1 Introduction

Co-seismic landslide is one of the most destructive hazards associated with earthquake (Yin et al. 2009; Gorum et al. 2011; Xu et al. 2014). The indirect damage caused by co-seismic landslides is far greater than the direct effect of earthquake itself (Rodriguez et al. 1999; Keefer 2000; Ling et al. 2001, 2005; Huang and Li 2008; Duan et al. 2015). Therefore, estimating where and in what shaking conditions earthquakes are likely to trigger landslides is a key element in selecting the location of roads and towns (Wilson and Keefer 1985; Ambraseys and Menu 1988; Keefer and Wilson 1989; Wieczorek et al. 1993; Jibson and Keefer 1993; Ambraseys and Srbulov 1994; Khazai and Sitar 2000; Jibson et al. 2000; Romeo 2000; Decanini and Fabrizio 2001; Lveda et al. 2005; Kokusho and Ishizawa 2006; Bray and Travasarou 2007; Kokusho et al. 2009; Rodríguez-Peces et al. 2014; Huang and Xiong 2017; Huang et al. 2018). Newmark (1965) proposed the landslide model by modeling the landslide as a rigid block with friction that slides on an inclined plane, and the block will begin to slide against the friction of the sliding surface when the seismic intensity exceeds the seismic strength of the slope. Newmark displacement analysis requires two types of data: (1) the critical acceleration (ac) of the slope, which is simply the base acceleration required to overcome shear resistance and initiate sliding; (2) the earthquake acceleration time history, which represents earthquake action. Since the earthquake acceleration time history can only be obtained after the earthquake, the Newmark model cannot be used for the prediction of landslides.

In the past 30 years, researchers used peak acceleration (PGA) and other parameters to replace the earthquake acceleration time history data in the Newmark model and established a series of regression equations to predict the Newmark displacements of the slope (Ambraseys and Menu 1988; Jibson 1993; Milesa and Ho 1999; Al-Homoud and Tahtamoni 2000; Romeo 2000; Shang and Lee 2011; Rajabi et al. 2011; Lee et al. 2012; Ma and Xu 2018). The relation curve with the critical acceleration ratio (ac/amax) and the permanent displacement (DN) as the independent variable and corresponding variables, respectively, is called the Jibson permanent displacement method. Since the peak acceleration of the earthquake can be determined by using the national zonation map of ground-motion peak acceleration. The critical acceleration is an inherent attribute of the slope. The Jibson method is practically applicable to the prediction of the permanent displacement of seismically triggered landslides. Recently, the Jibson method has been used internationally and has received recognition. For example, Chen et al. (2011) used the Jibson method to carry out mapping of earthquake-triggered landslides in the Yingxiu area.

However, the accuracy of the Jibson displacement prediction method has not been examined. Through analyzing the Newmark sliding-block model, we found that this model ignores the attenuation effect of the shear strength on the sliding surface during an earthquake, resulting in the calculated value of the Jibson method being smaller than the actual earthquake value. This inference was verified by real seismic data from the Wenchuan earthquake. Therefore, Jibson method was modified by adding attenuation coefficients to the shear strength in this paper.

Ultimately, a hazard zoning method based on the modified Jibson method was established to map the spatial distribution of landslides by referring to the rock failure model (Jaeger and Cook 1969). The relevant data of the Wenchuan earthquake were used to establish the hazard zoning method, and the correctness of the zoning method was tested by the seismic data of the Lushan earthquake.

2 Newmark model theory

2.1 Newmark sliding-block model

The potential landslide is modeled as a block resting on a plane inclined at an angle (α) from the horizontal (Newmark 1965). The block has a base acceleration required to overcome and initiate sliding, called the critical acceleration (ac). The model calculates the permanent displacement of the block as it is subjected to a base acceleration (a), which representing the earthquake shaking (Fig. 1).

Fig. 1
figure 1

Sliding-block model of Newmark analysis

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The Newmark displacement can be calculated by integrating the seismic acceleration-time history beyond the critical acceleration for twice (Fig. 2), and the equation is expressed as

$$D_{\text{N}} = \iint_{t} {\left( {a(t) - a_{\text{c}} } \right){\text{d}}t{\text{d}}t}$$
(1)

where DN is the Newmark permanent displacement, and a(t) is the acceleration time history.

