PAST APPLICATIONS OF NEWMARK'S METHOD

     Newmark's method has been applied rigorously in a variety of ways to slope-stability problems. Most applications have dealt with the seismic performance of dams and embankments (Yegian and others, 1991; Makdisi and Seed, 1978), which was Newmark's (1965) original intent. Newmark's method also has been successfully applied to landslides in natural slopes (Wilson and Keefer, 1983). Several simplified approaches have been proposed for applying Newmark's method; these involve developing empirical relationships to predict slope displacement as a function of critical acceleration and one or more measures of earthquake shaking. Many such studies plot displacement against critical acceleration ratio-the ratio of critical acceleration to PGA (Franklin and Chang, 1977; Makdisi and Seed, 1978; Ambraseys and Menu, 1988). Other studies have related critical acceleration ratio to some normalized form of displacement: Yegian and others (1991) calculated exceedance probabilities for displacement normalized by PGA, the equivalent number of earthquake shaking cycles, and the square of the period of the base motion; Lin and Whitman (1986) used simple, artificial ground-motion wave forms (triangular, rectangular, or sinusoidal) to relate critical acceleration ratio to displacement normalized by PGA. Jibson (1993) related Newmark displacement to critical acceleration and Arias intensity, an earthquake-shaking measure described below. Miles and Ho (1999) and Miles and Keefer (2000) compared results from these simplified methods with their own method of rigorously integrating artificially generated strong-motion time histories.

     Wilson and Keefer (1983) applied Newmark's method to a landslide triggered by the 1979 Coyote Creek, California, earthquake. The slide occurred near a strong-motion instrument, and the landslide displacement predicted in the Newmark analysis using the record from that instrument agreed well with the observed displacement. This method of using real acceleration-time histories to predict displacements in natural slopes has been applied to experimentally predict and map seismic slope stability in San Mateo County, California (Wieczorek and others, 1985). It has been adapted to back-analyze shaking conditions required to trigger landslides formed in the Mississippi Valley during the 1811-12 New Madrid earthquakes (Jibson and Keefer, 1993) and to reconstruct failure conditions of other seismically triggered landslides (Jibson 1996; Jibson and Harp, 1996). It also forms the basis for constructing digital seismic landslide hazard maps in southern California (Jibson and others, 1998, 2001).

TYPES OF SLIDING-BLOCK ANALYSIS CURRENTLY IN USE

     Since Newmark first introduced his analytical method in 1965, several variations have been proposed that are designed to yield more accurate estimates of slope displacement by modeling the dynamic slope response more rigorously. This, of course, involves a trade-off. One great advantage of Newmark's method is it's theoretical and practical simplicity. This simplicity, however, is the result of many assumptions that limit the accuracy of the results in many cases. Chief among these limitations is the assumption of rigidity--that the landslide block is perfectly rigid and experiences no internal deformation. This assumption is reasonable for relatively thin landslides in stiff or brittle materials, but it introduces significant errors as landslides become thicker and material becomes softer. More sophisticated methods do a better job of modeling the dynamic elastic response of the landslide material and thus yield more accurate displacement estimates, but again, there is a trade-off in the complexity of the analysis and the difficulty in acquiring the needed input parameters.

     At present, analytical procedures for estimating permanent co-seismic landslide displacements can be grouped into three types: rigid-block, decoupled, and coupled.

     Rigid-block analysis. Rigid-block analysis is the analysis first developed by Newmark (1965), which was described in detail below in the section "Conducting a Newmark Analysis". To briefly summarize, it treats the potential landslide block as a rigid mass (no internal deformation) that slides in a perfectly plastic manner on an inclined plane. Thus, the mass experiences no permanent displacement until the base acceleration exceeds the critical (yield) acceleration of the block; when the base acceleration exceeds the critical acceleration, the block begins to move downslope. Displacements are estimated by double-integrating the parts of an acceleration-time history that lie above the critical acceleration.

