Investigation on the Stability of Fissured Slopes Reinforced with Anchor Cables under Seismic Action

Based on the upper bound theorem of limit analysis (UBLA) combined with the pseudostatic methods, this paper elaborates on a calculated procedure for evaluating ﬁssured slope stability under seismic conditions reinforced with prestressed anchor cables. An existing simple slope case is presented as a case study in this work. The comparison is given to verify that the solution derived from this study is correct and feasible. By means of a numerical optimization procedure, the critical location of the crack is determined from the best upper bound solutions. The results demonstrate a signiﬁcant inﬂuence of the depth of crack and seismic acceleration coeﬃcient on the critical location distribution of the cracks. Meanwhile, the axial force of anchor cables is investigated via parametric studies. It is shown that the variation of the crack depth has little eﬀect on the axial force of anchor cables. Moreover, this paper also illustrates the variation in the axial force of anchor cables under the impact of ﬁve marked factors (crack depth, anchor arrangement, anchor inclination angle, slope angle, and seismic conditions). Finally, the required critical length of the free section of anchor cables is determined to ensure the stability of ﬁssured slopes subjected to seismic action.


Introduction
Slope instability is a significant problem in engineering practice and will cause heavy casualties and huge economic losses worldwide [1][2][3]. In the past decades, stability analysis of slopes is one of the most fundamental and written-about topics in geotechnical research, which is classic but continues to be active in the literature [4][5][6][7]. Due to the development of soil tension, cracks are often found at the crest of cohesive soil slopes; Taylor [8] said that ". . .. . .the action within the tension zone is a subject that is worthy of much study," whereas the analyses of slopes with crack are relatively scarce. Baker [9] indicated that the tension cracks at the posterior position of the slide surface are induced by the self-weight of soil and other external loads. Moreover, the literature on seismic-induced slope instability has shown that fissured slopes experience seismic activity [10][11][12]. us, determining the stability of the fissured slopes under seismic action is a particularly important engineering issue in the seismic zone. To stabilize dangerous slopes, anchor cables can be considered as an efficient reinforcement measure in slope engineering [13,14]. Compared with traditional retaining structures, reinforcement systems involving permanent grouted anchors may provide better technical and economic advantages [15]. Furthermore, under prestressed anchor cables, the strength and bearing capacity of the slope would be fully utilized to effectively improve slope stability, which have been widely used to reinforce high and steep slopes [16]. e existing methods for stability analysis of fissured slopes can be categorized as follows: (i) Limit equilibrium (LE) methods. Terzaghi and Peck [17] derived the estimated formula of tension cracks at the crest of slopes when the forces between soil mass slices applied. Spencer [18], Law and Lumb [19], and Zhu et al. [20] also used the framework of slices method to investigate the slopes tension stability problems. (ii) Variational LE method. Baker [9] employed variational extremization for the assessment of the impact of the tensile crack of the cohesive soil slope. Since then, Leshchinsky [21] and Li et al. [22] also extended this variational LE method to obtain the factor of safety for fissured slopes. Zhou et al. [23] presented an extension of the variational analysis of slope stability with a tension crack considering seismicity. (iii) Numerical methods. Antão et al. [24] attempted the finite-element upper bound limit analysis, where the presence of a crack of specified location and depth was included in the stability analysis of a slope reinforced with sheet pile. Liu and Zhao [25] conducted a numerical limit analysis on a wet slope with a tension crack in the upper slope surface using the rigid finite-element method (RFEM). Another powerful approach for stability analysis of fissured slopes is the upper bound theorem of limit analysis (UBLA). In many situations, upper bound theorems have proven to be more robust and efficient than the conventional LE method [25]. Based on the UBLA method, Utili [26] investigated the effect of location and depth of vertical cracks on slope stability. Michalowski [5] considered the effect of the crack-opening process on the stability of homogeneous slopes. Recently, the UBLA and the pseudostatic method were used to examine the stability of fissured slopes subjected to seismic loading [11,12,27]. As to slope reinforced with anchor cables, Li et al. [15] and Zhao et al. [28] proposed the UBLA method to consider the reinforcing of one row of anchors on slope stabilization under conditions of a rotational failure mechanism. In addition, Yan et al. [29] and Su et al. [30] also applied the UBLA method for evaluating slope stability considering the effect of anchor cables and seismic loading.
Based on the above analysis, the purpose of this paper is to extend the UBLA method of slope stability with tension crack by Michalowski [5] and Utili and Abd [11] into slopes reinforced with prestressed anchor cables and seismic condition. A rotational failure mechanism (log-spiral surface) was considered to find the critical results [4,[31][32][33][34]. By comparison and analysis of existing slope examples, the correctness and feasibility of the proposed UBLA method have been verified. e calculation can be used to work out the critical location of the crack. A typical case is presented to determine the effect of crack depth, anchor arrangement, anchor inclination angle, slope angle, and seismic conditions on the axial force of anchor cables. Furthermore, analysis and discussion are conducted regarding the required critical length of the free section of anchor cables.

