# The Stress Calculation Methods of Antislide Structures with Continuous Ladders

Authors: Gong Chen; Qiang Xu
Journal: Advances in Civil Engineering
 (2020)

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## Abstract

In antislide structures with continuous ladders (ASCLs), horizontal and vertical reinforced concrete antislide members connected in continuous ladders, head to tail, are set along the slip surfaces of slopes. The antislide members are connected with each other and anchored in the solid bedrock from the sliding mass to the sliding zone to resist the landslide thrust and replace the soft materials in the sliding zone. The effects of ASCLs, which are complex and hyperstatic mechanical systems, are calculated by using different numerical simulation software programs and compared with engineering practice experience. However, these effects are uncertain and the use of other analysis methods is required to verify them. In this paper, first, the antislide mechanism of these structures was proposed. Second, the slip surfaces were taken as boundaries, and the ASCL of the Houzishi landslides was taken as an example. Third, the stress models of the structures and load effects were simplified, and then, an ASCL stress calculation method was established to obtain the expressions for structural stress analysis by using the displacement method of structural mechanics, elastic foundation beam method, and boundary constraints. A comparison of the results of the structural stress from the analytical methods and numerical simulation methods indicated that the whole displacement of the structures exhibited a domino effect, which was downwards to the right. The trends of the structural stress determined with the analytical methods and numerical simulation methods were similar. The ultimate results of the analytical methods and the ultimate results of the numerical simulation methods were also similar. The conclusions proposed that the ultimate results of the analytical methods exhibited a hysteretic effect, unlike the ultimate results of the numerical simulation methods. The ultimate results of the analytical methods and numerical simulation methods were adopted for the design of structural stress based on the principle of internal stress envelope diagrams.


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## Body

## 1. Introduction

### 1.1. Summary of the Structures

Antislide structures with continuous ladders (ASCLs) are complex and hyperstatic mechanical systems. The horizontal and vertical reinforced concrete antislide members connected in continuous ladders, head to tail, are set along the slip surface of a slope. The antislide members are connected with each other and anchored in the solid bedrock from the sliding mass to the sliding zone to resist the landslide thrust and replace the soft materials in the sliding zone [1].The longitudinal profile of one ASCL along the slip surface of a slope is shown in Figure1. Many ASCLs are connected by using horizontal binding beams that are similar to the binding beams in the frame structures to increase the global stability and antisliding capacity. The plane layout of an ASCL is shown in Figure 2 [1]. Due to the horizontal and vertical reinforced concrete antislide members that are connected in head to tail continuous ladders and the slip surface that the ASCLs pass through at a large depth, the construction technology of ASCLs is different from other traditional antislide structures. First, the vertical guidance holes are excavated. Second, the horizontal major holes (horizontal binding beams) are excavated. Third, the horizontal holes (horizontal reinforced concrete antislide members) are excavated alternately from both ends of the horizontal major holes. Finally, the vertical holes (vertical reinforced concrete antislide members) are excavated at the end of the horizontal holes; in addition, the horizontal and vertical reinforced concrete antislide members are gradually constructed [2].Figure 1
The longitudinal profile of one ASCL along the slip surface of a slope. The zone above the sliding zone is a sliding body while the zone below the sliding zone is a slip bed. Binding beams connect the ASCLs along the horizontal direction.Figure 2
The plane layout of an ASCL. ASCLs are constructed along the longitudinal direction. Binding beams connect the ASCLs along the horizontal direction.ASCLs are applicable to landslides where the rock masses above and below a slip surfaces are integrated, the structures in the sliding bodies are small, the positions of the slip surfaces are clear and deep, and the landslide thrust is large. The global stability of the rock is enhanced by strengthening the mechanical properties of the geotechnical materials applied in the sliding zone or around it. ASCLs have been successfully used in the Houzishi landslides in Fengjie County of Chongqing city, as the most complex geohazard governance in the Three Gorges Reservoir Area in terms of sequential bedrock landslides [2].


### 1.2. Literature Review

Zou et al. [3] simulated and calculated the antisliding effects and structural stress of an ASCL by using Fast Lagrangian Analysis of Continua (FLAC) software. The research showed that the safety and stability of landslides increase and that good geohazard governance could be obtained by using an ASCL. Fu et al. [4] researched the landslide stabilization effects of an ASCL, which were simulated and calculated by using the finite difference strength reduction method, according to the design and construction of a landslide treatment. Ma [5] used FLAC to simulate and calculate the axial force, the shear force, and the bending moment of an ASCL. The research showed that not only the stress state of the ASCL met the design requirements but also the safety and stability of landslides were improved. Zheng [1] concluded that axial force, shear force, and bending moment are borne by an ASCL and that the deformation of an ASCL is smaller when antislide piles are also implemented by using FLAC to compare stress and the deformation results. In addition, Zheng and Hong [2] proposed that the internal antislide mechanism of an ASCL effectively decreased the shearing deformation of the sliding zone, the development of the plastic zone, and the global displacement of the sliding body and effectively increased the resistance of the geotechnical materials; additionally, they identified that the external antislide mechanism of an ASCL improved the mechanical properties of the geotechnical materials and developed a global antislide function by using FLAC to simulate and calculate the effects of the ASCL on landsliding.Based on the assumption of ideal elastic-plastic material, Ito and Matsui [6] obtained the relation between the strength parameters and the sliding resistance force provided by the single antislide pile. Rajashree and Sundaravedivelu [7] carried out the analysis of laterally loaded piles in soft clay, idealising the pile as beam elements and the soil by nonlinear inelastic spring elements modelled with elastoplastic subelements. Halabe and Jaina [8] analysed the single piles under pure lateral loads and discussed the influence of related parameters. Hassiotis et al. [9] proposed a methodology for the design of slopes reinforced with a single row of piles. Abbas et al. [10] carried out the deformation behavior of piles was related to the section shape and slenderness ratio of piles. Frank and Pouget [11] proposed the excavation or increases in the driving force of landslides behind piles led to the deformation of piles and even pile damage. Kourkoulis et al. [12] used a hybrid method for the analysis and design of slope-stabilizing piles. Nian et al. [13] performed 3D numerical analysis for typical examples of slopes reinforced with antislide piles using the strength reduction FEM with consideration of the interaction of the pile, soil, and slope. In landslide controlling, Lirer [14] proposed the antislide piles played an important role and coordinate deformation with the surrounding soil to improve the stability of the whole slope. Ashour and Ardalan [15] presented a new procedure for the analysis of slope stabilization using piles. Shooshpasha and Amirdehi [16] studied the stability analysis of slopes reinforced with one row of free head piles by using the shear strength reduction method with the software of Abaqus. Tehrani et al. [17] proposed the calculation of pile displacement was achieved under certain loads. Kahyaoglu et al. [18] proposed the equation of pile deformation, as proposed under different lateral load patterns, was related to the elastic modulus and moment of inertia of the pile; the cantilever pile length affected the distributions of slope pressure above the slip surface. Vega-Posada Carlos et al. [19] developed a simplified analytical approach to analyze soil-structure interaction of beam-column elements (i.e., beams, columns, and piles) with generalized end-boundary conditions on a homogeneous or nonhomogeneous Pasternak foundation. Aqoub et al. [20] proposed that the transfer of loads to the piles was increased during the monotonic loading stage but at a lower rate with increasing the embankment height. Belato et al. [21] researched on the performance of semiempirical methods based on the standard penetration test (SPT) for the prediction of bearing capacity already disseminated in the practice of Brazilian Foundation Engineering. Pratap and Chatterjee [22] observed that the maximum bending moment increased and more mobilization of earth pressure taken place with increase in horizontal seismic acceleration coefficients, magnitude of uniform surcharge, and embedded depth and decrease in the distance of surcharge from the top of the wall in loose sand. Naphol et al. [23] researched an experimental investigation of the properties of CFG. Fattah et al. [24] presented a series of model experiments conducted on single pile embedded in saturated and unsaturated expansive soil. Amir et al. [25] researched that the group reduction factor was considered as a parameter commonly used in spring models created from pile groups to consider the group effects in soil-pile interaction analysis.