     Decoupled analysis. More sophisticated analyses were developed to account for the fact that potential landslide masses are not rigid bodies but deform internally when subjected to seismic shaking. The most commonly used of such analyses was developed by Makdisi and Seed (1978) and estimates the effect of dynamic response on permanent slip in a two-step procedure: (1) Perform a dynamic analysis of the slope (using programs such as QUAD4 or SHAKE) assuming no failure surface; estimate acceleration time histories at several points within the slope and develop an average acceleration time history for the slope mass above the potential failure surface. (2) Use this average time history as the input in a rigid-block analysis and estimate the permanent displacement. This approach is commonly referred to as a decoupled analysis because the computation of the dynamic response and the plastic slip are performed independently. Decoupled analysis thus does not take into account the effects of slip on the ground motion.

     Coupled analysis. In a coupled analysis, the dynamic response of the slide mass and the permanent displacement are modeled together so that the effect of plastic slip on the ground motions is accounted for. Lin and Whitman (1983) pointed out that the assumptions of the decoupled analysis introduce errors in the estimation of total slip and compared results for coupled and decoupled analyses. They showed that, in general, decoupled analysis yielded conservative results that were within about 20 percent of the coupled results. More recently, Rathje and Bray (2000) compared results from rigid-block analysis with linear and non-linear coupled and decoupled analyses.

WHICH ANALYSIS SHOULD BE USED?

     Selecting which of these types of analyses to conduct is critical in getting accurate displacement estimates. The current state of knowledge suggests that the best basis for this selection is the period ratio, T_s/T_m, the ratio of the fundamental site period (T_s) to the mean period of the earthquake shaking (T_m) (Rathje and Bray, 1999; 2000). The fundamental site period can be estimated as

Ts = 4h/Vs

(1)

where h is the maximum vertical distance between the ground surface and slip surface used to estimate the yield acceleration and V_s is the shear-wave velocity of the materials above the slip surface. Mean period of the earthquake shaking was defined by Rathje and others (1998) as the inverse of the weighted average frequency over a frequency range of 0.25 to 20 Hz. Mean period can be estimated for rock site conditions as a function of earthquake magnitude (M) and source distance (r, in km) as follows:

ln(Tm) = ln(0.411 + 0.0837(M - 6) + 0.00208r)	for M 7.25	(2a)
ln(Tm) = ln(0.516 + 0.00208r)	for 7.25 M 8.0	(2b)

     As a general rule, coupled analysis gives good results for all conditions, but, of course, it is the most complex to conduct. The following table provides general guidelines for selecting between rigid-block and decoupled analysis in the terms of the period ratio.

Rigid-block analysis is thus appropriate for analyzing thin, stiff landslides having a period ratio of 0.2 or less. Between 0.2 and 1, rigid-block analysis yields unconservative results and should not be used. For period ratios between 1 and 2, rigid-block analysis yields conservative results, but decoupled analysis give results closer to results from coupled analysis, which is considered the most accurate result. For period ratios greater than 2, rigid-block analysis yields highly over-conservative results that significantly overestimate displacement.

CONDUCTING A NEWMARK ANALYSIS

     Before describing how to apply Newmark's method, the limiting assumptions need to be stated. Newmark's method treats a landslide as a rigid-plastic body, that is, the mass does not deform internally, experiences no permanent displacement at accelerations below the critical or yield level, and deforms plastically along a discrete basal shear surface when the critical acceleration is exceeded. Thus, Newmark's method is best applied to coherent landslides such as translational block slides and rotational slumps. Other limiting assumptions commonly are imposed for simplicity but are not required by the analysis:

     1. The static and dynamic shearing resistance of the soil are taken to be the same (Newmark, 1965; Chang and others, 1984).

     2. The effects of dynamic pore pressure are neglected. This assumption generally is valid for compacted or overconsolidated clays and very dense or dry sands (Newmark, 1965; Makdisi and Seed, 1978).

