1d nonlinear numerical methods - civil.utah.edubartlett/cveen7330/1d nonlinear numerical... ·...
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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
© Steven F. Bartlett, 2011
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
3:32 PM
1D Nonlinear Numerical Methods Page 2
© Steven F. Bartlett, 2011
Homework Assignment #5 (cont.)
Solution (Excel) for uniform Vs = 80 m/s and 10 damping5.
Solution (FLAC)6.
Nonlinear MethodsSunday, August 14, 2011
3:32 PM
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© Steven F. Bartlett, 2011
Homework Assignment #5 (cont.)
Solution (Excel) (first 5 time steps)5.
Nonlinear MethodsSunday, August 14, 20113:32 PM
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© Steven F. Bartlett, 2011
EQL Method
Nonlinear Methods
Comparison of 1D Equivalent Liner vs. 1D Nonlinear MethodsSunday, August 14, 2011
3:32 PM
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© Steven F. Bartlett, 2011
Target Spectrum for Comparisons
EQL vs NL ComparisonsSunday, August 14, 2011
3:32 PM
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© Steven F. Bartlett, 2011
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
3:32 PM
1D Nonlinear Numerical Methods Page 7
© Steven F. Bartlett, 2011
Fundamental Equation of Motion
Lumped Mass System used in DeepSoilSunday, August 14, 2011
3:32 PM
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© Steven F. Bartlett, 2011
Modified Soil Hyperbolic Model used in DeepSoil
DEEPSoil - Hyperbolic ModelSunday, August 14, 2011
3:32 PM
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© Steven F. Bartlett, 2011
Introducing Pressure Dependency (Important for Deep Sediments)
DEEPSoil (cont.)Sunday, August 14, 2011
3:32 PM
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© Steven F. Bartlett, 2011
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
3:32 PM
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© Steven F. Bartlett, 2011
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
3:32 PM
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© Steven F. Bartlett, 2011
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
3:32 PM
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© Steven F. Bartlett, 2011
DARENDELI, 2001
Effects of Mean Effective Stress on Shear Modulus and Damping CurvesSunday, August 14, 2011
3:32 PM
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© Steven F. Bartlett, 2011
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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© Steven F. Bartlett, 2011
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
3:32 PM
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© Steven F. Bartlett, 2011
DARENDELI, 2001
Effects of Plasticity on Shear Modulus and Damping CurvesSunday, August 14, 2011
3:32 PM
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© Steven F. Bartlett, 2011
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
Shearing Strain (%) PI = 0 % PI = 15 % PI = 30 % PI = 50 % PI = 100 %1.00E-05 0.804 0.997 1.191 1.450 2.0962.20E-05 0.808 1.000 1.193 1.451 2.0974.84E-05 0.820 1.008 1.199 1.456 2.1001.00E-04 0.839 1.021 1.209 1.464 2.1052.20E-04 0.884 1.053 1.234 1.482 2.1174.84E-04 0.982 1.122 1.287 1.523 2.1431.00E-03 1.174 1.257 1.392 1.603 2.1932.20E-03 1.602 1.562 1.628 1.786 2.3094.84E-03 2.474 2.198 2.128 2.175 2.5601.00E-02 3.953 3.317 3.028 2.888 3.0292.20E-02 6.579 5.440 4.803 4.343 4.0294.84E-02 10.184 8.650 7.664 6.824 5.8761.00E-01 13.788 12.217 11.092 10.024 8.5412.20E-01 17.199 15.951 14.966 13.941 12.2794.84E-01 19.565 18.829 18.185 17.458 16.1321.00E+00 20.716 20.460 20.178 19.815 19.069
