Stability analysis of gravity dams under uncertainty using the fuzzy sets theory and a many-objective GA

Abstract

Uncertainties in the analysis and design parameters of a gravity dam are spread out over the system and make the stability safety factors uncertain too. To analyze the effects of such uncertainties on the dam performance, a conceptual model based on the fuzzy sets theory is introduced here. The input uncertainties are simulated by triangular fuzzy numbers and introduced to the governing equations. To evaluate the extreme values of the dam safety factors, a many objective optimization problem is formed. To solve the problem efficiently, a many-objective genetic algorithm (MOGA) is coupled to the dam stability analysis model. The model is able to estimate all extreme values of the dam safety factors in only one single optimization run. An example gravity dam with and without uncertainty is analyzed respectively by the traditional and the fuzzy approaches. It is found that small input uncertainties can highly influence the analysis responses and, the proposed method can efficiently capture all safety factors uncertainties. For example, the simulations revealed that only ± 10% uncertainty in the dam design parameters would lead to about −346 to +146 % uncertainty in the stability safety factors.

Keywords

Uncertainty stability analysis fuzzy sets theory gravity dam safety factor

Abbreviations

SPF

Standard project flood

OBE

Operating basis earthquake

MCE

Maximum credible earthquake

PMF

Probable maximum flood

Factor of safety

U.D.

Upstream Direction

D.D.

Downstream Direction

P.E.L.

Pre-Earthquake Level

Load Condition

N.A.

Not Applicable

1 Introduction

For designing a new gravity dam or assessing an existing one, the stability analysis is often required. Accordingly, the stability and performance of the structure are evaluated through computing some safety factors. Any decision on the design or rehabilitation of the structure is made based on the evaluated responses. The simulations accuracy is highly dependent on the accuracy of the mathematical model in one side, and the precision of input variables in another side. The latter is a major concern with mathematical simulations and can significantly affect the results reliability. In general, the design parameters of a gravity dam are finally decided based on engineering judgments and previous experiences. This issue leads to appear uncertainty in the results of structural analysis and design. Uncertainty is a general concept that reflects our lack of sureness about something or someone, ranging from just short of complete sureness to an almost complete lack of conviction about an outcome [1]. Stability analysis of gravity dams is of those civil engineering problems that are significantly under the influences of uncertainties. The well-known regulations for designing the gravity dams have been issued by the U.S. Army Corps of Engineers (USACE), the U.S. Bureau of Reclamation (USBR) and the U.S. Federal Energy Regulatory Commission (FERC) these regulations suggest some safety factors for stability of concrete gravity dams.

Also, for handling common uncertainties due to the material properties and loading conditions several design codes have been so far introduced based on the engineering judgments and experiences. Nevertheless, the traditional structural analysis models are not able to evaluate the effects of input uncertainties on the system responses explicitly. To investigate how the input uncertainties are spread out over the system, the exiting analysis models are required to be developed and equipped with other computational models.

Main parameters in evaluating a gravity dam safety factors include the geometry of dam body, material properties, loading conditions and geotechnical properties of the foundation. For the structural analysis of a dam, all these parameters are assumed constant with “crisp” values. However, in reality, because of lack of information, imprecise measurements, and stochastic nature of some data the aforementioned parameters cannot be quantified precisely. As a result, they would introduce uncertainties to the simulations results. In practice, the uncertainties of input data as well as the simulation model are considered indirectly by conservative values of safety factors. However, for more reliable designs the input uncertainties should be directly incorporated into the structural analysis of dam [2]. For this purpose, different approaches have been so far applied to the analysis of engineering systems. They vary from simple engineering judgments to the sophisticated statistical or intelligent models. For a long time, the theory of statistics and probability was the predominant approach for handling uncertainties in simulations [3]. The Monte Carlo simulation method has been one of the most popular approaches for representing Another approach is the application of fuzzy sets theory and fuzzy logic originally introduced by Zadeh [7]. Principles of the theory were then developed by him and his colleagues as well as by Mamdani and Assilian [8], Sugeno [9], Zimmerman [10], Buckley [11], Pedrycz [12],Kandel [13], Bit et al. [14] and Bardossy and Duckstein [15]. The fuzzy sets theory was initially intended to be an extension of dual logic and/or classical sets theory [16] however, during the last decades; the concept of fuzziness has been highly developed in the direction of a powerful fuzzy mathematics. At present, the fuzzy approach can, in some sense, be considered as the most general method for uncertainty analysis of engineering systems [17]. In the field of civil engineering, many investigations have exploited the fuzzy sets theory for handling vague data in analysis and design of structures. Some successful applications of fuzzy sets theory in civil engineering works are; risk analysis of water pollution in hydro-systems [18, 19], flood routing and management [20, 21], analysis of water supply pipe networks [3, 22], safety analysis of marine systems [23, 24], structural analysis [25 –27], coastal engineering [28, 29] and geotechnical engineering [30].

