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State of Art in
Design and Analysis of
Concrete Gravity Dams
Dr. Bakenaz A. Zeidan Faculty of Engineering
Tanta University, Egypt
1/24/2015 Dr. Bakenaz A. Zeidan
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Faculty of Engineering – Tanta University
2014
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DESIGN AND ANALYSIS OF CONCRETE
GRAVITY DAMS
Presentation Outline
Introduction to Gravity Dams
About Concrete Gravity Dams
Cases of Loading on Gravity Dams
Theoretical Approach Gravity Dams
Modeling of Gravity Dam
Analysis of Gravity Dams
Safety Criteria for Gravity Dams
Recent Trends in Gravity Dams
Summing up
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INTRODUCTION
Many concrete gravity dams have been in service for over 50
years.
Older existing dams may fail to meet revised safety criteria
and structural rehabilitation.
The identified causes of failure, based on a study of over 1600
dams [1] are: foundation problems (40%), inadequate spillway
(23%), poor construction (12%), uneven settlement (10%),
and high pore pressure (5%), acts of war (3%), embankment
slips (2%), defective materials (2%), incorrect operation (2%),
and earthquakes (1%).
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INTRODUCTION
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Classification of Dams Worldwide
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INTRODUCTION
STAGE I: PLANNING STUDIES
• Geography, geology, hydrology, construction materials,
STAGE II: Design
• Dam profile, loads determination, stability analysis, stress analysis, safety criteria
STAGE III:
Construction, operation, and maintenance
•Channel diversion, foundation treatment, concrete curing, construction joints, instrumentation, operation, maintenance
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PLANNING STUDIES
5/13/2013
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DR. BAKENAZ ZEDAN
TOPOGRAPHIC SURVEYS
FOUNDATION STUDIES
MATERIALS AND CONSTRUCTION
FACILITIES
HYDROLOGIC STUDIES
RESERVOIR OPERAION
STUDY
INTRODUCTION
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Choice of Dam Geometry &
Material Properties
• H, B, hu, hd, hs, γc, γw, γs, α ….
Determination of Acting
Loads
• W , P , P , c u d
W Ps, Ws H, w
V, Phd , …….
Stability & Stress Analysis
• FSo, FSs, σheel, σtoe ,
σ1. σ2, σmax.
σmin, qmax ,…
DESIGN
STAGES
INTRODUCTION
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INTRODUCTION
STABILITY ANALYSIS
Allowable F.O.S. against overturning
Allowable F.O.S. against forward
sliding
STRESS ANALYSIS
σmax ≤ max. allowable compression stress
for dam concrete
σmax ≤ max. allowable bearing stress for
dam foundation
σmin ≥ 0 .0 no tension is allowed
qmax ≤ max. allowable shear stress for dam
concrete
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SAFETY CRITERIA
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INTRODUCTION C
ON
ST
RU
CT
ION
Channel diversion,
Foundation Treatment
Concrete Curing,
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INTRODUCTION C
ON
ST
RU
CT
ION
Instrumentation
Operation
Maintenance
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CONCRETE GRAVITY DAMS
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CONCRETE GRAVITY DAMS
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Figure 1: Different types of concrete dams (2).
Types of gravity dams: Gravity dams
Buttress dams Arch dams
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CONCRETE GRAVITY DAMS
Basic Definitions
Length of the dam
Structural height of the dam
Max. base width of the dam
Toe and Heel
Hydraulic height of the dam
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CONCRETE GRAVITY DAMS
Dam Concrete Static Properties (USBR)
• Strength
• Elastic Properties
• Thermal Properties
Dam Concrete Dynamic Properties
• Strength
• Elastic Properties
• Average Properties
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CONCRETE GRAVITY DAMS
Average Properties (USBR)
Compressive strength- 3,000 to 5,000 Ibs/in2 (20.7 to 34.5 MPa)
Tensile strength- 5 to 6 % of the compressive strength
Shear strength: Cohesion-about 10% of the compressive strength
Coefficient of internal friction- 1.0
Poisson’s ratio- 0.2
Instantaneous modulus of elasticity- 5.0 x 106 lbs/in2 (34.5 GPa)
Sustained modulus of elasticity- 3.0 x 106 lbs/in2 (20.7 GPa)
Coefficient of thermal expansion- 5.0 x 10-6/“F (9.0 x l0-6PC)
Unit weight- 150 Ibs/ft3 (2402.8 kg/m3)
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CONCRETE GRAVITY DAMS
Foundation Properties
• Deformation Modulus
• Shear Strength
• Pore Pressure and Permeability
• Treatment
• Compressive and Tensile Strength
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CONCRETE GRAVITY DAMS Criteria- Foundation data required for the analysis of a gravity dam (4):
The deformation modulus of each type of material
within the loaded area of the foundation.
