1D Nonlinear Numerical Methods - civil.utah.edubartlett/CVEEN7330/1D Nonlinear Numerical... · Lecture Notes Pp. 275 -280 Kramer DEEPSOIL.pdf 2001 Darendeli, Ch. 10 Reading Assignment

Lecture Notes○

Pp. 275 - 280 Kramer○

DEEPSOIL.pdf○

2001 Darendeli, Ch. 10○

Reading Assignment

DeepSoil User's Manual○

2001 Darendeli○

Other Materials

Homework Assignment #5

Plot the scaled acceleration time historya.Plot the scaled response spectrum b.

Obtain the scaled Matahina Dam, New Zealand record from the course website and plot the following: (10 points)

1.

For sands, Darendeli, 2001 curvesa.For silts, use Darendeli, 2001 with PI = 0b.For clays, use Darendeli, 2001 curves with PI = 20c.Treat layer 18 as a clay with PI = 20 and use Darendeli, 2001 curvesd.Treat layer 19 as a sand and use Darendeli, 2001 curvese.For the bedrock velocity, use the velocity corresponding to the deepest Vs measurement in the soil profile with 2 percent damping

f.

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)

2.

Response spectrum summary a.Acceleration time histories for layer 1 b.pga profilec.

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)

3.

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).

4.

(SEE NEXT PG.)

© Steven F. Bartlett, 2011

1D Nonlinear Numerical MethodsSunday, August 14, 2011

3:32 PM

1D Nonlinear Numerical Methods Page 1


Homework Assignment #5 (cont.)

Varying thicknessi.Varying unit weightii.Varying shear modulusiii.

Heterogeneous layersa.

Dampingb.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:

c.

Layer # layer thickness unit weight Vs Damping

(m) kN/m 3̂ (m/s)

1 1 19 150 5

2 1 19 170 5

3 1 19 190 5

4 0.5 20 150 5

5 1 20 150 5

6 0.5 20 150 5

7 2 20 150 5

8 1 21 170 5

9 1 21 170 5

10 1 21 170 5

Poisson ratio = 0.35

v(t) = A cos( t + )

A = 0.3

6.283

0.000

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).

6.

Modify the finite difference spreadsheet provided on the course website to include (20 points):

5.

Nonlinear MethodsSunday, August 14, 2011

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Solution (Excel) for uniform Vs = 80 m/s and 10 damping5.

Solution (FLAC)6.

Nonlinear MethodsSunday, August 14, 2011

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Solution (Excel) (first 5 time steps)5.

Nonlinear MethodsSunday, August 14, 20113:32 PM



EQL Method

Nonlinear Methods

Comparison of 1D Equivalent Liner vs. 1D Nonlinear MethodsSunday, August 14, 2011

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Target Spectrum for Comparisons

EQL vs NL ComparisonsSunday, August 14, 2011

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Nonlinear Results (DEEPSoil at Surface from 5 km Convolution

EQL (Shake) Results at Surface from 5 km Convolution

EQL vs NL Comparisons (cont.)Sunday, August 14, 2011

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Fundamental Equation of Motion

Lumped Mass System used in DeepSoilSunday, August 14, 2011

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Modified Soil Hyperbolic Model used in DeepSoil

DEEPSoil - Hyperbolic ModelSunday, August 14, 2011

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Introducing Pressure Dependency (Important for Deep Sediments)

DEEPSoil (cont.)Sunday, August 14, 2011

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Incorporating Pressure Dependency in Damping

[K] = stiffness matrix small strain

viscous damping

hysteretic damping incorporated by the hysteretic behavior of the soil

DeepSoil (cont.)Sunday, August 14, 2011

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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.

DEEPSoil (cont.)Sunday, August 14, 2011

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As part of various research projects [including the SRS (Savannah River Site) Project AA891070. EPRI (Electric Power Research Institute) Project 3302. 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.