### 1.3. Research Significance

The effects of ASCLs, which are complex and hyperstatic mechanical systems, are calculated and contrasted by using different numerical simulation software or engineering practice experience. However, these effects remain uncertain, and the use of other analysis methods is required to verify them. There has been no research on the stress calculation methods for ASCLs in the relevant codes, professional books, or research literature in China, such as the Design code for geohazard prevention (DB 50/5029-2004) [26], Specification of design and construction for landslide stabilization (DZ/T 0219-2006) [27], Design code for engineered slopes in hydropower projects and water resources (DL/T 5353-2006) [28], Design code for engineered slopes in water resources and hydropower projects (SL 386-2007) [29], and Engineering design and examples of the new type of supporting structures [30]. Applied research on ASCLs has been performed on the construction plan, construction technology, numerical simulation, monitoring, etc., but basic research on ASCLs is lacking.Based on the above research background, scientific references are provided not only for the specifications and structural design of an ASCL but also to fill the gaps in the basic research of these structures, starting with the antislide mechanism and the stress calculation method of ASCLs.


## 1.1. Summary of the Structures

Antislide structures with continuous ladders (ASCLs) are complex and hyperstatic mechanical systems. The horizontal and vertical reinforced concrete antislide members connected in continuous ladders, head to tail, are set along the slip surface of a slope. The antislide members are connected with each other and anchored in the solid bedrock from the sliding mass to the sliding zone to resist the landslide thrust and replace the soft materials in the sliding zone [1].The longitudinal profile of one ASCL along the slip surface of a slope is shown in Figure1. Many ASCLs are connected by using horizontal binding beams that are similar to the binding beams in the frame structures to increase the global stability and antisliding capacity. The plane layout of an ASCL is shown in Figure 2 [1]. Due to the horizontal and vertical reinforced concrete antislide members that are connected in head to tail continuous ladders and the slip surface that the ASCLs pass through at a large depth, the construction technology of ASCLs is different from other traditional antislide structures. First, the vertical guidance holes are excavated. Second, the horizontal major holes (horizontal binding beams) are excavated. Third, the horizontal holes (horizontal reinforced concrete antislide members) are excavated alternately from both ends of the horizontal major holes. Finally, the vertical holes (vertical reinforced concrete antislide members) are excavated at the end of the horizontal holes; in addition, the horizontal and vertical reinforced concrete antislide members are gradually constructed [2].Figure 1
The longitudinal profile of one ASCL along the slip surface of a slope. The zone above the sliding zone is a sliding body while the zone below the sliding zone is a slip bed. Binding beams connect the ASCLs along the horizontal direction.Figure 2
The plane layout of an ASCL. ASCLs are constructed along the longitudinal direction. Binding beams connect the ASCLs along the horizontal direction.ASCLs are applicable to landslides where the rock masses above and below a slip surfaces are integrated, the structures in the sliding bodies are small, the positions of the slip surfaces are clear and deep, and the landslide thrust is large. The global stability of the rock is enhanced by strengthening the mechanical properties of the geotechnical materials applied in the sliding zone or around it. ASCLs have been successfully used in the Houzishi landslides in Fengjie County of Chongqing city, as the most complex geohazard governance in the Three Gorges Reservoir Area in terms of sequential bedrock landslides [2].


## 1.2. Literature Review

Zou et al. [3] simulated and calculated the antisliding effects and structural stress of an ASCL by using Fast Lagrangian Analysis of Continua (FLAC) software. The research showed that the safety and stability of landslides increase and that good geohazard governance could be obtained by using an ASCL. Fu et al. [4] researched the landslide stabilization effects of an ASCL, which were simulated and calculated by using the finite difference strength reduction method, according to the design and construction of a landslide treatment. Ma [5] used FLAC to simulate and calculate the axial force, the shear force, and the bending moment of an ASCL. The research showed that not only the stress state of the ASCL met the design requirements but also the safety and stability of landslides were improved. Zheng [1] concluded that axial force, shear force, and bending moment are borne by an ASCL and that the deformation of an ASCL is smaller when antislide piles are also implemented by using FLAC to compare stress and the deformation results. In addition, Zheng and Hong [2] proposed that the internal antislide mechanism of an ASCL effectively decreased the shearing deformation of the sliding zone, the development of the plastic zone, and the global displacement of the sliding body and effectively increased the resistance of the geotechnical materials; additionally, they identified that the external antislide mechanism of an ASCL improved the mechanical properties of the geotechnical materials and developed a global antislide function by using FLAC to simulate and calculate the effects of the ASCL on landsliding.Based on the assumption of ideal elastic-plastic material, Ito and Matsui [6] obtained the relation between the strength parameters and the sliding resistance force provided by the single antislide pile. Rajashree and Sundaravedivelu [7] carried out the analysis of laterally loaded piles in soft clay, idealising the pile as beam elements and the soil by nonlinear inelastic spring elements modelled with elastoplastic subelements. Halabe and Jaina [8] analysed the single piles under pure lateral loads and discussed the influence of related parameters. Hassiotis et al. [9] proposed a methodology for the design of slopes reinforced with a single row of piles. Abbas et al. [10] carried out the deformation behavior of piles was related to the section shape and slenderness ratio of piles. Frank and Pouget [11] proposed the excavation or increases in the driving force of landslides behind piles led to the deformation of piles and even pile damage. Kourkoulis et al. [12] used a hybrid method for the analysis and design of slope-stabilizing piles. Nian et al. [13] performed 3D numerical analysis for typical examples of slopes reinforced with antislide piles using the strength reduction FEM with consideration of the interaction of the pile, soil, and slope. In landslide controlling, Lirer [14] proposed the antislide piles played an important role and coordinate deformation with the surrounding soil to improve the stability of the whole slope. Ashour and Ardalan [15] presented a new procedure for the analysis of slope stabilization using piles. Shooshpasha and Amirdehi [16] studied the stability analysis of slopes reinforced with one row of free head piles by using the shear strength reduction method with the software of Abaqus. Tehrani et al. [17] proposed the calculation of pile displacement was achieved under certain loads. Kahyaoglu et al. [18] proposed the equation of pile deformation, as proposed under different lateral load patterns, was related to the elastic modulus and moment of inertia of the pile; the cantilever pile length affected the distributions of slope pressure above the slip surface. Vega-Posada Carlos et al. [19] developed a simplified analytical approach to analyze soil-structure interaction of beam-column elements (i.e., beams, columns, and piles) with generalized end-boundary conditions on a homogeneous or nonhomogeneous Pasternak foundation. Aqoub et al. [20] proposed that the transfer of loads to the piles was increased during the monotonic loading stage but at a lower rate with increasing the embankment height. Belato et al. [21] researched on the performance of semiempirical methods based on the standard penetration test (SPT) for the prediction of bearing capacity already disseminated in the practice of Brazilian Foundation Engineering. Pratap and Chatterjee [22] observed that the maximum bending moment increased and more mobilization of earth pressure taken place with increase in horizontal seismic acceleration coefficients, magnitude of uniform surcharge, and embedded depth and decrease in the distance of surcharge from the top of the wall in loose sand. Naphol et al. [23] researched an experimental investigation of the properties of CFG. Fattah et al. [24] presented a series of model experiments conducted on single pile embedded in saturated and unsaturated expansive soil. Amir et al. [25] researched that the group reduction factor was considered as a parameter commonly used in spring models created from pile groups to consider the group effects in soil-pile interaction analysis.


