Seismic Analysis of Concrete Dams
Workshop2017 USSD Annual Conference
April 6-7, 2017
Session #8 – Accuracy of the Analysis
Introduction
• The goal of the analysis
• Define response of the structure to the operating conditions
• Determine:• Displacements, stresses, strains,• Damage of material,• Predict operational life of structure• etc.
Engineering Interest in Analysis
Seismic Analysis of Concrete Dams Workshop2017 USSD Annual ConferenceApril 6-7, 2017
• Geometry
• Material properties
• Loads
Real Structure vs Physical Model
Seismic Analysis of Concrete Dams Workshop2017 USSD Annual ConferenceApril 6-7, 2017
Mathematical Model
Seismic Analysis of Concrete Dams Workshop2017 USSD Annual ConferenceApril 6-7, 2017
Boundary Value Problem (BVP) ‐ nonlinearities• Equations of Motion (for reservoir) , i,j,k =1,2,3
• Geometric Equations , , , ,
• Constitutive Equationswhere: u, ε, σ ‐ displacement, strain, stress tensorsρ, b, ‐mass density and body forceC ‐material coefficients matrix
• Boundary Conditions (BC) – Γ– 0 Γ– Γσ, Γu are parts of the boundary
• Initial Conditions. – , , ,
• Wave (acoustic) equation (for reservoir)
• Fluid‐structure interface where: p – dynamic pressurenF, nS, ‐ outward to fluid and structural domainsC – fluid coefficients
Solving Complex Mathematical Problems
Seismic Analysis of Concrete Dams Workshop2017 USSD Annual ConferenceApril 6-7, 2017
It makes no sense to attempt to solve a BVP without a precise knowledge of what the problem is.
Tom Hughes
Seismic Analysis of Concrete Dams Workshop2017 USSD Annual ConferenceApril 6-7, 2017
Flow Chart for Analysis of Concrete Dams
Seismic Analysis of Concrete Dams Workshop2017 USSD Annual ConferenceApril 6-7, 2017
Real Object (dam, reservoir, foundation, etc.)
Real Object (dam, reservoir, foundation, etc.)
Physical Model (assumed geometry, loads, materials)
Mathematical Model (system of equations)
Analytical or Numerical Solutions(selection of representative model)
Analysis of the Model (Computer Program)
Interpretation of Analysis Results(displ., strains, stresses, damage)
Verification of the Results (convergence & accuracy,
verification with test results)
Results Presentation(the analysis completed)
Analytical vs Numerical Solutions
Seismic Analysis of Concrete Dams Workshop2017 USSD Annual ConferenceApril 6-7, 2017
BVP for Westergaard Model• Equation of Motion
• Geometric Equation12
• Constitutive Law for Reservoir
u, v are horizontal and vertical displacementsp is the pressure in the reservoir
• Analytical Solution at u/s Face8 1
sin 2, , …
116
Model Assumptions:• Upstream face is straight and vertical at Γσ.• The dam is considered to be a rigid block.• The reservoir is infinite in length at Γp.• Small motions are assumed during earthquake.• The problem is defined in 2D space.• The period of free vibration of the dam, T0, is significantly smaller than the period of vibration, T, of the earthquake (resonance is not expected).
• Horizontal harmonic dam motion applied to dam
Analytical vs Numerical Solution
Seismic Analysis of Concrete Dams Workshop2017 USSD Annual ConferenceApril 6-7, 2017
The dynamic response of a system beyond its linear elastic range (and for complex geometry)
is generally not amenable to the analytical solution.
Numerical methods are therefore essential in the analysis of nonlinear systems.
(A.K. Chopra)
Find the perimeter of the circle (Greeks determined the parameter by measuring the segment lines)
(by J.N. Reddy)
Basic Concept of Numerical Analysis
Seismic Analysis of Concrete Dams Workshop2017 USSD Annual ConferenceApril 6-7, 2017
Accuracy
Convergence
Evaluation of Numerical Methods
Seismic Analysis of Concrete Dams Workshop2017 USSD Annual ConferenceApril 6-7, 2017
Basic Terminology: • Stability: Numerical errors which are generated during the solution of discretized equations should not be magnified
• Consistency: The discretization of a PDE should become exact as the mesh size tends to zero
consistency + stability = convergence
• Convergence: The numerical solution should approach the exact solution of the partial differential equations (PDE) and converge to it as the mesh size tends to zero. Three convergence criteria:
• Displacements• Out‐of‐balance forces• Incremental internal energy
Evaluation of Accuracy
Seismic Analysis of Concrete Dams Workshop2017 USSD Annual ConferenceApril 6-7, 2017
Basic Terminology
Error: The difference between the exact solution and the approximate one.
(Zienkiewicz, Taylor)
error = u – u*
u is the exact solution (displacement, strain, stress)u* is the corresponding numerical solution
Error is inherent in any numerical solutionof the equation of motion
(A.K. Chopra)
Computational Errors
Seismic Analysis of Concrete Dams Workshop2017 USSD Annual ConferenceApril 6-7, 2017
• Discretization in space• Numerical integration• Constitutive relations• Solution of dynamic response - error related to:
• numerical integration of equations of motions• a few modes used in a mode superposition analysis
• Iteration procedures • Round-off error (precision arithmetic of the computer)
Sources of Computational Errors
Seismic Analysis of Concrete Dams Workshop2017 USSD Annual ConferenceApril 6-7, 2017
• Develop practical procedures for evaluation of nonlinear solutions describing dam‐foundation‐reservoir interaction for seismic loads.
• Initiate a discussion at USSD forum on:• Accuracy of the numerical solutions for dam‐foundation‐reservoir interaction for seismic loads
• Efficiency of the advanced analysis
Objectives of the Workshop Session
Seismic Analysis of Concrete Dams Workshop2017 USSD Annual ConferenceApril 6-7, 2017
• What level of mathematical model complexity is required to accurately predict behavior of the concrete dams during earthquake?
• Is the complex analysis (dynamic solution with nonlinearities) able to accurately predict behavior of concrete dams?
• Can such models be used to quantify responses of dams?
Workshop Discussion - Questions
Seismic Analysis of Concrete Dams Workshop2017 USSD Annual ConferenceApril 6-7, 2017
• Confidence in analysis results developed in dam/reservoir/foundation analysis for seismic loads.
• Primary sources of errors in the solution of nonlinear problems for dynamic loads.
• Accuracy versus cost of the analysis.
Workshop Discussion
Seismic Analysis of Concrete Dams Workshop2017 USSD Annual ConferenceApril 6-7, 2017
• Challenges in interpretation of the non‐linear analysis results in the engineering practice.
• Is Finite Element method commonly used in the numerical analysis of dam‐reservoir‐foundation system for seismic loads?
Workshop Discussion - Questions
Seismic Analysis of Concrete Dams Workshop2017 USSD Annual ConferenceApril 6-7, 2017