- Research
- Open Access
Risk analysis of gravity dam instability using credibility theory Monte Carlo simulation model
- Cao Xin^{1, 2, 3}Email author and
- Gu Chongshi^{1, 2, 3}
- Received: 19 January 2016
- Accepted: 2 June 2016
- Published: 18 June 2016
Abstract
Risk analysis of gravity dam stability involves complicated uncertainty in many design parameters and measured data. Stability failure risk ratio described jointly by probability and possibility has deficiency in characterization of influence of fuzzy factors and representation of the likelihood of risk occurrence in practical engineering. In this article, credibility theory is applied into stability failure risk analysis of gravity dam. Stability of gravity dam is viewed as a hybrid event considering both fuzziness and randomness of failure criterion, design parameters and measured data. Credibility distribution function is conducted as a novel way to represent uncertainty of influence factors of gravity dam stability. And combining with Monte Carlo simulation, corresponding calculation method and procedure are proposed. Based on a dam section, a detailed application of the modeling approach on risk calculation of both dam foundation and double sliding surfaces is provided. The results show that, the present method is feasible to be applied on analysis of stability failure risk for gravity dams. The risk assessment obtained can reflect influence of both sorts of uncertainty, and is suitable as an index value.
Keywords
- Gravity dam
- Anti-sliding stability
- Risk analysis
- Fuzzy sets
- Credibility theory
Background
Risk analysis of gravity dam anti-sliding instability involves factors of complicated uncertainty. In traditional analysis method, main influencing factors of gravity dam instability (e.g. various imposed loads, material properties and geometric parameters) are all viewed as random variables (Peyras et al. 2012). Now researchers gradually realize that fuzziness is significant in risk analysis of gravity dam instability, including: fuzzy stability failure limit state (Su et al. 2009), fuzzy mechanical parameters of dam foundation (Hua and Jian 2003), fuzzy safety monitoring data of dam and dam foundation (de la Canal and Ferraris 2013; Gu et al. 2011) etc. Therefore, problems of calculating precise failure risk ratio of gravity dam instability influenced by complicated uncertainties and determining corresponding indexes and standards need to be solved.
Since gravity dam involves factors of complicated uncertainty during operation, researchers have applied possibility theory (Zadeh 1965, 1996) into stability failure risk analysis of gravity dam (Su and Wen 2013; Haghighi and Ayati 2016; Li et al. 2011; Sadeghi et al. 2010). Corresponding models have been established to consider both randomness and fuzziness of influence factors. However, in current studies, fuzziness and randomness of factors influencing gravity dam stability are studied separately, which makes the gained risk assessments are described jointly by probability measure and possibility measure. For example, a certain failure risk probability interval is \([\tilde{R}_{L} ,\tilde{R}_{U} ]\) with possibility \(\alpha = 0.5.\) Given that probability and possibility are independent, this description fails to synthetically reflect the contribution of fuzzy factors to risk assessments, and is difficult to clearly characterize the likelihood of risk occurrence in practical engineering.
Actually, fuzziness and randomness of influence factors have no essential difference in stability failure risk analysis of gravity dam. They shall be processed in the same frame. Therefore, this article views gravity dam anti-sliding stability failure as a hybrid event, uses credibility theory (Liu 2006) to consider both randomness and fuzziness of failure criterion, design parameters and measured data simultaneously and establishes credibility stability failure risk analysis model of gravity dam to objectively reflect both sorts of uncertainty. Combining present analysis model with Monte Carlo simulation, calculation method and procedure are proposed to analyze the risk of anti-sliding stability of gravity dam.
Instability credibility risk ratio of gravity dam
Instability risk calculation of gravity dam based on Monte Carlo simulation can be divided into four steps: (1) determine the instability model function of gravity dam; (2) identify and quantify uncertainty of model factors; (3) simulate; (4) analyze results and calculate instability risk ratio.
Instability risk calculation mode of gravity dam
Gravity dam instability is regarded as a random fuzzy event. Fuzziness in the calculation model comes from model factors and failure criterion. There will be three conditions: failure criterion is determined while the model contains both random and fuzzy factors; failure criterion is fuzzy and the model contains random factors only; failure criterion is fuzzy and the model contains both random and fuzzy factors.
- (1)State function is random fuzzy variable and failure criterion is expressed by the limit state function. In other words, when \(Z < K,\) the structure failed and the failure criterion is \(\eta \left( Z \right) = \left\{ {\begin{array}{*{20}l} {1,} \hfill & {{\text{if}}\, Z < K} \hfill \\ {0,} \hfill & { {\text{if}}\, Z \ge K} \hfill \\ \end{array} } \right.\). Then, the credibility risk ratio is degraded into:$$\tilde{P}(Z < K) = \mathop \int \nolimits_{ - \infty }^{K} \varphi \left( Z \right)dZ = \varPhi \left( K \right).$$(2)
Instability model function of gravity dam
According to equal safety coefficient method, when ABC and BCD reach the limit state at the same time, the double sliding surface model will surface buckling failure. In other words, the limit state function meets \(Z = Z_{1} = Z_{2} .\) Based on this implicit function, thrust Q and the limit state function value Z can be calculated.
