EARTHQUAKE-INDUCED TRANSIENT GROUND STRAINS AND ROTATIONS FROM DENSE SEISMIC ARRAYS

Transcription

1 EARTHQUAKE-INDUCED TRANSIENT GROUND STRAINS AND ROTATIONS FROM DENSE SEISMIC ARRAYS A Dissertation Submitted in Partial Fulfilment of the Requirements for the Master Degree in Engineering Seismology By Chiara SMERZINI Supervisor: Prof. Roberto PAOLUCCI May, 2008 Istituto Universitario di Studi Superiori di Pavia Università degli Studi di Pavia

2 The dissertation entitled Earthquake-induced transient ground strains and rotations from dense seismic arrays, by Chiara Smerzini, has been approved in partial fulfilment of the requirements for the Master Degree in Engineering Seismology. Roberto PAOLUCCI Jonathan P. STEWART

3 Abstract ABSTRACT The aim of this work is to illustrate an empirical procedure for evaluating transient ground strains and rotations based on the records obtained by two dense seismic networks, namely the Parkway Valley, New Zealand, and the UPSAR, California, arrays. Due to the substantial lack of directs measurements of strains and rotations it is common practice to derive them indirectly relying on simplified relationships with restricting assumptions. The dynamic behavior of the surface ground strain tensor is investigated, introducing invariant measures of peak ground strains: the highest and lowest principal strains, HPS(t) and LPS(t), respectively. The computed peak ground strains, PGSs, show an important dependence on azimuth, by a factor of about two. Furthermore, a clear linear relationship of the PGS as a function of the most common measures of ground motion severity, such as peak ground acceleration PGA, velocity PGV and displacement PGD is found whatever the parameter. The results of the interpolation procedure are in reasonable agreement with other published relationships, arrayderived estimates and with some of the few available direct records of strain in buried pipelines and tunnels. Nonetheless, such relationships point out the limitations of the simplified approaches used in the engineering practice, which uniquely consider the wave passage effect on the ground strain evaluation and tend to underestimate significantly the observations. A similar linear trend is found when PGV data are correlated with Peak Ground Rotation, PGR, estimates. Provided the scarce knowledge concerning the issue of rotational ground motions, more attention is devoted to a broad synthesis of maximum rotation estimates from different sources. Although the empirical estimates for PGR are in reasonable agreement with other array-derived estimates, synthetics obtained by 3D Spectral Element simulations and individual data retrieved from literature. Nevertheless, the comparison between synthetics and experimental data shows some crucial differences, such as the potential dependence on site effects, which should be checked against a larger number of experimental data and numerical simulations. Keywords: underground structures, dense seismic arrays, interpolation, Peak Ground Strain, Peak Ground Rotation, empirical correlations, Spectral Element Method

4 Acknowledgments ACKNOWLEDGEMENTS I wish to thank Prof. R. Paolucci for carefully supervising this work, for his teachings for his useful suggestions throughout the work. I am very grateful also to Prof. J.P. Stewart for his fruitful suggestions and Prof. F.J Sánchez-Sesma for his teachings and sparkly suggestions during my 4 months visit at the Universidad Nacional Autónoma de México, Distrito Federal. I thank also M. Stupazzini and L. Scandella for their useful comments. I wish to thank W. R. Stephenson, G. Q. Wang, P. Spudich and J. Fletcher for kindly providing some data of the Parkway Valley and UPSAR arrays.

5 Index TABLE OF CONTENTS Page ABSTRACT... i ACKNOWLEDGEMENTS... ii TABLE OF CONTENTS... iii LIST OF FIGURES...v LIST OF TABLES... xiii 1 INTRODUCTION Preface Scope and organization of the work Objectives Thesis outline TRANSIENT SEISMIC GROUND STRAINS Introduction Peak Ground Strain (PGS) estimates based on plane wave propagation assumption Ground strains from body waves Ground strains from surface waves PGS evaluation in the technical guidelines Ground strain estimates from recorded data: single- and multiple-station approaches Single station estimates Multi station estimates: the seismo-geodetic approach Numerical/analytical ground strain evaluation Near field strains: Bouchon & Aki s work (1982) Empirical attenuation relationships for PGS estimation Trifunac & Lee (1996) Abrahamson (2003) Scandella and Paolucci (2006) Spatial incoherency and soil heterogeneity effects ROTATIONAL SEISMIC GROUND MOTIONS: THEORY, OBSERVATIONS AND IMPLICATIONS...31 iii

6 Index 3.1 Introduction Theoretical considerations Rotational components of motions under incident plane waves Rotations due to a double-couple point source Effects of rotations on the response of inertial sensors Effects of rotations on building response Summary of observations SPATIAL INTERPOLATION OF DISPLACEMENT RECORDS FROM DENSE SEISMIC NETWORKS Selection of records from dense seismic networks: Parkway Valley, New Zealand, and UPSAR, California The Parkway Valley (New Zealand) temporary network The UPSAR(California) array Spatial interpolation of displacement records Interpolation technique Validation of the interpolation procedure THE SURFACE GROUND STRAIN TENSOR: EMPIRICAL EVALUATION AND DISCUSSION OF ITS PROPERTIES Goals and organization of the chapter Evaluation of strain tensor at the ground surface Definition of invariant ground strain measures: Highest Principal Strain (HPS) & Low Principal Strain (LPS) Significant features of the axial-shearing strain state Empirical relationships for peak ground strain evaluation Dependence on azimuth Dependence on the peak parameters of ground motion SEISMIC GROUND ROTATIONS: ARRAY-DERIVED ESTIMATES, NUMERICAL SIMULATIONS AND POSSIBLE EFFECTS ON STRUCTURAL RESPONSE Main topics and organization of the chapter Surface ground rotations obtained by the interpolation procedure Empirical correlation for PGR Synthesis of PGR estimates from different sources Effect of coupled tilt and translational motion on structural response CONCLUSIONS Ground strains Ground rotations REFERENCES APPENDIX A...I iv

7 Index LIST OF FIGURES Page Figure 2.1 Damage to San Francisco transmission and distribution system (SFPCU, then SVWC) during the devastating 1906 earthquake (April 18, MW=7.9). On the left side highlighted are details of the Pilarcitos pipe destroyed by the earthquake (see the corresponding location on the right-side map). From Eidinger et al. (2006)...7 Figure 2.2 Left: propagation path of seismic waves traveling upwards towards the ground surface in increasingly soft layers of geological materials. Right: apparent propagation velocity V a for body waves. γ S indicates the angle of incidence and v(x,z,t) is the S-wave in-plane particle motion...11 Figure 2.3 Comparison between single station strain estimates from seismic recorded data (dashed lines9 and direct measures from strainmeters, at Piñon Flat Observatory in California during three regional earthquakes (see text for further details): A) Northridge; B) Little Skull Mountain; C) St. George. Radial ε rr, tangential ε θθ and shear ε rθ strains are compared. For the tangential strains only the observed quantities are reported, since form they should vanish theoretically. From Gomberg & Agnew (1996) Figure 2.4 Contour lines of horizontal strain factors Logχ ( PGV / V s ) of strong motions in Los Angeles basin during the 1996 Northridge earthquake. From Trifunac et al. (1996)...16 Figure 2.5 Components of the surface displacement gradient D estimated for the 23 Oct 1993 M6.6 earthquake in the Valley of Mexico from three relatively densely spaced stations (ROMA: A, B and C). From Bodin et al. (1997)...19 Figure 2.6 Comparison of dynamic displacement gradient tensor components (as defined in the text) using single-station (solid lines) and seismo-geodetic (dashed lines) approaches for the eastern portion (composed of 4 stations) of the Geyokcha array in the frequency bands Hz (left) and Hz. The maximum separation distance within the sub-array was 400 m. The data refer to an earthquake which occurred on 23 Sept 1993 with epicentral distance R~64 km, V a ~2.72 km/s and azimuth ξ~325. From Gomberg et al. (1999) Figure 2.7 Comparison of horizontal peak strains (in µ) estimated by single-station approach and those calculated from array-derived horizontal gradients. From Chung (2007)...20 Figure 2.8 Examples of longitudinal strains (solid lines) and horizontal ground velocities (dashed lines) time histories obtained by Bouchon and Aki (1982) by simulating near-field differential v

8 Index motions induced by a dip-slip fault embedded in layered media. The source model and the dynamic properties of the layered earth structure are shown in the bottom side of the figure..21 Figure 2.9 Comparison in logarithmic scale between the radial ε max rr and shear r ϑ strain factors computed by SYNACC program, on the y-axis, and those predicted by eqq. (2.14) and (2.15) on the x- axis. The results for two different geological models are reported: on the left β 1 =500m/s; on the right β 1 =100m/s in the uppermost layer. In both graphs superimposed are the 45 straight lines for comparison purpose...23 Figure 2.10 Magnitude dependence of the strain-velocity relationship Log( εa max ) vs. Log( PGV a / β1 ) for the synthetic accelerograms computed by Trifunac & Lee (1996) for a selected geological model...24 Figure 2.11 Simplified open basin model referred to by Scandella & Paolucci (2006) for a parametric numerical study of the PGS a -PGV relationship in the presence of strong lateral discontinuities. From Scandella & Paolucci (2006) Figure 2.12 Functions F 1 and F 2 in eq. (2.17) calibrated on 2D numerical analyses, for various dipping angles α. From Scandella & Paolucci (2006) Figure 2.13 Examples of empirical spatial coherency models found in literature for separation distance ξ=500 m. The following parameter have been used: for Luco & Wong model, α= ; for Harichandran & Vanmarcke, A=0.736, a=0.147, b=2.78, k=5210 m, f 0 =1.09Hz, determined from data of event 20 at the SMART-1 array in Taiwan); for Hao et al., only longitudinal distances are considered ξ=ξ l and ξ t =0 (i.e. the stations are located along a line connecting the epicenter), β 1 = , a 1 = ,b 1 = and c 1 = , determined from event 30 at the SMART-1 array Figure 2.14 Comparison of actual strain time histories (ε) and strain estimate time histories (v/v a ) obtained from forward modeling simulation using the Luco & Wong s model. The following case studies are considered: A) α = sec/m and V a = 1500 m/s; B) α = sec/m and V a = 1500 m/s and C) α = sec/m and V a = 500 m/s. Adapted from Zerva (1992)...29 Figure 2.15 Variation of PGS as a function of the apparent propagation velocity V a, ranging from 0 to (i.e. standing waves), for two different values of the incoherence parameters α of Luco and Wong s model: α = s/m and α = s/m. Note the increasing α, motions looses coherence with separation distance. Superimposed are the vertical lines indicating the critical apparent propagation velocity, V a cr, corresponding to the selected values of α. Adapted from Zerva (1992)...30 Figure 3.1 Coordinate system and notation for the analytical derivations summarized in Table 3.1. ω y represents rocking (or tilting) while ω z represents torsional motion Figure 3.2 Geometry and notation used to derive rotations for a double-couple point source Figure 3.3 A) Non-vanishing rotation component ω θ and ω φ as a function of time at a given point in an infinite ideal medium due to a double-couple point source with M W = 5, as given by eq. (3.7) and (3.8). The observation point is located at (5 km, π/8, π/8) in spherical coordinates and the source time function rare is given by a Gaussian ~ exp(-t 2 /T 2 ) with T ~ 0.1 sec (being the rise time τ R ~ 0.4 sec) and maximum amplitude equal to the selected M 0 (recall relation M W = ε max vi

9 Index 2/3Log 10 (M 0 ) -6). The properties of the medium are: ρ = 2800 kg/m 3, β = 3 km/s and ν = ¼ (Poisson s solid) Figure 3.4 A) Peak Ground Rotation (PGR, in rad) as a function of M W calculated at R = 5 km by applying the closed-form solution of eqs. (3.7) and (3.8). The same model as in Figure 3.3 is considered. B) Comparison of the PGRs obtained at R=5 km (filled dots) with those at R=50km (filled triangles). Superimposed are the straight lines connecting the logarithm of peak rotations and moment magnitude...37 Figure 3.5 Sketch of three transducers in SMA-1 accelerograph with the corresponding oscillation directions. Angles α i describe the deflection of the i th pendulum. From Trifunac & Todorovska (2001) Figure 3.6 Fourier amplitude spectra of ground acceleration {d 2 X i /dt 2 }, normalized rocking {φ i g}, and normalized torsional acceleration {d 2 φ i /dt 2 g} at a site on basement rock for earthquake magnitudes M =4, 5, 6 and 7, source at depth H = 0 (surface faulting) and epicentral distance R = 10 km. The small shaded areas denote the limiting constant values of φ(ω) for ω 0. From Trifunac & Todorovska (2001) Figure 3.7 Log-Log frequency response of rotations and translations for a horizontal pendulum. Seismic recordings are the combination of these three factors. From Pillet & Virieux (2007).40 Figure 3.8 Chi-Chi M7.6 earthquake (Taiwan), recorded at the TCU06 accelerometric station. A) recorded accelerations signals; B) velocity time histories obtained by integration of A): according to Pillet & & Virieux (2007) the slope of uncorrected velocity time historiy is a measure of the static part of tilt input motion; C) corrected displacement traces after removing the spurios tilt terms. From Pillet & Virieux (2007) Figure 3.9 Comparison of Single Degree Of Freedom, SDOF, oscillator response in terms of displacement (in c)) due to pure translational input motion (reported in a)) and coupled tilttranslational motion (tilting input motion used in the analysis is illustrated in b)). From Kalkan & Graizer (2007) Figure 3.10 A) Torsional response (in terms of rotational displacement SD, pseudo-velocity PSV and pseudo-acceleration PSA) and Fourier spectra (dashed line) for the synthetics illustrated in B). 5 values of damping ratios and periods ranging from 0.04 s to 15 s are considered. The following earthquake parameters are considered: M = 6.5, R = 10 km and soft site conditions. From Lee & Trifunac (1985)...44 Figure 3.11 Ratio of rotational to translational response spectrum for damping ratio ζ=5% adapted from Lee & Trifunac(1985). From Castellani & Zembaty (1996) Figure 3.12 PGV h vs. PGR z retrieved from literature (see Table 3.2 for further details)...48 Figure 3.13 PGV h vs. distance R retrieved from literature (see Table 3.2 for further details). Data with R>100 km have been omitted...48 Figure 3.14 Rotational field (see also Table 3.3) produced by the buried 30-km-long strike-slip fault at the site locations illustrated below. See also Figure 2.8 in the previous chapter for further details about the source process vii

10 Index Figure 3.15 PGR as function of distance obtained by Bouchon & Aki (1982) through semi-analytical simulations of a Strike Slip (SS) fault of unitary slip and magnitude M W = 6. From Oliveira & Bolt (1989) Figure 4.1 Location of Parkway Valley. From Stephenson (2007) Figure 4.2 Left: location of recording stations of the temporary Parkway Valley array, used for this study; the box denotes the area considered for the spatial interpolation procedure, illustrated in the following section. Right: relative location of Parkway Valley and the selected seismic events (event #1 and #2)...52 Figure 4.3 Left: location of recording stations of the UPSAR; right: map including the UPSAR and the epicentres of the M W 6 Parkfield and M W 6.5 San Simeon earthquakes. From Wang et al. (2006) Figure 4.4 Example of cross-section (b), shown in (a), passing through the array site. Tsm=Santa Margherita formation (upper Miocene), Tm=Monterey formation (middle Miocene), Tv=volcanic rocks of Lang Canyon (lower Miocene), Tt=Temblor formation (lower Miocene), KJf=Franciscan assemblage (Cretaceous and Jurassic), QT p =Paso Robles formation (Pleistocene and Pliocene), and TP r =Pancho Rico formation (upper Miocene). Highlighted is the Paso Robles formation outcropping at the UPSAR location. From Fletcher at al. (1992)..54 Figure m resolution topography showing the location of the UPSAR seismograph stations. The map covers an area of approximately 1 km x 1 km. From Wang et al. (2006) Figure 4.6 Example of comparison between the B. Spline and Delaunay algorithms (see text for more details). Left: comparison in terms of contour lines of horizontal displacement (denoted with SV, in mm*100) obtained with Delaunay (top) and B. Spline technique (bottom), for t = 16.3 sec. Right: the same contour lines illustrated on the left hand side are overlapped. The small stars indicate the available receivers. Parkway event #1 is considered...58 Figure 4.7 Snapshots of interpolated ground displacement for t=16.3 sec during event Parkway #1: on the left horizontal in-plane component u(x,y), on the right vertical out-of-plane component, w(x,y) Figure 4.8 Sequence of snapshots of interpolated vertical ground displacement w(x,y) during event Parkway #1. The displacement has been band-pass filtered between 1 and 2 Hz to clearly catch the fundamental resonance frequency of the transverse cross-section of the Parkway Valley. 59 Figure 4.9 Comparison of recorded (thick line) EW displacement ( in a) ) and velocity ( in b) ) at station 09 of Parkway Valley array for event 1 with the one obtained by B. Spline interpolation (thin line), when in the interpolation procedure the same receiver is omitted. c) Comparison of the corresponding velocity Fourier amplitude spectra...60 Figure 4.10 Same as in Figure 4.9 for event 2. Station 06 is considered herein...61 Figure 4.11 Comparison of recorded (thick line) EW displacement ( in a) ) and velocity ( in b) ) at station 02 of the UPSAR array during the Parkfield earthquake with the one obtained by B. Spline interpolation (thin line), when in the interpolation procedure the same receiver is omitted. c) Comparison of the corresponding velocity Fourier amplitude spectra Figure 4.12 Same as in Figure 4.11 for the San Simeon earthquake, omitting the receiver viii

11 Index Figure 4.13 Comparison of the observed displacement records both in time (up) and frequency (bottom) domain with those derived by the two interpolation techniques: B. Spline (thin grey line) and Delaunay (thick dashed line). The same test depicted in Figure 4.9 is considered herein...62 Figure 4.14 A) Observed vs. predicted peak ground displacement (PGD), using the proposed interpolation procedure, for Parkway Valley event 1 (left) and for the UPSAR San Simeon earthquake (right). B) Same comparison but in terms of Peak Ground Velocity (PGV)...63 Figure 5.1 Definition of invariant measures of the ground strain tensors using the Mohr circle representation: HPS(t) and LPS(t). The former is, at each time interval, the maximum principal strain, while the latter is the minimum, under the assumption of a plane stress-strain state...67 Figure 5.2 Time variations of the highest principal strain, HPS(t), and the lowest principal strain, LPS(t), calculated at receivers 10 (up) and 15 (bottom) for the Parkway event # Figure 5.3 Same as in Figure 5.2 but for the Parkway event # Figure 5.4 Same as in Figure 5.2 but for the Parkfield event and for receivers 02 (up) and 11 (bottom) of the UPSAR networks Figure 5.5 Same as in Figure 5.4 but for the San Simeon event...71 Figure 5.6 Sketch of the Mohr circle representation corresponding to purely shear conditions Figure 5.7 Comparison between the principal strains induced by purely shear waves (left) or by surface (Rayleigh) waves (right) n Figure 5.8 Time variations of the normalized first invariant I ( ) 1 t, defined in eq. (5.10), at some representative stations at both Parkway Valley (up) and UPSAR (bottom) arrays. Namely receiver 9, for the Parkway valley (event #1 and #2), and receiver 6, for the UPSAR (Parkfield and San Simeon earthquakes) are considered...72 n Figure 5.9 Time variations of I ( ) 1 t at 4 representative receivers of the UPSAR array during the San Simeon earthquake. Superimposed are the vertical lines corresponding to the P- and S- wave arrivals, as estimated by Wang et al. (2006)...73 Figure 5.10 Right: principal strain directions, computed at the Parkway Valley (event #1) at receiver 10 (A) and 15 (B), with the corresponding trajectories of displacement particle motion (hodograms). The length of the arrows is proportional to the frequency of occurrence of the strain principal directions within each angular sector (from 0 to 180, measured from the horizontal axis counter clockwise). Left: EW and NS displacement time histories from which the illustrated hodograms are calculated. Highlighted by horizontal lines is the time window (namely from 16 to 20 sec) over which both principal strain directions and hodograms are calculated...75 Figure 5.11 Same as in Figure 5.10 for the event #2 at the Parkway array Figure 5.12 Same as in Figure 5.10 for the Parkfield EQ: receiver 02 and 11 are considered here and the principal strain directions are calculated in the time window from 5.0 to 10.0 sec...77 Figure 5.13 Same as Figure 5.10 for the San Simeon EQ: a larger time window from 10.0 to 20.0 sec is considered...78 ix

12 Index Figure 5.14 Radar diagrams showing, for all earthquakes under consideration, the azimuth-dependence of ψ (m/s) coefficient of eq. (5.11) along a set of prescribed azimuths θ. The rightmost hand side sketch illustrates the definition of radial strains ε θθ, from which the left radar graphs are derived, and of transverse strains ε ρρ, perpendicular to the formers Figure 5.15 Correlation of observed HPS max - PGV pairs for the 4 earthquakes under consideration. The solid line HPS max = PGV/φ is the LS best fit line extended to all the HPS max -PGV pairs corresponding to the median value φ =963 m/s, while the two dotted lines are associated to φ - =671 m/s and φ + =1382 m/s corresponding to the 16 and 84 percentile, respectively (the standard deviation between the observed data and the best-fitting line is around 0.16) Figure 5.16 A) PGS vs. PGV pairs, either obtained by our spatial interpolation procedure, or through direct strain gauge measurements in Japan (Iwamoto et al., 1988). Superimposed are two representative lines defined by equation 1a (radial strain) of Trifunac and Lee (1996) for two combinations of the parameters R (epicentral distance, in km) and Vs (shear-wave velocity in the uppermost 50 m, in km/s). Such equation reads as: log 10 PGS = ( Vs) + ( Vs) R + [(1-0.19Vs) + ( Vs) R] Log 10 (PGV/Vs). B) LS best-fit line (solid line) extended to both our experimental datasets, including Spudich & Fletcher (2008) estimates, and strain measurements performed in Japan; the dashed lines correspond to +/- standard deviation σ = Figure 5.17 A) Same as in Figure 5.15 but in terms of PGA (in m/s 2 ), for the four earthquakes under study; B) comparison with the direct strain gauge measurements from Japan (Nakamura et al., 1981 and JSCE, 1977); note that the LS line fitting the extended dataset, including also the direct Japanese measurements, would be characterized by nearly the same slope...85 Figure 5.18 Same as in Figure 5.17 B) but superimposed are the median LS best-fit line extended to the entire dataset (including the direct strain measurements in buried pipelines and tunnels) and the two LS line corresponding to +/- σ=0.30. Notice that the median value of the coefficient of proportionality between PGS and PGA is almost the same as that derived by our dataset only (see Figure 5.17 A) )...86 Figure 5.19 Correlation of observed PGS(i.e. HPS max ) - PGD pairs for the 4 earthquakes under consideration. The solid line PGS=PGD γ /10 δ (see eq. (5.15)) is the LS best fit line corresponding to the median value γ = 0.79 and δ = -4.26, while the two dotted lines are associated to the 16 and 84 percentile, respectively (the standard deviation σ between the observed data and the best-fitting line is around 0.19)...87 Figure 5.20 Comparison of the observed PGS-PGD pairs with those calculated according to eq. (2.16), recommended by Abrahamson (2003) Figure 5.21 Comparison of the observed PGV-PGS correlation for the 4 earthquakes under consideration with the commonly adopted values of PGS for seismic design of underground structures Figure 5.22 PGS/PGV ratio as a function of the apparent propagation velocity V a (in m/s). There is a critical V a above which spatial incoherence effects dominate rather than wave passage. Adapted from Zerva (2000)...90 x

13 Index Figure 6.1 Time histories (left) and corresponding Fourier amplitude spectra (right) of torsional acceleration (up), velocity (middle) and displacement (bottom) calculated at station 16 during the M4.9 Parkway # Figure 6.2 Same as in Figure 6.1 but for M4.2 Parkway #2 event, receiver Figure 6.3 Same as in Figure 6.1 but for M6.0 Parkfield earthquake, receiver Figure 6.4 Same as in Figure 6.1 but for M6.5 San Simeon earthquake, receiver Figure 6.5 PGV vs. PGR z pairs, obtained through the spatial interpolation procedure for all the 4 earthquakes under consideration. Superimposed are the best-fit LS line corresponding to the median value χ = 2120 m/s (solid line) and the two best-fit lines (dashed line) associated with the 16 and 84 percentile, χ- = 1434 m/s and χ+ = 3130 m/s, respectively...96 Figure 6.6 Comparison of the PGV vs. PGR z pairs, obtained by our spatial interpolation procedure, with the results obtained by Spudich & Fletcher (see Table 6.1) for the same Parkfield event (main shocks) and three aftershocks (M5.1, M4.9 and M4.7). The classification into broadband and sub-arrays estimates refers to different procedures developed by the authors. However, for comparison purposes with our results, the broadband estimates have to be taken into account. For comparison purposed superimposed is the LS best-fit lines calculated on our data...98 Figure 6.7 PGA vs. PGRz pairs, for the 4 experimental datasets under study, along with the median fitted straight line, defined by eq. (6.6) Figure 6.8 Grenoble basin: example of 3D simulations by GeoELSE. Maps of the peak values of torsion (PGω z = PGR z ) for a selected earthquake scenario (M W =6, neutral directivity, linearelastic behavior; the location of the hypocenter is indicated on the upper left corner of the graph). On the right-hand-side map the location, where the maximum rotation is experienced, is highlighted. From Stupazzini et al. (in preparation) Figure 6.9 Synthetic values of PGV h vs. PGR z in logarithmic scale obtained from SE simulation of Grenoble basin with reference to events with M6.0 and M4.5, neutral directivity and linear visco-elastic soil behavior. Superimposed are the individual data retrieved from literature, listed in Table 3.2 (in Chapter 3). Data from 18 to 21, summarized in Table 6.3, are plotted in terms of average value (filled circle) and their minimum and maximum value (denoted by bars). The straight line of eq. (6.5) is also superimposed for comparison purposes (dashed line). Adapted from Stupazzini et al. (in preparation) Figure 6.10 Three degree-of-freedom model considered in the analysis of the effects of coupled tilt and rotational ground motion on structural response Figure 6.11 Horizontal (up) and tilt (bottom) ground accelerations for the event # 5 in Table 6.6 (Grenoble basin) considered as input for the study of the 3dof model. They are synthetic time histories obtained by the Spectral Element simulation of the seismic response of Grenoble basin for a realistic earthquake scenario (see previous section) Figure 6.12 Relative displacement, velocity and absolute acceleration of the 3dof system, when either pure translational (thin lines) or coupled tilt-translational (thick lines) base excitations are considered, for the event num. 5 in Table 6.6. The corresponding Fourier Amplitude spectra are also illustrated on the right-hand-side xi

14 Index Figure 6.13 A) Peak relative displacement of the structure under coupled tilt-translational excitation, d max tilt, over the peak relative displacement under pure translational motion, d max trans as function of the ratio (in s/m) between the peak ground rotation, PGR, and the peak ground translational velocity, PGV. The 5 case studies listed in Table 6.6, numbered from 1 to 5, are considered. B) Same as in A) but in terms of the ratio of the peak absolute acceleration of the structure under coupled tilt-translational excitation, a max tilt, over the peak absolute acceleration under pure translational motion, a max trans Figure A. 1 San Fernando and Los Angeles regions showing those areas characterized by more than 6 pipe break per km. From Trifunac & Torodovska (1997)... II Figure A. 2 Evidences of slope failures at tunnel portal (left) and conduct break (right) during the 1999 Chi-Chi earthquake. From Hashash et al. (2001)....III Figure A. 3 Reported damage of the Duzce s water network. From Alexoudi et al. (2007)...III Figure A. 4 Comparison of different damage relationships for pipelines as function of PGA as prediction parameter. The functional forms RR = f(pga) of the relations plotted here are synthesized in Table A VI Figure A. 5As in Figure A. 4 for PGV. The fragility curves are listed in Table A VIII xii

