Effect of Pre-Yielding Elasticity on Sliding Triggered by Near-Fault Motions Modeled as Idealized Wavelets

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1 Effect of Pre-Yielding Elasticity on Sliding Triggered by Near-Fault Motions Modeled as Idealized Wavelets E. Garini, G. Gazetas, N. Gerolymos Soil Mechanics Laboratory, National Technical University of Athens, Greece ABSTRACT: The influence of elastic pre-yielding on the response of a mass resting on an inclined plane is investigated in this paper. The ultimate shearing capacity of the interface obeys Coulomb s friction law. The slope is subjected to near-fault triggering by two types of idealized wavelets: (i) a Ricker wavelet, representative of forward directivity affected motions, containing strong long-period acceleration pulses, and (ii) an one-cycle sinusoidal wavelet, representative of fling-affected motions, containing an one-sided velocity pulse with an ensuing permanent displacement. The asymmetric sliding response is analyzed and the effect of a number of parameters is explored. They include: the critical acceleration ratio, a C /a H, the excitation frequency, f o, the changing polarity of excitation, and the magnitude of elastic pre-yielding displacement, dy. INTRODUCTION Several applications in geotechnical earthquake engineering require an understanding of the dynamic sliding response of a block of mass m supported on seismically vibrating base through an asymmetric frictional contact. Ιn his 9 seminal Rankine Lecture, Newmark proposed that the seismic performance of earth dams and embankments be evaluated in terms of permanent deformations which occur whenever the inertia forces on a potential slide mass are large enough to overcome the frictional resistance at the failure surface. He proposed the analog of a rigid block on inclined plane for a simple way of analytically obtaining approximate estimates of these deformations. Newmark s analog has seen numerous applications and extensions: seismic deformation analysis of earth dams and embankments, displacements associated with landslides, seismic deformation of landfills with geosynthetic liners, seismic settlement of surface foundations, movements of wedges in rock slopes, and even potential sliding of concrete gravity dams. The extension of the analog by Richards & Elms (979) to gravity retaining walls has met worldwide acceptance, and has found its way into seismic codes of practice. Several other generalized applications have also appeared. A numerical study has been recently presented by the authors (Garini et al 7, Gazetas et al 9) for a rigid block supported through a rigid-plastic frictional contact surface on an inclined plane, and subjected to slope-parallel excitation. The latter was described with nearfault seismic records strongly influenced by forward-directivity or fling-step effects. Our study had consistently and repeatedly revealed a profound sensitivity of both maximum and residual slippage: (i) on the sequence and even the details of the pulses contained in the excitation, and (ii) on the polarity of shaking. A few of the findings contradicted some of the prevailing beliefs that have emanated from statistical correlation studies in literature. However, all these finding were based on the extreme assumption of a perfectly rigid-plastic interface. Since in most realistic systems some pre-sliding elasticity is unavoidable, this paper investigates an elastic-plastic support interface. Fig. illustrates the problem studied herein.

2 (a) D(t) μ =f(d) m β A(t) (b) (c) T(t) T(t) mg(μcosβ sinβ) mg(μcosβ sinβ) D(t) dy D(t) mg(μcosβ + sinβ) mg(μcosβ + sinβ) Figure. (a) The problem studied in the paper sliding on inclined plane undergoing excitation parallel to the slope, (b) ideally rigid-plastic behavior of the interface as studied by Garini et al (7), and (c) elastic-perfectly plastic sliding response studied here. DIRECTIVITY AND FLING IN NEAR-FAULT MOTIONS Earlier studies of Newmark-sliding were based on records available in the late 97 s and 98 s. Very few of those motions were near-fault records from large magnitude (M >.) events. Today, however, such near-fault records are known to often contain either long-period high-amplitude acceleration pulses, or large residual displacements the outcome, respectively, either of the coherent arrival of seismic waves when the fault rupture propagates towards the site, or of tectonic permanent displacement (offset) of the earth in the proximity of the seismogenic fault rupture. The terms forward-rupture directivity and fling step have been given to the two phenomena (Singh, 988; Somerville et al, 99; Abrahamson, ; and Bolt, ). Fig. illustrates in idealized form some fundamental characteristics of these two types of near-fault motions. For strike slip earthquakes, the signature of forward rupture directivity appears in direction normal to the fault; whereas, the fling step is significant in the parallel component of motion in close proximity to the fault, especially if the latter emerges on the surface with a large static offset. The two phenomena (and directivity in particular) have been the subject of seismological (theoretical and instrumental) as well as earthquake engineering research. Two idealized motions ( wavelets ) are used as excitation in this paper to represent in a simple way typical directivity and fling affected ground motions. They are the one-cycle sinusoidal and the Ricker wavelets: the former modeling a typical fling affected motion, and the latter a directivity affected motion. Four characteristic frequencies are utilized for each of these motions as follows: Richer: f o =. Hz,.7 Hz,. Hz, and, Hz Sinusoidal: f o =.7 Hz,. Hz,. Hz,.7 Hz

