DISPLACEMENT ANALYSIS OF SUBMARINE SLOPES USING ENHANCED NEWMARK METHOD

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1 DISPLACEMENT ANALYSIS OF SUBMARINE SLOPES USING ENHANCED NEWMARK METHOD N. ZANGENEH and R. POPESCU Faclt of Engineering & Applied Science, Memorial Universit, St. John s, Newfondland, Canada A1B 3X5 Abstract The Newmark method for predicting seismicall indced displacement of slopes is enhanced b introdcing of the ield (threshold) acceleration on the real soil strength, acconting for the effects of pore water pressre. Seismicall indced pore water pressre bild-p and its dissipation after the earthqake are calclated as a fnction of soil properties and grond motion characteristics, based on well-recognized practical methods. The proposed model provides more realistic predictions of slope displacements. It is validated based on centrifge experimental reslts. Kewords: Newmark method, sbmarine slopes, pore water pressre bild-p and dissipation, seismic loading. 1. Introdction Displacement analsis is a more rational alternative to psedo-static seismic analsis of slope stabilit. Newmark (1965) introdced a limit-eqilibrim-based displacement analsis method that predicts the displacements of an infinite slope dring an earthqake based on a soil strength dependent ield acceleration and prel kinematic criteria. As opposed to the psedo-static method of slope stabilit analsis, which provides a factor of safet for some ver short time instants dring an earthqake, the Newmark method can provide a prediction of slope performance based on the total displacement at the end of shaking. In man applications of the Newmark method, the ield acceleration is assmed constant dring the earthqake. When appling the method to satrated granlar soils, however, de to the bild-p of excess pore water pressre (EPWP), soil strength and conseqentl the ield acceleration will decrease. In addition, after the end of shaking the generated pore pressres start to dissipate with time, which reslts in increasing soil strength and ield acceleration. In this paper, the effects of EPWP bild-p/dissipation are investigated, and a procedre for calclating permanent displacements of sbmarine slopes sbjected to seismic loads is introdced. The method is based on the algorithm proposed b Newmark, and it ses state-ofpractice methods for estimating EPWP bild-p and dissipation. The reslts of the analsis are also verified sing centrifge test reslts.. Analsis Procedre Non-cohesive soils ma experience significant pore water pressre bild-p de to cclic or earthqake loading. In the limit, it can lead to a state of zero effective stress 193

2 194 Zangeneh and Popesc and soil liqefaction. Therefore, in case of non-cohesive deposits, a total stress analsis is not appropriate and ma give highl over-conservative reslts. Instead, an effective stress approach shold be sed to consider the effects of EPWP and changes in soil shear strength..1 YIELD ACCELERATION Figre 1. Psedo-static analsis of an infinite sbmarine slope. For stabilit analsis of an infinite slope, the failre srface is assmed as a plane parallel to the slope (Fig. 1). The factor of safet ( FS ) is expressed b the ratio of available soil shear strength ( τ ) to the shear stress developed on the failre plane (τ ): f τ f FS = (1) τ in which, soil shear strength at failre is expressed in terms of effective parameters according to the Mohr-Colomb failre criterion: τ = c + ( σ ) tanφ () f where c is the soil effective cohesion, φ is the effective internal friction angle, σ is the total stress (normal to the failre srface), and is the total (hdrostatic + excess) pore water pressre. Therefore, the factor of safet can be written as follows: c + ( γ d cos β e kγd sin β cos β ) tanφ FS = (3) γ d sin β cos β + kγd cos β where with reference to Figre 1, γ is the effective (or boant) nit weight of soil, d the depth of failre plane, β is the slope angle, e is the EPWP (in excess of hdrostatic) generated de to earthqake, and k is the seismic coefficient defined as the ratio between the horizontal earthqake acceleration and the gravitational acceleration (g).