Basic Assumptions.
To formulate the problem, the analyses presented in this work were developed under the following basic assumptions: (1) e fissured slope can be considered as isotropic, homogeneous, and dry soil.
(2) e soil-anchor-reinforced system can be studied in plane-strain conditions. (3) e tensile crack always appears at the top of the slope and the crack depth is distributed vertically along the height of the slope. (4) e failure surface passes through the toe of the fissured slope and the sliding soil mass moves as a rigid body. (5) e seismic action of the fissured slope sliding soil mass is simplified to pseudostatic load.

Failure Mechanism of Fissured Slopes.
As previously assumed, an isotropic and homogeneous fissured slope reinforced with a row of anchors is considered in the present study. e failure mechanism with the log-spiral line beneath the toe of the fissured slope is shown in Figure 1, where log-spiral line AD is the failure surface of the fissured slope, and H and H 1 are fissured slope height and crack depth, respectively. L 1 and L 2 are a horizontal distance from point B to point C′ and a horizontal distance from the crack to point C′, respectively. e parameters X E and α are the horizontal distance from the anchor head to the toe of the slope and the inclination angle of anchors, respectively. e parameters β and θ d are the slope angle and the angle related to the cracks, respectively. e parameters θ 0 and θ h are angles of the failure mechanism. e parameters r 0 and r h are polar diameters of the log-spiral line. e fissured slope is represented as a rigid body that slides along the log-spiral surface with center O and angular velocity ω. From Figure 1, the geometric relationship of the failure mode can be expressed as where ψ � tanφ m /F S . e strength reduction technique is widely used in the stability analysis of safety factor F S , based on the linear Mohr-Coulomb (M-C) failure criterion [35,36]. Similarly, c m � c/F S . e parameters c m and φ m are the mobilized cohesion and internal friction angle for the fissured slope to attain the limit state of stability, respectively.
Due to the assumption of the failure mechanisms through the bottom of the crack, knowledge on the known depth of the existing crack, H 1 , brings in an additional equation about θ d , which can be determined by the unknown θ 0 and θ h and the geometric relationship in Figure 1.