The effects of joints, shears, and faults obtained by
direct (testing) or indirect (reduction factor) methods.
An effective deformation modulus.
The effective deformation moduli.
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GRAVITY DAM LOADS
Factors to be considered as contributing to the loading combinations for a gravity dam are: Reservoir & tail water loads Temperature Internal hydrostatic pressure Dead weight Wind Wave Ice Silt Earthquake
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GRAVITY DAM LOADS
Static Loads:
Dead weight
Reservoir hydrostatic pressure
Tail water hydrostatic pressure
Uplift pressure
Sand and silt
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GRAVITY DAM LOADS
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Reduced Uplift Extreme Uplift
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GRAVITY DAM LOADS
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Seismic Loads
Dam Body
Horizontal Inertia/ Seismic Forces
Vertical Inertia/ Seismic Forces
Reservoir Body
Hydrodynamic Pressures
in Excess to Hydrostatic Pressures
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GRAVITY DAM LOADS
Earthquake Excitation
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TAFT GROUND MOTION, 1952
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GRAVITY DAM LOADS
Seismic Loads
Horizontal and vertical accelerations are generated due to earthquake excitations which are not equal, horizontal being of greater intensity than vertical.
Earthquake acceleration ϋg is usually designated as a fraction of the acceleration due to gravity g and is expressed as:
• ϋg = α⋅g
• where α is called the Seismic Coefficient and
αh :Horizontal seismic coefficient = 1.5 α
αv : Vertical seismic coefficient = 0.75 α
Seismic force = M. ϋg = M. α⋅g = W. α
Horizontal inertia force H= W. αh
Vertical inertia force V = W. αv
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GRAVITY DAM LOADS • Seismic Loads (Chopra 2012)
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Nonlinear α
Constant α
Linear α
Spectrum
approximate
simplified
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GRAVITY DAM LOADS
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Seismic Loads
H
V
W
H
Inertia forces due to earthquakes
Horizontal inertia force H= W. αh
Vertical inertia force V = W. αv
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GRAVITY DAM LOADS
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Hydrodynamic Pressure Westergard Equation (1933)
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GRAVITY DAM LOADS
Hydrodynamic Pressure Westergard Equation (1933)
The hydrodynamic pressure generated due to the horizontal movement of the water body in the
reservoir during earthquakes may to be calculated by:
P= Cs. γw.α.h
where:
P: Hydrodynamic Pressure
in KN/m2 depth y below reservoir surface
Cs : is a shape factor
γw : unit weight of reservoir water in KN/m3
α : seismic coefficient
h: reservoir depth (m)
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GRAVITY DAM LOADS Load Combinations
1-Usual (Normal) 2-Unusual (Maximum) 3- Extreme ( Earthquake)
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Stresses in Koyna Dam (Thailand) due to
Earthquake
Principal Stresses in a concrete gravity Dam
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GRAVITY DAM LOADS
▫ Stability Criteria Accounts for :
Sliding stability
Tension stress
Compressive stress
Displacement
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Figure 10: Simple dam model showing critical areas for compressive (blue), tensile (green) and sliding (red).
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GRAVITY DAM LOADS
• Factors of Safety (USBR)
(1) Compressive stress.-
The maximum allowable compressive stress should in no case exceed:
” 1,500 lbs/in2 (10.3 MPa for “Usual Loading Combinations”.
“2,250 lbs/in2 (15.5 MPa) for “Unusual Loading Combinations”.
A safety factor greater than 1 for “Extreme Loading Combinations”.
Safety factors of 4.0, 2.7, and 1.3 should be used in determining
allowable compressive stresses in the foundation for “Usual,”
“Unusual,” and “Extreme Loading Combinations,”
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GRAVITY DAM LOADS
• Factors of Safety (USBR)
(2) Tensile stress
The minimum allowable compressive stress computed without internal
hydrostatic pressure should
σz = p. γ. h – (ft/s)
where:
σz = minimum allowable stress at the face
p = a reduction factor to account for drains
γ = unit weight of water
h = depth below water surface
ft = tensile strength of concrete at lift surfaces
s = safety factor.
