Non-Linear Hyperbolic Model & Parameter Selection

1. Non-Linear Hyperbolic Model & Parameter Selection Short Course on Computational Geotechnics + Dynamics Boulder, Colorado January 5-8, 2004 Stein Sture Professor of Civil Engineering University of Colorado at Boulder

2. Non-Linear Hyperbolic Model & Parameter Selection Computational Geotechnics Contents Introduction Stiffness Modulus Triaxial Data Plasticity HS-Cap-Model Simulation of Oedometer and Triaxial Tests on Loose and Dense Sands Summary

3. Non-Linear Hyperbolic Model & Parameter Selection Computational Geotechnics Introduction Hardening Soils Most soils behave in a nonlinear behavior soon after application of shear stress. Elastic-plastic hardening is a common technique, also used in PLAXIS. Usage of the Soft Soil model with creep Creep is usually of greater significance in soft soils. Hyperbolic stress strain response curve of Hardening Soil model

4. Non-Linear Hyperbolic Model & Parameter Selection Computational Geotechnics Stiffness Modulus Elastic unloading and reloading (Ohde, 1939) We use the two elastic parameters nur and Eur Initial (primary) loading Definition of E50 in a standard drained triaxial experiment

5. Non-Linear Hyperbolic Model & Parameter Selection Computational Geotechnics Stiffness Modulus Oedometer tests Definition of the normalized oedometric stiffness Values for m from oedometer test versus initial porosity n0 Normalized oedometer modulus versus initial porosity n0

6. Non-Linear Hyperbolic Model & Parameter Selection Computational Geotechnics Stiffness Modulus Normalized oedometric stiffness for various soil classed (von Soos, 1991)

7. Non-Linear Hyperbolic Model & Parameter Selection Computational Geotechnics Stiffness Modulus Values for m obtained from triaxial test versus initial porosity n0 Normalized triaxial modulus versus initial porosity n0

8. Non-Linear Hyperbolic Model & Parameter Selection Computational Geotechnics Stiffness Modulus Summary of data for sand: Vermeer & Schanz (1997) Comparison of normalized stiffness moduli from oedometer and Triaxial test Engineering practice: mostly data on Eoed Test data:

9. Non-Linear Hyperbolic Model & Parameter Selection Computational Geotechnics Triaxial Data on p 21p Equi-g lines (Tatsuoka, 1972) for dense Toyoura Sand Yield and failure surfaces for the Hardening Soil model

10. Non-Linear Hyperbolic Model & Parameter Selection Computational Geotechnics Plasticity Yield and hardening functions 3D extension In order to extent the model to general 3D states in terms of stress, we use a modified expression for in terms of and the mobilized angle of internal friction where

11. Non-Linear Hyperbolic Model & Parameter Selection Computational Geotechnics Plasticity Plastic potential and flow rule with

12. Non-Linear Hyperbolic Model & Parameter Selection Computational Geotechnics Plasticity Flow rule with Primary soil parameters and standard PLAXIS settings

13. Non-Linear Hyperbolic Model & Parameter Selection Computational Geotechnics Plasticity Hardening soil response in drained triaxial experiments Results of drained loading: stress-strain relation (s3 = 100 kPa) Results of drained loading: axial-volumetric strain relation (s3 = 100 kPa)

14. Non-Linear Hyperbolic Model & Parameter Selection Computational Geotechnics Plasticity Undrained hardening soil analysis Method A: switch to drained Input: Method B: switch to undrained Input:

15. Non-Linear Hyperbolic Model & Parameter Selection Computational Geotechnics Eu 1.4 E50 2cu Plasticity Interesting in case you have data on Cu and not no C’ and ’ Assume E50 = 0.7 Eu and use graph by Duncan & Buchignani (1976) to estimate Eu

16. Non-Linear Hyperbolic Model & Parameter Selection Computational Geotechnics Plasticity Hardening soil response in undrained triaxial tests Results of undrained triaxial loading: stress-strain relations (s3 = 100 kPa) Results of undrained triaxial loading: p-q diagram (s3 = 100 kPa)

17. Non-Linear Hyperbolic Model & Parameter Selection Computational Geotechnics HS-Cap-Model Cap yield surface Flow rule (Associated flow) Hardening law For isotropic compression we assume with

18. Non-Linear Hyperbolic Model & Parameter Selection Computational Geotechnics HS-Cap-Model For isotropic compression we have q = 0 and it follows from For the determination of, we have another consistency condition:

19. Non-Linear Hyperbolic Model & Parameter Selection Computational Geotechnics HS-Cap-Model Additional parameters The extra input parameters are K0 (=1-sinf) and Eoed/E50 (=1.0) The two auxiliary material parameter M and Kc/Ks are determined iteratively from the simulation of an oedometer test. There are no direct input parameters. The user should not be too concerned about these parameters.

20. Non-Linear Hyperbolic Model & Parameter Selection Computational Geotechnics 1 2 3 HS-Cap-Model Graphical presentation of HS-Cap-Model Yield surfaces of the extended HS model in p-q space (left) and in the deviatoric plane (right)

21. 1 = 2 = 3 Non-Linear Hyperbolic Model & Parameter Selection Computational Geotechnics HS-Cap-Model Yield surfaces of the extended HS model in principal stress space

22. Non-Linear Hyperbolic Model & Parameter Selection Computational Geotechnics Simulation of Oedometer and Triaxial Tests on Loose and Dense Sands Comparison of calculated () and measured triaxial tests on loose Hostun Sand Comparison of calculated () and measured oedometer tests on loose Hostun Sand

23. Non-Linear Hyperbolic Model & Parameter Selection Computational Geotechnics Simulation of Oedometer and Triaxial Tests on Loose and Dense Sands Comparison of calculated () and measured triaxial tests on dense Hostun Sand Comparison of calculated () and measured oedometer tests on dense Hostun Sand

24. Non-Linear Hyperbolic Model & Parameter Selection Computational Geotechnics Summary • Main characteristics • Pressure dependent stiffness • Isotropic shear hardening • Ultimate Mohr-Coulomb failure condition • Non-associated plastic flow • Additional cap hardening HS-model versus MC-model As in Mohr-Coulomb model Normalized primary loading stiffness Unloading / reloading Poisson’s ratio Normalized unloading / reloading stiffness Power in stiffness laws Failure ratio