1D Ground Response Analysis - Civil Engineeringcivil.utah.edu/~bartlett/CVEEN6330/5330l8.pdfThree...

© Steven F. Bartlett, 2011

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.

1.

Three classes of dynamic soil models:2.a) equivalent linear (SHAKE and DEEPSOIL)b) cyclic nonlinear (DEEPSOIL)c) advanced constitutive (DEEPSOIL and FLAC)

Vertically 1-D propagation of shear waves in a multi-layered system is assumed in EQL method.

a.

EQL method produces an approximation to the nonlinear response of soils under earthquake loading, but is very efficient computationally.

b.

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.

c.

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.

d.

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.

e.

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.

f.

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.

g.

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.

3.

1D Ground Response AnalysisWednesday, August 17, 2011

12:45 PM

Lecture 8 - Ground Response Analyses


EQL Method

Nonlinear Methods

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

3:32 PM



Equivalent liner approximation to the viscoelastic model

Definition of Damping

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)

○

Note:

Gmax = Vs2

Note that the equivalentlinear method does not follow the actual hysteresis loops.

Equivalent Linear Method (EQL) and Shear Modulus and DampingWednesday, August 17, 2011

12:45 PM



Reduction of Secant Shear Modulus as a Function of Shear Strain

Shear Modulus Degradation Curve

EQL - Shear Modulus and Damping (cont)Wednesday, August 17, 2011

12:45 PM



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

EQL - Shear Modulus Degradation Curves (Sands)Wednesday, August 17, 2011

12:45 PM



EQL - Shear Modulus Degradation Curves (Clays)Wednesday, August 17, 2011

12:45 PM



Effects of Confining Stress on Damping

EQL - Damping Curves for SandsWednesday, August 17, 2011

12:45 PM



The width (i.e., area) of the hysteresis loops shown by a cyclic loaded soil increases with○

Soils dissipate (damp) elastic energy by slippage of grains with respect to each other.

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

Like the modulus reduction behavior, damping is influenced by the plasticity of the soil.○

the level of cyclic shear strain, hence, damping increase with increasing cyclic shear strain.

EQL - Damping Curves for ClaysWednesday, August 17, 201112:45 PM



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.

EQL - Iterating to Obtain Strain-Compatible PropertiesWednesday, August 17, 2011

12:45 PM



Note that for each successive iteration the error for the shear modulus and damping decreases.

EQL - Iterating to Obtain Strain-Compatible Properties (cont.)Wednesday, August 17, 2011

12:45 PM



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. Becausethe 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.

EQL Method and Transfer FunctionsWednesday, August 17, 2011

12:45 PM



Transfer Function for Single Soil Layer on Rock

EQL - Transfer Functions for Single LayerWednesday, August 17, 2011

12:45 PM



(from ProSHAKE user's manual)

EQL - Transfer Functions for Multiple LayersWednesday, August 17, 2011

12:45 PM



(From ProSHAKE user's manual)

EQL - MATLAB EXAMPLEWednesday, August 17, 2011

12:45 PM



Input Ground Motions Soil Inputs

Results

EQL Analysis

Ground Response Analysis - Flow Chart for DesignWednesday, August 17, 2011

12:45 PM



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)

pga = 0.65 g from attenuation relation

This example usesattenuation relations

Selection of Input Ground MotionWednesday, August 17, 2011

12:45 PM



Note that pgavalue has been changed to 0.65 g using Deepsoil.

Rename andsave thisrecord.

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?

Important question:

Scaling of Input Record to Target SpectrumWednesday, August 17, 2011

12:45 PM



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

Soil InputsWednesday, August 17, 2011

12:45 PM



t = H1/Vs1 + H2/Vs2 + H3/Vs3

t = 10/1000 + 30/1500 + 40/2000

t = 0.05 s

Calculate the total travel time through the layered system○

Vs = (10+30+40)/0.05s

Vs = 1600 ft/s

Vs = H/t○

fo = 1600/[4[(10+30+40)]

fo = 5 Hz

fo = Vs/4H○

To = 0.2 s (compare with previous page)

To = 1 / fo○

Soil Inputs - Calculation of the Fundamental Period of Soil ColumnWednesday, August 17, 2011

12:45 PM



Damping ratio only required for elastic analyses

Use total unit weight for EQL method

Water table information not requiredfor EQL method

Soil Inputs (cont.)Wednesday, August 17, 2011

12:45 PM



Soil Inputs (cont.)Wednesday, August 17, 2011

12:45 PM



Acceleration time history at surface (pga value is about 0.87 g)

Analysis ResultsWednesday, August 17, 2011

12:45 PM



Comparison of input response spectrum (black) with surface soil spectrum (blue)

Surface soil

Shear strain time history

Analysis Results (cont.)Wednesday, August 17, 201112:45 PM


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•

To see the shear modulus anddamping properties for each layer, select the Materials Properties button

To exit from this screen, select next

Hmax = Vs/(4 * Cut off frequency)Hmax = maximum sublayer thicknessCut off frequency = max. frequency of propagated wave (use about 20 Hz).

DEEP SOIL HELP - Step 2aThursday, February 28, 2013

6:17 AM


Select the Material Type for Each Layer•Select the Target Curve for Each Layer•Select Use Discrete Points•Select Calculate Curves•

To exit, select the last damping value,then strike the tab key followed by the enter key

DEEP SOIL HELP - Step 2a - Material PropertiesThursday, February 28, 2013

6:17 AM


Nothing to do on this screen, the EQL method does not require shear strength•

The information on this screenis not needed for the EQL methodand is ignored during the analysis.

DEEP SOIL HELP - Step 2a - Shear StrengthThursday, February 28, 2013

6:17 AM


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.

•

NEHRP Site Class B

DEEP SOIL HELP - Step 2b - Bedrock PropertiesThursday, February 28, 2013

6:17 AM


No changes required on this screen•

This means that the average shear strainis 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.

DEEP SOIL HELP - Step 3 - Analysis TypeThursday, 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.•

DEEP SOIL HELP - Step 4 - Selection of Time HistoryThursday, February 28, 2013

6:17 AM


DEEP SOIL HELP - Step 5 - AnalysisThursday, February 28, 20136:17 AM


DEEP SOIL HELP -Step 6 - ResultsThursday, February 28, 2013

6:17 AM



BlankWednesday, August 17, 2011

12:45 PM


1D Ground Response Analysis - Civil Engineeringcivil.utah.edu/~bartlett/CVEEN6330/5330l8.pdfThree...

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