## 1.3. Research Significance

The effects of ASCLs, which are complex and hyperstatic mechanical systems, are calculated and contrasted by using different numerical simulation software or engineering practice experience. However, these effects remain uncertain, and the use of other analysis methods is required to verify them. There has been no research on the stress calculation methods for ASCLs in the relevant codes, professional books, or research literature in China, such as the Design code for geohazard prevention (DB 50/5029-2004) [26], Specification of design and construction for landslide stabilization (DZ/T 0219-2006) [27], Design code for engineered slopes in hydropower projects and water resources (DL/T 5353-2006) [28], Design code for engineered slopes in water resources and hydropower projects (SL 386-2007) [29], and Engineering design and examples of the new type of supporting structures [30]. Applied research on ASCLs has been performed on the construction plan, construction technology, numerical simulation, monitoring, etc., but basic research on ASCLs is lacking.Based on the above research background, scientific references are provided not only for the specifications and structural design of an ASCL but also to fill the gaps in the basic research of these structures, starting with the antislide mechanism and the stress calculation method of ASCLs.


## 2. Theoretical Models

### 2.1. Antislide Mechanism of an ASCL

Assuming a slip surface as a boundary in the ASCL, the resultant force due to the thrust, which is distributed to the horizontal and vertical antislide members above the slip surfaces, and the sliding resistance in front of the antislide members, which can resist part of the landslide thrust, impact the horizontal and vertical antislide members above the slip surfaces: these forces include the axial force, the shear force, and the bending moment. Due to the connections of the antislide members, the effects on the slip surfaces are transmitted to the solid bedrock by the anchoring effects between the antislide members below the slip surfaces and the slip bed.


### 2.2. Basic Ideas

Above the slip bed, which consists of solid bedrock, the sliding body may slide along a slip surface of the slope. Assuming a slip surface is a boundary in the ASCL, the horizontal and vertical antislide members above the slip surfaces are simplified to be the nonstatic structures that are fastened on the slip surfaces, so the resultant force due to the thrust and the sliding resistance in front of the antislide members are simplified to be the external load; therefore, the structures above the slip surfaces can be analysed by using the displacement method of structural mechanics.Compared with the slip bed, which is simplified to be the foundation, the internal stress of the horizontal and vertical antislide members below the slip surfaces, which are embedded in the slip bed and simplified to be the structures in the foundation, can be calculated and analysed by using elastic foundation beam methods.


### 2.3. Simplification of the Mechanical Model

According to the mechanical design and construction technology of an ASCL, the combination of the horizontal and vertical antislide members are simplified to be rigid frames due to the connection nodes, which are simplified as rigid nodes. As mentioned before, the effects of the ASCL, which are complex and hyperstatic mechanical systems, are difficult to calculate and analyze. Taking the shape of the ASCL of the Houzishi landslides as an example, the three-dimensional (3D) structures are simplified to be two-dimensional (2D) structures, which are taken as one ASCL in the landslides. This ASCL is shown in Figure3.Figure 3
One ASCL in a landslide. The zone above the sliding zone is a sliding body while the zone below the sliding zone is a slip bed. Assuming a slip surface as a boundary in the ASCL, the ASCL is disintegrated into the section above the slip surface and the section below the slip surface. The ends of the antislide members and the intersection on the slip surface are numbered in English letters.


### 2.4. Simplification of the Load

If a sliding body contains a complete rigid layer, nondisturbed stiff clay, or similar rock-like material, the landslide thrust is assumed to have a rectangular distribution. If a sliding body contains gravelly soil or rocky soil, the landslide thrust is assumed to have a triangular distribution. If a sliding body contains materials with geotechnical properties between those of the abovementioned materials, the landslide thrust is assumed to have a trapezoidal distribution, which can be represented as the superimposition of a rectangular distribution and a triangular distribution. While the sliding bodies in front of the structures and above the slip surfaces may slide, hypothetically, the upper part of the structures does not bear the resistance. However, while the sliding bodies in front of the structures and above the slip surfaces are fundamentally stable, the upper part of the structures bears the resistance, which is assumed to be equal to or less than the excess sliding force and the passive earth pressure of the slip bodies in front of the structures, as the resistance in front of the structures is assumed to be equal to the minimum of the excess sliding force and the passive earth pressure of the slip bodies in front of the structures. In structural design, the resistance distribution of the structures is usually adopted to match the landslide thrust distribution or a parabolic curve. The bearing load of the structures above the slip surfaces is controlled by the landslide thrust and the resistance in front of the structures, and the distribution type is usually rectangular, triangular, or trapezoidal [31].The major loads on the ASCL are due to the landslide thrust, the resistance in front of the structures, and the anchoring effects between the antislide members and the slip bed, while the minor loads on the ASCL are due to the gravity of the antislide members, the friction and end-bearing forces of the antislide members, and geotechnical factors. When the structural stress is calculated and analysed, the minor loads and the axial deformation of the structures should be neglected due to the short length of the horizontal and vertical antislide members. As mentioned before, the ASCL is applicable to landslides where the rock masses above and below a slip surface are integrated, the structures in the sliding bodies are small, the positions of the slip surfaces are clear and deep, and the foundation coefficient above the slip surfaces changes minimally. The resultant force due to the thrust and the sliding resistance in front of the antislide members is simplified to be a trapezoidal load distributed across the antislide members above the slip surface, which can be decomposed into a rectangular load and a triangular load. The trapezoidal load near the slip surfaces is strong, whereas the trapezoidal load far from the slip surfaces is weak. The simplified figure of the load effects of the ASCL above the slip surfaces is shown in Figure4.Figure 4
Simplified figure of the load effects of the ASCL above the slip surfaces: (a) simplified figure of the load effects of BCD; (b) simplified figure of the load effects of FGH; (c) simplified figure of the load effects of JKLNO.
(a)(b)(c)


### 2.5. Boundary Conditions below the Slip Surfaces

The section size and partitioned length above and below the slip surfaces, which should be equal or approximately equal, are relevant to the geological conditions, the location of the structures, etc. The horizontal and vertical antislide members may be partly rigid members or partly elastic members due to the complex geological conditions and the layout of the structures. The bottom boundary conditions of the antislide members (AB, LM, and OP) below the slip surfaces are considered to be the hinge support due to the complete layer of the slip bed and the shallow anchoring depth in Figure 3. Because the bottoms of the rigid members form the hinge support, the members rotate around their bottoms. Because the bottoms of the elastic members form the hinge support, the effects of the structures should be calculated and analysed by considering the hinge support, which is the boundary condition at the bottom of the structures.Ignoring the axial deformation of the antimembers, their combinations in the ASCL above the slip surfaces are presented byDEF and HIJ in Figure 3. If DEF is the rigid combination, DE is a vertical rigid member and EF is a horizontal rigid member. Furthermore, while DEF (rigid combination) bears the axial force, shear force, and bending moment on the slip surfaces, DE will rotate around some node, and EF will rotate similarly around some node. Thus, DEF (rigid combination) can only rotate around Node E, which is the only common joint between DE and EF. The structural stress of DEF (rigid combination) is shown in Figure 5.Figure 5
The structural stress ofDEF (rigid combination). Due to the shear force and bending moment of the ends in the rigid members, the angular displacement of the ends will occur, but no lateral deflection deformation will occur.Otherwise, ifDEF is the elastic combination, DE is the vertical elastic member and EF is the horizontal elastic member. Node E cannot move along the horizontal and vertical directions due to the DE and EF, which constrains the horizontal and vertical displacement of Node E. Therefore, DEF (elastic combination) rotates around Node E, but no horizontal and vertical displacement occurs, and the rigid Node E maintains its right angle. The structural stress of DEF (elastic combination) is shown in Figure 6.Figure 6
Structural stress ofDEF (elastic combination). Due to the shear force and bending moment of the ends in the elastic members, angular displacement and lateral deflection deformations of the ends will occur.The internal stress at NodeE of DEF is shown in Figure 7, the boundary condition at Node E of DEF is shown in formula (1), and the equilibrium condition of the internal stress at Node E of DEF is shown in formula (2).(1)xE=yE=0,∠DED′=∠FEF′,∠DEF=∠D′EF′=π2,where xE is the horizontal displacement of Node E, yE is the vertical displacement of Node E, ∠DED′ is the intersection angle between DE and D′E, ∠FEF′ is the intersection angle between FE and F′E, ∠DEF is the intersection angle between DE and FE, and ∠D′EF′ is the intersection angle between D′E and F′E.(2)∑Fx=0,FQED+FNEF=0,∑Fy=0,FNED−FQEF=0,∑ME=0,MED+MEF=0,where ∑Fx is the resultant force at Node E along the x-direction, ∑Fy is the resultant force at Node E along the y-direction, ∑ME is the resultant moment at Node E, FQED is the shear force at Node E along the ED direction, FQEF is the shear force at Node E along the EF direction, FNED is the axial force of DE, FNEF is the axial force of EF, MED is the resultant moment at Node E along the ED direction, and MEF is the resultant moment at Node E along the EF direction.Figure 7
The internal stress at NodeE of DEF. The resultant force at Node E of DEF along the x-direction and y-direction is 0, while the resultant moment at Node E of DEF is 0.


