The lack of evaluation standard for safety coefficient based on finite element method (FEM) limits the wide application of FEM in roller compacted concrete dam (RCCD). In this paper, the strength reserve factor (SRF) method is adopted to simulate gradual failure and possible unstable modes of RCCD system. The entropy theory and catastrophe theory are used to obtain the ultimate bearing resistance and failure criterion of the RCCD. The most dangerous sliding plane for RCCD failure is found using the Latin hypercube sampling (LHS) and auxiliary analysis of partial least squares regression (PLSR). Finally a method for determining the evaluation standard of RCCD safety coefficient based on FEM is put forward using least squares support vector machines (LSSVM) and particle swarm optimization (PSO). The proposed method is applied to safety coefficient analysis of the Longtan RCCD in China. The calculation shows that RCCD failure is closely related to RCCD interface strength, and the Longtan RCCD is safe in the design condition. Considering RCCD failure characteristic and combining the advantages of several excellent algorithms, the proposed method determines the evaluation standard for safety coefficient of RCCD based on FEM for the first time and can be popularized to any RCCD.
Roller compacted concrete dams (RCCDs) are now constructed with great frequency in the world. Over the past two decades, many high RCCDs have been built, such as the 130 m high Baise dam and the 200.5 m high Guangzhao dam in China. Currently the highest dam of this type is Longtan Dam at 216 m in China, with DiamerBhasha Dam planned at 272 m in Pakistan [
The RCCD is mainly composed of the body and interface. The interface plays an important role as the weak point of the stability of the RCCD. The stability and safety of a RCCD to guard against sliding along the dam interface are key factors that must be addressed in the design stage of the RCCD. Generally speaking, analysis methods for the RCCD stability should include the rigidbody limiting equilibrium method (LEM) and finite element method (FEM). LEM is normally adopted before considering FEM. Ministry of Water Resources [
When the dam is in critical state, small perturbation will lead the dam away from the equilibrium state into the instability quickly. According to the dissipative structure theory, in the course of dam failure, some abnormal deformations appear certainly with the reduction of system entropy [
RCCD failure is formed under a variety of combined factors. The sliding mechanism and failure mode are determined by the mechanical parameters (deformation and strength parameters) of RCCD interface. The sensitivity analysis of effect factors for the stability of RCCD interface can determine the major control plane of the RCCD, namely, the most dangerous sliding plane which has great significance for RCCD antisliding stability analysis. In this paper, sensitivity analysis for RCCD interface stability based on the RCCD failure criterion with a hybrid algorithm is proposed. Latin hypercube sampling (LHS) was introduced by McKay et al. [
This paper reports the analyses of evaluation standard for safety coefficient of the Longtan RCCD based on FEM. The result shows that the Longtan RCCD is safe in the design condition.
The transversely isotropic and elastoplastic model is used for the constitutive relationship of RCCD material. The isotropic and elastoplastic model is used for the constitutive relationship of the rock material of the dam foundation. The most typical DruckerPrager (DP) criterion [
Zienkiewicz and Pande [
With the gradual increasing of the reduction coefficient
The process of RCCD failure is the evolution process from the “disorder” to the “order.” Because of the correspondence between the system entropy and the system disorder degree, the safety state of RCCD can be easily judged by calculating the entropy value of RCCD in this process [
The information entropy is a measure of the disorder associated with a random variable. The concept of information entropy was introduced by Shannon [
In the process of RCCD failure, the dam interface is main control plane. Therefore, the interface entropy is mainly researched here. The total displacement
With the reduction of the strength parameters by SRF method, the degree of order increases, and the entropy decreases. When the entropy decreases to certain value, the instable interface will be completely damaged.
Because the entropy
The derivation of (
When
Equation (
When
When
Through (
The RCCD failure criterion is defined as follows.
If
If
If
In this paper, the LHS method is adopted to generate the combinations of various mechanical parameters, and the elastoplastic finite element analysis is carried out based on the entropy catastrophe criterion; in the end, the most dangerous sliding plane is obtained by the sensitivity analysis of the mechanical parameters using the auxiliary analysis of PLSR.
