Evaluation of seismic reliability of gravity dam-reservoir-inhomogeneous foundation coupled system

Abstract

The seismic performance of gravity dam-reservoir-foundation coupled system is investigated utilizing probabilistic approach. In this research, the uncertainties associated with modeling parameters are incorporated in nonlinear response history simulations to realistically quantify their effects on the seismic performance of the system. The methodology is applied to Pine Flat gravity dam and the foundation is considered to be inhomogeneous assuming a constant spatial geometry but with various rock material properties. The sources of uncertainty are taken into account in the reliability analysis using Latin Hypercube Sampling procedure. The effects of the deconvolution process, number of samples, and foundation inhomogeneity are investigated.

Keywords

gravity dams / dam-reservoir-foundation interaction / seismic reliability / inhomogeneous foundation / earthquake deconvolution

Introduction

Seismic damages imposed to Koyna dam, Hsinfengkiang dam, Sefid-Rud dam, and Shih-Gang dam have proven that earthquakes may damage or trigger the failure of constructed concrete dams. So, the seismic safety of existing concrete dams and the risks posed by earthquakes have been a growing issue in the last decades because many of the dams have been designed with outdated analysis procedures and with seismic loading assumptions that are now inadequate [¹]. The main potential seismic failure modes of concrete gravity dams are tensile cracking and sliding along jointed sections specifically at the dam-foundation interface [²,³]. Gravity dams are usually evaluated using deterministic analysis methods; however, probabilistic and reliability methods can be perceived, due to the sources of uncertainty presented in earthquake ground motions and in parameters describing the structural system, as complementary to deterministic analyses to support the decision-making process.

Therefore, the seismic reliability assessment is employed as a useful tool in dam safety. This method of analysis for concrete dams is in its early development stage and examples are scarce [¹,⁴–⁸]. It requires an analytical or numerical model that robustly captures the nonlinear structural behavior and explicit consideration of important sources of uncertainty. These uncertainties may be aleatoric (inherent randomness) or epistemic (lack of knowledge). In a nonlinear seismic time-history analysis, a primary source of modeling uncertainty lies in the definition of the analysis model parameters as compared to the components’ actual behavior [⁹,¹⁰]. Application of seismic reliability analysis to gravity dams requires identification of potential failure modes presented as limit-state (performance) functions and prediction of the conditional probability of limit-state exceedance under different earthquake events. This exceedance probability (EP) can be rigorously estimated using statistical techniques in probability analysis [¹¹].

Concrete gravity dams are in continuous interaction with their reservoir and foundation rock or soil. These media can significantly influence the seismic response of gravity dams during earthquakes [¹²,¹³]. Since the foundation rock is a natural medium, there is large uncertainty related to the foundation. Especially there may be various kinds of rocks with spatially varying properties separated by joints, fissures, and faults within the foundation [¹⁴]. The importance of the dam-foundation interaction and also inhomogeneity of the foundation on the seismic response of gravity dams have been previously studied [¹⁵–¹⁹]. The dam-foundation interaction can be appropriately modeled in a numerical finite-element analysis by including inertia, stiffness, and damping of the rock, assigning proper boundary conditions to the far-end boundaries of the foundation, and selecting proper earthquake input mechanism. Different boundary conditions and earthquake input mechanisms have been used to consider the effect of the local rock conditions on the earthquake response of dam-reservoir-foundation coupled systems [²⁰–²²].

In this study, an approach is utilized to evaluate the seismic reliability of gravity dam-reservoir-foundation coupled systems. In this approach, the uncertainties associated with modeling parameters are incorporated in nonlinear response history simulations to realistically quantify their effects on the seismic performance of the system. As an application example, the Pine Flat gravity dam is selected and the finite-element method is used to numerically model the dam-reservoir-foundation system including a base joint between the dam and the foundation. The foundation is considered to be inhomogeneous assuming a constant spatial geometry but with various rock material properties. The sources of uncertainty related to the material properties and other modeling parameters are taken into account in the reliability analysis using a sampling procedure to generate statistically significant 1000, 2000, and 4000 samples of the model. The system is analyzed under specific ground motions with different return periods according to operating basis earthquake (OBE), maximum design earthquake (MDE), and maximum credible earthquake (MCE) records of the dam site. The deconvolved-base-rock input model is utilized as earthquake input mechanism. Three different performance functions are defined and evaluated under the selected earthquakes. The effects of the deconvolution process, number of samples, and foundation inhomogeneity are investigated. Finally, the obtained results are compared with the first-order second-moment (FOSM) reliability method.

Utilized methodology

The utilized approach for seismic reliability assessment utilizes the performance-based earthquake engineering procedure combining robust numerical methods for obtaining the behavior of concrete dams with mathematical models of structural reliability analysis. In this procedure, the probability of exceeding various limit-states of the structural performance is computed employing a probabilistic framework through nonlinear time-history simulation [²³]. In summary, the uncertainty is the main framework of the analyses in this investigation and the finite element analysis is performed as the subset (Fig. 1). The reliability methods are implemented in uncertainty framework using MATLAB in order to generate sample matrix. The deterministic model (finite element model) is called externally by MATLAB and after the finite element analyses are completed, the responses enter the post-processing stage.

This methodology is illustrated more specifically in Fig. 2. In the first step, the case-study is selected. Then, proper structural performance functions and related limit-states should be defined. Several different metrics can be used to quantify the structural performance through limit-state functions. In the third step, suitable earthquake ground motions are picked out. They could be actual or artificial records, selected based on various design levels under seismic effects, such as definitions of OBE, MDE, or MCE according to concrete dam guidelines [²⁴]. Random variables along with their correlations and probability distribution functions are determined in the fourth step. The random variables are uncertain parameters whose values cannot be exactly determined; they are based upon scarce data coming from similar published cases [¹¹].