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THEORETICAL APPROACH
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THEORETICAL APPROACH
Governing Equations The well-known Helmholtz equation governing the pressure p Zienkiewicz (2000) , Chopra (1967):
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Ω (1)
where P is the acoustic hydrodynamic pressure; t is time and ∂ is the two-dimensional Laplace operator and C is the speed of pressure wave given by:
where
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THEORETICAL APPROACH Boundary Conditions Dam-Reservoir Boundary
Reservoir-Foundation Boundary
Reservoir-Far-End Boundary
Free-Surface Boundary
P(x, y, z, t) = 0
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MODELING OF CONCETE GRAVITY DAMS
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GRAVITY DAM MODELING
Mathematical Modeling
Analytical Modeling
Physical Modeling
Experimental setup
Numerical Modeling
Deterministic Modeling
Stochastic Modeling
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MODELING OF CONCETE GRAVITY DAMS
ALALYTICAL MODELING
EXIMENTAL MODELING
NUMERICAL MODELING
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MODELING OF CONCETE GRAVITY DAMS
• FEM Deterministic Modeling
• Monte Carlo Simulation
Probabilistic Modeling
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FINITE ELEMENT MODELING
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FE Mesh
Once a dam has been modeled in FEM, it is possible to experiment and change details about it without the need to restart the whole process.
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FINITE ELEMENT MODELING FEM Formulation: Zienkiewicz (2000)
The standard Galerkin’s Finite Element technique in which the structure displacement vector is discretized as:
u= Nu 𝒖 , …………………….p= Np 𝒑
where 𝒖 and 𝒑 are the nodal parameters of each field and Nu and Np are appropriate shape functions. The discrete equations of the structure dynamic response following Galerkin method reads
M 𝒖 + C 𝒖 +K 𝒖 – Q 𝒑 + f =0 (7) In which M, C, K and f refer to mass matrix, damping matrix, stiffness matrix of the structure and prescribed
force vector respectively, where 𝒖 , 𝒖 and 𝒖 are displacement, velocity and acceleration vectors respectively. Standard Galerkin’s discretization applied to the fluid Equation (1) and its boundary conditions leads to [7]
S 𝒑 + ξ 𝒑 + H 𝒑 + QT 𝒖 + q = 0 (9)
in which S, ξ, H and q are pseudo fluid mass matrix, pseudo fluid damping matrix, pseudo fluid stiffness matrix and prescribed flux vector respectively . Q is a transform matrix and 𝒑 , 𝒑 and 𝒑 are nodal pressure vector, the first and second order derivatives of nodal pressure vector with respect to time, respectively.
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FINITE ELEMENT MODELING
FEM Formulation: Zienkiewicz (2000)
The coupled equation of the fluid-structure-foundation system based on Equations (7) and (9) subjected to earthquake ground motion can be presented as follows:
In which represents the nodal ground acceleration vector.
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+
(11)
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FINITE ELEMENT MODELING
• Dam - Reservoir –Foundation Coupling System
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FINITE ELEMENT MODELING
Dam-Reservoir Coupling System
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Interface Elements
• Imposing Line elements between fluid and concrete elements
Coincide nodes
• Coupling coincide nodes on the interface
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FINITE ELEMENT MODELING
Dam - Foundation Coupling System
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Fixed Soil Foundation
Massless Soil Foundation
Mass soil Foundation
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FINITE ELEMENT MODELING
Assumptions:
Only the displacements in the direction normal to the interface are assumed to be compatible in the structure as well as the fluid.
The fluid is generally assumed to be linear-elastic, incompressible, irrotational and nonviscous.
2-D finite element model is implemented.
Absorption is considered at reservoir bottom.
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FINITE ELEMENT MODELING
Assumptions:
A length of 2 to 3 times reservoir depth is recommended along with Summerfield boundary conditions.
The depth of foundation is taken about 1.5 the dam base width into account in the calculations.
The dam and foundation materials are assumed to be linear-elastic, homogeneous and isotropic.
The effect of foundation flexibility is considered as ratios i.e. modulus of elasticity of foundation to modulus of elasticity of dam Ef/Ec.
