On the seismic stability analysis of reinforced rock slope and optimization of prestressed cables

Abstract

The evaluation of the seismic stability of high rock slopes is of vital importance to ensure the safe operation of the hydropower stations. In this paper, an equivalent pseudo-static force analysis based on the finite element method is developed to evaluate the seismic stability of reinforced rock slopes where the prestressed cables are modeled by the bar elements applied with nodal forces and bounded only at the anchored parts. The method is applied to analyze a high rock slope in south-west China and the optimization of cables. The stabilization effects of prestressed cables on the seismic stability of the slope are studied, the simulations of the concrete heading are discussed and the potential failure modes of the shear concrete plug are compared. Based on this, the optimization of cables is studied including the anchor spacing and inclined angles.

Keywords

high rock slope / reinforced system / optimization / prestressed cable / seismicity

Introduction

The exploitation of hydroelectricity in China has been growing rapidly in the past few decades, especially with the construction of some large-scale hydropower stations in western part of China incluing the well known Three Gorges dam, Ertan hydropower station, Jinping hydropower station, and most recently Houziyan hydropower station in Yangtze River basin and etc. Site locations of hydropower stations are normally located in the areas of steep rock for the height difference and flow speed of river required to generate hydroelectricity. Thus the stability of those steep and high rock slopes has become an outstanding issue, particularly in south-west China where preexisting faults or joints are inclined along the slope. Many rock slopes in hydraulic engineering are located in a complex geological body with preexisting joints that are subjected to construction disturbances. Therefore how to analyze the slope stability for the safe operation of the power station using an effective and suitable reinforced system has become a topic of great research and engineering significances in hydropower development.

For the slope in hydroelectricity engineering, the commonly used reinforced structures include cables, anchor bolts, headings and shear concrete plugs. In slope engineering, cables are the most widely used type of reinforcement structures to reduce the disturbance from excavation to the slope stability. Cables require relatively short installation time and can ensure safety with a satisfied cost effective performance. They have been used in many hydropower stations in China such as Jingping and Lijiaxia hydropower stations. Concrete headings and shear concrete plugs are constructed to cut through the joint surfaces and work together with cables to improve the slope stability. For instance, Jingping hydropower station is mainly reinforced by the prestressed cable and the shear concrete plug, supplied by the anchor bolt and the concrete lattice column.

Over several decades, various methods have been proposed to analyzed the slope stability, including the limit equilibrium method (LEM), which has been greatly improved by Janbu [1], Sarma [2], Morgenstern and Price [3], Donald and Chen [4], the strength reduction method [5,6] and some methods combining genetic algorithms and probabilistic theories [7,8]. Much research has also been carried out on the reinforcement mechanisms in the slope stability from both by theoretical approach and by investigations and onsite tests. Michalowski [9] conducted a limit analysis approach to determine the amount of reinforcement necessary to prevent collapse of a uniformly reinforced slope. Based on the kinematic approach of the yield design theory using two translational failure mechanisms, a stability analysis of reinforced jointed rock slope by Siad [10] was performed to evaluate passive fully grouted rock bolts. Ding et al. [11] modeled reinforcement mechanisms of the prestressed cable by the numerical test. Li et al. [12] discussed the design principle and method of the prestressed cable for the support of rock slopes. Zhu et al. [13] studied the reinforcement technology of deep unloaded joints in the left bank of Jinping hydropower staltion and its numerical simulation. All the researches improve our understanding to the reinforcement in the slope stability.

Considering the vital influence of the earthquake on the slope stability, Chehade et al. [14] conducted a nonlinear global dynamic analysis, taking important parameters in seismic analysis into account, of reinforced slopes stability under seismic loading. Adapting the Pseudo-dynamic method, Eskandarinejad and Shafiee [15] analyzed the seismic stability of a reinforced slope considering non-associated flow rule and Bray and Travasarou [16] investigated the pseu-dostatic coefficient in the slope stability evaluation. Moreover, together with vital hydraulic engineering in China now, Liu et al. [17] analyzed stability of the Dagangshan dam abutment slope under seismicity and Luo et al. [18] conducted the quasi-static analysis of the seismic stability of reinforced rock slopes under surcharge and water pressure conditions. The reinforced slope stability under seismic has been greatly improved in theory and practice, however, due to the complexity of slope engineering, theoretical researches on reinforcement of rock slopes still need further development. Simultaneously, despite the great volume of research work, reports of optimization and analysis of reinforcements, including prestressed cables, headings and shear concrete plugs, appear to be rare.