3 Displacement Analsis of Sbmarine Slopes Using Newmark Method 195 In this std, onl fll satrated soils are taken into accont. The boant (or effective) weight of the sliding block, W, is sed in Eqation (3) to calclate the normal effective stress. However, becase it is assmed that dring a seismic event, the soil behavior is mostl ndrained, the inertial force of the earthqake is applied to both soil particles and pore water. Ths, the inertial force is eqal tok W. B setting the factor of safet eqal to 1, the ield acceleration coefficient at each time instant t for downslope sliding can be obtained as follows: d c + [ γ d cos β φ γ e ( t)]tan d sin β cos β k ( t) = (4) γd cos β + γd sin β cos β tanφ r σ / v or, with = e 0 (the ratio between EPWP and initial effective vertical stress) and σ v 0 = γ d cos β : d c + γ d cos β[1 r φ γ ( t)]tan d sin β cos β k ( t) = (5) γd cos β + γd sin β cos β tanφ The ield acceleration coefficient is defined here as k = a / g, where a is the ield acceleration and g is the acceleration of gravit. For the case of ver mild slopes, it ma be worth considering also the possibilit of seismicall indced pslope sliding. The ield acceleration coefficient for pslope sliding is: c γ d cos β[1 r φ γ ( t)]tan d sin β cos β k ( t) = (6) γd cos β γd sin β cos β tanφ The sliding block downslope a d (t) and pslope a (t) accelerations can be calclated b b sing the following eqations: cos( φ d d β ) a ( t) = ( k( t) k ( t) ) g (7) b cos( φ ) cos( φ + β ) a ( t) = ( k( t) k ( t) ) g (8) b cos( φ ) where k( t) g represents the seismic acceleration time histor. Finall, the slope displacement can be compted b integrating twice the block acceleration based on the direction of motion. sat. ESTIMATION OF EXCESS PORE PRESSURE BUILD-UP According to Seed and Idriss (198), the rate of pore pressre development in ndrained cclic simple shear tests on most granlar soils, falls within a fairl narrow range when plotted in the normalized form shown in Figre.

4 196 Zangeneh and Popesc EPWP Ratio (r) Average (α = 0.7) Figre. Rate of pore water pressre bild-p in cclic simple shear tests (Seedet al. 1975). Crves sch as those shown in Figre can be expressed b the following relation: 1 α N e eq r = = arcsin σ π (9) 0 NL v N eq (N eq /N L ) where is the nmber of eqivalent stress ccles applied to the sample p to a certain moment, NL is the nmber of stress ccles reqired to prodce liqefaction, and α is called the pore pressre bild-p parameter. For a real acceleration time histor, the nmber of eqivalent stress ccles, N, can be calclated based on a procedre eq introdced b Seed (1975). B varing the vale of α, Eqation (9) can fit a large palette of ndrained pore water pressre generation crves, as shown in Fig. 3. α= EPWP Ratio (r) (N eq /N L ) Figre 3. Rate of pore pressre generation for different vales of α (Seed and Idriss 198). The reslts of a tpical Newmark analsis, as described before, which acconts for the decrease of ield acceleration de to EPWP bild-p is shown in Figre 4. As the

5 Displacement Analsis of Sbmarine Slopes Using Newmark Method 197 earthqake indces a gradal increase in pore pressre, the ield accelerations decrease gradall. In this particlar example, one shold note that if no redction in the ield acceleration were considered, the permanent displacement wold be mch smaller, and therefore, non-conservative. Also note that in Figre 4, two different permanent displacements are calclated, one of which considers the possibilit of pslope sliding that is reasonable for nearl flat sbmarine slopes. Displacement (withot pslope) Displacement (with pslope) Effect of Bild-p Downslope Yield Acceleration Earthqake Acceleration Upslope Yield Acceleration Figre 4. Slope displacements considering the effect of bild-p. Downslope Yield Acceleration Indefinite Displacement Constant Negative Yield Acceleration Upslope Yield Acceleration Earthqake Acceleration Figre 5. Slope displacement ignoring the effect of excess pore water pressre dissipation.

