A Case Study of Using Numerical Analysis to Assess the Slope Stability of National Freeways in Northern Taiwan ^†

Abstract

Taiwan is located at a junction of tectonic plates, which results in frequent earthquakes. Its terrain is mostly hilly, and its rainfall ranks among the highest in the world. Each of these elements affects the stability of slopes in various regions of Taiwan. Several slopes along Taiwan’s Freeway 1 and 5 have experienced landslides and rockfalls. It is imperative that the slope stability of these national freeways be analyzed to avoid future slope collapses brought on by precipitation or other outside factors. Thus, three sites on Taiwan’s Freeway 1 and 5 were chosen for numerical slope stability analysis in this study. PLAXIS 2D CE (Version: 24.02.00.1144) finite element software was used in this study to simulate and analyze the safety of freeway slope protection projects. Displacements induced by normal and high groundwater levels were discussed. Moreover, a pseudo-static study of slope displacements under seismic conditions was performed. According to the results of the numerical study, the force operating on the slope was centered on the sliding surface when the groundwater level was normal, and it extended to the top when the groundwater level was high. By comparison, under seismic conditions, the force acting on the slope extended to the whole slope. Furthermore, the slope safety factor of Site 1 was greater than the design specification value in three different scenarios. This confirms that the slope protection project at Site 1 is effective.

1. Introduction

Many nations have constructed substantial road networks and residential communities in hilly and mountainous terrain as a result of the global population increasing and socioeconomic development. However, landslides occasionally occur in these regions, and they can be extremely dangerous for people and property, particularly in places with steep terrain and considerable rainfall [1,2,3]. Thus, geotechnical researchers have focused increasingly on slope stability assessments. Duncan [4] stated that finite element methods have been used to analyze a large number of dams, as well as other embankments and slopes. Kardani et al. [5] proposed a hybrid superposition integration method based on finite element analysis and field data to improve slope stability prediction. Ahmadi et al. [6] conducted a reliability analysis of stiffened soil’s internal and external stability under static and seismic loads. Chen et al. [7] proposed an example of numerical analysis of slope stability on a national expressway in Northern Taiwan (This article is a revised and expanded version of a paper entitled “Case Studies on Numerical Analysis of Slope Stability of National Expressway in Northern Taiwan”, presented at ACEM22/Structures22, 16–19 Au-gust 2022, Seoul, Republic of Korea). Li et al. [8] engaged in a characterization and stability analysis of rock-mass discontinuities in layered slopes, taking the Fushun West Open-Pit Mine as an example.

In mountainous regions, landslides rank among the most dangerous natural calamities [9,10]. Taiwan’s landscape is primarily made up of hills and mountains, which make up almost two-thirds of the island’s overall area. As a result, safety in the event of landslides is a major concern for the appropriate government bodies. Since the construction of National Freeway 1, the demand for motorways has grown. The freeway segment between Keelung and Xizhi was constructed 46 years ago, and shotcrete has been used to cover the bedrock slope’s surface. The shotcrete covering has been repaired and maintained, but aging has led to cracks and plant growth on the surface. On 13 January 2011, a severe landslide occurred on the slope of the southbound lane of Taiwan’s Freeway 1, with a mileage of 4K + 820. It is worth noting that on National Freeway 5, several slopes are classified as high-risk according to the “Freeway Maintenance Manual” [11].

Slope stability is an important and complex issue in road engineering which is related to soil properties, soil density, soil pore water pressure, excavation methods, load effects, etc. In view of the above, locations with a high risk of landslides on the slopes of Taiwan’s Freeways 1 and 5 were selected in this work as case studies because severe landslides have occurred near these locations. The slope displacement and safety factor at normal and high groundwater levels were simulated using the PLAXIS 2D CE finite element analysis tool, and variations in acceleration, speed, and displacement in slope models were examined using pseudo-static analysis in seismic circumstances. In short, the aim of this study was to provide numerical simulation data to government departments as a reference for evaluating the effectiveness of freeway slope protection projects. In the future, the numerical simulation results might help prevent the collapse of freeway slopes when severe rains or other conditions impair their stability.

2. Literature Review

2.1. Factors Affecting Slope Stability

Under specific geological background circumstances, slope instability typically arises as a result of the excitation of initiating events like intense earthquakes and heavy rains [12]. A hillslope’s stability can be affected by a variety of causes. Geological material, geological structure, topographic and environmental conditions, and engineering aspects are the four main causes of landslides according to Hung [13].

• Geological material: The slope stratum is mainly composed of single or multiple geological materials. The cementation, particle size, and composition of the geological materials directly affect the stability of the slope.
• Geological structure: Weak planes (such as strike and dip angle) and slope surfaces (such as dip slope, anti-dip slope, and cross-dip) are the main factors affecting slope stability. Other geological structures, such as faults, joints, folds, layers, and other discontinuous surfaces, transform the slope’s rock and soil into discontinued or fragmented stones. These structures reduce rock and soil strength and increase weathering, affecting the slope’s stability.
• Topographic and environmental conditions: Topography includes the slope’s general form and rough surface terrain. While slope terrain has its own height and gradient, surface terrain relates to the characteristics of the geological structure. Groundwater, earthquakes, and rainfall are examples of environmental elements that affect slope stability.
• Engineering aspects: Slope stability is affected directly or indirectly by human factors such as freeways, tunnel excavation, blasting, overdevelopment of slopes, substandard building or site selection, poor slope drainage systems, and inadequate maintenance of protective structures.