This paper is based on a high rock slope of a hydropower station under construction in south-west China and the reinforced system design of the high rock slope is analyzed. Through the combined use of the LEM and the finite element method (FEM), the reinforced effects of shear concrete plug, heading and prestressed cable are studied. Topics like the 2D simulation method of the heading, the possible failure modes of the shear concrete plug, the modeling of the prestressed cable in rock slopes under seismicity and the optimization of cable arrangement are also discussed. The research results cannot only provide references for the design and construction of slope engineering, but also help to improve the understanding of the reinforced system.

Engineering background

Geomorgraphics and geostructures of the rock slope

The hydropower station is located in the area between south-east of Qinghai-Tibet Plateau and Szechwan Basin in China. The hydro power station junction comprises the concrete face rockfill dam (CFRD), the flood discharge tunnel, the hydroelectricity generation system, water release structure and diversion structure. The 223.5 m high water retaining structure is mainly the CFRD. The reservoir flood discharging structures are composed of the spillway tunnel, the deep-hole discharge tunnel, the special discharge tunnel and the unloading discharge tunnel. The spillway tunnel and unloading discharge tunnel are situated in the right bank, and the deep-hole discharge tunnel and special discharge tunnel in the left bank. As shown in Fig. 1, the deep-hole discharge tunnel lies in the outside of the mountain, and the special discharge tunnel in the inside. The natural inlet slope inclination of the deep-hold discharge tunnel and special discharge tunnel is between 35° and 40°. The excavation width and the maximum excavation height are 135 and 97 m respectively.

Engineering geological condition

The attitude of the discharge tunnel inlet slope is N20°E/NW. Most bedrock of the inlet slope is exposed, scattered by the diluvia layer with thickness varying between 1 to 5m. The bedrock is composed of the dolomitic limestone and metamorphic limestone of the 10th level in Early-Devonian (D11) with the laminiplantation of the gray heavy layer to huge heavy layer containing medium and thin layers. The attitude of the rock bed is N50°E/NW∠40-55°. The flood foundation lies in the downside of the slight weathering and weak unloading rock mass, mainly belonging to the III type rock mass. According to the geological investigation, the discharge tunnel inlet does not pass through the regional faults, of which the geological structures are secondary faults, compresso-crushed zones and joint fissures. The geological description of secondary faults and compresso-crushed zones affecting the slope safety are showed in Table 1. The major joint surfaces in the slope are J1 located at N50°E/NW∠45°, J2 at N40-70°E/SE∠55-85°, J3 at SN/W∠35° and J4 at N30-a60°W/NE∠60°-75°. The natural slope is stable. Under disturbances of construction, however, the combination of these structural surfaces can lead to sphenoid slips and failure of the inlet slope.

Reinforced system design of flood tunnel inlet slope

Preliminary analysis of slope safety

In the preliminary analysis of the slope safety, a characteristic cross section of slope is selected for 2D analysis as shown in Fig. 2. In this part, LEM and FEM are combined together to assess the slope stability in nature and under excavation without support. The potential failure modes and their safety factors are showed in Table 2.

As shown in Table 2, the VI sliding mode is the controlling failure mode comprising joint surfaces F12-1 and g10-1. The safety factor of mode VI is 1.079 by Morgenstern-Price method, which is larger than the design safety factor (1.05) of the natural slope. For considering the stress strain relation of rocks, the safety factor, slightly higher than the safety factor by Morgenstern-Price method, is 1.257 by FEM. The analysis result shows that the natural slope is safe.