Risk identification of model factors
Uncertainty of model factors has to be analyzed when discussing instability process of gravity dam using the opinion of mixed uncertainty. Gravity dam, a complicated structural system, will suffer various concentrated forces and distributed forces (e.g. dead load, hydraulic pressure and seepage pressure of dam foundation) during construction and operation. Man causes of gravity dam instability include flood, earthquakes, seepage of dam foundation, material aging, and so on. The following text analyzes uncertainty of water level and material parameters.
Water level is a random variable. Upstream water level is related with pondage, reservoir inflow and discharge flow, while downstream water level is mainly influenced by discharge flow. Reservoir inflow is affected by rainfall and shows high uncertainty. Discharge flow is controlled by hydraulic parameters and also shows certain uncertainty. Additionally, hydrometry has system and observation errors, which shows some uncertainty. Dam water level during the service period is estimated through statistical analysis of complete monitoring sequences, which could use the classic frequency approach to process it uncertainty. Upstream and downstream water levels can be viewed as random variables (Salmon and Hartford 1995) which obey a certain probability distribution.
Material parameters involved in the instability risk analysis of gravity dam include volume weights of dam and rocks as well as shearing friction coefficient and cohesion of dam foundation and slip surface. Volume weight of dam concrete was determined by test. Sample size met requirement of statistical analysis and was treated as a random variable. Uncertainty of mechanical parameters of rocks in dam foundation includes: (1) uncertainty of engineering geological investigation; (2) uncertainty of engineering geological rock group and rock structure. Fuzziness of mechanical parameters of rocks, statistical variables that combines survey crew experiences and rock grouping, has more important significance.
Uncertainty of failure criterion
Random fuzzy simulation
In credibility stability failure analysis model of gravity dam, some variables \(\left( {H_{1} ,\,H_{2} ,\,\gamma_{c} ,\,\alpha_{2} } \right)\) were vested with a random distribution and others \(\left( {f^{{\prime }} ,\,c^{{\prime }} ,\,\alpha_{1} ,\,f_{1}^{'} ,\,c_{1}^{'} ,\,f_{2}^{'} ,\,c_{2}^{'} ,\;\alpha ,\;U_{1} ,\;U_{2} ,\;U_{3} } \right)\) were given with a possibility distribution. If the model contains k random variables \(\left( {X_{1} ,\;X_{2} , \ldots ,\;X_{k} } \right)\) and \(n - k\) fuzzy variables \(\left( {X_{k + 1} ,\;X_{k + 2} ,\; \ldots ,\;X_{n} } \right).\)
- 1.
Generate a random number in \(\left[ {0, 1} \right]\) to every random variable. The variable value which takes this random number as the probability distribution is used as one sample \(\left( {p^{{x_{1} }} ,p^{{x_{2} }} , \ldots ,p^{{x_{k} }} } \right)_{i} .\)
- 2.
Choose one \(\alpha\)-cut and the membership function of the jth sampling is recorded as \(\mu_{j}\).
- 3.
Generate maximum and minimum of the response function under this horizontal cut set, \(u_{ij}\) and \(l_{ij} .\)
- 4.
Choose another \(\alpha\)-cut and repeat Step 2 and Step 3.
- 5.
Choose another random probability distribution sequence and repeat Step 2–4.
