Seismic Analysis of Concrete Dams Workshop - · PDF fileSeismic Analysis of Concrete Dams Workshop 2017 USSD Annual Conference April 6-7, 2017 Session #8 – Accuracy of the Analysis

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


Mathematical Model


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


It makes no sense to attempt to solve a BVP without a precise knowledge of what the problem is.

Tom Hughes


Flow Chart for Analysis of Concrete Dams


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


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


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


Accuracy

Convergence

Evaluation of Numerical Methods


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


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


• 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


• 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


• 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


• 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


• 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


Documents

Seismic Analysis of Concrete Dams Workshop - · PDF fileSeismic Analysis of Concrete Dams Workshop 2017 USSD Annual Conference April 6-7, 2017 Session #8 – Accuracy of the Analysis