The slope excavation procedure is simulated by the FEM divided into 5 steps, as showed in Fig. 3. Step 0 shows the stability of the natural slope. The slope is then cut to the height of 1847.5 m in step 1. Then in step 2, 3, 4, the slope is gradually excavated to the heights of 1826.28 m, 1806.28 and 1776 m respectively. The result is shown in Table 3. In the first two excavation steps, with a reduction in sliding forces, the safety factor of the Mode VI increases from 1.257 to 1.268. In step 3, the slope safety factor is reduced to 1.237 and satisfies the safety requirement. In step 4, due to the excavation of the slope toe and the exposure of joint surface as a hanging structure, the safety factor reduces to 0.666. The slope excavation at step 4 without supports is also analyzed by the LEM and the safety factor obtained for this case are 0.658 and 0.659 respectively by Morgenstern-Price method and Spencer method. The analysis results show that the safety factors of the slope during excavation without reinforcements are below the minimal requirement of 1.05, thus it needs to be supported.

Reinforced system scheme

According to the analysis in section 3.1, additional reinforcements are needed to ensure the slope excavation safety. As shown in Figs. 4-6, the reinforced system of the slope is composed of prestressed cables, heading and shear concrete plug. There are two shear concrete plugs with the size of 9 m (width) × 7 m (height) cutting through the slip surfaces F12-1 and g10-1 at the height of 1847.50 m. The heading is respectively installed on the longitudinal axis of the special deep flood tunnel, the special discharge tunnel and the middle of these two flood tunnels. The arrangement of the prestressed cables is shown in Fig. 7. Above the excavation line of the slope, 3 rows of prestressed cables with anchorage force of 2500 kN are installed. On the slanted part of the slope, 16 rows of prestressed cables with anchorage force of 3000 kN. In the third level, 2 rows of prestressed cables with anchorage force of 2000 kN are installed on the vertical face of the slope. For all of the cables, the line spacing is 4 m with the anchor angle of 5°.

Evaluation of stabilization effects of the reinforced system

In the assessment of the reinforced effect, the prestressed cables and the shear concrete plugs are mainly considered by 2D FEM. The analysis on the stabilizing effects of the headings will be introduced later. The FEM simulation of the excavations and reinforcements can be divided into 5 construction steps. Step 0 is the natural slope stability analysis. Step 1 is to install 3 rows prestressed cables each row (anchorage force= 2500 kN) upon the excavation line of the slope. step 2 is to excavate the slope to the height of 1847.5 m and 6 rows prestressed cables (anchorage force= 3000kN) are installed. Step 3 is to excavate the slope to the height of 1826.28 m and 5 rows prestressed cables (anchorage force= 3000 kN) are installed. Step 4 is to excavate the slope to the height of 1806.28 m and 5 rows prestressed cables (anchorage force= 3000 kN) are installed. Step 5 is to excavate the slope to the water inlet soleplate with the height of 1776 m and 2 rows prestressed cables (anchorage force= 2000 kN) are installed.

Besides, the operation condition and the sudden water drop condition are also simulated. For the operational condition, the water level increases to 1842 m and the water pressure on the control joints is modeled as shown in Fig. 8. For the case of sudden drop of water table, the water level decreases by 10 and 20m in the operation condition, while the water pressure in the control joints stays unchanged.

As shown in Table 4, the prestressed cable can obviously improve the slope stability and the safety factor after excavation construction is 0.970, which is less than the required safety factor 1.05 and it needs to install the shear concrete plug. With the combined support of the cable and shear concrete plug, the safety factor after excavation construction improves to 1.139 and safety factor under the operation condition increases to 1.272, satisfying the safety requirement. The installation of shear concrete plug can improve the safety factor of sliding modes by about 15-20%. In the normal operation condition, the water impoundment can produce pressures on the slope surface and help to improve the stability of the slope, which increases the slope safety factor by 25-30% compared to the construction condition. For the sudden drawdown condition, the slope safety factor will reduce with the drawdown of the water level. When the water level decreases by 20m, the slope safety factor reduces by 10%.

