Geotechnical Earthquake Engineering is a special discipline in our profession. It requires understanding of the terminology and methods that are necessary to safely design structures against earthquakes as well as understanding of the features of soil behaviour under dynamic and cyclic loading, including liquefaction. Important backgrounds on geotechnical earthquake engineering and liquefaction can be found in the books by Kramer (1996) and Idriss & Boulanger (2008).
PLAXIS 2D/3D Ultimate includes the essential facilities for the numerical modelling of earthquakes and liquefaction, such as:
• Calculation types ‘Dynamic’ and ‘Dynamic with consolidation’
• Constitutive models for cyclic loading and liquefaction
• Viscous, free-field and compliant base boundaries that represent the far field and prevent spurious wave reflections at the model boundaries
• Input and pre-processing of dynamic loads, dynamic displacements, velocities and accelerations.
• Efficient tool to perform one-dimensional site response analysis.
• Post-processing tool for response analysis (part of the Curves manager in Output)
In this series of blogs, the focus is on constitutive models. It is important to realise that advanced models for static loading are not necessarily good for dynamic loading (and vice versa). Since dynamic calculation phases are generally preceded by static phases, you may need to ‘switch’ the material datasets of the soil layers at the start of your dynamic calculation.
Dynamic calculations require other features of soil behaviour to be included in a constitutive model than static calculations. These are mostly related with the effects of cyclic loading, such as:
• Strain-dependent stiffness (modulus reduction) and regain of small-strain stiffness upon load reversal; strain-dependent hysteretic damping
• Degradation of stiffness and strength in cyclic loading
• Accumulation of (plastic) strains in cyclic loading
• Build-up of pore pressures in undrained cyclic loading
• Decrease of stiffness and increase of (shear) strain when reaching liquefaction
Despite the advantages of advanced constitutive models, such as the Hardening Soil model and the Soft Soil model, as discussed in my previous blogs, these models do not capture the features as mentioned above. Only the Hardening Soil model with small-strain stiffness includes the first feature and may be used in dynamic calculations for non-liquefiable soils.
The first model for cyclic loading and liquefaction that PLAXIS implemented was the UBC3D-PLAXIS Liquefaction Model. This is a 3D implementation of the two-dimensional UBCSand model, as originally developed by Beatty & Byrne (1998) at the University of British Columbia (UBC). UBC3D-PLM has remarkable similarities with the Hardening Soil model, with the extra feature that it accumulates plastic strains in cyclic loading. In combination with undrained behaviour, it builds-up pore pressures, which may lead to liquefaction after a certain number of cycles (depending on its parameter values). In addition to an extensive validation by PLAXIS and various universities, NTUA in Athens, Greece, published a procedure for the determination and calibration of model parameters (Anthi & Gerolymos, 2019)
A few years after the first model, we implemented the PM4Sand model, following the original implementation (Version 3) by Boulanger & Ziotopoulou (2015). The PLAXIS implementation (Vilhar et al., 2018) gives very similar results as the original model, but our model is probably more efficient since it is implemented in an implicit finite element environment.
PM4Sand can be regarded as an advanced cyclic loading and liquefaction model. However, it is (still) a two-dimensional model, just like the original. There is a clear procedure to determine the model parameters. The main parameter, hp0, defines the number of cycles to arrive at the point of liquefaction for a given cyclic stress ratio. It can be calibrated from the results of a series of cyclic Direct Simple Shear tests or from correlations with Cone Penetration Test or Standard Penetration Test data. The PLAXIS SoilTest facility allows for a convenient calibration of the hp0 parameter (see illustration).
Image: Undrained cyclic DSS test simulation with PM4Sand model using the PLAXIS SoilTest facility
Although dynamic analysis is generally more complicated than static analysis, I trust that with this information you are triggered and feel more confident to start your earthquake and liquefaction modelling. In general, dynamic calculations require different constitutive models than static calculations. PLAXIS 2D/3D Ultimate provides all necessary facilities for dynamic analysis, as well as two soil constitutive models that can be used specifically for geotechnical earthquake analysis and liquefaction evaluation. Calibration of model parameters can be conveniently performed using the PLAXIS SoilTest facility.
Kramer SL (1996), “Geotechnical Earthquake Engineering”. Prentice Hall.
Idriss & Boulanger (2008), “Soil Liquefaction During Earthquakes”. Earthquake Engineering Research Institute (EERI).
Beaty M, Byrne P (1998), “An Effective Stress Model for Predicting Liquefaction Behaviour of Sand”. Geotechnical Earthquake Engineering and Soil Dynamics III, ASCE Geotechnical Special Publication, No. 75, 766-777.
Anthi M, Gerolymos N (2019), “A Calibration Procedure for Sand Plasticity Modeling in Earthquake Engineering: Application to TA-GER, UBCSAND and PM4SAND”. In: Proc. 7th Int. Conf. on Earthquake Geotechnical Engineering, Rome. CRC Press.
Boulanger & Ziotopoulou (2015), “PM4Sand (Version 3) – A Sand Plasticity Model for Earthquake Engineering Applications”. Report No. UCD/CGM-15/01, Center for Geotechnical Modeling, Dept. of Civil and Environmental Engineering, University of California Davis, CA.
Vilhar G, Laera A, Foria F, Gupta A, Brinkgreve RBJ (2018), “Implementation, Validation and Application of PM4Sand Model in PLAXIS”. In: Proc. GEESD V, ASCE Geotechnical Special Publication No. 292, 200-211.
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