**5. The shear-band slope stability analysis method**

Considering the lateral compression of tectonic plates, Hsu [3] simulated the formation of shear failure bands, as shown in **Figure 7**. **Figure 7a** and **b** demonstrate that when the strain goes deep into plastic range, the localization of deformations of the tectonic plate will appear due to the loss of ellipticity caused by strain softening and further develop shear failure bands and shear-band tilting slopes.

Since the amount of shear banding accumulates from all previous tectonic earthquakes, the degree of brittle fracture in rocks of multi-step shear-band tilting slopes (**Figure 13**) will increase and large-scale shear failure of slopes will occur.

The Tsaoling area had three shear failures before the 1999 Jiji earthquake, in 1861, 1941, and 1979 [13]. Given that slope shear failure continued to occur at the same location, it can be deduced that the existing slope stability analysis methods did not consider the shear banding effect. Therefore, the Tsaoling landslide area had a total length of 2324.5 m and an average thickness of 24 m as shown in **Figure 13**. Although Hung et al. [14] obtained the shear strength parameters for the sliding plane, since the shear failure plane shown in **Figure 13** is the interface between silty sandstone and shale, the shear strength parameters they obtained were low because only the horizontal ground vibration was considered. To improve this, a new shear-band slope stability analysis method is provided below.

**Figure 14.** *Shear banding that causes the hanging wall to rise [12].*

#### **5.1 The shear-band slope stability analysis method**

**Figure 14** shows a sliding block with two sliding surfaces, the lower sliding surface has a relatively gentle slope and the upper sliding surface has a relatively steep slope. Since the relatively steep sliding surface is caused by the uplifting action of shear banding, **Figure 14** shows that the shear textures within an overall shear band include principal displacement shear D, thrust shear P, Riedel shear R, conjugate Riedel shear R', and compression textures S. Thus, the hanging wall shown in **Figure 14** will continue to be lifted during the shear banding of tectonic earthquakes, whereas the foot wall will remain stationary [12].

First, based on the consideration of the shear banding and ground vibration effects of tectonic earthquakes, a new slope stability analysis model as shown in **Figure 15** is presented; in which a shear banding force, *Ps*, is acting on the top of the upper sliding

**Figure 15.** *The proposed slope stability analysis model with dual shear failure planes [12].*

*Plasticity Model Required to Prevent Geotechnical Failures in Tectonic Earthquakes DOI: http://dx.doi.org/10.5772/intechopen.107223*

surface and is perpendicular to it. The horizontal and vertical ground vibration forces acting on the upper sliding block are *khW*<sup>1</sup> and - *kvW*1, respectively, and the horizontal and vertical ground vibration forces acting on the lower sliding block are *khW*<sup>2</sup> and �*kvW*2, respectively.

As for the shear-band tilting force *Ps*, due to the shear banding effect, it is assumed that tension cracks exist at both ends of the right side of the upper sliding block and the left side of the lower sliding block, and the relatively steep upper sliding surface was caused by the uplifting effect of shear banding from previous tectonic earthquakes. Under these conditions, the shear-band tilting force *Ps* will be approximately equal to half of the upper sliding block weight *W*1*=*2 [12].

Next, since the groundwater table is lower than the sliding surface when the Tsaoling landslide occurs, it will not affect the slope stability analysis results. Therefore, as shown in **Figure 13**, the required parameters or material properties for the slope stability analysis of the Tsaoling landslide area include:

For the upper sliding block, the sliding surface length *L*1*:*, inclination angle *α*1, weight *W1*, normal component of *W1* on the upper sliding surface *N*1, adhesion *cα*1, friction angle *δ*1, the resultant force from the friction and adhesion resistances *R*1, and shear banding force *Ps*.