Upper Bound eoretical Framework of Fissured Slopes
Reinforced with Anchors. e upper bound approach of limit analysis is used herein to analyze the stability of fissured slopes reinforced with one row of anchors, which provides a powerful mechanical tool for the stability analysis under the limit state. To solve slope stability problems, the use of limit analysis has almost exclusively concentrated on the kinematic theorem [4]; this is generally simpering to use than the static approach under certain assumptions [15]. Referring to Figure 1, the region ABCDA is represented as a rigid body rotating around the undefined center of rotation O with the soils below the log-spiral failure surface AD remaining at rest. Based on the UBLA, a slope will collapse if the rate of work done by external loads and body force exceeds the internal energy dissipation rate by soil cohesion. In this study, the work rate of the external force includes the work rate of gravity W r , the work rate of the horizontal seismic force W Eh , the vertical seismic force W Ev , and the work rate of the axial force provided by anchor W anchor . Note here that the work rate by the anchor is negative. e internal energy dissipation rate by soil cohesion D only takes place along the sliding surface in the present consideration.
According to the work of Chen [4] and Figure 1, the rate of external work for sliding region ABCDA, W r , can be calculated as the work of region ABC′ minus the work of region CDC′. Hence, the external work rate due to soil weight takes the form where c is the unit weight of soil mass and f 1 -f 6 are dimensionless expressions depending on the parameters θ 0 , θ h , θ d , and β. Figure 1: Basic definitions for a fissured slope reinforced with anchor cables subjected to seismic action. e horizontal and vertical pseudostatic forces are under consideration when calculating the rate of the seismic action, which is similar to that for determining the work rate of the external force due to the soil weight. A horizontal seismic acceleration coefficient, K h , and a vertical one, K v (K v � λK h ), are presented here. e directions of earthquake accelerations are not constant in most real situations [12]. e value of λ positive indicates vertical downward acceleration, whereas a negative value indicates vertical upward acceleration. e rate of external work attributing to vertical and horizontal pseudostatic forces can be expressed as To account for the reinforced effect of a row of anchors on the fissured slope, an axial force is considered to be applied to the soil sliding mass. According to the existing research by Li et al. [15], the axial force developing in anchor is due to the Earth pressure exerted against the structural facing of the bearing plates or the concrete pad to which the anchors are connected and prestressed, and the reinforcement mechanism consists of transferring the resisting tensile force of anchor to the slope structural facing. us, the work rate of the axial force provided by the anchor can be written as where T is the axial force exerted on the per unit width of sliding mass by the anchors and r F is the distance between the anchor head and the rotation center O. According to the studies of Zhao et al. [28] and Figure 1, the tensile force of anchor acts on the point F; thus, the value of r F is In equation (8), the parameter θ T is the angle which specifies the relative position of the anchor head and the rotation center, and the parameter α is the angle between the anchor and the horizontal.
Note that the crack prior exists at the top of the slope, which indicates that there is no internal dissipation of energy along the crack. us, the energy is just dissipated internally along the failure surface AD, which is found by multiplying the differential area rdθ/cosφ by c times the discontinuity in v cos φ, and the total internal dissipation rate of energy D can then be calculated as follows: where f d is a function involving the parameters θ 0 , θ h , and θ d .
Based on the failure mechanism and the UBLA mentioned above, the fissured slope stability reinforced with anchors can be addressed by the work-energy balance equation, which is defined as equating the external rate of work to the rate of internal energy dissipation. Expression of the rate of external work due to the soil weight W r , the horizontal seismic force W Eh , the vertical seismic force W Ev , the axial force provided by anchor W anchor , and the rate of internal energy dissipation D can be obtained as follows: Submitting equations (4)-(11) into equation (12) gives the normalized anchor axial force T f /rH 2 for the fissured slope under seismic conditions: where Referring to equation (13), for fissured slopes reinforced with anchors subjected to seismic action, a dimensionless parameter c/cHF S is introduced here to calculate the slope safety factor F S : In equations (13) and (15), the expressions of both T f and T fk correspond to the axial force of anchor, but their meanings used are different: the former is unknown and can be defined as the required value, whereas the latter is given to evaluate the stability of the fissured slope. Moreover, considering the parameters f 1 -f 12 all including F S , an implicit function method is used to solve equation (15) to obtain the safety factor F S .
By means of the mathematical optimization method, the minimum of T f /rH 2

Results and Discussion
In this section, to demonstrate the application of the method used, an idealized dry homogeneous slope example formerly examined by Cai and Ugai [37] is considered: a slope with height H � 8.0 m, slope angle β � 45°, and c � 20 kN/m 3 . e frictional cohesive soil properties φ � 20°and c � 12 kPa. One row of grouted anchors is arranged and has the initial design parameters X E � 4.0 m, α � 15°, and T fk � 40 kN/m.

Verification.
To validate the proposed analytical approach, two cases are illustrated to verify the reasonableness of the obtained solutions on slope stabilization. In Case 1, the arrangements X E of anchor cables range from 2.0 m to 6.0 m with the increments of 1.0 m. In this case, the parameter α is taken as 15°, and T fk � 40 kN/m. e calculated solutions of the factor of safety F S can be compared with results given by Deng et al. [16], as shown in Table 1. When different X E values are considered in anchor-reinforced slope analysis, it should be noted that the upper bound solutions proposed in this study are consistent with the simplified Bishop method and the LE method results of Deng et al. [16], with the maximum difference being less than 0.5%. In Case 2, the orientation angle α of anchor cables increases from 0 to 45°with an interval of 7.5°. In this case, the parameter X E is taken as 4.0 m, and T f � 40 kN/m. Li et al. [15] introduced the safety factors calculated by limit analysis and that by SLOPE/W. As can be seen in Figure 2, the trends of the three results are the same. In addition, the dotted blue lines, which represent the results of this study, are in good agreement with the solid lines, representing the results by limit analysis of Li et al. [15]. e two comparisons prove that the results derived from this study are reasonable for evaluating the stability of the Mathematical Problems in Engineering fissured slopes reinforced with anchor cables subjected to seismic action.

Determination of the Critical Location of Crack.
With the same arrangement of anchor (i.e., X E /H � 0.5), through the computations, the critical location of crack can be determined, as shown in Figure 3. e normalized location of the crack L/H is plotted against the normalized depth of crack H 1 /H for λ � −0.5, 0.5 and K h � 0.1, 0.3, and the required safety factor F S ranges from 1.2 to 1.6 with increments of 0.2. As shown in Figure 3, the critical crack location L/H decreases obviously with the increase in the depth of crack H 1 / H. It also can be seen that the higher the required safety factor of the slope F S is, the greater the value L/H will be, which means the farther the crack location is from the edge of the slope. Comparing Figures 3(a) with 3(b), it can be found that the critical crack location L/H increases with K h increasing. However, the critical crack location L/H decreases with λ increasing. It implies that, with an increase in the horizontal seismic acceleration and with a decrease in the vertical seismic acceleration, the critical location of the crack becomes deeper.

