Mathematical Problems in Engineering

Volume 2016, Article ID 4264627, 12 pages

http://dx.doi.org/10.1155/2016/4264627

## Application of Probabilistic Method to Stability Analysis of Gravity Dam Foundation over Multiple Sliding Planes

School of Hydraulic Engineering, Dalian University of Technology, No. 2 Linggong Road, Dalian 116024, China

Received 9 March 2016; Accepted 5 June 2016

Academic Editor: Egidijus R. Vaidogas

Copyright © 2016 Gang Wang and Zhenyue Ma. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The current challenge to the engineering profession is to carry out probabilistic methods in practice. The design point method in generalized random space (DPG method) associated with the method of divided difference can be utilized to deal with the complex problem of probability calculation of implicit performance function with nonnormal and correlated variables. For a practical concrete gravity dam, the suggested method is performed to calculate the instability probability of the dam foundation over multiple sliding places. The general conclusions drawn in the paper are identical to those in other research and the method is proved to be feasible, accurate, and efficient. As the same analysis principle, the method can also be used in other similar fields, such as in fields of slopes, earth-rock dams, levees, and embankments.

#### 1. Introduction

It is a trend to use the probabilistic method for evaluation of the risk of failure in almost all engineering fields, including some in geotechnical or structural engineering [1–3]. The current challenge to the geotechnical and structural engineering profession is to carry out probabilistic methods in practice [4]. Many researchers have focused on the research topics of reliability-based design and risk analysis and made progress in resolving the problem about the instability failure of slopes, dams, levees, embankments, and other geotechnical or structural engineering fields in recent years [1, 4, 5].

The Monte Carlo method (MC) is usually used to estimate the reliability index, . However this method is rarely adopted due to its huge calculation time [5, 6]. Besides MC, many other methods have been proposed for reliability analysis, such as the first-order reliability method (FORM) [1, 7], second-order reliability method (SORM) [8, 9], and some improved methods. In order to obtain , the partial derivatives of performance function, , are needed in these methods. But, in geotechnical or structural engineering, is usually implicit, and its partial derivatives are complex or difficult to be derived from implicit to explicit. Therefore, these conventional reliability methods only can be used to analyze small structures.

Always, the response surface methods (RSMs) are utilized to obtain the solution of for reliability problem with implicit for complex structures. Wong applied RSM to evaluate the reliability of a homogeneous slope [10]. Moore and Sa constructed confidence intervals about the difference in mean responses at the stationary point and alternate points based on the proposed delta method and F-projection method and compared coverage probabilities and interval widths [11]. Zheng and Das proposed an improved response surface method and applied that to the reliability analysis of a stiffened plate structure [12]. Guan and Melchers evaluated the effect of response surface parameter variation on structural reliability [13]. Gupta and Manohar used the response surface method to study the extremes of Von Mises stress in nonlinear structures under Gaussian excitations [14]. Wong et al. proposed an adaptive design approach to overcome the problem, which was that the solution of the reliability analysis initially diverged when the loading was applied in sequence in the nonlinear finite element (NLFE) analysis, and made several suggestions to improve the robustness of RSM [15]. Xu and Low used RSM to approximate the performance function of slope stability in slope reliability analysis, in which the response surface is taken as a bridge between standalone numerical packages and spreadsheet-based reliability analysis [16]. Cheng et al*.* presented a new artificial neural network (ANN) based response surface method in conjunction with the uniform design method for predicting failure probability of structures [17]. Gavin and Yau described the use of higher order polynomials in order to approximate the true limit state more accurately in contrast to recently proposed algorithms which focused on the positions of sample points to improve the accuracy of the quadratic stochastic response surface method (SRSM) [18]. Zou et al. presented an accurate and efficient MC for limit state-based reliability analysis at both component and system levels, using a response surface approximation of the failure indicator function [19]. Nguyen et al. proposed an adaptive construction of the numerical design, in which the response surface was fitted by the weighted regression technique, which allowed the fitting points to be weighted according to their distance from the true failure surface and their distance from the estimated design point [20]. Similar to support vector machine- (SVM-) based RSM, Samui et al. adopted relevance vector machine- (RVM-) based first-order second-moment method (FOSM) to build a RVM model to predict the implicit performance function and evaluate the partial derivatives with sufficient accuracy [21]. Tan et al. discussed similarities and differences between radial basis function networks (RBFN) based RSMs and SVM-based RSMs, which indicated that there is no significant difference between them, and then proposed two new sampling methods and a hybrid RSM to reduce the number of evaluations of the actual performance function [22, 23].