The present study intends to use the fuzzy sets theory and a many-objective GA for the stability analysis of gravity dams to evaluate how the input uncertainties affect the safety factors. For this purpose, a gravity dam simulation model is developed to calculate the stability and stress safety factors. The input uncertainties are represented by triangular fuzzy numbers and introduced to the governing equations. Using the α - cut approach, the fuzzy variables are discretized in limited number of membership function values. In each α - cut two objective functions to find the extreme values of each safety factor are introduced. Finally, a many-objective optimization problem to evaluate the minimum and maximum values of all safety factors in all α - cuts is formed. For solving the problem, a many-objective genetic algorithm (MOGA) is utilized. The proposed model is applied against an example gravity dam and the obtained results are discussed. In what follows, the basic notions, methodology and components of the model are described in details [4 –6].

2 Stability analysis of gravity dams

A concrete gravity dam is a solid structure made of concrete to store or deviate water for different purposes. The section of a gravity dam is approximately triangular in shape as depicted in Fig. 1a. It is so proportioned that the dam’s body weight resists all forces acting on it. The most important part of the analysis and design of a dam is to identify critical loadings and their combinations. The main forces acting on a dam are caused by (1) external water pressure, (2) temperature, (3) internal water pressure; i.e., pore pressure or uplift into the dam body and foundation, (4) weight of the structure, (5) ice pressure, (6) silt pressure, (7) earthquake, and (8) forces from gates or other appurtenant structures. The dam’s body weight and water pressure are directly calculated from the unit weight of the concrete and fluid respectively. However, the other forces need decisions on the expected reliability and safety factors. For this purpose, careful attentions must be paid to estimate the material properties and field data.

In this regard, engineering judgments and experiences are quite important. According to the USACE (1995) [31] provisions, there are seven critical loading conditions (Table 1) for stability analysis of a concrete gravity dam. These conditions are classified into the usual, unusual and extreme loadings as presented in Table 1. Also, the stability and stress criteria for concrete gravity dams for each loading condition are presented in Table 2.

The gravity dam stability analysis includes both overturning and sliding control. For each loading condition, the overturning stability is evaluated by applying all vertical (ΣV) and lateral forces and summing moments (ΣM) about the downstream toe (Fig. 1b). The resultant location along the base (x) is: $x = \frac{Σ M}{Σ V}$ (1)

The eccentricity (e) of the resultant force is: $e = x - \frac{B}{2}$ (2)

Where, B is the length of the section’s base. According to the USACE (1995) criteria, the overturning should be controlled for the resultant locations in Table 2. It is proved that, when the resultant of all forces acting above any horizontal plane through a dam intersects that plane outside the middle third, a non-compression zone will result. For usual loading conditions, the resultant should always lie within the middle third of the base for no tension to develop in the concrete. For unusual loading conditions, the resultant must remain within the middle half of the base. Also, for the extreme loading conditions, the resultant must remain sufficiently within the base to assure that the base pressures are within the prescribed limits. An eccentricity safety factor, FS_e, is defined as the following to control whether the resultant location in the base occurs in the aforementioned regions or not. ${FS}_{e} = \frac{α \frac{B}{2} - e}{α \frac{B}{2}} = \frac{α B - 2 e}{α B}$ (3)

Where, α is the fraction introduced in Table 2 to define the location of the resultant in the base. For usual loading conditions, α is 1/3 and for unusual and extreme loading conditions is respectively 1/2 and 1. When FS_e > 0 the above criteria are met for all loading conditions otherwise, when FS_e ≤ 0, the recommended criteria are violated. Also, the safety factor against the overturning is calculated by dividing the total passive moments ΣM_p by the total active moments ΣM_a about the toe (Fig. 1b) as the following. ${FS}_{o} = \frac{Σ M_{p}}{Σ M_{a}}$ (4)

The overturning safety factor FS_o must be greater than unity.

