Reliability and variance-based sensitivity analysis of arch dams during construction and reservoir impoundment

Abstract

The static performance of arch dams during construction and reservoir impoundment is assessed taking into account the effects of uncertainties presented in the model properties as well as the loading conditions. Dez arch dam is chosen as the case study; it is modeled along with its rock foundation using the finite element method considering the stage construction. Since previous studies concentrated on simplified models and approaches, comprehensive study of the arch dam model along with efficient and state-of-the-art uncertainty methods are incorporated in this investigation. The reliability method is performed to assess the safety level and the sensitivity analyses for identifying critical input factors and their interaction effects on the response of the dam. Global sensitivity analysis based on improved Latin hypercube sampling is employed in this study to indicate the influence of each random variable and their interaction on variance of the responses. Four levels of model advancement are considered for the dam-foundation system: 1) Monolithic dam without any joint founded on the homogeneous rock foundation, 2) monolithic dam founded on the inhomogeneous foundation including soft rock layers, 3) jointed dam including the peripheral and contraction joints founded on the homogeneous foundation, and 4) jointed dam founded on the inhomogeneous foundation. For each model, proper performance indices are defined through limit-state functions. In this manner, the effects of input parameters in each performance level of the dam are investigated. The outcome of this study is defining the importance of input factors in each stage and model based on the variance of the dam response. Moreover, the results of sampling are computed in order to assess the safety level of the dam in miscellaneous loading and modeling conditions.

Keywords

concrete arch dams / reliability / randomness / improved Latin hypercube sampling / variance-based sensitivity analysis / exceedance probability / Sobol′ index

Introduction

Structural reliability of concrete arch dams is one of the most important subjects because of disastrous consequences of their failure including not only economic losses but also human casualties. Their mass concrete body and rock foundation are continuously monitored even during the construction process and specifically during the reservoir impoundment. Those arch dams which are properly designed and constructed can survive common usual and infrequent unusual and extreme loading conditions [¹]. The usual loading mainly consists of the dam self-weight, the normal reservoir hydrostatic pressure, and the thermal loads [²]. The arch dams are in a permanent interaction with their rock foundation. There is a “peripheral joint” between the dam and the abutment rock. The rock is commonly considered as a homogenous domain in numerical analysis; nonetheless, it can be a very complex medium with high levels of uncertainty containing different layers, faults and fissures. The properties of the rock foundation may considerably alter the static and dynamic response of the dam-foundation system [³,⁴]. In particular, the presence of the soft rock layers may influence the response of the arch dams [⁵–⁹].

There is another type of joints within the arch dams’ body due to the stage-by-stage construction in individual cantilever monoliths. The monoliths are divided from each other by these joints called “contraction joints”. The contraction joints are usually grouted after each stage is erected up to the specific height. The monoliths act independently as cantilevers before the grouting process. Nevertheless, after the grout injection, they are integrated and the applied loads are transferred in cantilever and arch directions. The grout has negligible tensile strength with respect to the parent concrete material [²]. The contraction joints may be keyed or un-keyed; their behavior varies due to different templates of the shear keys. This specific construction type along with the different joint behavior changes the displacements and stress distribution within the dam body and its abutments; hence, they should be accurately modeled in the numerical analysis of arch dams [³].

The temperature fluctuations affect the arch dams more than other types of concrete dams such as gravity dams [¹⁰]. Specifically, the high amount of heat generated during the hydration of concrete must be controlled by techniques such as pre-cooling or post-cooling. The post-cooling scheme decreases the concrete temperature from the “hydration temperature” to the “grouting temperature” using the pipe-cooling. The grouting temperature, at which the grout injection is performed, is related to the mean annual temperature of the dam site [²]. The temperature change modifies the stress distribution within the dam body and should be considered during the design process.