To prevent road slopes from collapsing without warning, Taiwan’s Ministry of Transportation and Communications uses slope patrols, slope monitoring, and ground anchor inspections to closely monitor slope stability. According to the “Freeway Maintenance Manual” [11], slopes are divided into the following four categories: A, B, C, and D. The classification criteria and disposal countermeasures are described below.

• Class A slope: Slopes classified as class A exhibit clear indications of instability. Appropriate actions must be implemented right away, and cooperation with careful observation and monitoring is needed.
• Class B slope: Due to the possible indication of instability, additional patrols, monitoring, maintenance, reinforcing, and remediation are needed.
• Class C slope: The slope needs regular patrol or maintenance, with monitoring as necessary, even if there are no obvious signs of instability.
• Class D slope: Patrolling is still necessary, even though it is steady.

2.2. Prediction of Slope Stability

Predicting slope stability has been the subject of numerous studies over the last few decades. These research methodologies can be broadly categorized into three groups: the finite element technique (FEM), limit equilibrium method (LEM), and limit analysis method (LAM) [14]. Due to its ease of use, the LEM is one of the most popular approaches among them [15]. However, several idealized assumptions about the LEM lead to its imprecise response. Because it puts strict restrictions on limit state solutions [16] and offers a straightforward calculation procedure, the LAM is commonly used in slope stability [17]. However, it can be difficult to create theoretical formulations of upper or lower bounds when dealing with complicated slope structures and variable soil deposition [14].

Compared to the LEM and LAM, the FEM appears to be a more reliable and adaptable processing technique for slope performance modeling and evaluation [14,18,19,20]. Kafle et al. [21] constructed a finite element model to simulate the Bianjiazhai landslide near the Suofengying reservoir at the Wu River in Guizhou Province, China. Their results showed that sudden reservoir level fluctuations had a critical effect on the stability of the slope. In addition, a strength reduction analysis confirmed that the deformation behavior of the slope was mainly a shear-slip phenomenon, and the associated shear zones were identified. In order to accurately and efficiently simulate in situ stress prior to seismic loading, Kontoe et al. [22] used pseudo-static finite element analysis. They also investigated how sensitive the analysis findings were to the finite element mesh level.

The main metric used to assess slope stability is the safety factor [23], which is the ratio of the available shear strength to the equilibrium shear stress along a specific failure surface [4]. Assumptions on the form and location of the failure surface are not required when utilizing the FEM for a slope stability analysis. According to the FEM, failure occurs when the soil’s shear strength is not strong enough to support the sliding force of the slope, and it is feasible to model how displacement and slope failure develop. The FEM often employs the soil shear strength reduction technique to calculate the slope’s safety factor [24]. During the slope failure process, soil cohesion c and internal friction angle φ continue to decrease. According to the literature, no matter how the slope is formed, when it reaches the failure state, the reduced cohesion tends to have the same value. Accordingly, Zienkiewicz et al. [25] reduced the strength parameters (c, tanφ) of a slope step-by-step through a reduction factor of 1/N, and the reductions for the two parameters were equal. When reduced to the failure state, the N value at this time can be defined as the safety factor of the slope.

Giam and Donald [26] calculated the slope’s overall safety factor by examining the link between the node displacement and the change in soil strength characteristics (cohesion and friction angle) using the region close to the top of the neighboring slope toe as the chosen node. Ugai [27] determined the system’s overall safety factor iteratively by adopting a minimal system safety factor value for the material’s strength (c and φ). To determine whether the system was unstable, they then performed finite element analysis using the converted material strength. Brinkgreve and Bakker [28] used their developed PLAXIS finite element formula to calculate the slope’s safety factor and the corresponding sliding surface failure position in addition to the shear strength reduction technique. The idea of shear strength reduction was used by Matsui and San [29] to assess slope safety factors and the emergence of possible slip surface failure. Additionally, the shear strain in the failure process that extends from the slope’s base to its summit (i.e., total shear failure) was classified as slope failure.

Ugai and Leshchinsky [30] extended the two-dimensional problem to three dimensions. They used 20-node isoparametric solid elements and reduced integration with eight Gaussian integration points to conduct a three-dimensional vertical excavation slope simulation, taking into account the influence of pseudo-static seismic forces. The analysis results are compared with the results of the limit equilibrium method to verify their rationality. Griffiths and Lane [31] used the FEM to apply shear strength reduction to the analysis of slope stability, taking into account the effects of weak interlayers in the soil and groundwater. The occurrence of slope failure was determined using the non-convergence of the solution because slope failure is often accompanied by a rapid increase in displacement. The position and shape of the critical slip surface were represented by a deformed mesh diagram and node displacement vector diagram. In addition, the analysis results of the safety factor were compared with those of the limit equilibrium method. Based on the incremental method of elastic–plastic mechanics and the bilinear projection operator, Sun et al. [24] combined the strength reduction method with the φ-ν inequality and proposed a virtual element strength reduction method for slope stability analysis. The deformation of homogeneous and heterogeneous slopes with different strength reduction coefficients was solved, and the grid dependence of this method was discussed. Furthermore, numerical examples were used to verify the correctness and effectiveness of the proposed method.