A variety of approaches can be used to study the effects of the modeling uncertainties on the structural performance. The proposed methodology is based on sampling technique, i.e., Monte-Carlo reliability method [²⁵,²⁶]. The number of Monte-Carlo simulations performed should be large enough to capture the searched probability. Because the nonlinear time-history analysis of gravity dams is itself time-consuming, the direct Monte-Carlo procedure can become computationally prohibitive. As a remedy, the number of samples can be reduced employing variance reduction methods such as Latin hypercube sampling (LHS) [²⁷]. As LHS is an improved sampling approach, using small number of realizations generated by LHS can lead to reliable outcomes [²⁸]. In addition, there are other improved sampling strategies such as quasi-random Monte Carlo which can be applied based on the deterministic model [²⁹]. In this study, LHS is considered as the reliability method. Realizations of modeling random variables are generated in the first step, which is used as input for a simulation model. The model’s finite element matrices are assembled based on each sample properties in the next step.

Before nonlinear time-history analysis of the model samples, the selected free-field earthquake ground motions should be deconvolved (Step 7) considering the realization of the rock foundation in the absence of the dam, because the ground motion acceleration is applied at the base of the numerical model [³⁰–³⁵]. The deconvolution process is illustrated in Fig. 3. It is conducted in the frequency domain by computing the transfer function TF, of the foundation medium through its finite element analysis by applying the available free-field record a_ff(t), at the base and computing the acceleration time-history at the top of the foundation, a_top(t). The foundation’s TF is obtained as (Step 7a in Fig. 2)

(1)TF(\omega )={\displaystyle \frac{A_{\mathrm{top}}(\omega )}{A_{\mathrm{ff}}(\omega )}},

where A_top(ω) and A_ff(ω) are Fourier transforms of the computed top and the applied free-field records, respectively. Then, the inverse of the foundation’s TF is used to obtain Fourier transform of the required input base acceleration record, A_inp(ω):

(2)A_{\mathrm{inp}}(\omega )={\displaystyle \frac{A_{\mathrm{ff}}(\omega )}{TF(\omega )+\epsilon }},

where ε is a regularization parameter avoids dividing by very low values. The input deconvolved base record a_inp(t), that produces the free-field record at the top of the foundation is computed by the inverse Fourier transform of A_inp(ω) (Step 7b in Fig. 2). It is possible to use iterations to improve the results of the deconvolution process [³⁴]. In this deconvolution process, the delay time due to seismic wave propagation through the foundation is neglected; however, the so-called flat-box check [³⁶] has been used to check the quality of the deconvolution process. It is noteworthy that the foundation could be homogeneous or inhomogeneous but it should behave linearly so the deconvolution process can be applied. Also, because the properties of the foundation such as rock material properties may change in each sample, the deconvolution process should be repeated for each foundation sample to produce the correct free-field record at the top surface.

After completion of the deconvolution process, the finite element model of the dam-reservoir-foundation is analyzed under the deconvolved earthquake ground motion (Step 8); the required seismic responses are obtained (Step 9), and the limit-state (performance) functions are evaluated (Step 10). After evaluation of limit-state functions for all generated samples, the EP of limit-state P_f, and the corresponding reliability index b, can be computed as (Step 11)

(3)P_{\mathrm{f}}={\displaystyle \frac{N_{\mathrm{e}}}{N}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\Rightarrow \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\beta =-{\mathrm{\Phi }}^{-\mathrm{1}}(P_{\mathrm{f}}),

where N_e is the number of samples in which the limit-state is exceeded, N is the total number of samples, and F is the standard normal cumulative distribution function. The above process is repeated for all selected earthquake ground motions. If enough large number of earthquake records is considered, the aleatory randomness can be then evaluated (Step 12). This methodology is general and applicable to various case studies.

The second reliability method that applied in this study is FOSM method [³⁷]. This method is applied in order to propagate modeling uncertainties to quantify their effect on the conditional exceedance probabilities and to determine the contribution of each variable to the variance of the defined performance functions. In the FOSM method, the performance function is linearized using a Taylor series expansion about the mean values of the random variables. The variance of the response due to sources of modeling uncertainty is computed from the gradients of the performance function. The reliability index is determined by dividing the mean to the standard deviation of the performance function. However, the linear approximation may be problematic when the performance functions are highly nonlinear [⁹]. Also, importance measures that show the contribution of random variables in the variance of performance function can be calculated by the FOSM method. The sign of the importance measures indicates whether the random variable is a resistance or load parameter, i.e., whether an increase in the realization of the random variable yields a larger or smaller reliability index, respectively. The FOSM is one of the simplest reliability methods that requires very low computational cost, but it may provide an insufficient representation of the effects of modeling uncertainties [⁹].

Application example

Pine Flat gravity dam

The proposed methodology is applied to Pine Flat gravity dam as case-study. It is a 122 m high gravity dam located on the Kings River of central California in the United States. The tallest non-over-flow monolith of the dam is selected, as shown in Fig. 4(a), and numerically analyzed along with a portion of the full reservoir and the rock foundation using the finite element method. All components are modeled with eight-node continuum elements as illustrated in Fig. 4(b). Owing to the feature of probabilistic analysis which needs iteration and running the model over and over, the accuracy of mesh size has been evaluated in a way to use largest feasible size which does not ruin the accuracy. It is achieved by decreasing the mesh size of the model from a very coarse one in successive steps, and controlling the variation of target response (joint response and crest relative displacement) among them. The mesh is called to be converged when the difference between two successive steps is less than a tolerance value (say, 5%).