Structural analysis and design of arch dams have been traditionally based on deterministic methods and safety factors. These methods do not consider the randomness existing in the model, and the uncertainty is tackled using the safety factors for different loading conditions [¹¹]. The application of safety factors may result in considerable prediction errors stemming from several sources of uncertainty [¹²]. The deterministic methods estimate only the mean values and cannot predict the statistical variation [¹³]. However, utilizing reliability methods provides more comprehensive information regarding the risk levels revealed by a specific structure. In the reliability methods, the uncertainty of model parameters is presented by considering them as random variables [¹¹]. The loading scenarios and their randomness have to be defined as well as the performance indices for the model response. The performance indices can be presented as limit-state inequality defined by

(1)g(\mathit{X})=\epsilon -f(\mathit{X})\le \mathrm{0},

where g(X) is the limit-state function with the failure condition defined by the above inequality, f(X) is the response function of the model, e shows the threshold or decision variable defined for the performance indices, and X is the vector of random variables in the problem including geometry, materials, loads, etc. [¹⁴]. g(X) = 0 defines the limit-state surface. In the probabilistic modeling, the failure probability of each limit-state function, P_f[g(\mathit{X})\le \mathrm{0}], which is equal to the exceedance probability of that limit-state function, requires the quantification of the conditional probability of the system’s response for different given load events. It is described using the following equation [¹²]:

(2)P_f[g(\mathit{X})\le \mathrm{0}]={\displaystyle {\int }_{g(X)\le \mathrm{0}}h(\mathit{X})\mathrm{d}\mathit{X}},

where h(X) is the joint probability density function for the random variables.

Several different approaches of the probabilistic analysis have been developed as numerical tools for the reliability assessment of structures [¹⁵–²³]. The widely-used methods are categorized as: (a) The sampling methods such as Monte-Carlo simulation, importance sampling, directional sampling and Latin hypercube sampling; and (b) the approximation methods such as point-estimate method, mean-value first-order second-moment method, first-order reliability method and second-order reliability method. The reliability and probabilistic analysis of concrete dams has taken attention of researchers in recent years [²⁴–³¹]; nonetheless, these studies are very limited in both quantity and quality with respect to considerable amount of research in the field of other kinds of structures such as buildings and bridges [³²–³⁸]. In detail, one of the main drawbacks of previous studies in the field of dam engineering is the lack of utilizing novel uncertainty method. New approaches can assist to achieve more comprehensive, advanced, and reliable results. In addition, the past investigations only concentrated on limited source of randomness or comprised oversimplified models. To this end, in this research a comprehensive study is conducted on the concrete arch dam using the-state-of-the-art reliability and sensitivity approaches with a model advancement.

This investigation deals with the static performance of the arch dams during the construction and the reservoir impoundment in the uncertainty framework. As a case study, Dez arch dam is chosen and it is modeled along with its rock foundation using the finite element method. Four levels of model advancement are taken into account for the dam-foundation system: 1) The monolithic dam without any joint founded on the homogeneous rock foundation, 2) the monolithic dam founded on the inhomogeneous foundation including soft rock layers, 3) the jointed dam including the peripheral and contraction joints founded on the homogeneous foundation, and 4) the jointed dam founded on the inhomogeneous foundation. The specific performance indices are defined through limit-state functions for each model. As a result, the influence of the randomness of each parameter is determined on the performance of the dam. The reliability and sensitivity analysis are implemented to compute the exceedance probability of the defined limit-state functions and the relative importance of random variables for different models and conditions. The global sensitivity analysis based on improved Latin hypercube sampling (iLHS) is employed as the uncertainty method in this study. In order to obtain the effect of loading conditions in the results, each model is analyzed by having different levels of stages and conditions. In addition, the sensitivity of the structural performance of the arch dam during the construction and the reservoir impoundment is investigated.

Methodology of reliability and sensitivity analysis

In this research, the methods implemented for the reliability and sensitivity analysis are improved iLHS and variance-based sensitivity (VBSA). As the benchmark reliability method is the sampling, this method would have high computational cost depending on the number of samples. The finite element analysis of the arch dams itself suffers from high computational cost. There are approaches in the literature to overcome the issue of high computational cost [³⁹,⁴⁰]. In the current study, the iLHS proved to be an efficient, novel, and applicable method [⁴¹]. In addition, iLHS can be the foundation analysis for VBSA. Furthermore, first-order and total-order indices are computed in VBSA. These indices decompose the response variance into a sum of the contribution of each input factor and/or the combination of thereof. In the following, the description of the aforementioned methods is presented.

The crude sampling methods such as Monte-Carlo simulation is often computationally prohibitive because of a large number of samples required to reach an acceptable level of accuracy [⁴²]. As a result, using more efficient sampling methods is inevitable particularly when it comes to time-consuming deterministic model. Among these methods, iLHS is selected for this research because of its efficiency and novelty. The iLHS is a stratified Monte-Carlo sampling that leads to efficient estimation of the quantity of interest by reducing the variance of the classic Monte-Carlo [⁴³–⁴⁵]. Unfortunately, the appropriate sample size N, cannot be determined a priori to achieve a certain confidence level. However, using a relatively high N that is substantially larger than the number of random variables will result in reasonably accurate estimates for practical purposes. In this study, iLHS is performed for N = 1000 realizations of the dam-foundation system, a relatively high number that allows an acceptable level of accuracy.

