1D Nonlinear Numerical Methods

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1 1D Nonlinear Numerical Methods Page 1 1D Nonlinear Numerical Methods Reading Assignment Lecture Notes Pp Kramer DEEPSOIL.pdf 2001 Darendeli, Ch. 10 Other Materials DeepSoil User's Manual 2001 Darendeli Homework Assignment # Obtain the scaled Matahina Dam, New Zealand record from the course website and plot the following: (10 points) a. Plot the scaled acceleration time history b. Plot the scaled response spectrum Develop a soil profile for ground response analysis using soil properties for the I-15 project at 600 South Street (see attached) and the shear wave velocities found in SLC Vs profile.xls. (20 points) a. For sands, Darendeli, 2001 curves b. For silts, use Darendeli, 2001 with PI = 0 c. For clays, use Darendeli, 2001 curves with PI = 20 d. Treat layer 18 as a clay with PI = 20 and use Darendeli, 2001 curves e. Treat layer 19 as a sand and use Darendeli, 2001 curves f. For the bedrock velocity, use the velocity corresponding to the deepest Vs measurement in the soil profile with 2 percent damping Perform a site-specific, non-linear time domain ground response analysis for this soil profile using the pressure dependent hyperbolic model and Masing critera. Provide the following plots of the results: (15 points) a. Response spectrum summary b. Acceleration time histories for layer 1 c. pga profile Repeat problem 3 but perform a EQL analysis using the directions given in HW#3 problem 3. Plot a comparative plot of the response spectra using the spectrum from the nonlinear pressure dependent model (previous problem) versus the EQL pressure independent model (HW3 problem 4). (10 points). (SEE NEXT PG.)

2 1D Nonlinear Numerical Methods Page 2 Nonlinear Methods Homework Assignment #5 (cont.) 5. Modify the finite difference spreadsheet provided on the course website to include (20 points): a. Heterogeneous layers i. Varying thickness ii. Varying unit weight iii. Varying shear modulus b. Damping c. Given the information below, use the modified spreadsheet to perform a dynamic analysis for a duration of 2.0 s. Plot the response of the surface node versus time for verification: Layer # layer thickness unit weight Vs Damping (m) kn/m^3 (m/s) Poisson ratio = 0.35 v(t) = A cos( t + ) A = Verify your solution in 5 by performing an linear elastic analysis in DEEPSoil or FLAC for the same soil properties and velocity input (10 points).

3 1D Nonlinear Numerical Methods Page 3 Nonlinear Methods Homework Assignment #5 (cont.) 5. Solution (Excel) for uniform Vs = 80 m/s and 10 damping 6. Solution (FLAC)

4 1D Nonlinear Numerical Methods Page 4 Nonlinear Methods Homework Assignment #5 (cont.) 5. Solution (Excel) (first 5 time steps)

5 1D Nonlinear Numerical Methods Page 5 Comparison of 1D Equivalent Liner vs. 1D Nonlinear Methods EQL Method Nonlinear Methods

6 1D Nonlinear Numerical Methods Page 6 EQL vs NL Comparisons Target Spectrum for Comparisons

7 1D Nonlinear Numerical Methods Page 7 EQL vs NL Comparisons (cont.) EQL (Shake) Results at Surface from 5 km Convolution Nonlinear Results (DEEPSoil at Surface from 5 km Convolution

8 1D Nonlinear Numerical Methods Page 8 Lumped Mass System used in DeepSoil Fundamental Equation of Motion

9 1D Nonlinear Numerical Methods Page 9 DEEPSoil - Hyperbolic Model Modified Soil Hyperbolic Model used in DeepSoil

10 1D Nonlinear Numerical Methods Page 10 DEEPSoil (cont.) Introducing Pressure Dependency (Important for Deep Sediments)

stiffness matrix small strain viscous damping

11 1D Nonlinear Numerical Methods Page 11 DeepSoil (cont.) Incorporating Pressure Dependency in Damping [K] = stiffness matrix small strain viscous damping hysteretic damping incorporated by the hysteretic behavior of the soil

12 1D Nonlinear Numerical Methods Page 12 DEEPSoil (cont.) Pressure-dependent parameters b and d used to adjust curves in DEEPSoil. However, DARENDELI, 2001 has published newer curves based on confining pressure and PI. These are also incorporated in DEEPSoil.

