Lecture 8 - Ground Response Analyses Page 1 1D Ground Response Analysis 1. 2. 3. Dynamic behavior of soils is quite complex and requires models which characterize the important aspects of cyclic behavior, but need to be simple, rational models. Three classes of dynamic soil models: a) equivalent linear (SHAKE and DEEPSOIL) b) cyclic nonlinear (DEEPSOIL) c) advanced constitutive (DEEPSOIL and FLAC) The equivalent linear (EQL) method has been developed in the computer program SHAKE at the UC Berkeley. The EQL method is also available in DEEPSOIL. a. Vertically 1-D propagation of shear waves in a multi-layered system is assumed in EQL method. b. EQL method produces an approximation to the nonlinear response of soils under earthquake loading, but is very efficient computationally. c. In the EQL method, the nonlinear stress strain loop is approximated by a single equivalent linear secant shear modulus that is a function of the amount of shear strain. d. Iteration is required to determine the appropriate equivalent secant shear modulus Geq that is compatible with the amount of strain that develops during the modeling process. e. The equivalent damping is determined from strain-controlled laboratory tests and is defined as a function of the shear strain level and such damping is used in the modeling process. f. Because the EQL method is fundamentally a damped linear elastic method using strain compatible secant shear modulus and the associated damping, it cannot be used directly to solve problems involving permanent shear deformation because it does not calculate permanent strain. Because the EQL model does not follow the actual hysteresis loops, the final shear strain is zero after cycling has stopped with no residual permanent shear strain. g. Also, because it is a linear elastic model, there is no limiting value for the shear strength of the soil (no failure criterion required), so failure, or yielding, is not allowed in the model.
Lecture 8 - Ground Response Analyses Page 2 Comparison of 1D Equivalent Liner vs. 1D Nonlinear Methods Sunday, August 14, 2011 3:32 PM EQL Method Nonlinear Methods
Lecture 8 - Ground Response Analyses Page 3 Equivalent Linear Method (EQL) and Shear Modulus and Damping Equivalent liner approximation to the viscoelastic model Gmax = Vs 2 Definition of Damping Note that the equivalent linear method does not follow the actual hysteresis loops. Note: Gmax is calculated from geophysical tests Geq is the equivalent strain-compatible secant modulus that decreases as the level of strain increases. Damping is calculated from W (area of triangle) and W (area of hysteresis loop) (see above)
Lecture 8 - Ground Response Analyses Page 4 EQL - Shear Modulus and Damping (cont) Reduction of Secant Shear Modulus as a Function of Shear Strain Shear Modulus Degradation Curve
Lecture 8 - Ground Response Analyses Page 5 EQL - Shear Modulus Degradation Curves (Sands) Typical Shear Modulus Degradation Curve for Sand - Note that the shear modulus has been normalized on the y-axis by dividing by Gmax Effects of Confining Stress on Shear Modulus Degradation
Lecture 8 - Ground Response Analyses Page 6 EQL - Shear Modulus Degradation Curves (Clays)
Lecture 8 - Ground Response Analyses Page 7 EQL - Damping Curves for Sands Effects of Confining Stress on Damping
Lecture 8 - Ground Response Analyses Page 8 EQL - Damping Curves for Clays Soils dissipate (damp) elastic energy by slippage of grains with respect to each other. The width (i.e., area) of the hysteresis loops shown by a cyclic loaded soil increases with the level of cyclic shear strain, hence, damping increase with increasing cyclic shear strain. Like the modulus reduction behavior, damping is influenced by the plasticity of the soil. Damping ratios of highly plastic soils are lower than those of low plastic soils. Damping is also influenced by the effective confining stress, especially for low plastic soils. Damping decreases with increasing effective confining stress
Lecture 8 - Ground Response Analyses Page 9 EQL - Iterating to Obtain Strain-Compatible Properties The magnitude of the shear stress time history shown above is dependent on the strain-compatible modulus and damping values selected. However, the shear stresses and strains are unknown for each layer at the beginning of the analysis. Hence an initial guess of the strain-compatible moduli and damping properties is made for each layer and these values are kept constant during each individual run (i.e., moduli and damping do not change during each iteration). Subsequently, the EQL method solves for the shear stresses and strains in each layer using the assumed strain-compatible modulus and damping values. At the end of each run, the difference between the assumed modulus and damping values are compared with the values realized from the analyses. This process is repeated until the differences become small between the assumed and realized values. The EQL method iterates toward strain-compatible soil properties until the tolerance criterion is satisfied for all layers, or until the maximum number of iterations is reached, as specified by the user. Experience has shown that the results of many ground response analyses do not change much at tolerance levels below about 5% and this value is typically used for the convergence error. It is important to note the effective, or average shear stress and strain values achieved in each layer is used to calculate the strain-compatible properties for the next iteration. The effective values are taken to be some percentage of the maximum value. Often a factor of 0.65 is applied to the maximum value to represent the effective, or average shear strain value. This 0.65 factor was determined from statistical analyses of many shear stress time histories.