The iLHS utilizes stratified sampling leading to filling all the areas of the sample space. As this sampling procedure prevents replacement of sample points, the variance and as a result number of samples required is reduced. Hence, the computational efficiency of uncertainty analysis is considerably improved. This approach is recommended in nonlinear and complex model with high computational cost. The procedure of iLHS is to partition cumulative density function into N non-overlapping intervals having an equal probability. A value for the corresponding variable is randomly selected in each interval.

As the generated sample points in the classical Latin hypercube sampling may be close to the defined intervals’ margin, these samples do not cover the probable range of variables well. To overcome this problem, the optimum sampling approach is obtained from generated samples with respect to optimum midpoint distances. A number of valid candidate points are generated and distances to all points are calculated through this step-wise algorithm. Then, samples with minimum distance are selected. Utilizing the mentioned algorithm leads to more efficiently distributed points through the probable range of variables and it prevents to have samples with close distance to each other in the intervals’ margin [⁴²]. Hence, the mentioned approach is considered as iLHS. Although several samplings are required to satisfy minimum midpoint distance, the total computational cost of analysis remains approximately constant. The reason is that generating samples due to iLHS is not time-consuming and deterministic analysis has high computational cost.

Computing sensitivity measures for complex models and systems based on sampling is developed by many researchers [⁴⁶–⁵²]. The concept of the global sensitivity analysis is described as follows:

(3)y=f(x),x\in {\mathit{R}}^n,

where f, x and y are an arbitrary model, input parameter matrix and the output vector, respectively. The input parameters are defined on the n-dimensional unit cube Rⁿ as:

(4){\mathit{R}}^n=\{\mathit{x}:\mathrm{0}\le x_i\le \mathrm{1},i=\mathrm{1},\mathrm{2},\dots ,n\}.

The model function f is assumed to be square integrable. The concept of the variance-based sensitivity method is to investigate the influence of an input or a group of input parameters variance on the output variance of the model [⁵³–⁵⁷]. The contributions of each input parameter variance into the model variance are described using the following sensitivity indices which are called Sobol′ indices. The formulation is given in the following.

(5)S_i={\displaystyle \frac{Var[E(\mathit{y}|x_i)]}{Var(\mathit{y})}},

(6)S_{ij}={\displaystyle \frac{Var[E(\mathit{y}|x_i{x}_j)]}{Var(\mathit{y})}}-S_i-S_j,

where Var and E denote variance and expectation, respectively. These indices can be applied to any complex and arbitrary model functions. The second-order index S_ij expresses the model sensitivity to the interaction between the variables x_i and x_j (without the first order effects of x_i and x_j), and so on for the higher orders effects. The value of the aforementioned indices indicates the importance of the variable or the interaction of variables on the response of the model [⁵⁴,⁵⁵].

The total number of Sobol′ indices in a model with n input parameters is 2ⁿ‒1. As the number of indices increase, the computational cost of the analyses increase. In order to lower the time of analysis along with determining the interaction effects of input variables, the total sensitivity index of an input x_i is introduced as

(7)

where \ne irepresents all the non-ordered subsets of indices containing index i. Thus, {\displaystyle \sum _{l\in \ne i}{S}_l}is the sum of all sensitivity indices having i in their index. The estimation of the aforementioned indices can be performed by sampling-based simulation namely Latin hypercube sampling.

Case study: Numerical modeling, limit-state functions and random variables

In this research, Dez dam along with its rock foundation is selected as case study. It is 203 m high concrete arch dam with the crest length of 240 m located on Dez River in south-west part of Iran. Detailed description of this dam can be found in Ref. [⁵⁸]. The aerial view of the dam is shown in Fig. 1(a). The dam-foundation system is modeled by the finite element technique. The complicated nature of dam-foundation systems can be numerically modeled in various extents using deterministic structural analysis methods [⁵⁹–⁶²]. Here, four models of the Dez dam-foundation system are generated in an increasing order of complexity according to the constitutive model and the properties of the joints and the rock foundation. The models and the assumptions are tabulated in Table 1 along with the related random variables and limit-state functions (failure scenarios). They will be explained in detail in the following. The concrete and the rock materials in all models are assumed to be linearly elastic and isotropic with the material properties considered as random variables which are introduced in Table 2. The same finite element mesh is considered for all models as shown in Fig. 1(b).