Reinforced system optimization

Discussion on 2D simulation method of heading

The 2D simulation method and the reinforced effect of heading is an unsolved issue. In the slope reinforced system scheme, the concrete heading is respectively installed on the longitudinal axis of special deep flood tunnel, the special discharge tunnel and the middle of these two flood tunnels, with the cross section size of 5 m × 5.6 m. In the paper, the height of the shear concrete plug in the 2D simulation is calculated through the area equivalent principle: the total area of the three headings’ cross section divided by the distance of the two flood plugs is the equivalent height of heading, as shown in Fig. 9. Through this method, the equivalent height of heading is 1.8 m and the 2D FEM simulation schematic diagram is as shown in Fig. 10.

The results (Table 5) show that the headings can hardly produce the reinforced effect. After constructing the heading, the safety factor of the construction condition improves by 0.003. The safety factor variance of the operation condition and the sudden dropdown condition also appears to be the same.

To verify the 2D simulation method of the heading, a 3D FEM analysis on reinforced effect of the heading is also conducted. The 3D excavation model is simplified and excavation width is 160 m, as shown in Figs. 4-6 and 11. The distribution of the heading stress is as shown in Fig. 12 and the maximum principal stress varies between -1.14 and 4.7 MPa, which shows that the partial heading is in tension status and partially in press status. The stress in different positions of the heading conspicuously varies. At the conjunction of headings and shear concrete plugs, the stress concentration and high stress gradient appears. The upper part of heading is in press status, and the lower part is in tension status, whose tension stress has exceeded the concrete tension strength value. For the safety factor in 3D simulation (Table 6), the 3D constraint effect will greatly affect the slope safety and it also shows the 2D simulation tends to be conservative. Simultaneously, the result shows that the reinforced effect of the heading is not obvious, with the safety factor improving by 0.01. The 3D analysis result on heading accords with 2D result and testifies the reasonability of the 2D simulation method of heading.

Failure modes of shear concrete plug

When accessing the reinforced effect of the shear concrete plug, two potential failure modes should be considered. The first failure mode is shear fracture of shear concrete plug, as shown Fig. 13 (a). The second failure mode is the failure along the interface of the heading and rock mass, as shown Fig. 13 (b).

The safety factors of two potential failure modes in 2D FEM simulation are listed in Table 7. The safety factor of the second failure mode is smaller than the first failure mode, and the potential slope slip tends to be the second failure mode. In the 3D FEM simulation (Table 8), the safety factor of the second failure mode is 12.5% smaller than the first failure mode. In the design of shear concrete plug, these two failure modes should be checked to ensure slope safety.

The stabilization effects of prestressed cables in rock slope under seismicity and the optimization of cable arrangement

The arrangement and positioning of prestressed cables for seismicity slope stability is an important issue. In the following, we propose the use of truss element applied with nodal forces to analyze the reinforced slope stability under seismicity. Through the combination of FEM simulation and pseudo-static method, the prestressed cable internal force distribution and slope displacement is calculated. The safety factor of the slope control structural plane is also obtained by the self-produced procedure. Moreover, the disciplines how the factors containing the prestressed cable angle, value, length, spacing and the relationship with the slope structural planes influence the slope displacement and stability under the seismicity are also researched. Based on the above research, the optimized reinforced scheme is proposed and verified.

A model of prestressed cables in rock slope

1) Finite element model for the prestressed cables

The prestressed cable can be divided into two types, namely whole length bonding prestressed cable and free prestressed cable. For the free prestressed cable, the steel tendon is protected by both grease and plastic, and the stress at the free end can be adjusted to well accommodate the external impact and vibrations. With the wild use of non-boned cables, the free prestressed cable is frequently applied to slope engineering. In the following, we will focus on this type of cable. The free prestressed cable model can be equivalent to the truss element with applying forces on its nodes, as shown in Fig. 14 [8].

2) Seismicity effect

(1)F_i=a_h\xi G_{Ei}{a}_i/g,

here the seismic effects on the rock slope stability are modeled using the pseudo-static forces method. For rock slope analyzed here, the peak ground acceleration with a 10% probability of exceedance in 50 years is 0.182 g. The seismic attenuation coefficient is chosen as 0.25 and the elevation amplification factor along the height of the slope is 2.0. The equivalent seismic body forces are calculated according to Eq. (1) and are replaced by the equivalent uniformly distributed surface load.