Another crucial measure is the sliding safety factor FS_s that is to evaluate the resistance of the dam body against the sliding forces. The multiple-wedge analysis [31] is used for analyzing the sliding phenomenon along the base and within the foundation. The sliding safety factor FS_s is defined as the ratio of the maximum resisting shear T_F and the applied shear T along the slip plane at the service loading conditions. ${FS}_{s} = \frac{T_{F}}{T} = \frac{N tan (φ) + cL}{T}$ (5)

In which, N is the resultant of the forces normal to the sliding plane, φ is the soil foundation’s angle of internal friction, c is the cohesion intercept and L is the length of the base in compression for a unit strip of dam.

The stability of gravity dams should be also controlled against the over stressing. Stress analysis is performed to determine the distribution of stresses throughout the structure for static and dynamic loading conditions. The compressing stress in the toe is calculated using the composite stress equation defined for the unit width of the base as the following. $f = \frac{Σ V}{B} + \frac{MB / 2}{I}$ (6)

Where, $I = \frac{B^{3}}{12}$ is the base’s moment of inertia and, M = (ΣV) e is the moment of ΣV about the natural axis.

Substituting equations I and M in Equation (6), the compressing stress in the toe is obtained by the following equation. $f = \frac{Σ V}{B} (1 + \frac{6 e}{B})$ (7)

There is an allowable compressive stress for each loading condition as presented in Table 2. On this basis, the overstressing safety factor FS_T for the compressive stress in the toe is calculated as follows. ${FS}_{T} = \frac{fa}{f}$ (8)

Where, f_a is the allowable compressive stress in concrete for different loading conditions from Table 2. For a safe design FS_T must be greater than unity.

3 Uncertainty and fuzzy analysis

Uncertainties of an engineering system can be classified into the Aleatory and Epistemic uncertainties [32]. By definition, Aleatory means “dependent on luck or chance”. The source of Aleatory uncertainty is the natural variability arises from randomness events in the system parameters. For example, the inherent randomness in earthquake phenomenon is an Aleatory type of uncertainty. Epistemic means “dependent on human knowledge”. The epistemic uncertainty can be reduced in theory by increasing the knowledge and information about the system. The uncertainties resulted from inaccurate statistical analysis, an imprecise laboratory works and human measurements could be treated as epistemic type.

A gravity dam analysis may include significant uncertainties due to the inaccuracy of material properties and loading estimations, mathematical simulations, construction mistakes and stochastic events during the construction and operation. The stability safety factors, introduced above, represent the capacity of a gravity dam to resist different acting loads and loading conditions. However, because of uncertainties in the data the evaluated safety factors may include significant uncertainties. To have a reliable design, analyzing the system uncertainties and their effects on the dam safety factors is highly required. Fuzzy sets theory is used here for uncertainty analysis of gravity dams. In traditional dual logic, a statement can be true or false and nothing in between. In the fuzzy sets theory, a variable either belongs to a set or not. In the fuzzy logic a statement is not only ‘true’ or ‘false’ and partial truth is also accepted. Similarly, partial belonging to a set is possible.

Through the fuzzy sets theory, uncertainties are represented by fuzzy numbers. A fuzzy number N is a set defined on the universe of real numbers N ∈ R. For each variable x ∈ N, μ_N ∈ [0, 1] is called the grade of membership of x in N. If μ_N (x) =0 then, x is called “not included” in the fuzzy set and if μ_N (x) =1, then x is called fully included and if 0 < μ_N (x) <1, x is called the fuzzy member. A so-called α-cut operation (α ∈ μ_N) denoted by N_α is applied to the fuzzy numbers so that, each N_α is a crisp interval defined as [x_a,α, x_b,α] as shown in Fig. 2. If α = 0, the corresponding interval is called the ‘support’ indicated by the interval [x_a, x_b]. For α = 1, if the membership function is triangular, the interval reduces to one crisp value only, x_c, that is, the “most likely” value of N. This definition allows for identifying any crisp interval existing within the fuzzy set as a specific α-cut if the membership function μ_N is continuous and the fuzzy set is normalized and convex. The normalization condition implies that the maximum membership value is 1. $\exists x \in R, μ_{N} (x) = 1$ (9)

The convexity condition indicates that two arbitrary α and α′ intervals satisfy the following relation (N_α = [x_a,α, x_b,α]) : $α^{'} < α \Rightarrow x_{a, α^{'}} < x_{a, α^{'}}, x_{b, α^{'}} > x_{b, α}$ (10)