## 2.1. Antislide Mechanism of an ASCL

Assuming a slip surface as a boundary in the ASCL, the resultant force due to the thrust, which is distributed to the horizontal and vertical antislide members above the slip surfaces, and the sliding resistance in front of the antislide members, which can resist part of the landslide thrust, impact the horizontal and vertical antislide members above the slip surfaces: these forces include the axial force, the shear force, and the bending moment. Due to the connections of the antislide members, the effects on the slip surfaces are transmitted to the solid bedrock by the anchoring effects between the antislide members below the slip surfaces and the slip bed.


## 2.2. Basic Ideas

Above the slip bed, which consists of solid bedrock, the sliding body may slide along a slip surface of the slope. Assuming a slip surface is a boundary in the ASCL, the horizontal and vertical antislide members above the slip surfaces are simplified to be the nonstatic structures that are fastened on the slip surfaces, so the resultant force due to the thrust and the sliding resistance in front of the antislide members are simplified to be the external load; therefore, the structures above the slip surfaces can be analysed by using the displacement method of structural mechanics.Compared with the slip bed, which is simplified to be the foundation, the internal stress of the horizontal and vertical antislide members below the slip surfaces, which are embedded in the slip bed and simplified to be the structures in the foundation, can be calculated and analysed by using elastic foundation beam methods.


## 2.3. Simplification of the Mechanical Model

According to the mechanical design and construction technology of an ASCL, the combination of the horizontal and vertical antislide members are simplified to be rigid frames due to the connection nodes, which are simplified as rigid nodes. As mentioned before, the effects of the ASCL, which are complex and hyperstatic mechanical systems, are difficult to calculate and analyze. Taking the shape of the ASCL of the Houzishi landslides as an example, the three-dimensional (3D) structures are simplified to be two-dimensional (2D) structures, which are taken as one ASCL in the landslides. This ASCL is shown in Figure3.Figure 3
One ASCL in a landslide. The zone above the sliding zone is a sliding body while the zone below the sliding zone is a slip bed. Assuming a slip surface as a boundary in the ASCL, the ASCL is disintegrated into the section above the slip surface and the section below the slip surface. The ends of the antislide members and the intersection on the slip surface are numbered in English letters.


## 2.4. Simplification of the Load

If a sliding body contains a complete rigid layer, nondisturbed stiff clay, or similar rock-like material, the landslide thrust is assumed to have a rectangular distribution. If a sliding body contains gravelly soil or rocky soil, the landslide thrust is assumed to have a triangular distribution. If a sliding body contains materials with geotechnical properties between those of the abovementioned materials, the landslide thrust is assumed to have a trapezoidal distribution, which can be represented as the superimposition of a rectangular distribution and a triangular distribution. While the sliding bodies in front of the structures and above the slip surfaces may slide, hypothetically, the upper part of the structures does not bear the resistance. However, while the sliding bodies in front of the structures and above the slip surfaces are fundamentally stable, the upper part of the structures bears the resistance, which is assumed to be equal to or less than the excess sliding force and the passive earth pressure of the slip bodies in front of the structures, as the resistance in front of the structures is assumed to be equal to the minimum of the excess sliding force and the passive earth pressure of the slip bodies in front of the structures. In structural design, the resistance distribution of the structures is usually adopted to match the landslide thrust distribution or a parabolic curve. The bearing load of the structures above the slip surfaces is controlled by the landslide thrust and the resistance in front of the structures, and the distribution type is usually rectangular, triangular, or trapezoidal [31].The major loads on the ASCL are due to the landslide thrust, the resistance in front of the structures, and the anchoring effects between the antislide members and the slip bed, while the minor loads on the ASCL are due to the gravity of the antislide members, the friction and end-bearing forces of the antislide members, and geotechnical factors. When the structural stress is calculated and analysed, the minor loads and the axial deformation of the structures should be neglected due to the short length of the horizontal and vertical antislide members. As mentioned before, the ASCL is applicable to landslides where the rock masses above and below a slip surface are integrated, the structures in the sliding bodies are small, the positions of the slip surfaces are clear and deep, and the foundation coefficient above the slip surfaces changes minimally. The resultant force due to the thrust and the sliding resistance in front of the antislide members is simplified to be a trapezoidal load distributed across the antislide members above the slip surface, which can be decomposed into a rectangular load and a triangular load. The trapezoidal load near the slip surfaces is strong, whereas the trapezoidal load far from the slip surfaces is weak. The simplified figure of the load effects of the ASCL above the slip surfaces is shown in Figure4.Figure 4
Simplified figure of the load effects of the ASCL above the slip surfaces: (a) simplified figure of the load effects of BCD; (b) simplified figure of the load effects of FGH; (c) simplified figure of the load effects of JKLNO.
(a)(b)(c)


## 2.5. Boundary Conditions below the Slip Surfaces

The section size and partitioned length above and below the slip surfaces, which should be equal or approximately equal, are relevant to the geological conditions, the location of the structures, etc. The horizontal and vertical antislide members may be partly rigid members or partly elastic members due to the complex geological conditions and the layout of the structures. The bottom boundary conditions of the antislide members (AB, LM, and OP) below the slip surfaces are considered to be the hinge support due to the complete layer of the slip bed and the shallow anchoring depth in Figure 3. Because the bottoms of the rigid members form the hinge support, the members rotate around their bottoms. Because the bottoms of the elastic members form the hinge support, the effects of the structures should be calculated and analysed by considering the hinge support, which is the boundary condition at the bottom of the structures.Ignoring the axial deformation of the antimembers, their combinations in the ASCL above the slip surfaces are presented byDEF and HIJ in Figure 3. If DEF is the rigid combination, DE is a vertical rigid member and EF is a horizontal rigid member. Furthermore, while DEF (rigid combination) bears the axial force, shear force, and bending moment on the slip surfaces, DE will rotate around some node, and EF will rotate similarly around some node. Thus, DEF (rigid combination) can only rotate around Node E, which is the only common joint between DE and EF. The structural stress of DEF (rigid combination) is shown in Figure 5.Figure 5
The structural stress ofDEF (rigid combination). Due to the shear force and bending moment of the ends in the rigid members, the angular displacement of the ends will occur, but no lateral deflection deformation will occur.Otherwise, ifDEF is the elastic combination, DE is the vertical elastic member and EF is the horizontal elastic member. Node E cannot move along the horizontal and vertical directions due to the DE and EF, which constrains the horizontal and vertical displacement of Node E. Therefore, DEF (elastic combination) rotates around Node E, but no horizontal and vertical displacement occurs, and the rigid Node E maintains its right angle. The structural stress of DEF (elastic combination) is shown in Figure 6.Figure 6
Structural stress ofDEF (elastic combination). Due to the shear force and bending moment of the ends in the elastic members, angular displacement and lateral deflection deformations of the ends will occur.The internal stress at NodeE of DEF is shown in Figure 7, the boundary condition at Node E of DEF is shown in formula (1), and the equilibrium condition of the internal stress at Node E of DEF is shown in formula (2).(1)xE=yE=0,∠DED′=∠FEF′,∠DEF=∠D′EF′=π2,where xE is the horizontal displacement of Node E, yE is the vertical displacement of Node E, ∠DED′ is the intersection angle between DE and D′E, ∠FEF′ is the intersection angle between FE and F′E, ∠DEF is the intersection angle between DE and FE, and ∠D′EF′ is the intersection angle between D′E and F′E.(2)∑Fx=0,FQED+FNEF=0,∑Fy=0,FNED−FQEF=0,∑ME=0,MED+MEF=0,where ∑Fx is the resultant force at Node E along the x-direction, ∑Fy is the resultant force at Node E along the y-direction, ∑ME is the resultant moment at Node E, FQED is the shear force at Node E along the ED direction, FQEF is the shear force at Node E along the EF direction, FNED is the axial force of DE, FNEF is the axial force of EF, MED is the resultant moment at Node E along the ED direction, and MEF is the resultant moment at Node E along the EF direction.Figure 7
The internal stress at NodeE of DEF. The resultant force at Node E of DEF along the x-direction and y-direction is 0, while the resultant moment at Node E of DEF is 0.