3) Assessment method of slope stability

To assess the reinforced effect of the prestressed cables, the safety factor of the control slip surface, displacements of slope characteristic point and cable internal force are taken as indices to evaluate the slope stability after reinforcement. The safety factor is defined as the ratio of the sliding force and resistant force along the critical slip surface VI. The slope characteristic point is the peak of the lowest vertical step of the slope.

Principles of prestressed cable to slope stability under seismicity

1) Stabilization effects of prestressed cable to slope stability

Through the FEM simulation, the slope safety factor without the prestressed cable is 0.8988. The horizontal and vertical displacement of the slope characteristic point is -97.8 and -36.8 cm respectively. The slope displacement is too large and the safety factor cannot satisfy the requirement of the slope stability.

After the reinforcement of the prestressed cable, the safety factor increases to 1.1603, and the horizontal and vertical displacement at the characteristic point of the slope is -16.2 and -10.5 cm respectively. The maximum cable internal force of the first level cables is 458.3 and 458.3 kN for the second level, both are within the pullout limit of the cables. The maximum cable internal force of the third level is 548.6 kN and exceeds the cable pullout limit, meaning the some cables fail in level 3.

From the above analysis, it can be seen that the prestressed cables can greatly improve the slope stability and besides control the deformation of the slope. However, the cable internal force at the third level exceeds the cable pullout limit and should be redesigned.

2) Positioning angle of prestressed cable to slope stability

The anchor angel of the cable in original reinforced scheme is 5°. The slope stability is also studied when the anchor angel of cable is -5°, 0°, 10° and 15° respectively. The analysis result is listed in Table 9 and shown in Figs. 15-16.

With the increase of the anchor angle, the slope safety factor reduces, while the displacement of the slope characteristic point and the total cable internal force increase. The maximum cable internal force of the third level with different angles excesses the cable pullout limit. When the cable is 0°, the reinforced effect of the cable is best. The slope safety factor is 1.1913 and the cable internal force of different levels is smallest compared to other anchor angles. The horizontal and vertical displacement of the slope characteristic point is -16.1 and -10.5 cm respectively. Compared to the condition with the anchor angle 0°, the slope safety factor decreases by 10%, if the anchor angel is 15°, and the deformation of the slope characteristic point and the total cable internal force increases by 20%. When the anchor angel is -5°, the reinforced effect is slightly weaker than the condition when the anchor angle is 0°.

From the above analysis, the reinforced effect is the best when the anchor angle is 0°. Any deviation of the anchor angle from 0° will result in a decrease in the slope safety factor and an increase in the displacement at the characteristic point as well as the cable internal forces.

3) Influences of anchor length of prestressed cable on the slope stability

In the following, the influence of the cable length at different height levels with respect to the position of joint surfaces on the slope stability will be studied. Based on the original scheme, the slope stability is also studied by increasing or reducing the cable lengths at internals of 10 m. The structural surfaces that the cables pass through with variety of the anchor length are as shown in Table 10. The analysis result in different conditions is as shown Figs. 17-21.

For the free prestressed cable, the increase of anchor length, while the anchorage force stays unchanged, would weaken the constraint ability of the slope deformation. And the slope safety factor and the total cable internal force decrease and the displacement of the slope characteristic point increases.

For the first level cables, as the length of the cable increases, the capability of the cables to control the slope deformation reduces and the cable internal force reduces. The forces required to restrict the slope deformation are transmitted to the second and the third level cables. As introduced early, the critical slip surface goes through F12 and g10. The first level cable would not pass through the F12 and g10 structural surface respectively, when the cable length of the first level decreases by 10 and 20 m. In these conditions, the slope stability is weakened and the slope deformation increases, while the slope safety factor decreases. Especially for the condition that the length of the first level cable decreases by 20 m and doesn’t pass through the g10 structural surface, though the cable internal force of the first level decrease 260 kN, while the second level cable internal force and the total level cable internal force increase by about 700 and 500 kN respectively, the safety factor decreases and the slope deformation increases. It proves that the g10 structural surface has an important influence on the slope safety and the cable passing through this kind of structural surfaces would improve the slope stability, suffering a lager internal force simultaneously.