13 1D Nonlinear Numerical Methods Page 13 Shear Modulus and Damping Curves from DARENDELI, 2001 As part of various research projects [including the SRS (Savannah River Site) Project AA EPRI (Electric Power Research Institute) Project and ROSRINE (Resolution of Site Response Issues from the Northridge Earthquake) Project], numerous geotechnical sites were drilled and sampled. Intact soil samples over a depth range of several hundred meters were recovered from 20 of these sites. These soil samples were tested in the laboratory at The University of Texas at Austin (UTA) to characterize the materials dynamically. The presence of a database accumulated from testing these intact specimens motivated a re-evaluation of empirical curves employed in the state of practice. The weaknesses of empirical curves reported in the literature were identified and the necessity of developing an improved set of empirical curves was recognized. This study focused on developing the empirical framework that can be used to generate normalized modulus reduction and material damping curves. This framework is composed of simple equations. which incorporate the key parameters that control nonlinear soil behavior. The data collected over the past decade at The University of Texas at Austin are statistically analyzed using First-order. Second-moment Bayesian Method (FSBM). The effects of various parameters (such as confining pressure and soil plasticity on dynamic soil properties are evaluated and quantified within this framework. One of the most important aspects of this study is estimating not only the mean values of the empirical curves but also estimating the uncertainty associated with these values. This study provides the opportunity to handle uncertainty in the empirical estimates of dynamic soil properties within the probabilistic seismic hazard analysis framework. A refinement in site-specific probabilistic seismic hazard assessment is expected to materialize in the near future by incorporating the results of this study into the state of practice.

14 1D Nonlinear Numerical Methods Page 14 Effects of Mean Effective Stress on Shear Modulus and Damping Curves DARENDELI, 2001

15 1D Nonlinear Numerical Methods Page 15 Effects of Mean Effective Stress on Shear Modulus and Damping Curves (cont.) Shearing Strain (%) σo' = 0.25 atm σo' = 1.0 atm σo' = 4.0 atm σo' = 16 atm 1.00E E E E E E E E E E E E E E E E Shearing Strain (%) σo' = 0.25 atm σo' = 1.0 atm σo' = 4.0 atm σo' = 16 atm 1.00E E E E E E E E E E E E E E E E DARENDELI, 2001

16 1D Nonlinear Numerical Methods Page 16 Effects of Mean Effective Stress on Shear Modulus and Damping Curves (cont.) Curve 1 Curve 2 Curve 1 - Sand Darendeli, 2001 v' (psf) = OCR = 1 Ko = 0.4 N = 10 F = 1 Hz Curve 2 - Sand Darendeli, 2001 v' (psf) = 576 OCR = 1 Ko = 0.4 N = 10 F = 1 Hz Curve 2 Curve 1 DEEPSoil V4.0

17 1D Nonlinear Numerical Methods Page 17 Effects of Plasticity on Shear Modulus and Damping Curves DARENDELI, 2001

18 1D Nonlinear Numerical Methods Page 18 Effects of Plasticity on Shear Modulus and Damping Curves (cont.) Shearing Strain (%) PI = 0 % PI = 15 % PI = 30 % PI = 50 % PI = 100 % 1.00E E E E E E E E E E E E E E E E Shearing Strain (%) PI = 0 % PI = 15 % PI = 30 % PI = 50 % PI = 100 % 1.00E E E E E E E E E E E E E E E E DARENDELI, 2001

19 1D Nonlinear Numerical Methods Page 19 Shear Modulus and Damping Curves ( ' = 0.25 atm) DARENDELI, 2001

20 1D Nonlinear Numerical Methods Page 20 Shear Modulus and Damping Curves ( ' = 0.25 atm) Shearing Strain (%) PI = 0 % PI = 15 % PI = 30 % PI = 50 % PI = 100 % 1.00E E E E E E E E E E E E E E E E Shearing Strain (%) PI = 0 % PI = 15 % PI = 30 % PI = 50 % PI = 100 % 1.00E E E E E E E E E E E E E E E E DARENDELI, 2001

21 1D Nonlinear Numerical Methods Page 21 Shear Modulus and Damping Curves ( ' = 1 atm) DARENDELI, 2001

22 1D Nonlinear Numerical Methods Page 22 Shear Modulus and Damping Curves ( ' = 1 atm) Shearing Strain (%) PI = 0 % PI = 15 % PI = 30 % PI = 50 % PI = 100 % 1.00E E E E E E E E E E E E E E E E Shearing Strain (%) PI = 0 % PI = 15 % PI = 30 % PI = 50 % PI = 100 % 1.00E E E E E E E E E E E E E E E E DARENDELI, 2001

23 1D Nonlinear Numerical Methods Page 23 Shear Modulus and Damping Curves ( ' = 4 atm) DARENDELI, 2001

24 1D Nonlinear Numerical Methods Page 24 Shear Modulus and Damping Curves ( ' = 4 atm) Shearing Strain (%) PI = 0 % PI = 15 % PI = 30 % PI = 50 % PI = 100 % 1.00E E E E E E E E E E E E E E E E Shearing Strain (%) PI = 0 % PI = 15 % PI = 30 % PI = 50 % PI = 100 % 1.00E E E E E E E E E E E E E E E E DARENDELI, 2001