## 3. Analytical Calculations

### 3.1. Analytical Calculation of the Structural Stress above the Slip Surfaces

#### 3.1.1. Analytical Calculation of the Structural Stress ofBCD

The load decomposition ofBCD is shown in Figure 8, and the analytical calculation of the structural stress of BCD is shown in formulas (3) and (4). The analytical calculation of the structural stress of FGH is the same as that of BCD and is not repeated here.whereMBC is the bending moment at Node B along the BC direction, MCB is the bending moment at Node C along the CB direction, MDC is the bending moment at Node D along the DC direction, MCD is the bending moment at Node C along the CD direction, FQBC is the shear force at Node B along the BC direction, FQCB is the shear force at Node C along the CB direction, FQDC is the shear force at Node D along the DC direction, FQCD is the shear force at Node C along the CD direction, FNCB is the axial force of BC, FNCD is the axial force of CD, θC1 is the angular displacement of C1, θC2 is the angular displacement of C2, qBC is the rectangular load of the load decomposition of BC, qBC′ is the maximum of the triangular load of the load decomposition of BC, qCD is the rectangular load of the load decomposition of CD, qCD′ is the maximum of the triangular load of the load decomposition of CD, iBC is the linear stiffness of BC, iCD is the linear stiffness of CD, lBC is the length of BC, and lCD is the length of CD.Figure 8
The load decomposition ofBCD: (a) trapezoidal load; (b) rectangular load; (c) triangular load.(3)MBCMCBMDCMCDFQBCFQCBFQDCFQCD=2iBC2iBC−qBClBC212−qBC′lBC2204iBC4iBC−qBClBC212−qBC′lBC2302iCD2iCD−qCDlCD212−qCD′lCD2204iCD4iCD−qCDlCD212−qCD′lCD230−6iBClBC−6iBClBCqBClBC2−7qBC′lBC20−6iBClBC−6iBClBC−qBClBC2−3qBC′lBC20−6iCDlCD−6iCDlCDqCDlCD2−7qCD′lCD20−6iCDlCD−6iCDlCD−qCDlCD2−3qCD′lCD20θC1θC111,(4)θC1=−qBClBC2+qCDlCD248iBC+iCD,θC2=−qBC′lBC2+qCD′lCD2120iBC+iCD,FNCB=−FQCD,FNCD=FQCB,


#### 3.1.2. Analytical Calculation of the Structural Stress ofJKLNO

The load decomposition ofJKLNO is shown in Figure 9, and the analytical calculation of the structural stress of JKLNO is shown in formulas (5)∼(8).Figure 9
The load decomposition ofJKLNO: (a) trapezoidal load; (b) rectangular load; (c) triangular load.
(a)(b)(c)(5)MJKMKJMLKMKLMKNMNKMONMNOFQJKFQKJFQLKFQKLFQKNFQNKFQONFQNO=2iJK2iJK00−qJKlJK212−qJK′lJK2204iJK4iJK00qJKlJK212+qJK′lJK2302iKL2iKL00−qKLlKL212−qKL′lKL2204iKL4iKL00qKLlKL212+qKL′lKL2304iKN4iKN2iKN2iKN−qKNlKN212−qKN′lKN2202iKN2iKN4iKN4iKNqKNlKN212+qKN′lKN230002iNO2iNO−qNOlNO212−qNO′lNO220004iNO4iNOqNOlNO212+qNO′lNO230−6iJKlJK−6iJKlJK00qJKlJK2+7qJK′lJK20−6iJKlJK−6iJKlJK00−qJKlJK2−3qJK′lJK20−6iKLlKL−6iKLlKL00qKLlKL2+7qKL′lKL20−6iKLlKL−6iKLlKL00−qKLlKL2−3qKL′lKL20−6iKNlKN−6iKNlKN−6iKNlKN−6iKNlKNqKNlKN2+7qKN′lKN20−6iKNlKN−6iKNlKN−6iKNlKN−6iKNlKN−qKNlKN2−3qKN′lKN2000−6iNOlNO−6iNOlNOqNOlNO2+7qNO′lNO2000−6iNOlNO−6iNOlNO−qNOlNO2−3qNO′lNO20θK1θK2θN1θN21,(6)4iJK+4iKL+4iKN2iKN2iKN4iKN+4iNOθK1θN1=qKNlKN2−qKLlKL2−qJKlJK212−qKNlKN2+qNOlNO212,(7)4iJK+4iKL+4iKN2iKN2iKN4iKN+4iNOθK2θN2=3qKN′lKN2−2qKL′lKL2−2qJK′lJK260−qKN′lKN2+qNO′lNO230,(8)FNKJ=−FQNO−FQKL,FNKL=FQKJ−FQKN,FNKN=FNNK=−FQNO,FNNO=FQNK,


### 3.2. Analytical Calculation of the Structural Stress below the Slip Surfaces

To integrate the structural stress, the horizontal and vertical antislide members are both assumed to be elastic members. The analytical calculation of the structural stress below the slip surfaces is carried out as follows.


