Efficiency of scalar and vector intensity measures for seismic slope displacements

Abstract

Ground motion intensity measures are usually used to predict the earthquake-induced displacements in earth dams, soil slopes and soil structures. In this study, the efficiency of various single ground motion intensity measures (scalar IMs) or a combination of them (vector IMs) are investigated using the PEER-NGA strong motion database and an equivalent-linear sliding-mass model. Although no single intensity measure is efficient enough for all slope conditions, the spectral acceleration at 1.5 times of the initial slope period and Arias intensity of the input motion are found to be the most efficient scalar IMs for flexible slopes and stiff slopes respectively.

Vector IMs can incorporate different characteristics of the ground motion and thus significantly improve the efficiency over a wide range of slope conditions. Among various vector IMs considered, the spectral accelerations at multiple spectral periods achieve high efficiency for a wide range of slope conditions. This study provides useful guidance to the development of more efficient empirical prediction models as well as the ground motion selection criteria for time domain analysis of seismic slope displacements.

Keywords

seismic slope displacements / intensity measures / empirical prediction

Introduction

Realistic prediction of the permanent displacements in earth dams, soil slopes and soil structures under earthquake loading is an important topic for evaluating and mitigating seismic hazards. Since Newmark’s pioneering work on the rigid sliding block method [1], extensive research has been focused on finding the suitable ground motion parameters that can be reliably used to estimate the earthquake-induced displacements in earth dams, earth embankments and slopes. As earthquake records are complex transient time series, different ground motion intensity measures (IMs) can only represent certain aspects of ground motion characteristics. Some single intensity measures (termed scalar IMs), such as the peak ground acceleration (PGA), were often used in the Newmark-type rigid sliding block models [2-4]. Other scalar IMs such as the dominant period of the earthquake motion and Arias Intensity (IA) have also been used [3,5,6]. Using multiple intensity measures (termed vector IMs) has also been considered, for example in Ref. [7]. It should be noted that all of the above research assume that the slope behaves as a rigid-plastic material, and slope displacement is calculated by double integrating the part of the input acceleration that exceeds a critical value. The Newmark-type rigid sliding block model provides a simple index of dynamic slope performance. However, it is only appropriate for shallow landslides of stiff materials (e.g., rock blocks) that move along a well-defined slip surface.

To better understand the influence of soil nonlinearity on seismic slope response, Bray and his coworkers [8,9] have conducted numerical simulations using a simple nonlinear stick-slip model. In this model, the soil slope is simplified as a generalized single-degree-of-freedom system governed by the first modal shape of vibration. Nonlinear properties of soils are modeled using an equivalent-linear method, similar to the well-known SHAKE analysis [10], such that the stiffness and damping ratio of the system is modified to be compatible with the induced strain level during shaking. Irreversible permanent displacements would occur if the base acceleration exceeds a prescribed critical value. In this model, the dynamic characteristic is characterized by the initial (small-strain) period of the slope (T_s) and the strength of the earth slope is represented by the yield acceleration (K_y). Soil nonlinearity causes elongation of the slope period during shaking, and the most efficient scalar IM for the earthquake-induced displacements has been found to be the spectral acceleration at 1.5 times the initial period of the system. The standard deviation of the model is 0.67 (in natural log units) for all slope conditions [9].

The previous study focused on using a scalar IM to develop empirical predictive models for seismic slope displacements. Large uncertainties in existing methods demonstrate that no scalar IM can accurately predict slope displacement. Using vector IMs allows for a better representation of different aspects of ground motions and thus significantly improves the accuracy of slope displacement prediction. Yet, the efficiency of scalar and vector IMs has not yet been fully investigated. This study will systematically evaluate the efficiency of various scalar IMs and vector IMs in correlating with the seismic slope displacements considering different slope conditions. The results provide insight to improve the efficiency of empirical models as well as the ground motion selection methods for time domain analysis of seismic slope displacements.