Determination of the Axial Force of Anchor Cables.
Considering the effect of the crack depth H 1 /H on the designed stabilizing axial force of anchor cables, Figure 4 presents the changes in the normalized anchor axial force T f / rH 2 for various case parameters. According to Terzaghi eoretical Soil Mechanics [38], the depth of cracks was no more than half of the height of the slope where the slope angle is 90°, which was also applied to slope whose angle is less than 90°. Cousins [39] also obtained the same finding as above when the slope was under anhydrous conditions. us, herein set the depth of crack H 1 /H ranges from 0 to 0.5. It can be seen in each chart that the variation of the crack depth has little influence on the axial force of anchor cables. e curve is approximately symmetric in the range of H 1 / H � 0-0.5, and the maximum axial force of anchor cables is distributed in the range of H 1 /H � 0.2-0.3, which is illustrated by dotted lines delimiting an area of Figures 4(a) and 4(b). It is quite surprising that the difference between the maximum and minimum axial force of anchor cables is less than 10%. Besides, the charts in Figure 4 are suitable for designing the axial force of anchor cables for a fissured slope with certain safety requirements. Moreover, it is noted that the normalized axial force of anchor cables T f /rH 2 increases with λ and K h increasing. is is because the increase in the horizontal seismic force coefficient and the vertical downward seismic force direction will aggravate the fissured slope failure. So more anchor axial force of anchor cables is needed to stabilize the fissured slope. Figure 5 shows the relation between the designed stabilizing axial force of anchor cables and the arrangement of anchor (i.e., X E /H � 0.3-0.8) for different seismic conditions, and the dashed line indicates the slope with crack (i.e., H 1 / H � 0.2). From Figure 5, it is noted that the designed stabilizing axial force of anchor cables T f /rH 2 increases as the normalized arrangement of anchor X E /H increases. When X E /H � 0.3, the designed stabilizing anchor axial force of the slope with crack is close to that of the slope without crack. But with the increase in the value of X E /H, the gap between the required anchor axial force of the slope with crack and without crack becomes larger. What is noteworthy is that the solid line of the slope without crack does not exist at λ � −0.5 and F S � 1.1 in Figures 5(c)-5(d). ese are reasonable because when the direction of seismic force is vertical upward, the slope without crack may reach the safety factor F S � 1.1 without the need for anchor cables. However, the slope tends to be unstable when the seismic force is vertically downward, and the anchor cables are required for slope under the same safety factor.
To investigate the impact of the variability of anchor inclination angle on the designed stabilizing axial force of anchor cables, four different cases are also adopted. e results of the designed stabilizing axial force of anchor cables T f /rH 2 vary with the anchor inclination angle α under seismic action which are shown in Figure 6. It can be concluded that as the anchor inclination angle α increases,  Mathematical Problems in Engineering the designed stabilizing axial force of anchor cables T f /rH 2 increases. When the angle α is less than 20°and the required slope safety factor F S is less than 1.2, the increase in the amplitude of anchor axial force is small, but when the angle α is more than 20°and the required slope safety factor F S exceeds 1.2, the designed stabilizing axial force of anchor cables increases rapidly. It also can be seen from Figure 6 that, under the same safety factor, the designed stabilizing anchor axial force of the fissured slope (i.e., H 1 /H � 0.2) is significantly greater than that of the slope without cracks, which is consistent with the phenomenon in Figure 5. Meanwhile, the slope which has no crack may reach the safety factor F S � 1.1 without the requirement for anchor cables which is clearly shown in Figures 6(c) and 6(d).
To better illustrate the effect of slope angle on the anchor supporting the fissured slope, the relationship between the  axial force provided by anchor cables under seismic loading and the different slope angle β is plotted in Figure 7. As can be seen, a larger axial force is required when the slope angle β increases; especially when the slope angle exceeds 55°, the designed anchor cable axial force T f /rH 2 increases rapidly.
is means that the slope angle increase will increase the possibility of the slope collapse. us, the more axial force of anchor cables can be applied to maintain slope stability. It also can be seen from Figure 7 that anchor cable supporting is not always required for fissured slope at any angle β. Instead, the angle β should reach a certain value. For example, for a fissured slope seismic condition of λ � -0.5 and K h � 0.1 and the required safety factor F S � 1.2, the anchor cables are required to play their role when the slope angle β � 40°. Furthermore, with the increase of angle β, the gap of anchor axial force between the fissured slope and no crack slope is larger, which indicates that the steeper the fissured slope is, the more dangerous it is.
To completely study the impact of seismic conditions on the stability of the fissured slope, Figures 8(a) and 8(b) present the different values of T f /rH 2 by taking the seismic coefficient (i.e., K h and λ) as the x-coordinate. Considering slope with different required safety factors F S (F S � 1.2, 1.4, and 1.6) and slope with and without crack conditions, three pairs of variation lines are obtained in each figure. In Figure 8, the designed anchor cable axial force T f / rH 2 increases linearly with an increase in the horizontal seismic acceleration coefficient K h . It also can be seen that the value T f /rH 2 increases linearly as the seismic coefficient λ (λ � K v /K h ) increases, which indicates that a fissured slope under vertical downward seismic acceleration is more dangerous than that under vertical upward seismic acceleration.
is further confirms the results in Figures 4-7 about the influence of seismic condition on the anchor axial force.

