Kick-off Conference “Risk Management for Large Scale Infrastructures in the Romanian Bulgarian cross border area” Project Code 15.3.1.017

Probabilistic Analysis of Dams Aleksandar Zhelyazkov, Risk Engineering Ltd. Marin Kostov, Ph.D., Risk Engineering Ltd.

Overview PSA Procedure Case study 1 – Double-curvature arch dam Case study 2 – Embankment dam Conclusions

Identifying the hazards PSA Procedure Identifying the hazards Seismic hazard Flooding hazard, etc. Probabilistic analysis required for quantifying the hazard.

Obtaining the conditional failure probability Recognizing vulnerabilities: Deterministic analyses can be employed Conducting probabilistic analyses: probability distributions required for the main parameters (capacity, demand) Sampling methods Obtaining the fragility function – demand level vs. conditional probability of failure .

Obtaining the annual probability of failure The hazard curve and fragility curve are convoluted − P(f/x) . dβ (x) dx .dx which results in the annual probability of failure The probability of failure during the lifetime of the structure can be calculated (Poisson distribution) .

Case Study 1 – Double-curvature arch dam .

Description of the dam - height 130,5 m - crest length 459,4 m - crest thickness 8,8 m - base thickness 26,4 m - 17 separately erected 20 m wide cantilever blocks - contraction joints – 10 cm shear key locks - spillway with four sections .

Deterministic analysis for MCE level - Failure and damage scenarios are indentified 1) Damage on the downstream side of the wall; 2) Damage of the grouting curtain; 3) Damages due to the concrete compressive failure; 4) Damage due to the contraction joints opening;, etc. The estimated realistic failure scenarios are: 5) Sliding of the wall at the contact “concrete – rock”; 6) Sliding of the abutment at the contact “concrete – rock”; 7) Deep sliding of the abutment; 8) Global failure of the structure towards the upstream side and 9) Global failure of the structure towards the downstream side. .

Capacity analysis - Non-linear static analysis – push-over .

Probabilistic analysis - Parameters are varied - The ratio of failure instances is determined - The results are fitted with a continuous cumulative distribution -> fragility function - The fragility curve and the hazard curve are convoluted - Sampling is required to account for epistemic uncertainty - The annual probability of failure or damage is obtained .

. № Damage scenarios Confidence Level 15% 50% 85% 1. Damages on the downstream side of the wall 4,89E-5 8,96E-5 1,64E-4 2. Damages of the grouting curtain 9,25E-5 1,76E-4 3. Damages due to the concrete compressive failure 1,11 E-5 2,48E-5 5,57E-5 4. Damages due to the contraction joints opening 9,33E-7 1,85E-6 3,7E-6   Failure scenario 5. Sliding of the wall at the contact “concrete – rock” 2,54E-7 3,32E-7 4,32E-7 6. Sliding of the abutment at the contact “concrete – rock” 2,31E-6 4,64E-6 9,32E-6 7. Deep sliding of the abutment 2,54 E-7 3,32Е-7 4,33Е-7 8 Global failure of the construction toward the upstream side 4,43E-7 1,74 E-6 6,83E-6 9 Global failure of the construction toward the downstream side

Case Study 2 – Embankment dam Description of the dam - Input as for the 2015 ICOLD Benchmark .

- Truncated normal distribution for friction angle - Lognormal distribution for cohesion - Overtopping fragility curve previously defined - Epistemic uncertainty introduced as variation of the means . Water Level vs. APF

Δ 𝐹 𝜑 = 𝐹 𝜑 + − 𝐹 𝜑 − ; Δ 𝐹 𝑐 = 𝐹 𝑐 + − 𝐹 𝑐 − Probabilistic analysis - Monte Carlo trials CPF= #FOS<1 #Trials ; 𝑁 𝑚𝑐 = [ 𝑑 2 4. 1−𝜀 2 ] 𝑚 ; - Taylor series method for estimation of epistemic uncertainty 𝜎= ( Δ 𝐹 𝜑 2 ) 2 + ( Δ 𝐹 𝑐 2 ) 2 Δ 𝐹 𝜑 = 𝐹 𝜑 + − 𝐹 𝜑 − ; Δ 𝐹 𝑐 = 𝐹 𝑐 + − 𝐹 𝑐 − .

- A family of fragility curves is obtained for the two failure - A family of fragility curves is obtained for the two failure scenarios - > conditional probability of failure - The fragility curves are combined – Common cause adjustment, Theorem of unimodal limits: max P i ≤P≤1− i=1 k 1− P i . Combined family of fragility curves

- Annual failure probability is calculated (convolution) - High annual probability of failure – risk mitigation measures required . Confidence level vs. AFP

Conclusions PSA is a key tool for decision-making PSA for case-study 1 – acceptable APF, vulnerabilities identified PSA for case-study 2 – high APF, mitigation measures required Need for further analyses may appear necessary – e.g. as a result of high uncertainty .

Aleksandar Zhelyazkov Structural Engineer, Risk Engineering Ltd., Contact information: Aleksandar Zhelyazkov Structural Engineer, Risk Engineering Ltd., Tel.: +359 87 9467861, E-mail: Aleksandar.Zhelyazkov@riskeng.bg The content of this material does not necessarily represent the official position of the European Union.