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Ground Response AnalysisPart - IILecture-261Transfer function Evaluation: casesUniform undamped soil on rigid rockUniform damped soil on rigid rockUniform undamped soil on elastic rockUniform damped soil on elastic rockLayered damped soil on elastic rockTransfer Function(Filter)2Uniform damped soil on elastic rockSoilRockGsrsGrrrHAsBsArBrzszrus (zs,t) = As ei(t kszs) + Bs ei(t + kszs)

ur (zr,t) = Ar ei(t - krzr) + Br ei(t + krzr)

At zs = 0 (ground surface)t(zs,t) = 0 As = Bs

3Uniform damped soil on elastic rockus (zs,t) = As ei(t kszs) + Bs ei(t + kszs)

Compatibility of displacements at the soil-rock boundary requiresus (zs=H) = ur (zr=0)

As [(eiksH + e-iksH) = Ar + Br

Continuity of stresses at the soil-rock boundary requirests(zs=H) = tr (zr=0)

t = G u/z

AsGsks (eiksH - e-iksH) = Grkr (Br Ar)4Uniform damped soil on elastic rock(Br Ar) = As (eiksH - e-iksH) Gsks/Grkr

Gsks/Grkr = az = Impedance ratio

(Br Ar) = As (eiksH - e-iksH) az

Br + Ar = As [(eiksH + e-iksH)]

Br = As [ eiksH (1+ az) + e-iksH(1- az) ]

Ar = As [ eiksH (1- az) + e-iksH(1+ az) ]

F(w) = As/Ar = 2/ [eiksH (1- az) + e-iksH(1+ az) ]

F(w) = 1/[cos ksH - i az sin ksH)]

5Uniform damped soil on elastic rock

Impedance ratio6Uniform damped soil on elastic rockNote:Even with no soil damping, resonance cannot occurWhy?Energy removed from soil layer by transmission into rockForm of radiation damping

7Layered damped soil on elastic rock

8Layered damped soil on elastic rockFor layer j

From equilibrium

From compatibility

9Layered damped soil on elastic rockFor layer j

From equilibrium

From compatibility

If we know the response at layer j (Aj and Bj are known), then we have two equations with two unknowns (Aj+1 and Bj+1)We can relate Aj+1 and Bj+1 to Aj and Bj by means of recursive relationships10Layered damped soil on elastic rockSolving for unknowns

Or relating the coefficients to those at the ground surface

11Layered damped soil on elastic rockThen the transfer function relating the motion in layer i to motion in layer j can be written as

If we know the motion at any layer, we can use this transfer function to compute the corresponding motion at any other layer.

The use of transfer function implies that a liner analysis is being performed.

12Equivalent linear analysisThe actual nonlinear hysteretic stress-strain behavior ofcyclically loaded soils can be approximated by equivalentlinear properties.

13Equivalent linear analysisAssume some initial strain and use this strain to estimate G and Use these G and values to compute the responseDetermine peak strain and effective strain Select properties based on updated strain level

14Equivalent linear analysisCompute response with new properties and determine resulting effective shear strainRepeat until the computed effective strains are consistent with assumed effective strains

15Nonlinear AnalysisSolve wave equation incrementally

Approximate partial derivatives

Finite difference form

16Nonlinear AnalysisSolve wave equation incrementallyThen

Velocity at time t+t can be calculated from velocity and shear stress at time t

17Nonlinear AnalysisStart with initial stiffness, Gmax

Compute response for small time step, t

Compute shear strain amplitude at end of time step

Use stress-strain model to find Gtan for next time step

Compute shear strain amplitude at end of next time step

Continue stepping through time for entire input motion18Nonlinear AnalysisNonlinear response is simulated in incrementally linear fashion

Material damping is taken care by hysteretic response

Approach requires good model for description of soil stress-strain behaviour19Nonlinear Stress-strain modelsTwo main types Cyclic nonlinear models Advanced constitutive models

Cyclic nonlinear modelsRequires: Backbone curve Unloading-reloading rules Pore pressure model20Nonlinear Stress-strain modelsAdvanced constitutive models123Require: Yield surface Hardening rule Failure surface Flow rule

21Nonlinear Stress-strain modelsCyclic nonlinear modelsAdvantagesRelatively simpleSmall number of parametersDisadvantagesSimplistic representation of soil behaviorCannot capture dilatancy effects

Advanced constitutive modelsAdvantagesCan better represent mechanics of yield, failureDisadvantagesMany parametersDifficult to calibrate22Comparison of Equivalent Linear and Non-Linear Site Response AnalysesInherent linearity can lead to spurious resonances in equivalent linear methodUse of effective shear strain can lead to overdamped or underdamped system, depending on nature of strain time historyEquivalent linear analyses much more efficient than Nonlinear analyses can be formulated in terms of effective stressesNonlinear analyses can predict permanent deformations23Comparison of Equivalent Linear and Non-Linear Site Response AnalysesNonlinear analyses require reliable stress-strain, or constitutive, modelsDifferences in computed response depend on the degree of nonlinearity in the actual soil response