The same loading conditions are applied to all four models. At the beginning, the models are loaded under the self-weight of the dam by modeling the stage-construction process in five layers each of them has about 40 m height [⁶³]. The construction stages (layers) are depicted in Fig. 1(c). After each stage of construction, the post-cooling process is modeled by analyzing the dam layer under the uniform temperature decrease of DT from the hydration temperature to the grouting temperature. The thermal stress is added up to the stress due to the weight of the dam body. This thermal stress is pertinent to DT, not to the absolute temperatures. Therefore, the hydration temperature is fixed at 25 °C, and DT is changed as a random variable to model different post-cooling temperature decreases. After the dam construction process is completed, the reservoir impoundment is modeled by analyzing the dam body under the hydrostatic pressure of the full reservoir with the level same as the dam crest level affecting on the dam’s upstream face.

Based on Table 1, Model 1 has monolithic (integrated) dam body without any joints founded on the homogeneous rock foundation. Its random variables are listed in Table 1. All random variables are related to the dam-foundation structural properties except the post-cooling temperature decrease (DT) which is related to the loading uncertainty. The only considered failure scenario for this model is the tensile overstressing which is quantified using the following limit-state function

(8)g_{\mathrm{1}}(X)=f_t-{\sigma }_t,

where f_t is the concrete tensile strength, and s_t is the peak value of the maximum principal (tensile) stress of the dam body. The performance index of this limit-state function can be in the form of

(9)

which represents the exceedance of the maximum tensile stress in one or more elements within the dam body from the threshold value f_t. The exceedance probability of this limit-state function is calculated using iLHS during the stage-construction of the dam and the reservoir filling. Also, the threshold value (decision variable) can be changed through a parametric study to obtain exceedance probability curve of the dam body. Additional analyses have shown that the compressive overstressing is not a critical limit-state for the arch dam models under the prescribed loading.

In Model 2, the monolithic dam model is founded on inhomogeneous rock foundation created by inserting a soft rock layer within the foundation. This soft rock layer, whose position is shown in Fig. 1(d), is a typical one in the foundation of arch dams [⁶⁴] that intersects with the dam body at about middle 1/3 height of the dam in both left and right abutments. The random variables are the same as Model 1; besides, the random variables related to the soft rock layer are involved. The same performance index as Model 1 is considered for Model 2. Investigating Model 2 with respect to Model 1 can reveal the effects of the presence of typical soft rock layer within the foundation on the performance of the dam-foundation system.

The peripheral and contraction joints are added in Model 3 which is founded on the homogeneous rock. Eighteen contraction joints are inserted within the dam body and a peripheral joint along the dam-foundation interface. The joints configuration is shown in Fig. 1(e). They are modeled as discrete cracks using contact definition between the joint surfaces in the normal and tangential directions. The normal behavior of the joints is determined using an exponential relation between the joint surface clearance (normal distance) and the transmitted normal stress. The considered relation is shown in Fig. 2(a). In the tangential direction, the sliding resistance is defined through the Mohr-Coulomb model as a function of the friction angle and the cohesion. In this model, the maximum transmitted shear stress, t_u, is limited to

(10){\tau }_u=m{\sigma }_N+c,

where s_N is the contact normal stress, m= tanj is the friction coefficient corresponding to the friction angle of j, and c is the cohesion. Beyond t_u, the joint surfaces will slide against each other. If the joint being in tension, then the shear stresses will be zero [⁶³]. The random variables considered in Model 3 are identical to Model 1 plus the friction coefficient of the joints. The assumption of c = 0 is assumed in the analysis. The performance indices of Model 3 are tensile overstressing defined by Eq. (8), and two other indices for the joint sliding and opening displacements along the peripheral and contraction joints. They are defined as

(11)

(12)

where d_s and d_o are the maximum sliding and opening displacements, respectively, the superscripts PJ and CJ represent the peripheral joint and the contraction joints, respectively, e_s and e_o are the sliding and opening threshold values. These performance indices control the exceedance of the maximum sliding and opening displacements along the peripheral and the contraction joints from the predefined thresholds. For different values of these thresholds, the related exceedance probability curves can be plotted in addition to the tensile overstressing curve mentioned earlier. The exceedance probability of each performance index is separately computed to investigate the dam performance. The limit-state g₁ is monitored during the dam construction and the reservoir impoundment. However, g₂ and g₃ are controlled only during the reservoir impoundment, because the joints are grouted after each stage of construction, hence they are in their initial position after the dam completion. In Model 4, the jointed dam Model 3 is located on the inhomogeneous rock foundation similar to Model 2. The same performance indices as Model 3 are considered for Model 4.