Ground motion intensity measures

Earthquake acceleration time histories are chosen from the PEER-NGA strong motion database [11] (http://peer.berkeley.edu/nga/). The database contains a total of 173 earthquakes from California, Japan, Taiwan and other seismic active regions, with a total of 3551 three-directional acceleration time histories. Only horizontal recordings from free-field conditions are used in the analysis, resulting in a total of 1560 pairs of ground motions of two horizontal directions. The equivalent-linear sliding mass model [8,9] is used in the present study. The detailed mathematical formulation of the model is summarized in Appendix. The permanent displacements of the sliding mass were computed using each of the as-recorded earthquake motions in the database. Two orthogonal recordings at the same station are treated as separate records. For each record, the permanent displacements were calculated by applying the record in the positive and the negative directions, and the maximum value of two directions was taken as the permanent displacement for that record. Figure 1(a) shows the computed permanent displacement using acceleration time history recorded during Superstition Hills Earthquake (1987) at USGS Station 286. The yield acceleration K_y is assumed to be 0.1 g and the site period T_s varies from 0 to 2 s. If the slope is rigid (T_s = 0 s), the permanent sliding displacement is 46 cm. The permanent displacement reaches approximately 90 cm if the site period falls into the range of 0.1–0.4 s due to the resonance of the site with the concentrated shaking energy at that period range (note the mean period of the record 0.38 s). As the slope becomes more flexible (T_s increases), the permanent displacement decrease to a negligible value if the site period is greater than 1 s. We assumed that the shear wave velocity of the soil is 250 m/s. During the shaking, the modulus reduction and damping ratio curves follow that for clays with plasticity index of 30 [12], as is illustrated in Fig. 1(b).

In this study, we consider several most commonly used intensity measures to characterize the ground motion time histories. The definitions of these IMs are summarized in Table 1. One may also refer to standard reference books for detailed explanation, e.g., Ref. [13]. The scalar IMs considered can be roughly categorized into the following groups: 1) IMs that are related to the peak amplitude of the ground motion time history, i.e., the peak ground acceleration (PGA); 2) IMs that are related to the frequency content of the ground motion, i.e., the spectral acceleration at initial site period, Sa(T_s), and the spectral acceleration at 1.5 times of initial site period, Sa(1.5T_s); 3) the integration of the spectral acceleration over a range of spectral periods such as acceleration spectrum intensity (ASI). 4) The time integration of the acceleration time history, representing certain kind of seismic energy, i.e., Arias intensity (IA) [14] and cumulative acceleration velocity (CAV); (5) the duration of the ground motion, i.e., the significant duration (D_5-95) [15], and 6) the mean period of the earthquake motion (T_m) [16]. These IMs represents different aspects of the ground motion characteristics and are commonly used in earthquake engineering design.

For each ground motion in the strong motion database, the seismic slope displacement can be computed. Figures 2 and 3 present the scatter plots of computed seismic slope displacements against the scalar IMs of each ground motion record. The slope resistance is assumed to be K_y = 0.1 g in this example. From these plots, it is evident that some IMs are better correlated with the displacements than the others. It may be desirable to use the best correlated IMs to develop empirical models such that increased accuracy and reduced uncertainty can be achieved. However, the predictive capacity of an IM changes with the properties of the slopes when comparing Fig. 2 (T_s = 0.1 s) with Fig. 3 (T_s = 1 s). In these cases, an IM that is closely correlated to the seismic displacements of stiff slopes (e.g., T_s = 0.1 s) may not be a good predictor for flexible slopes (e.g., T_s = 1 s). The objective of the present study is to evaluate the efficiency of these scalar IMs and their vector combinations under different slope conditions. Although it is out of the scope of this study, the results can provide significant insights to identify suitable IMs for use in empirical prediction models.

Efficiency of scalar intensity measures

Using 1560 pairs of ground motion records from the strong motion database and the idealized sliding mass model, earthquake-induced displacements are calculated for various slope conditions. The initial slope period, T_s, ranges from 0.01 s to 2 s. It should be noted that the initial periods of slopes usually fall between 0.2 s and 0.7 s, so there is little practical importance to consider slope periods greater than 2 s. Different levels of yield acceleration K_y is specified from 0.01 g to 0.5 g. For each (T_s, K_y) case, regression analysis was performed by assuming that the seismic displacements D (in natural log unit) are related to the scalar or vector IMs (in natural log unit) in the following quadratic form:

(1)\ln D=a+{\displaystyle \sum _{i=\mathrm{1}}^n{b}_i\ln I{M}_i+{\displaystyle \sum _{i=\mathrm{1}}^n{c}_i{(\ln I{M}_i)}^{\mathrm{2}}}}+ϵ·{\sigma }_{\ln D},

where IM_i (i = 1,2,…,n) represents a particular scalar IM that may be combined to form vector IMs, and n is the total number of IMs used (note n = 1 for the special case when a scalar IM is used). Parameters a, b_i, c_i are fitting parameters to be determined for given T_s and K_y. The standard deviation of residuals is {\sigma }_{\ln D}, and ϵ is a random variable following standard normal distribution. To evaluate the efficiency of IMs, the adjusted coefficients of determination (R_{adj}^{\mathrm{2}}) is used. The R_{adj}^{\mathrm{2}} adjusts for the number of regressors through modification of coefficients of determination R^{\mathrm{2}},