15 Index LIST OF TABLES Page Table 2.1 Longitudinal ground strain (ε a ) induced by seismic waves propagating along a buried extended structure as a function of peak ground velocity (PGV) or acceleration (PGA) and of the incident wave type. V P, V S and V R are the apparent propagation velocities of P-, S-, and Rayleigh waves respectively. φ is the angle of incidence of seismic waves with respect to the longitudinal axis of the structure; it is measured in the horizontal plane in the case of R-waves and in the vertical plane in the case of body waves...10 Table 2.2 As in Table 2.1 maximizing the strain with respect to φ Table 2.3 Main indications given by some technical guidelines for the seismic design of underground structures, referring to eq. (2.7)...13 Table 3.1 Closed-form solutions for tilts and torsion under the incidence of plane waves derived by Lee and Trifunac (1985 and 1987). ω y indicates tilt abut the horizontal y-axis while ω z indicates torsion, i.e. rotation about the z- axis. The following notation is used: the displacement field is u = [u, v, w] t i(r) ; A S amplitude of the incident (reflected) S wave; γ incidence angle; k α(β) = ω/α(β) wavenumber associated with P- (S-) waves; k = ω/c horizontal wavenumber and c phase velocity of Rayleigh or Love waves...33 Table 3.2 List of available data in literature: peak values of ground velocity (PGV h ), ground torsion (PGR z ) and torsion rate (PGRR z ). Additional information concern data type, the source parameters (magnitude, epicentral distance R and source mechanism) and type of soil. Data from Spudich & Fletcher (2008) correspond to broadband estimates (see Chapter 6 for further details). The legend is given below Table 3.3 Summary of results obtained by Bouchon and Aki (1982), in terms of longitudinal maximum strain (PGS a ) maximum tilt and torsion (PGR) and maximum rotation rate (PGRR). An illustrative example of the synthetic rotational time histories obtained by the authors is given by Figure Table 4.1 Geological model of soil column at Parkway Valley array, made available to us by W.R. Stephenson (personal communication, 2001 & 2006). Soil layers are reported from up to bottom...52 Table 4.2 Characteristics (magnitude, epicentral distance, available stations and peak ground motion values) of the earthquakes considered in this study. The peak values are calculated as the xiii

16 Index maximum absolute value of the horizontal components among all stations, i.e.: 2 2 PGV = max V ( t ) + V ( t )...55 t x y Table 4.3 Average interpolation error in predicting PGD and PGV (see Figure 4.14 for an illustrative example) for all receivers of the array and each earthquake of Table 4.2. The error is calculated as the absolute value of the difference between the observed and interpolated values, divided by the observed value Table 5.1 Comprehensive table of the empirical best-fitting relationships for PGS evaluation based on ground motion parameters PGV, PGA and PGD. For each correlation, the dataset, the coefficient of proportionality between PGS and the selected ground motion parameter in Log10 scale, the intercept in Log10 scale of the best-fit Least Square line and the standard deviation σ are summarized...89 Table 6.1 Data of PGV and PGR z according to Spudich & Fletcher (pers. Comm., 2008). Additional information about the earthquake magnitude and Joyner Boore distance R JB are listed. Highlithed in bold are the broadband estimates. Ar. 1-3(or 8-11, 5-12) f max 1.4(or 3.6) refers to strain and rotation estimates from sub-array 1-3 (or 8.11, 5-12) filtered in the (3.6) Hz band. Event refers to the main Parkfield shock Table 6.2 Comparison between the peak values of tilt, tilt rate, torsion and torsion rate obtained through our interpolation procedure and those calculated by Spudich & Fletcher (2008) for the M6.0 Parkfield event. Note that interpolation method refers to the mean values of the PGRs obtained within the array...98 Table 6.3 Peak values of Ground Velocity (PGV h ) and Ground Rotation (PGR z ) obtained from the proposed interpolation procedure to be complemented to Table 3.2 (Chapter 3). PGRs are given in terms of mean, maximum and minimum values Table 6.4 Structural model parameters. J is the sum of the centroidal moments of inertia of the building and the foundation and B is the width of the foundation Table 6.5 Equivalent spring and dashpot coefficients Table 6.6 Rotational and translational time histories considered as input for the analysis of the response of the simple model sketched in Figure Table A. 1 Pipeline fragility curves for Peak Ground Acceleration PGA found in literature. RR denotes repair rate per unite pipe length and PGA in measured in cm/s 2. Highlighted are the earthquake datasets considered for the fragility regression analyses...vi Table A. 2 Pipeline fragility curves for Peak Ground Velocity PGV developed by several authors. PGV is measured in cm/s... VII xiv

17 Chapter 1. Introduction 1 INTRODUCTION 1.1 Preface The present work focuses on the study of differential ground motions associated with earthquake shaking, referred to as dynamic displacement gradients as well. The potential effects of spatial variations of seismic motions on certain type of structures have long been recognized: certain damaging effects may result directly from dynamic deformations, such as strains, tilts and rotations, in addition to time variations of some translational components of ground motions. While the role of ground strains is widely accepted as strategic for damaging underground structures, such as subways, tunnels and pipelines, and, therefore, critical for their seismic design, the relevance of rotational ground motions for engineering applications is still subject of debate in the scientific community. Let us consider first the issue of ground strains from the engineering perspective. The seismic response of underground structures differs fundamentally from that of conventional aboveground structures. One of the basic differences is that the formers are subjected to spatially variable seismic ground motions, because of their considerable length. It is widely accepted indeed, both from theoretical and observational evidences, that the inertia effects for underground structures may be neglected and their seismic response is mainly based on the evaluation of the earthquake-induced deformations of the surrounding soil (see e.g. St John and Zarah, 1987; Hashash et al., 2001). Although it is evident the important role covered by transient ground strains in affecting the response of underground structures under seismic excitation, it is common practice to indirectly estimate the Peak Ground Strains (PGS) as a function of other parameters of ground motion severity, such as Peak Ground Acceleration (PGA), Velocity (PGV) or Displacement (PGD). Specifically, dating back to the pioneering studies of Newmark (1967), simplified guidelines for the design of pipelines and tunnels (e.g. ALA, 2001; AFPS, 2001; CEN, 2006) are based on the assumption that maximum ground strain can be reasonably estimated from the maximum particle velocity (PGV) divided by an adequate measure of the apparent wave propagation velocity. The main reason for this is the consistent lack of direct measures of ground strains during earthquakes. Strain gauges are, in fact, not generally available in the seismic networks developed all over the world. The obvious shortcoming of such an approach is that seismic strains are not observationally constrained and, as a consequence of this, they are subject to large uncertainties, partly due to the aleatory nature of the values of apparent wave propagation velocity (provided that the prevailing wave type is properly identified, which is not generally the case), and to the limitations themselves of the formulae, which are not applicable, for instance, in case of 1

18 Chapter 1. Introduction strong lateral discontinuities and may not reproduce spatial incoherence effects (Zerva, 1992; CUREE, 2004). These strong uncertainties are even reflected in the technical guidelines for seismic design of underground structure, which may lead to differences of at least a factor of two in the estimation of PGS, as recently discussed by Paolucci & Pitilakis (2007). On the other hand, the rotational component of earthquake-induced ground motions has been ignored for a long time, first because they were thought to be negligible and, second, because sensitive measuring devices were not available until quite recently. It is emblematic what Richter stated 50 years ago: Perfectly general motion would also involve rotations about three perpendicular axes, and three more instruments for these. Theory indicates, and observation confirms, that such rotations are negligible (C.F. Richter, 1958). This statement embodies what have been orienting scientific investigations and engineering applications so far. Nonetheless, nowadays some theoretical and observational evidences suggest that their quantitative knowledge may be of interest, primarily, from a seismological point of view and might have potential implications in engineering practice. From a seismological and geophysical point of view, understanding rotational motions has the following implications for: i) recovering the actual ground displacement history from double-integration of seismometer recordings, as the response of the instrument is actually polluted by rotational components (Graizer, 1991; Pillet and Virieux, 2007); ii) providing useful additional information about earthquake rupture process (Takeo & Ito, 1997; Takeo, 1998), especially in near-source regions, and subsurface properties (Wang et al., 2006). From the engineering standpoint, rotations might have significant effects on the response of high-rise buildings, as first pointed out by Newmark (1969) and Hart et al. (1975), or where soil-structure interaction effects play a relevant role. It is possible that some of these effects might be due to the asymmetry of the structure itself. Array-derived observations and numerical simulations suggest that the orders of magnitude of dynamic rotations induced by moderate to large earthquakes are too small to cause damage, except for large structures. Nevertheless, it is likely that in the near field (R <~25 km) of strong earthquakes (M>6.5) or, similarly, due to local unfavorable geological conditions, rotational motions approach large values to be potentially responsible of structural damage. Notice that, in the absence of direct measurements of rotations and tilts, they are indirectly deduced from seismometer data similarly to PGS estimates. A simple linear relation between Peak Ground Rotation PGR (or Peak Ground Rotation Rate, PGdω/dt=PGRR) and PGV (or PGA) is used, allowing one to determine, at least as a rough approximation, the amplitude of rotational components from the translational ones. The validity of such a simplified procedure is conditioned to the fulfillment of the assumptions of plane waves which propagates with constant velocity and without interference with other waves. 1.2 Scope and organization of the work The main scope of this work is to illustrate an experimentally-based method to estimate transient ground strains and, similarly, rotations from displacement records obtained by dense seismic networks, relying upon a suitable interpolation technique. Once the displacement wavefield is properly reconstructed by the spatial interpolation procedure, the displacement gradient tensor at the ground surface (namely small strain and rotation tensors) can be 2

19 Chapter 1. Introduction calculated by numerical differentiation of the displacements computed by interpolation. This approach can be seen as a variant of the so-called seismo-geodetic approach, i.e. multistation estimate of transient ground strains from closely spaced arrays, as carefully discussed by Zerva (2003) and applied by Spudich et al. (1995) and Bodin et al. (1997). This technique has been already applied by Smerzini et al. (2006) to evaluate dynamic strains during two weak ground motions recorded by the dense temporary network at the Parkway Valley (New Zealand) but here the application of the same method is extended to the strong motion obtained at the UPSAR array (California) during the San Simeon (M W = 6.5) and Parkfield (M W = 6.0) events. This allows, in fact, to cover a rather broad magnitude rang and to investigate some repeatable features of seismic ground strains and rotations. The application of the aforementioned procedure allows us to discuss the reliability and the limits of applications of the simplified relationships correlating PGS (and, secondarily, Peak Ground Rotation PGR) to other ground motion parameters, such as PGV, over a quite wide magnitude range (varying from around 4.0 to 6.5). Nevertheless, the range of PGVs is rather low, so that the results shown throughout this work apply to relatively weak shaking conditions. Furthermore, the estimation of the time-dependent strain tensor ε ( x,y,z = 0,t ) at the ground surface gives insight into noticeable features of the dynamic strain wavefield, such as the lowest and highest principal strains and the corresponding principal directions. Note that interest on rotational ground motion has been spreading relatively recently, such that a lot of uncertainties dominate the contemporary framework. Questions about the orders of magnitude of rotations likely to occur for a given earthquake scenario and potential their effects on structural response are not trivial. First, empirical estimates of earthquake-induced ground rotations are obtained from the application of the afore-mentioned empirical procedure. Subsequently, a significant effort has been devoted to compare some synthetic values of peak ground torsions PGR z (i.e. rotation about the vertical axis) obtained by means of 3D Spectral Element Method (Faccioli et al., 1997) simulations in the near-source region of the Grenoble (France) basin with both our empirical estimates and the available data retrieved from literature. These individual values include direct measurements, array-derived observations and numerical simulations. This comparison allows, first, to cross-check the applicability of the indirect techniques for rotation estimates and their agreement with more sophisticated numerical methods. Secondly, it contributes to construct a reference frame for the expected levels of ground rotations for realistic earthquake scenarios and their potential dependence on the magnitude, the epicentral distance the source mechanism and local geological conditions. Furthermore, a simplified engineering application is carried out to evaluate whether or not pronounced structural deformations can be ascribed to rotational input motions. Namely, using some representative rotation time histories, obtained both through the developed procedure and SE simulations, the response of a 3 degree of freedom system, subjected to either translational and coupled tilt-translational base excitation, is studied Objectives The main objectives can be summarized as follows: 3

20 Chapter 1. Introduction i. development of an observationally-based procedure for earthquake-induced transient ground strain evaluation from closely spaced seismic networks, based on a suitable spatial interpolation technique; ii. Estimation of surface ground strains and rotations for a set of 4 case studies, covering a rather wide range of magnitudes, epicentral distances and local soil conditions. iii. Investigation of the surface ground strain tensor defining new invariant measures of differential motion intensity, such as the principal strains and the corresponding principal directions. Their time variations and correlation with azimuth is discussed in order to identify potentially repeatable features of the dynamic deformation wavefield; iv. Derivation of new empirical relationships between either PGS or PGR and other ground motion parameters, such as PGV, PGA and PGD and comparison with other published results. v. Comparison of the our empirical estimates for PGRs with other published results (array-derived, numerical/analytical and direct measurements) and the synthetic PGRs computed by 3D Spectral Element simulations of the Grenoble basin, which include the complex effects of extended source, propagation path and local geological irregularities. vi. Example of a simple engineering application using a 4 degree-of-freedom (4dof) oscillator, incorporating soil-structure interaction effects, subjected to either pure translational and coupled translational-tilt input accelerations. 1.3 Thesis outline Chapter 2 addresses the topic of transient seismic ground strains as obtained from empirical techniques or analytical/numerical simulations. The advantages and limitations of evaluating maximum strains by simplified empirical correlations, recommended by most of the technical guidelines (ALA, 2001; Eurocode 8-CEN, 2006) for the design of underground structures are highlighted and discussed. The most commonly used methodologies for transient seismic assessment can be classified into: i) Empirical maximum strain evaluation; ii) Analytical and numerical seismic ground strain assessment; iii) seismic strains from recorded data. Furthermore peculiar emphasis is given to the issue of spatial variability of ground motion, which is a crucial factor affecting the magnitude of the observed strain wavefield. In Chapter 3 some original aspects regarding rotational ground motions and their implications in seismological and engineering applications are highlighted. Note that the choice to treat separately strains and rotations, though their similar physical meanings (they are, in fact, measure of displacement differential motions), is dictated by: i) the different level of knowledge about the two topics; ii) the different kinds of applications they are related to. In Chapter 4 an empirical procedure for evaluating transient ground strains from closely spaced seismic networks is illustrated, based on a suitable spatial interpolation technique. To this end, two arrays have been considered thanks to their relatively dense spacing: the 4

21 Chapter 1. Introduction Parkway Valley digital temporary array, located in New Zealand, and the UPSAR array, in California. Particular attention is devoted to the selection and validation of a suitable interpolation technique to obtain accurate estimates of the three-component displacement wavefield at the ground surface. In Chapter 5 the empirical evaluation of the time-dependent strain tensor at the ground surface is introduced. Some meaningful features of the surface ground tensor are discussed referring to the Mohr circle representation of strain conditions. Owing to this interpretation, new invariant strain parameters of the severity of differential ground motion are proposed. The issue of the dependence of these strain invariants on azimuth and its potential relationship with the prevailing wave propagation direction are carefully investigated. Thanks to the empirical procedure based on spatial interpolation new relationships for PGS estimation are shown and their dependence either on the particle velocity, or acceleration or displacement is discussed. Hence, the derived PGS correlations are compared both to available direct measures in pipelines or tunnels and to other published empirically- or numerically-based relationships. Finally Chapter 6 shows some preliminary investigations into some aspects of rotational ground motions, relying upon different tools. In particular, similarly to PGS estimates, new empirical relationships between PGV and PGR are derived from the application of the interpolation procedure and, then, compared with other recent published results. Attention is drawn on the synthesis of PGR estimates from various sources. A broad set of data, comprising, on one side, our empirical estimates and, on the other side, other published results, is compared with the synthetics generated by 3D numerical simulations by Spectral Element Method (SEM). 5

22 Chapter 2. Transient seismic ground strains 2 TRANSIENT SEISMIC GROUND STRAINS 2.1 Introduction It is widely recognized that, unlike above-ground structures where the inertial effects are of primary interest, the seismic design of underground structures, such as buried pipelines or tunnels, is unique in several ways. Their response is, in fact, dominated by the deformation of the surrounding soil and not the inertial properties of the tunnel structure itself (see e.g. St. John & Zarah, 1987). The critical issue of underground structure design is hence the proper understanding of the free-field ground motion. The quantification of ground strains is therefore critical to properly determine the input seismic motion for the analysis of such structures. For underground structures the seismic hazard can be classified into wave propagation hazard, i.e. transient ground strains and curvature induced by the passage of seismic waves, and permanent ground deformation, the latter including all the effects associated to ground failure phenomena (O Rourke & Liu, 1999). In a more general case, damage is caused by a combination of the two hazards. There have been different examples where pipeline damage was almost totally caused by wave propagation. For instance, most of the damage to water supply system which occurred during the Sept Michoacan earthquake (M W =8.0) has been attributed to wave propagation effects (see e.g. Rosenblueth & Meli, 1986). Water pipeline damage data from the 1999 Chi-Chi (Taiwan) earthquake underline the weight of earthquake-induced transient ground strains for the overall seismic hazard of pipelines: in this case transient ground strains were responsible of about half (~48%) of the total damage (Tromans, 2004). Starting from the 1906 M W =7.9 San Francisco earthquake, where several breaks and leaks in the pipeline system of the metropolitan area (in Figure 2.1) led to the largest urban fire loss in U.S. history, and moving to the 1999 Chi-Chi (Taiwan) and 1999 Kocaeli and Duzce (Turkey) earthquakes, damages to underground structures have been documented in several post-earthquake reconnaissance. Statistical analyses from catalogues of past strong motions have been used to estimate the expected damage by means of fragility relationships, in terms of repair rates per unit length of pipe as function of an appropriate measure of ground motion intensity, such as the Peak Ground Velocity, PGV, or Peak Ground Acceleration, PGA. A synthetic list of the most meaningful case histories of underground structure damage along with a brief description of the most widely used empirical damage relationships due to wave propagation is given in Appendix A. 6

23 Chapter 2. Transient seismic ground strains 30 Pilarcitos pipe P A C I F I C COLMA 30 Pilarcitos Pipeline Crystal Springs No 1 San Andreas No 1 Pilarcitos Alameda Reservoir Alameda pipe, slip joints pulled apart a few inches. Rapidly repaired. 2. two 8 blow-offs on 36 Alameda pipe. Rapidly repaired Crystal Spring No 1 broken at 4 spots, parallel to San Matteo Creek. Rapidly repaired, Liquefaction but not flowing to SF until 4 was repaired. Zones Crystal Spring No 1 broken due to 2,850 feet of wood trestle rfailure across 3 swamps.trestles and pipe repaired in 28 days Pilarcitos pipeline. About 29 breaks and 44 Crystal Springs No feet of collapsed wood trestle bridge. Colma Creek Alignement south of Colma was abandoned. 54 Crystal Springs No San Andreas No 1 pipeline. Roptured at Baden Crosing on wood trestle across Colma Creek. Repaired in 62 hours. 36 Alameda Pipeline S A N F R A N C I S C O B A Y San Andreas Fault Rupture O C E A N Figure 2.1 Damage to San Francisco transmission and distribution system (SFPCU, then SVWC) during the devastating 1906 earthquake (April 18, MW=7.9). On the left side highlighted are details of the Pilarcitos pipe destroyed by the earthquake (see the corresponding location on the rightside map). From Eidinger et al. (2006). Leaving aside the issue of permanent ground deformations, more emphasis is given herein to a better understanding of transient ground strains. In particular this chapter aims at illustrating the commonly used techniques for dynamic ground strain assessment, pointing out their principal advantages and shortcomings. The addressed approaches, carefully reviewed by Zerva (2003), can be summarized as follows: i. maximum strain estimation: using a simple traveling wave model peak ground strains estimates (PGS) are derived from the maximum particle velocity scaled by the inverse of an appropriate measure of the apparent wave propagation. ii. estimates from recorded data: single-station technique and multi-station approach; iii. analytical ground strain evaluation; iv. empirically- and numerically- based attenuation relationships for PGS. All the previous approaches suffer the intrinsic limitation of the lack direct measurements of ground strains to be used for validation purposes. From here on, Peak Ground Strain, Acceleration, Velocity, and Displacement will be referred to as PGS, PGA, PGV and PGD respectively. 7

24 Chapter 2. Transient seismic ground strains 2.2 Peak Ground Strain (PGS) estimates based on plane wave propagation assumption Owing to the lack of direct experimental measures of ground strains during earthquakes, the routine engineering practice is to indirectly estimate PGS, and the corresponding seismic action effects on the structure, based on certain approximations for both dynamic behavior of the structure and the traveling wave models. The simplified approach for maximum strain estimates is based on two fundamental assumptions. First, there is no relative motion between the ground and the underground structure, i.e. soil-structure interaction (SSI) effects play a negligible role (Newmark and Rosenblueth, 1971; St. John & Zarah, 1987). The second assumption is that the seismic excitation is given by a plane wave which propagates towards the ground surface with constant velocity without changing shape and interfering with other waves. Based on the above considerations, it can be proved that the PGS, measured in the direction of wave propagation, is simply related to the peak particle velocity PGV (in the same direction) by the relationship: PGS = PGV / V a (2.1) where V a is the apparent wave velocity, i.e. the propagation velocity along the ground surface. Eq. (2.1), dating back to more than 30 years ago (Newmark, 1967; Yeh, 1974), derives from a straightforward manipulation of the 1D solution of plane wave propagation equation in a homogenous unbounded medium. In particular, considering a generic plane wave propagating in the +x direction, the displacement u( x,t )can be expressed as: Differentiating once eq. (2.2) with respect to both space and time: u( x, t) = f ( x V a t ) (2.2) u( x, t) = f '( x Vat) x and u( x, t) = Va f '( x Vat) t (2.3) and substituting the terms above in the classical 1D wave equation, we obtain: u 1 u = x V t a (2.4) If, for simplicity, the displacement u( x,t ) is parallel to the x axis, eq. (2.4) gives directly PGS as function of PGV and V a. Newmark s approach has spread noticeably in engineering applications thanks to its simplicity of use. Nevertheless the validity of eq. (2.1) is conditioned to several limitations: i. it should not be applied when strong lateral discontinuities in soil material properties exist; ii. it should be applied for soil conditions where well-defined and undisturbed waves develop, such that they can be reasonably treated as propagating plane waves. 8

25 Chapter 2. Transient seismic ground strains iii. Provided that the previous two assumptions are satisfied, its practical application requires the knowledge of other parameters, such as the wave type, the angle of incidence with respect to tunnel axis and the values of the apparent wave velocity, parameters that can not be easily estimated in general. iv. It can not reproduce spatial incoherence effects, such as those due to near-source complex wavefields or due to the scattering caused by small scale heterogeneities, which may significantly increase the magnitude of strains. St. John & Zarah (1987) developed Newmark s approach further, deriving closed-form solutions for ground strains and curvatures due to longitudinal (P- ), shear (S- )- and Rayleigh (R- ) waves. These results provide a first-order evaluation of earthquake-induced ground strains and they might either overestimate or underestimate the actual ones, depending on the relative stiffness of the structure with respect to the surrounding soil. Referring to Hashash et al. (2001) for a comprehensive summary of different solutions applicable to various deformation modes (axial/shear strains and curvature) and prevailing wave patterns, we underline here those related to the estimation of longitudinal ground strains and curvatures (see Table 2.1). Axial strains are, in fact, commonly larger than shear strains, at least for shallow underground structures, typically down to 5 m where most of buried pipelines are located. On the other hand, the opposite becomes true (shear strains larger than axial ones) for increasing depth. For design purposes, the axial strains are usually more significant than curvatures, since the formers are proportional to V a -1 whilst the latter are proportional to V a -2. Table 2.2 gives the maximum axial strains and the corresponding value of φ, which is defined as the angle of incidence of seismic waves with respect to the longitudinal axis of the structure. For P- and R- waves the maximum axial strain occurs for φ = 0, i.e. when the pipe axis is parallel to the wave propagation direction, whereas for S- waves the most critical case occurs when the traveling waves propagate obliquely at 45 from tunnel axis. This expressions are of particular relevance to better understand the simplified formula recommended by many technical guidelines for underground structure design (see e.g. AFPS/AFTES, 2001; American Lifeline Alliance ALA, 2001). 9

26 Chapter 2. Transient seismic ground strains Table 2.1 Longitudinal ground strain (ε a ) induced by seismic waves propagating along a buried extended structure as a function of peak ground velocity (PGV) or acceleration (PGA) and of the incident wave type. V P, V S and V R are the apparent propagation velocities of P-, S-, and Rayleigh waves respectively. φ is the angle of incidence of seismic waves with respect to the longitudinal axis of the structure; it is measured in the horizontal plane in the case of R-waves and in the vertical plane in the case of body waves. Wave type Longitudinal strain Curvature P PGV ε a = V PGVS S ε a = sin φ cos φ V R (compressional component) S P PGV ε a = V R P R cos φ φ φ ρ = PGAP 2 sin cos V P 1 3 φ ρ = PGAS 2 cos V cos φ φ φ ρ = PGAR 2 sin cos V R S Table 2.2 As in Table 2.1 maximizing the strain with respect to φ. Wave type Maximum longitudinal strain PGV P PGS = V PGV S PGS = 2V PGV R PGS = V P S R S P R for φ = 0 for φ = 45 for φ = 0 As anticipated above, despite its simplicity, the most critical issue in such an approach is the determination of the prevailing wave type and a suitable measure of apparent wave propagation velocity V a, parameters which may not be properly constrained. As proposed by ASCE Oil and Gas Guideline (1984) and suggested by O Rourke & Deyoe (2004), a common rule of thumb considers that surface waves are significant at epicentral distances R more than about five focal depths D, while at shorter distances V a is the apparent S- wave velocity in the direction of the longitudinal axis of the structure. Hence, for instance, given R~400 km and D~18km - this is the case of the 1985 Michoacan earthquake (Mexico), quoted by O Rourke & Deyoe (2004) one would reasonably consider the phase velocity of surface waves. The determination of apparent wave velocity, typically ranging between 2.0 and 4-5 km/s (O Rourke & Deyoe, 2004), is even less straightforward. O Rourke and his co-workers developed some procedures for estimating appropriate values of V a. Referring to O Rourke & Liu (1999) for a more detailed description, we underline here some basic concepts regarding the evaluation of apparent wave velocity from either body (S- waves) and surface waves (Rwaves). Notice that in case of surface waves, Rayleigh waves are the most significant, since they induce axial strains of larger amplitude than those produced by the Love bending deformation. 10

27 Chapter 2. Transient seismic ground strains Ground strains from body waves As body waves approach ground surface from the source located at depth, they are reflected towards the normal according to the Snell s law 1, due to the decreasing S-wave velocity of the uppermost layers (see Figure 2.2, left). This is the reason why most applications, for sake of simplicity, assume a vertically incident S- wave; in such a case the apparent wave velocity, i.e. the velocity projected along the horizontal direction, would be infinite. In reality, body wave arrival is sub-vertical, forming small incident angle γ s with respect to the normal and giving rise to high values of V a. As graphically explained in Figure 2.2 (right), the apparent wave velocity V a is given by: VS V a = (2.5) sin( γ ) where V S is the shear velocity of the uppermost soil layers and γ S is the angle between the wave propagation direction, given by the wavenumber vector, and the vertical. Once again, despite the simplicity of eq. (2.5), the angle of incidence γ S is not an easy parameter to determine. O Rourke et al. (1982) developed a sophisticated technique for evaluating its values, involving the resolution of the eigenvalue problem associated to a time variant ground motion intensity tensor, defined on the basis of the ground accelerations in three mutually perpendicular directions. This technique has been applied by the authors to the 1971 San Fernando and 1979 Imperial Valley events, finding an average S- wave apparent velocity of about 3.4 km/s. More accurate evaluations of the apparent velocity values can be obtained by array data, when available (see Section for further details). In such a case, cross-correlation analysis and frequency-wavenumber spectra may be powerful techniques for determining the wave propagation azimuth and the wave velocity (or, more frequently, slowness) along the same direction. S Figure 2.2 Left: propagation path of seismic waves traveling upwards towards the ground surface in increasingly soft layers of geological materials. Right: apparent propagation velocity V a for body waves. γ S indicates the angle of incidence and v(x,z,t) is the S-wave in-plane particle motion. 1 V i+1 /sin(γ i+1 ) = V i /sin(γ i ) where V i and γ i denote the S- (or P- ) wave velocity of the i th layer. The relation gives the incidence angle at the top of the i+1 th layer, provided the angle of incidence at the i th layer. 11