25 1D Nonlinear Numerical Methods Page 25 Shear Modulus and Damping Curves ( ' = 16 atm) DARENDELI, 2001

26 1D Nonlinear Numerical Methods Page 26 Shear Modulus and Damping Curves ( ' = 16 atm) Shearing Strain (%) PI = 0 % PI = 15 % PI = 30 % PI = 50 % PI = 100 % 1.00E E E E E E E E E E E E E E E E Shearing Strain (%) PI = 0 % PI = 15 % PI = 30 % PI = 50 % PI = 100 % 1.00E E E E E E E E E E E E E E E E DARENDELI, 2001

27 1D Nonlinear Numerical Methods Page 27 Finite Difference Approach Wednesday, August 17, :45 PM Note that with this approach we can approximate the change of things that vary either in space or time, or both. In regards to time, we will use the forward differencing approach in formulating the finite difference approach.

28 1D Nonlinear Numerical Methods Page 28 Finite Difference Approach (cont.) Wednesday, August 17, :45 PM Finite difference calculation loop written with differential calculus

29 1D Nonlinear Numerical Methods Page 29 Finite Difference Approach (cont.) Finite difference calculation loop written with incremental approach

30 1D Nonlinear Numerical Methods Page 30 1D Finite Difference Solution for Wave Propagation Wednesday, August 17, :45 PM

31 1D Nonlinear Numerical Methods Page 31 1D Finite Difference Solution for Wave Propagation (cont.) Tuesday, March 04, :45 AM Steven F. Bartlett, 2014

32 1D Nonlinear Numerical Methods Page 32 1D Finite Difference Solution for Wave Propagation (cont.) Wednesday, August 17, :45 PM

33 1D Nonlinear Numerical Methods Page 33 1D Finite Difference Solution for Wave Propagation (cont.) Wednesday, March 05, :45 AM Steven F. Bartlett, 2014

34 1D Nonlinear Numerical Methods Page 34 1D Finite Difference Solution for Wave Propagation (cont.) Wednesday, August 17, :45 PM 1 Displacement of Top Node vs Time

35 1D Nonlinear Numerical Methods Page 35 1D Finite Difference Solution for Wave Propagation (cont.) Wednesday, August 17, :45 PM ;FLAC verification of solution without damping config dynamic extra 5 grid 1 10 model elastic ini y mul 1 ;set dy_damp rayl ; 5 percent damping at 5 hz fix y prop dens 2000 bulk 9.6E6 shear 3.2E6 def wave wave=amp*cos(omega*dytime) if dytime>=100 wave=0 endif end set amp=0.3 set omega = apply xvel 1 hist wave yvel=0 j=1 his 1 xdisp i 1 j 1 his 2 xdisp i 1 j 11 his 3 xvel i 1 j 1 his 4 dytime set dytime = 0 ;set dydt = ; Flac can calc automatically solve dytime 5.01 save model2.sav 'last project state'

36 1D Nonlinear Numerical Methods Page 36 Incorporating Damping Note that shear resistance has two components: elastic and damping.

37 1D Nonlinear Numerical Methods Page 37 Incorporating Damping (cont.) Stiffness due to viscous damping

38 1D Nonlinear Numerical Methods Page 38 Hysteretic Damping as Implemented in FLAC Background The equivalent-linear method (see Section 3.2) has been in use for many years to calculate the wave propagation (and response spectra) in soil and rock, at sites subjected to seismic excitation. The method does not capture directly any nonlinear effects because it assumes linearity during the solution process; strain-dependent modulus and damping functions are only taken into account in an average sense, in order to approximate some effects of nonlinearity (damping and material softening). Although fully nonlinear codes such as FLAC are capable in principle of modeling the correct physics, it has been difficult to convince designers and licensing authorities to accept fully nonlinear simulations. One reason is that the constitutive models available to FLAC are either too simple (e.g., an elastic/plastic model, which does not reproduce the continuous yielding seen in soils), or too complicated (e.g., the Wang model [Wang et al. 2001], which needs many parameters and a lengthy calibration process). Further, there is a need to accept directly the same degradation curves used by equivalent-linear methods (see Figure 3.23 for an example), to allow engineers to move easily from using these methods to using fully nonlinear methods.

39 1D Nonlinear Numerical Methods Page 39 Hysteretic Damping (cont.) Formulation Modulus degradation curves, as illustrated in Figure 3.23, imply a nonlinear stress/strain curve. If we assume an ideal soil, in which the stress depends only on the strain (not on the number of cycles, or time), we can derive an incremental constitutive relation from the degradation curve, described by τe/γ = Ms, where τe is the normalized shear stress, γ the shear strain and Ms the normalized secant modulus. τe = Msγ (elastic component) Mt = dτe / dγ = Ms + γ dms / dγ (elastic and viscous component) where Mt is the normalized tangent modulus. The incremental shear modulus in a nonlinear simulation is then given by G Mt, where G is the small-strain shear modulus of the material.