Shear Modulus and Damping Curves from DARENDELI, 2001 Sunday, August 14, 2011

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DARENDELI, 2001

Effects of Mean Effective Stress on Shear Modulus and Damping CurvesSunday, August 14, 2011

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Shearing Strain (%) σo' = 0.25 atm σo' = 1.0 atm σo' = 4.0 atm σo' = 16 atm1.00E-05 0.999 0.999 1.000 1.0002.20E-05 0.998 0.999 0.999 1.0004.84E-05 0.996 0.998 0.998 0.9991.00E-04 0.993 0.995 0.997 0.9982.20E-04 0.986 0.991 0.994 0.9964.84E-04 0.971 0.981 0.988 0.9921.00E-03 0.944 0.964 0.976 0.9852.20E-03 0.891 0.928 0.952 0.9694.84E-03 0.799 0.861 0.906 0.9381.00E-02 0.671 0.761 0.832 0.8852.20E-02 0.497 0.607 0.706 0.7894.84E-02 0.324 0.428 0.538 0.6451.00E-01 0.197 0.277 0.374 0.4822.20E-01 0.107 0.157 0.225 0.3114.84E-01 0.055 0.083 0.123 0.1791.00E+00 0.029 0.044 0.067 0.101

Shearing Strain (%) σo' = 0.25 atm σo' = 1.0 atm σo' = 4.0 atm σo' = 16 atm1.00E-05 1.201 0.804 0.539 0.3612.20E-05 1.207 0.808 0.541 0.3624.84E-05 1.226 0.820 0.548 0.3671.00E-04 1.257 0.839 0.560 0.3742.20E-04 1.330 0.884 0.588 0.3914.84E-04 1.487 0.982 0.649 0.4291.00E-03 1.792 1.174 0.769 0.5032.20E-03 2.458 1.602 1.039 0.6734.84E-03 3.762 2.474 1.607 1.0351.00E-02 5.821 3.953 2.618 1.7022.20E-02 9.097 6.579 4.572 3.0754.84E-02 12.993 10.184 7.621 5.4491.00E-01 16.376 13.788 11.134 8.5732.20E-01 19.181 17.199 14.946 12.4834.84E-01 20.829 19.565 17.990 16.0701.00E+00 21.393 20.716 19.792 18.528

DARENDELI, 2001

Effects of Mean Effective Stress on Shear Modulus and Damping Curves (cont.)Sunday, August 14, 2011

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Curve 1

Curve 2

Curve 1 - SandDarendeli, 2001

v' (psf) = 11357OCR = 1Ko = 0.4N = 10F = 1 Hz

Curve 2 - SandDarendeli, 2001

v' (psf) = 576OCR = 1Ko = 0.4N = 10F = 1 Hz

Curve 2

Curve 1

DEEPSoil V4.0

Effects of Mean Effective Stress on Shear Modulus and Damping Curves (cont.)Sunday, August 14, 2011

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DARENDELI, 2001

Effects of Plasticity on Shear Modulus and Damping CurvesSunday, August 14, 2011

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Shearing Strain (%) PI = 0 % PI = 15 % PI = 30 % PI = 50 % PI = 100 %1.00E-05 0.999 1.000 1.000 1.000 1.0002.20E-05 0.999 0.999 0.999 1.000 1.0004.84E-05 0.998 0.998 0.999 0.999 0.9991.00E-04 0.995 0.997 0.997 0.998 0.9992.20E-04 0.991 0.993 0.995 0.996 0.9974.84E-04 0.981 0.986 0.989 0.992 0.9941.00E-03 0.964 0.973 0.979 0.984 0.9892.20E-03 0.928 0.947 0.958 0.967 0.9784.84E-03 0.861 0.896 0.917 0.934 0.9561.00E-02 0.761 0.816 0.849 0.878 0.9172.20E-02 0.607 0.682 0.732 0.778 0.8434.84E-02 0.428 0.509 0.569 0.629 0.7221.00E-01 0.277 0.348 0.404 0.465 0.5712.20E-01 0.157 0.205 0.248 0.296 0.3924.84E-01 0.083 0.111 0.137 0.169 0.2381.00E+00 0.044 0.060 0.076 0.095 0.138


DARENDELI, 2001

Effects of Plasticity on Shear Modulus and Damping Curves (cont.)Sunday, August 14, 2011

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DARENDELI, 2001

Shear Modulus and Damping Curves (' = 0.25 atm)Sunday, August 14, 2011

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DARENDELI, 2001



Shear Modulus and Damping Curves (' = 0.25 atm)Sunday, August 14, 2011

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DARENDELI, 2001

Shear Modulus and Damping Curves (' = 1 atm)Sunday, August 14, 20113:32 PM



DARENDELI, 2001



Shear Modulus and Damping Curves (' = 1 atm)Sunday, August 14, 2011

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DARENDELI, 2001


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DARENDELI, 2001




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DARENDELI, 2001


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DARENDELI, 2001




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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.