One of the primary dynamic loads frequently observed on soil or rock slopes is seismic force [11]. Rock or soil slopes are subject to enormous dynamic stress, which has a significant impact on their stability. The pseudo-static analysis based on force equilibrium assumes that seismic force is a static inertial force operating laterally on the structure in structural stability assessment. Throughout the analysis, the seismic coefficient is progressively raised. The limit equilibrium approach, also known as the Mononobe–Okabe (M-O) method, was created in Japan in the 1920s by Okabe [32] and Mononobe and Matsuo [33]. It serves as the foundation for the most widely used cantilever and gravity structure analysis method. The M-O approach is also frequently employed for pseudo-static analysis. It calculates the dynamic soil pressure coefficient based on force equilibrium after transforming the seismic load into the inertial force of soil mass.

3. Overview of the Cases Studied

Three locations were selected for this investigation. Site 1 is found at the Freeway 1 Wudu exit (Figure 1), while Site 2 is located at Freeway 5 at a mileage of 7K + 082 (Figure 2), and Site 3 is located at Freeway 5 at a mileage of 7K + 410. The geological data and the overview of the slope protection project for each site are described below, and they act as a guide for the input values utilized in the subsequent numerical analysis.

The geological map of Site 1 is shown in Figure 3 [34]. The stratum on the slope belongs to the Nankang Formation, which is composed of thick layers of blue-gray, fine-grained gray sandstone and dark shale. Figure 4 displays Site 1’s drilling locations (BH6-1 to BH6-3) and core pictures [35]. Drilling-derived core pictures revealed that the rock disk in this region contains some joints, with laminae or thin shale layers serving as the primary weak surfaces.

Table 1. Uniaxial compression test results of Site 1 core.

Drilling Code	Depth (m)	Moisture Content (%)	Unit Weight (t/m3)	Compression Strength (MPa)	Strain (%)	Lithology
BH6-1	11.1	2.6	2.63	30.9	1.44	Sandstone
BH6-2	23.1	2.5	2.60	28.4	1.38	Sandstone
BH6-3	3.1	3.1	2.61	17.2	1.59	Sandstone

and

Table 2. Triaxial test results of Site 1 core.

Drilling Code	Depth (m)	Moisture Content (%)	Peak Strength		Residual Strength		Lithology
			cp (MPa)	φp (°)	cr (MPa)	φr (°)
BH6-1	3–4	3.3–5.5	0.70	29.6	0.34	28.8	Sandstone
BH6-2	12–13	2.5–6.5	9.09	53.8	1.26	44.2	Sandstone
BH6-3	4–5	5–9.3	2.50	44.6	1.21	37.5	Sandstone

display the drilled cores’ uniaxial compression and triaxial test results [35]. The slope is protected by four stages of RC curtain walls, free-styled grill beams, and grouted anchor bars, as shown in Figure 5 [35]. The grouted anchor bars are composed of three-meter-long #8 steel bars spaced three meters apart horizontally.

The geological map of Site 2 is shown in Figure 6 [34]. This stratum, which is a component of the Nankang Formation, is primarily composed of thick sandstone. The slope direction is northwest diagonal. Drilling data indicate that interbedded sandstone and shale are found beneath colluvial strata, which are located 4.9 m below the surface [34]. A total of 26 layers of 30 tons of prestressed ground anchors are used to protect the slope. The current ground anchors consist of a fourth-stage, 15-layer ground anchor; a free section of five meters; an anchor section of five meters; and a horizontal spacing of three meters, as shown in Figure 7 [36]. After that, eleven levels of ground anchors were placed, spaced three meters apart horizontally, with ten meters for the free section and five meters for the anchor section.

The geological map of Site 3 is shown in Figure 8 [33]. The formation here belongs to the Nankang Formation, which is composed of sandstone. The slope of Site 3 is approximately westward and is a diagonal slope. Drilling data show that the area from the surface to 4 m underground at Site 3 is colluvium, with siltstone underneath [33]. Ten layers of sixty tons of prestressed ground anchors are used to protect the slope. The current ground anchors are seven layers in length, with a free portion of fourteen meters, an anchor section of eleven meters, and a horizontal spacing of two and a half meters, as shown in Figure 9 [36]. Three layers of ground anchors were then placed, with an anchor section of 11 m, a free section measuring 14 m, and a horizontal spacing of 2.5 m.

4. Numerical Analysis Software and Methods Used

4.1. PLAXIS 2D Program

The PLAXIS 2D CE finite element analysis program, developed by Plaxis B.V., Delft, Netherlands, was used for numerical analysis and calculations in this study. The program was first developed in 1987 by Delft University of Technology in The Netherlands. The company was founded in 1993, and the first Windows-based version, PLAXIS 2D, was released in 1998. After years of research and promotion, a three-dimensional (3D) version was developed in 2001, and then PLAXIS 2D AE was developed in 2014 and updated to PLAXIS 2D CE in 2019.