6 198 Zangeneh and Popesc.3 ESTIMATION OF EXCESS PORE PRESSURE DISSIPATION After the end of shaking the excess pore water pressre starts dissipating. In cases where soil liqefies, the ield acceleration at the end of shaking is less than zero, which reslts in infinite post earthqake displacements (Figre 5). In realit, however, de to dissipation of pore pressre the soil regains part of its original shear strength and the ield acceleration increases and becomes positive (Figre 6). This reslts in limiting the displacements. Therefore, acconting for EPWP dissipation after the earthqake will provide better prediction of post-seismic displacements. Measred Displacement Downslope Yield Acceleration Effect of Dissipation Predicted Displacement Earthqake Acceleration Upslope Yield Acceleration Figre 6. Slope displacement considering the effect of excess pore water pressre dissipation. The one-dimensional consolidation theor can be applied to estimate the EPWP dissipation rate. In a soil laer with an distribtion of the initial excess pore pressre with depth [ i ( z)], the excess pore pressre at an time, e (t) at the depth d is (e.g. Craig 199): n= d 1 nπz nπz n π c t v t) = [( ( z)sin dz)(sin ) exp( )] i n 1 d = d d 4d ( e 0 where d is length of longest drainage path, and c is the coefficient of v consolidation: c = k /( m ) v v γ where k is soil permeabilit, m is the coefficient of w v volme compressibilit and γ is the nit weight of water. In this std, it is assmed w that the soil laer above the failre plane liqefies dring the earthqake, therefore, is a linear fnction of z, and Eqation (10) becomes: n= e = n= 1 (10) 4d 4d nπz n π Tv [ γ cos β( cosnπ + sinnπ )(sin )exp( )] (11) nπ n π d 4 i

7 Displacement Analsis of Sbmarine Slopes Using Newmark Method Calibration and Validation Using Centrifge Test Reslts To calibrate and verif the analsis procedre described in the previos section, the reslts of VELACS (Verification of Liqefaction Analsis b Centrifge Stdies, Arlandan and Scott, 1993) centrifge test for model performed b RPI (Figre 7) have been sed. This test simlates an infinite sbmarine slope with a depth of 10 m sbject to an earthqake with maximm acceleration of abot 0.g. The soil is a niform sand with relative densit Dr = 40%. The geomechanical soil properties were inferred b Popesc and Prevost (1993), based on reslts of laborator soil tests. 3.1 PORE PRESSURE BUILD-UP PARAMETER (α) Using the acceleration at levels of each pore pressre transdcer In this section, α is back calclated for two intermediate elevations where both EPWP and accelerations were recorded, namel (P6,AH4) and (P7,AH5) as shown in Figre 7. In Figres 8a and 8b, vales of α eqal to 0.5, 0.7,, and 4 are shown for points P6 and P7, respectivel. The EPWP parameter α corresponding to the best crve fit is α = 4. The vale of the nmber of ccles to liqefaction, N, is directl obtained from the L pore pressre records and the variation of the eqivalent nmber of ccles of the inpt motion with time Using the acceleration of the box (measred at the base of the model) The crve-fit procedre has been repeated for the same points at the same levels bt sing the centrifge box acceleration. In this case the best crve fit can be obtained for α = to 4 (Figre 9), which is close to the vales obtained in section Therefore, in real-life analses, one cold se the base (bedrock) seismic acceleration and still obtain acceptable prediction. Figre 7. VELACS Model # Configration.

8 00 Zangeneh and Popesc Recorded EPWP bild-p α=4 α= α=0.7 α=0.5 Recorded EPWP bild-p α=4 α= α=0.7 α=0.5 a) P6 b) P7 Figre 8. Calibration of α sing the acceleration at each level. Recorded EPWP bild-p Recorded EPWP bild-p α=4 α= α=4 α= α=0.7 α=0.7 α=0.5 α=0.5 a) P6 b) P7 Figre 9. Calibration of a sing the acceleration of the box. 3. CALIBRATING THE COEFFICIENT OF CONSOLIDATION The vales of EPWP after dissipation calclated sing Eqation (11) as well as the vales recorded in the centrifge test at point P6 are shown in Figre 10. Predicted sing c v=5.1 m /s Measred Figre 10. The predicted and measred excess pore water pressre vales dring the bildp and dissipation phases (at transdcer P6 see Figre 7).