     3. The critical acceleration is not strain dependent and thus remains constant throughout the analysis (Newmark, 1965; Makdisi and Seed, 1978; Chang and others, 1984; Ambraseys and Menu, 1988). The accompanying programs do not require this and allow for strain-dependant changes in critical acceleration.

     4. The upslope resistance to sliding is taken to be infinitely large such that upslope displacement is prohibited (Newmark, 1965; Chang and others, 1984; Ambraseys and Menu, 1988).

     The following sections outline the procedure for conducting a Newmark analysis and provide simple examples of its application.

Critical (Yield) Acceleration

     The first step in the analysis is to determine the critical or yield acceleration of the potential landslide. One way to do this is to use pseudostatic analysis, where critical acceleration is determined by iteratively employing different permanent horizontal earthquake accelerations in a static limit-equilibrium analysis until a factor of safety of 1.0 is achieved.

     Newmark (1965) simplified this approach by showing that the critical acceleration of a potential landslide is a simple function of the static factor of safety and the landslide geometry; it can be expressed as

ac = (FS - 1)g sin ,

(3)

where a_c is the critical acceleration in terms of g, the acceleration due to Earth's gravity; FS is the static factor of safety; and is the angle (herein called the thrust angle) from the horizontal that the center of mass of the potential landslide block first moves. Thus, determining the critical acceleration by this method requires knowing the static factor of safety and the thrust angle.

Factor of Safety

     As noted by Newmark (1965), modeling dynamic slope response requires undrained or total shear-strength parameters. During earthquakes, slope materials behave in an undrained manner because excess pore pressures induced by dynamic deformation of the soil column cannot dissipate during the brief duration of the shaking. Undrained strength also is called total strength because the contributions of friction, cohesion, and pore pressure are not differentiated, and the total strength is expressed as a single quantity.

     The factor of safety can be determined using any appropriate method that uses undrained or total shear strength. In materials whose drained and undrained behaviors are similar, drained or effective shear strengths can be used if undrained strengths are unavailable or difficult to measure. This allows great flexibility for users. For a rough estimate of displacement, a simple factor-of-safety analysis, perhaps of an infinite slope using estimated shear strength, could be used. On the other end of the spectrum, a highly detailed site study could be conducted to determine the factor of safety very accurately. Clearly, the accuracy of the safety factor, and the resulting predicted displacement, depends on the quality of the data and analysis, but the user determines what is appropriate.

Thrust Angle

     The thrust angle is the direction the center of gravity of the slide mass moves when displacement first occurs. For a planar slip surface parallel to the slope (an infinite slope), this angle is the slope angle. For simple planar block sliding, the thrust angle is the inclination of the basal shear surface. For circular rotational movement, Newmark (1965) showed that the thrust angle is the angle between the vertical and a line segment connecting the center of gravity of the slide mass and the center of the slip circle. For irregular shear surfaces, the thrust angle can be approximated visually, by estimating an "equivalent" circular surface, or by averaging the inclinations of line segments approximating the surface.

Calculation of Critical Acceleration

     Figure 2 illustrates a simple hypothetical slope and the critical failure surface having the lowest factor of safety (1.4) in undrained conditions. Newmark's (1965) geometric construction indicates a thrust angle of 30^o. According to Equation 3, a factor of safety of 1.4 and a thrust angle of 30^o would yield a critical acceleration of 0.20 g.

Figure 2. Model of hypothetical slope: heavy line is basal shear surface; FS is factor of safety; thrust angle is 30^o.

Earthquake Acceleration-Time History

     The most difficult aspect of conducting a Newmark analysis is selecting an input ground motion, and many ways of doing so have been proposed. Most studies have used some combination of the two approaches mentioned by Newmark: (1) scaling acceleration-time histories from actual earthquakes to a desired level of PGA (Franklin and Chang, 1977; Makdisi and Seed, 1978) and (2) using single or multiple cycles of artificial acceleration pulses having simple rectangular, triangular, or sinusoidal shapes (Lin and Whitman, 1986; Yegian and others, 1991). Both of these approaches yield useful results, but both also have inherent weaknesses. Scaling an acceleration-time history by simply expanding or contracting the acceleration scale does not accurately represent ground motion from earthquakes of different magnitudes or proximities because magnitude and source distance also affect the duration and predominant periods of shaking. And using simple artificial pulses of ground shaking is an unnecessary oversimplification in light of the current availability of digitized acceleration-time histories having a broad range of attributes.