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FINITE ELEMENT MODELING
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Material Properties Sample
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ANALYSIS OF CONCRETE GRAVITY DAMS Gravity Method 2-D rigid block dam. Linear base pressure distribution Simple dam geometry
FEM Method • 2-D, 3-D analysis • Complex dam geometry • Complex boundary conditions • linear/Nonlinear behaviour • Dam –reservoir interaction • Dam –foundation interaction • Crack analysis
Can be analyzed
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ANALYSIS OF CONCRETE GRAVITY DAMS
Modal Analysis and Natural Response
• The structural response of a material to different loads determines how it will be economically utilized in the design process.
• Earthquake is a major source of seismic forces that impinge on structures
• This necessitates the seismic analysis of concrete gravity dam
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ANALYSIS OF CONCRETE GRAVITY DAMS
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Modal Analysis and Natural Response
Mode shapes for a gravity dam with empty reservoir
Mode shapes for a gravity dam with full reservoir
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ANALYSIS OF CONCRETE GRAVITY DAMS
Dynamic Analysis
• Dynamic analysis refers to analysis of loads whose duration is short with the first period of vibration of the structure.
• Dynamic methods are appropriate to seismic loading
because of the oscillatory nature of earthquakes, and the subsequent structural responses.
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ANALYSIS OF CONCRETE GRAVITY DAMS
• Dynamic Analysis
The purpose of dynamic analysis is not to determine dam stability in a conventional sense, but rather to determine what damage will be caused during the earthquake, and then to determine if the dam can continue to resist the applied static loads in a damaged condition with possible loading changes due to increased uplift or silt liquefaction.
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ANALYSIS OF CONCRETE GRAVITY DAMS
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TAFT GROUND MOTION, 1952
Dynamic Analysis
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Dynamic Analysis
Factors to Be Considered in Dynamic Analysis:
1. Hydrodynamic and reservoir bottom absorption
effects upstream ground motion.
2. Hydrodynamic effects upstream ground motion.
3. Reservoir bottom absorption effects upstream ground
motion.
4. Hydrodynamic and reservoir bottom absorption effects
vertical ground motion.
5. Water compressibility effects upstream ground
motion.
6. Foundation interaction effects upstream ground motion.
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Dynamic Analysis
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Pseudo Dynamic Method (Quasi-static)
This procedure was developed by Pro. Anil Chopra as a hand calculated alternative to the more general analytical procedures which require computer programs
PINE FLAT DAM
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Dynamic Analysis
• Pseudo Dynamic Method (Quasi-static)
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Stress distribution in Pin Plate Dam after Chopra (2010)
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Dynamic Analysis
• Response Spectrum Analysis
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Dynamic Analysis Time History Analysis
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Dynamic Analysis
Time History Analysis
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DESIGN AND SAFETY CRITERIA
International Safety Regulation Codes United States Department of the Interior Bureau of Reclamation (USBR)
“Design Of Gravity Dams Design” Design Manual For Concrete Gravity
Dams, 1976
Federal Guidelines for Dam Safety-Earthquake Analyses and Design of
Dams-May 2005
US Army Corps of Engineers -Engineering And Design – Gravity Dam
Design – 2000
Dam Safety Code – 2008 -Australian Capital Territory
Egyptian Code for Hydraulic Structures (Part 7)
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DESIGN AND SAFETY CRITERIA
STABILITY ANALYSIS
Allowable F.O.S. against overturning
Allowable F.O.S. against forward
sliding
STRESS ANALYSIS
σmax ≤ max. allowable compression stress for
dam concrete
σmax ≤ max. allowable bearing stress for dam
foundation
σmin ≥ 0 .0 no tension is allowed
qmax ≤ max. allowable shear stress for dam
concrete
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DESIGN AND SAFETY CRITERIA Recommended Minimum Sliding Stability Safety Factors
Dams having a high or significant hazard potential.
Loading Condition Factor of Safety Usual 3.0 Unusual 2.0 Post Earthquake 1.3
Dams having a low hazard potential.