28 Chapter 2. Transient seismic ground strains Ground strains from surface waves When Rayleigh waves dominate seismic motions, V a coincides with the phase velocity V ph : V = λ f (2.6) ph λ and f being the wavelength and frequency, respectively. Contrary to body waves, the phase velocity is a function of the variation of V S with depth and also of frequency. Surface waves are generally responsible of the largest ground strains, since their phase velocity is smaller than body wave velocity. O Rourke et al. (1984) developed a simplified procedure in order to calculate the dispersion curve for horizontally layered profile with S- wave velocity increasing with depth. This procedure was applied to the case of a bridge with 21 m span between the abutments: the authors found V ph ~252 m/s and a peak particle ground velocity of 0.34 m/s, giving PGS~ and a maximum dislocation between the bridge abutments of around PGS*21 m = m PGS evaluation in the technical guidelines Based on the expressions summarized in Table 2.2, the technical guidelines and regulation codes for seismic design of underground structures generally recommend the determination of the maximum longitudinal strain along the direction of the structure, say x, by a formula that can be cast as follows: PGS x PGV kv = x a (2.7) where k represents a correction parameter to incorporate the effect of the angle of incidence φ with respect to the tunnel axis x. As discussed in the previous section, V a and k are usually dominated by significant uncertainties. This uncertainty is reflected in the practical rules provided by the technical guidelines themselves. A careful discussion of such an issue is given by Paolucci & Pitilakis (2007). For instance, ALA (2001) suggests to conservatively assume V a = 2.0 km/s and k = 2 for S- waves and k = 1 otherwise. Notice, in fact, that the value V a = 2.0 km/s constitutes the lower bound of the empirically acceptable range suggested by ASCE (1984) and O Rourke & Deyoe (2004). According to the French guidelines AFPS/AFTES (2001), V a should be chosen as min{v S, 1.0 km/s}, being V S the S- wave velocity averaged over a depth equal to the fundamental wavelength, and k = 2 to take into account the less favourable incidence mode. Contrary to these indications, Eurocode 8 (CEN, 2006) gives rather general indications, delegating the designers to undergo site-specific geophysical investigations for the determination of V a. When these are not available, k=1. Table 2.3 summarizes some inconsistencies in the technical indications for both k and V a from the previous seismic norms, highlighting the inconsistencies among them. 12

29 Chapter 2. Transient seismic ground strains Table 2.3 Main indications given by some technical guidelines for the seismic design of underground structures, referring to eq. (2.7). Guidelines Recommended k Recommended V a Eurocode k = 1 Further geophysical investigations needed American Lifeline Alliance ALA 2001 k = 2 for S -waves, k = 1 otherwise AFPS/AFTES 2001 k = km/s min{v S, 1.0 km/s} V S = S-wave velocity averaged over a depth equal to the fundamental wavelength 2.3 Ground strain estimates from recorded data: single- and multiple-station approaches Considering that strain gauges are not generally available in modern seismic networks, there is an obvious need to manipulate seismic records in order to achieve reliable estimates for earthquake-induced transient ground strains. As carefully illustrated by Zerva (2003), the methods for estimating transient ground strains from seismic data can be divided into two main categories: i) single station estimates; ii) multi-station estimates, typically from dense seismic arrays, alternatively referred to as seismo-geodetic approach. In the sequel the main features of the aforementioned techniques will be illustrated, quoting some examples of applications found in literature. Furthermore particular emphasis will be given to method ii), owing to its similarity with the procedure developed in this work Single station estimates The so-called single-station technique relies on seismometer data recorded at a certain location. As such an approach has fewer data than the seismo-geodetic approach illustrated in the following section, it requires a greater number of assumption regarding traveling wave and medium models. Basically, the a priori assumptions can be summarized as follows: i) seismic energy travels as plane waves; ii) the medium under consideration is laterally homogenous (i.e. 2D/3D configurations which may give rise to a larger strain wavefield, because of local irregularities, are neglected) or the scale of lateral variations is significantly larger than the dominant wavelength of seismic motions. In the latter case, though the medium is not homogenous on an absolute scale, it may be treated as locally homogenous compared to the prevailing wavelength; 13

30 Chapter 2. Transient seismic ground strains iii) apparent propagation velocity and azimuth is known. It is worth noticing that these hypotheses are the same behind the applicability of eq. (2.1) and that the third assumption leads to the same limitations and uncertainties underlined in the previous section. On the basis of the above assumptions, Gomberg & Agnew (1996) derived expressions relating dynamic strains, in particular radial ε rr, tangential ε θθ and shear ε rθ strains, to quantities measurable from seismograph stations. They considered shallow earthquakes, based on the further assumption that most of the energy at regional distances (from tens to hundreds of km) is carried by surface waves with periods of several to tens of seconds. At regional distances and for a single mode of Rayleigh and Love waves they approximated radial ε rr, tangential ε θθ and shear ε rθ strains as: ε rr 1 V R ph u t r ; ε θθ = 0 and ε rθ 1 2V L ph u t θ (2.8) with V R ph and V L ph equal to the Rayleigh and Love phase velocities, respectively. Notice that tangential strains ε θθ should vanish at regional distances due to plane wave hypothesis. Eq. (2.8) derives from a direct manipulation of (2.4) for Rayleigh and Love wave propagation. The same authors compared the strain estimates, obtained from eq. (2.8) with the direct measures from strainmeters at the Piñon Flat Observatory (PFO), in California, during three regional earthquakes: A) Northridge, California (R~206 km); B) St. George, Utah (R~470 km) and C) Little Skull Mountain, Nevada (R~345 km). Details about the analyzed datasets can be found in Gomberg & Agnew (1996). The comparison is depicted in Figure 2.3. The comparison is satisfactory, but the range of frequency under consideration islimited f= Hz, noticeably below frequencies of interest for engineering purposes. The most evident discrepancy found by Gomberg & Agnew concerns the non-zero tangential strains in contradiction with the theory (see eq. (2.8)). The authors suggest that this inconsistency might be due to three possible causes: a) errors in the evaluation of epicentral distance and azimuth, which may be varying with time; b) scattering/distortion effects on the local wavefield induced by irregular topography or material heterogeneities, making the assumption of planar wave propagation questionable; c) distortion of the source-receiver propagation path from the great circle. Furthermore the authors quantified the accuracy of single-station strain method in both amplitude and phase estimates using a sophisticated cross-spectral analysis technique. Their results highlight that strain estimates are affected by errors of around 10% and 20%, for phase and amplitude, respectively, for frequency not larger than ~0.15 Hz. For frequencies larger than Hz the accuracy of estimates dramatically decreases. The single station approach has been used even by Trifunac et al. (1996) and applied to the Northridge earthquake of 17 Jan 1994 (California) with certain variations. In particular, they verified that the peak ground strains in direction a can be approximated by: PGS a ~ PGV a A (2.9) V S 14

31 Chapter 2. Transient seismic ground strains where V S is the average S-wave velocity in the top 30 m of soil and A is a scaling factor, of the order of one, incorporating the effect of wave type, angle of incidence and soil conditions. Figure 2.4 shows the contour lines of the average regional horizontal strain factor, defined as the right-hand side of eq. (2.9) leaving aside the scaling factor A. The maximum strain horizontal strain factor they found was around In their dataset V S is typically >200 m/s. Assuming again that energy is predominantly carried by surface waves, they empirically estimate the scaling factors both for radial ε rr, shear ε rθ strains as: A~0.36 and A~0.22, respectively. Notice that these values of A are approximately valid for V S ~300 m/s and are site-specific (they have been determined by a local study in Imperial Valley, California), so that the results might not be applied to all locations of Los Angeles basin or other sites. Hence, considering A=0.36 and A=0.22 yields to PGS rr ~~ , for the radial component, and PGS rθ ~ , assuming the same maximum strain factor mentioned above. ε rr A) ε θθ ε rθ B) C) Figure 2.3 Comparison between single station strain estimates from seismic recorded data (dashed lines9 and direct measures from strainmeters, at Piñon Flat Observatory in California during three regional earthquakes (see text for further details): A) Northridge; B) Little Skull Mountain; C) St. George. Radial ε rr, tangential ε θθ and shear ε rθ strains are compared. For the tangential strains only the observed quantities are reported, since form they should vanish theoretically. From Gomberg & Agnew (1996). 15

32 Chapter 2. Transient seismic ground strains Figure 2.4 Contour lines of horizontal strain factors Logχ ( PGV / V s ) of strong motions in Los Angeles basin during the 1996 Northridge earthquake. From Trifunac et al. (1996) Multi station estimates: the seismo-geodetic approach An alternative approach is the so-called seismo-geodetic approach and relies on multiple data recorded by closely spaced seismograph arrays. The name of the method derives from the fact that it combines the techniques of earthquake engineering with those of geodesy. This method overcomes some of the limitations of the single-station method but it puts somehow new restrictions. Particular attention is given to the illustration of this approach since it reveals strong similarities with the work developed herein. In Chapter 4 both analogies and differences between the seismo-geodetic approach and our interpolation-based procedure for seismic strain evaluation from dense seismograph arrays will be discussed more carefully. The approach has been introduced first by Spudich et al. (1995) and Bodin et al. (1997): the former applied the procedure to strong motion data obtained at the UPSAR array at Parkfield, California, while the latter considered three relatively closely spaced stations in Mexico City. Spudich & Fletcher (2008) recently computed new array-derived estimates for the same UPSAR array. Referring to Bodin et al. (1997) for a detailed explanation of the rational behind this procedure, we limit here to illustrate the basic features. The method aims at evaluating the complete displacement wavefield at the ground surface expressed by means of the displacement gradient tensor D with time varying components u i( t ) / x j (i, j = 1, 2 and 3 to denote, e.g., north-south, east-west and up-down directions, respectively). D is calculated from ground displacements u i recorded by N three-component stations as follows: ui ui u = u u = x = u / x x x n n 0 n n n i i i j i j j j for i, j = 1,2 and 3 (2.10) 16

33 Chapter 2. Transient seismic ground strains where u n i is the differential displacement in the i th direction at the n th station with respect to a selected reference station, denoted with the apex 0, and x n j is the corresponding separation distance. Thus, from simple kinematic considerations, knowledge of the components of D allows one to calculate both strain and rotation as: ε ij 1 u i ui = + 2 x j x j and 1 u i ui θij = 2 x j x j (2.11) respectively. Introducing the stress-free boundary conditions 2 at the ground surface, considered to be flat, and assuming the Hooke s law for isotropic soil materials yields the following constrains: = u1 / x3 u1 / x 3 ; 2 / 3 = 2 / 3 u x u x (2.12a) ν λ 3 u3 / x3 = ( u3 / x3 + u3 / x3 ) = ( ε11 + ε 22) (2.13b) 1 ν λ + 2µ where λ and µ are the Lamé constants and ν the Poisson modulus. Therefore, eq. (2.12) reduces the number of independent displacement gradients in D : the six independent components can be found resolving eq. (2.10). The method obviously requires that at least 3 station recordings are available; a generalized least squares minimization scheme may be efficiently used if more than 3 are available. Figure 2.5 shows the six independent components of D at the ground surface for the 24 Oct 1993 M6.6 event in the Valley of Mexico, as found by Bodin et al. (1997) from the recordings obtained by three stations (ROMA A, B and C). The gradients are dominated by 2.5 sec surface waves and the horizontal gradients exceed the vertical ones by a factor of approximately 2-3. To determine the gradient tensor D from displacement records using eq. (2.10) the deformation field must be uniform within the networks at any time, i.e. the variations of displacement are linearly predictable, as pointed out by Bodin et al. (1997). That means that for a given network, a single strain estimate alone can be inferred. This assumption arises from plane wave propagation hypothesis (i.e. source complexity and possible wavefield distortion due to scattering and/or small scale heterogeneities are neglected at first order approximation) and it is fulfilled if the array is closely spaced enough. Specifically, the average separation distance of the array should not exceed one quarter of the dominant wavelength λ to have gradient estimates accurate to approximately 90%. Furthermore, as 0 γ 2 13 ε13 ε31 0 σ σ 0 = + = 1 u i ui n = where εij = + with the notation used by Bodin et γ 23 = ε 23 + ε32 = x j x j al. (1997). E 3 The Lamé constants are, in fact, defined as: µ =, i.e. shear modulus also denoted by G, and 2(1 + ν ) Eν λ =. (1 + ν ) (1 2 ν ) 17

34 Chapter 2. Transient seismic ground strains indicated by Gomberg et al. (1999), practical criteria require that: a) the processed records have a high signal to noise ratio (SNR) and b) the records are synchronized in time, since, when subtracting one seismogram to another one, the coherent part of the signal cancels out but noise does not. An interesting comparison between single- and multiple- station estimates, in terms of displacement gradients, can be found in Gomberg et al. (1999), applied to the case of Geyokcha, Turkmenistan small-aperture array. In particular, the authors compared the two approaches in the frequency bands Hz and Hz, analyzing 11 different seismic events at regional distances and investigating the possible causes of the differences. Figure 2.6 illustrates the comparison between the single station technique (solid lines) and the seismogeodetic approach (dashed line) in both frequencies bands. The agreement between the two methods is surprisingly good in the frequency range between 0.5 and 1.0 Hz, suggesting that the deformation field is mainly associated to plane coherent waves, as required by the singlestation methods. However, when information such as the apparent wave velocity and the azimuth of the source-receiver propagation path can not be easily determined, the multiplestation technique may be more suitable. On the other side, the agreement is totally unsatisfactory at larger frequencies (Figure 2.6, right), indicating that the displacement may not vary linearly within this frequency band and for small separation distances, probably due to the increasing lack of coherence of ground motion (Zerva, 1992). The violation of the hypothesis that displacement varies linearly within the array makes the seismo-geodetic method unstable. Similar results have been recently obtained by Chung (2007). In this study ground strains over a soft sediment-filled plain are estimated using two independent dense array data LLSST and SMART 1, in Taiwan, for a total number of 19 earthquakes. The comparison between the single-station approach and the calculated strains at the ground surface (see Figure 2.7) shows that the single station approach might lead to strong underestimation (of a factor of 5 for the event 17 in the same figure) when higher phase velocities associated to S- waves are used rather then the actual value of apparent velocities controlled by the arrival of surfaces waves rather than body waves. 18

35 Chapter 2. Transient seismic ground strains u E / x E u E / x N u / x Z E u N / x E u N / x N u / x Z N Figure 2.5 Components of the surface displacement gradient D estimated for the 23 Oct 1993 M6.6 earthquake in the Valley of Mexico from three relatively densely spaced stations (ROMA: A, B and C). From Bodin et al. (1997) Hz Hz single-station seismo-geodetic Figure 2.6 Comparison of dynamic displacement gradient tensor components (as defined in the text) using single-station (solid lines) and seismo-geodetic (dashed lines) approaches for the eastern portion (composed of 4 stations) of the Geyokcha array in the frequency bands Hz (left) and Hz. The maximum separation distance within the sub-array was 400 m. The data refer to an earthquake which occurred on 23 Sept 1993 with epicentral distance R~64 km, V a ~2.72 km/s and azimuth ξ~325. From Gomberg et al. (1999). 19

36 Chapter 2. Transient seismic ground strains Figure 2.7 Comparison of horizontal peak strains (in µ) estimated by single-station approach and those calculated from array-derived horizontal gradients. From Chung (2007). 2.4 Numerical/analytical ground strain evaluation Near field strains: Bouchon & Aki s work (1982) One of the first examples of numerical simulation of earthquake-induced ground strains, tilts and rotations was presented by Bouchon & Aki (1982), considering near-fault conditions, for both strike-slip and reverse focal mechanism, embedded in horizontally layered media. Given the complexity of the problem, topographic irregularities and lateral heterogeneities were neglected. The numerical simulations were carried out by means of the wavenumber method, an accurate numerical tool operating in the frequency domain for modeling seismic wave propagation originated by extended faults in flat layered crustal models (Bouchon & Aki, 1997; Bouchon, 2003). Different crustal models were considered in order to study the effect of low shallows wave velocities on the resulting strain field. Bouchon and Aki found that the presence of a low-velocity subsurface layer amplifies significantly the longitudinal strains, while decreases the tilts. Leaving aside the case of vertical strike-slip fault, we highlight here some of the basic results found for the dip-slip fault and a low velocity upper layer, to simulate the strain field radiated by the 1971 San Fernando (M W =6.6) earthquake. This is a particularly interesting case study as it caused extensive damage to freeway bridges located in the vicinity of the fault, which most likely have sustained large differential motions. The source geometry, corresponding to that of San Fernando earthquake, the adopted crustal model and the receiver layout are depicted in Figure 2.8. The width of the fractured zone is around 12 km and the rupture propagates upward with velocity of 2 km/s in the half-space and of 1 km/s in the 1 km wide sediment layer. The strain wavefield is computed at 11 receivers located at 5 km intervals on the ground surface. In Figure 2.8 the horizontal ground velocities are superimposed to the strain time histories resulting for 1 m fault slip at three of the considered receivers. This comparison clearly shows some interesting features of the strain-velocity relationship (2.1). Close to the fault tip the longitudinal strain reaches a value of and the 20

37 Chapter 2. Transient seismic ground strains horizontal velocities are of the order of 160 cm/s. The strain time histories at sites above and beyond the upper tip of the dip-slip fault resemble the corresponding horizontal ground velocity. For these stations the approximation (2.1) is, thus, numerically validated, yielding a phase velocity (assuming once again that the most significant contribution is given by surface waves) of around 2 km/s. These results are very similar to other simulations, not shown here for brevity. Furthermore the authors underlined that the phase velocity controlling strain estimates, based on (2.1), is associated to the rupture velocity and the shear wave velocity of the rock basement, rather than the shear velocity of the uppermost layers. However, it is worth remarking that the Bouchon & Aki conclusions apply to a synthetic wavefield in which most of the sources of spatial incoherence of earthquake ground motion are not accounted for, namely the 3D nature of the problem, the fault slip heterogeneity, the lateral variability of soil properties as well as their small-scale inhomogeneities. Therefore the results of this pioneering study may be biased to a non-conservative side, underestimating the spatial variability of earthquake ground motion and its corresponding measures. longitudinal strain horizontal velocity Figure 2.8 Examples of longitudinal strains (solid lines) and horizontal ground velocities (dashed lines) time histories obtained by Bouchon and Aki (1982) by simulating near-field differential motions induced by a dip-slip fault embedded in layered media. The source model and the dynamic properties of the layered earth structure are shown in the bottom side of the figure. 2.5 Empirically- and numerically- based relationships for PGS estimation In this section, we review some of the mostly used empirically- or numerically- based relationships for PGS evaluation which will be resumed later in the present work. 21

38 Chapter 2. Transient seismic ground strains Three meaningful relations, arising from different derivation and reasoning, are illustrated here: i) the numerically-based relation of Trifunac & Lee (1996) correlating peak radial ε rr and shear ε rθ strains to PGV; ii) iii) the empirically-based relationship found by Abrahamson (2003) between PGS and PGD; an improved version of the analytical simplified formula relating PGV and PGS derived by Scandella & Paolucci (2006) to take into account the presence of strong lateral discontinuities Trifunac & Lee (1996) Trifunac and his co-workers carried out a large number of numerical simulations by their code SYNACC, for epicentral distance R varying from 1 to 100 km, magnitude 5 < M < 7.5 and 4 different subsurface scaling models, in order to build up simple scaling correlations relating peak ground strains with peak ground velocity PGV, in both radial, transverse and up-down directions. The authors themselves cautioned that the simulations carried out are based on fundamental assumptions, which limit their application range: a) the input model consists of layered 1D geological structure, that leads to an obvious oversimplification of the actual inhomogeneities; b) near-source effects are neglected since waves are supposed to be polarized in a single direction; c) predominance of traveling surface waves has been considered. After showing that peak radial PGS rr, shear PGS rθ and vertical PGS z strains can be expressed as in (2.9), as a function of PGV and shear wave velocity β 1 in the top 50 m below the ground surface and a site-specific scaling factor A, they found the analytical max expression of the so-called strain factors ε a (i.e. right-hand side of eq. (2.9) leaving aside the factor A) with a denoting either radial, transverse or vertical direction. Leaving aside here the case of vertical strains, which will not be dealt with in the sequel, we report the expressions of the strain factors for the radial ε max rr and shear ε max r ϑ directions: ( ) ( ) Log = + + R + max ε rr β1 β1 ( β ) + ( β ) ( / β ) 1 1 R LogPGV rr 1 (2.14) max rθ ( ) ( ) Logε = β + + β R ( β ) + ( β ) ( / β ) 1 1 R LogPGV rθ 1 (2.15) with R (in km) = epicentral distance (1< R <100 km), β 1 (in km/sec) = average S- wave velocity in the top 50 m (0.05< β 1 <0.5 km/s). Figure 2.9 illustrates how eqq. (2.14) and (2.15) fit the synthetic peak strains, in both radial and shear direction. These relations seem to predict well maximum shear and radial strain, at least for the considered geological and numerical model, giving maximum errors between the predicted and the computed value not 22

39 Chapter 2. Transient seismic ground strains larger than approximately 5%. It is worth underling that the rationale behind the coefficients in eqq. (2.14) and (2.15), that multiply the factor Log( PGV a / β1 ), is to empirically correct β 1 to represent an equivalent phase velocity (i.e. V a of eq. (2.1) ). The authors found negligible dependence on earthquake magnitude M (see Figure 2.10) and only a weak dependence on epicentral distance D and local soil conditions expressed by the parameter β 1. They also showed that, for a wide range of magnitudes, source-site distances max and local geological conditions, the slope of Log( ε a ) vs. Log( PGV a / β1 ) is very closet to 1, justifying therefore eq. (2.1), at least numerically. Figure 2.9 Comparison in logarithmic scale between the radial ε max rr and shear r ϑ strain factors computed by SYNACC program, on the y-axis, and those predicted by eqq. (2.14) and (2.15) on the x- axis. The results for two different geological models are reported: on the left β 1 =500m/s; on the right β 1 =100m/s in the uppermost layer. In both graphs superimposed are the 45 straight lines for comparison purpose. ε max 23

40 Chapter 2. Transient seismic ground strains Figure 2.10 Magnitude dependence of the strain-velocity relationship Log( εa max ) vs. Log( PGV a / β1 ) for the synthetic accelerograms computed by Trifunac & Lee (1996) for a selected geological model Abrahamson (2003) Abrahamson (2003) derived from empirical evidences, relying on recordings from 5 dense arrays (namely: SMART 1, EPRI/Taipower LSST, EPRI Parkfield, Chiba and Garni) the following attenuation laws for PGS as function of PGD rather than PGV: M 5 5 a PGS / PGD( cm) = e / V / cm / cm (2.16) The terms at the right-hand side of eq. (2.16) stand for the three sources that are recognized to generate ground surface strains: i) wave passage effect: peak ground strain, induced by traveling waves along the M space, normalized with respect to PGD(in cm) are given by e / V a where M is the earthquake magnitude and V a = 2.0 km/s, according to Abrahamson s recommendations. For M = 7, a transient peak ground displacement of 1 cm would lead to peak strains due to wave passage of (using V a =2.0 km/s). The effect of magnitude on the ratio PGS/PGD or, equivalently PGS/PGV, is relatively poor, as found by Trifunac & Lee (1996). ii) iii) Spatial incoherence effect: this effect, carefully studied by Zerva (1992), is described by variations in phase angles and amplitudes of seismic motions as well as time arrival perturbations of waveforms. Based on empirically-derived spatial coherence functions and forward modeling of spatial incoherence effects (see e.g. Zerva & Zervas, 2002, for a detailed review) Abrahamson proposed a rule-ofthumb factor of Local variable site conditions: this effect leads to additional differential motions and, thus, strains, within the separation distances of interest. Based on empirical 24

41 Chapter 2. Transient seismic ground strains models quantifying the variability of ground motion as a function of spectral acceleration at 0.7 Hz (a standard deviation σ=3% is taken into account) at separation distance of 10 m, the same factor of is found as before. Such a factor is to be considered only when evidences of geological variable conditions and complex site effects (e.g. in alluvial valleys) are available. Notice that the lower is the magnitude, the more significant is the relative effect of spatial incoherence and site conditions. For example, for M = 4 the contribute coming from the wave passage effect is small (~10-6 ) compared to the sum of effects ii) and iii) (= ); on the other hand, for M = 6 all the three effects play a similar role in increasing surface strains Scandella and Paolucci (2006) To overcome some of the limitations of the simplified linear correlation (2.1), Scandella and Paolucci (2006) developed an analytical simplified formula for PGS as a function of PGV in the presence of strong lateral discontinuities, through a set of parametric analyses by 2D inplane numerical simulations of seismic wave propagation by a spectral element code (Faccioli et al., 1997; Stupazzini, 2004). A typical example of numerical results is shown in Figure 2.11, where it is clear that PGS a (= horizontal strain calculated on the ground surface) and PGV occur at the same spatial location (x denotes distance from the basin edge) only for low values of the dipping angle, (α < 10 ), while the larger is the dip angle the larger is the spatial distance between points where the maximum value of either PGS a or PGV occurs. The following empirical relation was found: 1 η PGV PGS = 1( /, α) + 2( /, α) a 1+ η F x L F x H V S (2.17) where V S is the shear wave velocity of the (homogeneous) basin, η is the soil-bedrock impedance ratio (η < 1) and the two functions F 1 and F 2 depend on the dip angle α, the normalized position x/l of the site with respect to the soil-bedrock contact. The geometrical quantities L and H = L tanα are clarified in Figure The variations of F 1 and F 2 as a function of x/l are plotted in Figure We refer the reader to Scandella & Paolucci (2006) for their exact expressions. The relation (2.17) was found to be satisfactory for several realistic cases, namely Thessaloniki (Greece) Catania (Italy) and Parkway Valley (New Zealand). Furthermore, comparing the results of eq. (2.17) with those obtained from numerical simulations on real case histories (see Scandella, 2006 for further details), used as a validation benchmark, it was found that V S,30, i.e. the average S- wave velocity in the uppermost 30 m, is a suitable approximation for V S to be used in (2.17). 25

42 Chapter 2. Transient seismic ground strains Figure 2.11 Simplified open basin model referred to by Scandella & Paolucci (2006) for a parametric numerical study of the PGS a -PGV relationship in the presence of strong lateral discontinuities. From Scandella & Paolucci (2006). Figure 2.12 Functions F 1 and F 2 in eq. (2.17) calibrated on 2D numerical analyses, for various dipping angles α. From Scandella & Paolucci (2006). 2.6 Spatial incoherency and soil heterogeneity effects It is worth underlying that eq. (2.1) and all techniques illustrated previously are expected to give only an lower bound of actual transient ground strains. Significant contributions, at low periods and small separation distances, may arise from spatial incoherence effects as well. In other words, nearly stochastic changes in amplitude and phase of ground motions, near-source effects and time arrival perturbations can strongly increase the amount of seismic ground strain, making these simplified strain estimates, based on a traveling wave model, unrealistic and uncertain within a factor of 2-3 (Bodin et al., 1997). Reasons of spatial variability are several (see e.g. Bard, 1995). The motion at a given site arises from different paths from the source and numerous scattered waves are diffracted by local heterogeneities whose wavelength is comparable to that of the dominant energy carried by seismic waves. The real case is therefore much more complex that the simple traveling wave assumption. The crustal media are, in fact, generally heterogeneous to various degrees, especially in the uppermost 26

43 Chapter 2. Transient seismic ground strains layers, enriching the complexity of the propagation path (such as multiple reflections, surface diffractions etc.) and even the seismic energy is commonly radiated by the fault with different azimuths, amplitudes and polarization. Leaving aside the details about the mathematical frame for the evaluation of the seismic spatial variability, we underline here that spatial coherency between the motions recorded at two stations i and k is expressed by means of the so-called coherency function γ ik ( ω). γ ik ( ω) is a complex number assuming values between 0 and 1 and it is a quantitative measure of the similarity in seismic wavefield at two discrete locations. For instance, if the recorded traces are hypothetically random white noise, i.e. uncorrelated realizations of a stochastic process, γ ( ω) ik 0. Several empirical coherence models have been proposed. These are commonly expressed as function of the circular frequency ω, separation distance ξ and, eventually, other site-specific parameters. Referring to Bard (1995), Abrahamson (1990; 1991) and Zerva & Zervas(2002) for a comprehensive overview, we present here some of the most relevant functional forms used to describe seismic spatial variability. It is expected that at low frequencies and short separation distances, the motions will be similar and, therefore, coherency will tend to unity as ω 0 and ξ 0 also. On the other hand, at large frequencies and long station separation distances, the motions will become uncorrelated, and coherency will tend to zero. In Figure 2.13 some of the developed expressions for spatial incoherency, based on SMART-1 data (Lotung, Taiwan), are shown as function of frequency and for ξ = 500 m. While the models of Harichandran & Vanmarcke (1986) and of Hao et al (1989) are rather close to each other over the range of frequency 0-10 rad/s, illustrated in Figure 2.13, the model developed by Luco & Wong (1986), which is probably the most quoted coherence model in literature (Zerva & Zervas, 2002), is characterized by higher values at relatively low frequencies (>~ 3/4 rad/s) and a faster decay at larger frequencies. Within this framework, Zerva (1992) studied carefully the contribution of coherency and wave propagation in seismic ground strains. For this purpose, the author used Luco and Wong s model (1986) 4 with two different values of the incoherence parameter α (α = sec/m and α = sec/m). Forward modeling of spatially incoherent fields, both in terms of velocity and strain, allowed her a direct comparison of actual strain time histories and simplified strain estimates, as they are currently used for design. These are, in fact, the ratio of particle velocity over a suitable measure of the apparent propagation velocity V a. Figure 2.14 presents the comparison of actual strain time histories and strain estimates for various values of the coherence drop parameters α and the apparent propagation velocity of motions. Figure 2.14A shows that, as intuitively expected, there is a good agreement between strain and strain estimates when seismic motions do not exhibit a significant loss of coherence (relatively small values of α parameter). That means that punctual strain can be accurately represented by simplified strain estimates, thus confirming eq. (2.1) in terms of peak values. 4 The expression of the frequently quoted spatial coherence model of Luco & Wong is: γ ω ξ α ω ξ coherent the seismic motions are) and ξ is the separation distance (, ) = exp where α (in s/m) is the incoherence parameter (the larger it is, the less ik 27