The Kings River basin is located within a complex geologic area containing pre-Cretaceous meta-sedimentary and meta-volcanic rocks that have been folded, faulted, and intruded by granitic rocks of three different ages [³⁸]. The dam is situated on hard metamorphic (meta-volcanic) rock consisting primarily of jointed amphibolite with scattered seams of calcite, quartz, and lesser occurrences of gypsum. Typically, rock at the dam site is hard, dark gray, fine-grained, and brittle. Twenty-two potential fault sources were identified at the Kings River basin but no major through-going or shear zones have been identified in this area [³⁸]. However, to assess the effects of foundation inhomogeneity on the seismic performance of the dam, three distinct rock regions are considered within the foundation as shown in Fig. 4. This illustrative configuration has been selected based on primary analyses showing its high influence on the dam seismic performance. No joint or fault is considered between the rock regions. The inertia, flexibility, and damping of the foundation are taken into account. The radiation damping is modeled using infinite elements at the bottom and lateral sides of the foundation as shown in Fig. 4(b) to avoid reflection of seismic waves back to the dam.

The reservoir is assumed to be full and the dynamic water-structure interaction is modeled using Eulerian-Lagrangian approach [³⁹,⁴⁰]. The transmitting boundary condition is assigned to the truncated far-end of the reservoir in the upstream direction. Linear elastic materials are used to model the behavior of the concrete, water and foundation rocks. The water’s density and bulk modulus are 1000 kg/m³ and 2.07 GPa, respectively. The commercial Abaqus software is used for the finite element analysis.

There is a single interface in the contact between the dam and the foundation. This base joint shown in Fig. 4(b) has no tensile strength which is common in concrete material, but it can mobilize shear strength up to some extent. The sliding resistance is defined through the Coulomb model as a function of the friction coefficient m [⁴¹]. The cohesion is neglected, but the uplift pressure, without drainage, assuming a linear variation form the upstream to the downstream face is considered. These assumptions are based on limiting the number of variables which leads to a better description of existing random variables effect on the model. Other failure planes, for example along lift lines, can be taken into account in the reliability analysis, but in this study, only one predefined sliding plane along the dam-foundation interface is considered. The model is first statically loaded under the self-weight of the dam and the hydrostatic pressure of the full reservoir. Then it is dynamically analyzed under the horizontal component of the deconvolved earthquake ground motions. A typical Rayleigh damping of 5% is used for the dam and the foundation [³⁶].

Performance functions

Multiple limit-states, related to structural failure modes, can be of interest for concrete gravity dams. As it was stated, the main potential seismic failure modes of gravity dams are tensile cracking and movement along the prescribed joint at the dam-foundation interface [⁴²]. Hence, in this study, three different performance functions are defined according to mentioned potential failure modes [⁴³]. The first one is tensile overstressing of the dam body which would result in tensile cracking. It is defined through subtracting the envelope maximum (tensile) principal stress, demanded within the dam body during the seismic analysis, from the tensile strength of concrete. The second and the third performance functions are related to the dam-foundation interface and are, respectively, sliding along the base joint, and the opening of the base joint in its upstream end adjacent to the reservoir. It is more comprehensive to define progressively more severe damage levels of the dam [⁴], however, owing to limited databases, there is a lack of available information about acceptable damage levels for dams’ performance [¹]. These damage levels are indeed the threshold values corresponding to each performance function. These performance functions are introduced in the following:

(4)g_{\mathrm{1}}(x)=t_{\mathrm{t}}-{\sigma }_{\mathrm{t}},

(5)g_{\mathrm{2}}(x)=t_{\mathrm{s}\mathrm{l}}-d_{\mathrm{s}\mathrm{l}},

(6) g_{\mathrm{3}}(x)=t_{\mathrm{o}\mathrm{p}}-d_{\mathrm{o}\mathrm{p}},

where g_{\mathrm{1}}(x), g_{\mathrm{2}}(x), and g_{\mathrm{3}}(x) are performance functions according to tensile overstressing, sliding, and opening of the base joint, respectively. Threshold values of tensile overstressing, sliding, and opening of base joints are denoted by t_{\mathrm{t}}, t_{\mathrm{s}\mathrm{l}}, and t_{\mathrm{o}\mathrm{p}}, respectively. The results of finite element models are incorporated in these performance functions as {\sigma }_{\mathrm{t}} for tensile stress responses, d_{\mathrm{s}\mathrm{l}} and d_{\mathrm{o}\mathrm{p}} for sliding and opening of the base joint, respectively.

Controlling earthquakes

For seismic analysis of gravity dams, the United States Army Corps of Engineers guidelines [²⁴] suggest return periods of 144-year, 950-year, and 10000-year for OBE, MDE, and MCE records in a common service life of 100 years. Probabilistic seismic hazard analysis of the Pine Flat dam site shows that the peak horizontal accelerations to be expected at the site are 0.18g, 0.27g, and 0.45g corresponding to return periods for the OBE, MDE, and MCE ground motions, respectively [³⁸]. They are proportional to hazard levels of 50%, 10%, and 1% in a service life of 100 years, respectively. The Kern County earthquake of 1952 recorded at Taft Lincoln School Tunnel is selected as the free-field ground acceleration. It is scaled to three increasing peak ground acceleration levels as stated above corresponding to the OBE, MDE, and MCE records of the site. Their annual EP is 0.69%, 0.11%, and 0.01%, respectively. The response spectra of the scaled records are shown in Fig. 5. It should be mentioned that studying the uncertainty related to the earthquake records, including the frequency content or duration, is beyond the scope of this research.