With the increase of the second level cable, its cable internal force decreases. The force to restrict the slope deformation is delivered to the first level and the third level cable. The slope deformation increases and the slope safety factor decreases. For the condition that the length of the first level cable decreases by 20 m, the maximum internal force of the third level cable is 475.257 kN and can satisfy the requirement of the cable pullout limit. The cable would pass through the f12 structural surface and is advantageous to the slope stability, when the cable length increase is from -10 to 0 m.

As the third level cable is cut by 10 m, the maximum cable internal forces of each level satisfy the requirement of pullout limit. In this condition, the third level cables will not pass through the weakly weathered structural surface and the cable internal forces decrease. However there is an increase of the cable internal forces observed at the first and second levels and reduction in the safety factors of the slope. With the third level cable being shortened by 20 m, the J3 structural surface are not cut through by the cables and the cable internal forces decrease dramatically. Consequently, the total cable internal force of the first and second levels increases by 400 kN and the cables fail. Therefore it is important to have J3 cut through by the cables.

From the above analysis, it can be seen that with the increase of cable length, the force to restrict the slope deformation is delivered to the neighboring levels of cables. The cable passing through the important structural surface would improve the slope safety and ameliorate the systemic cable stress. Considering the third level cable of the original scheme cannot satisfy the cable pullout limit, the length of the third level cable can be accreted or the length of second level cable can be reduced to ensure the slope safety.

4) Anchorage force of the prestressed cable to slope stability

Based on the original reinforced scheme, the slope stability is also studied when the anchorage forces of the three level cables are 2000, 2500 and 3000 kN (in 2D plane strain state are 500, 625 and 750 kN considering the spacing is 4 m) respectively. The analysis results under different conditions are listed in Tables 11-13.

A decrease in the anchorage force, with the cable length unchanged, will reduce the ability of the cables in constraining the slope deformation. The cable would sustain a lager internal force finally. When the three level cable anchorage forces are 500, 625 and 750 kN respectively, the slope safety factor is smaller than the original reinforced scheme. When the anchorage force of the three level cables is 500 kN, the slope safety decreases by 10% and the displacement of the slope characteristic point and the total cable internal force increases by 50% and 25% respectively, compared to the original reinforced scheme. The maximum cable internal force of the third level is 976. 824 kN and is almost twice the original reinforced scheme, which exceeds the cable pullout limit. When the anchorage force of the three level cables is 625 kN, the slope safety decreases by 5% and the displacement of the slope characteristic point and the total cable internal force increases by 30% and 12% respectively, compared to the original reinforced scheme. The maximum cable internal force of the third level is 914.149 kN and exceeds the cable pullout limit. When the anchorage force of the three level cables is 750 kN, the slope safety decreases fractionally and the horizontal displacement of the slope characteristic point increases by 1 cm and the vertical displacement remains the same, compared to the original reinforced scheme. The maximum cable internal force of the third level is 824.108 kN and exceeds the cable pullout limit. The total cable internal force is slightly less than the original reinforced scheme.

Since the above analysis results show the third level cables undertake a large portion of cable internal forces, in the following we will study the stabilization effects of the third level cables by varying the anchorage forces at 625, 750 and 1000 kN while the anchorage forces of the other two levels remain unchanged. The analysis results are shown in Tables 11-13. For anchorage force of 625 kN, the maximum cable force of third level is 914.149 kN which exceeds the cable pullout limit. When the third level cable anchorage force is 750 kN, the maximum cable internal force of the third level is 824.108 kN and also exceeds the cable pullout limit. When the third level cable anchorage force is 1000 kN, the maximum cable internal force of the third level is 818.748 kN and is within the cable pullout limit. The horizontal displacement of the slope characteristic point is less than the original reinforced scheme. From the above analysis, the increase of the cable anchorage force would well control the deformation of slope and ameliorate the cable internal force. To satisfy the cable pullout limit, the third level cable anchorage force should be increased.

5) The effects of anchor spacing on the slope stability

Based on the original reinforced scheme, the slope stability is also studied when the anchor spacing of the cable is 2 m × 2 m and 6 m × 6 m respectively. The analysis results in different conditions are listed in Tables 14-16.