The triangular and trapezoidal membership functions are very popular in engineering applications. In this study, the triangular membership function is used. To introduce the uncertainty to the design parameters it is considered that each parameter like x has a very likely crips value x_c that includes at most ±Δx uncertainty so that, for a certain α-cut we have, $x_{a, α} = x_{c} - Δ x_{α}$ (11.a)

$x_{b, α} = x_{c} - Δ x_{α}$ (11.b)

Where, Δx_α is the uncertanity at cut α and x_a,α and x_b,α are the lower and upper bounds of x at cut α respectively. Now, the problem of uncertainty analysis of concrete gravity dams is introduced as the following. This problem leads to a many-objective optimization problem which is then solved by a novel scheme of genetic algorithm.

For each design variable x, a most likely value x_c(the crisp value) is decided by engineering judgments, available data and technical references. For each design variable x, maximum uncertainty ±Δx is determined. Each design variable is represented by a triangular fuzzy number A = [x_a, x_c, c_b] with 2 * Δx support. The fuzzy numbers are discretized by a limited number of α-cuts from 0 to 1. As a result, for each variable at each α-cut, an interval is obtained.

According to each α-cut there would be an interval corresponding to each safety factor as a fuzzy response of the dam. The responses intervals show that how the input uncertainties are spread out over the system. To find safety factor fuzziness at each α-cut, it is required to determine the extreme values of that safety factor, the maximum and minimum values due to the input uncertainties. For this purpose, the safety factor is defined as a nonlinear function f (x) of x input uncertainties. At each α-cut, f (x) should be once maximized and once minimized. Imagine that, m types of safety factors are analyzed for the dam and, there are n input uncertainties represented by fuzzy numbers and discretized by number of s α-cuts. In each cut we will have an n-dimensional optimization problem with 2 * m objective function. In fact, the problem objective functions are the safety factors that should be both maximized and minimized. If one decides to consider all s α-cuts in the raised many-objective optimization the number of objective functions become 2m * s for each loading condition. It is worth noting that, generally, three loading conditions are considered for designing a gravity dam and, for each condition four safety factors are evaluated as previously discussed.

The above optimization problem is neither single nor traditional multiobjective since, the trade-off between the objective functions is not important. In this problem, all functions are independent but are required to be optimized simultaneously. One approach to solve this problem is to optimize each objective function separately by a single objective solver. If so, for every α-cut and for every safety factor, the optimization must be applied twice to find the upper and lower bounds of the fuzzy safety factor. By using this approach, for fuzzy analysis of all m safety factors in each α-cut the single-objective optimization must be 2m times applied to the problem. This issue makes the analysis computationally inefficient and burdensome to implement. On the other hand, the current problem has some particular features that help to solve it more efficiently if they are appropriately utilized. First, all objective functions, safety factors, have the same decision variables x (input uncertainties) and decision space. Second, the purpose of the optimization is only to determine the extreme values of the safety factors in each given α-cut and, the trade-off between them is not important. Third, all objectives are evaluated by the same function f (x). In fact, function f is the dam stability analysis model which computes all safety factors once it is run against a set of input uncertainties (x). In other words, every single structural analysis of dam contains all objective values in each given set of decision variables. Utilizing these characteristics in the optimization makes it possible to evaluate a candidate solution simultaneously for all objective functions. In approaches that the problem is treated as an iterative single-objective optimization, most computational efforts of the analysis model are wasted since, at each time only one objective function is optimized. Consideration of the above features implies that the current fuzzy analysis problem can be more efficiently solved if it is treated as a many-objective optimization problem.

To optimize many-objective problems without trade-off between the objective functions, Haghighi [33] introduced a novel scheme of many-objective GA (MOGA) and applied it to analyze random-based transient events in water supply pipe networks. To solve the introduced fuzzy analysis problem of gravity dams, this algorithm is also utilized here. First, based on the governing equations and principles introduced for the stability analysis of gravity dams, a standard simulation model is developed as a function of design variables including uncertainty. This model computes the dam safety factors. A fuzzification subroutine is added to the model that determines the fuzzy intervals of input variables according to each α-cut. Then, a many-objective GA is coupled to the simulation model to obtain the corresponding fuzzy intervals for the dam safety factors. The following section introduces the applied optimization model in details.