4.2. Soil Material Combination Law

The Mohr–Coulomb model is a common basic rock mass material model for numerical analysis in geological engineering [37,38]. Its stress–strain relationship covers linear elasticity and complete plasticity. The linear elasticity aspect follows Hooke’s law, and the required stiffness parameters are Young’s modulus E and Poisson’s ratio ν. The plasticity element follows the Mohr–Coulomb failure criterion, of which the maximum principal stress (${\sigma }_1^{\prime }$) vs. the minimum principal stress (${\sigma }_3^{\prime }$) is expressed in Equation (1), and observes the principle of non-associated plasticity. The strength parameters needed are cohesion c, angle of internal friction $\phi$, and angle of dilatancy ψ. The parameters required for rock masses in the Mohr–Coulomb model are listed in

Table 3. Model boundaries for each analysis site.

Site Code	Point A	Point B	Point C	Point D	L (m)	H (m)
Site 1	(0, 0)	(0, 60)	(180, 107.2)	(180, 0)	180	107.2
Site 2	(0, 0)	(0, 65)	(170, 142.5)	(170, 0)	170	142.5
Site 3	(0, 0)	(0, 48)	(160, 101.2)	(160, 0)	160	101.2

.

$${\sigma }_1^{\prime }={\displaystyle \frac{2c\mathrm{c}\mathrm{o}\mathrm{s}\phi }{1-\mathrm{s}\mathrm{i}\mathrm{n}\phi }}+{\displaystyle \frac{1+\mathrm{s}\mathrm{i}\mathrm{n}\phi }{1-\mathrm{s}\mathrm{i}\mathrm{n}\phi }}{\sigma }_3^{\prime }$$

(1)

4.3. Safety Calculation

“Safety calculation type” is an option in PLAXIS for calculating global safety factors. This option can be selected as a separate “Calculation type” in the “General tabsheet” [39]. In the safety method, the shear strength parameters tanφ and c, as well as the tensile strength of the soil, are gradually reduced until structural failure. In principle, the phi/c reduction process has no effect on the dilatancy angle ψ [39]. However, the dilatancy angle can never be greater than the friction angle. When the friction angle has reduced so much that it equals the given dilatancy angle, any further reduction in the friction angle results in the same reduction in the dilatancy angle [39].

The total multiplier safety factor, ΣMsf, is used to define the soil strength parameter values at a given stage of the analysis [39]:

$$\mathsf{\Sigma }Msf={\displaystyle \frac{\mathrm{t}\mathrm{a}\mathrm{n}{\phi }_{input}}{\mathrm{t}\mathrm{a}\mathrm{n}{\phi }_{reduced}}}={\displaystyle \frac{c_{input}}{c_{reduced}}}$$

(2)

where the parameters with the subscript “input” refer to the attributes entered in the material set, and the parameters with the subscript “reduce” refer to the reduced values used in the analysis. ΣMsf is set to 1.0 at the beginning of the calculation to set all material strengths to their input values.

The “Load advancement number of steps procedure” was used to perform safety calculations in this study. The incremental multiplier Msf was used to specify the increment of intensity reduction in the first calculation step. This increment was preset to 0.1, which is generally considered a good starting value. The strength parameter is automatically and continuously reduced until all additional steps have been performed. The number of steps in this study was set to 100. It is important to always check whether the last step resulted in a fully developed failure mechanism. If this is the case, the slope safety factor (SF) is given by the following [39]:

$$\mathrm{S}\mathrm{F}={\displaystyle \frac{\mathrm{a}\mathrm{v}\mathrm{a}\mathrm{i}\mathrm{l}\mathrm{a}\mathrm{b}\mathrm{l}\mathrm{e} \mathrm{s}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h}}{\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{t}\mathrm{h} \mathrm{a}\mathrm{t} \mathrm{f}\mathrm{a}\mathrm{i}\mathrm{l}\mathrm{u}\mathrm{r}\mathrm{e}}}=\mathrm{v}\mathrm{a}\mathrm{l}\mathrm{u}\mathrm{e} \mathrm{o}\mathrm{f} \mathsf{\Sigma }Msf \mathrm{a}\mathrm{t} \mathrm{f}\mathrm{a}\mathrm{i}\mathrm{l}\mathrm{u}\mathrm{r}\mathrm{e}$$

(3)

4.4. The Process of Establishing Numerical Models

The process of establishing a slope model using PLAXIS 2D software is as follows.

4.4.1. Model Boundary Conditions

The model boundary is the distance from the reference point (0, 0) in the PLAXSI 2D CE software, as shown in Figure 10. The positive direction of the X-axis is to the right of the reference point, and the negative direction is to the left. The positive direction of the Y-axis is above the reference point, and the negative direction is below the reference point. The model boundaries for each analysis site, that is, the coordinates of points A, B, C, and D in Figure 10, are shown in Table 3.