Lecture 8 - Ground Response Analyses Page 10 EQL - Iterating to Obtain Strain-Compatible Properties (cont.) Note that for each successive iteration the error for the shear modulus and damping decreases.
Lecture 8 - Ground Response Analyses Page 11 EQL Method and Transfer Functions 1. 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. 2. Define the transfer function (Eq. 10). The transfer function will have both real and imaginary parts. 3. 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. 4. Express the output motion in the time domain by means of an inverse Fourier transform. The EQL methods uses a Fast Fourier Transform (FFT) to convert the input motion (time domain) into a Fourier series (frequency domain). After computing the response in the frequency domain, it uses an inverse FFT to transform the solution back to the time domain. The FFT is a very efficient numerical procedure, but it requires the total number of acceleration values to be an integer power of 2 (e.g. 1024, 2048, 4096, etc.). Most computer programs will add the required number of trailing zero acceleration values to bring the total length to the number of terms you specify for the Fourier series. Because the Fourier series implies periodicity (it assumes that the total time history, including the trailing zeros, repeats itself indefinitely), you need to make sure you have enough trailing zeros to form a quiet zone sufficiently long to allow the response to die out before the next motion begins. The best results are usually obtained when the last third or more of the total time history is quiet.
Lecture 8 - Ground Response Analyses Page 12 EQL - Transfer Functions for Single Layer Transfer Function for Single Soil Layer on Rock
Lecture 8 - Ground Response Analyses Page 13 EQL - Transfer Functions for Multiple Layers (from ProSHAKE user's manual)
Lecture 8 - Ground Response Analyses Page 14 EQL - MATLAB EXAMPLE (From ProSHAKE user's manual)
Lecture 8 - Ground Response Analyses Page 15 Ground Response Analysis - Flow Chart for Design Input Ground Motions Soil Inputs Results EQL Analysis
Lecture 8 - Ground Response Analyses Page 16 Selection of Input Ground Motion This example uses attenuation relations pga = 0.65 g from attenuation relation Example of a design target spectrum for site class B soil (Vs = 2500 ft/s) developed from and attenuation relation (green and red) or from design code (i.e., MCEER/ATC-49)
Lecture 8 - Ground Response Analyses Page 17 Scaling of Input Record to Target Spectrum Note that pga value has been changed to 0.65 g using Deepsoil. Rename and save this record. Important question: Note in the above example we have scaled the Kobe record (input time history) to match the target spectrum at pga. Is this appropriate, or is there some other spectral value that could be used to scale the input time history?