4 Many-objective GA

For the new many-objective GA (MOGA) a simple real GA from Goldberg (1989) [34] is adopted as the main structure of the optimization. Then, it is developed by some special features and operators for the selection, pairing and elitism making it possible to optimize multiple objective functions in only one single simulation run. The problem of uncertainty analysis of gravity dams can be defined as a mathematical programming model as follows. $\begin{matrix} Find [\begin{matrix} x_{1, 1} & \dots & x_{1, n} \\ ⋮ & ⋱ & ⋮ \\ x_{U, 1} & \dots & x_{U, n} \end{matrix}] that optimizes (max or min) \\ F_{1 \times U} = {f_{1}, f_{2}, f_{3}, \dots, f_{U}} \\ Subject to, \\ x_{j}^{a, α} \leq x_{ij} \leq x_{j}^{b, α}; i = 1 to U and j = 1 to n \end{matrix}$ (12)

Where, U is the number of objective functions (safety factors), 2 * m for each α-cut, f is the vector of objective functions and, x is the vector the problem decision variables (input uncertainties). Also, $x_{j}^{a, α}$ and $x_{j}^{b, α}$ are respectively the lower and upper bounds of parameter x_j interval corresponding to the applied α-cut (Fig. 2). Each row i in the above matrix serves an objective function f_i. To solve this problem, the following scheme of GA is developed and coupled to the dam stability analysis model.

Similar to every GA, the MOGA starts with an initial population of solutions so-called chromosome in the GA terminology. The initial population has NP chromosomes randomly produced in [0, 1]. A GA population is a NP * n matrix containing normal values of decision variables r encoded in [0, 1]. Every row in the population is a solution alternative needs to be decoded first to be introduced to the simulation model as the following. $x_{j} = x_{j}^{a, α} + r_{i} \times (x_{j}^{b, α} - x_{j}^{a, α}) i = 1 to n$ (13)

The dam stability analysis model is run against every chromosome in the population. For every chromosome, all introduced safety factors are evaluated. This results in a fitness matrix with NP * U elements as the following, $Fitness matrix = [\begin{matrix} f_{1, 1} & \dots & f_{1, U} \\ ⋮ & ⋱ & ⋮ \\ f_{NP, 1} & \dots & f_{NP, U} \end{matrix}]$ (14) Each column is corresponding to a certain function f_i (i = 1 to U) and, each row is corresponding to a certain chromosome in the population.

With respect to each objective function i, each chromosome k (k = 1 to NP) is given a rank number R_i,k. For each objective function, the best chromosome rank number is one and the worst rank number is NP. On this basis, a ranking matrix is formed. Also, for each chromosome an averaged rank number ${\bar{R}}_{k}$ is calculated by averaging the row members of the ranking matrix. ${\bar{R}}_{k}$ averagely represents the fitness of chromosome k with respect to all objective functions.

$\begin{matrix} Ranking matrix & = & [\begin{matrix} R_{1, 1} & \dots & R_{1, U} \\ ⋮ & ⋱ & ⋮ \\ R_{NP, 1} & \dots & R_{NP, U} \end{matrix}] \\ \to \bar{R} = [\begin{matrix} {\bar{R}}_{1} \\ ⋮ \\ {\bar{R}}_{NP} \end{matrix}] \end{matrix}$ (15)

In this step, good chromosomes are identified and used for producing the new generation. To mark the good and bad chromosomes in the ranked population, the following algorithm is applied:

The best chromosome in each column with R = 1 is known as an elite chromosome and is transferred to the mating pool. Accordingly, the number of elites is at least equal to the number of objective functions. However, in some cases, it is possible to save a few more elite chromosomes for some objective functions. In fact, the number of elites N_e is a weighting factor to emphasis on some objective functions which are more intractable in optimization.

Some chromosomes are not elite but are relatively good in all objective functions. To identify good chromosomes, a user-defined threshold rank number R_good between 1 and NP is used. The chromosomes with the averaged rank $\bar{R} \geq R_{good}$ are marked “good” and transferred to the mating pool. By this criterion, NR_good chromosomes are selected from the ranked population. On this basis, there will be $\sum_{1}^{U} N_{e} + {NR}_{good}$ parents in the mating pool waiting for paring to generate the next generation. The rest of chromosomes are marked “bad” and discarded from the GA evolution.

