1D Equivalent Linear Method Page 1 1D Analysis - Simplified Methods Monday, February 13, 2017 2:32 PM Reading Assignment Lecture Notes Pp. 255-275 Kramer (EQL method) p. 562 Kramer (Trigonometric Notation - Fourier Series) Shake Theory.pdf Other Materials none Homework Assignment #3 1. 2. Download and install DEEPSOIL on your computer. (10 points) Obtain the Matahina Dam, New Zealand record from the PEER database (http://ngawest2.berkeley.edu/spectras/1121/searches/new) (20 points) a. To find this record, type: Matahina Dam in the Station Name Field on the input screen, then press the search records button. b. Display the time series of the selected record as unscaled. You should then be able to see the time histories for horizontal components 1 and 2 and the vertical component. c. Download this recording by selecting "download time series records (metadata+spectra+traces)." d. Use the horizontal time history from this station that has the largest velocity and displacement pluses as the fault normal component for the scaled record (Note: you do not have to perform a rotation to find the principal components, since we have already done this in a previous assignment; however this would normally be required. e. Scale this record so that peak ground acceleration is 0.65 g using simple scaling (i.e., no spectral matching required. f. Plot the scaled acceleration time history. g. Plot the scaled response spectrum of the scale record. Steven F. Bartlett, 2017
1D Equivalent Linear Method Page 2 1D Analysis - Simplified Methods Monday, February 13, 2017 2:32 PM 3. 4. Develop a soil profile for site-specific ground response analysis using soil properties for the I-15 project at 600 South Street (see attached) (20 points) a. Use the SCPT obtain velocities for depths less than 72 feet. Ensure that no layer is greater than 5 feet thick. Otherwise, subdivide the layer b. Use the downhole geophysics velocities from depths of 72 feet to 192 feet. Ensure that no layer is greater than 15 feet thick. Otherwise, subdivide the layer. c. For sands, use Seed and Idriss upper bound curves d. For silts, use Vucetic and Dorby curves with PI = 0 e. For clays, use Vucetic and Dorby curves with PI = 20 f. Treat the bottom layer of the soil log as a silty clay with PI = 20 and this layer extends to a depth of 200 feet. g. Below this layer, assume that the soil profile is a sand and has linearly increasing Vs measurements to a value of 1200 feet per second at a depth of 260 feet. Use 15-foot thick layers as a maximum thickness for this part of the profile h. For the elastic half space properties, use the velocity corresponding to the deepest Vs measurement in the vs profile with 2 percent damping Perform a site-specific, equivalent-linear (EQL) ground response analysis for this soil profile and provide the following plots: (10 points) a. Response spectrum summary b. Acceleration time histories for layer 1 c. pga profile d. Convergence check Steven F. Bartlett, 2017
1D Equivalent Linear Method Page 3 Homework Assignment Attachment Monday, February 13, 2017 2:32 PM
1D Equivalent Linear Method Page 4 Homework Assignment Attachment Monday, February 13, 2017 2:32 PM
1D Equivalent Linear Method Page 5 Homework Assignment Attachment Monday, February 13, 2017 2:32 PM
1D Equivalent Linear Method Page 6 1D Equivalent Linear Method 1. 2. 3. Dynamic behavior of soils is quite complex and requires models which capture the primary aspects of cyclic behavior, but these models need to be simple, rational models so they can be applied Three classes of 1D dynamic soil models: a) equivalent linear method b) cyclic nonlinear c) advanced constitutive models The equivalent linear method was developed a the U. of California at Berkeley and is incorporated in the program SHAKE. vertically 1-D propagation of shear waves in a multi-layered system is assumed in SHAKE. SHAKE produces an approximation to the nonlinear response of soils under earthquake loading, but is very efficient computationally. nonlinear stress strain loop is approximated by a single equivalent linear strain-compatible shear modulus that decreases with increasing shear strain Material damping is also estimated by a constant, strain-compatible value. The material properties for the model are usually developed from geotechnical laboratory testing, or estimated from typical values in literature. Limitations of EQL method (i.e., SHAKE) SHAKE cannot be used directly to solve problems involving ground deformation (linear model, which does not follow the hysteresis loop to model strain) final strain is zero (after cycling has stopped) because it is an elastic model no limiting value in shear strength, so failure does occur in the model failure of the soil has to be judged by the estimate of the maximum shear stress calculated by the model.