     Selecting a time history requires the user to know something of the shaking characteristics or design requirements pertinent to the situation of interest. Common design or hazard-assessment criteria include (1) a specified level of ground shaking, (2) a model earthquake of specified magnitude and location, or (3) an acceptable design amount of earthquake-triggered displacement.

Selecting a Time History for a Specified Level of Ground Shaking

     Criterion (1) is by far the simplest; it requires only that the user locate a sampling of digitized acceleration-time histories having the desired measure of earthquake shaking intensity near the specified level. PGA is a common measure of ground-shaking intensity, and digitized time histories having a wide variety of PGA's, even approaching 2 g, currently are available.

     A weakness of this approach is that PGA measures only a single point in an acceleration-time history and is thus a rather crude measure of shaking intensity. A more comprehensive and quantitative measure of total shaking intensity developed by Arias (1970) is useful in seismic hazard analysis and correlates well with the distribution of earthquake-induced landslides (Harp and Wilson, 1995; Jibson and others, 1998, 2001). Arias intensity is the integral over time of the square of the acceleration, expressed as

(4)

where I_a is Arias intensity in units of velocity, g is the acceleration of Earth's gravity, a(t) is the ground acceleration as a function of time, and T is the total duration of the strong motion. An Arias intensity thus can be calculated for each directional component of a strong-motion record. In cases where a given level of Arias intensity can be specified, selecting a strong-motion record of similar intensity is quite simple, and currently available records span a range of Arias intensities up to almost 25 m/s.

Selecting a Time History for a Specified Earthquake Magnitude and Location

     Criterion (2) can be somewhat more difficult. If acceleration-time histories exist for earthquakes of the desired magnitude that were recorded at appropriate distances, then they can be used. Satisfying both magnitude and distance requirements is often impossible, however, so it may be necessary to estimate shaking characteristics at the site of interest using published empirical or theoretical relationships that predict PGA, duration, and Arias intensity as a function of earthquake magnitude and source distance. Estimated shaking characteristics can then be compared with those from existing time histories to provide a basis for selecting appropriate records.

     An example of this procedure is from the Mississippi Valley, where large earthquakes occurred in 1811-12 but where no strong-motion records exist. The problem is to predict the performance of a slope in a moment-magnitude (M) 6.2 earthquake centered at least 8 km away. If no time histories for that magnitude and distance existed, shaking characteristics at the site would have to be estimated.

     PGA in this example can be estimated using the attenuation relationship of Nuttli and Herrmann (1984) for soil sites in the central United States:

log PGA = 0.57 + 0.50 mb - 0.83 log(R2 + h2)1/2 - 0.00069 R

(5)

where PGA is in centimeters per second squared, m_b is the body-wave magnitude, R is the epicentral distance in kilometers, and h is the focal depth in kilometers. A M-6.2 earthquake corresponds to m_b=5.8 (Heaton and others, 1986). For m_b=5.8, an epicentral distance of 8 km, and a minimum focal depth of 3 km, Equation 5 predicts a PGA of 491 cm/s² or 0.50 g. Methods for estimating strong-ground-shaking parameters in many different tectonic environments were published by the Seismological Society of America (1997).

     Estimating the Arias intensity at the site can be done in more than one way. Wilson and Keefer (1985) developed a relationship between Arias intensity, earthquake magnitude, and source distance:

log Ia = M - 2logR - 4.1,

(6)

where I_a is in meters per second, M is moment magnitude, and R is earthquake source distance in kilometers. For M=6.2 and R=8 km, Equation 6 predicts an Arias intensity at the site of 1.97 m/s.