Random variables

The random variables chosen are listed in Table 1 with their associated probability distribution function, mean, and coefficient of variation. The related domains are shown in Fig. 4(a). As limited material investigations are available for Pine Flat dam, most probability distributions are defined from empirical data of similar dams [⁴⁴–⁴⁶]. The uncertainty in modeling parameters is mainly considered to be epistemic because of this lack of knowledge. The elastic moduli of the rock regions are related to the dam’s one using the defined α ratios (Table 1). These ratios significantly affect the seismic response of gravity dams [⁴⁷]. For these ratios, the uniform distribution in the range of 0.25–1.50 is used because of large uncertainties due to lack of data [¹]. For the other random variables, some information is available from dam sites, so more common probability distributions such as the normal or the lognormal have been used. In particular, for the friction coefficient of the base joint, the normal distribution is used, as it is the distribution that fits best the available data [¹¹]. The random variables are all assumed to be uncorrelated. The Poisson’s ratios for the concrete and rock region are 0.20 and 0.33, respectively. The same material property is considered for the rock regions, but during the sampling procedure, the generated samples for these regions may be completely different that would lead to the inhomogeneous foundation.

Latin hypercube sampling

As it was stated, efficient number of samples of the dam-reservoir-foundation coupled system are derived using LHS. There is no predefined sample size N to achieve a certain confidence level, however, some formulas have been presented for various applications [⁴⁸,⁴⁹]. One of the simplest formulas suggested by Broding et al. [⁵⁰] is

(7)

where l is the confidence level, and P_f is the EP of limit-state. Assuming a confidence level of l = 98% and P_f = 10⁻³, which is a reasonable value in dam engineering [⁵¹], about 4000 samples are required based on this convenient formula. Although this formula sets the number of Monte-Carlo samples, it can be used as a first try for LHS approach. Therefore, to assess the number of samples, the LHS method is employed to obtain three different sets with 1000, 2000, and 4000 dam-reservoir-foundation samples by sampling the modeling parameters in Table 1. Each sample is then analyzed under the selected scaled ground motions. As there are three different earthquake records scaled to the given seismic intensities, i.e., OBE, MDE and MCE, the total number of simulations is \mathrm{3}\times (\mathrm{1000}+\mathrm{2000}+\mathrm{4000})=\mathrm{21000}. However, the same number of deconvolution processes is also conducted.

Results and interpretation

The histograms of maximum seismic responses of the three sample sets under the OBE, MDE, and MCE ground motions are illustrated in Fig. 6. Also shown is the best distribution fit using one of normal, lognormal, beta (with scaling if it is needed), and logistic fits, obtained by the Anderson-Darling test [⁵²–⁵⁴]. The distribution’s information, mean and standard deviation, for each histogram is presented in Table 2. Regardless of the number of samples, the same distribution fit with very similar mean and standard deviation is obtained. So only one fit for the 1000-sample set is shown in Fig. 6. As it is expected, when the applied earthquake becomes stronger, higher mean responses are generally captured. The mean responses are about 0.2, 0.7, and 4.8 MPa for tensile stress under the OBE, MDE, and MCE, respectively. However, even up to 8 MPa tensile stress may be observed under the MCE; it is because of linear behavior assumption for the concrete. The corresponding standard deviations also increase by increasing the earthquake intensity. So, it seems that the tensile stresses spread in a wider range. But, the coefficient of variation of the tensile stress is about 1.10, 0.38, and 0.23 under the OBE, MDE, and MCE, respectively. So, by increasing the earthquake intensity the lower dispersion of distribution is obtained.

About the base joint opening and sliding, the mean responses increase by increasing the earthquake intensity, but the standard deviation is the most under the MDE, then MCE, and then OBE. The coefficients of variation of the base joint opening are 0.57, 0.40, and 0.16 under the OBE, MDE, and MCE, respectively. They are 2.59, 0.75, and 0.26 for base joint sliding, respectively. Therefore, same as tensile stress, more earthquake intensity causes lower coefficient of variation. Rather high values of the coefficient of variation show that large uncertainties are controlling the results, no matter what number of samples is being used. Even with a low coefficient of variation, there is a small probability (i.e., less than 10%) for sliding more than 40 cm under the MCE. Based on the results obtained, it is important to increase the amount of information about the sources of uncertainty affecting the seismic responses. Increasing the amount of information would reduce the epistemic uncertainty on the estimated parameters of the statistical distributions of the variables, such as mean and variance, while natural variability cannot be reduced. However, epistemic uncertainty is present as long as future testing and experiments bring new data of the random variables; part of this uncertainty will remain aleatoric, as long as testing protocols include measurement errors [¹¹].

From the histograms, it is possible to compute the EP of the limit-states P_f, for varying threshold values using Eq. (3). The varying threshold values describe various damage levels. The obtained EP curves are plotted in Fig. 7 for the defined performance functions. As it is observed, the EP curves are very similar regardless of the number of samples used. Because approximately the same estimation of P_f is computed at each level of threshold for all performance functions, the number of simulations can be greatly reduced to prevent high computational cost. As the earthquake record becomes more intensive, the related EP curve will be shifted to the right side which shows higher exceedance probabilities. For example, considering threshold value of 2.25 MPa for the overstressing performance function, the exceedance probabilities would be 2%, 37%, and 93% for the OBE, MDE, and MCE, respectively. This threshold is a common value of the dynamic tensile strength of dam concrete [⁵⁵], so exceeding this value means tensile cracking of the dam body, however, it is not coincident with the total failure of the dam.