The probability distribution, mean and coefficient of variation of the selected random variables in the probabilistic and reliability analysis of Dez dam-foundation system, are given in Tables 2 and 3. The minimum and maximum cut-off values for each variable are also presented in Tables 2 and 3. The random variables are assumed to be uncorrelated. Some of the random variables have large uncertainties due to the lack of data which justify the use of probability distributions such as uniform with large coefficient of variation [⁶⁶]. This is the case for the concrete Poisson’s ratio (n_c) and coefficient of thermal expansion (a_c), the rock Poisson’s ratio (n_f, n_fr), and the post-cooling temperature decrease (DT). For the other random variables, the available information from dam sites and engineering judgment along with referring to other related researches leads to allocating the normal, lognormal and truncated lognormal probability distributions [³⁶,³⁸,⁶²,⁶⁴–⁶⁶].

The procedure how reliability framework combines with finite element models are illustrated in Fig. 3. First, probabilistic characteristics of input parameters goes into reliability framework and depending which reliability methods will be employed, the finite element model will be created due to the chosen approach. Then, the results of finite element analysis go back to reliability framework. This reciprocating relationship continues until all the required results will be obtained. At the end, target reliability and sensitivity results will be calculated in the reliability framework.

Exceedance probability of performance indices

The probability of exceeding the limit-state functions from the predefined threshold values for all models estimated through iLHS are presented in this section. The exceedance probabilities are computed using 1000 samples for each of four models. Implementing the iLHS with more samples is not recommended because of the high computational cost of the deterministic models.

Overstressing limit-state function

The exceedance probability of the tensile overstressing limit-state function in the four models considering the threshold value (the concrete tensile strength) of f_t = 3 MPa during all the construction stages and the reservoir impoundment are tabulated in Tables 4. In Models 1 and 2, the exceedance probability increases through the construction stages until the dam construction is completed. On the other hand, the exceedance probabilities decrease during the reservoir impoundment. Therefore, the reservoir impoundment causes the release in the tensile stress for the aforementioned models. By comparison of the exceedance probabilities calculated for Models 1 and 2, it is concluded that the presence of the soft layer in the foundation results in a decrease in the exceedance probability values and the most reduction is observed in Stage 3 which is adjacent to the soft layer. The minimum and maximum exceedance probabilities are detected in the construction Stages 1 and 5, respectively.

In Models 3 and 4, the trend in the exceedance probability values is in contradiction to the models without any joints. The exceedance probability is diminished through the construction stages. As the construction of the dam is finished and the reservoir impoundment commences, the exceedance probability increases. Although the exceedance probabilities are almost less than 1% and they are extremely low, it is observed that these values are a little higher in the model with inhomogeneous foundation than the homogeneous one. The least and most values of exceedance probability are determined for the construction Stages 5 and 6, respectively.

The comparison of the models including and excluding joints indicates that much lower exceedance probabilities are observed for the models with peripheral and contraction joints than the ones without joints. The reason is that the presence of the joints causes the release in the tensile stress and as a result, the computed exceedance probabilities are substantially decreased. Moreover, it is found that in all models, the soft layer does not change the results considerably.

The contours of the tensile overstressing exceedance probability on the dam’s upstream and downstream faces obtained for Models 1 and 2 are depicted in Fig. 4. Because of the extremely low probabilities observed in Models 3 and 4, the contours for these models cannot be used for an apt conclusion to be derived from. It can be concluded from Fig. 4 that during the stage-construction, the highest tensile overstressing exceedance probability is concentrated along the dam-foundation interface specifically near both upper ends. The reservoir impoundment lowers the exceedance probability in the mentioned parts and concentrates it at the bottom of the dam body on the downstream face. Nonetheless, the exceedance probability is below 5% for most parts of the dam during the construction and the reservoir filling. Therefore, using a concrete with more tensile strength can be prescribed for regions with higher overstressing failure probability. The foundation inhomogeneity slightly decreases the exceedance probabilities on both faces at the place of intersection of the soft rock layer.