1D Equivalent Linear Method Page 7 1D Wave Equation 1D Wave Equation for elastic material 1D Wave Equation for viscoelastic material Damping in a Visco-elastic material
1D Equivalent Linear Method Page 8 Visco-elastic model
1D Equivalent Linear Method Page 9 Visco-elastic model (cont.)
1D Equivalent Linear Method Page 10 Visco-elastic model (cont.)
1D Equivalent Linear Method Page 11 Equivalent Linear Method - Flow Chart 1. 2. 3. 4. 5. Express the input (rock outcrop) motion in the frequency domain as a Fourier series (as the sum of a series of sine waves of different amplitudes, frequencies, and phase angles). For an earthquake motion, this Fourier series will have both real and imaginary parts. Define the transfer function. The transfer function will have both real and imaginary parts. Compute the Fourier series of the output (ground surface) motion as the product of the Fourier series of the input (bedrock) motion and the transfer function. This Fourier series will also have both real and imaginary parts. Express the output motion in the time domain by means of an inverse Fourier transform. Calculate the shear strains from the displacement output of 4. Verify that the strain is compatible with the assumed shear modulus and damping values assumed. If not, iterate until strain compatible properties are obtained by changing the estimate of the effective shear modulus and associated damping.
1D Equivalent Linear Method Page 12 Fourier Transform
1D Equivalent Linear Method Page 13 Fourier Transform (cont.) Fourier amplitude spectrum from Seismosignal for the Matahina Dam, New Zealand record. The Fourier amplitude values (y-axis) are equal to the c n values in the above equation. In addition to a Fourier amplitude spectrum there is also a corresponding Fourier phase spectrum that gives the phase angle as a function of frequency. Unfortunately, Seismosignal does not provide this plot.
1D Equivalent Linear Method Page 14 Fourier Transform (cont.) Example The Fourier series can be used to match any periodic function, if enough terms are included. For example, lets use a Fourier series to generate a square function of the form: 2-2 A = 2 Tf = 1 6.283185 to= dt= 0.01 A = amplitude Tf = time of function (duration) Frequency (rad/s) dt = time step (s) Pasted from <file:///c:\users \sfbartlett\documents\my% 20Courses\6330 \Fourier_sqwave.xls> Blue line equals sum of series for 13 terms Other lines shows the individual terms.
1D Equivalent Linear Method Page 15 Fourier Transform (cont.) a1=4a/(n*pi) = 2.546479 a2 = 0 a3 = -0.84883 a4 = 0 a5 = 0.509296 a6 = 0 a7 = -0.36378 a8 = 0 a9 = 0.282942 a10= 0 a11= -0.2315 a12= 0 a13 0.195883 Amplitude of each of the terms in the series. For this case the even terms are not needed, so their Fourier amplitude is set to zero for the even terms. a n a n = 4A/(n*pi) Pasted from <file:///c:\users \sfbartlett\documents\my%20courses \6330\Fourier_sqwave.xls> Note because of space limitations only the first 0.18 s of the series is shown here. t 1st term 3rd term 5th term 7th term 9th term 11th term 13th term sum 0.00 2.55-0.85 0.51-0.36 0.28-0.23 0.20 2.09 0.01 2.54-0.83 0.48-0.33 0.24-0.18 0.13 2.06 0.02 2.53-0.79 0.41-0.23 0.12-0.04-0.01 1.98 0.03 2.50-0.72 0.30-0.09-0.04 0.11-0.15 1.92 0.04 2.47-0.62 0.16 0.07-0.18 0.22-0.19 1.91 0.05 2.42-0.50 0.00 0.21-0.27 0.22-0.12 1.97 0.06 2.37-0.36-0.16 0.32-0.27 0.12 0.04 2.05 0.07 2.30-0.21-0.30 0.36-0.19-0.03 0.17 2.10 0.08 2.23-0.05-0.41 0.34-0.05-0.17 0.19 2.07 0.09 2.15 0.11-0.48 0.25 0.10-0.23 0.09 1.99 0.10 2.06 0.26-0.51 0.11 0.23-0.19-0.06 1.91 0.11 1.96 0.41-0.48-0.05 0.28-0.06-0.18 1.89 0.12 1.86 0.54-0.41-0.19 0.25 0.10-0.18 1.95 0.13 1.74 0.65-0.30-0.31 0.14 0.21-0.07 2.06 0.14 1.62 0.74-0.16-0.36-0.02 0.22 0.08 2.14 0.15 1.50 0.81 0.00-0.35-0.17 0.14 0.19 2.11 0.16 1.36 0.84 0.16-0.27-0.26-0.01 0.17 1.99 0.17 1.23 0.85 0.30-0.13-0.28-0.16 0.05 1.85 0.18 1.08 0.82 0.41 0.02-0.21-0.23-0.10 1.80 Pasted from <file:///c:\users\sfbartlett\documents\my%20courses\6330\fourier_sqwave.xls>
1D Equivalent Linear Method Page 16 Fourier Transform (cont.)