DARENDELI, 2001
Effects of Plasticity on Shear Modulus and Damping Curves (cont.)Sunday, August 14, 2011
3:32 PM
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© Steven F. Bartlett, 2011
DARENDELI, 2001
Shear Modulus and Damping Curves (' = 0.25 atm)Sunday, August 14, 2011
3:32 PM
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© Steven F. Bartlett, 2011
DARENDELI, 2001
Shearing Strain (%) PI = 0 % PI = 15 % PI = 30 % PI = 50 % PI = 100 %1.00E-05 0.999 0.999 1.000 1.000 1.0002.20E-05 0.998 0.999 0.999 0.999 1.0004.84E-05 0.996 0.997 0.998 0.998 0.9991.00E-04 0.993 0.995 0.996 0.997 0.9982.20E-04 0.986 0.990 0.992 0.994 0.9964.84E-04 0.971 0.979 0.983 0.987 0.9911.00E-03 0.944 0.959 0.968 0.975 0.9832.20E-03 0.891 0.919 0.936 0.949 0.9664.84E-03 0.799 0.847 0.876 0.900 0.9321.00E-02 0.671 0.739 0.783 0.822 0.8762.20E-02 0.497 0.579 0.637 0.692 0.7744.84E-02 0.324 0.400 0.459 0.521 0.6251.00E-01 0.197 0.255 0.303 0.358 0.4612.20E-01 0.107 0.142 0.174 0.213 0.2934.84E-01 0.055 0.074 0.093 0.116 0.1671.00E+00 0.029 0.040 0.050 0.063 0.093
Shearing Strain (%) PI = 0 % PI = 15 % PI = 30 % PI = 50 % PI = 100 %1.00E-05 1.201 1.489 1.778 2.164 3.1292.20E-05 1.207 1.493 1.781 2.166 3.1314.84E-05 1.226 1.506 1.791 2.174 3.1361.00E-04 1.257 1.528 1.808 2.187 3.1442.20E-04 1.330 1.579 1.848 2.217 3.1634.84E-04 1.487 1.690 1.933 2.282 3.2041.00E-03 1.792 1.906 2.101 2.411 3.2862.20E-03 2.458 2.387 2.476 2.702 3.4724.84E-03 3.762 3.358 3.249 3.310 3.8681.00E-02 5.821 4.977 4.581 4.386 4.5932.20E-02 9.097 7.778 7.010 6.441 6.0704.84E-02 12.993 11.489 10.477 9.589 8.5791.00E-01 16.376 15.064 14.088 13.137 11.7982.20E-01 19.181 18.334 17.640 16.904 15.7164.84E-01 20.829 20.515 20.208 19.849 19.2131.00E+00 21.393 21.507 21.542 21.547 21.544
Shear Modulus and Damping Curves (' = 0.25 atm)Sunday, August 14, 2011
3:32 PM
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© Steven F. Bartlett, 2011
DARENDELI, 2001
Shear Modulus and Damping Curves (' = 1 atm)Sunday, August 14, 20113:32 PM
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© Steven F. Bartlett, 2011
DARENDELI, 2001
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
Shearing Strain (%) PI = 0 % PI = 15 % PI = 30 % PI = 50 % PI = 100 %1.00E-05 0.804 0.997 1.191 1.450 2.0962.20E-05 0.808 1.000 1.193 1.451 2.0974.84E-05 0.820 1.008 1.199 1.456 2.1001.00E-04 0.839 1.021 1.209 1.464 2.1052.20E-04 0.884 1.053 1.234 1.482 2.1174.84E-04 0.982 1.122 1.287 1.523 2.1431.00E-03 1.174 1.257 1.392 1.603 2.1932.20E-03 1.602 1.562 1.628 1.786 2.3094.84E-03 2.474 2.198 2.128 2.175 2.5601.00E-02 3.953 3.317 3.028 2.888 3.0292.20E-02 6.579 5.440 4.803 4.343 4.0294.84E-02 10.184 8.650 7.664 6.824 5.8761.00E-01 13.788 12.217 11.092 10.024 8.5412.20E-01 17.199 15.951 14.966 13.941 12.2794.84E-01 19.565 18.829 18.185 17.458 16.1321.00E+00 20.716 20.460 20.178 19.815 19.069
Shear Modulus and Damping Curves (' = 1 atm)Sunday, August 14, 2011
3:32 PM
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© Steven F. Bartlett, 2011
DARENDELI, 2001
Shear Modulus and Damping Curves (' = 4 atm)Sunday, August 14, 2011
3:32 PM
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© Steven F. Bartlett, 2011
DARENDELI, 2001
Shearing Strain (%) PI = 0 % PI = 15 % PI = 30 % PI = 50 % PI = 100 %1.00E-05 1.000 1.000 1.000 1.000 1.0002.20E-05 0.999 1.000 1.000 1.000 1.0004.84E-05 0.998 0.999 0.999 0.999 1.0001.00E-04 0.997 0.998 0.998 0.999 0.9992.20E-04 0.994 0.996 0.997 0.997 0.9984.84E-04 0.988 0.991 0.993 0.995 0.9961.00E-03 0.976 0.983 0.986 0.989 0.9932.20E-03 0.952 0.965 0.972 0.978 0.9864.84E-03 0.906 0.931 0.945 0.956 0.9711.00E-02 0.832 0.873 0.898 0.918 0.9452.20E-02 0.706 0.770 0.810 0.845 0.8934.84E-02 0.538 0.618 0.673 0.725 0.8021.00E-01 0.374 0.454 0.514 0.575 0.6752.20E-01 0.225 0.287 0.339 0.396 0.5014.84E-01 0.123 0.163 0.199 0.241 0.3271.00E+00 0.067 0.091 0.113 0.140 0.200