     Arias intensity also correlates closely with the combination of PGA and duration. R.C. Wilson (U.S. Geological Survey, unpublished data) developed an empirical equation using 43 strong-motion records to predict Arias intensity from PGA and a specific measure of duration:

Ia = 0.9(PGA2)(D5-95%),

(7)

where I_a is in meters per second, PGA is in g's, and D_5-95% is duration (hereafter called Dobry duration) in seconds, defined as the time required to build up the central 90 percent of the Arias intensity (Dobry and others, 1978). Estimating Arias intensities using this method requires an estimate of the duration of strong shaking. Dobry and others (1978) proposed an empirical relationship between duration and earthquake magnitude:

log D5-95% = 0.432M - 1.83,

(8)

where D_5-95% is Dobry duration in seconds and M is unspecified earthquake magnitude (probably local magnitude, M_L). In the magnitude range of interest, M_L values are generally identical to M values (Heaton and others, 1986), so M=6.2 yields a Dobry duration of 7.1 s. If this duration and the PGA of 0.50 g estimated above are used in Equation 7, an Arias intensity of 1.59 m/s is predicted, which agrees fairly well with that estimated by Equation 6.

     More recently, Abrahamson and Silva (1996) developed a much more rigorous equation to estimate median values of D_5-95%. For earthquake source distances greater than 10 km, the following equation is used:

(9a)

where M is magnitude and r is source distance. For distances less than 10 km, the following equation is used

(9b)

     These three indices of shaking intensity-PGA, duration, and Arias intensity-form a rational basis for selecting strong-motion records for analysis. Caution and judgment must be used in making these estimates, however, because the process of combining values from Equations 5-9, each of which has a range of possible error, compounds the uncertainty at each step. For this example, three records are chosen whose shaking characteristics reasonably match those estimated (Table 1). Selecting multiple records that span the range of estimated shaking characteristics provides a range of displacements whose significance the user can judge.

     The frequency content of earthquake shaking also significantly affects slope response. The mean period, defined above in equation 2, is the most useful measure of frequency content and can be used to identify earthquake records that are appropriate for specific analyses.

Selecting a Time History for a Specified Design Displacement

     Criterion (3) differs from the first two in that a limiting damage level (landslide displacement) is specified rather than the level of ground shaking. An example is to estimate the maximum level of ground shaking a slope having a critical acceleration of 0.20 g could experience without exceeding 10 cm of displacement.

     One approach to this problem is to simply iteratively analyze several strong-motion records to find those that yield about 10 cm of displacement at a_c=0.20 g. The magnitudes, source distances, focal depths, PGA's, Arias intensities, and durations of these records could then be examined to discern the approximate range of conditions the slope could withstand. Obviously, this approach could be time consuming, but it would produce a variety of possible threshold ground-shaking scenarios.

     An easier approach to this type of problem is to apply the simplified Newmark method discussed subsequently.

Calculating Newmark Displacement

     Once the critical acceleration of the landslide has been determined and the acceleration-time histories have been selected, Newmark displacement can be calculated by double integrating those parts of the strong-motion record that lie above the critical acceleration. Several methods for doing this, some rigorous and others highly simplified, have been proposed (Newmark, 1965; Makdisi and Seed, 1978; Chang and others, 1984; Ambraseys and Menu, 1988). Perhaps the most useful rigorous method was developed by Wilson and Keefer (1983). Figure 3A shows a strong-motion record having a hypothetical a_c of 0.2 g superimposed. To the left of point X, accelerations are less than a_c, and no displacement occurs. To the right of point X, those parts of the strong-motion record lying above a_c are integrated over time to derive a velocity profile of the block. Integration begins at point X (Figure 3AB), and the velocity increases to point Y, the maximum velocity for this pulse. Past point Y, the ground acceleration drops below a_c, but the block continues to move because of its inertia. Friction and ground motion in the opposite direction cause the block to decelerate until it stops at point Z. All pulses of ground motion exceeding a_c are integrated to yield a velocity profile (Figure 3B), which, in turn, is integrated to yield a cumulative displacement profile of the landslide block (Figure 3C).