The LHS technique is a constrained sampling technique whereby the input parameter range is divided into equiprobable nonoverlapping intervals. Let
Each element of
The basic steps of PLSR are given as follows [
The first principal component
It is not necessary to use all principal components to establish PLSR model, but a truncated way is adopted, which uses frontal principal components,
The principal component
The ability that the factor
The steps of determining the most dangerous sliding plane of the RCCD are shown as follows.
The deformation and strength parameters of the RCCD are taken as the effect factors, and the ranges of the factors are determined.
The combinations of the effect factors are established by the LHS.
The elastoplastic finite element model is established, and the safety coefficients of the dam failure corresponding to the combinations in Step
The sensitivities of mechanical parameters to the whole stability of the dam are determined using the auxiliary analysis of PLSR to analyze the combinations of the effect factors and corresponding safety coefficients of the dam failure. The interface of the most sensitive parameter is the most dangerous sliding plane.
Compared with conventional concrete dam, RCCD has obvious rolled characteristic and relatively weak construction interface. A lot of studies show that at the time of the RCCD failure, an interface is yielded completely, and the sliding channel appears [
Typical failure process of RCCD interface.
In the process of the RCCD failure, the quasielastic state is an important critical point. Before this point, the dam is mainly in elastic state and it is stable as a whole. After this point, the yield area rapidly expands until the dam failure. Therefore, the evaluation standard for RCCD safety coefficient based on the SRF method is given as
If the allowable value
Before determining [
Least squares support vector machine (LSSVM) is a modified version of standard SVM, in which analytical solutions can be obtained solving linear equations instead of a quadratic programming (QP) problem. Given a training data set
The SVM inequality constraint is translated into equality constraint as follows:
The solution of LSSVM regression will be obtained by constructing Lagrangian function as follows:
The solution of (
The Elimination of
LSSVM function can be expressed as follows:
In order to avoid “dimension disaster” in highdimensional feature space, kernel
The kernel function has the following choices: polynomial function, radial basis function (RBF), and sigmoid function. In this paper, RBF is selected as follows:
Main parameters of LSSVM using RBF are regularization parameter
Specific steps are shown as follows.
Determine the ranges of the regularization parameter
Normalize training set and use the cross validation of 1fold to obtain the the best combination of the parameters without affecting model accuracy.
Enlarge the ranges of the parameters obtained in Step
Compared to general grid search method, improved grid search method not only can reduce training volume, but also can improve forecast accuracy of the model.
PSO is a heuristic global optimization algorithm and has been broadly applied in optimization problems. PSO is developed on a very simple theoretical framework and can be implemented easily with only primitive mathematical operators [
The inertia weight
The basic steps of determining the [
The most dangerous sliding plane of the RCCD is determined.
The elastoplastic model is established, and the quasielastic strength reserve coefficient
According to the test result, the ranges of all factors are determined. On this basis, as for the factors which have an unambiguous effect on the dam stability, their values are the maximums or minimums. As for ambiguous factors, LHS is used to construct the combinations of various factors.
The quasielastic strength reserve coefficients corresponding to the combinations constructed in Step
The combinations of various factors and the corresponding quasielastic strength reserve coefficients in Step
The LSSVM model established in Step
According to (
The Longtan dam, the highest RCCD in the world currently, was built on the Hongshui River in China. Longtan RCCD was constructed in two phases. In the first phase, normal water level is 375 m, and the elevation of dam top is 382 m. In the second phase, normal water level is 400 m, and the elevation of dam top is 406.5 m. In this paper,
Material partitions of Longtan
The FEM grid of Longtan
In this paper, RCC is regarded as transversely isotropic material. According to the material partitions shown in Figure
Ranges of interface elastic modulus.
Elastic modulus (GPa) 