Post-processing of simulated results
Application
Basic parameters
Basic random variables and fuzzy variables of dam body and foundation
Variables | Variable type | Parameter |
---|---|---|
Upstream depth \(H_{1} \left( {\text{m}} \right)\) | Random | \(\mu_{{H_{1} }} = 118,\;\sigma_{{H_{1} }} = 2.2\) |
Downstream depth \(H_{2} \left( {\text{m}} \right)\) | Random | \(\mu_{{H_{1} }} = 8,\;\sigma_{{H_{1} }} = 0.8\) |
Volume weight of dam concrete \(\gamma_{c} \left( {{\text{kN/m}}^{ 3} } \right)\) | Random | \(\mu_{{f^{{\prime }} }} = 24.0, \;\sigma_{{f^{{\prime }} }} = 0.5\) |
Shearing friction coefficient of dam foundation surface \(f^{{\prime }}\) | Fuzzy | \(\left[ {0.9,1,1,1} \right]\) |
Shearing cohesion of dam foundation surface \(c^{{\prime }} \left( {\text{MPa}} \right)\) | Fuzzy | \(\left[ {0.8,0.9,1.0} \right]\) |
Uplift pressure reduction coefficient of upstream curtain \(\alpha_{1}\) | Fuzzy | \(\left[ {0.16,0.20,0.38} \right]\) |
Uplift pressure reduction coefficient of downstream curtain \(\alpha_{2}\) | Random | \(\mu_{{\alpha_{2} }} = 0.30,\;\sigma_{{\alpha_{2} }} = 0.03\) |
Basic parameters and fuzzy distribution parameters of basement
Stratum | Parameters of AB (\(c_{1}\)/MPa) | Parameters of BC (\(c_{2}\)/MPa) | Density γ (kN/m^{3}) | α | β | ||
---|---|---|---|---|---|---|---|
T _{3} ^{2-3} | \(f_{1}\) | 0.35 | \(f_{2}\) | [0.93, 0.98, 1.05] | 26 | [21°11′, 21°53′, 22°44′] | 34°26′ |
\(c_{1}\) | 0.1 | \(c_{2}\) | [0.90, 1.00, 1.10] | ||||
T _{3} ^{2-5} | \(f_{1}\) | 0.35 | \(f_{2}\) | [0.93, 0.98, 1.05] | 27 | [21°35′, 23°26′, 25°21′] | 35°15′ |
\(c_{1}\) | 0.1 | \(c_{2}\) | [0.90, 1.00, 1.10] | ||||
T _{3} ^{2-6-1} | \(f_{1}\) | 0.35 | \(f_{2}\) | [0.93, 0.98, 1.05] | 27 | [19°1′, 19°49′, 20°52′] | 34°14′ |
\(c_{1}\) | 0.1 | \(c_{2}\) | [0.90, 1.00, 1.10] | ||||
JC2-3 | \(f_{1}\) | [0.35, 0.40, 0.44] | \(f_{2}\) | [0.93, 0.98, 1.05] | 26 | 16°36′ | 29°5′ |
\(c_{1}\) | [0.10, 0.12, 0.13] | \(c_{2}\) | [0.90, 1.00, 1.10] | ||||
JC2-4 | \(f_{1}\) | [0.35, 0.40, 0.44] | \(f_{2}\) | [0.93, 0.98, 1.05] | 26 | 19°18′ | 33°1′ |
\(c_{1}\) | [0.10, 0.12, 0.13] | \(c_{2}\) | [0.90, 1.00, 1.10] | ||||
JC2-5 | \(f_{1}\) | [0.35, 0.40, 0.44] | \(f_{2}\) | [0.93, 0.98, 1.05] | 26 | 21°40′ | 34°19′ |
\(c_{1}\) | [0.10, 0.12, 0.13] | \(c_{2}\) | [0.90, 1.00, 1.10] | ||||
JC2-6 | \(f_{1}\) | [0.35, 0.40, 0.44] | \(f_{2}\) | [0.93, 0.98, 1.05] | 27 | 15°46′ | 38°51′ |
\(c_{1}\) | [0.10, 0.12, 0.13] | \(c_{2}\) | [0.90, 1.00, 1.10] | ||||
JC2-7 | \(f_{1}\) | [0.35, 0.40, 0.44] | \(f_{2}\) | [0.93, 0.98, 1.05] | 27 | 16°2′ | 36°12′ |
\(c_{1}\) | [0.10, 0.12, 0.13] | \(c_{2}\) | [0.90, 1.00, 1.10] |
Result analysis
Credibility instability risk ratios of dam foundation and hypothetical sliding channels
Slip surface | \(\tilde{P}\) |
---|---|
Dam base | 2.25 × 10^{−5} |
JC2-3 | 5.64 × 10^{−6} |
T _{3} ^{2-6-1} | 2.68 × 10^{−5} |
JC2-4 | 6.61 × 10 ^{ −5 } |
JC2-5 | 3.42 × 10^{−5} |
T _{3} ^{2-5} | 5.08 × 10^{−6} |
JC2-6 | 0 |
JC2-7 | 0 |
T _{3} ^{2-3} | 0 |
Conclusions
Stability failure of gravity dam involves complicated uncertainty. In this article, after adequately considering uncertainty of various factors, a method to analyze stability failure risk of gravity dam and corresponding calculation procedure are proposed. Conclusions are drawn as follows.
To overcome the deficiency of existing risk analysis method that considers randomness and fuzziness separately, this article applies the credibility theory into dam failure and establishes a credibility stability failure risk analysis model of gravity dam to integrate randomness and fuzziness together. Based on Monte Carlo simulation, a mixed algorithm is combined with post-processing of credibility risk analysis mode. And general calculating method is adopted. Stability failure of gravity dam are viewed as hybrid events. Inputs of this calculating method are a series of probability distribution and fuzzy sets. By introducing credibility measure, fuzziness in hybrid variables is mapped from possibility space into the probability space, so risk ratio obtained is represented by a determinate real number.