Fig. 2
figure 2

Demonstration of the Newmark-analysis algorithm. (a) Earthquake acceleration-time history with critical acceleration (horizontal dashed line) of 0.20 g superimposed. (b) Velocity of landslide block versus time. (c) Displacement of landslide block versus time

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Newmark (1965) defines the critical acceleration of the block as a parameter containing the static factor of safety and slope geometry expressed as

$$a_{\text{c}} = (F_{\text{s}} - 1)g\sin \theta$$
(2)

where ac is the critical acceleration; g is the acceleration due to Earth’s gravity; Fs is the static factor of safety; θ is the angle from the horizontal that the center of the potential slide block first moves, and it can be generally approximated as the slope angle.

The static factor of safety (Fs) in this condition is expressed as

$$F_{\text{s}} = \frac{{(c^{{\prime }} )}}{\gamma t\sin a} + \frac{{\tan (\varphi^{{\prime }} )}}{\tan a} - \frac{{m\gamma_{w} \tan (\varphi^{{\prime }} )}}{\gamma \tan a}$$
(3)

where φ′ is the effective friction angle; c′ is the effective cohesion; α is the slope angle; γ is the material unit weight; γw is the unit weight of water; t is the slope-normal thickness of the failure slab; and m is the proportion of the saturated slab thickness in the slope. The first term on the left side represents the cohesive component of the strength, the second term accounts for the frictional component, and the third term represents the reduction in frictional strength due to pore pressure.

The Newmark model cannot be used for predicting the permanent displacement, because the acceleration time history (a(t) in Eq. 1) can only be obtained using the seismological station after the earthquake.

2.2 Jibson method of permanent displacement prediction

Jibson (1993) used the peak ground acceleration (PGA) and other parameters that can be obtained before the earthquake to establish a new predicting method for the permanent displacement. In 2007, Jibson collected 2270 strong vibration records from 30 earthquakes around the world and re-established four regression equations based on the parameters of Arias intensity, critical acceleration, and critical acceleration ratio. Among these displacement prediction equations, the equation (Jibson 2007) with the critical acceleration ratio (ac/amax) as the parameter has the best fit. It is expressed as:

$$\log D_{\text{N}} = 0.215 + \log \left[ {\left( {1 - \frac{{a_{\text{c}} }}{{a_{\hbox{max} } }}} \right)^{2.341} \left( {\frac{{a_{\text{c}} }}{{a_{\hbox{max} } }}} \right)^{ - 1.438} } \right]$$
(4)

where amax is the peak ground acceleration, and ac is the critical acceleration of the slope.

2.3 Theoretical defect of the Newmark sliding-block model

The Newmark model assumes that the critical acceleration of the block remains unchanged during the process of an earthquake, which ignores the dynamic attenuation of the shear strength on the sliding surface and causes the predicted value of the Jibson method to be smaller than the displacement caused by an actual earthquake. In fact, under a static load, the static cohesion and internal friction angle are taken as the shear strength parameters of the slope. However, under the dynamic load, the dynamic cohesion and internal friction angle should be taken as the shear strength parameters of the slope.

3 Modification of the Jibson method

The dynamic cohesion and internal friction angle (characterized by multiplying the static cohesion and internal friction angle by their respective attenuation coefficients) were used to calculate the critical acceleration under the dynamic load. In this way, the Newmark model was modified to improve the prediction accuracy of the Jibson method.

In this study, we mainly focus on the stability of highway slope in the Wenchuan earthquake. According to the field investigation, it was found that the highway slope is soil–rock medium, and the ratio of soil to rock is approximately 7:3. Therefore, we focused on the soil–rock medium (the ratio of soil to rock is 7:3) in this project. Also, the dynamic triaxial test results show that the attenuation coefficients of cohesion and internal friction angle of specimens with different soil–rock ratios vary from 0.55 to 0.7 and from 0.75 to 0.93, respectively (Wang et al. 2012). Since the highway slope is soil–rock medium and the ratio of soil to rock is approximately 7:3, thus, we determined that the attenuation coefficients of the cohesion and internal friction angle are 0.6 and 0.85, respectively. In order to improve the applicability of this method, further research will be conducted on the attenuation coefficients for a different soil and rock in the future.

Based on the analysis above, two attenuation coefficients were added into Eq. 3 to modify the static factor of safety in the Newmark model and expressed as

$$F_{\text{s}} = \frac{{(0.6c^{{\prime }} )}}{\gamma t\sin a} + \frac{{\tan (0.85\varphi^{{\prime }} )}}{\tan a} - \frac{{m\gamma_{w} \tan (0.85\varphi^{{\prime }} )}}{\gamma \tan a} \,$$
(5)

Result showed that the critical acceleration (ac) is 0.7 times the previous one after taking the modified Fs into Eq. 2, and the modified equation of the Jibson method was proposed and expressed as

$$\log D_{\text{N}} = 0.215 + \log \left[ {\left( {1 - \frac{{0.7a_{\text{c}} }}{{a_{\hbox{max} } }}} \right)^{2.341} \left( {\frac{{0.7a_{\text{c}} }}{{a_{\hbox{max} } }}} \right)^{ - 1.438} } \right]$$
(6)

The attenuation coefficients can be adjusted appropriately for areas with less seismic intensity.