Lecture 8 - Ground Response Analyses Page 18 Soil Inputs Soil total unit weight Soil type Plastic index (for cohesive soils) Vs measurement in layer Appropriate shear modulus reduction curve Appropriate damping curve fo = Vs/4H
Lecture 8 - Ground Response Analyses Page 19 Soil Inputs - Calculation of the Fundamental Period of Soil Column Calculate the total travel time through the layered system t = H1/Vs1 + H2/Vs2 + H3/Vs3 t = 10/1000 + 30/1500 + 40/2000 t = 0.05 s Vs = H/t Vs = (10+30+40)/0.05s Vs = 1600 ft/s fo = Vs/4H fo = 1600/[4[(10+30+40)] fo = 5 Hz To = 1 / fo To = 0.2 s (compare with previous page)
Lecture 8 - Ground Response Analyses Page 20 Soil Inputs (cont.) Use total unit weight for EQL method Damping ratio only required for elastic analyses Water table information not required for EQL method
Lecture 8 - Ground Response Analyses Page 21 Soil Inputs (cont.)
Lecture 8 - Ground Response Analyses Page 22 Analysis Results Acceleration time history at surface (pga value is about 0.87 g)
Lecture 8 - Ground Response Analyses Page 23 Analysis Results (cont.) Surface soil Comparison of input response spectrum (black) with surface soil spectrum (blue) Shear strain time history
Lecture 8 - Ground Response Analyses Page 24 DEEP SOIL HELP - Step 2a Thursday, February 28, 2013 6:17 AM To see the shear modulus and damping properties for each layer, select the Materials Properties button Hmax = Vs/(4 * Cut off frequency) Hmax = maximum sublayer thickness Cut off frequency = max. frequency of propagated wave (use about 20 Hz). To exit from this screen, select next Fill out soil properties in spreadsheet box in upper right Include layer name Unit weight should be total unit weight for total stress analysis Set the water table location Use the Material Properties Button to further define dynamic properties for each soil layer
Lecture 8 - Ground Response Analyses Page 25 DEEP SOIL HELP - Step 2a - Material Properties Thursday, February 28, 2013 6:17 AM To exit, select the last damping value, then strike the tab key followed by the enter key Select the Material Type for Each Layer Select the Target Curve for Each Layer Select Use Discrete Points Select Calculate Curves
Lecture 8 - Ground Response Analyses Page 26 DEEP SOIL HELP - Step 2a - Shear Strength Thursday, February 28, 2013 6:17 AM The information on this screen is not needed for the EQL method and is ignored during the analysis. Nothing to do on this screen, the EQL method does not require shear strength
Lecture 8 - Ground Response Analyses Page 27 DEEP SOIL HELP - Step 2b - Bedrock Properties Thursday, February 28, 2013 6:17 AM NEHRP Site Class B Define the rock properties in this screen, usually elastic half-space selection is most appropriate. The shear wave velocity used in on this screen (2500 ft/s) should be consistent with the value used in developing the target design spectrum.
Lecture 8 - Ground Response Analyses Page 28 DEEP SOIL HELP - Step 3 - Analysis Type Thursday, February 28, 2013 6:17 AM This means that the average shear strain is about 65 percent of the peak shear strain. This value was determined from statistical analyses of several time histories, but is an approximation. Some research have showed that this ratio is also a function of earthquake magnitude. However, for the purposes of this class, we will use 0.65. No changes required on this screen
Lecture 8 - Ground Response Analyses Page 29 DEEP SOIL HELP - Step 4 - Selection of Time History Thursday, February 28, 2013 6:17 AM Select the layers for where output is desired. Layer 1 is the surface and should always be selected. Select the time history used for the analysis. This will be placed in the base as an outcropping rock motion. Press the analysis button to start the computer run.
Lecture 8 - Ground Response Analyses Page 30 DEEP SOIL HELP - Step 5 - Analysis Thursday, February 28, 2013 6:17 AM
Lecture 8 - Ground Response Analyses Page 31 DEEP SOIL HELP -Step 6 - Results Thursday, February 28, 2013 6:17 AM
Lecture 8 - Ground Response Analyses Page 32 DEEP SOIL HELP -Step 6 - Results Thursday, February 28, 2013 6:17 AM
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