#### 3.2.1. Analytical Calculation of the Structural Stress of AB (Elastic Member)

The initial effects and stress diagram ofAB (elastic member) are shown in Figure 10. Node B on the slip surfaces is taken as the starting point, and the analytical calculation of the displacement and internal stress of the section, which is x m distance from Node B along the BA direction, is shown in formulas (9) and (10). The analytical calculation of the structural stress of LM and OP is the same as that of AB and is not repeated here.whereMJK is the bending moment at Node J along the JK direction, MKJ is the bending moment at Node K along the KJ direction, MLK is the bending moment at Node L along the LK direction, MKL is the bending moment at Node K along the KL direction, MKN is the bending moment at Node K along the KN direction, MNK is the bending moment at Node N along the NK direction, MON is the bending moment at Node O along the ON direction, MNO is the bending moment at Node N along the NO direction, FQJK is the shear force at Node J along the JK direction, FQKJ is the shear force at Node K along the KJ direction, FQLK is the shear force at Node L along the LK direction, FQKL is the shear force at Node K along the KL direction, FQKN is the shear force at Node K along the KN direction, FQNK is the shear force at Node N along the NK direction, FQON is the shear force at Node O along ON direction, FQNO is the shear force at Node N along the NO direction, FNKJ is the axial force of JK, FNKL is the axial force of KL, FNKN is the axial force of KN, FNNO is the axial force of NO, θK1 is the angular displacement of K1, θK2 is the angular displacement of K2, θN1 is the angular displacement of N1, θN2 is the angular displacement of N2, qJK is the rectangular load of the load decomposition of JK, qJK′ is the maximum of the triangular load of the load decomposition of JK, qKN is the rectangular load of the load decomposition of KN, qKN′ is the maximum of the triangular load of the load decomposition of KN, qKL is the rectangular load of the load decomposition of KL, qKL′ is the maximum of the triangular load of the load decomposition of KL, qNO is the rectangular load of the load decomposition of NO, qNO′ is the maximum of the triangular load of the load decomposition of NO, iJK is the linear stiffness of JK, iKL is the linear stiffness of KL, iKN is the linear stiffness of KN, iNO is the linear stiffness of NO, lJK is the length of JK, lKL is the length of KL, lKN is the length of KN, and lNO is the length of NO.Figure 10
The initial effects and stress diagram ofAB (elastic member): (a) the initial effects of AB (elastic member); (b) the stress diagram of AB (elastic member).(9)yx=φ1AByBA+φ2ABφBAβAB+φ3ABMBAEABIABβAB2+φ4ABFQBAEABIABβAB2,φx=−4βABφ4AByAB+φ1ABφBA+φ2ABMBAEABIABβAB+φ3ABFQBAEABIABβAB2,Mx=EABIAB−4βAB2φ3AByBA+−4βABφ4ABφBA+φ1ABMBAEABIAB+φ2ABFQBAEABIABβAB,Qx=EABIAB−4βAB2φ2AByBA+−4βAB2φ3ABφBA+−4βABφ4ABMBAEABIAB+φ1ABFQBAEABIABβAB,wherey (x) is the vertical displacement of the section that is x m distance from Node B along the BA direction, φ (x) is the angular displacement of the section that is x m distance from Node B along the BA direction, M (x) is the bending moment of the section that is x m distance from Node B along the BA direction, Q (x) is the shear force of the section that is x m distance from Node B along the BA direction, yBA is the initial vertical displacement of Node B, φBA is the initial angular displacement of Node B, MBA is the initial bending moment of Node B, FQBA is the initial shear force of Node B, EAB is the elastic coefficient of AB, IAB is the section inertial moment of AB, and βAB is the deformation coefficient of AB.As mentioned before, the boundary condition of NodeA is shown in formula (11) considering the bottom of AB (elastic member), which is a hinge support. By calculating formula (11), formula (12) is obtained. Furthermore, formula (12) is plugged into formula (9) to obtain the structural effects of AB (elastic member).(10)φ1AB=cosβABxchβABx,φ2AB=sinβABxchβABx+cosβABxshβABx2,φ3AB=sinβABxshβABx2,φ4AB=sinβABxchβABx−cosβABxshβABx4,shβABx=eβABx−e−βABx2,chβABx=eβABx+e−βABx2,βAB=KVBPAB4EABIAB1/4,where KV is the vertical foundation coefficient of the slip bed, BP(AB) is the calculation width of AB, and e is the natural exponent that is equal to 2.718281…(11)MAB=EABIAB−4βAB2φ3AByBA+−4βABφ4ABφBA+φ1ABMBAEABIAB+φ2ABFQBAEABIABβAB|x=lAB=0,yA=φ1AByBA+φ2ABφBAβAB+φ3ABMBAEABIABβAB2+φ4ABFQBAEABIABβAB3|x=lAB=0,(12)yBA=MBAEABIABβAB24φ3ABφ4AB+φ1ABφ2AB4φ2ABφ3AB−4φ1ABφ4AB+FQBAEABIABβAB34φ4AB2+φ2AB24φ2ABφ3AB−4φ1ABφ4AB|x=lAB,φBA=−MBAEABIABβABφ1AB2+4φ3AB24φ2ABφ3AB−4φ1ABφ4AB−FQBAEABIABβAB24φ3ABφ4AB+φ1ABφ2AB4φ2ABφ3AB−4φ1ABφ4AB|x=lAB,where lAB is the length of AB, MAB is the bending moment at Node A along the AB direction, and yA is the vertical displacement of Node A.


#### 3.2.2. Analytical Calculation of the Structural Stress ofDEF (Elastic Combination)

The boundary conditions at NodeE of DEF (elastic combination) are shown in formulas (13)∼(15), and the equilibrium condition of the internal stress at Node E of DEF (elastic combination) is shown in formula (16). According to the boundary conditions, the initial effects of DE and EF are solved. The effects of the members are obtained considering their initial effects, which are plugged into the displacement and internal stress calculations of the members. The analytical calculation of the structural stress of the HIJ is the same as that of the DEF and is not repeated here:(13)xED=φ1DExDE+φ2DEφDEβDE+φ3DEMDEEDEIDEβDE2+φ4DEFQDEEDEIDEβDE3|y=lDE=0,yEF=φ1EFyFE+φ2EFφFEβEF+φ3EFMFEEEFIEFβEF2+φ4EFFQFEEEFIEFβEF3|x=lEF=0,where xED is the horizontal displacement at Node E along the ED direction, xDE is the initial horizontal displacement at Node D, φDE is the initial angular displacement at Node D, MDE is the initial bending moment at Node D, FQDE is the initial shear force at Node D, EDE is the elastic coefficient of DE, IDE is the section inertial moment of DE, βDE is the deformation coefficient of DE, lDE is the length of DE, yEF is the vertical displacement at Node E along the EF direction, φFE is the initial angular displacement at Node F, MFE is the initial bending moment at Node F, FQFE is the initial shear force at Node F, EEF is the elastic coefficient of EF, IEF is the section inertial moment of EF, βEF is the deformation coefficient of EF, and lEF is the length of EF.(14)φED=φEF,φED=−4βDEφ4DExDE+φ1DEφDE+φ2DEMDEEDEIDEβDE+φ3DEFQDEEDEIDEβDE2|y=lDE,φEF=−4βEFφ4EFyFE+φ1EFφFE+φ2EFMFEEEFIEFβEF+φ3EFFQFEEEFIEFβEF2|x=lEF,where φED is the angular displacement at Node E along the ED direction and φEF is the angular displacement at Node E along the EF direction.(15)∑ME=0,MED+MEF=0,MED=EDEIDE−4βDE2φ3DExDE+−4βDEφ4DEφDE+φ1DEMDEEDEIDE+φ2DEFQDEEDEIDEβDE|y=lDE,MEF=EEFIEF−4βEF2φ3EFyFE+−4βEFφ4EFφFE+φ1EFMFEEEFIEF+φ2EFFQFEEEFIEFβEF|x=lEF,where ∑ME is the resultant moment at Node E, MED is the bending moment at Node E along the ED direction, and MEF is the bending moment at Node E along the EF direction.(16)∑Fx=0,FQED+FNEF=0,∑Fy=0,FNED−FQEF=0,FQED=EDEIDE−4βDE3φ2DExDE+−4βDE2φ3DEφDE+−4βDEφ4DEMDEEDEIDE+φ1DEFQDEEDEIDE|y=lDE,FQEF=EEFIEF−4βEF3φ2EFyFE+−4βEF2φ3EFφFE+−4βEFφ4EFMFEEEFIEF+φ1EFFQFEEEFIEF|x=lEF,where ∑Fx is the resultant force at Node E along the x-direction, ∑Fy is the resultant force at Node E along the y-direction, FQED is the shear force at Node E along the ED direction, FQEF is the shear force at Node E along the EF direction, FNED is the axial force of DE, FNEF is the axial force of EF, MED is the bending moment at Node E along the ED direction, and MEF is the bending moment at Node E along the EF direction.