Determination of the Required Critical Length of Free
Section of Anchor Cables. In order to ensure the stability of the fissured slope, the anchoring section of anchor cables needs to fully enter the sliding bed. us, it is important to determine the required critical length of the free section of anchor cables. Figure 9 investigates the influence of the arrangement of anchor X E /H on the required critical length of free section of anchor cables L EF /H for the slope with and without cracks. An interesting phenomenon shows that when the anchor is in front of a specific position, the required critical length of the free section of anchor cables will increase with the increase of the depth of the slope crack. But beyond this position, the opposite phenomenon will occur,    Mathematical Problems in Engineering and the increase in the crack depth will lead to a decrease in the required critical length of the free section of anchor cables. For example, in Figure 9(a) where λ � 0.5, K h � 0.1, and F S � 1.2, the specific position of anchor X E /H � 0.47, and in Figure 9(b) where λ � 0.5, K h � 0.3, and F S � 1.2, the specific position of anchor X E /H � 0.4. e possible reason for this phenomenon is that, with the increase of crack depth, the location of the crack is close to the edge of the slope, as shown in Figure 3, resulting in changes in the critical slip surface. It is also observed from Figure 9 that, under the same depth of the crack, the greater the safety factor, the larger the required critical length of free section of anchor cables and the smaller the specific position of the anchor cables. In addition, the values of K h have considerable influence on the required critical length of the free section of anchor cables, but the values of λ have little influence on it.

Conclusions
e UBLA combined with the pseudostatic method is applied to investigate the stability of fissured slopes under seismic load reinforced with anchor cables. An existing simple example is that a homogeneous slope reinforced with anchors is presented in this work as a case study. Based on a rotational failure mechanism and a numerical optimization procedure, the proposed method is adopted to obtain the critical location of the crack. en, to better stabilize the fissured slope, five factors (i.e., crack depth, H 1 /H, anchor arrangement, X E /H, anchor inclination angle, α, slope angle, β, and seismic conditions, K h and λ) are considered to determine the axial force of anchor cables. Moreover, the required critical length of the free section of anchor cables is determined to ensure the stability of the fissured slope. Besides, based on the results of parametric studies, the following conclusions can be drawn: (1) e critical location of crack L/H increases (i.e., the crack is away from the edge of the slope) as the required safety factor of the slope F S and the horizontal seismic acceleration coefficient K h increases, but it decreases with increases in the depth of crack H 1 /H and the vertical seismic acceleration K v (K v � λK h ). (2) e variation of the crack depth has little effect on the axial force of anchor cables. It is found from Figures 4-8 that the normalized axial force of anchor cables T f /rH 2 increases with λ and K h increasing, which implies that more anchor axial force of anchor cables is needed to stabilize the fissured slope, due to the fact that the increase of the horizontal seismic force coefficient and the vertical downward seismic force direction aggravate the fissured slope failure. It is also undoubted that the higher the required safety factor of the slope F S is, the greater the axial force of the anchor cables T f is. Moreover, an increase in the arrangement of anchor X E , the anchor inclination angle α, and the slope angle β could result in a greater axial force of anchor cables which is more needed to stabilize the fissured slope under seismic conditions. (3) When the arrangement of anchor X E is in front of a specific position, the required critical length of the free section of anchor cables L EF will increase with the increase of the depth of crack H 1 . However, when the arrangement of the anchor is beyond this position, L EF shows an opposite tendency. In addition, the horizontal seismic acceleration coefficient K h has considerable influence on the required critical length of free section of anchor cables L EF , but the seismic coefficient λ has little influence on it.
Data Availability e generated or analyzed data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that there are no conflicts of interest related to this paper.