4.4.2. Soil Parameters and Groundwater Levels of Soil Layers

Based on the drilling report and slope profile, the soil layer was established using the “create borehole” method, as shown in Figure 11. The soil parameters and groundwater level were set, as shown in Figure 12. Different soil layers were modeled using the fifteen node triangular elements.

4.4.3. Structural Settings and Material Parameters

The corresponding simulation elements were set according to the site structure. The slope protection structure included grouted anchor bars, lattice beams, RC curtain walls, stiffened retaining walls, and ground anchors. The grouted anchor bars, the stiffened grid in the stiffened retaining wall, and the anchor section of the ground anchor were simulated with embedded beam elements. The lattice beam, RC curtain wall, and RC panel of the stiffened retaining wall were simulated using plate elements. The road surface was simulated with the plate element, and the free segment of the ground anchor was simulated by using the node-to-node anchor elements.

4.4.4. Generation of Mesh

After the model was established and the structure was assigned material properties, the model was divided into finite elements to perform calculations. The accuracy of the model simulation is related to the mesh density. Increasing the density of the mesh increases the computation time of the model.

4.4.5. Hydraulic Conditions

In PLAXIS effective stress analysis, the total stress is composed of the effective stress of soil and the total stress of water. The total stress of water consists of static water pressure, seepage water pressure, and excess pore water pressure. Static water pressure and seepage water pressure are generated through the set water level difference. Excess pore water pressure is generated by structural loading under undrained conditions.

5. Numerical Analysis Results and Discussion

Two aspects were the focus of this study: the displacement of the slope at normal and high groundwater levels and the displacement of the slope model while it is subjected to seismic loading during pseudo-static analysis.

5.1. Normal Groundwater Level and High Groundwater Level

The “normal” groundwater level employed in the slope stability analysis was the average groundwater level as established by survey monitoring wells [34]. The “high” groundwater level was estimated to be two-thirds of the total level.

5.2. Seismic Loading in the Pseudo-Static Analysis

The seismic loading utilized in the pseudo-static analysis was taken from the Seismic Design Specifications for Freeway Bridges published by the Ministry of Transportation and Communications of the Republic of China in Taiwan [40]. The case was located on a type 1 rock bed in the Shiding District of New Taipei City; hence, the seismic coefficient used in the pseudo-static analysis was as follows:

$$k_h=0.5\times 0.4S_{DS}=0.5\times 0.4\times 0.7 \mathrm{g}=0.14 \mathrm{g}$$

(4)

$$k_v=0.5 \left(k_h\right)=0.07 \mathrm{g}$$

(5)

where $S_{DS}$ is the designed short-period horizontal seismic acceleration of the site, $k_h$ is horizontal seismic acceleration, and $k_v$ is vertical seismic acceleration.

5.3. PLAXIS Analysis

PLAXIS 2D was used to build slope simulation models, with the Mohr–Coulomb model used for soil layers. The soil parameters used in the model were the unsaturated unit weight ${\gamma }_{\mathrm{u}\mathrm{n}\mathrm{s}\mathrm{a}\mathrm{t}}$, saturated unit weight ${\gamma }_{\mathrm{s}\mathrm{a}\mathrm{t}}$, Young’s modulus E, Poisson’s ratio ν, cohesion c, internal friction angle φ, and dilatancy angle ψ. The model consisted of grouted anchor bars, free-styled grill beams, RC curtain walls, free and fixed lengths of the ground anchor, and a typical retaining wall. The ground anchor design is based on the freeway’s slope engineering design specifications [41]. The side friction resistance of the front and rear ends of grouted anchor bars is based on the calculation formula proposed by Das in 2007 [42]. The calculation formula is as follows:

$$\tau ={\displaystyle \frac{P}{A_s}}={\displaystyle \frac{P}{\pi \times D\times L}}$$

(6)

where τ is the side friction resistance (kN/m), A_s is the lateral area of ground anchor, P is the preforce (kN/m), D is the diameter of ground anchor (m), L is the length of ground anchor (m).

5.3.1. Case Simulation of Site 1

Table 4. Soil parameters of Site 1.

Soil Layer	Parameters
	${\mathit{\gamma }}_{\mathbf{u}\mathbf{n}\mathbf{s}\mathbf{a}\mathbf{t}}$ (kN/m3)	${\mathit{\gamma }}_{\mathbf{s}\mathbf{a}\mathbf{t}}$ (kN/m3)	E (kN/m2)	ν	c (kN/m2)	φ (°)	ψ (°)
Weathered rock layer	22.0	22.5	2.5 × 104	0.25	10	28	0
Sandstone layer	25.2	25.7	4 × 106	0.25	150	26	0

displays the soil properties for Site 1,

Table 5. Material parameters of grouted anchor bars of Site 1.

Simulation Elements	Young’s Modulus (kN/m2)	Unit Weight (kN/m3)	Diameter (cm)	Horizontal Spacing (m)	Front End Side Friction Resistance (kN/m)	Rear End Side Friction Resistance (kN/m)
Embedded beam row element	2.5 × 107	10	25	3	509	509

and

Table 6. Material parameters of free-form lattice beams and RC curtain walls of Site 1.