Figure 3. Illustration of the Newmark algorithm, adapted from Wilson and Keefer (1983). A, earthquake acceleration-time history with critical acceleration (dashed line) of 0.2 g superimposed; B, velocity of the landslide versus time; C, displacement of landslide versus time. Points X, Y, and Z are discussed in the text.

     The algorithm of Wilson and Keefer (1983) permits both downslope and upslope displacement by using the thrust angle to explicitly account for the asymmetrical resistance to downslope and upslope sliding. If pseudostatic yield acceleration is used and the thrust angle is not readily obtainable, the program can be simplified to prohibit upslope displacement. This prohibition was justified by Newmark (1965), as well as others (Franklin and Chang, 1977; Chang and others, 1984; Lin and Whitman, 1986; Ambraseys and Menu, 1988,), because a_c in the upslope direction is generally so much greater than a_c in the downslope direction that it can be assumed to be infinitely large. In most cases, the upslope a_c is greater than the PGA, and no error is introduced by prohibiting upslope displacement. The Newmark algorithm we use in the accompanying program is based on that of Wilson and Keefer but prohibits upslope displacement.

     Integration programs for calculating Newmark displacement can be customized to accept acceleration-time histories in either of two formats: successive pairs of time and acceleration values (time files) or a single string of acceleration values sampled at a constant time interval. The latter is the simpler approach and insures that the integration is performed consistently throughout the time history. The accompanying programs are designed for files of acceleration values at a constant time interval, but utility programs to convert time files to acceleration files are included.

     Digitized strong-motion records can be obtained in several ways. Analog strong-motion records can be manually digitized to obtain a data file of time/acceleration pairs. Such a file can be used in a Newmark integration program that accepts paired data, or it can be resampled at a constant time interval by a simple linear interpolation program. Also, strong-motion records from many worldwide earthquakes are available in digital format from the National Oceanic and Atmospheric Administration's National Geophysical Data Center in Boulder, Colorado.

A SIMPLIFIED NEWMARK METHOD

     The previous sections have described how to rigorously conduct a Newmark analysis. Although this approach is straightforward, many of its aspects are difficult for many users, and thus a simplified approach for estimating Newmark displacements would be helpful.

     As discussed above, previous studies have proposed general relationships between Newmark displacement and some normalized parameter(s) of critical acceleration (Franklin and Chang, 1977; Makdisi and Seed, 1978; Lin and Whitman, 1986; Ambraseys and Menu, 1988; Yegian and others, 1991). Any of these that include parameters appropriate to a problem of interest can be applied with relative ease. Most depend directly on PGA, which, as noted, is a widely used but rather crude measure of shaking intensity. Therefore, a simplified method based on Arias intensity, a better measure of shaking intensity, was proposed by Jibson (1993).

     Jibson's (1993) simplified Newmark method used an empirical regression equation to estimate Newmark displacement as a function of shaking intensity and critical acceleration; the model was calibrated by conducting Newmark analyses on 11 strong-motion records. Jibson and others (1998, 2001) slightly modified the functional form of that equation to make the critical-acceleration term logarithmic and used a much larger group of strong-motion records--280 recording stations in 13 earthquakes (Table 2)--to develop a new regression equation. (With this larger data set, a logarithmic critical-acceleration term yielded a much better fit than a linear term.) They analyzed both of the horizontal components of acceleration from 275 of the recordings and a single component from the remaining 5, which yielded 555 single-component records. For each record, they determined the Arias (1970) intensity; then, for each record, they conducted a rigorous Newmark analysis for several values of critical acceleration, ranging from 0.02 g to 0.40 g. The resulting Newmark displacements were regressed on two predictor variables: critical acceleration and Arias intensity. The resulting regression equation is

log Dn = 1.521 log Ia - 1.993 log ac -1.546 0.375,

(10)

where D_n is Newmark displacement in centimeters, I_a is Arias intensity in meters per second, and a_c is critical acceleration is g's (Jibson and others, 1998, 2001). The regression equation is well constrained (R² = 83%) with a very high level of statistical significance (>99%); the model standard deviation is 0.375. This model yields the mean Newmark displacement when the last term in the equation is ignored; the variation about this mean results from the stochastic nature of earthquake ground shaking. Thus, two strong-motion records having identical Arias intensities can produce different Newmark displacements for slopes having the same critical acceleration. Therefore, Equation 10 yields a range of displacements that must be interpreted with considerable judgment.