For the lower sliding block, the sliding surface length *L*2*:*, inclination angle *α*2, weight *W*2, normal component of *W*<sup>2</sup> on the lower sliding surface *N*2, adhesion *cα*2, friction angle *δ*2, and the resultant from the friction and adhesion resistances *R*<sup>2</sup>

As for the vertical interface between the upper and lower sliding blocks, under static equilibrium the active earth pressure of the upper sliding block *Pa*<sup>1</sup> and its angle of intersection with the horizontal plane *ϕ*<sup>1</sup> need to equal the active earth pressure of the lower sliding block *Pa*<sup>2</sup> and its angle of intersection with the horizontal plane *ϕ*2, respectively; therefore, the factor of safety that satisfies the relationships of *Pa*<sup>1</sup> ¼ *Pa*<sup>2</sup> ¼ *Pa* and *ϕ*<sup>1</sup> ¼ *ϕ*<sup>2</sup> ¼ *ϕ* is the factor of safety *FS* desired in the analysis.

For the forces acting on the upper and lower sliding blocks shown in **Figure 16** under static equilibrium, the closed force polygons of all forces acting on the upper sliding block and lower sliding block are presented in **Figure 16a** and **b**, respectively.

**Figure 16.** *The closed force polygons for the upper and the lower sliding blocks [12].*

For the upper sliding block, the closed force polygon shown in**Figure 16a** can be used as supplemented by the horizontal and vertical force balances (*i.e.*, <sup>P</sup>*Fh* <sup>¼</sup> 0 andP*Fv* <sup>¼</sup> 0). The active earth pressure for the upper sliding block *Pa*<sup>1</sup> is thus given in Eq. 18 as:

$$P\_{a1} = A^\* \left/ \left[ \cos \phi\_m + \sin \phi\_m \tan \left( a\_1 - \delta\_{1m} \right) \right] \tag{18}$$

where

$$A^\* = \left[W\_1(\mathbf{1} - k\_v) - P\_\prime \cos a\_1 - c\_{\alpha 1m} L\_1 \sin a\_1\right] \tan \left(a\_1 - \delta\_{1m}\right) + W\_1 k\_h + P\_\prime \sin a\_1 \quad \text{(19)}$$
  $-c\_{a1m} L\_1 \cos a\_1$ 

Next, for the lower sliding block, the closed force polygon shown in **Figure 16b** can be used as supplemented by the horizontal and vertical force balances of <sup>P</sup>*Fh* <sup>¼</sup> <sup>0</sup> and <sup>P</sup>*Fv* <sup>¼</sup> 0, respectively. The active earth pressure of the lower sliding block *Pa*<sup>2</sup> is given by Eq. 20 as:

$$P\_{a2} = B^\* \left/ \left[ \cos \phi\_m + \sin \phi\_m \tan \left( a\_2 - \delta\_{2m} \right) \right] \right. \tag{20}$$

where

$$B^\* = -[W\_2(1 - k\_v) - c\_{a2m}L\_2 \sin a\_2] \tan \left(a\_2 - \delta\_{2m}\right) - W\_2 k\_h + c\_{a2m}L\_2 \cos a\_2 \tag{21}$$

Since the force balance conditions for the upper and lower sliding blocks need to be met at the same time, a trial-and-error method can be used by first assuming a factor of safety *FSa*. Subsequently, Eq. 18 can be used to calculate the active earth pressure acting on the upper sliding block *Pa*1, and then Eq. 20 can be used to calculate the active earth pressure acting on the lower sliding block *Pa*2. Finally, after the calculated value of *Pa*<sup>1</sup> � *Pa*<sup>2</sup> becomes less than the error tolerance *ε*, the slope stability factor of safety *FS* is then set equal to *FSa*. Based on this trial-and-error method [12], a calculation procedure was used to obtain the slope stability factor of safety *FS*.

#### **5.2 Case study**

The large-scale Tsaoling landslide induced by the 1999 Jiji earthquake was selected as a case study for the slope stability analysis. The back analyses of the slope stability were conducted under the following two conditions:

Condition 1: Considering the effects of shear banding and the horizontal and vertical ground vibration of the tectonic earthquake [12].

Condition 2: Only considering the effect of the horizontal ground vibration of the tectonic earthquake [14].