(2)R_{adj}^{\mathrm{2}}=\mathrm{1}-(\mathrm{1}-R^{\mathrm{2}}){\displaystyle \frac{N-\mathrm{1}}{N-p-\mathrm{1}}},

(3)R^{\mathrm{2}}=\mathrm{1}-{\displaystyle \frac{{\displaystyle \sum _{k=\mathrm{1}}^N{\left(\ln {\widehat{D}}^{(k)}-\ln D^{(k)}\right)}^{\mathrm{2}}}}{{\displaystyle \sum _{k=\mathrm{1}}^N{\left(\ln {\widehat{D}}^{(k)}-{\displaystyle \frac{\mathrm{1}}{N}}{\displaystyle \sum _{i=\mathrm{1}}^N\ln {\widehat{D}}^{(k)}}\right)}^{\mathrm{2}}}}},

where p is the total number of regressors (p = 2n, not counting the constant term), N is the sample size. \ln {\widehat{D}}^{(k)} and \ln D^{(k)} are the-k^th observed and predicted values. R^{\mathrm{2}} is the ratio of explained variation to the total variation of the data, and it measures the predictive power of the empirical regression. R^{\mathrm{2}} ranges from 0 (no predictive power) to 1 (completely predictive). Correspondingly, the maximum value of R_{adj}^{\mathrm{2}} is 1, and it can become a negative number. The value of R_{adj}^{\mathrm{2}} is a good indicator for the efficiency of IMs, and it increases only if new term improves the model more than that would be expected by chance. In general, a larger value of R_{adj}^{\mathrm{2}} implies a smaller standard deviation of residuals, thus more efficient in empirical prediction.

Using selected scalar IMs as predictors, contours of R_{adj}^{\mathrm{2}} values were computed for slopes of different T_s and K_y are shown in Fig. 4. It is worth mentioning that the analysis is based only on seismic displacements that are greater than 1 cm. Displacements smaller than 1 cm usually causes little engineering consequence. However, the scatter of these data in log space is much larger than the others. Therefore, data elimination is necessary to avoid empirical correlation being controlled by these small-valued data. Furthermore, R_{adj}^{\mathrm{2}} will not be calculated if the number of data are less than 30 in order to guarantee that the sample size is big enough for statistical analysis. The efficiency of each scalar IM is summarized as follows:

By definition, the spectral acceleration at the initial site period, Sa(T_s), represents the peak acceleration of a single-degree-of-freedom visco-elastic oscillator of period T_s subjected to the earthquake shaking. Due to nonlinearity of the sliding mass system, Sa(T_s) is no longer an effective intensity measure, as shown in Fig. 4(a), for most cases. Instead, elongation of the slope vibration period makes the spectral acceleration at an elongated period a better IM. In an earlier study [9], Sa(1.5T_s) was proposed as the most efficient scalar IM. As shown in Fig. 4(b), using Sa(1.5Ts) can indeed significantly improve the prediction (R_{adj}^{\mathrm{2}}>\mathrm{0.7}) especially for slopes with site periods longer than 0.2 s. However, the efficiency of Sa(1.5T_s) greatly decreases () if the slope becomes stiffer (T_s<0.2 s), where the efficiency is only slightly higher than that of Sa(T_s).

Compared with the spectral acceleration, PGA has much improved efficiency for stiff slope cases (T_s<0.2 s). It is most efficient when T_s is within 0.1 s to 0.3 s, and the efficiency decreases when T_s increases. Similar behavior can be observed from ASI, which is the integrated spectral acceleration over the spectral periods from 0.1 s to 0.5 s. The efficiency of ASI only slightly decreases with increasing K_y when T_s is within 0.1s to 0.3 s.

It is interesting to compare the efficiency of IA and CAV side by side. IA is derived from time integration of the square of the entire acceleration time history, and CAV is time integration of the absolute value of acceleration time history. Both of them implicitly consider the amplitude and the duration of the ground motion. However, as shown in Figs. 4(e) and (f), the overall efficiency of IA is much better than that of CAV since high efficiency is achieved for a wider range of slope conditions. CAV is only marginally better than IA in the cases of very small K_y and long slope period T_s. Among all scalar IMs that have been considered so far, IA is the most efficient IM for stiff slope cases.

Similarity in the efficiency of PGA and IA can be attributed to their strong correlation (correlation coefficient between IA and PGA is 0.8). However, we also emphasize the improved efficiency of IA for the cases of a stiff sliding mass with weak sliding resistance (i.e., T_s<0.1 s and K_y<0.2 g). Under these conditions, the cumulative quantity of the entire time history correlates with the sliding displacement more effectively than an instantaneous spike in the time sequence. One might attribute the improvement of IA to its implicit consideration of duration in the formulation. However, further investigation reveals that the significant duration, D_5-95, has virtually no correlation with the seismic displacements for all slope conditions ( for most cases, refer to Figs. 2 and 3 for two special cases). Similarly, the mean period T_m of the ground motion has almost no predictive power at all.