     Thus, Newmark displacement, an index of seismic slope performance, can be estimated as a function of critical acceleration (dynamic slope stability) and Arias intensity (ground-shaking intensity). The accompanying program package includes a program to conduct a simplified Newmark analysis using Equation 10.

     Equation 10 can be applied to the example summarized in Table 1. For the lower estimated Arias intensity of 1.59 m/s and a critical acceleration of 0.2 g, the mean value from Equation 10 is 1.4 cm, and the range bracketing two standard deviations is 0.6-3.4 cm. For the higher value of Arias intensity of 1.97 m/s, Equation 10 yields a mean value of 2.0 cm and a range of 0.8-4.7 cm. Displacements calculated from the three selected strong-motion records fall within this range; thus, the simplified Newmark method presented here yields reasonable results.

     Equation 10 can be applied to estimate the dynamic performance of any slope of known critical acceleration because it is derived from generic values of critical acceleration that are not site specific. Thus, several types of hazard analyses for earthquake-triggered landslides can be developed:

     1. If the Arias intensity at a site can be specified, and if the critical acceleration of the slope can be determined, then the Newmark displacement can be estimated.

     2. If critical displacement can be estimated and the critical acceleration of the slope is known, then the threshold Arias intensity that will cause slope failure can be estimated.

     3. If a critical displacement and Arias intensity can be estimated, then the threshold critical acceleration below which slope failure will occur can be estimated.

INTERPRETING NEWMARK DISPLACEMENTS

     The significance of Newmark displacements must be judged by their probable effect on a potential landslide. Wieczorek and others (1985) used 5 cm as the critical displacement leading to macroscopic ground cracking and general failure of landslides in San Mateo County, Calif.; Keefer and Wilson (1989) used 10 cm as the critical displacement for coherent landslides in southern California; and Jibson and Keefer (1993) used this 5-10 cm range as the critical displacement for landslides in the Mississippi Valley. In most soils, displacements in this range cause ground cracking, and previously undeformed soils can lose some of their peak shear strength and end up in a weakened or residual-strength condition. In such a case of strength loss, a static stability analysis in residual-strength conditions can be performed to determine the slope stability after earthquake shaking ceases.

     Blake and others (2002) make the following recommendations for application of sliding-block analysis in southern California:

• For slip surfaces intersecting stiff improvements (i.e., buildings, pools, etc.) computed median displacements should be less than 5 cm.
• For slip surfaces occurring in ductile (non-strain-softening) soil that do not intersect engineered improvements (i.e, landscaped areas, patios), computed median displacements should be less than 15 cm.
• For slip surfaces in soils with significant strain softening (i.e., sensitivity > 2), if a_c was calculated from peak strengths, displacements as large as 15 cm could trigger strength reductions, which in turn could destabilize the slope. For such cases, the design should either be performed using residual strengths allowing median displacements less than 15 cm, or using peak strengths allowing median displacements less than 5 cm.

     Any level of critical displacement can be used according to the parameters of the problem under study and the characteristics of the landslide material. Highly ductile materials may be able to accommodate more displacement without general failure; brittle materials might accommodate less displacement. What constitutes "failure" may vary according to the needs of the user. Results of laboratory shear-strength tests can be interpreted to estimate the strain necessary to reach residual strength.

