# Seismic Safety Assessment of Arch Dams Using an ETA-Based Method with Control of Tensile and Compressive Damage

^{1}

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## Abstract

**:**

## 1. Introduction

#### 1.1. Framework and Motivation

#### 1.2. Objetives and Contributions

## 2. On the Seismic Safety of Concrete Dams

#### 2.1. Models for Non-Linear Dynamic Analysis of Dam–Reservoir-Foundation Systems

**Figure 1.**Representation of an arch dam–reservoir-foundation system. Possible interaction effects for non-linear dynamic analysis.

#### 2.2. Ground Motion

**Figure 2.**Seismic acceleration time histories: (

**a**) design spectrum compatible generated acceleration time history; (

**b**) intensifying dynamic excitation produced based on an ETEF.

#### 2.3. Methods of Analysis

#### 2.4. Seismic Design and Performance Criteria

## 3. ETA-Based Method for Seismic Safety Assessment of Arch Dams

^{+}; a

_{d}

^{+}) and the other with compressive damage (t

^{−}; a

_{d}

^{−}), which correspond to the duration or the respective acceleration level of an intensifying seismic load that the dam can withstand without presenting unacceptable levels of damage. In practice, the endurance limits are determined for two excitation levels, a

_{d}

^{+}and a

_{d}

^{−}, which correspond to the maximum acceleration values of the intensifying seismic action that originate acceptable damage states according to specific criteria defined for tensile, and compressive damage, respectively. In this way, it is expected for concrete cracking under tensions to become excessive after a

_{d}

^{+}(repair interventions required), and for compressive damages to increase until concrete crushing occurs in key areas of the dam after a

_{d}

^{−}(collapse scenario).

_{OBE}< a

_{d}

^{+}, and for the SEE, if a

_{SEE}< a

_{d}

^{−}.

- Phase 1: Development of the 3D finite element model of the dam–reservoir–foundation system and generation the intensifying seismic load.
- Phase 2: Perform the non-linear seismic calculations and output the non-linear response results in multiple time steps (response time histories, deformed shapes with joint movements, principal stresses fields, tensile and compressive damage).
- Phase 3: Estimation of the performance endurance limits and comparison with reference ground motion parameters (OBE and SEE) for the seismic safety assessment.

**Figure 3.**Schematic representation of the proposed methodology for seismic safety assessment of dams based on ETA.

## 4. Used Finite Element Program (DamDySSA)

#### 4.1. Dynamic Behavior of the Dam–Reservoir-Foundation System: Finite Element Formulation

#### 4.2. Non-Linear Time-Stepping Method for Non-Linear Seismic Analysis

^{+}for damage under tension and d

^{−}for damage under compression [82,83] (Figure 6b).

## 5. Results: Seismic Safety Assessment of Arch Dams

#### 5.1. Case Study I: Cabril Dam (130 m-High)

#### 5.1.1. Dam Description and Finite Element Mesh

^{2}and an effective storage of about 615 million m

^{3}. The reservoir level usually ranges from a minimum at el. 265 m to the normal level (NWL) at el. 295 m.

_{w}= 1440 m/s. These material properties have been validated based on experimental results from vibration monitoring data under ambient/operational conditions and during seismic events [15,16,17]. For non-linear seismic analysis, all vertical contraction joints were incorporated into the dam mesh, using appropriate normal and shear stiffness values, null cohesion, and a 30° friction angle. The concrete constitutive damage law was adopted for all dam elements, with tensile strength f

_{t}= 3 MPa and compressive strength f

_{c}= −30 MPa.

#### 5.1.2. Non-Linear Seismic Analysis and Seismic Safety Assessment

**Figure 10.**Seismic safety assessment of Cabril Dam: evolution of tensile and compressive damage for increasing acceleration levels.

#### 5.2. Case Study II: Cahora Bassa Dam (170 m-High)

#### 5.2.1. Dam Description and Finite Element Mesh

**Figure 11.**Case study II: Cahora Bassa Dam. Location and seismic hazard zones. Upstream, cross-section and plan views. Variation of the reservoir level over time.

_{w}= 1500 m/s. All material properties have been calibrated using dynamic experimental data, including modal parameters estimated from measured vibrations and seismic response results [15,16,17]. In order to simulate the non-linear structural response, vertical contraction joints were introduced in the dam body, considering calibrated stiffness values, null cohesion, and a 30° friction angle; the non-linear behavior of concrete is reproduced using the constitutive damage law with tensile strength f

_{t}= 3 MPa and compressive strength f

_{c}= −30 MPa for all dam elements.

#### 5.2.2. Non-Linear Seismic Analysis and Seismic Safety Assessment

**Figure 14.**Seismic safety assessment of Cahora Bassa Dam: evolution of tensile and compressive damage for increasing acceleration levels.

#### 5.3. Discussion

## 6. Conclusions and Future Research

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 5.**Types of finite elements used for discretization of the dam–reservoir-foundation system and of dam discontinuities.

**Figure 6.**(

**a**) Non-linear joint model with cohesion and (

**b**) Concrete constitutive model with two independent damage variables.

**Figure 7.**Case study I: Cabril Dam. Location and seismic hazard zones. Upstream, cross-section and plan views. Variation of the reservoir level over time.

**Figure 9.**Non-linear seismic analysis of Cabril Dam. Deformed shape and principal stresses for t = 5.4 s, radial displacement envelope at the central section (until t = 5.4 s), and displacement time history at the crest central point.

**Figure 12.**Finite element mesh and material properties used for dynamic analysis of Cahora Bassa Dam.

**Figure 13.**Non-linear seismic analysis of Cahora Bassa Dam. Deformed shape and principal stresses for t = 3.2 s, radial displacement envelope for the lateral cantilever (until t = 3.2 s), and displacement time history at the crest of the lateral cantilever.

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**MDPI and ACS Style**

Alegre, A.; Oliveira, S.; Mendes, P.; Proença, J.; Ramos, R.; Carvalho, E. Seismic Safety Assessment of Arch Dams Using an ETA-Based Method with Control of Tensile and Compressive Damage. *Water* **2022**, *14*, 3835.
https://doi.org/10.3390/w14233835

**AMA Style**

Alegre A, Oliveira S, Mendes P, Proença J, Ramos R, Carvalho E. Seismic Safety Assessment of Arch Dams Using an ETA-Based Method with Control of Tensile and Compressive Damage. *Water*. 2022; 14(23):3835.
https://doi.org/10.3390/w14233835

**Chicago/Turabian Style**

Alegre, André, Sérgio Oliveira, Paulo Mendes, Jorge Proença, Rafael Ramos, and Ezequiel Carvalho. 2022. "Seismic Safety Assessment of Arch Dams Using an ETA-Based Method with Control of Tensile and Compressive Damage" *Water* 14, no. 23: 3835.
https://doi.org/10.3390/w14233835