# Advances on the Failure Analysis of the Dam—Foundation Interface of Concrete Dams

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- Water levels in the reservoir higher than design water levels;
- Uplift pressures higher than design;
- Lack of knowledge on shear strength parameters along the failure surface;
- Degradation/aging phenomena in the dam-foundation contact, with loss of shear strength.

- Approach I—Parameter average with no data;
- Approach II—Parameter average with data;
- Approach III—Parameter average with variance reduction factor;
- Approach IV—Spatially variable strength.

## 2. Sliding Strength Models and Uncertainties

#### 2.1. The Mohr-Coulomb Strength Model

- Cohesion acts only on zones subjected to compressive effective stresses;
- If normal effective tensile stresses exceeds the tensile strength of the contact, a crack will develop, starting at the upstream end. The value of uplift inside the crack will be constant and equal to the value of uplift at the upstream end.

_{n}is the total normal stress, u is the pore pressure, tanφ is the local friction coefficient and coh is the local cohesion.

- As water level rises, the uplift increases, reducing frictional strength;
- As water level rises, due to the increase in the overturning moment, tensile stresses may develop at the upstream toe leading to cracking and therefore reducing the area of the contact plane under compressive stresses. This fact has two consequences: loss of cohesive strength and increase in shear stresses, which may reach the peak shear strength locally.

#### 2.2. Uncertainties in the Strength Model

## 3. Modeling Approaches

- Approach I—Parameter average with no data;
- Approach II—Parameter average with data;
- Approach III—Parameter average with variance reduction factor;
- Approach IV—Spatially variable strength.

#### 3.1. Approach I—Parameter Average with No Data

_{contact}, and cohesion, coh

_{contact}, for the whole contact plane. In the absence of specific, on-site data, characteristic values of strength parameters are proposed by the engineer based on his/her own experience or published data of dams with similar characteristics and geology features. Then, a factor of safety against sliding is calculated. The characteristic values shall be selected as cautious estimates of the value affecting the occurrence of the limit state, thus the characteristic value is normally a prudent estimate of the average value on the failure surface, not a particular fractile of the test results. According to Eurocode 7 [23], the calculated probability of a worse value governing the occurrence of the limit state under consideration should not be greater than 5% [24].

_{contact}~ N(μ[tanφ

_{contact}]; SD[tanφ

_{contact}])

_{contact}~ N(μ[coh

_{contact}]; SD[coh

_{contact}])

_{contact}] is the mean and SD[tanφ

_{contact}] is the standard deviation of the friction coefficient for the whole contact plane, while μ[coh

_{contact}] is the mean and SD[coh

_{contact}] is the standard deviation of cohesion for the whole contact plane.

_{local}] is the mean and SD[tanφ

_{local}] is the standard deviation of local friction coefficient, while μ[coh

_{local}] is the mean and SD[coh

_{local}]) is the standard deviation of local cohesion. These probabilistic parameters are the same at each point of the failure surface. If probabilistic normal distributions are assumed for local friction coefficient, tanφ

_{local}, and for local cohesion, coh

_{local}, and assuming that they are statistically independent, uncorrelated random values, then we have Equations (7) and (8).

_{local}~ N(μ[tanφ

_{local}]; SD[tanφ

_{local}])

_{local}~ N(μ[coh

_{local}]; SD[coh

_{local}])

_{contact}] = μ[tanφ

_{local}]

_{contact}] = SD[tanφ

_{local}]

_{contact}] = μ[coh

_{local}]

_{contact}] = SD[coh

_{local}]

#### 3.2. Approach II—Parameter Average with Data

#### 3.3. Parameter Average with Variance Reduction Factor

_{S}

^{2}is the variance reduction factor, considering the spatial extent of the governing failure mechanism; COV

_{inher}is the coefficient of variation of the parameter inherent variability; COV

_{meas}is the coefficient of variation of the measurement errors; COV

_{trans}is the coefficient of variation of the transformation errors; and COV

_{stat}is the coefficient of variation of the statistical parameters.

_{X}

^{2}is the variance reduction factor in the horizontal direction X, which is the upstream-downstream direction, and Г

_{Y}

^{2}is the variance reduction factor in the horizontal direction Y, which the left-right direction of the failure surface. The variance reduction factor in a particular direction, i, is calculated using Equations (23) and (24) which represent a simplification of Vanmarcke’s Equations [28].

_{i}is the Scale of Fluctuation of the strength parameter in the direction i and L

_{i}is the extent of the failure mechanism in the direction i. If SOF

_{i}> L

_{i}Equation (23) is used. If L

_{i}≤ SOF

_{i}then Equation (24) is used.