However, RSM and its improved methods mentioned above are relatively complicated and need much more computing cost, since they have a lot of iterating calculations at different design points associated with numerical method (i.e., finite element method) to fit limit state curved surface, which is represented by . In this paper, a novel design point approach in generalized random space (DPG method) associated with the method of divided difference is proposed and applied to analyze the probability of gravity dam foundation instability over multiple sliding planes. In the method, implicit performance function with nonnormal and correlated variables is considered, and iterating calculation for is performed in generalized random space directly. The whole procedure for reliability analysis does not need much preparation mathematically and is relatively simple for the complex engineering problems.

#### 2. Basic Principle of Computing Instability Probability of Gravity Dam Foundation

Due to the complexity of foundation stability against sliding of gravity dam, for simplification, the instability probability is commonly evaluated using models of single or dual sliding plane(s) for the dam foundation [24]. However, instability of foundation over multiple sliding planes is a general case occurring in bedrock under most engineering geological conditions for a gravity dam. Therefore, it is necessary to apply a model of multiple sliding planes to analysis on instability probability of gravity dam foundation, but there is limited existing research regarding this case. In traditional deterministic analysis, equal safety coefficient method (namely, equal- method) is used to analyze and assess the stability failure of dam foundation. In the method, factor of safety, , is regarded as an unknown number in nonlinear equations and needs to be solved by iterative calculating numerically, similar to the application of the methods of Bishop, Spencer, and Janbu as well, which are widely used for stability analysis on slope. The reliability index or failure probability is calculated by implicit performance function, , associated with the factor of safety . Accounting for this, the conventional FORM and its improved methods (such as JC method which combines the method of equivalent normalization transformation with FORM and is recommended by the Joint Committee of Structural Safety) cannot be directly applied since the partial derivatives of with respect to the variable cannot be evaluated directly [1, 25]. Hence, if a partial derivative of is calculated by divided difference mathematically, the value of the derivative at any design point will be worked out easily, and the limit state curved surface will not be fitted completely as RSM [1, 25]. Usually, the JC method associated with the method of divided difference could calculate design points by iteration on the actual limit state curved surface so that it could avoid fitting the overall curved surface. The method can provide sufficient accuracy and efficiency to calculate probability of failure as prone to dealing with complex problems for large projects, which has been verified by many examples in some literatures. So it is a good idea to use the similar method to calculate instability probability of gravity dam foundation over multiple sliding planes.

##### 2.1. Uncertainty in the Analysis

Probabilistic methods can reveal the contributions of different components to the uncertainty in the analysis on the probability of failure of dam foundation for gravity dam. The uncertainty related to instability of dam foundation could be classified as follows: (1) the uncertainty of hydrographic and hydraulic parameters; (2) the uncertainty of mechanical parameters of rocks and soils; (3) the uncertainty of seismic excitation; (4) other uncertainty, underlying the process of dam design, construction, operation, management, and so on.

The second type of uncertainty above is studied mainly in the paper, and others are ignored by deterministic engineering applications just like in conventional analysis for dam stability against sliding. The uncertainty of mechanical parameters of rocks and soils might be enslaved to analysis methods, due to subjective or objective conditions, derived from the lack of knowledge, and so on. Always, the laboratory and* in situ* measured data of the properties of rocks or soils are inadequate. So there is demonstrable distinction between the measured values from the statistics of the small sampling and the true values of the actual physical and mechanical properties existing in real world.

Many of studies have shown that friction coefficient and cohesion force against shear fracture of rocks or discontinuities are two important uncertainty parameters influencing the foundation stability of gravity dam. Firstly, by a great number of data of field measurements and their statistical back-analyses, there are obvious scatter in the spatial variation of the two shear strength parameters not only for rock mass and faults of different projects but also for different types of rock mass and faults at different locations in the same project. Chen et al. indicated that the variability of is nearly at the same level, but the variability of is not so by the study on the variability of the shear strength parameters of the foundation rock mass and faults at several dam sites in China [26, 27]. Secondly, the probability distributions and negative correlation of the two parameters cannot be ignored. The random parameters might be normally distributed or ln-normally distributed.

In our probabilistic analysis, the two parameters against shear fracture, and , are to be regarded as random variables following some distributions, such as normal or ln-normal distribution, and considering their variability and correlation.

##### 2.2. Performance Function for Computing Instability Probability

###### 2.2.1. Equivalence Safety Factor Method for Foundation Instability over Multiple Sliding Planes

The sliding patterns of gravity dam foundation could be divided by slip paths over single, dual, and multiple plane(s). The last one is the general case for analysis of foundation instability of gravity dam by assuming the potential failure surface as shown in Figure 1. There are slip planes of sliding wedges from upstream to downstream in the foundation. The th wedge is subjected to (a) the vertical normal stress and the horizontal shear stress induced by action of dam or other upside loads along the topside of the wedge, (b) dead load , (c) force and uplift normal to the th slip plane, (d) resistance provided by the th wedge, (e) angle of inclination of the interface between the th and th wedges, and (f) angle of inclination , the length , and the shear strength parameters and of the th slip plane.