Compared with the original model, the modified Jibson model considers the attenuation of shear strength of rock and soil under earthquakes and introduces attenuation coefficient. It perfects the Newmark model in terms of its mechanics.

4 Verification of modification effect using real seismic data

4.1 Data

The Wenchuan earthquake is one of the most devastating earthquakes recorded in the era of modern instrumentation, and it provides special conditions for testing the accuracy of the modified Jibson method. The process of the verification requires two aspects of data: (1) the real displacements (DN) of slopes and (2) the ac and PGA of slopes used for calculating the predicted displacements. Exactly, the Wenchuan earthquake provides special conditions for studying the Jibson method. The survey samples are mainly from the slopes (containing DN, ac, and PGA) along the roads in seismic areas, and the distribution of these slopes is shown in Fig. 3. The displacement of each slope was acquired by manual field measurement (Figs. 4 and 5).The critical acceleration of each slope was calculated through the slope parameters, and the PGA of each slope was obtained using the ground-motion attenuation model with the east–west acceleration time history recorded by the Wolong station during the Wenchuan earthquake (Cui et al. 2011). The information of DN, ac, and PGA is listed in Table 1.

Fig. 3
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Distribution of displacement collection points

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Fig. 4
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A typical landslide along the highway

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Fig. 5
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Slippage from the concrete retaining wall’s bottom

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Table 1 Data of slope displacement acquisition points
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4.2 Processing

In order to test the accuracy of the modified Jibson method (Fig. 6), the measured displacements of the slopes were put in the same coordinate axis with the prediction curves of the traditional Jibson method and the modified Jibson method. It was concluded that the modified equation agreed better with the measured value than the traditional equation in the region with large seismic intensities (Fig. 6). However, the modified displacement value showed a larger trend than the measured value in the region with low seismic intensities. It can be explained that the attenuation coefficients of the internal friction angle and the cohesion force were set to be the fixed values in Eq. 6 for regions with large earthquake intensities, which means the attenuation effect was considered too strong for regions with low seismic intensities. Obviously, landslides triggered by strong earthquakes are the focus of our attention. In summary, the accuracy of the modified Jibson method was superior to that of the traditional method in regions with large earthquake intensities.

Fig. 6
figure 6

Comparison between predicted and measured values

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5 Hazard zoning for seismically triggered landslides based on a modified Jibson method

5.1 Relationship between Newmark displacement and areal density of landslides

Jibson used the Weibull curve to fit two parameters (Newmark permanent displacement and the probability of slope failure) with the data of 1994 Northridge earthquake and obtained the equation for the probability of slope failure (Jibson 2000). However, the areal density of landslides (the proportion of landslide area per square kilometer) can be more intuitive than the probability of slope failure for seismic landslides. Therefore, the Weibull curve was used to fit the areal density of landslides and Newmark permanent displacements. Eventually, an equation for predicting the areal density of landslides was established.

The Wenchuan earthquake (magnitude 8.0) presents all of the data required for detailed regional analyses. Firstly, the northern part of the area from Maoxian to Beichuan County was chosen as the study area (Fig. 7). This area contains seismic intensity zones of IX, X, and XI. Moreover, the lithology is diverse, which facilitates the collection of representative permanent displacements and the areal density of landslides. Regions with slopes below 10° do not have the topographical conditions to trigger landslides and have been removed from the study area. Secondly, ArcGIS was used to collect the real areal density of earthquake-triggered landslides in the study area. Based on the collection of geological, topographical information (Wang et al. 2013), and PGA information in the study area, the modified Jibson method was used to calculate the slope permanent displacements. Two sets of data (areal density of landslides and permanent displacements) were projected to the same layer in ArcGIS, which facilitated their corresponding collection (Fig. 8). Since the permanent displacements calculated by the modified Jibson method are discrete values, ten sets of data were obtained. The average areal density of landslides corresponding to each set of displacement values was counted, respectively, and two sets of data were shown in Table 2.