Minimum  12.19  9.75  10.56  8.45  9.75  7.80  12.20  9.75 
Maximum  25.50  20.40  22.10  17.68  20.40  16.32  25.50  20.40 
Ranges of interface strength parameters.
Strength parameter 









Minimum  0.82  1.17  0.73  1.04  0.70  0.66  0.86  0.83 
Maximum  1.26  2.04  1.12  1.80  1.08  1.14  1.32  1.44 
Sample and calculation results of mechanical parameters.
Number 


















1  12.41  10.28  12.29  10.14  11.70  9.36  17.30  14.90  1.08  1.71  0.98  1.56  0.97  1.08  1.30  1.41  1.82 
2  12.86  10.64  14.6  11.99  14.54  12.20  23.95  20.22  0.84  1.33  0.84  1.33  0.9  0.94  1.07  1.21  1.57 
3  13.30  11.70  16.14  13.83  17.03  14.47  15.53  14.54  1.11  1.36  0.88  1.08  0.74  0.96  0.94  1.19  1.55 
4  13.74  12.41  18.06  15.68  19.51  7.94  21.73  19.87  0.9  1.56  0.96  1.58  1.02  1.02  0.88  1.33  1.72 
5  14.63  14.54  11.91  11.68  15.61  14.76  25.28  13.83  1.21  1.71  0.93  1.61  0.95  1.04  1.14  1.37  1.85 
6  19.07  9.93  17.68  9.22  16.67  8.23  21.73  17.38  1.16  1.88  1.04  1.71  0.86  1.12  1.04  1.41  1.89 
7  19.95  10.28  19.22  11.07  18.80  11.07  13.75  11.70  0.96  1.53  0.81  1.43  0.71  0.89  1.07  1.10  1.61 
8  21.73  13.48  14.6  8.60  17.74  11.35  23.06  11.35  1.05  1.94  0.96  1.20  1.05  0.92  1.14  1.12  1.72 
9  22.62  20.22  20.75  16.30  18.45  14.19  20.85  15.25  1.02  1.5  0.85  1.28  0.80  0.81  0.88  1.02  1.65 
10  23.66  17.03  12.29  8.60  19.16  13.91  24.39  15.96  0.93  2.03  1.04  1.51  0.87  0.96  1.05  1.15  1.73 
11  23.50  18.45  16.14  12.30  13.12  9.65  22.18  15.61  0.97  1.29  0.74  1.76  1.04  0.86  0.99  1.06  1.79 
12  25.28  19.51  18.45  14.14  15.61  11.92  14.20  9.93  1.24  1.85  1.01  1.53  0.92  0.78  0.94  0.86  1.76 
The ranges of the elastic modulus of the RCC bodies, conventional concrete, and dam foundation rock.
Elastic modulus (GPa) 









minimum  24.38  19.50  23.44  18.75  22.50  18.00  21.38  11.20 
maximum  40.95  32.76  39.38  31.50  37.80  30.24  35.91  20.80 
Using LHS to generate the combinations of the interface mechanical parameters and considering the horizontal elastic modulus bigger than the vertical, the sample results of the mechanical parameters are obtained in Table
Outer loads are mainly the weight load, water load, and uplift pressure. In this example, the upstream water level is the normal water level 375.00 m, and the downstream water level is 225.00 m. The strength reserve coefficients
On this basis, using the auxiliary analysis of PLSR, the important indexes of the mechanical parameters are obtained, and the sensibilities of the mechanical parameters to the
The sensibilities of the mechanical parameters to the strength reserve coefficient of the dam failure.
Through the above analysis, it is shown that the RCC2 interface is the most dangerous sliding plane of the RCCD. The quasielastic strength reserve coefficient of the RCC2 interface
According to the previous analysis, the deterioration of strength parameters and seepage condition and the high water level will make the quasielastic strength reserve coefficient become smaller. So the strength parameters are chosen to be the minimums, the seepage condition is without drainage in dam foundation, the uplift pressure reduction factor
Adopting the LHS and SRF method, the quasielastic strength reserve coefficients corresponding to the combinations of the above deformation parameters are calculated, and the final results are shown in Table
Sample and calculation results of mechanical parameters.
Number 
