The example demonstrates that the present method is effective for providing decision support to safety assessment of fuzzy stability of gravity dam. Sensibility analysis results show that credibility risk ratio can sensitively reflect variance change of fuzzy factors. Risk ratio described by credibility measure reflects both randomness and fuzziness and agrees with description habit of traditional probability risk ratio. If there are indexes and standards on credibility risk ratio, credibility theory could become an effective representation tool of fuzzy-random stability failure risk analysis of gravity dam.
Declarations
Authors’ contributions
CX designed the research, drafted the manuscript, reviewed the literature, prepared the Matlab coding for CTMCS model and analyzed the data. GC provided guidance, conducted final editing and proofreading. Both authors read and approved the final manuscript.
Acknowledgements
This research has been partially supported by National Natural Science Foundation of China (SN: 51139001, 51479054) and Research Fund for the Doctoral Program of Higher Education of China (SN: 20020294005, 20120094130003).
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- Altarejos-García L, Escuder-Bueno I, Serrano-Lombillo A (2012) Methodology for estimating the probability of failure by sliding in concrete gravity dams in the context of risk analysis. Struct Saf 36–37:1–13 View ArticleGoogle Scholar
- Baudrit CD, Guyonnet D, Dubois D (2005) Postprocessing the hybrid method for addressing uncertainty in risk assessments. J Environ Eng (N Y) 131:1750–1754View ArticleGoogle Scholar
- Baudrit C, Dubois D, Guyonnet D (2006) Joint propagation and exploitation of probabilistic and possibilistic information in risk assessment. IEEE Trans Fuzzy Syst 14:593–608View ArticleGoogle Scholar
- de la Canal M, Ferraris I (2013) Risk analysis holistic approach as a base for decision making under uncertainties. Chem Eng Trans 33:193–198Google Scholar
- Gu C, Li Z, Xu B (2011) Abnormality diagnosis of cracks in the concrete dam based on dynamical structure mutation. Sci China Technol Sci 54:1930–1939View ArticleGoogle Scholar
- Haghighi A, Ayati AH (2016) Stability analysis of gravity dams under uncertainty using the fuzzy sets theory and a many-objective GA. J Intell Fuzzy Syst 30:1857–1868View ArticleGoogle Scholar
- Hua JH, Jian HH (2003) Analysis of fuzzy-random reliability of slope stability. Rock Soil Mech 24:657–660Google Scholar
- Li X, Liu B (2009a) Foundation of credibilistic logic. Fuzzy Optim Decis Mak 8:91–102View ArticleGoogle Scholar
- Li X, Liu B (2009b) Chance measure for hybrid events with fuzziness and randomness. Soft Comput 13:105–115View ArticleGoogle Scholar
- Li H, Li J, Kang F (2011) Risk analysis of dam based on artificial bee colony algorithm with fuzzy c-means clustering. Can J Civ Eng 38:483–492View ArticleGoogle Scholar
- Liu B (2006) A survey of credibility theory. Fuzzy Optim Decis Mak 5:387–408View ArticleGoogle Scholar
- Ma F, Wu Z (2001) Application of the fuzzy comprehensive appraisal methods in the dam safety monitoring. Water Resour Power 19:59–62Google Scholar
- Peyras L, Carvajal C, Felix H, Bacconnet C, Royet P, Becue J-P, Boissier D (2012) Probability-based assessment of dam safety using combined risk analysis and reliability methods–application to hazards studies. Eur J Environ Civ Eng 16:795–817View ArticleGoogle Scholar
- Sadeghi N, Fayek AR, Pedrycz W (2010) Fuzzy Monte Carlo simulation and risk assessment in construction. Comput Aided Civ Inf 25:238–252View ArticleGoogle Scholar
- Salmon GM, Hartford DND (1995) Risk analysis for dam safety—part II. Int J Rock Mech Min 7(32):342AGoogle Scholar
- Shlyakhtenko D (2005) A free analogue of Shannon’s problem on monotonicity of entropy. Adv Math (N Y) 208:824–833View ArticleGoogle Scholar
- Su H, Wen Z (2013) Interval risk analysis for gravity dam instability. Eng Fail Anal 33:83–96View ArticleGoogle Scholar
- Su H, Wen Z, Hu J, Wu Z (2009) Evaluation model for service life of dam based on time-varying risk probability. Sci China Ser E 52:1966–1973View ArticleGoogle Scholar
- Zadeh LA (1965) Fuzzy sets. Inform Control 8:338–353View ArticleGoogle Scholar
- Zadeh LA (1996) Fuzzy logic and the calculi of fuzzy rules and fuzzy graphs: a precis. Mult Valued Log 1:264Google Scholar