44 Chapter 2. Transient seismic ground strains The equivalence of actual strains and strain estimates for this value of α is due to the fact that strains are predominantly related to wave passage, since motions basically propagate unchanged on the ground surface. On the other hand, for spatially incoherent motions (Figure 2.14 B), that is frequent especially in near-source regions, maximum strain estimates PGV/V a are able to catch only half of the actual PGS. Consequently strain estimates provide unconservative results when incoherence effects are significant. Only if the V a is reduced from 1500 m/s to 500 m/s (Figure 2.14 B and C, respectively) strain estimate approaches actual strain achieving a satisfactory agreement. However it is apparent that V a ~ 500 m/s falls clearly in the range of surface wave rather than body wave propagation velocities. Another interesting observation concerning Zerva s work (1992) is the following. She pointed out that there is a critical apparent propagation velocity V a cr (which is a function of α), above which the maximum strain is nearly constant and mainly governed by incoherence of motions. On the other hand, provided V a < V a cr, maximum seismic strains tend to the inverse of apparent propagation velocity with decreasing values of the apparent propagation of motions. Figure 2.15 shows the variations of maximum strain as a function of V a for α = and s/m; the threshold associated to V a cr is superimposed in the aforementioned plot. The following comments arise from the analysis of Figure 2.15: i. the variations of maximum strain with V a show the same patterns, regardless of the degree of spatial incoherency of seismic motions. ii. For α = s/m, V a cr turns out to be ~ 1000 m/s, while for α = s/m, V a cr ~ 3000 m/s. iii. For V a > V a cr PGSs are essentially controlled by the incoherency of motions, remaining constant. They assume values significantly higher than the estimate PGV/V a. On the other hand, for V a < V a cr PGSs are controlled mainly by wave passage and increase with decreasing V a, approaching the estimate PGV/V a. iv. It is expected that wave effects, when V a < V a c and strain estimates generally apply satisfactorily, are more relevant in those regions where surface wave (with low propagation velocity) dominate rather than body waves (typically in near-source regions and with larger propagation velocity). Furthermore, it has been demonstrated by observational evidences (Nakajima et al., 1998) and empirical studies that local subsurface soil conditions may contribute to additional seismic strains: ground strains will be generally higher in soft soils than in stiff soils. This effect consists of variations of ground motion amplitudes within the separation distances of interest and, based on empirical attenuation relationships (see Paragraph 2.5), may give arise to around per unit of PGD (in cm). This means that for a transient peak ground displacement of 1 cm the part of strains related solely to site amplification effects is (Abrahamson, 2003). 28

45 Chapter 2. Transient seismic ground strains Figure 2.13 Examples of empirical spatial coherency models found in literature for separation distance ξ=500 m. The following parameter have been used: for Luco & Wong model, α= ; for Harichandran & Vanmarcke, A=0.736, a=0.147, b=2.78, k=5210 m, f 0 =1.09Hz, determined from data of event 20 at the SMART-1 array in Taiwan); for Hao et al., only longitudinal distances are considered ξ=ξ l and ξ t =0 (i.e. the stations are located along a line connecting the epicenter), β 1 = , a 1 = ,b 1 = and c 1 = , determined from event 30 at the SMART-1 array. Strain ε(t) Strain estimate ( v(t)/v a ) Figure 2.14 Comparison of actual strain time histories (ε) and strain estimate time histories (v/v a ) obtained from forward modeling simulation using the Luco & Wong s model. The following case studies are considered: A) α = sec/m and V a = 1500 m/s; B) α = sec/m and V a = 1500 m/s and C) α = sec/m and V a = 500 m/s. Adapted from Zerva (1992). 29

46 Chapter 2. Transient seismic ground strains Figure 2.15 Variation of PGS as a function of the apparent propagation velocity V a, ranging from 0 to (i.e. standing waves), for two different values of the incoherence parameters α of Luco and Wong s model: α = s/m and α = s/m. Note the increasing α, motions looses coherence with separation distance. Superimposed are the vertical lines indicating the critical apparent propagation velocity, V a cr, corresponding to the selected values of α. Adapted from Zerva (1992). 30

47 Chapter 3. Rotational ground motions: theory, observations and implications 3 ROTATIONAL SEISMIC GROUND MOTIONS: THEORY, OBSERVATIONS AND IMPLICATIONS 3.1 Introduction Perfectly general motion would also involve rotations about three perpendicular axes, and three more instruments for these. Theory indicates, and observation confirms, that such rotations are negligible (Richter, 1958). Richter s sentence embodies what the scientific and engineering community has been thinking for a long time about rotational motions induced by seismic waves. The reason for this is essentially double: on one side, rotational components of motion were considered to be small enough to have negligible effects on man-made structures and, on the other side, until quite recently, the related measuring devices were not sensitive enough. Earthquake-induced rotational motions can be divided into two components: tilt (alternatively called rocking by many authors) and torsion. The former denotes rotation about one of the horizontal axis (x- or y- axis), whilst the latter indicates the rotation about the vertical z-axis. From here on we will use this notation: ω x and ω y for tilt; ω z for torsion. Recent investigations have pointed out that tilting and torsional motions may be relevant close to the earthquake source (see e.g. the detailed review performed by Igel et al., 2005; Cochard et al., 2006) and for surface waves (Lee & Trifunac, 1985; 1987). Based on the theory of defects, Takeo & Ito (1997) demonstrated that abrupt changes of fault slip and/or tensile fracture can generate rotational seismic waves. There are few observational evidences of how significant rotations may be in near source regions (see e.g. Nigbor, 1994; Takeo, 1998 and Huang, 2006). These data show that rotations may be of one up to two orders of magnitude greater than expected from the classical linear theory. Furthermore, several studies, based partly on numerical simulations (Bouhcon & Aki, 1982; Castellani & Boffi, 1986) and partly on finite-differencing of dense seismic arrays (see e.g. Niazi, 1986; Oliviera & Bolt,1989; Singh et al., 1997; Huang, 2003; Ghayamghamian & Nouri, 2007 and the very recent work of Spudich et al., 2008) indicated that rotational components may be non-negligible near the fault plane. The rotational component of ground motions have been studied also theoretically, using, on one side, kinematic source models (Aki & Richards, 2002) and, on the other side, elastodynamics theory of plane wave propagation (Trifunac, 1982; Lee & Trifunac, 1985 and 1987). 31

48 Chapter 3. Rotational ground motions: theory, observations and implications The potential implications of rotational ground motions are basically the following: i. in seismology, they can provide useful constrains to correct the response of the seismometers, which is polluted by rotations, even when they are expected to be very small and, therefore, of negligible interest from an engineering point of view. As reviewed first by Graizer (2005 and 2006) and, subsequently, by Pillet & Virieux (2007), recovery of permanent displacement or long-period far-field wavefield requires an accurate estimate of rotations. Furthermore they can provide accurate data for arrival times of S waves, they might be a powerful indicator of the local velocity structure (e.g. the presence of alluvial basin as studied by Wang et al., 2006) and, in near field regions, they might provide further constrains for the source rupture process (Takeo & Ito, 1997). ii. From the engineering point of view, rotations may be responsible of damage in highrise building and in those structures where soil-structure interaction effects are significant. For recent reviews of the potential effects of rotational ground motions on structures we address the reader to Trifunac (2006) and Kalkan & Graizer (2007). However, these potential benefits in seismology and their importance in engineering practice are still under discussion. The addressed questions, derived basically by the lack of quantitative and consolidated measures, are the following (see e.g. Nigbor, 1994): a) what are the magnitudes of rotations for a realistic earthquake scenario both in near- and far- field? And are they relevant in terms of earthquake damage potential? b) What is the order of errors in evaluating displacement, velocity and accelerations due to the neglected rotational components of motion? This section is organized in the following way. First, the fundamental theory for rotational components of earthquake-induced ground motions is illustrated; second, the main studies on the possible effects of rotation on inertial sensor and building response are highlighted and, finally, the main approaches and observations undertaken so far are summarized and briefly discussed. 3.2 Theoretical considerations Rotational components of motions under incident plane waves Similar considerations to Newmark s development for estimating strains (see Paragraph 2.2 in the previous chapter) can be done for the rotational components of motions. Based on the elastic wave theory, first Trifunac (1982) and, subsequently, Lee and Trifunac (1985; 1987) provided analytical solutions for estimating rocking and torsional components of motion at the ground surface of an ideal half-space associated with harmonic plane P-SV, SH, Rayleigh and Love waves. They also developed a method from computing synthetic torsion and rocking accelerograms from the corresponding translational components. The result of 32

49 Chapter 3. Rotational ground motions: theory, observations and implications their work is summarized in Table 3.1. The coordinate system they refer to in deriving these solutions is depicted in Figure 3.1. Table 3.1 Closed-form solutions for tilts and torsion under the incidence of plane waves derived by Lee and Trifunac (1985 and 1987). ω y indicates tilt abut the horizontal y-axis while ω z indicates torsion, i.e. rotation about the z- axis. The following notation is used: the displacement field is u = [u, v, w] t ; A S i(r) amplitude of the incident (reflected) S wave; γ incidence angle; k α(β) = ω/α(β) wavenumber associated with P- (S-) waves; k = ω/c horizontal wavenumber and c phase velocity of Rayleigh or Love waves. Wave Amplitude of Non-vanishing rotations type rotation (*) P ω y = i r [ i ( k x sin t )] ω α γ ω y β ω A S k β e 2 = w α Va SV ikβ i r [ i( kβ x sin γ ωt )] SV ω y = ωy ω ( AS + AS ) e 2 = w Va i [ i( kβ x sinγ ωt )] ωz 1 ω SH ωz = AS ikβ sinγe = v 2 Va ω R ω y = ω y ω ikw( x,z,t ) = i w( x,z,t ) C = w Va i i ω Love ω z = ωz 1 ω kv( x,z,t ) = i v( x,z,t ) 2 2 C = v 2 Va (*) normalized with respect to the amplitude of displacement PGV / PGR V a α β V a 2 V a V a 2 V a Figure 3.1 Coordinate system and notation for the analytical derivations summarized in Table 3.1. ω y represents rocking (or tilting) while ω z represents torsional motion. 33

50 Chapter 3. Rotational ground motions: theory, observations and implications Note that expressions for ω y and ω z both show a linear dependence on the circular frequency ω, through the wavenumber k. Furthermore, as intuitively expected, greater rotations are associated with S- and surface (R-) waves. More generally, based on Table 3.1, rocking and torsional components at ground surface of an ideal half-space induced by harmonic plane waves (P, SV, R and Love waves) can be written as follows: ω 1 u j ω i = A u j = A (3.1) V V t a a with y j = if z i = z i = y and β / α A = 1 for 1 / 2 P SV, R SH,Love wave incidence. Note that the relation between rocking ω y and velocity in the case of P, SV and Rayleigh waves is a consequence of fundamental principles of elastodynamics. Introducing the freesurface condition, i.e. τ = 0 at z = 0, into the equation of motion, we can easily derive: such that: xz u w + z x u = 0 w = x z = 0 z= 0 z z= 0 z = 0 z= 0 z z= 0 (3.2) 1 w u u w ω y = curl[ u( x,t )] = = = (3.3) 2 x z x Therefore for a harmonically propagating plane wave of arbitrary amplitude, peak values of rocking motion at ground surface are given by: ω ( x,z = 0,t ) = ikw( x,z =,t ) (3.4) y 0 Considering the maximum values of (3.4), and knowing the horizontal (apparent) velocity of propagation allows one to have a first-order approximation estimation of the peak rocking values (PGR y ) from the measurements of the transversal velocity field and its peaks (PGV h ): PGR y 1 = PGVz (3.5) V a The same considerations as for rocking motions apply to torsional components leading to: PGR z 1 = PGVh (3.6) 2V a Note the obvious similarity of eq. (3.5) and (3.6) with eq. (2.1) of Chapter 2, except the factor of 2 at the denominator of eq. (3.6). Analytical considerations suggest, therefore, that the ratio of peak ground strain over peak ground torsion, PGS/PGR z, is equal to 2, provided that PGV is measured along the same direction. The order of magnitude of rotations and strains is hence 34

51 Chapter 3. Rotational ground motions: theory, observations and implications expected to be the same, at least under the assumption of propagating plane waves in homogenous media. As underlined in the previous section, the applicability of eqq. (3.5) and (3.6) is obviously limited by the fulfillment of the same approximations of homogenous plane wave propagation and, when it applies, by the accuracy of the estimates of the prevailing wave type and the apparent velocity Rotations due to a double-couple point source Aki & Richards (2002) give expressions for the displacement field u ( x;t ) generated by a double-couple point source (i.e. point source shear dislocation) in an unbounded, homogeneous, isotropic and elastic medium. We refer to the geometry and notations depicted in Figure 3.2: the dislocation is in the x-y plane with slip along x. Applying the curl operator to the given expressions of u ( x;t ) it is possible to demonstrate that rotations are given by: R A 3 3 M 0 1 ω( x;t ) = M 0( t r / β ) + ( t r / β ) + 8πρ β r β r t 4 β r R where A is the rotational radiation pattern defined as: 2 M 0 2 t ( t r / β ) (3.7) A R = cosθ sinφθˆ + cosφ cos 2θ ˆ φ (3.8) As intuitively expected, rotations are zero at the arrival of P- wavefront, starting only at the S- wave arrival, even in the near-field region. Note, moreover, that eq. (3.7) includes terms proportional to 1/r 3 (near-field terms), to 1/r 2 (intermediate field) and 1/r (far-field). Figure 3.3A depicts the non-vanishing components of rotations (rotation filed has no radial components) as a function of time at a given point in the near-field region (R = 5 km) of an infinite, homogenous and isotropic medium (mass density ρ = 2800 kg/m 3, shear wave velocity β = 3 km/s, Poisson s ratio ν = 0.25) due to a double-couple point source with M 0 ~ Nm, i.e. M W = 5.0. The observation points is located at (R,θ,φ) = (5 km,π/8, π /8), in spherical coordinates and the source time function rate (illustrated in Figure 3.3B is given by 2 2 Gaussian dependence of the form f ( t ) = exp( t / T ) where T is considered to be about 1/4 the rise time, τ R. To estimate τ R, for a given magnitude, the scaling relation developed by Geller (1976) 1 has been used. The maximum displacement amplitude of f(t) is set equal to the selected M 0. Note that for M W = 5 the peak ground rotation at R = 5 km is about Finally, Figure 3.4A displays with filled dots the peak values of rotation (PGR, in rad) as a function of M W (in semi-log scale), at R = 5 km, obtained by applying the closed-form solution of eqs. (3.7) and (3.8), considering the same model of Figure 3.3. Note, for instance, that, passing from M = 5 to M = 6, PGR increases of about 1 order of magnitude, namely from ~ to ~ The logarithm of PGR is reasonably described by a straight line of 1 Log 10τ R = M S / 3 if M S < ; Log 10τ R = M S / 2 if M S For simplicity, M S is considered equal to M W. 35

52 Chapter 3. Rotational ground motions: theory, observations and implications expression: Log 10 PGR ~ 0.89M W 8.9 (superimposed in the graph). For comparison, in Figure 3.4 B PGRs vs. M W for R = 50 km (filled triangles), representative of far-field conditions, are shown together with the results for R = 5 km (filled dots, as in Figure 3.4A). In the former case the straight line connecting the logarithms of peak rotations with the magnitude turns out to be: Log 10 PGR = 0.82M W 9.6. A slight decrease of the coefficient of proportionality between Log 10 PGR and M W and a rather consistent increase of the corresponding intercept are observed, suggesting that, for large magnitudes, the difference in PGR between near and far field conditions might acquire a certain relevance. For instance, for M5 earthquake, passing from R = 5 to R = 50 km, PGR decreases by a factor of around 10, while, for M=7, it drops by a factor of about 17. These analytical considerations give some preliminary indications that rotational components may reasonably reach large values in the near-field regions for realistic earthquake scenarios (M >= 6 and extended source with variable slip distribution). Figure 3.2 Geometry and notation used to derive rotations for a double-couple point source. A) B) Figure 3.3 A) Non-vanishing rotation component ω θ and ω φ as a function of time at a given point in an infinite ideal medium due to a double-couple point source with M W = 5, as given by eq. (3.7) and (3.8). The observation point is located at (5 km, π/8, π/8) in spherical coordinates and the source time function rare is given by a Gaussian ~ exp(-t 2 /T 2 ) with T ~ 0.1 sec (being the rise time τ R ~ 0.4 sec) and maximum amplitude equal to the selected M 0 (recall relation M W = 36

53 Chapter 3. Rotational ground motions: theory, observations and implications 2/3Log 10 M 0-6). The properties of the medium are: ρ = 2800 kg/m 3, β = 3 km/s and ν = 0.25 (Poisson s solid). Log 10 PGR = 0.89*M W -8.9 A) Log 10 PGR = 0.89*M W -8.9 Log 10 PGR = 0.82*M W -9.6 B) Figure 3.4 A) Peak Ground Rotation (PGR, in rad) as a function of M W calculated at R = 5 km by applying the closed-form solution of eqs. (3.7) and (3.8). The same model as in Figure 3.3 is considered. B) Comparison of the PGRs obtained at R=5 km (filled dots) with those at R=50km (filled triangles). Superimposed are the straight lines connecting the logarithm of peak rotations and moment magnitude. 37

54 Chapter 3. Rotational ground motions: theory, observations and implications 3.3 Effects of rotations on the response of inertial sensors One of the classical challenges in strong motion record processing is the recovery of permanent displacement of ground or structures associated to faulting or to non-linear response. As carefully reviewed by Trifunac & Todorovska (2001), despite the remarkable increase of dynamic range and of resolution of strong motion digital instruments in the last 40 years (up to about 140 db nowadays) the determination of zero-frequency static displacements is still a crucial issue. There are in fact physical constraints in the definition of the zero baseline and, similarly, in the correct identification of the forcing function. The most accurate representation of this includes, in fact, not only translational terms, which are routinely considered in modeling the dynamic behavior of strong motion transducers, but also terms proportional to torsion and tilt of the instrument. Therefore the determination of static displacements can not avoid the accurate estimation of all the 6 components of ground motion, 3 translations and 3 rotations (see e.g. Graizer, 1989 and 2005; Trifunac & Todorovska, 2001 and Pillet & Virieux, 2007 for through presentations of the problem). To underline this point, we report herein the differential equation of small oscillations of a common horizontal pendulum motion (i.e. to model the dynamic behavior of a 3-transducers SMA-1 accelerograph, as schematically illustrated in Figure 3.5): y 1 2ω1ζ y1+ ω1 y1 = x1 gφ2 φ 3 r1 + x 2 α1 + (3.9) where y i = instrument displacement response; α i = angle of pendulum rotation; r i = length of pendulum arm; ω i = natural frequency; ζ i = fraction of the critical damping of the i th.. transducer (direction 1 corresponds to the longitudinal transducer L of Figure 3.5); x 1 = ground acceleration in the i th direction; φ i = rotation of the ground surface about the x i axis and g = gravitational acceleration. The second and third terms of the right-hand-side of eq. (3.9) represent, respectively, contributions from tilting and from angular acceleration to the instrument response. The last term represents, instead, the contribution from cross-axis sensitivity. In order to compare the spectral behavior of translational and rotational motions, Figure 3.6 illustrates the smoothed Fourier spectra of the driving forces of eq. (3.9), as calculated by Trifunac and Todorovska (2001). The authors compute the rotational spectra from the corresponding Fourier spectra of velocity owing to relationship (3.1) transferred to the frequency domain. φ (ω ) is computed from the Fourier spectrum of translation assuming the following values of phase velocity: c 3, 3, 3, 1, 0.3 and 0.1 km/s.. at frequencies f = 0.001, 0.01, 0.1, 1, 10 and 100 Hz. The spectra for rotational acceleration φ ( ω ) are then calculated 2 by multiplying φ (ω ) by ω. We note that.. for M>4 the spectra φ (ω ) approach a constant for ω 0, while the contributions of φ ( ω ) are significant only at high frequencies.. with maximum values at around 10 Hz. Furthermore the spectral amplitudes of φ ( ω ) are larger than those of φ (ω ) only for f > 10 Hz. According to the authors the plateau of φ (ω ) at low frequencies may be reasonably explained by rotational components responsible of residual displacements. Similar conclusions are done by Graizer (1989; 2005) and Pillet & Virieux (2007). These authors underlined that tilt contributions may be noticeable at very long periods. 38

55 Chapter 3. Rotational ground motions: theory, observations and implications Note that seismic recordings are the combination of the three contributions depicted in Figure 3.7. Therefore, while the effects of angular acceleration and cross-axis sensitivity are commonly negligible for modern accelerometers (they may be considerable for long-arm pendulum with natural frequencies of around 25 Hz), the tilting terms can not be neglected and control the low frequency part of pendulum response. In conclusion, in a typical triaxial instrument, the response of two horizontal sensors (L and T of Figure 3.5) is the sum of two simultaneous contributions: horizontal translation and tilt. That means that double-integration of eq. (3.9), when the last two terms in the right-hand-side are neglected, would yield displacement and double-integrated rotations. An example is given in Figure 3.8: Pillet & Virieux (2007) argued that the slopes of horizontal velocities (in B), calculated by single integration from the acceleration recordings during the Chi-Chi M earthquake, are measure of tilt. From these slopes, of ~ 4.08 cm/s 2 and 3.44 cm/s 2 for the NS and EW components, respectively, the authors found tilt of ~5.44 mrad for N40 azimuth. Figure 3.5 Sketch of three transducers in SMA-1 accelerograph with the corresponding oscillation directions. Angles α i describe the deflection of the i th pendulum. From Trifunac & Todorovska (2001). 39

56 Chapter 3. Rotational ground motions: theory, observations and implications Figure 3.6 Fourier amplitude spectra of ground acceleration {d 2 X i /dt 2 }, normalized rocking {φ i g}, and normalized torsional acceleration {d 2 φ i /dt 2 g} at a site on basement rock for earthquake magnitudes M =4, 5, 6 and 7, source at depth H = 0 (surface faulting) and epicentral distance R = 10 km. The small shaded areas denote the limiting constant values of φ(ω) for ω 0. From Trifunac & Todorovska (2001). Figure 3.7 Log-Log frequency response of rotations and translations for a horizontal pendulum. Seismic recordings are the combination of these three factors. From Pillet & Virieux (2007). 40

57 Chapter 3. Rotational ground motions: theory, observations and implications A B C Figure 3.8 Chi-Chi M7.6 earthquake (Taiwan), recorded at the TCU06 accelerometric station. A) recorded accelerations signals; B) velocity time histories obtained by integration of A): according to Pillet & & Virieux (2007) the slope of uncorrected velocity time historiy is a measure of the static part of tilt input motion; C) corrected displacement traces after removing the spurios tilt terms. From Pillet & Virieux (2007). 3.4 Effects of rotations on building response There are some evidences of structural damage caused by rotational ground motion. It is worth separating a torsional response due to the asymmetry in the structure with a torsional response directly associated with earthquake-induced ground shaking. This summary mainly focuses on this latter aspect. Torsional response of tall buildings in Los Angeles, during the San Fernando earthquake in 1971 could be attributed to torsional input excitations (Hart et al., 1975), while rotational differential motions may have been responsible of the collapse of bridges during the San Fernando (1971), Miyagi-ken-Oki (1978) and Northridge (1994) earthquakes (Ghayamghamia & Nouri, 2007). Newmark (1969) was the first to show the possible effects of rotational components of input motion on buildings. According to Stratta and Griswold (1976) a relative horizontal rotation (rocking) of about rad between adjacent columns supporting a rigid slab, could cause a spread footing to fail at its base. Later, Gupta & Trifunac (1987b; 1989; 1990b and 1991) studied simple models subjected to both torsional and rocking excitations for various combinations of soil and structural conditions. An interesting result is that rocking excitation gets significant for tall structures supported by soft soil deposits, while torsional excitations can dominate in the response of long and stiff structures supported by soft soils. Other studies showed that the separation of the effects of rocking excitation and the rocking 41

58 Chapter 3. Rotational ground motions: theory, observations and implications associated with soil-structure interaction are essential for interpretation of the observed interstory drifts in full-scale structures (e.g., Todorovska & Trifunac, 1992a,b; 1993). Analyses of building response highlight some critical issues. Three representative examples, found in literature, are given herein: i. For a well-documented 14-story reinforced concrete structure in Los Angeles (Hollywood Storage building), the asymmetry of the foundation and strong torsional excitation carried mainly by surface waves propagating along the longitudinal axis of the building caused large torsional response (Trifunac et al., 2001a). ii. Based on strong motion records at the Pacoima Dam-upper left abutment during the 1994 Northridge earthquake, where a residual tilt of about 3.1 in the N40 direction was measured, Kalkan and Graizer (2006) showed that tilt component has significant influence on structural response, when the latter is modeled as a single degree-offreedom (dof) subjected to coupled tilt and translational motion. Coupling of these driving forces results in a significant increase (up to a factor of 2) of the residual displacement, as illustrated in Figure 3.9. Moreover when tilt motion is applied together with significant translations, secondary P- effects may lead to dynamic instability up to collapse of the structure. All these effects are even more pronounced when non linear soil effects are likely to interact. For these reasons, according to Trifunac and his collaborators, a proper and accurate identification of these contributions may be crucial: without proper consideration of these contributions the observed drifts may be erroneously assumed to result completely from relative displacement of structures, and this can lead to false confidence that the current design methods are conservative (IWGoRS, 2006, Menlo Park, California). A few attempts to estimate rotational response spectra based either on array observations or numerical simulations have been done so far (Lee and Trifunac, 1985 and 1987; Castellani & Boffi, 1986; Castellani & Zembaty, 1996). In Boffi & Castellani (1985) the rotational response spectrum is defined as the ratio of the maximum reaction at the base and the moment of inertia of an oscillator with a single rotational degree of freedom with natural period T and with base excitation with time dependence given by d 2 φ g (t)/dt 2 (i.e. ground tilt acceleration). In other words, rotational response spectra are the maximum response, φ max, of single rotational dof systems of specified natural periods and dampings when subjected to the torsional accelerograms as input. As illustrative example Figure 3.10A depicts the torsional response spectra, in terms of displacement SD (in rad), relative pseudo velocity PSV (rad/sec) and relative pseudo acceleration PSA (rad/sec 2 ), in the period range from 0.04 s to 15 sec with 5 damping ratio values when subject to corresponding torsional accelerations, represented in Figure 3.10B. These are synthetic accelerograms derived from the corresponding translations, under the assumption of harmonic plane wave incidence in elastic halfspace, for M=6.5 earthquake at epicentral distance R = 10 km and on soft site conditions. The authors pointed out that the ratio between torsion (rocking) and horizontal (vertical) response is large at high frequencies and decrease almost linearly in the low frequency range. This linear trend is well approximated by the ratio of rotational to translational Fourier spectrum, owing to eq. (3.1). 42