Finite Difference ApproachWednesday, August 17, 2011

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Finite difference calculation loop written with differential calculus

Finite Difference Approach (cont.)Wednesday, August 17, 2011

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Finite difference calculation loop written with incremental approach

Finite Difference Approach (cont.)Sunday, August 14, 2011

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1D Finite Difference Solution for Wave Propagation Wednesday, August 17, 2011

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1D Finite Difference Solution for Wave Propagation (cont.)Tuesday, March 04, 2014

11:45 AM



1D Finite Difference Solution for Wave Propagation (cont.)Wednesday, August 17, 2011

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1D Finite Difference Solution for Wave Propagation (cont.)Wednesday, March 05, 2014

11:45 AM



-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Displacement of Top Node vs Time


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;FLAC verification of solution without dampingconfig dynamic extra 5grid 1 10model elasticini y mul 1;set dy_damp rayl 0.05 5; 5 percent damping at 5 hzfix yprop dens 2000 bulk 9.6E6 shear 3.2E6def wave wave=amp*cos(omega*dytime)

wave=0 if dytime>=100

endifendset amp=0.3set omega = 6.283apply xvel 1 hist wave yvel=0 j=1his 1 xdisp i 1 j 1his 2 xdisp i 1 j 11his 3 xvel i 1 j 1his 4 dytimeset dytime = 0;set dydt = 0.0002; Flac can calc automaticallysolve dytime 5.01save model2.sav 'last project state'


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Note that shear resistance has two components: elastic and damping.

Incorporating DampingSunday, August 14, 2011

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Stiffness due to viscous damping

Incorporating Damping (cont.)Sunday, August 14, 2011

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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.

Hysteretic Damping as Implemented in FLACSunday, August 14, 2011

3:32 PM



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.

Hysteretic Damping (cont.)Sunday, August 14, 2011

3:32 PM



FLAC code for single zone model with hysteretic damping

conf dyn ext 5grid 1 1model elasprop dens 1000 shear 5e8 bulk 10e8fix x yset dydt 1e-4ini dy_damp hyst default -3.5 1.3his sxy i 1 j 1his xdis i 1 j 2his nstep 1ini xvel 1e-2 j=2cyc 1000ini xvel mul -1cyc 250ini xvel mul -1cyc 500

Hysteretic Damping (cont.)Sunday, August 14, 2011

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Hysteretic Damping - Types of Tangent-Modulus FunctionsSunday, August 14, 2011

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Default Model (cont)

Hysteretic Damping - Tangent-Modulus Functions (cont.)Sunday, August 14, 2011

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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 dydef 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(0.5 + 1.0 / (timestep * 10.0)) nstep2 = nstep1 * 2 nstep3 = nstep1 + nstep2 nstep5 = nstep1 + 2 * nstep2endsetup;gri 1 1;m mohrm elasticprop den 1000 sh givenShear bu givenBulk cohesion = 50e3fix x yini xvel setVel j=2set dydt 1e-4ini dy_damp hyst default -3.325 0.823; hysteretic dampinghis sxy i 1 j 1his xdis i 1 j 2his nstep 1cyc nstep1ini xv mul -1cyc nstep2ini xv mul -1cyc nstep2his write 1 vs 2 tab 1


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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)+'%')endHLoopsave singleelement.sav 'last project state'


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Rayleigh DampingSunday, August 14, 2011

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Rayleigh Damping (cont.)Sunday, August 14, 2011

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Hysteretic vs Rayleigh Damping ComparisonSunday, August 14, 2011

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BlankSunday, August 14, 2011

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Documents

1D Nonlinear Numerical Methods - civil.utah.edubartlett/CVEEN7330/1D Nonlinear Numerical... · Lecture Notes Pp. 275 -280 Kramer DEEPSOIL.pdf 2001 Darendeli, Ch. 10 Reading Assignment