_{meas}≅ 0. If a well-established model is used to transform measured test results into the required parameter, then COV

_{trans}≅ 0. Assuming that the probabilistic parameters that describe the statistical distribution are known, then COV

_{stat}≅ 0.

#### 3.4. Spatially Variable Strength

^{′}

_{n}(i,j) is the average normal effective stress acting on cell (i,j), tanφ(i,j) is the friction coefficient of cell (i,j), coh(i,j) is the cohesion of cell (i,j) and δ(i,j) is the indicator function defined by Equation (26).

_{S}, is given by Equation (28).

_{S}is the sum of M random variables normally distributed, it will be also normally distributed. Under the hypothesis of no spatial correlation in strength parameters, which means a null scale of fluctuation, the expected value, E[R

_{S}], and the variance, VAR[R

_{S}], can be estimated by Equations (29) and (30).

_{fail}, defined by Equation (32), we can use the normal probability distribution of R

_{S}to estimate the probability of failure.

_{S}, and, in accordance, a set of different N values of the factor of safety, which is defined by Equation (33).

_{i}< 1, N

_{f}, and the total number of sets, N. Obviously, to be able to capture low orders of magnitude of the probability of failure, a large value of N is needed. Another strategy to approximate the probability of failure is to calculate a shorter number of random fields, retrieve the FS

_{i}values, and calculate the expected value and the standard deviation of the sample of FS. Then, assuming that FS is normally distributed, the probability of failure can be calculated according to Equation (32). This latter approach is followed in this paper.

## 4. Application to a Case Study

#### 4.1. Case Study Dam

^{3}. Data for material properties for dam-foundation contact are given in Table 1. Tensile strength in the contact is assumed to be zero.

^{−4}·year

^{−1}.

Sample | Friction Coefficient | Cohesion (MPa) |
---|---|---|

1 | 1.00 | 0.5 |

2 | 0.75 | 0.3 |

3 | 1.03 | 0.3 |

4 | 1.00 | 0.7 |

5 | 1.15 | 0.8 |

6 | 1.33 | 0.2 |

7 | 1.38 | 0.6 |

8 | 1.00 | 0.0 |

9 | 1.15 | 0.1 |

10 | 1.73 | 0.2 |

11 | 1.96 | 0.2 |

12 | 1.88 | 0.4 |

13 | 1.73 | 0.7 |

14 | 1.48 | 0.1 |

15 | 1.88 | 0.4 |

Combination | Water Level (m) | Annual Exceedance Probability (AEP) (year^{−1}) | Drains, K | Prob(K) | Combination Probability (year^{−1}) |
---|---|---|---|---|---|

N°1 | 80 | 10^{−4} | 0.33 | 0.9 | 9.00 × 10^{−5} |

N°2 | 80 | 10^{−4} | 1.00 | 0.1 | 1.00 × 10^{−5} |

#### 4.2. Approach I. Parameter Average with no Data

_{local}~ N(μ[tanφ

_{local}] = 1.36; SD[tanφ

_{local}]) = 0.39; coh

_{local}~ N(μ[coh

_{local}] = 0.37 MPa; SD[coh

_{local}] = 0.25 MPa). Both normal distributions have been tested with the Kolmogorov-Smirnov test and the skewness test, and the result is that the null hypothesis: i.e., the sample comes from a population normally distributed, cannot be rejected.

_{contact}]

_{5%}= 0.72 and [coh

_{contact}]

_{5%}= 0 MPa. Cohesion is assumed to be null as the 5% fractile corresponds with negative values without physical meaning. Case (b) uses the mean values.

μ[tanφ_{local}] | SD[tanφ_{local}] | μ[coh_{local}] | SD[coh_{local}] |
---|---|---|---|

1.36 | 0.39 | 0.37 | 0.25 |

μ[tanφ_{contact}] | SD[tanφ_{contact}] | μ[coh_{contact}] | SD[coh_{contact}] |
---|---|---|---|

1.36 | 0.39 | 0.37 | 0.25 |

Combination n° | Water Level (m) | K Drains | FS (a) | FS (b) | Conditional Probability of Failure | Combination Probability | Probability of Failure |
---|---|---|---|---|---|---|---|

1 | 80 | 0.33 | 1.15 | 2.53 | 5.17 × 10^{−2} | 9.00 × 10^{−5} | 4.65 × 10^{−6} |

2 | 80 | 1.00 | 0.69 | 1.51 | 3.20 × 10^{−1} | 1.00 × 10^{−5} | 3.19 × 10^{−6} |

#### 4.3. Approach II. Parameter Average with Variable Amount of Data

Number of Data | μ[tanφ_{contact}] | SD[tanφ_{contact}] | μ[coh_{contact}] | SD[coh_{contact}] |
---|---|---|---|---|