59 Chapter 3. Rotational ground motions: theory, observations and implications Castellani & Zembaty (1996) developed further ratios between rotational and translational spectrum, on the basis of strong motion array data. The principal results are the following: i) when referred to near-field conditions the ratio can be of one order of magnitude higher than in far-field conditions (see Figure 3.11); ii) in the same conditions, the ratios are comparable with the orders of magnitude prescribed by Eurocode 8 Part II (CEN, 2002), at least for soft soil conditions. In this framework, note that Eurocode 8 is probably the only code worldwide that state explicitly that translation and rocking excitations could be coupled for the design of long structures, such as bridges, and tall slender towers. Specifically, based on the assumption of monochromatic plane waves with constant shape and velocity in homogenous media, Eurocode 8 prescribes the following rotational response spectra about the i th axis: ϕ ϕ ϕ S i (T ) = ωs i (T ) / c, with c = S-wave velocity, ω = 2π / T circular frequency and S i (T ) being the site-dependent response spectra, in terms of displacement, along direction i. Note that the expression above assumes that rotational response spectra can be derived by the v d corresponding velocity horizontal response spectra S i (T ) = ωs i (T ), scaled by the inverse of the S- wave velocity. It basically derives from a straightforward manipulation of eq. (3.1) and, therefore, it suffers from the same drawbacks highlighted previously. Finally, it is worth underlining that the amount of rotational excitation likely to occur for a given earthquake scenario is not obvious. Figure 3.9 Comparison of single degree-of-freedom (dof) oscillator response in terms of displacement (in c)) due to pure translational input motion (reported in a)) and coupled tilt-translational motion (tilting input motion used in the analysis is illustrated in b)). From Kalkan & Graizer (2007). 43

60 Chapter 3. Rotational ground motions: theory, observations and implications A) Response Spectra SD, PSV, PSA Fourier Spectrum ~ B) ~ ~ Figure 3.10 A) Torsional response (in terms of rotational displacement SD, pseudo-velocity PSV and pseudo-acceleration PSA) and Fourier spectra (dashed line) for the synthetics illustrated in B). 5 values of damping ratios and periods ranging from 0.04 s to 15 s are considered. The following earthquake parameters are considered: M = 6.5, R = 10 km and soft site conditions. From Lee & Trifunac (1985). 44

61 Chapter 3. Rotational ground motions: theory, observations and implications Figure 3.11 Ratio of rotational to translational response spectrum for damping ratio ζ=5% adapted from Lee & Trifunac(1985). From Castellani & Zembaty (1996). 3.5 Summary of observations As discussed in the previous chapter, the techniques for estimating rotational ground motions are basically the same as those used for the determination of strains (array-derived and numerical simulations), owing to their intrinsic similarities Provided the substantial lack of direct measurements of rotations and the limited acquaintance with earthquake-induced rotational ground motions, we have considered interesting to synthesize in a comprehensive way most of the available data in literature. Table 3.2 presents a summary of most of the available data in literature in terms of peak ground torsion (denoted with PGR z ) and peak torsion rate (PGRR z ). Additional information concerns the data type (measured, derived from array recordings or numerical), the source parameters (magnitude M, epicentral distance R and source mechanism) and type of soil (a very simplified classification was assumed between soft and stiff soil, considering Vs 30 = 300 m/s as threshold to roughly distinguish between soft and stiff conditions). Two graphical representations of the data listed in Table 3.2 are given in Figure 3.12 and Figure 3.13: in the former, the PGV h -PGR z pairs are plotted, highlighting the different nature of the estimates (either experimental, numerical/analytical or direct measurements), while in the latter PGR z are plotted as a function of distance R and for 4 different class of magnitudes (M<5, 5<=M<6, 6<=M<6, M>=7). It is apparent the scarcity of direct observations. As far as available from published data, only three direct measurements of ground rotations have been reported: 1) and 2) 1997 Ito events in Taiwan, studied by Takeo (1998); 3) measurement of an explosive source by Nigbor (1994). Only the first one is interesting for earthquake engineering purposes. The measured values of rotation (or rotation rate if the former is not available) seem to disagree with the experimental/numerical ones, being larger of at least one order of magnitude. This has been attributed by many authors to the possible complexity and spatial variability of near-fault 45

62 Chapter 3. Rotational ground motions: theory, observations and implications ground motion, complexity which is difficult captured by numerical simulations and arrayderived observations (see e.g. Huang, 2003; Fletcher & Spudich, 2007 for a careful discussion about the inconsistencies between synthetic and observed values of ground seismic rotations). These methods, though reliable, suffer from band limitations in the frequency domain. Let us consider a few comments about the data summarized in Table 3.2: i. in the pioneering work of Bouchon & Aki (1982), strains, tilts and rotations are simulated in the vicinity (R~1 km) of both a strike-slip (SS) and thrust (T) fault generating M~6 through a semi-analytical method. As illustrated in Figure 3.14 and Table 3.3 they found that the maximum rotational motions induced by a buried 30-kmlong SS fault are: PGR rad and corresponding velocity PGRR rad/sec. For comparison, also the maximum strains are reported. Note that large rotational components are restricted to the zone around the fault and decay rapidly in the direction transverse to the fault. Figure 3.15, adapted from Oliveira & Bolt (1989) underlines the rapid decay of rotation with distance from fault supporting the suggestion that their significance from an engineering point of view may be limited to near-fault regions. Furthermore the authors found a good correlation between strain/rotational ground motion and velocity time histories. The vertical and horizontal apparent phase velocities, which turn out to be the scaling factor between ground velocities and shear, rocking and torsional motion, respectively, are close to 2 km/s, suggesting that they are controlled by rupture velocity and basement rock shear velocity rather than by near-surface velocity. ii. At ground surface the gradients corresponding to deformations across vertical planes seem to dominate at least by a factor of 2 (Bodin et al., 1997). This is the reason why torsional components are considered to be of greater amplitude than the corresponding rocking components. In this framework Takeo (1998) points out that tilts seem to be dominated by relatively higher frequency than torsions and ω z > ω x, ω y (he observes that the frequency content of tilts is slightly smaller than the Nyquist frequency of the recording system = 10 Hz, making a quantitative and accurate estimation of tilts rather difficult). iii. There are few investigations about the variability of rotational motions with depth. Singh et al. (1997) and Bodin et al. (1997) have noted that vertical gradient displacements tend to depend significantly on depth variations (especially within the depth range 0<z<30 m) reaching peak values greater than those observed at ground surface at least by a factor of 10. Moderate (M~6.6) and large (M 7.5) events tend to generate, respectively, peak vertical gradients of ~ and ~ at depth between 0 and 30 m. Therefore according to their investigations surface differential displacements underestimate what happens at shallow depths. Trying to summarize the results available from published data, seismic ground rotations are likely to be considerable in near-source regions (R <~20 km) and for magnitude greater than 6.0. In such conditions they might assume values of ~ 10-4 to

63 Chapter 3. Rotational ground motions: theory, observations and implications Table 3.2 List of available data in literature: peak values of ground velocity (PGV h ), ground torsion (PGR z ) and torsion rate (PGRR z ). Additional information concern data type, the source parameters (magnitude, epicentral distance R and source mechanism) and type of soil. Data from Spudich & Fletcher (2008) correspond to broadband estimates (see Chapter 6 for further details). The legend is given below. Reference # Bouchon & Aki (1982) Lee & Trifunac (1985) Niazi (1987) Oliviera & Bolt (1989) Castellani & Boffi (1989) Nigbor (1994) Bodin et al.; Singh et al. (1997) Data type M w EQ. parameters R [km] Source mech. Type of soil PGV h [m/s] PGR z [rad] SS SS N.A SS N.A N.A N.A N.A N.A SS kton 1 Expl R R Takeo SS N.A. (1998) SS N.A. Huang (2003) Spudich & Fletcher (2008) T SS SS SS SS LEGEND PGV h = Peak Ground horizontal Velocity calculated as PGV = max 2 2 V ( t ) + V ( t ) PGR z = Peak Ground Rotation about the vertical axis (i.e. peak torsion) R = distance from the fault (note that the values listed above refer to different definitions) Data type: 1 = array-derived. h t x y 47

64 Chapter 3. Rotational ground motions: theory, observations and implications 2 = numerical/semi-analytical 3 = measured Source Mechanism: SS = strike-slip T = thrust R = reverse Soil Type: Rough distinction between soft and stiff soil where basing on V S,30 : soft if V S,30 < 300 m/s 1 = stiff 2 = soft Figure 3.12 PGV h vs. PGR z retrieved from literature (see Table 3.2 for further details). Figure 3.13 PGV h vs. distance R retrieved from literature (see Table 3.2 for further details). Data with R>100 km have been omitted. 48

65 Chapter 3. Rotational ground motions: theory, observations and implications Figure 3.14 Rotational field (see also Table 3.3) produced by the buried 30-km-long strike-slip fault at the site locations illustrated below. See also Figure 2.8 in the previous chapter for further details about the source process. Table 3.3 Summary of results obtained by Bouchon and Aki (1982), in terms of longitudinal maximum strain (PGS a ) maximum tilt and torsion (PGR) and maximum rotation rate (PGRR). An illustrative example of the synthetic rotational time histories obtained by the authors is given by Figure Source mechanism M 0 [Nm] PGS a (-) Strike-slip (1966 Parkfield EQ.) Thrust (San Fernando EQ.) PGR [rad] torsion tilt PGRR [rad/sec] N.A. N.A. Figure 3.15 PGR as function of distance obtained by Bouchon & Aki (1982) through semi-analytical simulations of a Strike Slip (SS) fault of unitary slip and magnitude M W = 6. From Oliveira & Bolt (1989). 49

66 Chapter 4. Spatial interpolation of displacement records from dense seismic networks 4 SPATIAL INTERPOLATION OF DISPLACEMENT RECORDS FROM DENSE SEISMIC NETWORKS In this section an empirical procedure for evaluating transient ground strains from closely spaced seismic networks is illustrated, based on a suitable spatial interpolation technique. To this end, two arrays have been considered thanks to their relatively dense spacing: the Parkway Valley digital temporary array, located in New Zealand, and the UPSAR array, in California. Particular attention is devoted to the selection and validation of a suitable interpolation technique to obtain accurate estimates of the three-component displacement wavefield at the ground surface. 4.1 Selection of records from dense seismic networks: Parkway Valley, New Zealand, and UPSAR, California For the purpose of this work, outlined in the previous Section, we have considered two seismic networks, characterized by remarkably close spacing between adjacent stations: the Parkway Valley (New Zealand) and UPSAR (California) arrays. Below are summarized the main features of these networks The Parkway Valley (New Zealand) temporary network In this section we consider the weak motion records of a dense seismograph network temporally installed in Parkway Valley, Wainuiomata, New Zealand. This is a shallow alluvial valley that has already been the object of several detailed studies on the description of seismic ground motion and the quantification of complex site effects, partly based on the analysis of the available weak motion records (see e.g. Chávez-García et al., 1999 and 2002; Stephenson, 2000 and 2007; Paolucci et al., 2000) and partly derived from numerical modelling and simulations (see e.g.. Chávez-García, 2003; Paolucci & Faccioli, 2003). A detailed analysis was also carried out by Yu et al. (2003) in order to evaluate the proper criteria for selection of rock reference stations for site amplification studies. The valley of Parkway is located in the Northern Island of New Zealand, at short distance from Wellington (see Figure 4.1). It is included in highly indurated graywacke of the Torlesse conglomerate formation and filled with variable layers of mixed muds, gravels and sands, as pinted out in Table 4.1. It is not a true closed basin, being open to South; the length of the basin, considered as the extension of experimental area, is around 1.3 km, whereas its width is of the order of 0.4 km. 50

67 Chapter 4. Spatial interpolation of displacement records from dense seismic networks In order to capture details of the surface displacement wavefield and of its spatial variability, Parkway Valley was instrumented with a dense digital temporary array operating from 1 Aug to 12 Oct, The network consisted of 24 autonomous stations equipped with 1-Hz velocity meters coupled with three-component digital EARSS seismometer (Gledhill et al., 1991). Of the 24 stations, four stations (from 22 to 25) were located on the weathered greywacke surrounding the basin, while the other 19 were installed on the soft alluvial sediments filling the basin. The raw velocity signals were passed through a 30 Hz, 18 db/octave anti-aliasing filter and digitized at 100 samples per second. The time traces were recorded independently by each receiver from the official time broadcast in New Zealand, with a maximun error of 10 msec in absolute time (Chávez-García, 1999). The receivers were installed at noticeable short distances from each other with a relative separation distances within the basin ranging approximately from 23 to 354 m and average spacing of about 40 m. This is the most attractive features of Parkway Valley for the purpose of our work. The network recorded both regional and local earthquakes, providing a high-quality data set with a surprising high signal-to-noise ratio (Yu, 1996). Among the six events, made available to us by W. R. Stephenson, the ones characterized by the largest number of recording stations (17 operating receivers in both cases) have been selected (denoted with Parkway #1 and #2 in Table 4.2). The spatial distribution of the corresponding epicentres is illustrated in Figure 4.2. These are weak motion events, since the maximum local magnitude is 4.9, inducing PGVs that reach the order of mm/s for the strongest event (briefly called event 1 or, alternatively, Parkway #1). Figure 4.2 (left) shows the layout of Parkway valley array: only the stations which have been effectively used for the spatial interpolation procedure, illustrated in the following section, are shown. Further details about the temporary digital network of Parkway Valley and the available datasets can be found in Smerzini et al. (2006). Figure 4.1 Location of Parkway Valley. From Stephenson (2007). 51

68 Chapter 4. Spatial interpolation of displacement records from dense seismic networks Figure 4.2 Left: location of recording stations of the temporary Parkway Valley array, used for this study; the box denotes the area considered for the spatial interpolation procedure, illustrated in the following section. Right: relative location of Parkway Valley and the selected seismic events (event #1 and #2). Table 4.1 Geological model of soil column at Parkway Valley array, made available to us by W.R. Stephenson (personal communication, 2001 & 2006). Soil layers are reported from up to bottom. Soil material V P [m/sec] V S [m/sec] Density [kg/m3] Thickness [m] swamp swamp swamp alluvium deposits ex-lake deposits old alluvium rock basement half-space The UPSAR (California) array The second case study refers to the U.S. Geological Survey Dense Seismograph Array (UPSAR), a 14-station seismic array located at the southern end ( N and W) of the 40-km-long section of the San Andreas fault that broke in the 1966 Parkfield earthquake, California (see right map in Figure 4.3). It is a dense array consisting of 14 irregularly spaced seismograph stations with an aperture of around 1 km, installed and designed in the late 1980s in order to optimally portray the rupture front of the anticipated Parkfield earthquake (Fletcher et al., 1992; Fletcher et al., 2006). The stations are deployed over an area of approximately 0.45 km 2. Each station consists of a three-component L-22 velocity transducer with a natural frequency of 2 Hz, coupled with three-component accelerometers with a sampling frequency f s =200 Hz. Even in this case, 52

69 Chapter 4. Spatial interpolation of displacement records from dense seismic networks synchronization was carefully checked across the whole array, enabling the use of relative differences. The inter-station array spacing of the UPSAR is not as ideal for this study as that of Parkway Valley, since the separation distance between two adjacent sensors is significantly larger, ranging from 25 up to 960 m, with an average value of around 136 m. From the geological point of view, there are also remarkable differences between the two array settings. Contrary to the 2D/3D geological configuration of Parkway basin, the UPSAR array is located in more uniform and horizontally embedded geological environment. The rock basement is estimated to lie at around 1.5 km depth, while the Paso Robles formation outcrops, exhibiting an average S- wave velocity of ~ 400 m/s and an average P- wave velocity of ~ 950 m/s in the top 60 meters. A schematic cross-section passing through the array location is illustrated in Figure 4.4. These velocities are, on average, larger than those at Parkway Valley, suggesting stiffer local site conditions and, therefore, lower amplification factors. Besides, the UPSAR ground motion records might be affected by moderate topographic effects (Wang et al., 2006; Fletcher et al., 2006), since the stations are situated over hilltops, with a 43-m elevation difference between the lowest station (01) and the highest one (08), as illustrated in Figure 4.5. In view of the application of the interpolation procedure, described in the sequel, we considered the ground motions detected by the UPSAR during both the December 22, 2003, San Simeon M W 6.5, and the September 28, 2004, Parkfield M W 6.0 earthquakes, with epicentral distances R = 65 km and R = 11.6 km, respectively. Records of Parkfield are available at the web site while records for the San Simeon earthquake were kindly provided by Wang G.Q. For the two selected events 11 station recordings are available, in terms of acceleration time histories (see Table 4.2 for further details). Figure 4.3 Left: location of recording stations of the UPSAR; right: map including the UPSAR and the epicentres of the M W 6 Parkfield and M W 6.5 San Simeon earthquakes. From Wang et al. (2006). 53

70 Chapter 4. Spatial interpolation of displacement records from dense seismic networks Figure 4.4 Example of cross-section (b), shown in (a), passing through the array site. Tsm=Santa Margherita formation (upper Miocene), Tm=Monterey formation (middle Miocene), Tv=volcanic rocks of Lang Canyon (lower Miocene), Tt=Temblor formation (lower Miocene), KJf=Franciscan assemblage (Cretaceous and Jurassic), QT p =Paso Robles formation (Pleistocene and Pliocene), and TP r =Pancho Rico formation (upper Miocene). Highlighted is the Paso Robles formation outcropping at the UPSAR location. From Fletcher at al. (1992). Figure m resolution topography showing the location of the UPSAR seismograph stations. The map covers an area of approximately 1 km x 1 km. From Wang et al. (2006). 54

71 Chapter 4. Spatial interpolation of displacement records from dense seismic networks Table 4.2 Characteristics (magnitude, epicentral distance, available stations and peak ground motion values) of the earthquakes considered in this study. The peak values are calculated as the maximum absolute value of the horizontal components among all stations, i.e.: PGV = max V ( t ) + V ( t ) 2 2 t x y Earthquake M W Epicentral distance # of available PGA PGV PGD [km] records [cm/s 2 ] [cm/s] [cm] Parkway # Parkway # Parkfield San Simeon Spatial interpolation of displacement records The empirical evaluation of surface ground strains has been carried out through the following steps: i. selection of a suitable interpolation technique of the available displacement recordings, upon an appropriate pre-processing; ii. validation of the interpolation procedure to check the order of accuracy and, thus, reliability of the obtained results; iii. evaluation of the continuous three-component displacement wavefield at the ground surface, from which the time-dependent strain tensor can be determined applying a classical central finite difference scheme. The available datasets (summarized in Table 4.2) have been pre-processed first by operating a standard baseline correction to remove the constant drift present in most accelerograms, then by high-pass filtering (namely, a third order Butterworth digital filter has been applied to the original recorded data) with cut-off frequencies f c =0.1 Hz, for Parkway Valley and Parkfield records, and f c =0.05 Hz for San Simeon. The choice of such corner frequencies has been dictated by the frequency content of the input seismic motions. Furthermore, it was checked that, by applying such cut-off frequencies, the displacement time histories resulting from double integration of the filtered accelerograms were not affected by any remarkable baseline drift. This would, in fact, eventually lead to unrealistic estimates of the differential displacements and, therefore, strains. Since it was possible to remove such drifts using relatively low cut-off frequencies, compared to the earthquake magnitudes involved, such preprocessing is not considered to have altered significantly the accuracy of the spatial interpolation of displacements for ground strain evaluation. Indeed, the low-frequency part of ground motion is generally the most coherent one and poorly contributes to transient ground strains. Due to matters of computational efficiency, before being input in the interpolation procedure, the records were decimated to a time step t=0.01 s, for Parkway events, and t=0.02 s for 55

72 Chapter 4. Spatial interpolation of displacement records from dense seismic networks the UPSAR data. Recalling the fundamental theorem of sampling (i.e. Nyquist theorem), this operation does not compromise the data up to the corresponding Nyquist frequencies of 50 Hz and 25 Hz, respectively, which are significantly larger than the frequency range dominating the corresponding ground motions. Before starting describing the used interpolation technique, it is worth illustrating the basic similarities and differences between our empirical method and the seismo-geodetic approach, mentioned in Chapter 2 (see Paragraph 2.3.2). Although our method belongs to the previous family of array-derived estimates, there are few meaningful differences to point out. Firstly, while the seismo-geodetic approach is formulated in terms of differential displacements between the recorded displacements and a selected reference station, such that it can be resolved as a generalized least-squares problem, our procedure begins developing an appropriate spatial interpolation from the displacement recordings, obtained at the stations of interest. Once, the interpolation is carried out, the application of a classical finite difference scheme to the interpolated wavefield allows us to obtain the ground strain tensor at the surface. Secondly, the a priori assumption behind the seismo-geodetic approach of linear variations of displacements within the array, therefore of uniform strain, does not apply to our procedure. In particular the use of high order interpolating functions allows one to recover the spatial variations of strain field within the array, at least at long periods Interpolation technique The spatial interpolation aims at constructing the continuous time-dependent displacement wavefield at the ground surface based on the available records at discrete locations. Through a combined space and time discretization, it is therefore possible to estimate the two horizontal displacement functions at ground surface u(x,y,z=0,t), v(x,y,z=0,t) and the vertical one w(x,y,z=0,t). In all cases, the interpolation procedure begins by defining a quadrangular interpolation grid having a constant spatial step x = y =5 m and including all the recording stations: namely, 130x130 nodes for Parkway #1 and #2; 160x160 nodes for Parkfield and San Simeon earthquakes. A peculiar convergence test (not reported herein) was made to ensure that the selected spatial step was small enough to obtain results independent on the spatial resolution of the numerical grid. The spatial interpolation was carried out by two different algorithms, implemented in Matlab: 1. Biharmonic Spline method (Sandwell, 1987), referred to in the sequel as B. Spline; 2. Delaunay triangulation technique (Barber et al., 1997). The former 1) consists basically in finding the interpolating curve (in one dimension) or the surface (in two dimensions, as in our case) as a linear combination of Green s functions of the biharmonic operator 6 centered at each data point. The corresponding coefficients are found by solving a linear system of equations. There may be fewer modal parameters than unknowns so that the overdetermined linear system may be solved through a least squares algorithm. 6 As illustrative example, for the 1D case, the Green function satisfying the biharmonic operator 4 4 dφ / dx 6 δ ( ) = x turns out to be: 3 φ ( x) = x. 56

73 Chapter 4. Spatial interpolation of displacement records from dense seismic networks Although this method might be rather slow, it reveals much more flexible because both slopes and values can be adjusted to better fit irregularly spaced data points. Algorithm 2), linked to the determination of the Voronoi diagram, starts from the partition of the plane into different triangular regions, in which the prediction of the weight of the interior points is interpolated from that of the vertices of the triangle that contains them (Barber et al., 1996). The comparison of the two interpolation procedures led us to prefer the B. Spline algorithm, as it allows to determine the displacement field also outside of the Delaunay triangles, where the triangulation cannot be applied, as illustrated in Figure 4.6 as illustrative example (Parkway #1 is considered). From Figure 4.6 it also apparent that, owing to the partition into triangular regions, the Delaunay technique tends to distort the interpolated wavefield at the edges of these regions, making it less appealing. On the other hand, the most significant shortcoming of B. Spline interpolation is that it tends to produce spurious oscillations at the corners of the interpolation grid, where the number of measure points drastically decreases. This is clear in Figure 4.6 where the B. Spline algorithm predicts a fictitious amplification at the bottom left corner, where there is no station to constrain the interpolating displacement surface. This aspect has been accounted for in choosing the size of the interpolation grid. As final check, a more quantitative comparison of the performance of the two interpolation method will be illustrated in the following section (see Figure 4.13) within the framework of the validation of the interpolation procedure itself. As representative examples, Figure 4.7 shows some snapshots of interpolated displacement, obtained through the B. Spline algorithm for, for t = 16.3 sec, during the most significant phase of motion, referring to Parkway #1. The displacement wavefield obtained by the aforementioned interpolation algorithm reveals realistic and able to reproduce the seismic pattern of ground motion and its fundamental frequencies. Considering the same case study, Figure 4.8 depicts a sequence vertical displacement, w(x,y), snapshots, band-pass filtered between 1 and 2 Hz, that clearly shows the complex site effects which occur at Parkway valley due to strong lateral heterogeneities. Referring to Paolucci et al. (2000) for a careful investigation of the complex site effects occurring at Parkway Valley, we underline herein that Figure 4.8 seems to confirm the well-known pattern of the 2D in-plane first vibration mode of an alluvial valley (Bard and Bouchon, 1985), with vertical displacements oscillating out of phase if they lie within the opposite side of the basin. The choice of the abovementioned band-pass filter was dictated by the resonance frequency of a representative transverse cross-section of Parkway Valley, known to be at around 1.6 Hz. 57

74 Chapter 4. Spatial interpolation of displacement records from dense seismic networks sv[mm*100] 0 stations stazioni 3950 north [m] north [m] north [m] east [m] east [m] Figure 4.6 Example of comparison between the B. Spline and Delaunay algorithms (see text for more details). Left: comparison in terms of contour lines of horizontal displacement (denoted with SV, in mm*100) obtained with Delaunay (top) and B. Spline technique (bottom), for t = 16.3 sec. Right: the same contour lines illustrated on the left hand side are overlapped. The small stars indicate the available receivers. Parkway event #1 is considered. Figure 4.7 Snapshots of interpolated ground displacement for t=16.3 sec during event Parkway #1: on the left horizontal in-plane component u(x,y), on the right vertical out-of-plane component, w(x,y). 58

75 Chapter 4. Spatial interpolation of displacement records from dense seismic networks 5-5 Figure 4.8 Sequence of snapshots of interpolated vertical ground displacement w(x,y) during event Parkway #1. The displacement has been band-pass filtered between 1 and 2 Hz to clearly catch the fundamental resonance frequency of the transverse cross-section of the Parkway Valley Validation of the interpolation procedure To quantitatively evaluate the performance of the proposed spatial interpolation technique, a validation procedure was built and applied systematically to the various stations of the two networks. In this procedure we first omit one receiver from the computation (say e.g. station J); then, after calculating through interpolation the displacement and velocity time histories at the same station, the recorded and interpolated displacements/velocities at receiver J are compared, both in time and frequency domain. Figure 4.9 and Figure 4.10 depict two representative examples of validation check for Parkway Valley (event 1 and event 2, respectively) carried out omitting in the interpolation procedure the receivers 09 and 06, respectively, located relatively far from the nearest stations. Similar examples for the Parkfield and San Simeon earthquakes recorded by the UPSAR micro-array are reported in Figure 4.11 and Figure 4.12, referring in this case to stations 02 and 08, respectively. The agreement turns out to be satisfactory both in frequency and time domains, supporting the accuracy of the chosen interpolation method in reproducing faithfully the observed punctual 59

76 Chapter 4. Spatial interpolation of displacement records from dense seismic networks recordings and capturing the prevailing waveforms and frequencies. It is noteworthy that, for the Parkway array, it has been noted that the accuracy of the interpolation decreases in the coda of signals (see for instance Figure 4.9 from time t >~ 17.5 sec) probably due to an increasing lack of coherence in the records in the final phases of ground motion. Furthermore, for the UPSAR array, the accuracy of the interpolation deteriorates for frequency larger than Hz (Figure 4.11) due to a much larger inter-station spacing. A practical rule to estimate the effect of the separation distance between stations for ground strain evaluations was proposed by Bodin et al. (1997), based on theoretical arguments introduced by Lomnitz (1997), who suggested a 90% accuracy of the array-based gradient estimates if the characteristic array dimension is less than approximately ¼ of the dominant wavelength. Referring to Fletcher et al. (2006), who estimated from the Parkfield earthquake records at UPSAR an apparent wave propagation V a = 2.5 km/s, we can infer that the actual displacement wavefield can be reproduced with a sufficient accuracy up to f max = V a / 4 ~2.5 Hz, where = 250 m is a representative measure of the separation distance between stations. This "rule of thumb is in reasonable agreement with our findings. Finally, we depict in Figure 4.13 the comparison in time (up) and frequency (bottom) domain between some representative results of the two interpolation algorithms, Delaunay vs. B. Spline, considering the same test illustrated in Figure 4.9. This check highlights the better accuracy of the B. Spline scheme in predicting the observations, in agreement with the findings of other tests not shown here. Figure 4.9 Comparison of recorded (thick line) EW displacement ( in a) ) and velocity ( in b) ) at station 09 of Parkway Valley array for event 1 with the one obtained by B. Spline interpolation (thin line), when in the interpolation procedure the same receiver is omitted. c) Comparison of the corresponding velocity Fourier amplitude spectra. 60

77 Chapter 4. Spatial interpolation of displacement records from dense seismic networks Figure 4.10 Same as in Figure 4.9 for event 2. Station 06 is considered herein. Figure 4.11 Comparison of recorded (thick line) EW displacement ( in a) ) and velocity ( in b) ) at station 02 of the UPSAR array during the Parkfield earthquake with the one obtained by B. Spline interpolation (thin line), when in the interpolation procedure the same receiver is omitted. c) Comparison of the corresponding velocity Fourier amplitude spectra. 61