The base joint opening is observed even for low seismic intensities. Because the opening is measured at the upstream end of the base joint adjacent to the reservoir, it may result in water to penetrate inside the base joint that endangers the dam stability. The probability of maximum opening to exceed 1 cm is 0, 29%, and 97% under the OBE, MDE, and MCE, respectively. The base joint at dam heel position will probably open up to even 4 cm under the MCE. Minor, moderate, and severe damage will be imposed on the dam’s drain system for incipient, 2.5 and 5 cm base sliding, respectively [⁵⁶]. The moderate and severe damage probabilities are 16% and 8% under the OBE, respectively, while they are 53% and 29% under the MDE, and more than 90% under the MCE. A sliding displacement of 15cm would cause unacceptable differential movements with the adjacent monoliths and could, eventually, cause loss of reservoir control [¹]. The probability of exceeding from this high sliding is 3%, 8%, and 50% for the OBE, MDE, and MCE, respectively. However, the threshold values of sliding displacement can be defined as a function of the drain diameter and the deformability of inter-monolith water-stops.

All in all, considering two seismic failure modes of tensile overstressing and movement along joints, the dam will undergo severe damage level with the probability more than 90% under the MCE ground motion, while this probability is below 5% for the OBE. Approaching coincidently to two failure modes will probably cause the total failure of the dam under intense earthquake ground shaking; however, the dam will safely survive light earthquakes in order of OBE. It should be noted that the computed conditional probabilities do not provide much information on dam total safety because it has to be multiplied by the probability of the seismic loadings [¹¹]. In this way, the total annualized probability of failure is computed by summing the products of the probability of the seismic event by the conditional probability of failure for all potential failure modes. The complete seismic risk of the dam can be assessed by integrating the total annualized probability of failure in terms of the failure consequences. The obtained individual risk can then be meaningfully compared with published guidelines [⁵⁷,⁵⁸]; however, it is beyond the scope of this paper.

Sensitivity to earthquake deconvolution process

To investigate the effects of the deconvolution process, the 1000-sample set is re-analyzed under the OBE, MDE, and MCE deconvolved using the foundation with mean values for the random variables of the rock regions. The deconvolution process is done only once for each earthquake, and it is not repeated for every sample. So, the same ground motion is applied to all samples. The new computed EP curves for all performance functions are compared with those obtained considering a deconvolution for each sample (Section 4) in Fig. 8 to identify the effects of the deconvolution process. The difference between the EP curves increases by increasing the earthquake intensity. So, the deconvolution process is more important for larger earthquakes. However, the maximum difference between the estimated P_f values in all threshold levels is up to 12% for the defined performance functions. Excluding the deconvolution process for each sample flattens the EP curve, and the mean values are shifted into lower values. It shows that the obtained results are more sensitive when the deconvolution process is considered for each sample.

In the next step, the 1000-sample set is re-analyzed under the MDE which is applied without any deconvolution in the free-field condition to the model base. The resulted EP curves are illustrated in Fig. 9 against the EP curves obtained considering the deconvolution of the MDE. Ignoring the deconvolution and applying the ground motion in the free-field condition totally underestimates the exceedance probabilities in entire threshold range for all failure modes. The difference is more for lower threshold values. It is expected that magnifying the earthquake intensity would increase this underestimation. Hence, the earthquake records should be deconvolved when analyzing dam-reservoir-foundation systems considering inertia and inhomogeneity of the rock.

Importance of foundation inhomogeneity

Considering a homogeneous foundation with the density of 2600 kg/m³ and elastic moduli of 26.25 GPa, i.e., the mean values of rock properties in Table 1, the 1000-sample set of the dam-reservoir-foundation system with three random variables for the concrete and the base joint (Table 1) is generated and analyzed under the deconvolved MDE record. The obtained histograms are not presented but the best distribution fits along with their mean and standard deviation are reported in Table 3. Comparing Table 3 with the related part in Table 2 shows that assuming the foundation as homogeneous, the mean responses are shifted into the lower values while the standard deviations are increased which results in a higher coefficient of variation (CoV). So, considering the foundation inhomogeneity would result is lower dispersal of the seismic responses. Final column shows CoV assuming inhomogeneous foundation.

The computed EP curves are shown in Fig. 10 and compared with those assuming inhomogeneous foundation (Section 4). The assumption of homogeneous foundation means a reduction of random variables from nine to three. Excluding sources of uncertainty related to the foundation causes the EP curves to be flattened. It is in agreement with the conventional expectation that the effect of modeling uncertainty is to sharpen the EP curves [⁹]. Although at first glance it might seem that the difference is at most below 10% for all performance functions, noticing to the relative difference would be considerable in some cases. For instance, in tensile stress diagram, at some points there are almost 50% relative differences. But in the base joint sliding graph even relative differences can be considered to zero percent.

Comparison with FOSM method

In this section, the exceedance probabilities are derived through the FOSM method [⁵⁹–⁶¹] by considering all of the random variables listed in Table 1. As the FOSM method demands much lower computational cost with respect to the LHS, the purpose of this section is evaluating the efficiency of this method. The FOSM’s EP curves are compared with those obtained using the LHS in Fig. 11. All results are computed excluding the deconvolution process for each sample, but the records are deconvolved once considering the mean parameters of the foundation as described in Section 5. It can be seen in Fig. 11 that the FOSM method would result is more flattened EP curves, however, the shape of the curves is similar between the two methods. The difference would be more approaching to the curves tail, i.e., higher threshold values, due to increase in the dispersion in this range of thresholds.