## 3.1. Analytical Calculation of the Structural Stress above the Slip Surfaces

### 3.1.1. Analytical Calculation of the Structural Stress ofBCD

The load decomposition ofBCD is shown in Figure 8, and the analytical calculation of the structural stress of BCD is shown in formulas (3) and (4). The analytical calculation of the structural stress of FGH is the same as that of BCD and is not repeated here.whereMBC is the bending moment at Node B along the BC direction, MCB is the bending moment at Node C along the CB direction, MDC is the bending moment at Node D along the DC direction, MCD is the bending moment at Node C along the CD direction, FQBC is the shear force at Node B along the BC direction, FQCB is the shear force at Node C along the CB direction, FQDC is the shear force at Node D along the DC direction, FQCD is the shear force at Node C along the CD direction, FNCB is the axial force of BC, FNCD is the axial force of CD, θC1 is the angular displacement of C1, θC2 is the angular displacement of C2, qBC is the rectangular load of the load decomposition of BC, qBC′ is the maximum of the triangular load of the load decomposition of BC, qCD is the rectangular load of the load decomposition of CD, qCD′ is the maximum of the triangular load of the load decomposition of CD, iBC is the linear stiffness of BC, iCD is the linear stiffness of CD, lBC is the length of BC, and lCD is the length of CD.Figure 8
The load decomposition ofBCD: (a) trapezoidal load; (b) rectangular load; (c) triangular load.(3)MBCMCBMDCMCDFQBCFQCBFQDCFQCD=2iBC2iBC−qBClBC212−qBC′lBC2204iBC4iBC−qBClBC212−qBC′lBC2302iCD2iCD−qCDlCD212−qCD′lCD2204iCD4iCD−qCDlCD212−qCD′lCD230−6iBClBC−6iBClBCqBClBC2−7qBC′lBC20−6iBClBC−6iBClBC−qBClBC2−3qBC′lBC20−6iCDlCD−6iCDlCDqCDlCD2−7qCD′lCD20−6iCDlCD−6iCDlCD−qCDlCD2−3qCD′lCD20θC1θC111,(4)θC1=−qBClBC2+qCDlCD248iBC+iCD,θC2=−qBC′lBC2+qCD′lCD2120iBC+iCD,FNCB=−FQCD,FNCD=FQCB,


### 3.1.2. Analytical Calculation of the Structural Stress ofJKLNO

The load decomposition ofJKLNO is shown in Figure 9, and the analytical calculation of the structural stress of JKLNO is shown in formulas (5)∼(8).Figure 9
The load decomposition ofJKLNO: (a) trapezoidal load; (b) rectangular load; (c) triangular load.
(a)(b)(c)(5)MJKMKJMLKMKLMKNMNKMONMNOFQJKFQKJFQLKFQKLFQKNFQNKFQONFQNO=2iJK2iJK00−qJKlJK212−qJK′lJK2204iJK4iJK00qJKlJK212+qJK′lJK2302iKL2iKL00−qKLlKL212−qKL′lKL2204iKL4iKL00qKLlKL212+qKL′lKL2304iKN4iKN2iKN2iKN−qKNlKN212−qKN′lKN2202iKN2iKN4iKN4iKNqKNlKN212+qKN′lKN230002iNO2iNO−qNOlNO212−qNO′lNO220004iNO4iNOqNOlNO212+qNO′lNO230−6iJKlJK−6iJKlJK00qJKlJK2+7qJK′lJK20−6iJKlJK−6iJKlJK00−qJKlJK2−3qJK′lJK20−6iKLlKL−6iKLlKL00qKLlKL2+7qKL′lKL20−6iKLlKL−6iKLlKL00−qKLlKL2−3qKL′lKL20−6iKNlKN−6iKNlKN−6iKNlKN−6iKNlKNqKNlKN2+7qKN′lKN20−6iKNlKN−6iKNlKN−6iKNlKN−6iKNlKN−qKNlKN2−3qKN′lKN2000−6iNOlNO−6iNOlNOqNOlNO2+7qNO′lNO2000−6iNOlNO−6iNOlNO−qNOlNO2−3qNO′lNO20θK1θK2θN1θN21,(6)4iJK+4iKL+4iKN2iKN2iKN4iKN+4iNOθK1θN1=qKNlKN2−qKLlKL2−qJKlJK212−qKNlKN2+qNOlNO212,(7)4iJK+4iKL+4iKN2iKN2iKN4iKN+4iNOθK2θN2=3qKN′lKN2−2qKL′lKL2−2qJK′lJK260−qKN′lKN2+qNO′lNO230,(8)FNKJ=−FQNO−FQKL,FNKL=FQKJ−FQKN,FNKN=FNNK=−FQNO,FNNO=FQNK,


## 3.1.1. Analytical Calculation of the Structural Stress ofBCD

The load decomposition ofBCD is shown in Figure 8, and the analytical calculation of the structural stress of BCD is shown in formulas (3) and (4). The analytical calculation of the structural stress of FGH is the same as that of BCD and is not repeated here.whereMBC is the bending moment at Node B along the BC direction, MCB is the bending moment at Node C along the CB direction, MDC is the bending moment at Node D along the DC direction, MCD is the bending moment at Node C along the CD direction, FQBC is the shear force at Node B along the BC direction, FQCB is the shear force at Node C along the CB direction, FQDC is the shear force at Node D along the DC direction, FQCD is the shear force at Node C along the CD direction, FNCB is the axial force of BC, FNCD is the axial force of CD, θC1 is the angular displacement of C1, θC2 is the angular displacement of C2, qBC is the rectangular load of the load decomposition of BC, qBC′ is the maximum of the triangular load of the load decomposition of BC, qCD is the rectangular load of the load decomposition of CD, qCD′ is the maximum of the triangular load of the load decomposition of CD, iBC is the linear stiffness of BC, iCD is the linear stiffness of CD, lBC is the length of BC, and lCD is the length of CD.Figure 8
The load decomposition ofBCD: (a) trapezoidal load; (b) rectangular load; (c) triangular load.(3)MBCMCBMDCMCDFQBCFQCBFQDCFQCD=2iBC2iBC−qBClBC212−qBC′lBC2204iBC4iBC−qBClBC212−qBC′lBC2302iCD2iCD−qCDlCD212−qCD′lCD2204iCD4iCD−qCDlCD212−qCD′lCD230−6iBClBC−6iBClBCqBClBC2−7qBC′lBC20−6iBClBC−6iBClBC−qBClBC2−3qBC′lBC20−6iCDlCD−6iCDlCDqCDlCD2−7qCD′lCD20−6iCDlCD−6iCDlCD−qCDlCD2−3qCD′lCD20θC1θC111,(4)θC1=−qBClBC2+qCDlCD248iBC+iCD,θC2=−qBC′lBC2+qCD′lCD2120iBC+iCD,FNCB=−FQCD,FNCD=FQCB,