Fig. 7
figure 7

Study area

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Fig. 8
figure 8

Displacement and areal density data

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Table 2 Newmark displacement calculation and landslide surface density
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The Weibull curve was used to fit two sets of data, and the relationship curve is shown in Fig. 9. The formula is as follows:

$$p = 0.301\left( {1 - \exp \left( { - 0.00018D_{N}^{2.56} } \right)} \right)\;R^{2} = 0.98$$
(7)
Fig. 9
figure 9

Landslide density curve

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5.2 Study on hazard zoning for seismically triggered landslides

Based on the trend of the areal density curve, a criterion of classification with areal density as an index was proposed for earthquake-triggered landslide areas, as shown in Table 3.

Table 3 Earthquake-triggered landslide hazard classification system
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On the basis of analyzing the trend of the areal density curve, we divided the severity of seismically triggered landslides into three levels: (1) mild danger zone: the displacement ranged from 0 to 10 cm (corresponding areal density of landslides: 0–0.03). No structural damage occurred on the slope, and the shear strength was approximately the same before the earthquake; (2) moderate danger zone: the displacement ranged from 10 to 40 cm (corresponding areal density of landslides: 0.03–0.27). Slopes suffered the structural damage, but there was still residual strength. Landslides were widely distributed; and (3) severe danger zone: the displacement value was more than 40 cm (corresponding areal density of landslides: 0.27–0.30). Slopes were prone to disintegration, and landslides were distributed densely.

5.3 Rationality verification of the hazard zoning method with the data of the Lushan earthquake

The Lushan earthquake presents an ideal case to test the accuracy of the hazard zoning method. The epicenter is located at 30°18′N, 103°E with the intensity of IX, and a depth of 13 km. The Lushan earthquake triggered more than 1600 landslides, which provides favorable conditions for the rationality test of hazard zoning method. The scope of the study area is from 30°10′0′′N to 30°20′0′′N latitudes and 102°50′0′′E–103°10′0′′E longitudes, covering an area of 422 km2, and including part of the VIII-degree zone and the entire IX-degree zone.

In order to test the rationality of the risk zoning method, we compared the real distribution of the areal density of landslides in the study area with the predicted distribution. As mentioned before, the calculation of the modified Newmark displacements in the study area required two aspects of data: the information of ac and the PGA. The shear strength parameters for ac are shown in Table 4, and the PGA information can be obtained from the zoning map of ground-motion peak acceleration in China. The results showed that the Newmark displacements in study area ranged from 0 to 40 cm (Fig. 10). The displacements were 20–40 cm in the center of IX-degree zone; 20 cm or less at the junction of VIII and IX-degree zone; and less than 10 cm in the VIII-degree zone. On the basis of the displacements, the areal density of landslides in study area was obtained using Eq. 7. The study area was divided into two types of danger zone (the mild danger zone and the moderate danger zone) according to distribution of the areal density of landslides. The hazard zoning results and real earthquake-triggered landslides in the study area are both shown in Fig. 11. The average areal density of landslides was 0.013 in the mild danger zone and 0.21 in the moderate danger zone, respectively, which fell within the areal density interval of the hazard zoning method. Comparing with the phenomenon that more than 60% of the landslides in Lushan earthquake were concentrated in the very low hazard area in the traditional Newmark model calculated by Ma and Xu (2018), the hazard zoning method based on the modified Newmark model showed a good trend that the areal density of landslides increased rapidly with the increase in the hazard level. Therefore, the rationality of this seismic landslide hazard zoning method was verified using the data of the Lushan earthquake.

Table 4 Rock parameters in the study area
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Fig. 10
figure 10

Prediction value of Newmark displacement

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Fig. 11
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Distribution of real landslide points of Lushan earthquake

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6 Conclusions

The Newmark model was modified by adding attenuation coefficients to the effective internal friction angle and cohesion (0.85 and 0.6, respectively), and the critical acceleration in this model is 0.7 times it before. The data of the Wenchuan earthquake showed that the accuracy of Jibson predicting method was significantly improved after the modification. Based on the modified Jibson method, a hazard zoning method for earthquake-triggered landslides with areal density as the zoning index was established. The severity of seismic landslides was divided into three levels according to the areal density (mild danger zone: 0–0.03; moderate danger zone: 0.03–0.27; and severe danger zone: 0.27–0.3). The rationality of this hazard zoning method has been verified by the data of the Lushan earthquake. The hazard zoning method adopted the macroscopic index of areal density, which not only contains information on the deformation and damage of slopes, but also reflects the changes in the pregnancy environment of mountain disasters after the earthquake. Therefore, the hazard zoning method can provide a scientific basis for the construction of major projects in mountainous areas with a high seismic intensity.