Linear & Non-linear SSI Calculations: Methods used in EDF to determine raft uplift

Transcription

1 Linear & Non-linear SSI Calculations: Methods used in EDF to determine raft uplift Pauline BILLION Frédéric ALLAIN Georges DEVESA Nicolas HUMBERT Ilie PETRE-LAZAR OECD workshop, 8th oct OECD Workshop EDF Impedance Calculations

2 Table of Contents 1. Introduction 2. Methods 2.1. An equivalent non-linear post-processing based on a linear method 2.2. A linear method treating local non-linearities 2.3. A non-linear transitory method 3. Comparisons between methods 4. Conclusions 2

3 Introduction Raft uplift: difficult phenomenon to observe & to quantify Occurs promptly Displacements and rotations values involved are small But the phenomenon needs to be accounted for in NPPs design in order to accurately evaluate the behavior of the nuclear buildings during earthquakes. A detailed model of the foundation raft helps determining the uplifted area more precisely. 3

4 Review of methods used in EDF An equivalent non-linear post-processing based on a linear method A linear method treating local non-linearities (FEM) A non-linear transitory method (FEM) 4

5 An equivalent non-linear post-processing based on a linear method (1/3) Method used in EDF in the 1980s. The M-θ relation is integrated in the post-processing stage of the calculation. Method founded on the energy equivalence principles: The response of a non-linear system can be estimated stating that the energy communicated to the non-linear system is equal to the energy that would be calculated with a linear model. Linear Nonlinear 5

6 An equivalent non-linear post-processing based on a linear method (2/3) Simple method that determines uplifted area from few parameters: Maximal moments Vertical accelerations Simplified geometry Uplift occurs when soil springs are no longer in compression. 6

7 An equivalent non-linear post-processing based on a linear method (3/3) Using a linear behaviour law: θ 1 and M 1 W 1 Energy equivalence principle: W 1 = W 2 θ 2 S Modified version of this method taking into account the rotational stiffness of the soil changes in the formulas Advantages & Drawbacks of the method: +: Quick, conservative (used for design in the 1980s) & simple - : Over-estimation of the uplifted area explained by not taking into account rotational stiffness 7

8 Review of methods used in EDF An equivalent non-linear post-processing based on a linear method A linear method treating local non-linearities (FEM) A non-linear transitory method (FEM) 8

9 A linear method treating local non-linearities Method used in EDF since the 1990s. Method treated with the resolution of: Linear operators of the structure (K and M) Local non-linearities coming from the M-θ relation (integrated in the calculation) Calculation: Modal dynamic transitory analysis Advantages & Drawbacks of the method: +: Linear calculations, small computation time - : No real uplifted area (if no criteria stated) 9

10 Review of methods used in EDF An equivalent non-linear post-processing based on a linear method A linear method treating local non-linearities (FEM) A non-linear transitory method (FEM) 1 0

11 A non-linear transitory method (1/3) Method currently used in EDF. 2 softwares: Code_Aster + ProMISS3D (linked with Aster). 2 sorts of springs in the model: Linear springs representing the soil (6 springs: springs mat (SSI analyses)) Non-linear springs representing the raft/soil contact (under each node of the raft) Surface meshes Non-linear springs Surface meshes Soil springs 1 1

12 A non-linear transitory method (2/3) A 2-step analysis: Imposing the gravity to the structure (static calculations) While maintaining gravity, imposing the seismic loadings to the structure (dynamic calculations) 1 2

13 A non-linear transitory method (3/3) The contact springs allow the uplift to occur. Contact is broken as soon as the non-linear springs are not compressed anymore. All the nodes of the raft are potentially submitted to uplift At each instant of the computation, the area is tested for uplift. Advantages & Drawbacks of the method: +: More physical representation of the phenomenon - : Great computation time, complexity, spring mat 1 3

14 Comparison Uplifted area determined with 3 methods v. ZPA (g): Linear method Non-linear method Energy equivalence method original modified version (rotational stiffness) Uplifted area determined with 3 methods v. accelerograms (g) 14

15 80% 70% Uplifted area determined with 4 methods v. ZPA (g) Energy equivalence method (original) 60% 50% 40% 30% Energy equivalence method (modified) 20% 10% Linear & Non-linear 0% ZPA (g) 15

16 Comparison Uplifted area determined with 3 methods v. ZPA (g) Energy equivalence method is always greater (margins) Modified energy equivalence method: results are closer to non-linear calculations results Linear calculation similar to non-linear if modal base is complete/rich 80% 60% Energy equivalence method (original) 40% Energy equivalence method (modified) 16 20% 0% Linear & Non-linear ZPA (g)

17 Comparison Uplifted area determined with 3 methods v. ZPA (g) Uplifted area determined with 3 methods v. accelerograms (g) 17

18 60% 50% Energy equivalence method (original) 40% 30% 20% Energy equivalence method (modified) 10% Linear & Non-linear 0% Sets A B C D E F 18

19 Comparison Uplifted area determined with 3 methods v. accelerograms (g) 6 sets of accelerograms are compared: 2 directions only: X and Z Accelerograms derived from EUR spectra (ZPA=0.25g) Energy equivalence method: over-estimation of the results (for the 6 sets) Great influence of the sets on results 60% Energy equivalence method (original) 50% 40% 30% Energy equivalence method (modified) 20% 10% Linear & Non-linear 19 0%

20 Conclusions Several methods developed in the last decades: linear and nonlinear methods. Design approach: use of simple methods (static and linear ones) to help understand and check hypotheses. Margins can be exhibited. Then advanced methodologies may be used. It is admitted that direct methods (linear and non-linear) estimate realistically the uplifting phenomenon. But great computation time, establishment of a criterion, The energy equivalence method is not time consuming and simple. But it overestimated the area. Improvements to this method can be brought by accounting for the rotational stiffness. Non-linear methods are considered closer to reality but the computation time is great. Methods taking into account attaching forces (between concrete and bedrock) are considered as best estimate but require more knowledge, more studies and better education (to apply them). 20

21 21 Thank You for Your Attention!

22 Appendixes 22