In this step, the couples are selected. Using the tournament selection method, the parents are picked up from the mating pool. For each parent, two chromosomes x and y are randomly selected. x wins the tournament if one of the following conditions is met otherwise, y wins.

x is an elite chromosome and y is not.

x and y are both elite but x has a better averaged rank number $({\bar{R}}_{x} < {\bar{R}}_{y})$ .

None of x and y are elite but x has a better averaged rank number.

The parents are paired and number of $NP - (\sum_{1}^{U} N_{e} + {NR}_{good})$ new offsprings are generated. In this study, the blend crossover method (BLX-α) proposed by Eshelman and Shaffer (1993) [35] is adopted for paring the parents. Every couple produces two children and the new generation as a combination of children and parents is created.

Finally, a few genes in the new population, except the elite chromosomes, are randomly mutated. The mutation ratio μ is a critical parameter to every GA and needs to be initially calibrated.

The stopping criteria are checked for all objective functions. If all functions converged, the optimization is terminated otherwise, the algorithm goes back to step 2 where, the previous population is replaced by the new one and the search is continued.

Also, for more clarification of the proposed algorithm Fig. 3 presents the connections between the optimization model, the dam stability analysis model and the fuzzy analysis model for each applied α-cut.

5 Example

To investigate the model, an example gravity dam is analyzed and designed here according to the USACE (1995) regulations. The most likely design parameters of the dam are presented in Table 3. The normal water level of the dam’s reservoir is considered equal to the standard project flood level. The geometry parameters, material properties, loading values and conditions and physical coefficients of the dam include uncertainty as shown in Table 3. First, the stability analysis of the dam is carried out using the traditional approach without considering uncertainty of input parameters. Then the stability of dam is evaluated considering the input uncertainties using the proposed fuzzy approach. Comparing the results of the aforementioned models reveal that how the input uncertainties are spread out over the dam design and influence its stability.

Traditional method: For stability analysis of a gravity dam the most likely values (crisp values) of design variables are decided based on the engineering judgments and experiences as well as the field data and experiments. For the current case study, these crisp design values are summarized in column 6 of Table 3. Through the traditional method, the dam simulation model is run once resulting in the crisp values of all safety factors. The evaluated safety factors are checked with the safe limits and standard design criteria and accordingly, the stability of dam is judged. The results of this simulation are presented in Table 4. As seen, in all analysis conditions, the safety factors are within the allowable ranges. Accordingly, the traditional approach results in that the dam in different terms of stability is safe.

Fuzzy method: In reality, the design variables cannot be precisely estimated and therefore, contain uncertainty. To investigate that how the uncertainties influence the dam stability, the safety factors are analyzed by the fuzzy method too. The current analysis indicates that in what extent the input uncertainties affect the crisp safety factors calculated by the traditional method. The input uncertainties are represented by isosceles triangular fuzzy numbers. In this example, at most ±10% uncertainty (Δx = 0.1x_c) is introduced to each design parameter presented (Table 3). According to the introduced algorithm as well as the analysis process in Fig. 3, the input fuzzy numbers are discretized by a limited number of α-cuts and the corresponding input intervals as well as the fuzzy safety factors intervals are obtained by the many-objective GA. The input fuzzy variables are discretized in 11α-cuts consisting of α = 0 (the fuzzy number support), 0.1, 0.2 … 0.9 and 1 (the crisp or most likely value). In this problem there are 23 input fuzzy variables and 4 safety factors. For each α-cut it is aimed at estimating the extreme values of each safety factor. Therefore, for each α-cut we need to solve a 23-dimensional optimization problem with 8 objective functions (the maximum and minimum values of 4 safety factors). For better management of the solution procedure, each α-cut with 8 objective functions is solved by the MOGA separately. Also, the model is applied to all loading conditions and the corresponding fuzzy safety factors are evaluated by the proposed algorithm.

To perform the optimization, first, the MOGA parameters are calibrated through some initial trial-and-errors. Accordingly, the population size is decided 100, the mutation ratio is determined 5% and the maximum number of generations is considered to be 1000. The fuzzy safety factors for different loading conditions are illustrated in Fig. 4. This figure shows that how important the fuzzy analysis could be to evaluate the stability of a gravity dam under uncertainties. The supports of the evaluated fuzzy safety factors in α = 0 indicate the maximum uncertainties may appear in the responses of the dam. It is found that the critical combinations of only ±10% input uncertainties could seriously threat the stability of the dam so that, as summarized in Fig. 5, several safety factors would experience more than ±100% uncertainty and violate the allowable ranges. For instance, the input uncertainties have resulted in about –346 to +146% uncertainty in the stability safety factors and –59 to +134 % in the stress safety factor. Similar results are also concluded from Fig. 5 for other safety factors in different loading conditions.