9 Displacement Analsis of Sbmarine Slopes Using Newmark Method 01 The best-fit crve of post-earthqake EPWP dissipation was obtained for c v = 5.7 m/s. A vale of c v = 5.1 m/s 3 was calclated sing a vale k = m / s for soil permeabilit (Popesc & Prevost 1993) and compting the coefficient of volme compressibilit as m v = 1/ B. B is the low strain blk modls of the soil that is a fnction of the average effective confining stress dring the dissipation phase. 3.3 SLOPE DISPLACEMENTS CONSIDERING DISSIPATION Finall, b appling all the previosl described procedres the permanent displacement of the slope is calclated and shown in Figre 11. The predicted vale is satisfactoril close to the measred vale. Measred Displacement Downslope Yield Acceleration Predicted Displacement Earthqake Acceleration Upslope Yield Acceleration Figre 11. The predicted and measred permanent displacements considering EPWP bildp and dissipation effects (at transdcer P6 see Figre 7). 4. Model Limitations and Frther Research This paper presents a std in progress that needs frther addressing of a few aspects: (1) the acceleration applied at the base of the block was that recorded in the experiment at that location (or level), and () the nmber of ccles to liqefaction was estimated from the measrements sing the time at which r reached one in the centrifge model. These two elements are nknown when making a prediction of slope displacements. The were sed in the crrent std to check the proposed method for calclating excess pore water pressre bild-p and dissipation. In the next phase, for a given seismic motion the nmber of ccles to liqefaction will be estimated based on information on soil strength and maximm seismic acceleration, and sing a method compatible with the crrent gidelines for liqefaction strength assessment (Yod et al.

10 0 Zangeneh and Popesc 000) sing in-sit tests reslts (e.g. CPT). The Newmark model assmes that slope displacements are concentrated in a narrow band, i.e. below a moving rigid block. This is a good assmption for laered soils having a weaker laer sandwitched between more resistant soils. In homogeneos soils, sch as that in the VELACS model #, the downslope displacements are distribted with depth, with maximm vales at the soil srface. For sch sitation, the rigid block assmption is a limitation of the Newmark model. This can be mitigated b either: (1) considering the flexibilit of the moving block (e.g. Rathje and Bra, 000) and acconting for continos softening, or () considering a "stack of rigid blocks". 5. Smmar and Conclsions The original Newmark model has been enhanced b appling state-of-practice methods of estimating the EPWP bild-p dring seismic events and dissipation after the earthqake to obtain more realistic predictions of permanent slope displacements. The reslts have been calibrated, and validated based on centrifge test reslts. The reslts show that the proposed procedre is promising, especiall for risk assessment, involving a large nmber of analses and reqiring a reliable and time effective algorithm. The method has some limitations that will be mitigated as discssed in section References Craig, R.F Soil Mechanics, Chapman & Hall, London Newmark, N.M., Effects of earthqakes on dams and embankments. 5 th. Rankine Lectre, Géotechniqe, 15(): Popesc, R. and Prevost, J.H Centrifge validation of a nmerical model for dnamic soil liqefaction. Soil dnamics and Earthqake Engineering, 1: Rathje, E. M., and Bra, J. D Nonlinear Copled Seismic Sliding Analsis of Earth Strctre. Jornal of Geotechnical and Geoenvironmental Engineering, 16(11): Seed, H.B. and Idriss, I.M On the importance of dissipation effects in evalating pore pressre changes de to cclic loading. Soil Mechanics - Transient and Cclic Loads, eds., Pande, N., Zienkiewics, O.C.: Seed, H. B., Idriss, I. M., Makdisi, F., and Banerjee, N Representation of irreglar stress-time histor b eqivalent niform stress series in liqefaction analses. Report No. EERC 75-9, Earthqake Engineering Research Center, Universit of California, Berkele. Yod, T.L., Idriss, I.M., Andrs, R.D., Arango, I., Castro, G., Christian, J.T., Dobr, R., Finn, W.D.L., Harder, L.F., Hnes, M.E., Ishihara, K., Koester, J.P., Liao, S.S.C., Marcson, W.F., Martin, G.R., Mitchell, J.K., Moriwaki, Y., Power, M.S., Robertson, P.K., Seed, R.B., and Stokoe, K.H. 000, Liqefaction Resistance of Soils: Smmar report from the 1996 NCEER and 1998 NCEER/NSF workshops on Evalation of liqefaction resistance of soils, Jornal of Geotechnical and Geoenvironmental Engineering, Vol. 17(10):