From **Figure 13**, points A to K can be roughly divided into five shear-band tilting slope steps: ABC, CDE, EFG, GHI, and IJK [12]. Each step in the shear-band tilting slope has a relatively gentle slope segment, AB, CD, EF, GH, and IJ, and a relatively steep slope segment, BC, DE, FG, HI, and JK. Using the sliding failure mechanism of multi-step shear-band tilting slopes, it was found that after sliding failure of the first step, each of the following steps would have a sliding failure in succession.

For the slope stability analysis of the Tsaoling landslide area, the coordinates, horizontal distance, elevation difference, and inclination angle of each line segment shown in **Figure 13** were first recorded. Then, for the shear failure planes of the five non-shear banding zones AB, CD, EF, GH, and IJ shown in **Figure 13**, the unit weight


*Plasticity Model Required to Prevent Geotechnical Failures in Tectonic Earthquakes DOI: http://dx.doi.org/10.5772/intechopen.107223*

#### **Table 2.**

*The adopted kh and kv corresponding to each PGA for the five step sliding blocks [12].*

for the corresponding sliding blocks was 24.52 kN*=*m3, whereas for the five shear banding zones BC, DE, FG, HI, and JK it was 20.60 kN*=*m3 . From the distribution map of peak acceleration for the landslide area in Tsaoling, the peak ground acceleration (PGA) for the five step shear failure blocks in the Tsaoling landslide area can be obtained from PGA distribution map of the 921 Jiji earthquake reported by Earthquake Prediction Center, Central Weather Bureau, Taiwan [15]. The corresponding relationship between the PGA and seismic acceleration coefficients was provided by the Ministry of Economic Affairs [16]. The adopted horizontal seismic acceleration coefficient *kh* and vertical seismic acceleration coefficient *kv* corresponding to each PGA are given in **Table 2**.

In the slope stability analysis for the five stepped shear-band tilting slopes, the shear banding forces *Ps*, the horizontal ground vibration forces of the upper sliding block, the vertical ground vibration forces of the upper sliding block, the horizontal ground vibration forces of the lower sliding block, and the vertical ground vibration forces of the lower sliding block are given in **Table 3**.

The shear resistance strength parameters obtained through back-calculation under Condition 1 are given in **Table 4**.

The shear resistance strength parameters obtained through back-calculation under Condition 2 are given in **Table 5**.

The obtained safety factors using the shear resistance strength parameters of the sliding surface obtained from the back analysis under different conditions for the shear-band slope stability analysis are given in **Table 6**.


#### **Table 3.**

*The shear banding forces and ground vibration forces used in the case study for the shear-band tilting slopes [12].*


#### **Table 4.**

*Shear resistance strength parameters obtained through back analyses under condition 1 [12].*


#### **Table 5.**

*Shear resistance strength parameters obtained through back analyses under condition 2 [14].*


#### **Table 6.**

*Shear-band slope stability analyses as obtained from the back-calculation results under conditions 1 and 2 [12].*

For multi-step slope failure induced by the effects of shear banding and ground vibration of a tectonic earthquake, when the shear banding effect is ignored and only the horizontal ground vibration effect is considered, low shear strength parameters of the sliding plane are obtained by back analysis. When the shear-band slope stability analysis is carried out using the shear strength parameters obtained from the back analysis, only the back analysis results that consider the effects of the shear banding and the horizontal and vertical ground vibrations can obtain a factor of safety that meets the actual requirements.

*Plasticity Model Required to Prevent Geotechnical Failures in Tectonic Earthquakes DOI: http://dx.doi.org/10.5772/intechopen.107223*

For the Tsaoling landslide that occurred during the 921 Jiji earthquake, when the back analysis considered the effects of the shear banding and the horizontal and vertical ground vibration effects, **Table 6** shows that the shear strength parameters obtained from the back analysis of the five-step shear-band slopes resulted in the calculated safety factors being consistent with the actual safety factor of 1.0; however, when the back analysis only considered the horizontal ground vibration effect, **Table 6** shows that the shear strength parameters obtained from the back analysis of the five-step shear-band slopes resulted in the calculated safety factors being much lower than the actual safety factor of 1.0 [12].