6 Conclusion

The construction and operation of gravity dams are affected by several uncertainties. For a safe design, consideration of input uncertainties in the structural analysis of the dam is essential. This study introduced a fuzzy model for stability analysis of the gravity dams under uncertainties. The model was developed based on the fuzzy sets theory, a standard simulation model for stability analysis of dam and a many-objective optimization. In the proposed model, the input uncertainties are represented by fuzzy numbers with a triangular membership function. A limited number of α-cuts are considered to break the problem into a sequence of optimization sub-problems. For each α-cut the fuzzy input variables are introduced to the model in form of an interval. Consequently, the dam stability analysis in each α-cut results in a many objective optimization problem. To solve the problem efficiently, a many-objective genetic algorithm was developed and coupled to the stability analysis model. Once the optimization is run the extreme values of all safety factors are evaluated. Repeating the optimization for all α-cuts would result in the fuzzy numbers of safety factors.

To investigate the model, it was applied against an example gravity dam. The dam was first design using the traditional approach according to the USACE (1995) regulations. The traditional stability analysis resulted in that the dam is safe in all terms of safety factors in all loading conditions. Then, to show that how the fuzzy model works and how the input uncertainties could spread out over the dam performance, ±10% uncertainty was introduced to the design parameters. The results obtained by the model manifest that small uncertainties in the input variables are critically superposed in the design process and would lead to large uncertainties in the dam safety factors. Accordingly, the dam apparently safe when no uncertainty is considered may considerably fails when the input uncertainties are included in the analysis. Another finding is that the dam safety factors do not behave monotonically with the input uncertainties. In other words, to estimate the extreme values of the safety factors, the application of optimization is necessary. Although it is possible to solve the problem using an iterative-single-objective optimization, solving that by using the many-objective genetic algorithm would significantly decrease the load of computations and exploits the structural analysis model more efficiently.

References

Sayers

P.G.

, Simm

, Hawkes

J.D.

, Ramsbottom

P.J.

, Meadowcroft

D.M.

, Hall

I.C.

and Risk

J.W.

, Performance and uncertainty in flood and coastal defence - a review, 2002.

Verhoosel

C.V.

, Steven

T.P.S.

, Hulshoff

and Gutiérrez

M.A.

, Uncertainty and reliability analysis of fluid structure stability boundaries, AIAA Journal 47(1) (2009).

Haghighi

and Asl

A.Z.

, Uncertainty analysis of water supply networks using the fuzzy set theory and NSGA-II, Engineering Applications of Artificial Intelligence 32 (2014), 270–282.

Jing

, et al., Monte carlo simulation–aided analytic hierarchy process approach: Case study of assessing preferred non-point-source pollution control best management practices, Journal of Environmental Engineering 139(5) (2012), 618–626.

Janssen

, Monte-Carlo based uncertainty analysis: Sampling efficiency and sampling convergence, Reliability Engineering & System Safety 109 (2013), 123–132.

Seifollahi-Aghmiuni

, et al., Effects of pipe roughness uncertainty on water distribution network performance during its operational period, Water Resources Management 27(5) (2013), 1581–1599.

Zadeh

L.A.

, Fuzzy sets, Information and Control 8 (1965), 338–353.

Mamdani

E.H.

and Assilian

, An experiment in linguistic synthesis with a fuzzy logic controller, International Journal of Man-Machine Studies 7(1) (1975), 1–13.

Sugeno

, Industrial Applications of Fuzzy Control. Elsevier Science Inc., (1985), p. 269.

10.

Zimmerman

H.J.

, Fuzzy Set Theory and Its Application. Boston: Dordrecht, 1985.

11.

Buckley , Fuzzy programming and the multicriteria decision problem. In: Kacrzyk

(Ed.), Optimization Models Using Fuzzy Sets and Possibility Theory. Boston: Reidel Publishing Co., 1987.

12.

Pedrycz

, Fuzzy Control and Fuzzy Systems. New York: Wiley, 1989.

13.

Kandel

, Fuzzy Expert System CRC, FL: Boca Ratan, 1992.

14.

Bit

A.K.