3 The response spectra of corresponding pairs of these motions are approximately matched to give almost the same maximum spectral acceleration at nearly the same dominant period. For instance, Fig. illustrates the spectral matching of two of the pairs. Fault strike Site Directivity pulse (Normal component of Displacement) Fling step (Parallel component of Displacement) General Example: Forward Directivity General Example: Fling Step Α(t) : m/s (Ricker wavelet) (One-cycle Sinus pulse ) 7 9 V(t) : m/s ΔV V. m/s 7 9 D(t) : m D maxl =. m..7 D residual = D residual =.9 m 7 9 Figure. Explanatory sketch of the forward-directivity and fling-step phenomena as reflected in the displacement records; and examples of simple wavelets bearing the signature of the two effects. RESULTS: THE EFFECT OF PRE-YIELDING ELASTICITY ON SLIDING Fig. depicts the asymmetric sliding response in terms of acceleration, velocity, and displacement time-histories for a mass of m = Mgr on a slope of inclination β = o. The system is subjected to a Ricker wavelet of peak acceleration g and central frequency. Hz. The right hand-side portrays the time-histories corresponding to a rigid-plastic interface with critical ratio of a C /a H =.. Notice that the mass is moving in unison with the base as long as the critical acceleration a C is not exceeded. Whenever base acceleration exceeds the critical

4 acceleration, the block slides either downward (usually) or upward (rarely). As a result, a residual yielding displacement of. m occurs. The left hand-side of Fig. illustrates the elastic-plastic response of the same mass-base system. Here, the block can displace elastically up to. m before yielding. As a consequence, a phase difference between input excitation and induced mass response occurs. Notice this phase shift in Fig. in the acceleration and velocity time-histories (see arrows). It is emphasized that pre-yielding displacement exists in both the upward and downward direction. Thus, even though no uphill sliding happens after sec in Fig., an elastic upward displacement does take place reversal of accumulated slippage. In addition, the block does not rest after the last sliding period but continues uphill and downhill elastic oscillations after the end of triggering. The force-displacement hysteresis loop in Fig. shows the two yielding events and the accompanying elastic branches.. g Ricker, f o =. Hz,. Hz g g One-cycle Sinus, f o =.7 Hz,. Hz g Ricker One-cycle Sinus S A : g T : s Figure. The two idealized time histories used as base excitation, with their response acceleration spectra S A :g T. (Peak ground acceleration: g). COMPILATION OF RESULTS All our numerical results with Ricker wavelets as base excitations are compiled in Figs and 7. Fig. depicts the permanent slippage with respect to the critical acceleration ratio, a C /a H, for four values of pre-yielding deformation, dy, and four excitation frequencies, f o. As expected, when the acceleration ratio, a C /a H, increases the induced slippage decreases. This general trend is independent of the existence or not of the pre-yielding displacement. Notice that the existence of pre-yielding elasticity may lead to larger or smaller permanent displacements, depending on f o and a C /a H. Furthermore, in Fig.7 observe the influence of frequency on sliding displacement. For frequency, f o =. Hz, as the elastic region dy increases the slippage D also increases. The

5 same is valid for f o =.7 Hz. However, when the frequency, f o, increases up to. Hz the response changes. Observe that while slippage becomes greater for pre-yielding region between and., for larger values of dy the sliding response decreases, in the case of. Hz. The behavior is more complicated when the frequency takes the value of. Hz. α C / α Η =. o Ground Sliding block dy =. m dy = m Α : m/ s V : m/ s m D : m... m st slide nd slide.9 m.. Figure. Acceleration, velocity, and displacement time histories for a maximum elastic deformation of dy =. m are presented at the left, and for dy = are presented at the right (excitation: Ricker wavelet of frequency f o =. Hz). EFFECT OF CHANGING POLARITY The next two Figures (Figs 8 and 9) address a most astonishing effect: that of the reversal in polarity (i.e., change from + to direction in which the excitation is applied). [This is the same as having two identical slopes, one opposite to the other ( across the street so to speak), subjected to the same excitation, as sketched at the top of Figs. 8 and 9.] A few researchers and only in recent years (Kramer & Lindwall, ; Fardis et al, ) appear to have published on the importance of the polarity of shaking. This has much to do

6 st slide nd slide T : kn D : m Initial elastic deformation induced by the m g sinβ component Figure. Force-displacement response for the case of an elasto-plastic sliding system with dy =. m, β = o, a C /a H =. and a Ricker excitation of. Hz frequency. Notice that yielding occurs only in one direction, as it was expected.. g m = Mgr o g 8 Ricker. Hz. Ricker.7 Hz Slippage : m 9 dy = dy =. m dy =. m dy =. m.... dy = dy =. m dy =. m dy =. m.. Slippage : m..8. Ricker. Hz dy = dy =. m dy =. m dy =. m...8. Ricker. Hz dy = dy =. m dy =. m dy =. m α C /α H α C /α H Figure. Influence of maximum elastic deformation, dy, on asymmetric sliding response triggered by Ricker wavelets of maximum acceleration g and of different frequencies.