- | 1.36 | 0.39 | 0.37 | 0.25 |

5 | 1.36 | 0.18 | 0.37 | 0.11 |

10 | 1.36 | 0.12 | 0.37 | 0.08 |

15 | 1.36 | 0.10 | 0.37 | 0.06 |

Number of Data | [tanφ_{contact}]_{5%} | [coh_{contact}]_{5%} |
---|---|---|

- | 0.72 | 0 |

5 | 1.08 | 0.19 |

10 | 1.16 | 0.24 |

15 | 1.20 | 0.26 |

Combination n° | Water Level (m) | K Drains | FS (a) | FS (b) | Conditional Probability of Failure | Combination Probability | Probability of Failure |
---|---|---|---|---|---|---|---|

1 | 80 | 0.33 | 1.84 | 2.53 | 3.00 × 10^{−4} | 9.00 × 10^{−5} | 2.70 × 10^{−8} |

2 | 80 | 1.00 | 1.10 | 1.51 | 1.58 × 10^{−1} | 1.00 × 10^{−5} | 1.58 × 10^{−6} |

Combination n° | Water Level (m) | K Drains | FS (a) | FS (b) | Conditional Probability of Failure | Combination Probability | Probability of Failure |
---|---|---|---|---|---|---|---|

1 | 80 | 0.33 | 2.04 | 2.53 | <10^{−5} | 9.00 × 10^{−5} | <9.00 × 10^{−10} |

2 | 80 | 1.00 | 1.21 | 1.51 | 6.93 × 10^{−2} | 1.00 × 10^{−5} | 6.93 × 10^{−7} |

Combination n° | Water Level (m) | K Drains | FS (a) | FS (b) | Conditional Probability of Failure | Combination Probability | Probability of Failure |
---|---|---|---|---|---|---|---|

1 | 80 | 0.33 | 2.12 | 2.53 | <10^{−5} | 9.00 × 10^{−5} | <9.00 × 10^{−10} |

2 | 80 | 1.00 | 1.27 | 1.51 | 3.85 × 10^{−2} | 1.00 × 10^{−5} | 3.85 × 10^{−7} |

#### 4.4. Approach III. Parameter Average Using Variance Reduction Factor for Spatial Variability

SOF (m) | Lx (m) | Ly (m) | Гx^{2} | Гy^{2} | Гs^{2} |
---|---|---|---|---|---|

10 | 60 | 15 | 0.16 | 0.52 | 0.08 |

20 | 60 | 15 | 0.30 | 0.75 | 0.22 |

**Table 12.**Total coefficient of variation including scale of fluctuation and spatial extent of governing mechanism.

SOF (m) | Friction Coefficient | Cohesion (MPa) | ||||||
---|---|---|---|---|---|---|---|---|

Mean | SD | COV | COV_{TOTAL} | Mean | SD | COV | COV_{TOTAL} | |

10 | 1.36 | 0.39 | 0.29 | 0.08 | 0.37 | 0.25 | 0.68 | 0.19 |

20 | 1.36 | 0.39 | 0.29 | 0.14 | 0.37 | 0.25 | 0.68 | 0.32 |

SOF (m) | µ[tanφ_{contact}] | SD[tanφ_{contact}] | µ[coh_{contact}] | SD[coh_{contact}] |
---|---|---|---|---|

0 | 1.36 | 0.39 | 0.37 | 0.25 |

10 | 1.36 | 0.11 | 0.37 | 0.07 |

20 | 1.36 | 0.19 | 0.37 | 0.12 |

SOF (m) | [tanφ_{contact}]_{5%} | [coh_{contact}]_{5%} |
---|---|---|

0 | 0.72 | 0 |

10 | 1.18 | 0.25 |

20 | 1.05 | 0.17 |

Combination n° | Water Level (m) | K Drains | FS Case (a) | FS Case (b) | Conditional Probability of Failure | Combination Probability | Probability of Failure |
---|---|---|---|---|---|---|---|

1 | 80 | 0.33 | 2.14 | 2.53 | <10^{−5} | 9.00 × 10^{−5} | <9.00 × 10^{−10} |

2 | 80 | 1.00 | 1.28 | 1.51 | 4.48 × 10^{−2} | 1.00 × 10^{−5} | 4.48 × 10^{−7} |

Combination n° | Water Level (m) | K Drains | FS Case (a) | FS Case (b) | Conditional Probability of Failure | Combination Probability | Probability of Failure |
---|---|---|---|---|---|---|---|