The exceedance probabilities are the highest under the MCE, then MDE, and then OBE. The FOSM method will not lead to 100% EP for the base joint opening even under the MDE for zero threshold. The probability of developing tensile stresses within the dam body is about 90% under the OBE. It is concluded that for higher threshold values the simpler the reliability method used, the higher the value of the EP obtained. But in general, opposite of what was observed before [¹¹], the simpler reliability method may overestimate or underestimate the EP with respect to the more precise technique. So, it may be necessary to perform costly and time-demanding analysis methods for seismic safety evaluation of gravity dams in the context of risk analysis.

The contours of EP of the tensile overstressing performance function under the MDE from the FOSM method are depicted in Fig. 12 for varying threshold values (concrete tensile strength). The same legend is employed for all contours. As the concrete tensile strength increases, the EP of failure decreases. Excluding the deconvolution process generally results in higher EP in entire dam body specifically for lower thresholds. The spatial variation of EP shows that highest probability belongs to the upper parts of the dam on both upstream and downstream faces, specifically near the dam neck. Due to the presence of the base joint, the probability of failure along the dam base, particularly near the dam heel which is one of the most prone areas to cracking [⁴¹], is low as compared with other areas. Considering the common tensile strength of 2.25 MPa, the most vulnerable areas to cracking concentrate around the dam neck on both opposite faces of the dam. The downstream face is more vulnerable than the upstream face. The concentration of vulnerable areas is more considerable when the deconvolution process is excluded. In this situation, the probability of tensile cracking for the lower half of the dam section is below 30%.

The FOSM importance measures for each random variable is computed and presented in Table 4 for the defined performance functions including the deconvolution process. No specific trend can be observed in the importance measures. Their value and sign change not only through performance functions but also from record to record. It shows that the importance measures obtained from approximate differentiation-based methods such as FOSM cannot be reliably used in a nonlinear dynamic analysis of dam-reservoir-massed foundation coupled systems. This shortcoming caused by the method itself in addition to the nonlinear behavior of the model.

Conclusions

In this study, a methodology was proposed to evaluate the seismic reliability of gravity dam-reservoir-foundation coupled systems. The uncertainties associated with modeling parameters were incorporated in nonlinear time history analysis to realistically quantify their effects on the seismic performance of the system. The methodology was applied to Pine Flat gravity dam which was numerically modeled with a base joint between the dam and the foundation. The foundation was considered to be inhomogeneous with a constant spatial geometry but with various rock material properties. The material properties and the base joint friction coefficient were considered as random variables; they were probabilistically investigated employing Monte-Carlo simulation with LHS to generate statistically significant 1000, 2000, and 4000 samples of the model. The system was analyzed under the OBE, MDE, and MCE records of the dam site. The deconvolved-base-rock input model was utilized as earthquake input mechanism. Three different performance functions related to seismic failure modes of the gravity dam were defined and evaluated under the selected earthquakes. The same distribution fit with very similar parameters and approximately the same EP curves were obtained for seismic output results regardless of the number of samples. So, in this type of analysis, the number of simulations can be reduced in order of four, from 4000 to 1000, with the same estimation of probability. However, this may not be the case if other types of nonlinearities are introduced in the model. Higher mean responses were obtained under larger earthquake, but the coefficient of variation decreased by increasing the earthquake intensity. However, high values of the coefficient of variation showed that large uncertainties are controlling the results. It was shown that considering two seismic failure modes of tensile overstressing and movement along joints, the dam will undergo severe damage level with the probability more than 90% under the MCE, while this probability is below 5% for the OBE. Approaching coincidently to two failure modes will probably cause the total failure of the dam under intense earthquake ground shakings; however, the dam will safely survive light earthquakes in order of OBE.

About the deconvolution process, it was found that it is more important for larger earthquakes. Excluding the deconvolution process for each sample flattened the EP curve, and the mean values were shifted into lower values. The obtained results were more sensitive when the deconvolution process was considered for each sample. Ignoring the deconvolution and applying the ground motion in the free-field condition totally underestimated the exceedance probabilities for all failure modes. Hence, the earthquake records should be deconvolved when analyzing dam-reservoir-foundation systems considering inertia and inhomogeneity of the rock. Assuming the foundation as homogeneous resulted in lower mean seismic responses but a higher coefficient of variation. Excluding sources of uncertainty related to the foundation caused the EP curves to be flattened. However, the difference was at most below 10% for all performance functions but relative differences at some points were outstanding. As a result, the uncertainties related to the inhomogeneity of rock foundation should not be estimated especially when the inertia and stiffness of the foundation is taken into account. Applying the FOSM method resulted in more flattened EP curves; the difference was more for higher threshold values. It was shown that the simpler reliability method may overestimate or underestimate the EP with respect to the more precise technique. So, the FOSM method cannot be reliably used in a nonlinear dynamic analysis of dam-reservoir-massed foundation coupled systems. It is necessary to perform costly and time-demanding analysis methods for seismic safety evaluation of gravity dams in the context of risk analysis.