40 1D Nonlinear Numerical Methods Page 40 Hysteretic Damping (cont.) FLAC code for single zone model with hysteretic damping conf dyn ext 5 grid 1 1 model elas prop dens 1000 shear 5e8 bulk 10e8 fix x y set dydt 1e-4 ini dy_damp hyst default his sxy i 1 j 1 his xdis i 1 j 2 his nstep 1 ini xvel 1e-2 j=2 cyc 1000 ini xvel mul -1 cyc 250 ini xvel mul -1 cyc 500

41 1D Nonlinear Numerical Methods Page 41 Hysteretic Damping - Types of Tangent-Modulus Functions

42 1D Nonlinear Numerical Methods Page 42 Hysteretic Damping - Tangent-Modulus Functions (cont.) Default Model (cont)

43 1D Nonlinear Numerical Methods Page 43 Hysteretic Damping - Tangent-Modulus Functions (cont.)

44 1D Nonlinear Numerical Methods Page 44 Hysteretic Damping - Tangent-Modulus Functions (cont.)

45 1D Nonlinear Numerical Methods Page 45 Hysteretic Damping - Tangent-Modulus Functions (cont.)

46 1D Nonlinear Numerical Methods Page 46 Hysteretic Damping - Tangent-Modulus Functions (cont.)

47 1D Nonlinear Numerical Methods Page 47 Hysteretic Damping - Tangent-Modulus Functions (cont.) The parameters for the various tangent-modulus functions can be changed to fit or types of modulus reduction and damping data. To judge the fit of the function parameters to the experimental data, the following FLAC subroutine can be used. conf dy def setup givenshear = 1e8 ; shear modulus CycStrain = 0.01 ; cyclic strain (%) / 10 ;---- derived.. setvel = 0.1 * min(1.0,cycstrain/0.1) givenbulk = 2.0 * givenshear timestep = min(1e-4,1e-5 / CycStrain) nstep1 = int( / (timestep * 10.0)) nstep2 = nstep1 * 2 nstep3 = nstep1 + nstep2 nstep5 = nstep1 + 2 * nstep2 end setup ; gri 1 1 ;m mohr m elastic prop den 1000 sh givenshear bu givenbulk cohesion = 50e3 fix x y ini xvel setvel j=2 set dydt 1e-4 ini dy_damp hyst default ; hysteretic damping his sxy i 1 j 1 his xdis i 1 j 2 his nstep 1 cyc nstep1 ini xv mul -1 cyc nstep2 ini xv mul -1 cyc nstep2 his write 1 vs 2 tab 1

48 1D Nonlinear Numerical Methods Page 48 Hysteretic Damping - Tangent-Modulus Functions (cont.) def HLoop emax = 0.0 emin = 0.0 tmax = 0.0 tmin = 0.0 loop n (1,nstep5) emax = max(xtable(1,n),emax) emin = min(xtable(1,n),emin) tmax = max(ytable(1,n),tmax) tmin = min(ytable(1,n),tmin) endloop slope = ((tmax - tmin) / (emax - emin)) / givenshear oo = out(' strain = '+string(emax*100.0)+'% G/Gmax = '+string(slope)) Tbase = ytable(1,nstep3) Lsum = 0.0 loop n (nstep1,nstep3-1) meant = (ytable(1,n) + ytable(1,n+1)) / 2.0 Lsum = Lsum + (xtable(1,n)-xtable(1,n+1)) * (meant - Tbase) endloop Usum = 0.0 loop n (nstep3,nstep5-1) meant = (ytable(1,n) + ytable(1,n+1)) / 2.0 Usum = Usum + (xtable(1,n+1)-xtable(1,n)) * (meant - Tbase) endloop Wdiff = Usum - Lsum Senergy = 0.5 * xtable(1,nstep1) * ytable(1,nstep1) Drat = Wdiff / (Senergy * 4.0 * pi) oo = out(' damping ratio = '+string(drat*100.0)+'%') end HLoop save singleelement.sav 'last project state'

49 1D Nonlinear Numerical Methods Page 49 Rayleigh Damping

50 1D Nonlinear Numerical Methods Page 50 Rayleigh Damping (cont.)

51 1D Nonlinear Numerical Methods Page 51 Hysteretic vs Rayleigh Damping Comparison

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CVEEN 7330 Modeling Exercise 2c

CVEEN 7330 Modeling Exercise 2c Table of Contents Table of Contents... 1 Objectives:... 2 FLAC Input:... 2 DEEPSOIL INPUTS:... 5 Required Outputs from FLAC:... 6 Required Output from DEEPSOIL:... 6 Additional