The above analysis shows that the efficiency of scalar IMs varies with slope conditions, and there is no single scalar IM that is efficient to predict seismic displacements for all slope conditions. The finding is consistent with a previous study conducted for some special cases [17]. Using vector IMs allows for different characteristics of the ground motions to be considered collectively. Therefore, it is expected that the overall predictive efficiency can be improved.

Efficiency of vector intensity measures

Since Sa(1.5T_s) and IA are the most efficient scalar IMs for flexible slope cases (T_s>0.2 s) and stiff slope cases (T_s<0.2 s) respectively, the vector IM that incorporates both of them (termed as “Sa(1.5T_s) + IA”) can improve the overall efficiency for all slope conditions. As is shown in Fig. 5 (a), R_{adj}^{\mathrm{2}}>\mathrm{0.8} is achieved for slope conditions with K_y<0.1 g and T_s<2 s. The “Sa(1.5T_s) + IA” scheme has obvious advantage comparing with “Sa(1.5T_s) + CAV” and “Sa(1.5T_s) + PGA” combinations. It should be noted that Bray and Travasarou [9] recommended using Sa(1.5T_s) for T_s>0.05 s and PGA for T_s<0.05 s in their empirical model. However, their approach is obviously less efficient than using the “Sa(1.5T_s) + IA” scheme.

One limitation of using Sa(1.5T_s) as a predictor is that one has to have the prior knowledge of the slope period. For some cases, it is difficult to estimate the slope period accurately. The situation could result in increased uncertainty in seismic displacement prediction using these property-dependent IMs. It is therefore more desirable to have property-independent IMs so that they can be used for any slope condition. A promising option is to use the ordinates of spectral acceleration from short period to long period range to form the vector IMs, called “Sa-Vector”, The Sa-Vector samples a wide range of frequency content of the ground motions and describes the overall characteristics of the ground motions.

The efficiency of Sa-Vectors is evaluated using three and four spectral ordinates, namely, Sa(T = 0, 0.2, 1 s) and Sa(T = 0, 0.2, 1, 2 s), shown in Figs. 5(d) and (e). Using the Sa-Vectors, high efficiency (R_{adj}^{\mathrm{2}}=\mathrm{0.8}-\mathrm{0.9}) is achieved for T_s<1 s. The efficiency gradually decreases to 0.6 if K_y increases from 0 to 0.4 g. It is interesting to notice that inclusion of the spectral ordinate at 2s, Sa(T = 2s), into the Sa-Vector can significantly improve the efficiency for the range of 1 s<T_s<2 s. The efficiency of the Sa-Vector can be further improved by using more spectral acceleration ordinates. Figure 5(f) shows R_{adj}^{\mathrm{2}} contours of “Sa(16T),” a Sa-Vector consisted of 16 spectral ordinates. The 16 periods are evenly spaced in log between 0.01 and 10 s. Using Sa(16T) results in R_{adj}^{\mathrm{2}} that is greater than 0.85 for all slope conditions.

Conclusions

The paper systematically studied the efficiency of various scalar IMs and vector IMs in predicting seismic slope displacements using a simplified sliding mass model. Although no single IM is found to be satisfactorily efficient for all slope conditions, Sa(1.5T_s) and IA have been identified as the most efficient scalar IMs for flexible and stiff slopes, respectively.

Using vector IMs can significant improve the accuracy of seismic displacement prediction and extend the high efficiency to a wider range of slope conditions. Among various vector IMs considered, “Sa(1.5 T_s) + IA” can be an effective choice if only two IMs should be used. If more IMs to be used, the Sa-vector that incorporates at least four spectral ordinates, such as Sa(T = 0, 0.2, 1, 2 s), is recommended.

The study provides useful guidance to the development of more efficient empirical prediction models as well as the ground motion selection criteria for time-domain seismic slope analysis. To represent the real seismic hazard at a site, the selected ground motions should capture the variability of the most efficient IMs. Among various vector IMs considered, the spectral accelerations at multiple spectral periods achieve high efficiency for a wide range of slope conditions. The desirable property of Sa-Vector implies that it is important for the input ground motions to capture the variability of the spectrum accelerations at multiple spectral periods. For this purpose, a ground motion selection method recently proposed [18] can be useful in time-domain seismic slope analysis.

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