1 | 80 | 0.33 | 1.81 | 2.53 | 7.00 × 10^{−4} | 9.00 × 10^{−5} | 6.30 × 10^{−8} |

2 | 80 | 1.00 | 1.08 | 1.51 | 1.64 × 10^{−1} | 1.00 × 10^{−5} | 1.64 × 10^{−6} |

#### 4.5. Approach IV. Spatially Variable Strength

**Table 17.**Factors of safety results for spatially varied strength parameters with scale of fluctuation of 0 m.

Combination n° | Random Field Realization (SOF = 0 m) | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Mean | SD | COV | |

1 | 2.60 | 2.69 | 2.67 | 2.65 | 2.64 | 2.65 | 2.67 | 2.63 | 2.62 | 2.61 | 2.64 | 0.03 | 0.01 |

2 | 2.04 | 2.10 | 2.10 | 2.08 | 2.06 | 2.06 | 2.09 | 2.05 | 2.07 | 2.04 | 2.07 | 0.02 | 0.01 |

**Table 18.**Factors of safety results for spatially varied strength parameters with scale of fluctuation of 10 m.

Combination n° | Random Field Realization (SOF = 10 m) | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Mean | SD | COV | |

1 | 2.85 | 2.73 | 2.28 | 2.56 | 2.62 | 3.25 | 2.77 | 3.08 | 3.04 | 2.44 | 2.76 | 0.30 | 0.11 |

2 | 2.31 | 2.10 | 1.75 | 2.16 | 2.03 | 2.69 | 2.14 | 2.55 | 2.42 | 1.83 | 2.20 | 0.30 | 0.14 |

**Table 19.**Factors of safety results for spatially varied strength parameters with scale of fluctuation of 20 m.

Combination n° | Random Field Realization (SOF = 20 m) | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | Mean | SD | COV | |

1 | 2.65 | 2.84 | 2.32 | 2.10 | 1.77 | 2.77 | 2.63 | 2.75 | 2.40 | 1.45 | 2.37 | 0.47 | 0.20 |

2 | 2.04 | 2.26 | 1.80 | 1.54 | 1.17 | 2.25 | 2.17 | 2.37 | 1.89 | 0.95 | 1.84 | 0.48 | 0.26 |

## 5. Discussion of Results

- | Combination 1 | Combination 2 | |||||||
---|---|---|---|---|---|---|---|---|---|

FS (a) | FS (b) | Conditional Probability of Failure | Probabi-lity of Failure | FS (a) | FS (b) | Conditional Probability of Failure | Probabi-lity of Failure | ||

I: Parameter average (no data) | 1.15 | 2.53 | 5.17 × 10^{−2} | 4.65 × 10^{−6} | 0.69 | 1.51 | 3.20 × 10^{−1} | 3.19 × 10^{−6} | |

II: Parameter average (with N data) | N = 5 | 1.84 | 2.53 | 3.00 × 10^{−4} | 2.70 × 10^{−8} | 1.10 | 1.51 | 1.58 × 10^{−1} | 1.58 × 10^{−6} |

N = 10 | 2.04 | 2.53 | < 10^{−5} | < 9 × 10^{−10} | 1.21 | 1.51 | 6.93 × 10^{−2} | 6.93 × 10^{−7} | |

N = 15 | 2.12 | 2.53 | < 10^{−5} | < 9 × 10^{−10} | 1.27 | 1.51 | 3.85 × 10^{−2} | 3.85 × 10^{−7} | |

III: Parameter average (var reduction) | SOF = 10 | 2.14 | 2.53 | < 10^{−5} | < 9 × 10^{−10} | 1.28 | 1.51 | 4.48 × 10^{−2} | 4.48 × 10^{−7} |

SOF = 20 | 1.81 | 2.53 | 7.00 × 10^{−4} | 6.30 × 10^{−8} | 1.08 | 1.51 | 1.64 × 10^{−1} | 1.64 × 10^{−6} | |

IV: Spatially variable strength | SOF = 0 | 2.59 | 2.64 | ≈0 | ≈0 | 2.04 | 2.07 | ≈0 | ≈0 |

SOF = 10 | 2.27 | 2.76 | ≈0 | ≈0 | 1.71 | 2.20 | 3.10 × 10^{−5} | 3.1 × 10^{−10} | |

SOF = 20 | 1.60 | 2.37 | 1.66 × 10^{−3} | 1.49 × 10^{−7} | 1.05 | 1.84 | 4.06 × 10^{−2} | 4.06 × 10^{−7} |

_{cond}= 5.17 × 10

^{−2}for Combination 1 and FS = 3.20 × 10

^{−1}for Combination 2, while for SOF = 20 m we get P

_{cond}= 7.00 × 10

^{−4}for Combination 1 and P

_{cond}= 1.64 × 10

^{−1}for Combination 2 and for SOF = 10 m we get P

_{cond}< 10

^{−5}for Combination 1 and P

_{cond}= 4.48 × 10

^{−2}for Combination 2.