Stiff sitesResults quite similarWeak input motions

Soft sitesNonlinear analysis preferableLiquefiable sitesStrong input motions2425Numerical CodesDimensionsOSEquivalent LinearNonlinear1-DDOSDyneq, Shake91AMPLE, DESRA, DMOD, FLIP, SUMDES, TESSWindowsShakeEdit, ProShake, Shake2000, EERACyberQuake, DeepSoil, NERA, FLAC, DMOD20002-D / 3-DDOSFLUSH, QUAD4/QUAD4M, TLUSHDYNAFLOW, TARA-3, FLIP, VERSAT, DYSAC2, LIQCA, OpenSeesWindowsQUAKE/W, SASSI2000FLAC, PLAXISNumerous computer programs are available for ground response analysis.

Based on the computational procedure, number of dimensions, and operating system, these codes can be classified as follows.26SHAKESHAKE is one of the widely used numerical programs for ground response analysis.

Program SHAKE computes the responses in a system of homogeneous, visco-elastic layers of infinite horizontal extent subjected to vertically travelling shear waves. The program is based on the continuous solution to the wave equationadapted for use with transient motions through the fast Fourier transform algorithm.

The nonlinearity of the shear modulus and damping is accountedfor by the use of equivalent linear soil properties using an iterative procedure to obtain values for modulus and damping compatible with the effective strains in each layer.

The strain dependence of modulus and damping is accounted for by an equivalent linear procedure based on an average effective strain level computed for each layer. The program is able to handle systems with variation in both moduli and damping, and takes into account the effect of the elastic base.

Compute the time history of acceleration at the surface of the linear elastic soil deposit shown in Figure (a) in response to the E-W component of the Gilroy No.1(rock) motion shown in the Figure (a).Figure (a) Soil DepositExample Problem27Solution Computation of the ground surface motion from the bedrock motion can be accomplished in the following series of steps:- 1. Obtain the time history of acceleration of the input motion. In this case the input motion is the E-W component of the Gilroy No.1 (rock) motion shown in Figure (b) below. The Gilroy No.1 record consists of 2000 acceleration values at 0.02-sec intervals.

Figure b. E-W component of the Gilroy No.1 (rock) motion28- 2. Compute the Fourier series of the bedrock (input) motion. The Fourier series is complex valued; its one-sided Fourier amplitude spectrum is shown in Figure (c) below. The Fourier amplitude spectrum is defined for frequencies up to 1/2Dt=25Hz, but most of the energy in the bedrock motion is at frequencies less than 5 to 10 Hz.

29- 3. Compute the transfer function that relates the ground surface (output) motion to thebed-rock ( input) motion. The transfer function for the case of undamped soil is real valued [Figure (d)]. The transfer function has values of 1 below frequencies of about 10 Hz. However, at frequencies that approach the fundamental frequency of the soil deposit (fo=vs/4H=26.25 Hz), the transfer function begins to take on large values.

30- 4. Compute the Fourier series of the ground surface (output) motion as the product of the transfer function the Fourier series of the bedrock (input) motion. At frequencies less than 5 to 10 Hz, the Fourier spectrum of the ground surface motion is virtually the same as that of the bedrock motion. Although the transfer function indicates that frequencies above 20 Hz or so will be amplified strongly, the input motion is weak in that frequency range. The one-sided amplitude spectrum is shown in Figure (e). Examination of this Fourier amplitude spectrum indicates that the ground surface motion has somewhat more high frequency motion, but is generally similar to that of the bedrock motion.

31- 5. Obtain the time history of the ground surface motion by inverting its Fourier series. Figure (f).

32- Because the transfer function related to the computation of free surface motion from bedrock motion, an important problem of practical interest involves the computation of bedrock motion from a known free surface motion deconvolutionDECONVOLUTION33Exercise ProblemsFor the case of a uniform layer of undamped soil overlying rigid bedrock, develop a transfer function that relates shear stress, (z=H/2, t), to the bedrock acceleration, Plot the modulus of the transfer function from kH=0 to kH=2

An acceleration reduction factor can be defined as the ratio of the peak acceleration at depth z, to the peak ground surface acceleration, i.e.

For the case of a uniform layer of undamped soil overlying rigid bedrock, develop an expression for the reduction factor as a function of the thickness and shear wave velocity of the soil layer, and the frequency of the input motion

3435Kramer (1996) Geotechnical Earthquake Engineering, Prentice Hall.Villaverde, R. (2009) Fundamental Concepts of Earthquake Engineering , CRC Press.Towhata, T. (2008) Geotechnical Earthquake Engineering, Springer.SHAKE2000 Users manual.

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