard deviation time histories of dam crest acceleration and panel dynamic stress (tensile stress is positive) based on the Generalized Probability Density Evolution Method (GPDEM) and the Monte Carlo Method (MCM) is illustrated in Fig. 2.19. By comparing 377 GPDEM samples with 5000 MCM samples, the mean and standard deviation obtained from 377 GPDEM closely match those from 5000 MCM. This demonstrates the effectiveness and accuracy of the GPDEM method, which requires fewer finite element samples and computation time.

4 multiline graphs plot 2 almost overlapping fluctuating trends each for 377 G P D E M and 5000 M C M. A. Mean in meters per second square versus time. B. Standard deviation in meters per second square versus time. C. Stress in megapascals versus time. D. Standard deviations in megapascals versus time.

However, the fluctuations in the mean and standard deviation time histories also indicate significant differences in various physical quantities for different seismic responses. These differences arise from the stochastic nature of seismic motion. Therefore, a comprehensive analysis of the seismic response of high concrete faced rockfill dams is needed from the perspective of random vibration. Nevertheless, it's worth noting that there is still a slight difference between the results of 377 GPDEM samples and 5000 MCM samples, as seen from the second-order statistical values.

A series of deterministic seismic response analyses were used to obtain the physical quantities of acceleration and panel stress, which were then inserted into the GPDEM equation. The probability information was obtained using the Total Variation Diminishing (TVD) scheme of finite difference method. Figure 2.20 shows the cumulative distribution functions at two representative time instances. It can be observed that the cumulative distribution functions obtained from 377 GPDEM samples and 5000 MCM samples fit well, confirming the high precision and effectiveness of GPDEM. It's also noticeable that the cumulative distribution functions change over time, indicating the significant influence of seismic motion on the seismic response of panel-stacked rockfill dams under the coupling effect of nonlinear rockfill behavior and stochastic seismic excitation. Therefore, the seismic response of high concrete faced rockfill dams requires analysis from a stochastic perspective.

2 multiline graphs plot 2 sigmoid trends each labeled 377 G P D E M and 5000 M C M. The trends form a crisscross shape. A. C D F in % versus acceleration in meters per second square. B. C D F in % versus stress in megapascals. 3.00 s and 10.00 s are marked on each graph.

Figure 2.21 presents the cumulative distribution functions of maximum acceleration and maximum stress obtained based on the virtual random process and solving the GDEE using the SUPG scheme of the finite element method. The comparison between the results of 377 GPDEM samples and 5000 MCM samples further validates the accuracy and efficiency of the GPDEM method. Additionally, due to the influence of the probability density evolution process, an improved level of accuracy is achieved.

2 multiline graphs plot 2 overlapping sigmoid trends each for 377 G P D E M and 5000 M C M. A. C D F in % versus maximum acceleration in meters per second square. B. C D F in % versus maximum stress in megapascals.

By comparing the failure probabilities or reliabilities, as well as the mean and standard deviation of responses, the results obtained from several hundred GPDEM simulations are within the same order of magnitude as those obtained from tens of thousands of Monte Carlo simulations. However, the efficiency of GPDEM is dozens or even several tens of times higher than the Monte Carlo method. This indicates the effectiveness and efficiency of this approach for large-scale geotechnical engineering stochastic and probabilistic analysis. In conclusion, considering the high accuracy and efficiency of the Generalized Probability Density Evolution Method, along with its strong theoretical foundation, it can be effectively employed for performance-based seismic safety assessment of high concrete faced rockfill dams, with the expectation of achieving favorable results.