The exceedance probability curves for varying threshold values (concrete tensile strengths) of the overstressing performance index are illustrated in Fig. 5. In Model 1, the probabilities increase through the construction stages; the curves are close to each other for the first 4 stages but for Stages 5 and 6, they are far away from the others. In this model, a trend in which the curves shifted to the right (higher exceedance probability value) during construction stages is observed. As the reservoir is being filled, the exceedance probability is dropped (meaning the curve is shifted to left). The aforementioned results of Model 1 are the same for Model 2 except that due to the foundation inhomogeneity, the curves are close to each other for the first 3 stages. In models without joints, the curves of Stages 1 and 5 perform as the lower and upper bound for all other stages, respectively. As the curves of Model 1 and 2 are compared, it is concluded that the curves for Stages 1, 2, 4 and reservoir impoundment are approximately identical. Hence, the assuming inhomogeneous foundation does not influence the result of these stages. On the other hand, the most difference between these curves is observed for Stage 3. The preceding conclusion is also observed from exceedance probability values presented in Tables 4 which was mentioned earlier. It is noteworthy that the figure incorporating both Models 1 and 2 is not presented in order to prohibit repetition of the results. Moreover, the exceedance probability for the threshold value of 7 MPa is almost zero for Stages 1 to 4 in these models.

Contrarily to Models 1 and 2, the exceedance probability curves of Models 3 and 4 shift to the left as the construction stages are completed. Afterward, the probabilities increase as the reservoir impoundment stage initiates. It is observed that Stage 5 and reservoir impoundment have the least and the highest probability values, respectively. Moreover, the comparison of models with and without joints indicates that the exceedance probabilities of jointed models are significantly lower than the ones excluding the joints. In addition to the impact of construction and peripheral joints on the results, the presence of the soft layer also alters the results in some manner. The exceedance probability curves for Stages 1, 2 and 4 belonging to Models 3 and 4 are almost identical as it is observed in comparison of Models 1 and 2. However, the curves of Stage 3 and reservoir impoundment imply broad difference relative to other curves. On this account, the soft layer which is adjacent to the 3rd stage of construction affects the obtained results of not only Stage 3 but also the reservoir impoundment stage as well.

The histograms of the maximum tensile stress obtained from the 1000 iLHS analyses are plotted in Fig. 6. For the models without the joints, the standard deviation of Model 2 is less than Model 1. Moreover, for Models 1 and 2, the histogram is shifted to the right side (stresses increase) as the construction stages completed, however it is shifted to the left side (stresses decrease) when the reservoir is being filled. For the models including the joints, the stresses are much less than the models without the joints. The trends during the construction stages are in contrary to what was observed for the models without the joints.

Joints sliding and opening limit-state functions

The exceedance probability of the joints sliding and opening limit-state functions during the reservoir impoundment for the threshold values of 2 and 0.5 mm, respectively, are presented in Table 5. These probabilities are meant just for Models 3 and 4 which include the joints. Considering the defined thresholds, the probability of the joint opening is higher than the joint sliding. The presence of the soft rock layer increases the exceedance probability for both limit-state functions.

The exceedance probability curves for the sliding and opening limit-state functions of Models 3 and 4 are presented in Fig. 7. For the same threshold value, the sliding has higher exceedance probability than the opening. In addition, Model 4 shows higher exceedance probability than Model 3 which shows the increasing effect of the soft rock layer within the dam foundation.

Sensitivity analysis

The sensitivity analysis results for all the models are determined using variance-based sensitivity approach. The ranking of random variables based on their influence on the variance of the dam responses are defined using Sobol′ indices. In this investigation, the first-order and total order sensitivity indices are calculated. This section is dedicated to the results of sensitivity analysis obtained for overstressing, joint opening and sliding limit-state functions.