^{−3}. Comparing this results with those obtained for SOF = 20 m under Approach III, that give P

_{cond}= 7.00 × 10

^{−4}, higher probabilities of failure are obtained with the more rigorous Approach IV, which may indicate that the simplified Approach III may be a little optimistic and does not provide conservative results. On the other hand, for Combination 2, the expected value of the factor of safety is µ[FS] = 1.84 and the standard deviation is SD[FS] = 0.48. The range of values obtained is [0.95–2.37] which shows important variability as well. The estimated conditional probability of failure is Prob(FS < 1) = 4.06 × 10

^{−2}. Comparing again this results with those obtained with Approach III, P

_{cond}= 1.64 × 10

^{−1}, slightly lower probabilities of failure are obtained with the more rigorous Approach IV. The former comparison shows that no clear conclusions can be drawn using Approach III as it may under or over-estimate probabilities of failure.

^{−9}which is probably lower than the figure obtained with Approach III, that gives P

_{cond}< 10

^{−5}. For Combination 2, the expected value of the factor of safety is µ[FS] = 2.20 and the standard deviation is SD[FS] = 0.30. The range of values obtained is [1.75–2.69]. The estimated conditional probability of failure is Prob(FS < 1) = 3.10 × 10

^{−5}, which is lower than the value of 4.48 × 10

^{−2}obtained with Approach III.

**Figure 4.**Factor of safety vs. conditional probability of failure using 5% fractile for strength characteristic values.

_{cond}= 0.5) has been included. It is important to stress the fact that the shown probabilities are conditional probabilities, not total probabilities. This Figure shows the ability of the factor of safety to capture the conditional probability of failure in a consistent way, for different hypothesis of available information and scale of fluctuations of the strength parameters, thus providing a reference to interpret the probabilistic results of a risk analysis applied to a dam. The authors have been using similar plots to help dam engineers not familiar with risk concepts in the interpretation of risk analysis.

**Figure 5.**Factor of safety vs. Conditional probability of failure using 5% fractile for strength characteristic values, and including 5% fractile of factor of safety distribution for Approach IV.

## 6. Conclusions

- Why are failures less frequent than predicted? Typical coefficients of variation for soil engineering properties are reported to be on the order of 20%–30%. Presuming a mean factor of safety of 1.5, corresponding reliability indices (β) are about 1.67, implying probabilities of failure of about 0.05. These are an order of magnitude larger than the observed frequency of adverse performance.
- What is the actual variability of soil and rock properties? Variations in soil engineering data involve at least: (1) actual variability from one point to another; and (2) noise. In addition, there are at least two bias errors that creep into assessments: (3) statistical error due to limited number of observations; and (4) model error due to the approximate nature of our mathematical descriptions of soil behaviour.
- What are the effects of spatial correlation? Geological materials arrive at their present configurations by a geologic process that follows physical principles. Therefore, their physical properties exhibit spatial correlation. While there have been successes in describing spatial correlation statistically and in modelling spatially correlated variables, the techniques for dealing with spatial correlation are difficult to implement, they are poorly understood in practice, thus their consequences are often ignored.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Altarejos-García, L.; Escuder-Bueno, I.; Morales-Torres, A. Advances on the Failure Analysis of the Dam—Foundation Interface of Concrete Dams. *Materials* **2015**, *8*, 8255-8278.
https://doi.org/10.3390/ma8125442

**AMA Style**

Altarejos-García L, Escuder-Bueno I, Morales-Torres A. Advances on the Failure Analysis of the Dam—Foundation Interface of Concrete Dams. *Materials*. 2015; 8(12):8255-8278.
https://doi.org/10.3390/ma8125442

**Chicago/Turabian Style**

Altarejos-García, Luis, Ignacio Escuder-Bueno, and Adrián Morales-Torres. 2015. "Advances on the Failure Analysis of the Dam—Foundation Interface of Concrete Dams" *Materials* 8, no. 12: 8255-8278.
https://doi.org/10.3390/ma8125442