Input Ground Motion Characterization for Spatially Variable Soil and its Influence on Soil Structure Interaction Assessment

Transcription

1 Input Ground Motion Characterization for Spatially Variable Soil and its Influence on Soil Structure Interaction Assessment Nicholas Simos, PhD, PE & Carl Costantino, PhD, PE Energy Sciences and Technology Dept. Brookhaven National Laboratory

2 GOAL Link randomness of earthquake input & variability in site properties to: Variance of ground response Coherence of ground response Ground Motion Cross Correlation Critical power spectral matrix for structures

3 RANDOMNESS IN SEISMIC INPUT [3x3] Cross Spectral Matrix Representation of INPUT Motion at Bedrock provides stochastic representation: -Family of earthquakes with same energy -Variance Stationary or WEAKLY Stationary earthquake

4 Similar thinking when a structure is involved (site + SSI) Hstr(w) HSSI(w) PSDinput(w)

5 SITE VARIABILITY Why the need for Probabilistic/Stochastic Treatment? *Soil properties exhibit considerable variation even within homogeneous layers *Deterministic approaches based on either the expected value or the equivalent value, or conservative extremes may be misleading Q: In the larger scheme of things, does variability in the supporting soil really matter, or is it just an academic exercise? A: YES, seismic input on a structure is a predominantly local effect

6 R R PROBABILISTIC MODEL (POROSLAM CODE) [ ( x) ] = 0 E φ ( ξ ) = E[ ϕ( x) ϕ( x ξ )] ϕϕ + R ϕϕ 2 ξ ( ξ ) = exp δ ξ ξ ξ = exp + δ δ ( ) ϕϕ 1 [R] = covariance matrix providing correlation characteristics {λ} T = [L] { } T ξ = x ij =distance separating two fundamental volumes { } = random (Gaussian) δ = correlation distance (open to interpretation ONLY real data vector ( -n +n can exactly reveal it] { } = vector of spatially correlated site property LL T = R Use of MONTE CARLO to generate the statistics of the output

7 {λ} T = [L ij ] { } T

8 Realization - 1 Realization -2

9 1.8 Real Rocking Compliances - Surface Layer Over Half-Space Vs_nominal Vs_nom-3sigma Vs_nom+3sigma Frequency (dimensionless) 1.4 Real Part of Rocking Compliances - Surface Layer Overlaying Half-Space Vs_nom Vs_var1 Vs_var2 Vs_var3 Re[Cmm] Freq. (dimensionless)

10 1.2 Radiation of Rocking Compliances - Surface Layer Overlying Half Space Vs_nom Vs_nom-3sigma Vs_nom+3sigma Im[Cmm] Freq. (dimensionless) 1.2 Radiation Component or Rocking Compliances - Surface Layer Overlaying Half-Space Vs_nom Vs_var1 Vs_var2 Vs_var3 Im[Cmm]

11 NEED TO ESTABLISH PROBABILISTIC SOIL MODELS REFLECTING REAL SITES AND TRUE CORRELATIO PARAMETERS Thus, focus on a well-studied site (e.g., SRS site) and work with available of large number of deep borings/cone penetrometers GOAL Best estimate and variation of shear wave velocity profile across entire site APPROACH Assessment of layer SWV from median and correlation with adjacent layers END GOAL Generation of frequency dependent site amplification factors defining surface spectra

12 APPLICATIONS WHERE VARIABILITY MAY HAVE MORE PRONOUNCED EFFECTS Pile Interaction Analysis Liquefaction in deposits & Liquefaction propagation HOW ABOUT A TYPICAL CASE WHERE SITE PROPERTY DATA HAVE BEEN OBTAINED (possibly contaminated)? Process of Gaussian field interpolation may be applied (M. Hoshiya & J. Shiba, Comp. Stoch. Mech.,1995) to ensure that the measured data are accounted for in the probabilistic site model.

13 Random/Stochastic Representation of Input Cross-Correlated PSD Matrix Analytic form of a PSD

14 Estimation of Cross-Correlated PSD Matrix For a bi-variate process in which there is a joint Gaussian density for the random phase variables Φ 1 and Φ 2 (distributed uniformly over the interval [0,2π]), standard deviation σ and correlation coefficient ρ, cross spectral densities S 12 (ω 1, ω 2 ) can be derived from the relation below, where and h =-1 for ω 1 ω 2 0 h = 1 for ω 1 ω 2 <0.

15 LINKING INPUT-SITE-STRUCTURE

16 POROSLAM & Direct GEneration of Spectra (DIGES) POROSLAM 2-D FEA code solving Biot s Eqns for two-phase material Transmitting boundary capacity Wave propagation Impedance calculation Pore pressure generation Earth Dam safety analysis Liquefaction (up-to initiation) Probabilistic Analysis (Monte Carlo on seismic input) DIGES Stochastic input representation Direct transfer to 3-D discrete structural model COMPLETE Probabilistic Analysis (input + site) = Work in Progress

17 ARE CROSS-SPECTRAL TERMS IMPORTANT, ANYWAY? Let s look at some real data (Kobe, 1995)

18 MULTIVARIATE PROCESS WITH CROSS-CORRELATION BETWEEN THE VARIATES (H1;H2;V components)

19 MULTIVARIATE PROCESS WITH CROSS- CORRELATION BETWEEN THE VARIATES (H1;H2;V components) AND STRUCTURAL RESPONSE

20 FOCUS ON SITE RESPONSE Assume BEDROCK motion coherent but cross-correlated (say H & V components) Transfer Function Hhh(w) and Hvv(w) are needed for motion at the site surface. WHILE the PSD-expressed does not need to be varied, the Total Transfer (combination of Hhh and Hvv) function for the site needs to be established for every realization of the spatial variability vector governing the site property being varied!! Newman expansion/perturbation method may be used to limit matrix inversion cost. However, for every realization, through, The cross-spectral matrix at each surface location is derived!!1 Cross-correlation of response motion for every location and motion COHERENCE between locations are now available Because of several random realizations of the site, even a VARIANCE of these key quantities can be established!!

21 SITE RESPONSE

22 WHAT DO YOU DO WITH ALL THE VARIANCES? In Regression Analysis of Strong Motion Data, Joyner & Boore the variability in ground-motion data was delineated between earthquakes, recording sites and records themselves. As in the study above, through the process in this work information on the variability (or our understanding of it) of different earthquakes (families expressed in PSD form) at the same locations, or variance associated with an individual observation can be established in looking at the site response. How is that important? Recall how we establish RESPONSE SPECTRA based on random vibration: ys;p = rs;p y(s) Where rs;p is a peak factor and y(s) is the standard deviation of a 1-DOF system response

23 SUMMARY SINCE LOCAL SITE RESPONSE IS OF PARAMOUNT IMPORTANCE, UNDERSTANDING UNCERTAINTIES IN WHAT WE SEE (or think we see) IS CRUCIAL. BENCHMARKING OF RANDOMIZED PROCESSES & METHODOLOGIES WITH FIELD OBSERVATIONS WILL PROVIDE AN EXCELLENT TOOL TO USE IN THAT DIRECTION. IT HAS BEEN STATED THAT OUR CURRENT APPROACH IN DEFINING EARTHQUAKE INPUT HAS BEEN VERY CONSERVATIVE (Response Spectra). HOPEFULLY AN APPROACH BASED ON Power Spectra Description of Earthquakes AND SITE UNCERTAINTY INCLUSION MORE REALISTIC INPUT CAN BE GENERATED.