78 Chapter 4. Spatial interpolation of displacement records from dense seismic networks Figure 4.12 Same as in Figure 4.11 for the San Simeon earthquake, omitting the receiver 08. Figure 4.13 Comparison of the observed displacement records both in time (up) and frequency (bottom) domain with those derived by the two interpolation techniques: B. Spline (thin grey line) and Delaunay (thick dashed line). The same test depicted in Figure 4.9 is considered herein. Since we will deal in the following with peak values of ground motion and the previous highfrequency limitation of the interpolation may significantly affect such estimates, we have further extended the validation procedure to check the accuracy of the predicted peak values of ground motion at each receiver of the arrays under study. Therefore, each receiver has been selectively omitted from the interpolation calculation, and the predicted value of either PGV or PGD has been compared with the observed one. 62

79 Chapter 4. Spatial interpolation of displacement records from dense seismic networks As illustrative example, we illustrate in Figure 4.14 the observed vs. predicted PGDs, in A) and PGVs, in B), for two of the earthquakes considered (Parkway #1 and San Simeon) while in Table 4.3 a summary of results for all the earthquakes is reported, in terms of average interpolation error in predicting either PGV or PGD. As can be expected based on the higher coherency of the displacement wavefield with respect to the velocity one, the results in terms of PGD are better than for PGV, but no systematic tendency was found of the interpolated values to over- or under-predict the observed ones. At both arrays the error ranges from ~ 18 % to ~ 35 % in terms of PGV and from ~ 5 % to ~ 15 % in terms of PGD. No similar approach can be proposed to estimate the level of error of the PGS evaluations, because there is no direct observation of ground strain: as a reasonable guess, the error in terms of PGS is expected to be similar to that in terms of PGV, since both are obtained by first order derivatives of the displacement wavefield. Figure 4.14 A) Observed vs. predicted peak ground displacement (PGD), using the proposed interpolation procedure, for Parkway Valley event 1 (left) and for the UPSAR San Simeon earthquake (right). B) Same comparison but in terms of Peak Ground Velocity (PGV). 63

80 Chapter 4. Spatial interpolation of displacement records from dense seismic networks Table 4.3 Average interpolation error in predicting PGD and PGV (see Figure 4.14 for an illustrative example) for all receivers of the array and each earthquake of Table 4.2. The error is calculated as the absolute value of the difference between the observed and interpolated values, divided by the observed value. Error type Parkway Parkway UPSAR UPSAR [%] event 1 event 2 Parkfield San Simeon ε PGD ε PGV

81 Chapter 5. The surface ground strain tensor: empirical evaluation and discussion of its properties 5 THE SURFACE GROUND STRAIN TENSOR: EMPIRICAL EVALUATION AND DISCUSSION OF ITS PROPERTIES 5.1 Goals and organization of the chapter After having acquired confidence on the accuracy achieved by our spatial interpolation technique in reproducing the observed displacements and velocities, we proceed with the evaluation of the in-plane transient strain tensor at ground surface during the selected earthquakes. This section will address the following topics: i) estimation of the time-dependent small strain tensor by central finite difference scheme; ii) iii) interpretation of some meaningful characteristics of the strain wavefield thanks to the tensorial representation, such as the dependence on azimuth of appropriately defined invariant measures function of principal strains and the corresponding inplane directions; derivation of new empirical relationships between PGS and the most relevant parameters for ground motion severity, such as PGV, PGA and PGD. The same procedure was applied to estimate the ground surface rotations and, subsequently, to investigate the correlation between PGR and PGV. For this we refer the reader to the next chapter, dedicated to the discussion of rotational ground motion. 5.2 Evaluation of strain tensor at the ground surface The evaluation of the three-component transient displacement fields by spatial interpolation allows us to determine the time-dependent small-strain tensor at ground surface, by the kinematic relationships: u 1 u v ε x = ε xy = + 2 ε (,, = 0, ) = x y x x y z t 1 u v v ε = + ε = xy y 2 y x y (5.1) 65

82 Chapter 5. The surface ground strain tensor: empirical evaluation and discussion of its properties Attention was limited to the strain tensor components in the horizontal plane, because no information was available on the displacement variability with depth to evaluate the three missing components ε z, ε xz and ε yz. It is noteworthy that in the simplest case of a flat ground surface (as in the case of Parkway Valley), it can be easily shown that the stress-free boundary condition 7 leads to the further conditions: i) τ xz = 0 ε xz = 0 ii) τ yz = 0 ε yz = 0 (5.2) and, under the assumption of linear-elastic isotropic behaviour (that may be considered reasonable for the Parkway Valley weak earthquakes; recall, for instance, that the maximal displacement amplitude is solely of around 0.40 mm for the strongest event #1): ν ν iii) σ z = 0 ε z = ( σ x + σ y ) = ( ε x + ε y ) E ν 1 (5.3) Notice that τ ij denotes the shear stress acting along the direction j, normal to i, while σ i denotes the normal isotropic stress acting along the direction I (the repeated index is herein, in fact, omitted). Therefore, under such an assumption, the strain component varying with depth can be assessed univocally by the two axial components. Once u(x,y,t) and v(x,y,t) have been calculated by spatial interpolation, the calculation is carried out through a 2 nd order centered finite difference scheme: u u v v 1 u u v v ε ε ε 2 x 2 y 2 2 y 2 x j j+ 1 j 1 j j+ 1 j 1 j j+ 1 j 1 j+ 1 j 1 x = ; y = ; xy = + (5.4) where j denotes the generic node of the computational grid. The determination of every component of the strain tensor at each node of the computational domain allows the calculation of the strain time histories at the positions corresponding to receivers of the arrays Definition of invariant ground strain measures: Highest Principal Strain (HPS) & Low Principal Strain (LPS) The tensorial representation of ground strain allows us to obtain in a straightforward way the principal values in the plane, i.e., the highest principal strain, denoted in the following as HPS(t), the lowest principal strain, LPS(t), and the principal directions where they occur. We underline that, although the stress-strain pattern is not plane, we have limited our study to ground strain tensor in the plane x-y because, firstly, z is a principal direction in our case and, secondly, no information is available about the behavior of displacement with depth. We recall herein that the solution of the generalized eigenvalue problem: 7 n with n = [ 0 0 1] σ = 0, if the z-axis is oriented upward. 66

83 Chapter 5. The surface ground strain tensor: empirical evaluation and discussion of its properties ε E = e E for i=1,2 (5.5) i i provides the eigenvectors E i along which the shear deformation vanishes, i.e. the principal strain directions, and the corresponding eigenvalues e i, i.e. the actual amplitude of the axial strains in the principal reference system. Hence, at each time interval t and for a given position (x, y), we end up with e 1 (t) and e 2 (t) such that: i HPS(t) = max{e 1, e 2 }; LPS(t) = min{e 1, e 2 } (5.6) Figure 5.1 provides a graphical representation of eq. (5.6), using the Mohr circle representation of the strain tensor in the plane. Notice that, once the eigenvectors and eigenvalues are calculated, the first and second strain invariants, I 1 and I 2 respectively, may also be evaluated as follows: I 1 (t)=hps(t)+lps(t); I 2 (t)=hps(t)*lps(t) (5.7) Furthermore, based on HPS(t), we can define the corresponding maximum value, as: [ ] HPS = max HPS( t ) (5.8) max that will be considered in the following sections as a suitable invariant measure of ground motion severity in terms of strain. It is noteworthy that, similarly to eq. (5.8), the maximum value of the quantity LPS(t), may be also defined, as follows: t [ ] LPS = max LPS( t ) (5.9) max Nevertheless we have focused our attention predominantly on the ground strain parameter (5.8) rather than (5.9), since the former can be reasonably regarded as the most severe invariant strain measure, hence considered suitable for conservative estimates and analyses. t Figure 5.1 Definition of invariant measures of the ground strain tensors using the Mohr circle representation: HPS(t) and LPS(t). The former is, at each time interval, the maximum principal strain, while the latter is the minimum, under the assumption of a plane stress-strain state. 67

84 Chapter 5. The surface ground strain tensor: empirical evaluation and discussion of its properties Significant features of the axial-shearing strain state The tensorial representation of the ground strain pattern along with the definition of new invariant measures of its severity allow us to investigate some meaningful features of the axial-shearing state of deformation induced by seismic motions. A very interesting feature of the HPS and LPS time histories is that they tend to be systematically out-of-phase, as shown in Figures from 5.2 to 5.5 for all networks and events considered. For each array, the time variations of HPS and LPS are illustrated for two different stations in order to highlight the consistency of results, regardless of the particular location at ground surface. Recalling the basic concepts behind the Mohr circle representation of the strain tensor, and that the pure shear strain condition is characterized by opposite values of HPS and LPS, as graphically explained in Figure 5.6, such a pattern suggests a strong contribution of shear waves. Such an evidence led us to investigate further some features of the ground strain invariants, such as the first invariant, the principal strain directions and their potential correlation with 1 both the direction of propagation and of particle motion. To this end, the first invariant I n( t ), normalized with respect to HPS, was considered as a potentially interesting parameter useful to discriminate the prevailing wave type: 1 HPS(t ) + LPS( t ) LPS( t ) I n( t ) = = 1+ HPS(t ) HPS( t ) (5.10) As a matter of fact, keeping in mind the Mohr circle representation, when the shearing 1 contribution is dominant (S- or Love waves), I n is expected to approach 0, whilst the latter should be approximately equal to 1 when P- or Rayleigh waves dominate ground motions, as illustrated by the corresponding sketches in Figure 5.7. In the bottom side of the figure the principal directions within the plane direction of displacement (x-axis) vs. direction of wave propagation (y-axis) are depicted with thick dashed lines. For instance, under purely shear conditions, the principal strain directions are rotated by an angle of 45 with respect to the direction of the particle motion, while Rayleigh wave would induce ground strains mainly aligned along the direction of displacements, at least theoretically. 1 Unfortunately, the results of test on time variations of I n as a parameter to discriminate prevailing ground motions were controversial. As an example, Figure 5.8 illustrates some representative time histories of the normalized first invariant, as defined by eq. ((5.10)), at selected receivers of both the Parkway Valley (up) and UPSAR (bottom) arrays. Namely receiver 9, for the Parkway valley (event #1 and #2), and receiver 6, for the UPSAR (Parkfield and San Simeon earthquakes) are considered. Some comments arise from the analysis of Figure 5.8. First, it is apparent that there is no clear evidence of the correlation 1 between the variation of I n and the principal P- and S- phases, such that the latter tends to oscillate irregularly around the average value of about This feature is preserved 1 rather well passing from one earthquake to another. Nevertheless, the average value of I n( t ) calculated at UPSAR array turns out to be slightly larger than the one calculated at Parkway Valley, regardless of the earthquake considered. Finally, no clear repeatable features are observed considering the same receiver but different seismic events. 68

85 Chapter 5. The surface ground strain tensor: empirical evaluation and discussion of its properties In order to investigate better the potential usefulness of this parameter, principal strains induced by the San Simeon earthquake, for which detailed analyses about particle motion are available in literature, is investigated. For this purpose, Figure 5.9 depicts the time variations 1 of I n at 4 representative receivers of the UPSAR array during the San Simeon earthquake; the superimposed vertical lines denote the P- and S- wave arrivals (approximately at 4 and 10 sec, 1 respectively) as estimated by Wang et al. (2006). As noted before, I n does not show any meaningful correlation with the main P- and S- contributions. Only in few cases, we can 1 observed a slight decreasing trend in the behavior of I n when the contribution of the shear waves is prevailing (see e.g. receiver 11). To clarify the possibility to extract useful information from this parameter more datasets should be studied and compared. Figure 5.2 Time variations of the highest principal strain, HPS(t), and the lowest principal strain, LPS(t), calculated at receivers 10 (up) and 15 (bottom) for the Parkway event #1. 69

86 Chapter 5. The surface ground strain tensor: empirical evaluation and discussion of its properties Figure 5.3 Same as in Figure 5.2 but for the Parkway event #2. Figure 5.4 Same as in Figure 5.2 but for the Parkfield event and for receivers 02 (up) and 11 (bottom) of the UPSAR networks. 70

87 Chapter 5. The surface ground strain tensor: empirical evaluation and discussion of its properties Figure 5.5 Same as in Figure 5.4 but for the San Simeon event. Figure 5.6 Sketch of the Mohr circle representation corresponding to purely shear conditions. 71

88 Chapter 5. The surface ground strain tensor: empirical evaluation and discussion of its properties Figure 5.7 Comparison between the principal strains induced by purely shear waves (left) or by surface (Rayleigh) waves (right). Figure 5.8 Time variations of the normalized first invariant 1 ( ) n I t, defined in eq. (5.10), at some representative stations at both Parkway Valley (up) and UPSAR (bottom) arrays. Namely 72

89 Chapter 5. The surface ground strain tensor: empirical evaluation and discussion of its properties receiver 9, for the Parkway valley (event #1 and #2), and receiver 6, for the UPSAR (Parkfield and San Simeon earthquakes) are considered. n Figure 5.9 Time variations of I ( ) 1 t at 4 representative receivers of the UPSAR array during the San Simeon earthquake. Superimposed are the vertical lines corresponding to the P- and S- wave arrivals, as estimated by Wang et al. (2006). As further investigation of the potential information contained in the tensor representation (5.1) of the strain field, the time-space variations of the principal strain directions, associated with HPS, were considered. Specifically, the frequency of occurrence of the principal strain directions at selected stations of the arrays was analyzed and correlated with the polarization of displacement particle motions. Figures from 5.10 to 5.13 illustrate the strain principal directions calculated at selected receivers for every earthquake under study in appropriate time windows centered in the most intense phase of motion. The length of each arrow in the plots is proportional to the frequency of occurrence of the corresponding principal strain direction within an angular sector of amplitude of 9, from 0 to 180, measured from the horizontal axis counter clockwise. Superimposed on the plots are also the trajectories of the particle motion, in terms of both displacement and velocity, calculated within the same time window as for the principal directions. That is to say that, though the adopted scale is arbitrary, the longest arrow in the plot denotes the principal direction which occurs most frequently in the corresponding angular sector. 73

90 Chapter 5. The surface ground strain tensor: empirical evaluation and discussion of its properties For each case study the following considerations can be made: i. Parkway Valley event #1: the principal strains show two preferred angular positions of around 45 and 135 with respect to the horizontal axis, respectively, regardless of the selected receiver. Keeping in mind that the HPS and LPS tend to be nearly out-ofphase suggesting a quasi-purely shear conditions in the most severe phase of motion (see Figure 5.2), one would reasonably expect that the principal strain directions align along a direction rotated of approximately 45 with respect to the direction of displacement particle motion, which turns out to polarized mainly along the horizontal direction (as sketched in the left hand side of Figure 5.7). Furthermore, past studies have pointed out that motion within the valley is strongly polarized along the EW direction, probably owing to the occurrence of 2D resonance phenomena (Paolucci et al., 2000). Principal strain directions at 45 seem then to confirm the prevalence of shear strains at ground surface. ii. Parkway Valley event #2: as observed for the event #1 respectively, it is possible to identify two preferred directions at around 45 and 135 (see bottom side of Figure 5.3), even though the distribution of principal strain directions seems more homogeneous, especially for receiver 10 (up side of Figure 5.3). iii. Parkfield: the results seem to be rather contradictory, since the most frequent principal strain directions turn out to be parallel with respect to the particle motion for the considered time window. Consider for instance the principal strains calculated at receiver 2 (up side of Figure 5.4): they concentrate mainly at the angular positions of 45 /135 while the particle displacement clearly polarizes along the diagonal of the first quadrant. This feature would suggest a predominance of surface waves (as illustrated on the right-hand side of Figure 5.7) that is not actually found (see, e.g., Fletcher et al., 2006). iv. San Simeon: as observed for the Parkway events, while the particle motion polarizes along an almost vertical direction, the principal strains tend to occur the most along 45 /135. The principal strain directions calculated at stations 10 and 15 of Parkway Valley array over the same phase of motion does not seem to maintain passing from event #1 to event #2, they rather show peculiar features depending on the considered event. On the other side, at the UPSAR the principal strain directions do not change significantly passing from Parkfield to San Simeon earthquake (compare Figure 5.12 with Figure 5.13). Provided the limited amount of available case studies, these controversial observations can not be helpful to derive general indications about the possible correlation (or absence of correlation) between the space-time distribution of principal strains at a given position and the seismic event. While for the Parkway events #1 (Figure 5.10) and #2 (Figure 5.11) and the San Simeon earthquake (Figure 5.13) the comparison of the direction of displacement particle motion with the principal strain directions seems to indicate coherently the dominance of shear strain conditions, for the Parkfield event (Figure 5.12) the principal strain directions turn out to be in 74

91 Chapter 5. The surface ground strain tensor: empirical evaluation and discussion of its properties most cases parallel to particle motion. This would suggest the prevalence Rayleigh wave contribution, which is not actually found for the selected earthquake. The reason for these discrepancies in the spatial distribution of the principal strain directions is not totally clear yet to the author and the occurrence of effects related to complex rupture pattern in the near field could probably be an explanation. A more systematic analysis of the principal strain directions along with a precise identification of the P-, S- and R- arrival might be useful in explaining the found inconsistencies. Figure 5.10 Right: principal strain directions, computed at the Parkway Valley (event #1) at receiver 10 (A) and 15 (B), with the corresponding trajectories of displacement particle motion (hodograms). The length of the arrows is proportional to the frequency of occurrence of the strain principal directions within each angular sector (from 0 to 180, measured from the horizontal axis counter clockwise). Left: EW and NS displacement time histories from which the illustrated hodograms are calculated. Highlighted by horizontal lines is the time window 75

92 Chapter 5. The surface ground strain tensor: empirical evaluation and discussion of its properties (namely from 16 to 20 sec) over which both principal strain directions and hodograms are calculated. Figure 5.11 Same as in Figure 5.10 for the event #2 at the Parkway array. 76

93 Chapter 5. The surface ground strain tensor: empirical evaluation and discussion of its properties Figure 5.12 Same as in Figure 5.10 for the Parkfield EQ: receiver 02 and 11 are considered here and the principal strain directions are calculated in the time window from 5.0 to 10.0 sec. 77

94 Chapter 5. The surface ground strain tensor: empirical evaluation and discussion of its properties Figure 5.13 Same as Figure 5.10 for the San Simeon EQ: a larger time window from 10.0 to 20.0 sec is considered. 5.3 Empirical relationships for peak ground strain evaluation After having acquired confidence with the most relevant features of the in-plane ground strain tensor at ground surface during the selected earthquakes, we now touch on some important issues meaningful for the practical evaluation of peak ground strains, with reference to the classical and simplest approach adopted in the common engineering practice, as highlighted in Chapter 2. We will discuss the following topics: i) dependence of the invariant ground strain measures with respect to azimuth and its potential relationships with the dominant wave propagation direction; ii) dependence of the peak parameters of ground motion with respect to acceleration, velocity and displacement 78

95 Chapter 5. The surface ground strain tensor: empirical evaluation and discussion of its properties iii) comparison of the obtained empirical relationships for assessing the maximal strains with both direct strain gauge measurements performed in buried pipelines and tunnels (e.g. Nakamura et al, 1981 and Iwamoto et al., 1988) and other published empirical correlations, such as that developed by Abrahamson (2003) Dependence on azimuth Consider first the issue regarding the azimuth dependence of surface ground strains. For this purpose, we have first calculated, by rotation of the strain tensor, the axial peak ground strain (PGS θ ) along a generic azimuth θ, at each receiver of the network for each earthquake, together with the corresponding peak ground velocity (PGV θ ), projected along the same direction. For each azimuth, we have interpolated the observed PGS θ -PGV θ pairs and estimated the ψ parameter of the least-square best-fit line: PGSθ = PGV θ / ψ (5.11) In Figure 5.14 we show through suitable radar plots the variation of ψ with θ for all the four seismic events under consideration. To obtain these plots, the strain and velocity components along a prescribed direction θ have been evaluated at each station of the array: ψ is therefore calculated as the inverse of the slope of the least-squares best-fit line connecting the PGS θ and PGV pairs along that direction. Although the results for the event 2 of Parkway Valley turn out to be rather incoherent with respect to the others, they are presented however for sake of completeness. They reveal to be affected by large, and probably unrealistic, fluctuations, likely due to the very small magnitude of peak values of strain (about 10-7 ), giving rise to unrealistically high values of ψ of the order of Leaving aside Parkway event 2, it is noteworthy that the variations of ψ with θ range from about 1380 to 2000 m/s for Parkway Valley event 1, from 1330 to 2250 m/s for Parkfield, and from 1260 to 2060 m/s for San Simeon. Therefore, we can deduce a factor of about 2 of variability of the ψ factor in eq. (5.11), and hence of the V a factor in eq. (2.1), due to the azimuth dependence. Even though the results are not completely clear in this respect, the inclination of the radar plots in Figure 5.14 seems to be correlated to the prevailing direction of wave propagation, as would be the case for earthquake ground motion propagating as plane waves. We underline firstly that the direction corresponding to the minimum value of ψ is associated to the direction where the maximum peak strains occur, i.e. the direction of the maximal differential displacement. This direction was found to be rather close to the prevailing direction of wave propagation, specifically: (i) for Parkway #1 the approximately NS axis of the valley, as estimated by Chávez-García et al. (2002); (ii) for Parkfield the 230 azimuth estimated by Fletcher et al. (2006); (iii) for San Simeon the 80 azimuth suggested by Wang et al. (2006). 79

96 Chapter 5. The surface ground strain tensor: empirical evaluation and discussion of its properties Figure 5.14 Radar diagrams showing, for all earthquakes under consideration, the azimuth-dependence of ψ (m/s) coefficient of eq. (5.11) along a set of prescribed azimuths θ. The rightmost hand side sketch illustrates the definition of radial strains ε θθ, from which the left radar graphs are derived, and of transverse strains ε ρρ, perpendicular to the formers Dependence on the peak parameters of ground motion As a second important issue, we address now the correlation of peak ground strains with the most common measures of ground motion severity, such as PGA, PGV and PGD. Owing to the previously discussed dependence on azimuth of the peak ground strain, we have decided to use for this purpose an invariant measure of this parameter, namely, the maximum value of the in-plane principal strain HPS max defined by eq. (5.8). For ease of notation in the figures shown in the sequel we have denoted HPS max as PGS. This should be kept in mind for comparison purposes: while the experimental PGS values refer to the peak strain calculated along a prescribed direction (e.g., the longitudinal direction of the pipe where the strainmeter was installed), our results provide the maximum PGS with respect to every horizontal direction. (a) Dependence on Peak Ground Velocity A rather interesting result is illustrated in Figure 5.15, where HPS max as a function of the largest absolute value of ground velocity PGV is shown for each earthquake and each station of the array. Note that PGV is not in general measured along the same direction as HPS max, 80

97 Chapter 5. The surface ground strain tensor: empirical evaluation and discussion of its properties but both HPS max and PGV are invariant measures of ground motion severity. The notable feature of Figure 5.15 is that all data tend to be aligned, with a relatively small dispersion and irrespective of the earthquake magnitude, distance, site conditions or prevailing wave type, along the Least Squares (LS) line with equation α β Log HPS = α Log PGV β i.e. HPS = PGV / 10 (5.12) 10 max 10 where α ~ and β = If the parameter α is forced to be unity, coherently to relationship (1.1), the best fit line turns out to be: max HPS = PGV / ϕ (PGV in m/s) (5.13) max where ϕ = 963 m/s is the median value, while 671 m/s and 1382 m/s correspond to the 16 and 84 percentile, respectively. The solid line superimposed in Figure 5.15 is the best-fit LS line extended to all the HPS max -PGV pairs corresponding to the median value ϕ = 963 m/s, while the two dashed lines are associated to ϕ =671 m/s and ϕ + =1382 m/s, being ±σ=0.16 the standard deviation characterizing the dispersion of our datasets. Repeated in Figure 5.16 are the PGS vs. PGV pairs (we recall herein that the former is the same as HPS max ) obtained by the spatial interpolation procedure together with: 1) direct strain gauge measurements performed in two buried pipelines in the town of Hachinohe during the 1978 Miyagiken-Oki earthquake (M7.8), Japan (Iwamoto et al., 1988); 2) the estimates recently derived by Spudich & Fletcher (Pers. Comm., 2008) for the same M6.0 Parkfield event and three aftershocks (in order of decreasing magnitude, M=5.1, M4.9 and M4.7). Basically, they applied the so-called seismo-geodetic approach (explained in Chapter 2) to the UPSAR recordings in order to derive both strains, tilts and torsions. Referring to Spudich & Fletcher (2008) we underline here that we have considered, for each earthquake, the broadband estimate of the maximum horizontal shear strain. Under the hypothesis of almost purely shear strain conditions (which is moreover confirmed by the very small first strain invariants found by the authors themselves), this should be very close to the values of HPS max. A similar trend is suggested by the experimental datasets, although our PGS values tend to be larger: this can be reasonably explained if one takes into account that, as noted above, we have considered the maximum principal value of ground strains, while 1) the strain gauge measurements were obtained along the longitudinal axis of the pipe, so that they may not provide the largest value of PGS among all possible spatial directions and 2) Spudich & Fletcher (2008) consider an uniform value of maximum horizontal strain within the entire array which is not maximized with respect to azimuth. As a further term of reference, we have also plotted in Figure 5.16 two representative lines defined by equation 1a of Trifunac and Lee (1996), providing PGS (radial strain) vs. PGV calculated by a large set of numerical simulations, as a function of the average S- wave velocity in the top 50 m (V s ) and of the epicentral distance (D). Trifunac and Lee recognized a weak dependence of the results on these parameters, and also pointed out that the dependence on earthquake magnitude was negligible. Although our data are only based on four earthquakes recorded by two arrays, so that they are not sufficient to derive general and well constrained indications on the 81

98 Chapter 5. The surface ground strain tensor: empirical evaluation and discussion of its properties dependence of ground strains on the previous parameters, our findings tend to support a weak dependence of the PGS-PGV relationship on magnitude, distance and site conditions. It is worth highlighting that the choice of fitting our datasets alone (as reported in Figure 5.15) without including also the direct and numerically-derive data depends on the fact that the various datasets are intrinsically inhomogeneous, such that a common best-fit LS line might be misleading. Furthermore, another relevant observation concerns the interpretation of the PGV/PGS ratio. Assuming the PGV/PGS ratio as the actual horizontal wave propagation velocity V a of the prevailing wave type (either apparent velocity of body waves or phase velocity of surface waves) is misleading. For example, the average PGV/PGS ratio ~ 1000 m/s deduced from eq. (5.13) is about 2.5 times smaller than the apparent wave propagation velocity V a = 2500 m/s estimated at the UPSAR array by spatial cross-correlation analyses of the Parkfield earthquake records (Fletcher et al., 2006). Similarly, Bodin et al. (1997) deduced from a micro-array in Mexico City a PGV/PGS ratio 3 times smaller than the prevailing phase velocity of surface waves. Similar conclusions, although derived from a different approach based on the statistical analysis of the effect of random variability of soil properties, were obtained by Zerva and Harada (1997). These authors found that the PGV/PGS ratio is poorly dependent on the apparent wave propagation velocity V a, and mainly governed by loss of coherency of the displacement wavefield, for V a larger than a certain threshold. We incidentally note that the rms PGV/PGS ratio calculated by Zerva and Harada (see Fig. 7 of the abovementioned paper) approaches 1000 m/s, close to the value derived from eq. (5.13). Further details about this aspect can be found in the end of this section (see Figure 5.22). Figure 5.15 Correlation of observed HPS max - PGV pairs for the 4 earthquakes under consideration. The solid line HPS max = PGV/φ is the LS best fit line extended to all the HPS max -PGV pairs corresponding to the median value φ =963 m/s, while the two dotted lines are associated to φ - =671 m/s and φ + =1382 m/s corresponding to the 16 and 84 percentile, respectively (the standard deviation between the observed data and the best-fitting line is around 0.16). 82

99 Chapter 5. The surface ground strain tensor: empirical evaluation and discussion of its properties Figure 5.16 A) PGS vs. PGV pairs, either obtained by our spatial interpolation procedure, or through direct strain gauge measurements in Japan (Iwamoto et al., 1988). Superimposed are two representative lines defined by equation 1a (radial strain) of Trifunac and Lee (1996) for two combinations of the parameters R (epicentral distance, in km) and Vs (shear-wave velocity in the uppermost 50 m, in km/s). Such equation reads as: log 10 PGS = ( Vs) + ( Vs) R + [(1-0.19Vs) + ( Vs) R] Log 10 (PGV/Vs). B) LS best-fit line (solid line) extended to both our experimental datasets, including Spudich & Fletcher (2008) estimates, and strain measurements performed in Japan; the dashed lines correspond to +/- standard deviation σ = (b) Dependence on Peak Ground Acceleration Moving now to the issue of the dependence upon PGA, Figure 5.17 illustrates: A) the observed PGS-PGA pairs obtained from our datasets along with the median best-fit LS line, obtained when the coefficient of proportionality between log 10 (PGS) and log 10 (PGV) is forced to be unity. The lines corresponding to the 16 and 84 percentile are also shown for sake of completeness; B) superimposed to our datasets are two representative direct measurements sets from Japan, obtained on buried pipelines (Nakamura et al, 1981, quoted by Trifunac and Lee, 1996) and tunnels (JSCE, 1977). 83