1D Equivalent Linear Method Page 17 Transfer Functions Development of Transfer Function - Function to relate base rock motion to surface soil motion.
1D Equivalent Linear Method Page 18 Transfer Functions (cont.)
1D Equivalent Linear Method Page 19 Transfer Functions (cont.) Transfer function for 2-layer system (rock and soil) The same process can be used to calculate the transfer functions for a multiple layer system.
1D Equivalent Linear Method Page 20 Transfer Functions (cont.) For more details, see Shake Theory.pdf
1D Equivalent Linear Method Page 21 Transfer Functions (cont.) a. b. c. d. e. Period function (earthquake acceleration time history) Fast Fourier transform (FFT) yield Fourier series with 2 n terms Each term of the Fourier series is inputted into transfer function. The transfer function is used to calculate the soil response for each layer (i.e., complex response) and is represented for each term in the series. The complex response with all it terms is converted back into a single response by use an inverse Fast Fourier transform (IFFT). Once this is completed, the program checks to see if the G (shear modulus) and D damping are consistent with those assumed at the beginning of the analysis, if not then the program adjust the input G and D values and recalculates the associated strain until convergence is achieved.
1D Equivalent Linear Method Page 22 Iteration to Determine Strain Compatible Properties Goal of Equivalent Linear Analysis is to determine values of Gsec and equivalent damping that are consistent for each soil layer with the level of strain produced in that layer. These are called strain compatible properties. Note that the transfer functions develop on the previous pages are only valid only for an elastic material and prescribed damping However, a nonlinear system can be express by using the secant shear modulus, Gsec and equivalent damping Hysteretic behavior approximated by Gsec and equivalent damping Equivalent damping is the damping ratio that produces the same energy loss in a single cycle as the equivalent actual hysteresis loop Earthquakes produce earthquake motion that is highly irregular with a peak amplitude that may only be approached in a few spikes in the record. As a result, it is common practice to characterize the effective strain level of a transient record as 50 to 70 percent of the peak value, based on statistical analysis of the number of significant cycles in earthquake records and a comparison of their peaks with the maximum peak. Usually a value of 0.65 is used for the effective strain level in practice. The results, however are not very sensitive to this assumed value.
1D Equivalent Linear Method Page 23 Development of Input Motion - Time Domain http://peer.berkeley.edu/peer_ground_motion_database/
1D Equivalent Linear Method Page 24 Development of Input Motion - Time Domain (cont.) Search Criteria for Earthquake Records
1D Equivalent Linear Method Page 25 Development of Input Motion - Time Domain (cont.) Tuesday, March 04, 2014 2:32 PM Search Results Steven F. Bartlett, 2014
1D Equivalent Linear Method Page 26 Development of Input Motion - Time Domain (cont.) Tuesday, March 04, 2014 2:32 PM Double click on individual record to show its components 0.65 g target Note that the fault normal component of the Matahina Dam, New Zealand has a response spectrum that is generally above the target spectrum. It would be a good candidate for analysis, if we are only using 1 record. Fault normal component Steven F. Bartlett, 2014
1D Equivalent Linear Method Page 27 Development of Input Motion - Time Domain (cont.) Note that the PEER website does appear to scale properly, so the time history was adjusted to that pga = 0.65 g Fault normal component of the Matahina Dam, New Zealand
1D Equivalent Linear Method Page 28 Development of Input Motion - Time Domain (cont.) Tuesday, March 04, 2014 2:32 PM Fault normal component of the Matahina Dam, New Zealand scaled to 0.65 pga using Seismosignal Steven F. Bartlett, 2014
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