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

3. 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 Computational Geotechnics Non-Linear Hyperbolic Model & Parameter Selection

4. Stiffness Modulus Definition of E 50 in a standard drained triaxial experiment Elastic unloading and reloading (Ohde, 1939) We use the two elastic parameters ur and E ur Initial (primary) loading Computational Geotechnics Non-Linear Hyperbolic Model & Parameter Selection

5. Stiffness Modulus Definition of the normalized oedometric stiffness Values for m from oedometer test versus initial porosity n 0 Normalized oedometer modulus versus initial porosity n 0 Oedometer tests Computational Geotechnics Non-Linear Hyperbolic Model & Parameter Selection

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

7. Stiffness Modulus Values for m obtained from triaxial test versus initial porosity n 0 Normalized triaxial modulus versus initial porosity n 0 Computational Geotechnics Non-Linear Hyperbolic Model & Parameter Selection

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

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

10. 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 Computational Geotechnics Non-Linear Hyperbolic Model & Parameter Selection

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

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

13. Plasticity Results of drained loading: stress-strain relation (  3 = 100 kPa) Results of drained loading: axial-volumetric strain relation (  3 = 100 kPa) Hardening soil response in drained triaxial experiments Computational Geotechnics Non-Linear Hyperbolic Model & Parameter Selection

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

15. 2c u E u  1.4 E 50 Plasticity Interesting in case you have data on C u and not no C’ and  ’ Assume E 50 = 0.7 E u and use graph by Duncan & Buchignani (1976) to estimate E u Computational Geotechnics Non-Linear Hyperbolic Model & Parameter Selection

16. Plasticity Results of undrained triaxial loading: stress-strain relations (  3 = 100 kPa) Results of undrained triaxial loading: p-q diagram (  3 = 100 kPa) Hardening soil response in undrained triaxial tests Computational Geotechnics Non-Linear Hyperbolic Model & Parameter Selection

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

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

19. HS-Cap-Model Additional parameters The extra input parameters are K 0 (=1-sin  ) and E oed /E 50 (=1.0) The two auxiliary material parameter M and K c /K s 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. Computational Geotechnics Non-Linear Hyperbolic Model & Parameter Selection

20. HS-Cap-Model 1 23 Graphical presentation of HS-Cap-Model Yield surfaces of the extended HS model in p-q space (left) and in the deviatoric plane (right) Computational Geotechnics Non-Linear Hyperbolic Model & Parameter Selection

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

22. 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 Computational Geotechnics Non-Linear Hyperbolic Model & Parameter Selection

23. 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 Computational Geotechnics Non-Linear Hyperbolic Model & Parameter Selection

24. 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 Computational Geotechnics Non-Linear Hyperbolic Model & Parameter Selection