Overstressing limit-state function

The first order sensitivity index (S_i) and the total sensitivity index (S_Ti) are calculated for each model and for all construction and reservoir impoundment stages. The results are illustrated in Fig. 8. In Model 1, the most important variables are determined to be DT, E_c and a_c. The least influential input parameters are the Poisson’s ratio of concrete and foundation rock. By comparing the results of construction stages based on S_i, DT is the most important variable for the first four construction stages and also the reservoir-filling stage. In Stage 5, which is the final stage of construction, E_c becomes the most important variable. On the other hand, the comparison of the results based on S_Ti indicates that when the interaction of each variable with the others is incorporated, DT is the most important parameters for all construction stages. The difference between S_i and S_Ti results of Stage 5 implies that although contribution of E_c in the response variance is greater than DT, the effect of DT interactions is much greater than E_c. Except for the mentioned difference, the trend of both S_i and S_Ti during construction and reservoir impoundment stages are approximately the same.

In Model 2, the variables having the highest effect on the variance of the response are DT, E_c and a_c. In addition to these variables, E_fr for the first three stages and r_c for the last three stages become slightly important. The discrepancy in the trend of S_i is observed in Stage 5 as what was observed in Model 1. In this stage, the ranking of parameters E_c and DT is changed. Furthermore, the Poisson’s ratio of all materials (concrete, foundation rock and soft layer foundation) has the least influence in all stages. Moreover, the comparison of Models 1 and 2 sensitivity indices implies that the trend of the sensitivity measures for these models are similar. Therefore, it is concluded that assuming inhomogeneity for the foundation in the models without joints does not alter the sensitivity measures significantly.

The Sobol′ indices computed for the jointed models show more complicated trend as the dam goes through construction stages than models without joints. In these models, the variables j_p, r_c, E_c, E_f, DT, and a_c are the most important variables based on both S_i and S_Ti indices. However, the ranking of the aforementioned variables differs in some of the stages. For instance, the difference in Sobol′ index values are observed for the first stage of construction and also the reservoir impoundment stage from other stages. Nonetheless, the ranking of the important variables is the same for the other stages in both Models 3 and 4 regarding the first order and total sensitivity measures. In Model 3, the comparison of S_i and S_Ti indices makes it clear that j_p shows the highest effect in the variance when it comes to interaction with other random variables. In Models 3 and 4, j_p and r_c individually have almost the same contribution in response variance but total contribution of j_p is higher than r_c. In Model 3, the least important variables are j_c and n_f. It can be concluded that the friction angle of contraction joints is not influential at all. Contrarily, the peripheral joint’s friction angle plays an important role in the variance of the response. Although the Poisson’s ratio of materials has the least index values in all models, however Poisson’s ratio of concrete shows a slight contribution to the response variance in the jointed models in the construction Stages 4 and 5. In Model 4, the least important variables are j_c, n_f, n_fr, and E_fr.

Moreover, the results of Models 3 and 4 indicate that the inhomogeneity assumption does not alter the results and their trend considerably similar to what was observed in Models 1 and 2. As a deduction, the presence of the soft rock layer is not of paramount importance in all models in sensitivity analysis. The comparison of models including and excluding joints implies that thermal-related variables have more contribution in the variance of responses in models without joint than the models with contraction and peripheral joints.

Joints sliding and opening limit-state functions

The first order and the total sensitivity measures of the joints’ sliding and opening limit-state functions for Models 3 and 4 are presented in Fig. 9. According to the indices computed for Model 3, j_p is the most important variable in both opening and sliding of the defined joints. After that, it comes to r_c and E_c. The fourth important parameter is DT. It is showed that based on the sliding limit-state, r_c is more important than E_c. Contrarily, the indices calculated for E_c is higher than r_c in the opening limit-state function. It can be concluded that the ranking of these two variables is changed for different joints limit-state. The variables n_c and n_f are the least influential variables as what was observed in the previous results. The variables j_p and j_c have a slight contribution to the variance of the sliding and opening responses. By comparing S_i and S_Ti, it can be concluded that the ranking of the variables does not change as the higher order of sensitivity measures are considered in the analysis.

Furthermore, the Sobol′ indices for Model 4 shows similar conclusion as Model 3. All the aforementioned determination of Model 3 can be applied to Model 4. The indices of parameters related to the soft rock layer indicate low values. Accordingly, the presence of the soft rock layer does not have an impact on the variance of the joints’ sliding and opening responses.