100 Chapter 5. The surface ground strain tensor: empirical evaluation and discussion of its properties Analogously to the PGS-PGV relationship, the LS approximation of the data derived by spatial interpolation shows that the coefficient of proportionality between log 10 PGS and log 10 PGA is very close to 1, so that a simple linear relationship of the form PGS = λ PGA (5.14) can be proposed. A coefficient λ ~ , where PGA is measured in m/s 2, was found suitable both to model our data alone, and including the Japanese measurements as well. Specifically, while the standard deviation varies significantly from ~ 0.17 to ~ 0.30 if the Japanese data are included, the median value of the coefficient of proportionality between PGS and PGA remains nearly the same, as reported in Figure Due to the inhomogeneity of the experimental set of PGS values, as underlined previously, we have compared the dispersion of the relationships considering our data alone. In this case the standard deviation associated to both the PGV and PGA best-fit lines is σ ~ 0.17, showing that, contrary to expectations, there is no significant variation in the dispersion of data when selecting either PGV or PGA, as a predictive parameter for PGS. 84

101 Chapter 5. The surface ground strain tensor: empirical evaluation and discussion of its properties Figure 5.17 A) Same as in Figure 5.15 but in terms of PGA (in m/s 2 ), for the four earthquakes under study; B) comparison with the direct strain gauge measurements from Japan (Nakamura et al., 1981 and JSCE, 1977); note that the LS line fitting the extended dataset, including also the direct Japanese measurements, would be characterized by nearly the same slope. 85

102 Chapter 5. The surface ground strain tensor: empirical evaluation and discussion of its properties Figure 5.18 Same as in Figure 5.17 B) but superimposed are the median LS best-fit line extended to the entire dataset (including the direct strain measurements in buried pipelines and tunnels) and the two LS line corresponding to +/- σ=0.30. Notice that the median value of the coefficient of proportionality between PGS and PGA is almost the same as that derived by our dataset only (see Figure 5.17 A) ). (c) Dependence on Peak Ground Displacement Unlike the previous PGS-PGV and PGS-PGA relationships, which were suitably fitted by a linear trend, the observed PGS-PGD was not found to enjoy the same property. As shown in Figure 5.19, the PGS-PGD pairs tend to align along a LS best-fit line of equation: Log PGS = γ Log PGD δ (5.15) with γ = 0.79, δ = 4.26 and PGD expressed in cm. The standard deviation σ between the observed data and the line of eq. (5.15) turns out to be approximately equal to Forcing γ = 1, as for eqq. (5.13) and (5.14) would have led to significantly underestimate ground strains for the low magnitude Parkway earthquakes and overestimate the UPSAR dataset, at larger values of peak displacements (accordingly, σ would approach a value of about 0.35, significantly greater than 0.19). This may be an indication supporting the magnitudedependence of the PGS-PGD correlation, as suggested by Abrahamson (2003). The author proposed the following relationship: PGS / PGD( cm ) = e / V (5.16) ( M ) 5 5 a At the right-hand side of eq. (2.16), the wave passage effect is quantified by the first term, V a being the apparent wave propagation velocity, while the other two terms incorporate empirically the effect of spatial incoherence and of local site amplification, respectively. We have applied eq. (2.16) to the four earthquakes considered in this study, using V a = (cm/s), which was the value used by Abrahamson to best fit his dataset. Furthermore, as a measure of PGD to be input in eq. (2.16) we have considered the average for each earthquake 86

103 Chapter 5. The surface ground strain tensor: empirical evaluation and discussion of its properties dataset, while the additional term referring to the site conditions effect was considered only for the Parkway Valley data and neglected for the recordings of the UPSAR array. A good agreement is obtained as illustrated in Figure 5.20, in spite of the empirical nature of the terms in eq. (2.16). If the additional terms in eq. (2.16) related to the spatial incoherence and site effects were not considered, the wave passage effect alone would have considerably underestimated the observed results and, hence, would be inappropriate to properly evaluate the peak ground strains. Figure 5.19 Correlation of observed PGS(i.e. HPS max ) - PGD pairs for the 4 earthquakes under consideration. The solid line PGS=PGD γ /10 δ (see eq. (5.15)) is the LS best fit line corresponding to the median value γ = 0.79 and δ = -4.26, while the two dotted lines are associated to the 16 and 84 percentile, respectively (the standard deviation σ between the observed data and the best-fitting line is around 0.19). Figure 5.20 Comparison of the observed PGS-PGD pairs with those calculated according to eq. (2.16), recommended by Abrahamson (2003). 87

104 Chapter 5. The surface ground strain tensor: empirical evaluation and discussion of its properties (d) Final remarks Table 5.1 presents a comprehensive summary of all the empirical best-fitting correlations obtained for PGS evaluating based on the various ground motion parameters: PGV, PGA and PGD. For each correlation, the dataset on which is derived, the coefficient of proportionality between PGS and either PGV, PGA or PGD along with the intercept in logarithmic scale of the best-fit LS curve and the corresponding standard deviation are shown. Note that both LS regressions obtained either forcing the coefficient of proportionality to be 1 or not are summarized in order to underline the originally observed (linear or not) trend of the data. From the analysis of the empirical correlations illustrated above the following concluding remarks arise: i) a linear trend was found between Log 10 PGS as a function of either Log 10 PGV or Log 10 PGA, with a remarkably low dispersion (σ ~ 0.16). It is remarkable that the coefficient of proportionality in the log-log plane is very close to 1.0, leading to a simple linear correlation between PGS and the other ground motion parameters. ii) iii) iv) The superposition of our results with the ones obtained by direct strain gauge measurements in buried pipelines and tunnels in Japan points out a similar linear trend of maximum ground strains as a function of either PGV or PGA. This is further indirect confirmation of the reliability of our calculations. The comparison with other array-derived estimates in terms of PGS for the same UPSAR array turns out to be rather satisfactory. The correlation of PGS with PGD was found to be weaker than with PGV and PGA, probably due to the larger dependence of PGD on magnitude. When comparing our results with the magnitude-dependent PGS-PGD relationship proposed by Abrahamson (2003), the agreement was found to be quite satisfactory. v) As a final consideration, Figure 5.21 shows the comparison of the observed PGV- PGS correlation, calibrated on the 4 earthquake under study, with the commonly adopted PGS values for the seismic design of underground structures (see Paragraph in Chapter 2 for further details). It is apparent that the commonly adopted PGS values, ranging from PGV/2000 up to PGV/4000, turn out to be significantly smaller than the ones derived empirically (~PGV/1000), by a factor of at least 2-4. This suggests the seismic design recommendations to be unconservative. This inconsistency may be explained on the basis of the analyses carried out by Zerva (1992) and Zerva & Harada (1997), as anticipated previously. The authors found that there is a critical value of apparent velocity (see also Figure 2.15 in section 2), typically varying from ~1000 m/s to ~3000 m/s, above which seismic motions and, therefore, transient ground strains remain essentially constant, being poorly dependent on V a and mainly governed by spatial incoherence effects (see Figure 5.22, adapted from Zerva, 2000). In these conditions the PGV/PGS ratio tends to stabilize around the value of 1000 m/s, which turns out to be very close to the coefficient of proportionality of eq. (5.13). As a consequence of this, 88

105 Chapter 5. The surface ground strain tensor: empirical evaluation and discussion of its properties strains are much higher than those given by the simplified relationship of the form PGV/V a. In conclusion, the empirical analyses of this work suggest that the PGS values estimated by accounting for the wave passage effects alone are likely to be unconservative and this is may be due to loss of coherency of seismic ground motion, effect that is generally disregarded for seismic design of underground structures. Table 5.1 Comprehensive table of the empirical best-fitting relationships for PGS evaluation based on ground motion parameters PGV, PGA and PGD. For each correlation, the dataset, the coefficient of proportionality between PGS and the selected ground motion parameter in Log10 scale, the intercept in Log10 scale of the best-fit Least Square line and the standard deviation σ are summarized. Ground motion parameter PGV [m/s] PGA [m/s 2 ] PGD [cm] Dataset Coefficient of proportionality in Log 10 scale Intercept in Log 10 scale Standard Deviation σ Parkway #1 & # Parkfield, San Simeon Parkway #1 & #2 Parkfield, San Simeon + Iwamoto et al. (1988) + Spudich & Fletcher (2008) Parkway #1 & # Parkfield, San Simeon Parkway #1 & #2 Parkfield, San Simeon + Nakamura et al. (1981) + JSCE (1977) Parkway #1 & # Parkfield, San Simeon

106 Chapter 5. The surface ground strain tensor: empirical evaluation and discussion of its properties Figure 5.21 Comparison of the observed PGV-PGS correlation for the 4 earthquakes under consideration with the commonly adopted values of PGS for seismic design of underground structures. Figure 5.22 PGS/PGV ratio as a function of the apparent propagation velocity V a (in m/s). There is a critical V a above which spatial incoherence effects dominate rather than wave passage. Adapted from Zerva (2000). 90

107 Chapter 6. Seismic ground rotations: array-derived estimates, numerical simulations and possible effects on structural response 6 SEISMIC GROUND ROTATIONS: ARRAY-DERIVED ESTIMATES, NUMERICAL SIMULATIONS AND POSSIBLE EFFECTS ON STRUCTURAL RESPONSE 6.1 Main topics and organization of the chapter In this chapter the issue of rotational ground motions induced by earthquake excitations is investigated relying upon different tools and approaches. The following topics are discussed: i. analysis of surface ground rotations obtained by the application of the same interpolation procedure illustrated in the previous chapter. Similarly to what done for PGSs, an empirical correlation between PGV and PGR is derived and compared with other published array-derived data; ii. Comparison of the synthetic PGRs computed by complex 3D Spectral Eement simulations of the Grenoble basin (France) for different earthquake scenarios with the data computed at point i) and other published results retrieved from literature; iii. Investigation of the possible effects of coupled tilt and translational motion on structural response. To this end, a simple model of single degree-of-freedom (dof) structure resting on 2 dof foundation, subjected to both translational and rotational excitations, is used. 6.2 Surface ground rotations obtained by the interpolation procedure The application of the interpolation procedure, illustrated in the previous Chapter, enables to compute surface ground rotations. The same case studies are considered, namely: Parkway #1, Parkway #2, Parkfield and San Simeon events. Note that considering this broad dataset allows one to investigate the dependence of the amplitude of rotational ground motions on earthquake magnitude and epicentral distance, an issue that has deserved little attention so far. Similarly to what illustrated in Chapter 5 (section 5.2), simple compatibility considerations allow one to determine the small rotation tensor at the ground surface from the 3-component T u = u v w, as follows: displacement field [ ] 91

108 Chapter 6. Seismic ground rotations: array-derived estimates, numerical simulations and possible effects on structural response 0 ωz ωy θ ( x, y, z = 0, t ) = ωz 0 ω x (6.1) ω ω 0 y x 1 v w with ωx = 2 z y, 1 w u 1 u v ωy = and ωz = 2 x z 2 y x. In absence of further conditions, only the torsion ω z about the vertical axis can be calculated by a central finite difference scheme, since displacement variability with depth is not unknown. Nevertheless, taking into account the free-surface boundary condition (see Section 4.4, in particular eq. (4.2)) allows one to determine the two tilt components in the following way: γ γ xz yz u w w = 0 = ωy = z x x v w w = 0 = ωx = z y y (6.2) (6.3) As pointed out in the previous Chapter, conditions (6.2) and (6.3) assume that the ground surface is flat and apply, rigorously, only to Parkway valley array, as the UPSAR array is located in a hilly area. Even though approximated, these conditions have been applied as well to compute ω x and ω y, considering that the elevation differences between stations at the UPSAR array are much less than the prevailing wavelength. Referring, on one side, to Fletcher et al. (2006), who measured from the Parkfield earthquake records at UPSAR an apparent wave propagation V a >= 2.0 km/s, and, on the other side, to Wang et al. (2006), who found that motions at the array are dominated by frequencies within 3 Hz, the dominant wavelengths are greater than 600 m, value which is significantly larger than the maximum elevation difference between two stations ~ 40 m. As illustrative examples, Figure 6.1, Figure 6.2, Figure 6.3 and Figure 6.4 depict the time histories of ground torsional acceleration (in rad/s 2 ), velocity (in rad/s) and displacement (in rad) and the corresponding Fourier amplitude spectra for the 4 earthquakes under consideration, at selected receivers. It is noteworthy that: i. for low magnitude events, M < 5.0, and large epicentral distances (R ~ 81 km) maximum torsions do not exceed values of about 10-6 rad. In particular for M=4.2 and M=4.9 Parkway Valley events, PGRs are ~10-7 and ~10-6, respectively. ii. For larger earthquakes (M=6.0 and 6.5) in the near-source region (R ~ 10 km) rotations are of the order of This is a relatively small value, probably because amplification of seismic motions due to complex site effects plays a negligible role at the UPSAR area. 92

109 Chapter 6. Seismic ground rotations: array-derived estimates, numerical simulations and possible effects on structural response Figure 6.1 Time histories (left) and corresponding Fourier amplitude spectra (right) of torsional acceleration (up), velocity (middle) and displacement (bottom) calculated at station 16 during the M4.9 Parkway #1. Figure 6.2 Same as in Figure 6.1 but for M4.2 Parkway #2 event, receiver 6. 93

110 Chapter 6. Seismic ground rotations: array-derived estimates, numerical simulations and possible effects on structural response Figure 6.3 Same as in Figure 6.1 but for M6.0 Parkfield earthquake, receiver 3. Figure 6.4 Same as in Figure 6.1 but for M6.5 San Simeon earthquake, receiver

111 Chapter 6. Seismic ground rotations: array-derived estimates, numerical simulations and possible effects on structural response Empirical correlation for PGR (a) Dependence on PGV Similarly to what done for the strains, the correlation between Peak Ground Velocity PGV and Peak Ground Rotation PGR z (torsional component) is investigated. Specifically, based on the theoretical developments (Lee and Trifunac, 1987; Igel et al., 2005) we have preferred to correlate PGR z with the horizontal PGV calculated as square root of the squares of the two 2 2 horizontal peak motions, i.e. PGV = PGV + PGV. x A very similar linear trend as observed for the PGV-PGS pairs (see eq. (5.13) in Chapter 5) is found for the PGV-PGR z pairs. This is reasonably expected as both rotations and strains are differential displacement quantities. All datasets tend to be aligned along a straight line in loglog space with equation: y Log PGR = ηlog PGV ξ (6.4) 10 z 10 where η ~ 0.98 and ξ = As noted for both PGV- and PGA- vs. PGS correlations (see eq. (5.13) and (5.14)), the coefficient of proportionality between the maximum particle velocity and torsion is very close to 1. If the parameter η is forced to be unity, the LS best fit line turns out to be: PGR = PGV / χ (PGV in m/s) (6.5) z where χ = 2120 m/s is the median value, while 1434 m/s and 3130 m/s correspond to the 16 and 84 percentile, respectively, being σ = 0.17 the standard deviation. We incidentally note that, in their recent work, Spudich & Fletcher (2008) found a ratio of horizontal PGV (or PGA); predicted by the empirical ground motion relationship of Boore & Atkinson (2007), over PGR z (or peak ground torsion rate, PGRR z ) equal to 2c, with c = 1000 m/s. This scaling factor is very close to the parameter χ derived from eq. (6.5). Figure 6.5 displays the PGV-PGR z pairs for the 4 earthquakes under consideration. Superimposed is the best-fit LS line corresponding to the median value (solid line) and the two best-fit lines associated with χ = 1434 m/s and χ + =3130 m/s. Note that the coefficient of proportionality χ of eq. (6.5) is about twice the corresponding coefficient, ϕ = 963 m/s, relating PGV and PGS (see eq. (4.13) of previous Chapter). Coherently to the simple theoretical considerations highlighted in Chapter 2 and 3 (namely, eqs. (2.1) and (3.6)), the ratio PGR z /PGS turns out to be close to ½. However, this evidence does not necessarily imply that strains and torsions scale with the actual apparent propagation velocity, as also commented by Spudich & Fletcher (2008). Assuming the PGV/PGR z ratio as the actual V a, apart from a factor of 2, would be misleading. Consider, for instance, that half the median ratio PGV/PGR z ~ 2 km/s, inferred from eq. (6.5), is significantly smaller than the apparent wave propagation velocity V a ~ 2.5 km/s estimated at the UPSAR array by spatial crosscorrelation analyses of the Parkfield earthquake records (Fletcher et al., 2006). 95

112 Chapter 6. Seismic ground rotations: array-derived estimates, numerical simulations and possible effects on structural response Figure 6.6 illustrates the comparison of PGV-PGR z pairs, derived by our interpolation procedure, with the estimates obtained by Spudich & Fletcher (Pers. Comm., 2008) for the same M6.0 Parkfield event and three aftershocks (M=5.1, M4.9 and M4.7). The data retrieved by Spudich & Fletcher are reported in Table 6.1. Sub-array estimates refer to the results calculated considering selected sub-arrays within the UPSAR network and over different frequency bands, while the term broadband refer to a sort of averaged value, increased by a corrective factor to embody empirically higher frequency components into the estimates (Spudich & Flecther, 2008). For a coherent comparison with our method, the broadband estimates should be considered. First of all, it is interesting that the value of PGR z determined by Spudich & Fletcher for the Parkfield event ( ) is very close to the mean value of the set of pairs calculated by our interpolation procedure ( ). An extended comparison between the peak values of tilt, tilt rate, torsion and torsion rate is given in Table 6.2 (to be coherent with the work of Spudich 2 2 & Fletcher, the tilt is defined as = ωx + ω y ). This constitutes a further validation of our estimates. Furthermore also the broadband estimates associated with the aftershocks tend to align along the same line of eq. (6.5), supporting the hypothesis of a common relationship between PGR z and PGV, independent of earthquake magnitude and source to receiver distance. Dependence on site conditions should be checked against a larger number of case histories or numerical simulations, although the very different site conditions of Parkway Valley and UPSAR arrays tend to indicate a poor influence of site conditions as well. Figure 6.5 PGV vs. PGR z pairs, obtained through the spatial interpolation procedure for all the 4 earthquakes under consideration. Superimposed are the best-fit LS line corresponding to the median value χ = 2120 m/s (solid line) and the two best-fit lines (dashed line) associated with the 16 and 84 percentile, χ- = 1434 m/s and χ+ = 3130 m/s, respectively. 96

113 Chapter 6. Seismic ground rotations: array-derived estimates, numerical simulations and possible effects on structural response Table 6.1 Data of PGV and PGR z according to Spudich & Fletcher (pers. Comm., 2008). Additional information about the earthquake magnitude and Joyner Boore distance R JB are listed. Highlighted in bold are the broadband estimates. Ar. 1-3(or 8-11, 5-12) f max 1.4(or 3.6) refers to strain and rotation estimates from sub-array 1-3 (or 8.11, 5-12) filtered in the (3.6) Hz band. Event refers to the main Parkfield shock. Event name R JB [km] M w PGV h [m/s] PGR z [rad] Data type E-05 BroadBand E-05 Ar.1-3- f max E-05 Ar.1-3- f max E-05 Ar f max E-04 Ar f max E-05 Ar f max E-06 BroadBand E-06 Ar.1-3- f max E-06 Ar.1-3- f max E-06 Ar f max E-06 Ar f max E-06 Ar f max E-05 BroadBand E-06 Ar.1-3- f max E-06 Ar.1-3- f max E-06 Ar f max E-05 Ar f max E-06 Ar f max E-05 BroadBand E-06 Ar.1-3- f max E-06 Ar.1-3- f max E-06 Ar f max E-05 Ar f max E-06 Ar f max

114 Chapter 6. Seismic ground rotations: array-derived estimates, numerical simulations and possible effects on structural response Figure 6.6 Comparison of the PGV vs. PGR z pairs, obtained by our spatial interpolation procedure, with the results obtained by Spudich & Fletcher (see Table 6.1) for the same Parkfield event (main shocks) and three aftershocks (M5.1, M4.9 and M4.7). The classification into broadband and sub-arrays estimates refers to different procedures developed by the authors. However, for comparison purposes with our results, the broadband estimates have to be taken into account. For comparison purposed superimposed is the LS best-fit lines calculated on our data. Table 6.2 Comparison between the peak values of tilt, tilt rate, torsion and torsion rate obtained through our interpolation procedure and those calculated by Spudich & Fletcher (2008) for the M6.0 Parkfield event. Note that interpolation method refers to the mean values of the PGRs obtained within the array. PGR [rad] PGRR [rad/s] torsion tilt torsion tilt interpolation method (average) 8.98E E E E-04 Spudich & Fletcher (2008) 8.81E E E E-04 (a) Dependence on PGA After having discussed the dependence of peak ground torsion on PGV, we consider herein 2 2 the issue of the dependence on PGR z on PGA (calculated as PGAx + PGA y ). Figure 6.7 illustrates the experimental PGA vs. PGR z pairs, obtained by our interpolation procedure, along with the best-fitted straight line obtained by forcing the coefficient of proportionality between Log 10 PGA and Log 10 PGR z to be 1: PGR = µ PGA (PGA in m/s 2 ) with µ = (6.6) z Similarly to the PGV-PGR z relationship (in Figure 6.5) a simple linear correlation describes all datsets with satisfactory accuracy (σ ~ 0.19, only slightly larger than the dispersion of 98

115 Chapter 6. Seismic ground rotations: array-derived estimates, numerical simulations and possible effects on structural response relationship (6.5)). The comparison of eq. (6.6) with the equivalent correlation for PGS, provided by eq. (5.14), confirms that PGR z /PGS = µ/λ ~ ½. Figure 6.7 PGA vs. PGRz pairs, for the 4 experimental datasets under study, along with the median fitted straight line, defined by eq. (6.6). 6.3 Synthesis of PGR estimates from different sources In order to have a broader view of the issue of earthquake-induced ground rotations it has been considered interesting to compare PGR estimates coming from different sources: a) empirical estimates from the proposed interpolation procedure; b) various data retrieved from literature which include: b1) array-derived estimates, which suffer from limitations at high frequencies; b2) the available direct measurements and b3) numerical/analytical simulations. In this framework, particular attention has been devoted to the recent analyses carried out by Spudich & Fletcher (2008), due to the similarities with our work. c) synthetic rotational ground motions computed by means of the spectral element code GEO-Else (Stupazzini, 2004) in the framework of 3D numerical simulations of wave motion of the valley of Grenoble (France). The simulations include the simultaneous effects of seismic source, propagation path and local site effects. Without entering in details, the numerical simulations by Spectral Element Method SEM (Faccioli et al., 1997), are based on systematic analyses of the seismic rotational wavefields induced by M6.0 and M4.5 earthquakes inside the alluvial basin of Grenoble (France). The model of the valley is well-constrained since it was object of a recent benchmark (Chaljub, 2006) and the synthetic seismograms, obtained through different numerical techniques (FD vs. SEM vs. ADER-DG) agree reasonably well. The fault is located few kms far away from the 99

116 Chapter 6. Seismic ground rotations: array-derived estimates, numerical simulations and possible effects on structural response edge of the basin and the event is right lateral strike-slip. The maximum frequency of the SE simulation is around 2.0 Hz. Further details about the simulations can be found in Stupazzini et al. (2008). As illustrative example Figure 6.8 shows some representative results of the 3D simulations of seismic wave propagation in the Grenoble valley: the spatial distribution in the Grenoble area of peak values of torsion, PGR z (=PGω z ), and tilt, PGR h (=PGω h ), is illustrated for a selected earthquake scenario (M6, neutral directivity and linear-elastic behavior). Additional information of the maximum rotations experienced in the whole model is also shown. Peak rotation maps highlight the pattern and the order of magnitude of rotational ground motions for a realistic earthquake scenario, including the simultaneous effects of source directivity, propagation path and complex geo-morphological configurations. It is clear the significant amplification of rotational ground motions inside the alluvial basin due to the softer mechanical and dynamic properties. In the bedrock the order of magnitude of rotational ground motions is 10-5, while within the alluvial basin rotations are greater by, at least, one order of magnitude, in average, for comparable source-to-site distances. However, a proper quantification of such an amplification factor should be obtained by a more detailed analysis, disaggregating the simultaneous effects of geometrical attenuation, magnitude and source directivity. In this framework, it is meaningful to compare the synthetics PGR z -PGV pairs (Figure 6.9A) with the empirical data obtained from the proposed interpolation procedure (see Table 6.3) and various published results, summarized in Table 3.2 (in Chapter 3). It gives insights into the amplitude of peak rotational motions for a rather wide range of earthquake magnitude, epicentral distances and local site conditions. The comparison, limited herein to peak torsional motions for simplicity, is depicted in Figure 6.9B: a) data from 1 to 16 refer to the published results of Table 3.2 (the same numeration is used); b) the results provided by the recent study of Spudich & Fletcher (2008) are also superimposed. Note that for each event, numbered from 17 to 20, five estimates are provided: one broad-band and four sub-array estimates, as listed in Table 6.1 c) data from 21 to 24 are those obtained by application of the interpolation procedure for the 4 earthquakes studied throughout this work; contrary to the other data, they are plotted in terms of average value (filled circle) and their minimum and maximum value (denoted by bars); Some comments arise from the analysis of Figure 6.9. First of all, the synthetic PGV-PGR z pairs, calculated on outcropping bedrock and inside the alluvial basin, show roughly a linear trend in log-log space with a coefficient of proportionality close to approximately 1 but different intercepts. As first order approximation, we can deduce that the ratio of PGV over PGR z approaches a value ~ 8000 m/s for bedrock conditions, while it drops to a value of ~ 2000 m/s for the alluvial basin. The two straight lines of eq. PGω z= PGV /θ, with θ = 8 km/s (for bedrock) and θ = 2 km/s (alluvial), are superimposed in Figure 6.9A (thick and thin line, respectively). Note that the values of θ were chosen by eye rather than a least-squares linear regression, as we are interested in capturing only the overall trend. Therefore, passing from 100

117 Chapter 6. Seismic ground rotations: array-derived estimates, numerical simulations and possible effects on structural response soft alluvial conditions to bedrock, the PGV/PGR z ratio increases by a factor of about 4, in average. As recently observed by other authors (Wang et al., 2006), this parameter seems to be rather sensitive to local geological conditions, being clearly amplified inside the alluvial basin, and it might constitute an indicator to discriminate regions characterized by shallow soft soils and by outcropping bedrock. In this framework, a controversial aspect is the following. From the experimental point of view, we noted that the ratio of PGS (or, equivalently, 2PGR) over other strong motion parameters, such as PGV or PGA, tends to be fairly constant, irrespective of site condition, at least for the datasets under consideration. Contrary to that, numerical simulations show a clear distinction between the PGV/PGR z ratio calculated at outcropping bedrock and that calculated on soft sediments. This inconsistency may be due to the limited amount of experimental data considered in this study, not suitable to study the possible dependence of rotations on local site conditions in a systematic way. In this contribution, some evidences on the potential effect of local site conditions on rotational ground motions are suggested by empirical rotation estimates on soft local conditions (see points 4, 12 and 13). As a matter of fact, these estimates turn out to be fairly high for comparable level of translational ground velocity on relatively stiff conditions. The occurrence of seismic amplification of ground rotations due to complex site effects may be an explanation for the high values of PGR found for case histories 12 and 14, provided the very large epicentral distances (~300 km). These are, in fact, large magnitude events (M~6-7) recorded in the valley of Mexico, where very large amplifications of motion due to the local velocity structure are very likely to occur (Singh et al., 1997; Bodin et al., 1997). In conclusion, whether or not rotational ground motions are affected by complex site effects more significantly than do translational motions should be checked against a larger number of experimental data and numerical simulations. Array-derived estimates obtained from the present study are in reasonable agreement with the synthetics. For comparison purposes the straight line of eq. (6.5) is also shown in Figure 6.9B (thin line). Our empirical estimates seem to align along the same linear trend observed for the synthetic PGV-PGR z pairs inside the alluvial basin in log-log space. Furthermore, the estimates at Parkway Valley array, where significant basin-related amplification phenomena are expected to occur, are coherent with the synthetics obtained, for M4.5 event, in the area of the alluvial basin, rather than on the outcropping bedrock. The single available measurement of PGV-PGR z seems to be significantly larger than the simulated and array-derived estimates (point 11). Consider however that this data refer to an explosion rather than an earthquake record such that the comparison may be improper. Unfortunately measurements from Takeo (1998) can not be used for comparison purpose, since only rotation rates, measured during M~5.0 Ito-Japan events, are available. Nevertheless, also Spudich & Fletcher, in their recent work, noted a substantial disagreement between Takeo s measurements and empirical estimates. Direct measurements of rotation rates turns out to be greater, by factor of 5-60, than other estimates and the ratio of either PGR/PGV or PGRR/PGA are systemically higher than those of Parkfield and Chi-Chi (Huang, 2003) earthquakes. According to the authors the possible explanations for such a discrepancy may be basically two: 1) point measurements are affected by heterogeneities in the soil structure with dominant wavelengths of only few meters, that either array-derived procedures, averaged 101