Simulation Elements	Material Type	Axial Stiffness (kN/m)	Flexural Stiffness (kNm2/m)	Weight (kN/m/m)	ν
Plate element	Elasticity	7.53 × 106	$\mathrm{56,475}$	7.06	0.17

display the material parameters of Site 1’s slope protection structures, and Figure 13 displays Site 1’s analysis model. The blue line represents simulated free-form lattice beams, the yellow line represents simulated RC curtain walls, the blue arrows represent the pavement load, and the pink parts represent simulated grouted anchor bars. From top to bottom, the soil layer was separated into two layers: sandstone and worn rock.

The inhomogeneity of the geotechnical materials at Site 1 results in an uneven displacement field in the slope.

Table 7. Displacement of Site 1 in various scenarios.

scenarios	|u| (mm)	${\mathit{u}}_{\mathit{x}}$ (mm)		${\mathit{u}}_{\mathit{y}}$ (mm)
		Max	Min	Max	Min
Normal groundwater level	3.86	1.62	−0.30	3.56	−0.50
High groundwater level	3.26	1.69	−0.47	2.82	−1.13
Normal groundwater level and pseudo-static analysis	24.10	0	−20.54	2.46	−13.53

displays the displacement of Site 1 in each scenario. Contour line diagrams of the slope displacement in various scenarios are displayed in Figure 14, Figure 15 and Figure 16. Areas based on the same displacement present a series of spatial surfaces, namely, displacement isosurfaces [43]. The density of the isosurface represents the change in displacement [23]. The main effect of water pressure on the slope sliding surface is to reduce the normal pressure or force at the soil particle-to-particle contact. As seen in Figure 14, the forces acting on Site 1’s slope are focused on the slope’s sliding surface at normal groundwater levels. In comparison, Figure 15 shows that the forces acting on the slope rise to the top of the slope at high groundwater levels. This is because high water tables increase the pore water pressure, resulting in lower shear resistance in potential failure planes and thus increasing the risk of slope failure [21]. One of the most frequent major dynamic pressures on slopes is earthquakes. The rock and soil on the slope are under a great deal of dynamic stress [44]. Slope stability is thus significantly impacted by earthquakes. The pseudo-static analysis of Site 1 indicated that the force acting on the slope extended from the top to the sliding surface when seismic loading was added, as seen in Figure 16. It can be seen from the figure that the isosurfaces were densely distributed near the sliding surface. Furthermore, the closer to the free surface of the slope, the greater the magnitudes of the isosurface. It is worth noting that dense areas always represent localized surface failures. Overall, the above results are consistent with the existing literature [12,21,23].

The potential sliding surface based on safety analysis is particularly useful for observing the localization of deformations within the soil when failure occurs [39]. Figure 17, Figure 18 and Figure 19 show the potential sliding surface based on safety analysis in different scenarios at Site 1. Potential failure surfaces in a slope model can be identified through the density of contour lines [45]. Furthermore, Figure 20 shows the shading distribution diagram of the potential sliding surface based on safety analysis in Scenario 3 at Site 1. As shown in these figures, the potential failure surface of the Site 1 slope is obviously located at the top of the slope where there is no slope protection facility. It should be noted that the ultimate displacement increments following repeated iterations of the strength reduction method are shown in Figure 17, Figure 18, Figure 19 and Figure 20. The real displacement on the site is different from this one. It simply indicates that a greater displacement than in other places will occur following strength reduction.

The safety factor (SF) of a slope is the comparison of resistance and driving force, which can reflect the condition of the slope [29]. In slope stability analysis research, SF plays an important role in evaluating slope stability. Bowles proposed

Table 8. Safety factor and meanings of slope condition.

Safety Factor	Slope Condition	Meaning
SF < 1.07	Frequent occurrence of slope collapse	Labile
1.07 < SF < 1.25	Slope collapse sometimes occurs	Critical
SF > 1.25	Slope collapse events are rare	Stable

based on several landslide events [46]. As can be seen from the table, SF < 1.07 indicates frequent slope instability, 1.07 < SF < 1.25 indicates slope instability that has occurred, and SF > 1.25 indicates rare collapse events. Figure 21 shows Site 1’s ΣMsf curve in different scenarios. The ordinate value after the curve in the figure reaches the stable stage is the SF. As can be seen from

Table 9. Safety factors for Site 1 in various scenarios.

Scenarios	Safety Factor
	Simulation Result	Design Specification
Scenario 1: normal groundwater level	1.627	≧1.5
Scenario 2: high groundwater level	1.626	≧1.2
Scenario 3: normal groundwater level and pseudo-static analysis	1.223	≧1.1

Note: Initial value of safety factor for Site 1 at normal groundwater level without slope protection facilities = 1.286.

, the SF of Scenario 1 was 1.627, the SF of Scenario 2 was 1.626, and the SF of Scenario 3 was 1.223. According to the analysis results, in three different scenarios, the SF value of the slope of Site 1 was greater than the design specification value [11]. This shows that the slope protection project at Site 1 is working effectively. Furthermore, according to Bowles’ safety factor standard [46], the slope of Site 1 is stable in three different scenarios. Nonetheless, the stability study shows that the slope’s incremental displacement is mostly localized near the slope’s summit. Localized harm could result from this, so possible weak spots need to be addressed.