Conclusions

In this study, the structural performance of the concrete arch dams during the construction and the reservoir impoundment is investigated taking into account the effects of uncertainties of the model properties and the loading condition in the dam responses. As case study, the Dez arch dam is chosen and modeled along with its rock foundation using the finite element method. The dam-foundation system is refined through four models: (1) The monolithic dam without any joint founded on the homogeneous rock foundation, (2) the monolithic dam founded on the inhomogeneous foundation including typical soft rock layer, (3) the jointed dam including the peripheral and contraction joints founded on the homogeneous foundation, and (4) the jointed dam founded on the inhomogeneous foundation. The models are loaded under the self-weight of the dam considering the five-layer stage-construction, the thermal loading of the post-cooling process and the hydrostatic pressure during the reservoir impoundment. Three performance criteria are defined using the tensile overstressing, joints sliding and joints opening limit-state functions. The random variables are the material properties of the concrete, the rock, and the soft rock layer as well as the friction coefficient of the joints and the post-cooling temperature decrease. The reliability and sensitivity analysis are performed to compute the exceedance probability of the defined limit-state functions and the influence of individual random variables on the variance of the responses. For this purpose, iLHS along with VBSA is implemented in the uncertainty framework.

Regarding the overstressing limit-state function, in Model 1, the exceedance probability is increased through the construction stages; nonetheless, the reservoir impoundment reduces the exceedance probability. The same trend is observed for Model 2 except that the presence of the soft rock layer leads to lower exceedance probability with respect to Model 1. The most reduction in exceedance probability values is observed in Stage 3 adjacent to soft layer. The minimum and maximum exceedance probabilities in these two models are identified in the construction Stages 1 and 5, respectively. Inserting the joints in Models 3 and 4 substantially decreases the overstressing probabilities; the higher value of probability is observed during the reservoir impoundment. The exceedance probability of the joints sliding and opening limit-state functions indicates that the presence of the soft rock layer increases the exceedance probability for both sliding and opening limit-state functions.

In addition to exceedance probability, the Sobol′ indices are determined in order to rank the variables according to their impact on the variance of the dam responses. Similar to reliability analyses, the VBSA is performed for all limit-states, models and stages. The results of overstressing sensitivity analysis make it clear that DT, E_c and a_c are the most important parameters in models excluding joints based on both first-order and total effect indices. The assumption of inhomogeneity of foundation does not cause any changes in the conclusion of the most important variables. In these models, Poisson’s ratio of rock and concrete have almost no effect in the analyses results. Although there is a change in ranking of some parameters during construction and impoundment stages, the trends are almost the same for both S_i and S_Ti. It can be concluded from the comparison of S_i and S_Ti values that the variables that are individually important, are also the most important ones regarding their interaction effect.

Moreover, for jointed models the Sobol′ indices imply that j_p, r_c, E_c, E_f, DT, and a_c are the influential variables for overstressing limit-state. The peripheral joint friction angel is determined to be the most important parameter in the models including joints not only based on first-order index but also total-order sensitivity measure. Contrarily, the contraction joint friction angle is not of importance at all in these models. Likewise, Models 1 and 2, the results of models including joints does not alter with the presence of soft layer. In addition, the Poisson’s ratio of concrete and rock foundation almost has no effect on the variance of the responses.

The novel idea of utilizing the tensile overstressing exceedance probability contours is introduced in this research. These contours are depicted on the dam’s upstream and downstream faces obtained for Models 1 and 2 through all stages. These contours indicate that during the stage-construction, the highest tensile overstressing exceedance probability is concentrated along the dam-foundation interface specifically near both upper ends. In the reservoir-filling stage, the exceedance probabilities encounter reductions in the aforementioned parts and highest values concentrate at the bottom of the dam body on the downstream face. Furthermore, the exceedance probability is below 5% for most parts of the dam during the construction and the reservoir filling. It can be concluded that it is recommended to use a concrete with higher tensile strength in the zones having high exceedance probability values. The foundation inhomogeneity slightly decreases the exceedance probabilities on both faces at the place of intersection of the soft rock layer.

For both joints sliding and opening limit-state functions the Sobol′ indices are approximately the same. The sensitivity measures reveal that the presence of the soft rock layer does not significantly change the Sobol′ indices. For the joints sliding limit-state function for both jointed models, the random variables can be ranked according to their importance as j_p, r_c, E_c, and DT, which j_p is the most important one. For the joints opening limit-state function, the parameter j_p is the most important parameter and then E_c, r_c, and DT, respectively. As there is no difference in the ranking of variables based on first-order and total-order indices, it can be concluded that the interaction of variables does not affect the relative importance of variables in these limit-state function. In addition, there is slight difference between Sobol′ indices of variables in Model 3 and 4 indicating the insignificance of soft layer presence in sensitivity analysis.

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