References

[1] Bernier C, Padgett J E, Proulx J, Paultre P. Seismic fragility of concrete gravity dams with spatial variation of angle of friction: Case study. Journal of Structural Engineering, 2016, 142(5): 05015002

[2] USACE (US Army Corps of Engineers). Gravity Dam Design. EM-1110-2-2200. 1995

[3] ICOLD (International Commission on Large Dams). Design Features of Dams to Resist Seismic Ground Motion: Guidelines and Case Studies. Bulletin No. 120, 2001

[4] Ellingwood B, Tekie P B. Fragility analysis of concrete gravity dams. Journal of Infrastructure Systems, 2001, 7(2): 41–48

[5] Tekie P B, Ellingwood B. Seismic fragility assessment of concrete gravity dams. Earthquake Engineering & Structural Dynamics, 2003, 32(14): 2221–2240

[6] Lupoi A, Callari C. A probabilistic method for the seismic assessment of existing concrete gravity dams. Structure and Infrastructure Engineering, 2011, 8(10): 985–998

[7] Ghanaat Y, Patev R C, Chudgar A K. Seismic fragility analysis of concrete gravity dams. In: Proceedings of the 15th World Conference on Earthquake Engineering. Tokyo: International Association for Earthquake Engineering, 2012

[8] Alembagheri M, Seyedkazemi M. Seismic performance sensitivity and uncertainty analysis of gravity dams. Earthquake Engineering & Structural Dynamics, 2015, 44(1): 41–58

[9] Liel A B, Haselton C B, Deierlein G G, Baker J W. Incorporating modeling uncertainties in the assessment of seismic collapse risk of buildings. Structural Safety, 2009, 31(2): 197–211

[10] Huang Y, Xiong M. Probability density evolution method for seismic liquefaction performance analysis of earth dam. Earthquake Engineering & Structural Dynamics, 2017, 46(6): 925–943

[11] Altarejos-García L, Escuder-Bueno I, Serrano-Lombillo A, de Membrillera-Ortuño M G. Methodology for estimating the probability of failure by sliding in concrete gravity dams in the context of risk analysis. Structural Safety, 2012, 36–37: 1–13

[12] Fenves G, Chopra A K. Earthquake analysis of concrete gravity dams including reservoir bottom absorption and dam-water-foundation rock interaction. Earthquake Engineering & Structural Dynamics, 1984, 12(5): 663–680

[13] Alembagheri M, Ghaemian M. Seismic performance evaluation of a jointed arch dam. Structure and Infrastructure Engineering, 2016, 12(2): 256–274

[14] Bieniawski Z T. Engineering Rock Mass Classifications: A Complete Manual for Engineers and Geologists in Mining, Civil, and Petroleum Engineering. Hoboken: John Wiley & Sons, 1989

[15] Vaish A K, Chopra A K. Earthquake finite element analysis of structure-foundation systems. Journal of the Engineering Mechanics Division, 1974, 100(6): 1101–1116

[16] Chopra A K, Chakrabarti P. Earthquake analysis of concrete gravity dams including dam-water-foundation rock interaction. Earthquake Engineering & Structural Dynamics, 1981, 9(4): 363–383

[17] Goodman R E, Powell C. Investigations of blocks in foundations and abutments of concrete dams. Journal of Geotechnical and Geoenvironmental Engineering, 2003, 129(2): 105–116

[18] Chen S H, Qiang S, Shahrour I, Egger P. Composite element analysis of gravity dam on a complicated rock foundation. International Journal of Geomechanics, 2008, 8(5): 275–284

[19] Zhang L, Wang D, Zhang H, Wang W. Stability analysis of gravity dams on sloping layered rock foundation against deep slide. In: Proceedings of the 11th Biennial ASCE Aerospace Division International Conference on Engineering, Science, Construction, and Operations in Challenging Environments. ASCE, 2008

[20] Sanchez-Sesma F J. Site effects on strong ground motion. Soil Dynamics and Earthquake Engineering, 1987, 6(2): 124–132

[21] Léger P, Boughoufalah M. Earthquake input mechanisms for time-domain analysis of dam-foundation systems. Engineering Structures, 1989, 11(1): 37–46

[22] Bayraktar A, Hançer E, Akköse M. Influence of base-rock characteristics on the stochastic dynamic response of dam-reservoir-foundation systems. Engineering Structures, 2005, 27(10): 1498–1508

[23] Deierlein G G. Overview of a comprehensive framework for earthquake performance assessment. In: Fajfar P, Krawinkler H, eds. Performance-Based Seismic Design Concepts and Implementation Proceedings of an International Workshop. Bled: Pacific Earthquake Engineering Research Center, 2004, 15–26

[24] USACE (US Army Corps of Engineers). Time-History Dynamic Analysis of Concrete Hydraulic Structures. EM-1110-2-6051. 2003

[25] Helton J C, Davis F J. Latin hypercube sampling and the propagation of uncertainty in analyses of complex systems. Reliability Engineering & System Safety, 2003, 81(1): 23–69

[26] Rubinstein R Y, Kroese D P. Simulation and the Monte Carlo Method. Vol. 707. Hoboken: John Wiley and Sons, 2011

[27] Dalbey K R, Karystinos G N. Fast generation of space-filling Latin hypercube sample designs. In: Proceedings of the 13th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference. Fort Worth, 2010

[28] Hamdia K M, Msekh M A, Silani M, Vu-Bac N, Zhuang X, Nguyen-Thoi T, Rabczuk T. Uncertainty quantification of the fracture properties of polymeric nanocomposites based on phase field modeling. Composite Structures, 2015, 133: 1177–1190

[29] Vu-Bac N, Silani M, Lahmer T, Zhuang X, Rabczuk T. A unified framework for stochastic predictions of mechanical properties of polymeric nanocomposites. Computational Materials Science, 2015, 96: 520–535

[30] Reimer R B. Deconvolution of seismic response for linear systems. University of California at Berkeley Report CB/EERC-73/10. 1973

[31] Ghanaat Y, Hashimoto P S, Zuchat O, Kennedy R P. Seismic fragility of Muhleberg dam using nonlinear analysis with Latin Hypercube simulation. In: Proceedings of the 31st Annual USSD Conference. San Diego, 2011, 1197–1212