118 Chapter 6. Seismic ground rotations: array-derived estimates, numerical simulations and possible effects on structural response over tens/hundreds of meters, or low-frequency simulations, can not capture; 2) non-doublecouple sources, like the Ito events, are likely to generate higher rotational motions than do double-couple source mechanisms (i.e. with significant strike-slip components of motion) for comparable levels of translational motion. The agreement between array-derived estimates from different sources and synthetics is not optimal. Nevertheless empirical estimates tend to be approximately between the values simulated on bedrock and those inside the alluvial basin, probably because their frequency content is similar, not exceeding in general maximum frequency of 2 Hz. Provided that the maximum frequency of the considered SE simulations is about 2 Hz, a meaningful comparison with Spudich & Fletcher estimates involves the sub-arrays estimates filtered in the frequency band Hz (small filled triangles numbered from 17 to 20 in Figure 6.9B) rather than the broadband estimates (larger filled triangles). Referring the reader to Table 6.1 for the exact values of sub-array estimates, we observe that the agreement with Grenoble simulations is satisfactory: sub-array estimates tend to fall within the area corresponding to outcropping bedrock, coherently to the relatively stiff conditions at the UPSAR array. Figure 6.8 Grenoble basin: example of 3D simulations by GeoELSE. Maps of the peak values of torsion (PGω z = PGR z ) for a selected earthquake scenario (M W =6, neutral directivity, linear-elastic behavior; the location of the hypocenter is indicated on the upper left corner of the graph). On the right-hand-side map the location, where the maximum rotation is experienced, is highlighted. From Stupazzini et al. (in preparation). 102

119 Chapter 6. Seismic ground rotations: array-derived estimates, numerical simulations and possible effects on structural response Table 6.3 Peak values of Ground Velocity (PGV h ) and Ground Rotation (PGR z ) obtained from the proposed interpolation procedure to be complemented to Table 3.2 (Chapter 3). PGRs are given in terms of average, maximum and minimum values. Reference # Our work EQ. parameters M w R [km] Source mech. PGV h [m/s] PGR z [rad] N.A Min 8.79E-08 Mean 1.64E-07 Max 3.38E N.A Min 1.16E-06 Mean 2.25E-06 Max 3.33E SS SS Mean 8.98E-05 Min 4.07E-05 Max 1.51E-04 Mean 7.68E-05 Min 4.23E-05 Max 1.25E-04 Symbol A 103

120 Chapter 6. Seismic ground rotations: array-derived estimates, numerical simulations and possible effects on structural response B Figure 6.9 A) Synthetic values of PGV h vs. PGR z in logarithmic scale obtained from SE simulation of Grenoble basin with reference to events with M6.0 and M4.5, neutral directivity and linear visco-elastic soil behavior. Superimposed are the straight lines of eq. PGR z = PGV h /8000 and PGR z = PGV h /2000, describing the overall trend of the synthetics on outcropping bedrock and inside the soft deposits, respectively. B) The individual data retrieved from literature, listed in Table 3.2 (in Chapter 3) are compared with synthetics obtained in the Grneoble area. Data from 21 to 24, summarized in Table 6.3, are plotted in terms of average value (filled circle) and their minimum and maximum value (denoted by bars). The straight line of eq. (6.5) is also superimposed for comparison purposes (dashed line). Adapted from Stupazzini et al. (in preparation). 6.4 Effect of coupled tilt and translational motion on structural response A simple engineering application is shown in order to investigate the potential effects of coupled tilt and translational motions on structural response. The analyses starts from the consideration that dynamic response of structures subjected to earthquake induced base excitations are usually simplified in engineering practice by ignoring tilt and torsional component of ground motion. Recently a similar study has been carried out by Kalkan & Graizer (2007). Contrary to what done by these authors in the study presented herein soilstructure interaction effects are taken into account in the linear-elastic range. To this end, a simplified model of a single degree-of-freedom (dof) structure resting on a 2- dof foundation (=horizontal translation and tilt around its centre of mass) is considered. 104

121 Chapter 6. Seismic ground rotations: array-derived estimates, numerical simulations and possible effects on structural response Linear behavior of both the superstructure and the foundation is considered. The dynamic equations of motion are the following: M x + C x + K x = p (6.7a) where [ ] T 1 0 φ x = x x (6.7b) p = m1 x g m0 x g J φ g (6.7c) M m = m 0 0 J (6.7d) c1 c1 c1h C = c1 c1 + c0 c1h 2 c 1h c1h c1h + cr k1 k1 k1h K = k1 k1 + k0 k1h 2 k 1h k1h k1h + kr (6.7e) (6.7f) The terms in the equations above are explained in Figure In order to study a realistic case, where rotational motions might be intuitively relevant, i.e. tall structure over a flexible base, we have selected a representative pier of the high-speed railway (Gavi) of the TGV Mediterranean line. The relevant structural parameters are given in Table 6.4, while Table 6.5 provides the equivalent elastic spring and dashpot coefficients for modeling soil-structure interaction. As input for the analysis of the 3dof system, the following acceleration time histories have been considered as potentially representative of realistic coupled translational-rotational excitations: i. tilt motions at two receivers of UPSAR (namely, receivers 6 and 9) during the Parkfield event as obtained from the proposed interpolation procedure; ii. three representative rotation time histories obtained by SE simulation in the Grenoble valley, shortly illustrated in the previous section. Table 6.6 summarizes some representative engineering measures of both rotational and translational ground motion used as input for this simple analysis. As illustrative example, 105

122 Chapter 6. Seismic ground rotations: array-derived estimates, numerical simulations and possible effects on structural response Figure 6.11 depicts the synthetic translational (up) and tilt (bottom) time histories for event num. 5 in Table 6.6. The latter corresponds to the most severe excitation among the considered suite. Referring to the same input excitation, Figure 6.12 illustrates both time variations (up, A) and Fourier amplitude spectra (bottom, B) of relative displacement (x s =x 1 - x 0 - hφ), velocity and absolute acceleration (d 2 x s /dt 2 + d 2 x g /dt 2 ) of the structure when the latter is subjected to either pure translational motion (thick lines) or to coupled tilt and translation (dotted lines) base excitation. It is apparent that, even for the strongest base excitation, the effect of coupled translational-tilt motion on structural response turns out to be small, at least for the assumed structural model and for the limited suite under study (M<=6.0). Note that time variations of structural displacement do not show any significant difference passing from translational to coupled base excitations. For the same case illustrated in Figure 6.12, the difference ε (in percentage) between the peak displacement responses is ~ 4%, thus negligible for engineering purposes. On the other hand, spectral amplitudes show some differences for frequencies greater than about Hz (see Figure 6.12B). This is reasonable, since rotational ground motions are expected to affect mainly high frequency components. Nevertheless, such differences are nearly negligible from the engineering perspective. As comprehensive graph, Figure 6.13A shows the ratio of the peak relative displacement of the structure under coupled tilt-translational excitation, d max tilt, over the peak relative displacement under pure translational motion, d max trans, as function of the ratio between the peak ground rotation, PGR, and the peak ground translational velocity, PGV. The ratio PGR/PGV (in s/m) can be reasonably regarded as a possible representative parameter of the relevance of tilt excitations with respect to translational horizontal motions. Furthermore, provided the discussion of the previous sections, this parameter could be useful for identifying a possible threshold above which rotational excitations have a non-negligible influence on structural response. As anticipated previously, the maximum effects in terms of peak relative displacement turn out to be approximately 6% in the worst case (point 1 of Figure 6.13A), suggesting a negligible influence of rotational excitations on the structural response, in agreement with pioneering Richter s recommendation (1958), at least for the considered range of earthquake magnitudes and for PGV/PGR >~ 1000 m/s. As a general indication, Eurocode 8 Part II (CEN, 2002) confirms that the contribution of rotational components to the total response is in most cases small, i.e. do not increase the response more than 10%. Figure 6.13B depicts the same results as in Figure 6.13A but in terms of the ratio of the peak absolute acceleration of the structure under coupled tilt-translational excitation, a max tilt, over the peak absolute acceleration under pure translational motion, a max trans. The ratio a max tilt /a max trans turns out to be more affected by coupled tilt-translational motions, reaching a maximum value of around 11% (as before for input #1 of Table 6.6). This increase may be reasonably explained by the fact that accelerations are more affected by high frequency components of input excitations and, therefore, rotational motions, than do displacements. Furthermore, note that the considered range of PGV/PGR ratios, from approximately to 10 3 m/s, is likely representative of realistic earthquake-induced ground motions, at least for 4.0<M<6.5, provided the data analyzed in this work (see e.g. Figure 6.9). Note that Kalkan & Graizer (2007), in their recent study, obtained a much greater effect of rotational excitations on the structural response (see e.g. Figure 3.9), because they used a rotation rate time history of maximum amplitude ~ 15 degrees/sec, that is an exception rather than a common case. 106

123 Chapter 6. Seismic ground rotations: array-derived estimates, numerical simulations and possible effects on structural response As a final remark, such preliminary analyses do not allow one to draw general conclusion. It is not guaranteed, indeed, that tilt-torsional ground motions of significant amplitude, as those registered in near-field highly variable regions of strong earthquakes (M>=7) or influenced by complex site effects, remain negligible for earthquake engineering applications. Figure 6.10 Three degree-of-freedom model considered in the analysis of the effects of coupled translational and rotational ground motion on structural response. Table 6.4 Structural model parameters. J is the sum of the centroidal moments of inertia of the building and the foundation and B is the width of the foundation. m 1 [kg] k 1 [N/m] c 1 /c cr [%] J [kg m 2 ] h [m] m 0 [kg] B [m] 3.10E E E E E+05 7 Table 6.5 Equivalent spring and dashpot coefficients. k v [N/m] k 0 [N/m] k r [N/m] c 0 [Nsm] c r [Nsm] 2.00E E E E E

124 Chapter 6. Seismic ground rotations: array-derived estimates, numerical simulations and possible effects on structural response Table 6.6 Rotational and translational time histories considered as input for the analysis of the response of the simple model sketched in Figure Some engineering measures of earthquake-induced ground motion, such as PGA h, PGd 2 ω/dt 2, PGV h, PGD and PGR are listed. Event # PGA h Parkfiled (rec. 06) Parkfield (rec. 09) Grenoble A (rock) Grenoble B (basin) Grenoble C (basin) [m/s 2 ] PGd 2 ω/dt 2 [rad/s 2 ] PGV h [m/s] PGD [m] PGR [rad] Figure 6.11 Horizontal (up) and tilt (bottom) ground accelerations for the event # 5 in Table 6.6 (Grenoble basin) considered as input for the study of the 3dof model. They are synthetic time histories obtained by the Spectral Element simulation of the seismic response of Grenoble basin for a realistic earthquake scenario (see previous section). 108

125 Chapter 6. Seismic ground rotations: array-derived estimates, numerical simulations and possible effects on structural response A B Figure 6.12 Relative displacement, velocity and absolute acceleration of the 3dof system, when either pure translational (thin lines) or coupled tilt-translational (thick lines) base excitations are considered, for the event num. 5 in Table 6.6. The corresponding Fourier Amplitude spectra are also illustrated on the right-hand-side. 109

126 Chapter 6. Seismic ground rotations: array-derived estimates, numerical simulations and possible effects on structural response Figure 6.13 A) Peak relative displacement of the structure under coupled tilt-translational excitation, d tilt trans max, over the peak relative displacement under pure translational motion, d max as function of the ratio (in s/m) between the peak ground rotation, PGR, and the peak ground translational velocity, PGV. The 5 case studies listed in Table 6.6, numbered from 1 to 5, are considered. B) Same as in A) but in terms of the ratio of the peak absolute acceleration of the structure under coupled tilt-translational excitation, a tilt max, over the peak absolute acceleration under pure translational motion, a trans max. 110

127 Chapter 7. Conclusions 7 CONCLUSIONS In this work the evaluation of transient seismic ground strains and rotations has been investigated from both engineering and seismological perspectives. Earthquake-induced ground differential motions constitute a challenging issue owing to the substantial lack of direct measurements and their effects may be relevant for some kinds of applications. Although the role of ground strains is widely accepted as crucial for damaging underground structures, such as subways, tunnels and buried pipelines, seismic design of this kind of structures commonly relies on simplified relationships based on limiting assumptions of plane wave propagation in a homogenous ideal medium, which are not generally fulfilled in reality. On the other hand, the relevance of rotational ground motions for engineering applications is still subject of debate. Nonetheless a deeper understanding of the magnitude of ground rotations under realistic seismic excitations could have several implications from a seismological point of view, such as the possible recovery of permanent ground displacement from accelerometer recordings in near-source regions or additional information about earthquake rupture process. In this framework, an observationally-based procedure is proposed to calculate the in-plane strain and rotation tensors at ground surface, based on the spatial interpolation of the threecomponent displacement wavefield from records obtained by two dense seismic arrays, namely Parkway Valley (New Zealand) and the UPSAR array (California). Four earthquakes have been considered, covering a magnitude range from 4.2 to 6.5, a distance range from 11 km to about 80 km and different local site conditions. The accuracy of the interpolation procedure has first been explored by suitable validation checks that showed that the selected Biharmonic Spline technique allows one to obtain a reasonable approximation of the displacement wavefield up to approximately 2.5 Hz, in the least favorable case, i.e. larger separation distances. As for the peak values of ground motion, the estimated error was found to range between 5% and 15% in terms of PGD and between 20% to 35% in terms of PGV. Displacement gradients at the ground surface are hence determined applying a classical central finite difference scheme. Referring to seismic ground strains, a noticeable effort has been devoted to introduce invariant strain measures and to study their dependence with azimuth. Furthermore, new empirical correlations for Peak Ground Strain, PGS, as a function of other parameters of earthquake severity, such as Peak Ground Acceleration PGA, Peak Ground Velocity PGV and Peak Ground Displacement PGD, were determined. 111

128 Chapter 7. Conclusions On the other side, provided the limited acquaintance with seismic ground rotations, the topic is analyzed at a different level of detail. First, empirical estimates for transient ground rotations are obtained through the application of the proposed interpolation procedure. The latter is, in fact, capable of determining the 6 independent components of the displacement gradient tensor at the ground surface, once both free-surface condition and Hooke s law are introduced. Furthermore, PGRs obtained by the interpolation procedure are compared with: a) the synthetics computed by Spectral Element 3D simulation for a complex 3D model of both source process and geological configuration; b) the individual data retrieved from literature, comprising either array-based estimates, direct measurements or analytical/numerical simulations and c) some very recent empirical estimates. This rather broad comparison allows one to have a better understanding of the magnitude of rotational ground motions for a given earthquake scenario, depending on the magnitude, source-site distance and local site conditions. Owing to the lack of direct measurements and to the scarce research activity on this topic so far, this is an original outcome of this study. Below are summarized the most significant results and main indications regarding transient seismic strains and rotations. 7.1 Ground strains i. The determination of the horizontal strain tensor at the ground surface thanks to the interpolation procedure allows one to define invariant measures of peak ground strain, PGS, suitable to define its magnitude, independent of the specific direction along which it occurs. At a selected location, it is possible to calculate the highest principal strain HPS(t) and the lowest principal strain LPS(t). An interesting feature is that these two quantities tend to be approximately out-of-phase suggesting prevailing shear strain conditions at least during the most significant phase of motion. ii. The analysis of time-space variations of principal strain directions in relation with the prevailing direction of particle velocity did not lead to straightforward interpretations. Although one would expect that the principal strain directions to align along a direction rotated approximately 45 with respect to the prevailing direction of particle motion under shear strain conditions, this expectation was not clearly confirmed by all datasets. In particular the data obtained for the Parkfield earthquake are complex, probably owing to the occurrence of more complex effects which are not accounted for by the previous elementary considerations. iii. New empirical correlations between PGS and other peak values of ground motion, such as PGD, PGV and PGA, were derived for each case study. Their dependence on azimuth was found to affect the results by a factor of about two, at least based on the available datasets. Such observation roughly suggests that the largest values of strain tend to be aligned along the prevailing wave propagation directions, as it could be deduced from simple theoretical considerations, based on the assumption of plane wave propagation. iv. When the correlation is extended to all observed datasets, a clear linear trend of PGS as a function of either PGA, PGV or PGD is observed in log-log space with a 112

129 Chapter 7. Conclusions remarkably low dispersion (σ ~ 0.16) whatever the selected parameter. Furthermore, the low dispersion of results shows that there is a poor dependence on earthquake magnitude, source-to-site distance, wave type, as was also suggested by Trifunac and Lee (1996), based on a set of parametric numerical analyses. Whether or not such correlations depend on site conditions should be checked against a larger number of experimental data and numerical simulations. v. The observed PGV-PGS pairs tend to be aligned, irrespective of the earthquake magnitude, distance, site conditions or prevailing wave type, along the Least Squares (LS) best-fit lines of equation PGS~PGV/φ with φ~1000 m/s, in terms of median values (see eq. (5.13)). The comparison with direct strain gauge measurements performed in two buried pipelines in the town of Hachinohe during the 1978 Miyagiken-Oki earthquake (M7.8), Japan (Iwamoto et al., 1988), on one side, and, on the other side, the numerically-derived relations published by Trifunac & Lee (1996) confirms the same linear trend. vi. The superposition of results from the interpolation procedure, in terms of PGA-PGS pairs with the ones obtained by direct strain gauge measurements in buried pipelines (Nakamura et al, 1981, quoted by Trifunac and Lee, 1996) and tunnels in Japan (JSCE, 1977) is still satisfactory. This constitutes a further indirect confirmation of the reliability of our calculations. vii. The results are also in reasonable agreement with some published solutions relating PGS to PGD (Abrahamson, 2003). All these solutions highlight that wave passage effect is but one of the factors affecting earthquake-induced ground strains. The additional combined effect of spatial incoherence of ground motion and possible complex site effects increases significantly the PGS values with respect to the simplified evaluations used in the engineering practice, that tend to underestimate the observations. viii. As a further validation, the observed PGV-PGS pairs are in good agreement with the broadband estimates recently derived by Spudich & Fletcher (2008) for the Parkfield main shock and three aftershocks, based on the application of the so-called seismogeodetic approach. The value of PGS calculated independently by the authors for the Parkfield earthquake fits fairly well with our estimates for the same event. Moreover data corresponding to the aftershocks tend to be reproduced with reasonable agreement by the straight line of eq. (5.13), derived from our estimates alone. ix. As a final consideration, the commonly adopted PGS values for the seismic design of underground structures (PGV/2000 PGV/4000) turn out to be significantly smaller than the ones derived empirically (~PGV/1000), by a factor of at least 2-4, suggesting that seismic design recommendations may be unconservative. This discrepancy may be due to spatial incoherence effects, not accounted for in the simplified formulae used in engineering applications. Such an effect may yield maximum strain estimates significantly higher than those obtained by the simplified relationship PGV/V a, V a being the apparent wave propagation velocity. 113

130 Chapter 7. Conclusions 7.2 Ground rotations i. The same interpolation procedure used for the empirical evaluation of ground strains has been also applied to the 4 datasets considered throughout this study, to calculate rotations and tilts at ground surface. For low magnitude events (M=4.2 and 4.9) and large epicentral distances (R ~ 81 km) maximum torsions are ~ 10-7 and ~ 10-6, respectively, while for severe earthquakes (M=6.0 and 6.5) in the near-source region (R ~ 10 km) rotations are of the order of ii. The PGV-PGR z pairs tend to be aligned, with relative low dispersion (σ~0.16, as well as for maximum strain estimates), along a straight line of expression PGR~PGV/χ with χ ~ 2100 m/s (median value). The slope of the PGR z /PGV Least-Squares line (see eq. (6.5)) tends to be smaller by a factor of about two with respect to the corresponding slope of the PGS/PGV line (see eq. (5.13)), coherently to eqs. (2.1) and (3.6). Nevertheless, this evidence does not necessarily imply that strains and torsions scale with the actual apparent propagation velocity, as also commented by Spudich & Fletcher (2008). Assuming the PGV/PGR z ratio as the actual V a, apart from a factor of 2, would be misleading. Consider, for instance, that half the median ratio PGV/PGR z ~ 2 km/s, inferred from eq. (6.5), is significantly smaller than the apparent wave propagation velocity V a ~ 2.5 km/s estimated at the UPSAR array by spatial crosscorrelation analyses of the Parkfield earthquake records (Fletcher et al., 2006). iii. The comparison of PGV-PGR z pairs, derived by our interpolation procedure, with the estimates obtained by Spudich & Fletcher (Pers. Comm., 2008) is very satisfactory. For instance, the value of PGR z determined by Spudich & Fletcher for the Parkfield main shock ( ) is almost the same as the average value obtained by our interpolation procedure ( ). iv. A broad set of synthetics PGV-PGR z obtained through complex Spectral Element simulations by Stupazzini et al. (2008) for the Grenoble valley (France) is compared with the individual data retrieved from literature (including array-derived estimates, numerical/semi-analytical simulations and direct measurements) and with the empirical estimates as computed by the interpolation procedure. The comparison recovers a wide range of magnitudes, epicentral distance and site conditions. In log-log scale our empirical estimates seem roughly to align along the same linear trend observed for the synthetics PGV-PGR z pairs. In particular, the best-fitting line of eq. (6.5), calibrated on the experimental dataset alone, reproduces with reasonable agreement the overall trend of the synthetic PGR z s inside the alluvial basin. The agreement between array-derived estimates from different sources and synthetics is not optimal. Nevertheless empirical estimates tend to be approximately between the values simulated on bedrock and those inside the alluvial basin, probably because their frequency content is similar, not exceeding, in general, the maximum frequency of 2 Hz. Direct measurements of earthquake-induced rotations are very few such that it is difficult to evaluate the degree of accuracy and reliability of such estimates. However, they seem to provide higher values than both empirical and numerical estimates, as also underlined by other authors (Takeo, 1998; Spudich & Fletcher, 2008). 114

131 Chapter 7. Conclusions v. The SE numerical simulations for the Grenoble valley, which include the coupled effects of the seismic source, the propagation path between the fault and the observation points and local geo-morphological configurations, show a rather strong amplification of rotational ground motion inside the basin with respect to bedrock, by a factor of 10, in average. However, a proper quantification of such an amplification factor should be obtained by a more detailed analysis, disaggregating the simultaneous effects of geometrical attenuation, magnitude and source directivity.the synthetic PGV-PGR z pairs, subdivided into bedrock and soft soil conditions, roughly show a linear trend in log-log space with similar coefficient of proportionality, equal to approximately 1, but different intercepts. Roughly, this difference suggests that, passing from soft alluvial conditions to bedrock, the PGV/PGR z ratio increases by a factor of about 4, in average. vi. The dependence of rotations, specifically of the PGV/PGR z ratio, on complex site effects is controversial. From the experimental point of view, we noted that the ratio of PGS (or, equivalently, 2PGR) over other strong motion parameters, such as PGV or PGA, tends to be fairly constant, irrespective of site condition, at least for the datasets under consideration. On the other side, as commented above, deterministic simulations show a clear distinction between the PGV/PGR z ratio calculated at outcropping bedrock and that calculated on soft sediments by a factor of 4, in average. This discrepancy may be due to the limited amount of experimental data considered in this study, not suitable to study systematically the possible dependence of rotations on local site conditions. In this contribution, some evidences on the potential effect of local site conditions on rotational ground motions are suggested by empirical rotation estimates on soft local conditions from other sources. Whether or not rotational ground motions are affected by complex site effects more significantly than do translational motions should be checked, indeed, against a larger number of experimental data and numerical simulations. vii. A simple 3 degree-of-freedom model subjected to either purely translational or coupled tilt-translational base excitations is considered in order to study the potential effects of rotational motions on structural response. To this end, the analyses have been carried out for some representative tilt time histories, as the ones calculated at UPSAR, for the Parkfield event, through the interpolation procedure and the synthetics computed by the SE code for the Grenoble area. The effects on structural response, quantified as ratio of the peak relative displacement of the structure under coupled tilttranslational excitation, d max tilt, over the peak relative displacement under pure translational motion, d max trans, turn out to be negligible (<6%) from the engineering point of view, at least for the assumed model and for the considered range of translational/rotational excitations. When the ratio a max tilt /a max trans, a being the absolute structural acceleration, is considered the contribution of rotational motion to the structural response increases up to about 10%. Eurocode 8 Part II (CEN, 2002) confirms, as a general indication, that the contribution of rotational components to the total response is in most cases small, i.e. does not increase the response more than about 10%. However it is worth underlining that this analyses is only a starting point for further investigations. In conclusion, the role of rotational excitations for 115

132 Chapter 7. Conclusions earthquake damage potential is subject of debate and it should be further investigated with more complex structural models and a broader range of earthquake-induced rotational excitations. 116

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141 Appendix A APPENDIX A After presenting some meaningful case histories of underground structures and pipeline damage due to earthquakes, this appendix shows a synthetic overview of the most important damage relations used to estimate the likely amounts of damage to lifelines due to earthquakeinduced motions is shown San Francisco In the April (M W =7.9) earthquake the loss of water supply in the San Francisco distribution system, caused by the localized breaks of its buried pipelines was the major factor that led to the largest urban fire loss in U.S. history. Consider that the number of fires and/or explosions is estimated to be around 50 and 52 (Reed, 1906; Scawthorn & O Rourke, 1989). In this case, Contrary to the Chi-Chi earthquake data, damage statistics reveal that pipeline breaks were predominantly caused by liquefaction-induced failures: about 52% of the total pipeline breaks occurred, indeed, in the poorly extended liquefaction zones (O Rourke & Liu, 1999) Northridge, California The Jan Northridge earthquake (M W =6.4) provoked more than 1400 pipe breaks within solely the San Fernando area with deformation levels of the order from approximately and (Trifunac & Todorovska, 1997). Figure A. 1 depicts the areas characterized by a density of pipe breaks superior of 6 breaks/km both in (a) San Fernando region and (b) Los Angeles region Hyogoken-Nambu, Kobe (Japan) As a consequence of the 1995 Hyogoken-Nambu earthquake (M W =6.9) the collapse of the Daikai subway station, which had not been designed according to seismic resistance criteria, occurred Chi-Chi, Taiwan During the 1999 Chi-Chi earthquake (M W =7.5) the main damage occurred in tunnel portals due to soil instability phenomena as shown in Figure A. 2 (Hashash et al., 2001). One tunnel crossing the Chelungpu fault was destroyed because of 4 m fault offset Kocaeli and Düzce earthquakes, Turkey I

142 Appendix A The M W =7.1 earthquake of Nov 12, 1999, generated a right-lateral surface rupture of 40 km length on the so-called Düzce fault, which is a major northern branch of the North Anatolian Fault (NAF). At the time of the Düzce earthquake the excavation of the 3.3 km long twin Bolu tunnel, located along the Trans-European Motorway TEM and constructed according to the New Austrian Tunneling Method (NATM), was almost complete with about 30% of the final linear system in place (Ghasemi et al., 1999). The 12 Nov earthquake caused the collapse of both tunnels about m from the Elmalik portal. The collapse occurred in a section of the tunnel passing through clay weak gauge materials (Hashash et al., 2001). Figure A. 3 shows the map of the Duzce water network with the reported damage after the Kocaeli - Duzce sequence (from Alexoudi et al., 2007). Figure A. 1 San Fernando and Los Angeles regions showing those areas characterized by more than 6 pipe break per km. From Trifunac & Torodovska (1997). II

143 Appendix A Figure A. 2 Evidences of slope failures at tunnel portal (left) and conduct break (right) during the 1999 Chi-Chi earthquake. From Hashash et al. (2001). Figure A. 3 Reported damage of the Duzce s water network. From Alexoudi et al. (2007). Distinct damage relations exist in literature for permanent ground deformation and wave propagation effects. Provided the scope of the overall work, we will focus on the latter aspect. A more detailed review of damage relations due to wave propagation can be found in O Rourke & Liu (1999), ALA (2001) and Tromans (2004). III