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207 APPENDIX D-3 Calibration of CM Sub-Models General This study was commissioned to focus on the undrained response of saturated, relatively loose sands found in nature, up to and including soil liquefaction. The laboratory test-based calibration of nonlinear effective-stress constitutive models (i.e., âelement testsâ) has been conducted accordingly. However, in site response analysis, the drained and undrained response of soils other than potentially liquefiable sands cannot be ignored. âOther modelsâ discussed here exhibit nonlinear stress-strain response but lacked the capability of simulating excess pore water pressure (PWP) generation. This makes their general constitutive behavior considerably simpler than that of the effective-stress models (e.g., from MKZ, PM4SAND, UBCSAND, and UCSDSAND3 considered herein), but limits their application in effective-stress analysis. In total stress analysis, or in effective-stress analysis with low-intensity motion (i.e., because motion is of low intensity, excess PWP generation has a small or negligible effect on soil behavior), these âother modelsâ can be assigned to any or all soil layers. For more general effective-stress analysis, on the other hand, the use of these models is limited to non- liquefiable materials, with examples including silts, clays, dense sands and gravels, and/or soils above the water table. Relevant information about âother modelsâ is reproduced from the main study as Table D-1. Calibration of these models is briefly explained in subsequent sections. Table D-1. Other Constitutive Models Used in this Study. No. Model Name Relevant Information 1 S-I Total Number of Model Parameters: N/A (modulus reduction and damping curves input at 6 â 10 strain levels) Required/Default Parameters: N/A Generic Parameters Available: Y (for most soils) 2 LE-MC Total Number of Model Parameters: 8 Required/Default Parameters: 8/0 Generic Parameters Available: N 3 HSsmall Total Number of Model Parameters: 19 Required/Default Parameters: 19/0 Generic Parameters Available: N N/A = Not Applicable; N = No; Y = Yes.
208 S-I Model Modulus reduction and damping curves may be directly entered into equivalent-linear programs as a series of discrete values. They may be also selected from a database of such curves that accompany most equivalent-linear software. Because there is no constitutive law involved (modulus reduction and damping curves are developed by nonlinear regression of data from multiple sources or by engineering judgment), they do not qualify to be called âconstitutive models.â However, for practical reasons, especially when parameters of nonlinear constitutive models are developed by the fitting of these curves, they are referred to as the âSeed-Idriss (1970) Constitutive Modelâ and are abbreviated to herein as âS-I.â (a) Modulus Reduction Curve (b) Damping Curve Figure D-1. Development of Modulus Reduction and Damping Curves from Multiple Test Results (WLA Site â Unit Bâ : Silty Sand).
209 Given the above, there is no formal calibration of the S-I model. However, there is significant judgment associated with development of these curves. Figure D-1 illustrates such a process on an actual data set (WLA site silty sand, using data from others and new data generated in this study). One may simply develop curves by means of nonlinear regression, fit data using a nonlinear constitutive model which has a provision for curve fitting, or simply manually adjust the shape of a curve; in each of these procedures, preference may be given, or not, to a particular set of data. In the example below (Figure D-1), the modulus reduction curve was adjusted to come closer to modulus reduction data generated by in-situ testing (Cox et al., 2009). This data is not affected by sample disturbance, system compliance, and other factors that may impact the results of laboratory testing, but may have its own set of issues which are related to the inversion process. In other words, sources of potential error in modulus reduction are different but are likely less in aggregate. LE-MC Model The Linear Elastic - Mohr-Coulomb (LE-MC) model is a total stress model implemented in FLAC. It obeys linear elastic pre-yield behavior and perfectly plastic behavior following the yield. For dynamic problems, the behavior of this constitutive model can be modified by invoking the hysteretic damping feature with one of three options (i.e., âHardin,â âSig3,â or âSig4â); the âSig3â option was used for this study. The LE-MC constitutive model paired with âSig3â hysteretic damping essentially becomes a total stress 3-parameter hyperbolic model. The general form of the FLAC LE-MC model with âSig3â hysteretic damping is as follows: Equation (1) where M s is the value of the (secant) shear modulus multiplier at a given shear strain level, L is the logarithm of the current shear strain level, and a, b , and x 0, are user-provided inputs. In most cases, the three-parameter sigmoidal curve modelâs modulus reduction curve can be fitted âexactlyâ to that of the MKZ model, although the equation forms differ. Therefore, the calibration of the LE-MC model is identical to the calibration of its MKZ (Matasovic and Vucetic, 1993) counterpart. Calibration (i.e., fitting of the WLA modulus reduction and damping data shown in Figure D-1) with the LE-MC model is presented in Figure D-2.
210 (a) Modulus Reduction Curve (b) Damping Curve Figure D-2. Example Calibration of the LE-MC Model (FLAC). HSsmall Model The Hardening Soil with small-strain stiffness (HSsmall) is a total stress model implemented in PLAXIS. This model allows for the simulation of the nonlinear behavior of soil in a manner similar to the LE-MC model described above. The general form of the model (without cutoff applied) is reproduced from Laera and Brinkgreve (2015) as follows: Equation (2) where Gs is the value of the (secant) shear modulus at a given shear strain (γ), G0 is the initial shear modulus (same as Gmax), and γ0.7 is the strain at which the shear modulus reaches 70 percent of its initial value. The MKZ model reduces to the HSsmall model when its parameters are set as follows: β = 0.385 and s = 1.0. In other words, HSsmall is a three-parameter model out of which two parameters have fixed values. As such, it is more limited than its MKZ and LE-MC counterparts, which allows for more accurate fitting of modulus reduction and damping data. Calibration (i.e., fitting of the WLA modulus reduction and damping data shown in Figure D-1) with the HSsmall model is presented in Figure D- 3.
211 (a) Modulus Reduction Curve (b) Damping Curve Figure D-3. Example Calibration of the HSsmall Model (PLAXIS). References Cox, B.R., Stokoe, K.H., and Rathje, E.M. (2009), âAn In-Situ Test Method for Evaluating the Coupled Pore Pressure Generation and Nonlinear Shear Modulus Behavior of Liquefiable Soils,â G eotechn i cal T esti n g J ourn al , Vol. 32, No. 1, pp. 11-21. Laera, A. and Brinkgreve, R.B.J. (2015) âSite Response Analysis and Liquefaction Evaluation,â White Paper, Plaxis, Delft, The Netherlands, 42 pp. Matasovic, N. and Vucetic, M. (1993). âCyclic Characterization of Liquefiable Sands,â ASCE J ourn al of G eotechn i cal E n g i n eeri n g , Vol. 119, No. 11, pp. 1805 1822. Seed, H. B., and Idriss, I. M. (1970), âAnalyses of Ground Motions at Union Bay, Seattle During Earthquakes and Distant Nuclear Blasts,â B ul l eti n of the S ei sm ol og i cal S oci ety of A m eri ca, Vol. 60, No. 1, pp. 125-136.