When the anchor spacing of the cable is 2 m × 2 m, the slope safety increases by about 100% and the horizontal and vertical displacements of the slope characteristic point decreases by about 20% and 30% respectively, compared to the original reinforced scheme. The cable internal force of three levels conspicuously decreases and satisfies the cable pullout limit. When the anchor spacing of the cable is 6 m × 6 m, the slope safety decreases by about 10% and the horizontal and vertical displacements of the slope characteristic point increases by about 100% and 50% respectively, compared to the original reinforced scheme. The total cable internal force increases by 50% and the maximum cable internal force of three levels conspicuous increases, which exceed their cable pullout limit.

From the above analysis, the increase of the anchor spacing will help to control the deformation of slope and reduce the cable internal force, and the slope stability will be improved. To ensure the slope stability, anchor spacing can be increased properly.

Cable design optimization and 3D simulation verification

By applying the above findings, the original reinforced scheme is optimized as follows. First, the angle of the cables have been changed to level to the ground, i.e. 0°. Secondly, the spacing of the third level of cables is decreased to 2 m × 2 m, and the anchorage force is increased to 3000 kN, with its anchor length remains unchanged to satisfy the pullout strength.

The 2D plane strain state analysis shows the slope safety factor has been improved to 1.1755 after optimization and the horizontal and vertical displacements of the slope at the characteristic point decreases by 40% and 15% respectively. The maximum internal forces at different levels of cables are listed in Table 17. It can be seen that the maximum cable internal forces of the second and the third level are less than the original scheme. The maximum cable internal force of the third level is 746.774 kN and satisfies the requirement of the cable pullout limit.

To further validate the optimized reinforced scheme, the 3D slope stability simulation under seismicity is conducted, as shown in Figs. 4-6 and Fig. 22. The safety factor of the original and optimized scheme is 1.429 and 1.448 respectively, which is about 20% higher than the 2D slope analysis result and corresponds to practical situation. The cable internal forces of the two schemes are shown in Fig. 23. In the original reinforced scheme, the minimum cable internal force of the third level is 2020 kN and exceeds the cable pullout limit. While after optimization, only a small part of the cables exceeds the cable pullout limit and thus satisfy the slope safety requirement. The horizontal displacement of the slope characteristic point decreases by 32%. Therefore the optimized cable arrangement shows a better reinforced effect than the original scheme.

Discussion

In this paper, a simple and effective method using FE model and pseudo-static force method is proposed for analyzing seismic stability of rock slope and the reinforced effects of the cables. The initial analysis results show that the natural slope is safe and the slope under excavation without support is unsafe, which requires supporting structures. The slope reinforced by prestressed cables, shear concrete plug and concrete heading satisfies the stability requirement under both construction and operational conditions. First, the stabilization effects of the concrete heading are studied. Both a simplified 2D model and a full 3D model of the slope has been tested and compared. It is shown that the height of the concrete heading in 2D by the equivalent area of cross section criterion is feasible. The reinforced effect of the heading is almost negligible. Secondly, two typical failure modes of shear concrete plug, namely the shear failure passing through the concrete heading and the interface failure along the bonding between the heading and rock mass have been studied. It is found that potential failure mode of the inlet slope tends to bend round the shear plug. Finally, the effects of the positioning and arrangement of cables to the slope stability have been studied. The increase of cable length in one row of cables will result in the increase of internal forces in neighboring cables. A reduction in the anchor spacing helps to control the deformation of slope, reduces the internal cable force, and improve the overall slope stability substantially. For the slope stability, the reinforced effect is the best when the anchor angle is 0°. The deviation of the anchor angle from 0° results in a decrease in the safety level, an increase in the slope deformation as well as the cable internal forces. Based on these findings, an optimization scheme of cables has been proposed. Both 2D and 3D FE analysis results show that the optimized scheme can improve the slope safety factor, control the slope deformation and satisfy the cable pullout limit. It should be noted that in all test examples presented, the correctness of the analysis results is determined by the completeness and resolution of the geological information and the careful choice on the characteristic combination of joints as the potential slip surfaces.

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