5.3.2. Case Simulation of Site 2

Table 10. Soil parameters of Site 2.

Soil Layer	Parameters
	${\mathit{\gamma }}_{\mathbf{u}\mathbf{n}\mathbf{s}\mathbf{a}\mathbf{t}}$ (kN/m3)	${\mathit{\gamma }}_{\mathbf{s}\mathbf{a}\mathbf{t}}$ (kN/m3)	E (kN/m2)	ν	c (kN/m2)	φ (°)	ψ (°)
Colluvium	19.6	20.1	7 × 104	0.3	10	28	0
Weathered sandstone	23.5	24.0	3 × 105	0.3	50	30	0

displays the soil properties for Site 2,

Table 11. Material parameters of anchor section of ground anchor of Site 2.

Simulation Elements	Young’s Modulus (kN/m2)	Unit Weight (kN/m3)	Diameter (cm)	Horizontal Spacing (m)	Front End Side Friction Resistance (kN/m)	Rear End Side Friction Resistance (kN/m)
Embedded beam row element	2.5 × 107	10	20	3	95.5	95.5

and

Table 12. Material parameters of free section of ground anchor of Site 2.

Simulation Elements	Material Type	Axial Stiffness (kN/m)	Horizontal Spacing (m)	Pre-Force (kN/m)
Node-to-node anchor element	Elasticity	2 × 105	3	300

display the material parameters of Site 2’s slope protection structures, and Figure 22 displays Site 2’s analysis model. The road load is indicated by the blue arrow, the free segment of the ground anchor is indicated by the black line, and the anchor segment is indicated by the red line. From top to bottom, the soil layer is separated into worn sandstone and colluvium.

The displacement of Site 2 in each scenario is shown in

Table 13. Displacement of Site 2 in various scenarios.

scenario	|u| (mm)	${\mathit{u}}_{\mathit{x}}$ (mm)		${\mathit{u}}_{\mathit{y}}$ (mm)
		Max	Min	Max	Min
Normal groundwater level	75.12	17.51	−29.20	73.30	−30.05
High groundwater level	89.16	14.88	−47.95	56.64	−89.16
Normal groundwater level and pseudo-static analysis	80.32	0	−58.59	64.86	−47.62

. The contour line diagrams of the slope displacement in various scenarios are displayed in Figure 23, Figure 24 and Figure 25. Figure 23 shows that the forces operating on the slope were concentrated on the sliding surface and a portion of the slope top because Site 2 was at the normal groundwater level. With the high groundwater level, Figure 24 shows that the force acting on the slope moved up to the top of the slope. Figure 25 shows that the contour line diagrams of the slope displacement looked like those with a normal groundwater level when seismic conditions were added.

Figure 26, Figure 27 and Figure 28 show the potential sliding surface based on safety analysis in different scenarios at Site 2. In Scenario 1, the potential failure surface of slope was obviously located behind the slope protection facilities, as shown in Figure 26. In Scenario 2, the potential failure surface of slope was located on the lower slope of the road, as shown in Figure 27. In Scenario 3, the potential failure surfaces of the slope were located behind the slope protection facilities and on the lower slope of the road, as shown in Figure 28. Furthermore, Figure 29 shows the shading distribution diagram of the potential sliding surface based on safety analysis in Scenario 3 at Site 2.

Figure 30 shows Site 2’s ΣMsf curve in different scenarios. The SF was determined from the ordinate value after the curve in the figure reached the stable stage. As

Table 14. Safety factor for Site 2 in various scenarios.

Scenarios	Safety Factor
	Simulation Result	Design Specification
Scenario 1: Normal groundwater level	1.265	≧1.5
Scenario 2: High groundwater level	1.105	≧1.2
Scenario 3: Normal groundwater level and pseudo-static analysis	1.009	≧1.1

Note: Initial value of safety factor for Site 2 at normal groundwater level without slope protection facilities = 1.078.

shows, the SFs of Scenarios 1, 2, and 3 were 1.265, 1.105, and 1.009, respectively. According to the analysis results, in three different scenarios, the SF value of the slope of Site 2 was lower than the design specification value [11]. Moreover, according to Bowles’ safety factor standard [46], the slope of Site 2 in Scenarios 1, 2, and 3 was at a stable, critical, and labile level, respectively. This indicates that the effect of the slope protection project at Site 2 needs to be improved.

5.3.3. Case Simulation of Site 3

Table 15. Soil parameters of Site 3.

Soil Layer	Parameters
	${\mathit{\gamma }}_{\mathbf{u}\mathbf{n}\mathbf{s}\mathbf{a}\mathbf{t}}$ (kN/m3)	${\mathit{\gamma }}_{\mathbf{s}\mathbf{a}\mathbf{t}}$ (kN/m3)	E (kN/m2)	ν	c (kN/m2)	φ (°)	ψ (°)
Colluvial layer	18.6	19.1	7 × 104	0.3	10	28	0
Interbedded sandstone and shale	25.5	26.0	3 × 105	0.3	50	35	5

displays the soil properties for Site 3,

Table 16. Parameters of retaining wall materials of Site 3.