7 . g α C /α H... o g Ricker. Hz Ricker.7 Hz Slippage : m Slippage : m Ricker. Hz Ricker. Hz dy : m dy : m Figure 7. Effect of characteristic frequency, f o, and of α C /α Η ratio on maximum slippage with respect to elastic deformation, dy (Excitation: single Ricker wavelet). with the asymmetry of recorded motions, which is what accentuates the importance of polarity. It is mainly the near fault strong motions which are highly asymmetric due to the contained directivity and fling pulses. But few such motions had been recorded worldwide twenty years ago. Now a large number has become available. The sliding analysis of Fig. 8 is simple but most revealing. For a steep slope (β = ο ) and a yield acceleration ratio a C /a H =., we notice the following : When the first sinusoidal acceleration half-pulse is downward [as in Fig. 8(a) on the right] the block remains almost attached to the base. Only a mere.8 m uphill displacement takes place (see the upward slippage region enclosed by the dotted lines, starting at sec and ending at.8 sec). Even in this small.8 m deformation, the yielding part is particularly smaller than the elastic. The subsequent, second (and last), upward half-pulse acceleration of the base initiates an uninhibited downslope slippage of the block, which lasts for a long time after the excitation has terminated Δt. sec on t. sec. The result is a huge 7.7 m. In stark contrast, when the first sinusoidal acceleration half-pulse of the base is upward [as in Fig. 8(b) at the left] the block starts sliding downslope almost immediately. But it soon comes to a stop after about. seconds, as the upward base motion decelerates and then reverses. The resulting residual slip is only. m, almost. times smaller than the 7.7 m produced with the reverse motion! This effect of reversing the polarity of shaking is of profound importance, especially with fling type motions (as the sinus pulse studied above). It may not however be as dramatic with

8 directivity affected motions if they contain several competing cycles of pulses, as seen in Fig. 9 with the Ricker wavelet. + α C / α Η =. o o Ground Sliding block (b) (a) A(t) : m/s D(t) : m 8 7. m m T : kn D : m D : m Figure 8. Acceleration and displacement time histories for a stiff elasto-plastic system with yield displacement dy =. m. The third row of figures illustrates the force displacement response. (Excitation: one-cycle Sinus of.7 Hz frequency). CONCLUSIONS The solutions portrayed graphically in the paper (through the acceleration and velocity time histories of the base and mass) are easy to understand, offering considerable insight into the dynamics of asymmetric sliding when elastic pre-yielding is taken into account. The effects of elastoplastic yielding on the final accumulated displacement are: (a) the time shift in the

9 sliding time-histories of the mass, and (b) the increase of uphill deformation for accelerations smaller than the critical because of elasticity. Forward directivity and fling step affected motions, containing severe acceleration pulses and/or large velocity steps, may have an unpredictably-detrimental effect on residual slip, especially for small values of the critical acceleration. The unpredictability of asymmetric response arises from the sensitivity of the sliding on the sequence, duration, and details of motion. Changing the polarity of excitation (i.e., applying it in the + and then in the direction) has a significant effect on the accumulated slippage. + o o f o =.7 Hz, + f o =.7 Hz, f o =. Hz, + f o =. Hz, } } One-Cycle Sinus Ricker Slippage : m dy =. m dy =. m dy =. m α C /α H α C /α H α C /α H Figure 9. The impact of alternating excitation polarity on the induced yielding for both the Ricker and one-cycle sinus pulses. Three different values of elastic deformation, dy, are presented. 7 ACKNOWLEDGEMENT The research presented in this paper was financially supported by the Secretariat for Research and Technology of Greece, under the auspices of PENED Programme with Contract number ED78. We thank Dr. Ioannis Anastasopoulos for his thoughtful help in programming aspects of this study. REFERENCES Abrahamson, N. A. (), Effects of rupture directivity on probabilistic seismic hazard analysis, Proceedings, th International Conference on Seismic Zonation, Palm Springs, California, Earthquake Engineering Research Institute. Bolt, B.A. (), Earthquakes, W. H. Freeman & Co, Fifth Edition, New York. Boore, M. D. (), Effect of baseline corrections on displacements and response spectra for several Recordings of the 999 Chi-Chi, Taiwan, Earthquake, Bulletin of the Seismological Society of America, Vol. 9, No., pp Bray, J.D., and Rathje, E.M. (998), Earthquake-Induced Displacements of Solid-Waste Landfills, Journal of Geotechnical and Geoenvironmental Engineering, ASCE, Vol., pp. -. Constantinou, M. C., and Gazetas, G. (987), Probabilistic Seismic Sliding Deformations of Earth Dams and Slopes, Proceedings of the Specialty Conference on Probabilistic Mechanics and Structural Reliability, ASCE, Berkeley, pp. 8-.

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