Loading Condition Factor of Safety Usual 2.0 Unusual 1.25 Post Earthquake Greater than 1.0
Alternate Recommended Minimum Factors of Safety
Loading Condition Factor of Safety
Worst Static 1.5 Flood if Flood is PMF 1.3 Post Earthquake 1.3
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DESIGN AND SAFETY CRITERIA
Sliding Stability Safety Factors
Overturning Stability Safety Factors
Cracked Base Criteria
Safety Factor Evaluation
Foundation Stability
Construction Materials
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Recent Analysis Aspects
Fracture Analysis
Thermal Stress Analysis
Breach Analysis
Risk Analysis
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Recent Analysis Trends
Fracture Analysis
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Recent Analysis Trends
Thermal Stress Analysis
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Recent Analysis Trends
Breach Analysis
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Recent Analysis Trends
Risk Analysis
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RISK ANALYSIS OF CONCRETE GRAVITY DAMS
Risks under Normal Operations
Risks under Flood Loading
Risks under Earthquake Loading
Accounting for Uncertainty
Probabilistic Seismic Risk Assessment
•
•
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Relevant Case History
Austin (Bayless) Dam: 1911
Bouzey Dam: 1895
Koyna Dam: 1967
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SUMMING UP Gravity dams are very important structures.
The collapse of a gravity dam due to earthquake ground motion may cause an
extensive damage to property and life losses.
Therefore, the proper design of gravity dams is an important issue in dam
engineering.
An integral part of this procedure is to accurately estimate the dam earthquake
response.
The prediction of the actual response of a gravity dam subjected to earthquake is a
very complicated problem.
It depends on several factors such as dam-foundation interaction, dam-water
interaction, material model used and the analytical model employed.
In fluid-structure interaction one of the main problems is the identification of the
hydrodynamic pressure applied on the dam body during earthquake excitation.
The analysis of dam-reservoir system is complicated more than that of the dam
itself due to the difference between the characteristics of fluid and dam's concrete
on one side and the interaction between reservoir and dam on the other side.
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Main References USBR (1976) “Design Criteria For Concrete Arch And Gravity Dams”.
Westergard, H. M. (1933). Water pressure on dams during earthquakes.
TRANSACTIONS ASCE Vol.98.
Chopra A.K. (1967). Hydrodynamic Pressure on dams during earthquakes”
Proc .ASCE , EM6.
Chopra A.K . (1970). Earthquake response Analysis of concrete gravity dams.
Proc. ASCE, EM4.
Zienkiewicz, 0.C. and Taylor, R.L. (2000) “The Finite Element Method”; 5th Edition
McGraw-Hill.
Zeidan, B. A. (2014) "Seismic Analysis of Dam-Reservoir-Foundation
Interaction for Concrete Gravity Dams", International Symposium on Dams in
Environmental Global Challenges" ICOLD2014, Bali, Indonesia, June 1ST - 6TH, 2014
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Main References Fenves, G., And Chopra, A. K., (1985) “Effects Of Reservoir Bottom
Absorption And Dam-Water-Foundation Rock Interaction On
Frequency Response Functions For Concrete Gravity Dams”
Earthquake Engineering & Structural Dynamics, Vol. 13, 1985, Pp. 13-31.
Gaun F., Moore I.D. & Lin G. (1994) “Seismic Analysis of Reservoir-Dam-
Soil Systems in the Time Domain”, The 8th international conference on
Computer Methods and Advances in Geomechanics, Siriwardane & Zaman
(Eds), Vol. 2, 917-922.
Ghaemian M., Noorzad A. & Moghaddam R.M. (2005) “Foundation Effect
on Seismic Response of Arch Dams Including Dam-Reservoir
Interaction”, Europe Earthquake Engineering, 3, 49-57.
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Main References US. Army Corps of Engineers (USACE), (2003) “Time-History Dynamic Analysis of
Concrete Hydraulic Structures;” Chapter 2- Analytical Modeling of Concrete
Hydraulic Structures, Chapter 3-Time-History Numerical Solution Techniques”, EM 1110-
2-6051.
Lysmer J. & Kuhlemeyer R.L. (1969) “Finite Dynamic Model for Infinite Media”,
Journal of Engineering Mechanics Division, ASCE, 95 (EM4), 859-877.
Wilson E.L. (2000) “Three Dimensional Static and Dynamic Analysis of
Structures, A Physical Approach with Emphasis on Earthquake
Engineering”, 4th Ed., Computers and Structures Inc.
Wolf J. P. (1985) “Dynamic Soil-Structure Interaction”, Prentice Hall: Englewood
Cliffs, NJ.
Bakenaz A. Zeidan (2014) “Finite Element Modeling For Acoustic Reservoir-
Dam-Foundation Coupled System”, International Symposium on Dams in a Global
Environmental Challenges, ICOLD2014, Bali, Indonesia, 1-6 June, 2014.
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