, Biswal

M.P.

and Alam

S.S.

, Fuzzy programming approach to multicriteria decision making transportation problem, Fuzzy Sets Systems 50 (1992), 135–141.

15.

Bardossy

and Duckstein

, Fuzzy rule-based modeling with application to geophysical, EconomicBiological and Engineering System (1995), Ist Edition, CRC Press.

16.

Zimmermann

H.J.

, Fuzzy set theory, WIREs Comput Stat 2 (2010), 317–332.

17.

Liu

and Rao

Singiresu S.

, Fuzzy finite element approach for analysis of fiber-reinforced laminated composite beams, AIAA Journal 43(3) (2005), 651–661.

18.

Liu

, et al., Interval-parameter chance-constrained fuzzy multi-objective programming for water pollution control with sustainable wetland management, Procedia Environmental Sciences 13 (2012), 2316–2335.

19.

Liu

, et al., Fuzzy two-stage non-point source pollution management model for agricultural systems— A case study for the Lake Tai Basin: China, Agricultural Water Management 121 (2013), 27–41 .

20.

Hashemi

, et al., An extended compromise ratio model with an application to reservoir flood control operation under an interval-valued intuitionistic fuzzy environment, Applied Mathematical Modelling 38(14) (2014), 3495–3511.

21.

Lee

, Jun

K.S.

and Chung

E.-S.

, Robust spatial flood vulnerability assessment for Han River using fuzzy TOPSIS with α-cut level set, Expert Systems with Applications 41(2) (2014), 644–654.

22.

Haghighi

and Keramat

, A fuzzy approach for considering uncertainty in transient analysis of pipe networks, Journal of Hydroinformatics 14(4) (2012), 1024–1035.

23.

Acosta

, Wu

and Forrest

B.M.

, Fuzzy experts on recreational vessels: A risk modelling approach for marine invasions, Ecological Modelling 221(5) (2010), 850–863.

24.

Navas

J.M.

, Telfer

T.C.

and Ross

L.G.

, Spatial modeling of environmental vulnerability of marine finfish aquaculture using GIS-based neuro-fuzzy techniques, Marine Pollution Bulletin 62(8) (2011), 1786–1799.

25.

Valdebenito

, et al., American Society of Civil Engineers, pp, Approximation Concepts for Fuzzy Structural Analysis, in Vulnerability, Uncertainty, and Risk (2014), 135–144.

26.

Balu

A.S.

and Rao

B.N.

, Efficient Assessment of Structural Reliability in Presence of Random and Fuzzy Uncertainties, Journal of Mechanical Design, American Society of Mechanical Engineers 136(5) (2014), 051008–051008.

27.

Mullen

and Muhanna

, Bounds of structural response for all possible loading combinations, Journal of Structural Engineering 125(1) (1999), 98–106.

28.

Kim

, Kim

K.H.

and Shin

B.-S.

, Fuzzy model forecasting of offshore bar-shape profiles under high waves, Expert Systems with Applications 41(13) (2014), 5771–5779.

29.

Gray

S.R.J.

, et al., Are coastal managers detecting the problem? Assessing stakeholder perception of climate vulnerability using Fuzzy Cognitive Mapping, Ocean & Coastal Management 94 (2014), 74–89.

30.

Cheng

, et al., Evolutionary fuzzy neural inference system for decision making in geotechnical engineering, Journal of Computing in Civil Engineering 22(4) (2008), 272–280.

31.

USACE, Gravity Dam Design EM 1110-2-2200. US Army Corps of Engineers: Washington, DC, 1995.

32.

Ayyub

B.M.

and McCuen

R.H.

, Probability, statistics, and reliability for engineers and scientists, ed. 2. 2003, Chapman Hall/CRC,Boca Raton, Fla.: Chapman Hall/CRC.

33.

Haghighi

, Analysis of transient flow caused by fluctuating consumptions in pipe networks: A many-objective genetic algorithm approach, Water Resources Management 29(7) (2015), 2233–2248.

34.

Goldberg

D.E.

, Genetic Algorithms in Search, Optimization, and Machine Linearning. Reading, MA: Addison-Wesley, 1989.

35.

Eshelman

L.J.

and Schaffer

J.D.

, Real-Coded Genetic Algorithms and Interval-Schemata, in Foundations of Genetic Algorithms, Darrell

W.L.

, Eds., Elsevier, 1993, pp. 187–202.