[ 

1  23.50  10.71  14.37  9.10  19.23  10.44  23.50  16.46  31.84  22.02  25.19  23.98  30.61  18.86  30.53  14.56  1.22 
2  21.91  11.77  18.98  10.39  20.08  11.97  20.58  12.20  27.86  24.14  33.16  29.08  28.77  19.10  35.76  20.13  1.25 
3  18.98  14.12  13.91  8.91  15.18  13.34  19.51  16.03  38.80  24.41  29.66  28.31  23.27  18.61  22.11  18.78  1.24 
4  17.11  11.13  21.06  11.50  19.87  10.61  24.83  17.31  35.81  31.83  37.63  28.57  35.20  24.73  23.27  12.83  1.26 
5  14.72  10.07  12.06  11.31  18.59  13.51  25.10  13.48  35.15  23.88  26.79  23.47  27.24  24.00  26.47  19.94  1.26 
6  24.30  14.97  18.75  13.53  17.95  10.78  25.37  15.39  34.82  30.24  39.22  22.19  30.30  30.12  27.34  13.98  1.27 
7  20.04  17.10  11.60  10.20  11.77  8.57  17.12  16.67  39.46  30.77  36.67  27.04  30.92  28.89  28.79  19.36  1.25 
8  16.58  13.05  17.14  10.57  15.61  14.02  18.18  17.10  39.13  21.22  34.12  22.45  33.67  26.69  35.18  18.21  1.25 
9  13.65  11.35  18.52  16.66  18.38  13.85  13.93  12.63  37.80  23.35  29.34  21.17  36.12  27.18  30.82  11.30  1.28 
10  16.85  12.41  20.14  14.45  10.71  7.89  22.71  14.97  25.21  29.18  28.38  21.94  34.28  21.79  29.66  14.18  1.27 
11  20.58  14.54  11.83  11.13  16.89  12.83  16.06  11.56  32.17  29.45  28.06  20.66  35.81  21.55  27.92  12.64  1.24 
12  23.77  13.90  15.52  13.71  14.54  10.10  14.99  14.54  40.45  24.94  31.57  18.88  33.98  19.59  21.53  20.70  1.28 
On this basis, the nonlinear relation between the deformation parameters and the quasielastic strength reverse coefficients is determined using the LSSVM. For the LSSVM, the ranges of parameters
The stepwise regression (SWR) and LSSVM performances for training samples are shown in Table
Comparison between target values and estimated values from various algorithms for the training samples.
Criterion  MSE 

SWR 

LSSVM 

Comparison figure of training result of the algorithms.
PSO and Genetic Algorithm (GA) are applied to optimize the established LSSVM model. For the PSO,
Figure
Iteration process.
According to (
Evaluation standard for RCCD safety coefficient based on FEM is a problem requiring urgent solution. In this paper, the DP criterion, SRF method, information entropy, and catastrophe theory are applied to RCCD failure criterion. On this basis, the LHS and auxiliary analysis of PLSR are used to determine the most dangerous sliding plane of RCCD failure. Finally, LHS, LSSVM, and PSO are adopted to establish evaluation standard for RCCD safety coefficient based on SRF method. The proposed method combines the advantages of several excellent algorithms and can be popularized to any RCCD. The analysis of evaluation standard for Longtan RCCD safety coefficient based on SRF method is carried out. The calculation shows that the ultimate failure is controlled by the strength of RCC2 interface. In the design condition, the quasielastic strength reserve coefficient is 1.76 and the allowable value of the quasielastic strength reserve coefficient is 1.31. Due to
The authors do not have any conflict of interests regarding the content of the paper.
The present work is supported by the Special Fund Projects supported by Basic Scientific Research Operating Expenses of CentralLevel Public Academies and Institutes (no. CKSF2015022/GC, CKSF2014037/GC).