     Predicted Newmark displacements do not necessarily correspond directly to measurable slope movements in the field; rather, modeled displacements provide an index to correlate with field performance. For the Newmark method to be useful in a predictive sense, modeled displacements must be quantitatively correlated with field performance. In short, do larger predicted displacements relate to greater incidence of slope failure? Analysis of data from the Northridge earthquake facilitated answering this question; predicted Newmark displacements were compared directly with an inventory of landslide actually triggered. The results were then regressed using a Weibull model, which yielded the following equation (Jibson and others, 1998, 2001):

P(f) = 0.335[1 - exp(-0.048 Dn1.565)],

(11)

where P(f) is the probability of failure and D_n is Newmark displacement in centimeters. This equation can be used in any set of ground-shaking conditions to predict probability of slope failure as a function of predicted Newmark displacement. Because this model was calibrated using data from southern California, it may only be rigorously valid there. Jibson and others (1998, 2001) discuss appropriate application of this probability model. The accompanying program package has a program to estimate probability of failure based on Equation 11.

DISCUSSION

     Any idealized model is limited by its simplifying assumptions. The fundamental assumption of Newmark's model is that landslides behave as rigid-plastic materials; i.e., no displacement occurs below the critical acceleration, and displacement occurs at constant shearing resistance when the critical acceleration is exceeded. This assumption is reasonable for some types of landslides in some types of materials, but it certainly does not apply universally. Many slope materials are at least slightly sensitive--they lose some of their peak undrained shear strength as a function of strain. In such a case, Newmark's method would underestimate the actual displacement, because the strength loss during shear would reduce the critical acceleration as displacement occurs. For such materials, the Newmark displacement might be considered a minimum displacement and so would be unconservative.

     Some highly plastic, fine-grained soils behave as visco-plastic rather than rigid-plastic materials. The viscous response of these soils results in part from low permeability and high cohesion, and the result can be a radically dampened seismic response. Some active, slow moving landslides having factors of safety at or below 1.0 have experienced negligible inertial displacement even during large earthquakes (Jibson and others, 1994) because of viscous energy dissipation. In Newmark's method, displacement depends on the critical acceleration, which, in turn, depends on the static factor of safety. Therefore, a landslide at or very near static equilibrium should have a very low critical acceleration (theoretically, a_c=0 if FS=1) and thus should undergo large inertial displacements in virtually any earthquake. Thus, Newmark's method probably overestimates landslide displacements in visco-plastic materials.

     Generally, Newmark's method has considered static and dynamic shear strength to be the same and has ignored dynamic pore-pressure response; this has permitted use of static shear strengths, which are much more easily determined than dynamic strengths. For many soils, this assumption introduces little error, but static and dynamic strengths differ significantly for some soils. In such cases, dynamic shear-strength testing may be required, or static strengths can be adjusted by an empirical correction factor (Makdisi and Seed, 1978). Similarly, dynamic pore-pressure response, if considered significant, can be measured in dynamic tests or accounted for empirically by reducing the static shear strength.

     The accompanying programs make conducting large numbers of analyses almost trivial, and so the best approach for judging the likely performance of a slope is to select a large number of earthquake records, perhaps 50-200, that have a reasonable range of properties of interest and to then interpret the range of output displacements. Experience indicates that the results tend to be log-normally distributed, with a few records yielding very high displacements forming the right-hand tail of the distribution. Thus, mean displacements a virtually always greater than median displacements, and standard deviations are fairly high.

CONCLUSION

     Newmark's method is useful for characterizing seismic slope response. It presents a viable compromise between simplistic pseudostatic analysis and sophisticated finite-element modeling, and it can be applied to a variety of problems in seismic slope stability. The simplified method presented here provides an easy way to estimate ranges of possible displacement in cases where the seismic shaking intensity can be estimated. Probability of failure can also be estimated in certain situations on the basis of a model calibrated using data from the 1994 Northridge earthquake.

REFERENCES

Ambraseys, N.N., and Menu, J.M., 1988, Earthquake-induced ground displacements: Earthquake Engineering and Structural Dynamics, v. 16, p. 985-1006.

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