Shearing Strain (%) PI = 0 % PI = 15 % PI = 30 % PI = 50 % PI = 100 %1.00E-05 0.539 0.668 0.798 0.971 1.4042.20E-05 0.541 0.670 0.799 0.972 1.4054.84E-05 0.548 0.675 0.803 0.975 1.4071.00E-04 0.560 0.683 0.809 0.980 1.4102.20E-04 0.588 0.703 0.824 0.991 1.4174.84E-04 0.649 0.745 0.857 1.016 1.4331.00E-03 0.769 0.829 0.922 1.066 1.4642.20E-03 1.039 1.021 1.070 1.180 1.5374.84E-03 1.607 1.428 1.388 1.426 1.6931.00E-02 2.618 2.173 1.977 1.886 1.9912.20E-02 4.572 3.684 3.206 2.871 2.6484.84E-02 7.621 6.235 5.387 4.693 3.9341.00E-01 11.134 9.482 8.357 7.333 5.9722.20E-01 14.946 13.400 12.231 11.056 9.2264.84E-01 17.990 16.866 15.935 14.917 13.1181.00E+00 19.792 19.158 18.571 17.876 16.513
Shear Modulus and Damping Curves (' = 4 atm)Sunday, August 14, 2011
3:32 PM
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© Steven F. Bartlett, 2011
DARENDELI, 2001
Shear Modulus and Damping Curves (' = 16 atm)Sunday, August 14, 2011
3:32 PM
1D Nonlinear Numerical Methods Page 25
© Steven F. Bartlett, 2011
DARENDELI, 2001
Shearing Strain (%) PI = 0 % PI = 15 % PI = 30 % PI = 50 % PI = 100 %1.00E-05 1.000 1.000 1.000 1.000 1.0002.20E-05 1.000 1.000 1.000 1.000 1.0004.84E-05 0.999 0.999 0.999 1.000 1.0001.00E-04 0.998 0.999 0.999 0.999 0.9992.20E-04 0.996 0.997 0.998 0.998 0.9994.84E-04 0.992 0.994 0.996 0.997 0.9981.00E-03 0.985 0.989 0.991 0.993 0.9962.20E-03 0.969 0.977 0.982 0.986 0.9914.84E-03 0.938 0.954 0.964 0.972 0.9811.00E-02 0.885 0.915 0.932 0.946 0.9642.20E-02 0.789 0.839 0.869 0.895 0.9294.84E-02 0.645 0.716 0.763 0.804 0.8631.00E-01 0.482 0.564 0.623 0.679 0.7642.20E-01 0.311 0.386 0.444 0.506 0.6104.84E-01 0.179 0.233 0.279 0.331 0.4311.00E+00 0.101 0.135 0.166 0.203 0.280
Shearing Strain (%) PI = 0 % PI = 15 % PI = 30 % PI = 50 % PI = 100 %1.00E-05 0.361 0.448 0.534 0.650 0.9412.20E-05 0.362 0.449 0.535 0.651 0.9414.84E-05 0.367 0.452 0.538 0.653 0.9421.00E-04 0.374 0.457 0.541 0.656 0.9442.20E-04 0.391 0.469 0.551 0.663 0.9494.84E-04 0.429 0.495 0.571 0.678 0.9581.00E-03 0.503 0.547 0.611 0.709 0.9782.20E-03 0.673 0.667 0.704 0.780 1.0234.84E-03 1.035 0.924 0.903 0.934 1.1201.00E-02 1.702 1.407 1.281 1.227 1.3082.20E-02 3.075 2.433 2.100 1.871 1.7294.84E-02 5.449 4.318 3.659 3.138 2.5891.00E-01 8.573 7.021 6.022 5.151 4.0492.20E-01 12.483 10.780 9.557 8.381 6.6514.84E-01 16.070 14.619 13.472 12.268 10.2411.00E+00 18.528 17.522 16.655 15.677 13.847
Shear Modulus and Damping Curves (' = 16 atm)Sunday, August 14, 2011
3:32 PM
1D Nonlinear Numerical Methods Page 26
© Steven F. Bartlett, 2011
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
12:45 PM
1D Nonlinear Numerical Methods Page 27
© Steven F. Bartlett, 2011
Finite difference calculation loop written with differential calculus
Finite Difference Approach (cont.)Wednesday, August 17, 2011
12:45 PM
1D Nonlinear Numerical Methods Page 28
© Steven F. Bartlett, 2011
Finite difference calculation loop written with incremental approach
Finite Difference Approach (cont.)Sunday, August 14, 2011
3:32 PM
1D Nonlinear Numerical Methods Page 29
© Steven F. Bartlett, 2011
1D Finite Difference Solution for Wave Propagation Wednesday, August 17, 2011
12:45 PM
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© Steven F. Bartlett, 2014
1D Finite Difference Solution for Wave Propagation (cont.)Tuesday, March 04, 2014
11:45 AM
1D Nonlinear Numerical Methods Page 31
© Steven F. Bartlett, 2011