[32] Saouma V, Hansen E, Rajagopalan B. Statistical and 3D nonlinear finite element analysis of Schlegeis dam. In: Proceedings of the Sixth ICOLD Benchmark Workshop on Numerical Analysis of Dams. 2001, 17–19

[33] Cvijanovic V, Schultz M G. Nonlinear finite element model validation at lower crystal springs dam and Pacoima dam. Journal of Dam Safety, ASDSO, 2016, 14(1): 33–37

[34] Sooch G S, Bagchi A. A new iterative procedure for deconvolution of seismic ground motion in dam-reservoir-foundation systems. Journal of Applied Mathematics, 2014, 287605

[35] Bretas E M, Lemos J V, Lourenço P B. Seismic analysis of masonry gravity dams using the discrete element method: Implementation and application. Journal of Earthquake Engineering, 2016, 20(2): 157–184

[36] USBR (United States Bureau of Reclamation). State-of-practice for the nonlinear analysis of concrete dams at the Bureau of Reclamation. 2013

[37] Haukaas T, Der Kiureghian A. Finite element reliability and sensitivity methods for performance-based earthquake engineering. Peer, 2004, 266

[38] USBR (United States Bureau of Reclamation). Upper San Joaquin River Basin Storage Investigation, Raise Pine Flat Dam. Surface Storage Option Technical Appendix to the Phase 1 Investigation Report. 2003

[39] Alembagheri M, Estekanchi H E. Nonlinear analysis of aboveground anchored steel tanks using endurance time method. Asian Journal of Civil Engineering, 2011, 12(6): 731–750

[40] Alembagheri M. Dynamics of submerged intake towers including interaction with dam and foundation. Soil Dynamics and Earthquake Engineering, 2016, 84: 108–119

[41] Alembagheri M, Ghaemian M. Incremental dynamic analysis of concrete gravity dams including base and lift joints. Earthquake Engineering and Engineering Vibration, 2013, 12(1): 119–134

[42] Ghanaat Y. Failure modes approach to safety evaluation of dams. In: Proceedings of the 13th World Conference on Earthquake Engineering. Vancouver, 2004, 1115

[43] Hamdia K M, Silani M, Zhuang X, He P, Rabczuk T. Stochastic analysis of the fracture toughness of polymeric nanoparticle composites using polynomial chaos expansions. International Journal of Fracture, 2017, 206(2): 215–227

[44] Lo K Y, Lukajic B, Wang S, Ogawa T, Tsui K K. Evaluation of strength parameters of concrete-rock interface for dam safety assessment. In: Proceedings of Canadian Dam Safety Conference. Toronto, 1990, 71–94

[45] EPRI (Electrical Power Research Institute). Uplift Pressures, Shear Strengths, and Tensile Strengths for Stability Analysis of Concrete Gravity Dams. EPRI TR-100345. 1992

[46] Champagne K, Rivard P, Quirion M. Shear strength parameters of concrete gravity dams in Quebec. In: Proceedings of CDA 2013 Annual Conference. Toronto, 2012

[47] Lin G, Du J, Hu Z. Earthquake analysis of arch and gravity dams including the effects of foundation inhomogeneity. Frontiers of Architecture and Civil Engineering in China, 2007, 1(1): 41–50

[48] Melchers R E. Structural Reliability Analysis and Prediction. New York: John Wiley & Son, 1999

[49] Iman R L. Encyclopedia of quantitative risk analysis and assessment. New York: John Wiley & Son, 2008

[50] Broding W C, Diederich F W, Parker P S. Structural optimization and design based on a reliability design criterion. Journal of Spacecraft and Rockets, 1964, 1(1): 56–61

[51] Chauhan S S, Bowles D S. Dam safety risk assessment with uncertainty analysis. Ancold Bulletin, 2004, 73–88

[52] Anderson T W, Darling D A. A test of goodness of fit. Journal of the American Statistical Association, 1954, 49(268): 765–769

[53] Vu-Bac N, Lahmer T, Zhang Y, Zhuang X, Rabczuk T. Stochastic predictions of interfacial characteristic of polymeric nanocomposites (PNCs). Composites. Part B, Engineering, 2014, 59: 80–95

[54] Vu-Bac N, Lahmer T, Zhuang X, Nguyen-Thoi T, Rabczuk T. A software framework for probabilistic sensitivity analysis for computationally expensive models. Advances in Engineering Software, 2016, 100: 19–31

[55] Harris D W, Mohorovic C E, Dolen T P. Dynamic properties of mass concrete obtained from dam cores. Materials Journal, 2000, 97(3): 290–296

[56] Fenves G L, Chavez J W. Evaluation of earthquake induced sliding in gravity dams. In: Proceedings of the 11th World Conference on Earthquake Engineering. Tokyo, 1996

[57] USBR (United States Bureau of Reclamation). Guidelines for Achieving Public Protection in Dam Safety Decision Making. Technical Report. 2003

[58] Munger D F, Bowles D S, Boyer D D, Davis D W, Margo D A, Moser D A, Regan P J, Snortel N. Interim tolerable risk guidelines for US Army Corps of Engineers dams. In: Proceedings of the 29th Annual United States Society on Dams Conference. 2009, 1125–1141

[59] Vanmarcke E H. Matrix formulation of reliability analysis and reliability-based design. Computers & Structures, 1973, 3(4): 757–770

[60] Karamchandani A, Cornell C A. Sensitivity estimation within first and second order reliability methods. Structural Safety, 1992, 11(2): 95–107

[61] Liping W, Grandhi R V. Efficient safety index calculation for structural reliability analysis. Computers & Structures, 1994, 52(1): 103–111

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