Simulation Elements	Material Type	Axial Stiffness (kN/m)	Flexural Stiffness (kNm2/m)	Weight (kN/m/m)	ν
Plate element	Elasticity	7.53 × 106	$\mathrm{56,475}$	7.2	0.17

,

Table 17. Material parameters of anchor section of ground anchor of Site 3.

Simulation Elements	Young’s Modulus (kN/m2)	Unit Weight (kN/m3)	Diameter (cm)	Horizontal Spacing (m)	Front End Side Friction Resistance (kN/m)	Rear End Side Friction Resistance (kN/m)
Embedded beam row element	2.5 × 107	10	20	2.5	86.8	86.8

and

Table 18. Material parameters of free section of ground anchor of Site 3.

Simulation Elements	Material Type	Axial Stiffness (kN/m)	Horizontal Spacing (m)	Pre-Force (kN/m)
Node-to-node anchor element	Elasticity	2 × 105	2.5	600

display the material specifications of the slope protection structures at Site 3, and Figure 31 displays Site 3’s analysis model. The retaining wall is shown by the blue line, the ground anchor’s free section is shown by the black line, and the anchor segment is shown by the red line. From top to bottom, the soil layer is separated into a colluvial layer, and sandstone and shale that are interbedded.

Table 19. Displacement of Site 3 in various scenarios.

scenario	|u| (mm)	${\mathit{u}}_{\mathit{x}}$ (mm)		${\mathit{u}}_{\mathit{y}}$
		Max	Min	Max	Min
Normal groundwater level	16.41	2.70	−6.22	6.31	−3.80
High groundwater level	18.74	1.99	−12.83	4.19	−18.74
Normal groundwater level and pseudo-static analysis	37.45	0	−36.83	9.15	−25.42

displays Site 3’s displacement in each scenario. The contour line diagrams of the slope displacement in various scenarios are displayed in Figure 32, Figure 33 and Figure 34. Figure 32 indicates that the force acting on the slope was focused below the ground anchor since Site 3 was at the normal groundwater level. However, as Figure 33 illustrates, the force acting on the slope was focused at the top in the high groundwater level scenario. The pseudo-static analysis result seen in Figure 34 revealed that the force acting on the slope extended from the slope’s top to its toe when seismic conditions were included. The force acting on the slope was found to be centered beneath the ground anchors.

Figure 35, Figure 36 and Figure 37 show the potential sliding surface based on safety analysis in different scenarios at Site 2. In Scenario 1, the potential failure surfaces of the slope were mostly located behind the ground anchors, but some penetrated through the lower ground anchors, as shown in Figure 35. In Scenario 2, the potential failure surfaces of the slope were raised upward and penetrated through more ground anchors, as shown in Figure 36. In Scenario 3, the potential failure surfaces of the slope were raised more significantly upward, and the ground anchors were penetrated more, as shown in Figure 37. Furthermore, Figure 38 shows the shading distribution diagram of the potential sliding surface based on the safety analysis in Scenario 3 at Site 3.

Figure 39 shows Site 3’s ΣMsf curve in different scenarios. The SF was determined from the ordinate value after the curve in the figure reached the stable stage. As

Table 20. Safety factor for Site 3 in various scenarios.

Scenarios	Safety Factor
	Simulation Result	Design Specification
Scenario 1: Normal groundwater level	1.405	≧1.5
Scenario 2: High groundwater level	1.306	≧1.2
Scenario 3: Normal groundwater level and pseudo-static analysis	1.021	≧1.1

Note: Initial value of safety factor for Site 3 at normal groundwater level without slope protection facilities = 1.032.

shows, the SFs of Scenarios 1, 2, and 3 were 1.405, 1.306, and 1.021, respectively. According to the analysis results, in Scenario 1 and Scenario 3, the SF value of the slope of Site 3 was lower than the design specification value [11]. Moreover, according to Bowles’ safety factor standard [46], the slope of Site 2 in Scenarios 1, 2, and 3 was at a stable, stable, and labile level, respectively. This indicates that the effect of the slope protection project at Site 3 needs to be improved.

6. Conclusions

Due to Taiwan’s high-risk slopes on National Freeways 1 and 5, the PLAXIS 2D CE program was utilized in this study to simulate the slope stability of many existing slope protection schemes. The following conclusions were drawn from the numerical simulation results:

• Because the gradients at Sites 1 and 2 were similar, the simulation results showed that, with a normal groundwater level, the force acting on the slope was mostly located at the position of the sliding surface. In the high groundwater level analysis, the force acting on the slope extended to the top.
• Due to the superior strength of the soil, Site 1 saw comparatively little displacement at both normal and high groundwater levels during the analysis.
• The displacement at the top of the slope increased as the groundwater level rose, regardless of whether the slope analysis was conducted with a normal or high groundwater level.
• According to the analysis results, in three different scenarios, the SF value of the slope of Site 1 was greater than the design specification value. This indicates that the slope protection project at Site 1 is working effectively.
• According to the analysis results, the effect of the slope protection project at Sites 2 and 3 needs to be improved.