1D Finite Difference Solution for Wave Propagation (cont.)Wednesday, August 17, 2011
12:45 PM
1D Nonlinear Numerical Methods Page 32
© Steven F. Bartlett, 2014
1D Finite Difference Solution for Wave Propagation (cont.)Wednesday, March 05, 2014
11:45 AM
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© Steven F. Bartlett, 2011
-1
-0.8
-0.6
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0
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0.8
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0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Displacement of Top Node vs Time
1D Finite Difference Solution for Wave Propagation (cont.)Wednesday, August 17, 2011
12:45 PM
1D Nonlinear Numerical Methods Page 34
© Steven F. Bartlett, 2011
;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'
1D Finite Difference Solution for Wave Propagation (cont.)Wednesday, August 17, 2011
12:45 PM
1D Nonlinear Numerical Methods Page 35
© Steven F. Bartlett, 2011
Note that shear resistance has two components: elastic and damping.
Incorporating DampingSunday, August 14, 2011
3:32 PM
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© Steven F. Bartlett, 2011
Stiffness due to viscous damping
Incorporating Damping (cont.)Sunday, August 14, 2011
3:32 PM
1D Nonlinear Numerical Methods Page 37
© Steven F. Bartlett, 2011
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
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1D Nonlinear Numerical Methods Page 38
© Steven F. Bartlett, 2011
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
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1D Nonlinear Numerical Methods Page 39
© Steven F. Bartlett, 2011
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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1D Nonlinear Numerical Methods Page 40
© Steven F. Bartlett, 2011
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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© Steven F. Bartlett, 2011
Hysteretic Damping - Tangent-Modulus Functions (cont.)Sunday, August 14, 2011
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1D Nonlinear Numerical Methods Page 43
© Steven F. Bartlett, 2011
Hysteretic Damping - Tangent-Modulus Functions (cont.)Sunday, August 14, 2011
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1D Nonlinear Numerical Methods Page 44
© Steven F. Bartlett, 2011
Hysteretic Damping - Tangent-Modulus Functions (cont.)Sunday, August 14, 2011
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© Steven F. Bartlett, 2011
Hysteretic Damping - Tangent-Modulus Functions (cont.)Sunday, August 14, 2011
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© Steven F. Bartlett, 2011
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
Hysteretic Damping - Tangent-Modulus Functions (cont.)Sunday, August 14, 2011
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1D Nonlinear Numerical Methods Page 47
© Steven F. Bartlett, 2011
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'
Hysteretic Damping - Tangent-Modulus Functions (cont.)Sunday, August 14, 2011
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1D Nonlinear Numerical Methods Page 48
© Steven F. Bartlett, 2011
Rayleigh DampingSunday, August 14, 2011
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© Steven F. Bartlett, 2011
Rayleigh Damping (cont.)Sunday, August 14, 2011
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© Steven F. Bartlett, 2011
Hysteretic vs Rayleigh Damping ComparisonSunday, August 14, 2011
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© Steven F. Bartlett, 2011
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