## 3.1.2. Analytical Calculation of the Structural Stress ofJKLNO

The load decomposition ofJKLNO is shown in Figure 9, and the analytical calculation of the structural stress of JKLNO is shown in formulas (5)∼(8).Figure 9
The load decomposition ofJKLNO: (a) trapezoidal load; (b) rectangular load; (c) triangular load.
(a)(b)(c)(5)MJKMKJMLKMKLMKNMNKMONMNOFQJKFQKJFQLKFQKLFQKNFQNKFQONFQNO=2iJK2iJK00−qJKlJK212−qJK′lJK2204iJK4iJK00qJKlJK212+qJK′lJK2302iKL2iKL00−qKLlKL212−qKL′lKL2204iKL4iKL00qKLlKL212+qKL′lKL2304iKN4iKN2iKN2iKN−qKNlKN212−qKN′lKN2202iKN2iKN4iKN4iKNqKNlKN212+qKN′lKN230002iNO2iNO−qNOlNO212−qNO′lNO220004iNO4iNOqNOlNO212+qNO′lNO230−6iJKlJK−6iJKlJK00qJKlJK2+7qJK′lJK20−6iJKlJK−6iJKlJK00−qJKlJK2−3qJK′lJK20−6iKLlKL−6iKLlKL00qKLlKL2+7qKL′lKL20−6iKLlKL−6iKLlKL00−qKLlKL2−3qKL′lKL20−6iKNlKN−6iKNlKN−6iKNlKN−6iKNlKNqKNlKN2+7qKN′lKN20−6iKNlKN−6iKNlKN−6iKNlKN−6iKNlKN−qKNlKN2−3qKN′lKN2000−6iNOlNO−6iNOlNOqNOlNO2+7qNO′lNO2000−6iNOlNO−6iNOlNO−qNOlNO2−3qNO′lNO20θK1θK2θN1θN21,(6)4iJK+4iKL+4iKN2iKN2iKN4iKN+4iNOθK1θN1=qKNlKN2−qKLlKL2−qJKlJK212−qKNlKN2+qNOlNO212,(7)4iJK+4iKL+4iKN2iKN2iKN4iKN+4iNOθK2θN2=3qKN′lKN2−2qKL′lKL2−2qJK′lJK260−qKN′lKN2+qNO′lNO230,(8)FNKJ=−FQNO−FQKL,FNKL=FQKJ−FQKN,FNKN=FNNK=−FQNO,FNNO=FQNK,


## 3.2. Analytical Calculation of the Structural Stress below the Slip Surfaces

To integrate the structural stress, the horizontal and vertical antislide members are both assumed to be elastic members. The analytical calculation of the structural stress below the slip surfaces is carried out as follows.


### 3.2.1. Analytical Calculation of the Structural Stress of AB (Elastic Member)

The initial effects and stress diagram ofAB (elastic member) are shown in Figure 10. Node B on the slip surfaces is taken as the starting point, and the analytical calculation of the displacement and internal stress of the section, which is x m distance from Node B along the BA direction, is shown in formulas (9) and (10). The analytical calculation of the structural stress of LM and OP is the same as that of AB and is not repeated here.whereMJK is the bending moment at Node J along the JK direction, MKJ is the bending moment at Node K along the KJ direction, MLK is the bending moment at Node L along the LK direction, MKL is the bending moment at Node K along the KL direction, MKN is the bending moment at Node K along the KN direction, MNK is the bending moment at Node N along the NK direction, MON is the bending moment at Node O along the ON direction, MNO is the bending moment at Node N along the NO direction, FQJK is the shear force at Node J along the JK direction, FQKJ is the shear force at Node K along the KJ direction, FQLK is the shear force at Node L along the LK direction, FQKL is the shear force at Node K along the KL direction, FQKN is the shear force at Node K along the KN direction, FQNK is the shear force at Node N along the NK direction, FQON is the shear force at Node O along ON direction, FQNO is the shear force at Node N along the NO direction, FNKJ is the axial force of JK, FNKL is the axial force of KL, FNKN is the axial force of KN, FNNO is the axial force of NO, θK1 is the angular displacement of K1, θK2 is the angular displacement of K2, θN1 is the angular displacement of N1, θN2 is the angular displacement of N2, qJK is the rectangular load of the load decomposition of JK, qJK′ is the maximum of the triangular load of the load decomposition of JK, qKN is the rectangular load of the load decomposition of KN, qKN′ is the maximum of the triangular load of the load decomposition of KN, qKL is the rectangular load of the load decomposition of KL, qKL′ is the maximum of the triangular load of the load decomposition of KL, qNO is the rectangular load of the load decomposition of NO, qNO′ is the maximum of the triangular load of the load decomposition of NO, iJK is the linear stiffness of JK, iKL is the linear stiffness of KL, iKN is the linear stiffness of KN, iNO is the linear stiffness of NO, lJK is the length of JK, lKL is the length of KL, lKN is the length of KN, and lNO is the length of NO.Figure 10
The initial effects and stress diagram ofAB (elastic member): (a) the initial effects of AB (elastic member); (b) the stress diagram of AB (elastic member).(9)yx=φ1AByBA+φ2ABφBAβAB+φ3ABMBAEABIABβAB2+φ4ABFQBAEABIABβAB2,φx=−4βABφ4AByAB+φ1ABφBA+φ2ABMBAEABIABβAB+φ3ABFQBAEABIABβAB2,Mx=EABIAB−4βAB2φ3AByBA+−4βABφ4ABφBA+φ1ABMBAEABIAB+φ2ABFQBAEABIABβAB,Qx=EABIAB−4βAB2φ2AByBA+−4βAB2φ3ABφBA+−4βABφ4ABMBAEABIAB+φ1ABFQBAEABIABβAB,wherey (x) is the vertical displacement of the section that is x m distance from Node B along the BA direction, φ (x) is the angular displacement of the section that is x m distance from Node B along the BA direction, M (x) is the bending moment of the section that is x m distance from Node B along the BA direction, Q (x) is the shear force of the section that is x m distance from Node B along the BA direction, yBA is the initial vertical displacement of Node B, φBA is the initial angular displacement of Node B, MBA is the initial bending moment of Node B, FQBA is the initial shear force of Node B, EAB is the elastic coefficient of AB, IAB is the section inertial moment of AB, and βAB is the deformation coefficient of AB.As mentioned before, the boundary condition of NodeA is shown in formula (11) considering the bottom of AB (elastic member), which is a hinge support. By calculating formula (11), formula (12) is obtained. Furthermore, formula (12) is plugged into formula (9) to obtain the structural effects of AB (elastic member).(10)φ1AB=cosβABxchβABx,φ2AB=sinβABxchβABx+cosβABxshβABx2,φ3AB=sinβABxshβABx2,φ4AB=sinβABxchβABx−cosβABxshβABx4,shβABx=eβABx−e−βABx2,chβABx=eβABx+e−βABx2,βAB=KVBPAB4EABIAB1/4,where KV is the vertical foundation coefficient of the slip bed, BP(AB) is the calculation width of AB, and e is the natural exponent that is equal to 2.718281…(11)MAB=EABIAB−4βAB2φ3AByBA+−4βABφ4ABφBA+φ1ABMBAEABIAB+φ2ABFQBAEABIABβAB|x=lAB=0,yA=φ1AByBA+φ2ABφBAβAB+φ3ABMBAEABIABβAB2+φ4ABFQBAEABIABβAB3|x=lAB=0,(12)yBA=MBAEABIABβAB24φ3ABφ4AB+φ1ABφ2AB4φ2ABφ3AB−4φ1ABφ4AB+FQBAEABIABβAB34φ4AB2+φ2AB24φ2ABφ3AB−4φ1ABφ4AB|x=lAB,φBA=−MBAEABIABβABφ1AB2+4φ3AB24φ2ABφ3AB−4φ1ABφ4AB−FQBAEABIABβAB24φ3ABφ4AB+φ1ABφ2AB4φ2ABφ3AB−4φ1ABφ4AB|x=lAB,where lAB is the length of AB, MAB is the bending moment at Node A along the AB direction, and yA is the vertical displacement of Node A.


### 3.2.2. Analytical Calculation of the Structural Stress ofDEF (Elastic Combination)

The boundary conditions at NodeE of DEF (elastic combination) are shown in formulas (13)∼(15), and the equilibrium condition of the internal stress at Node E of DEF (elastic combination) is shown in formula (16). According to the boundary conditions, the initial effects of DE and EF are solved. The effects of the members are obtained considering their initial effects, which are plugged into the displacement and internal stress calculations of the members. The analytical calculation of the structural stress of the HIJ is the same as that of the DEF and is not repeated here:(13)xED=φ1DExDE+φ2DEφDEβDE+φ3DEMDEEDEIDEβDE2+φ4DEFQDEEDEIDEβDE3|y=lDE=0,yEF=φ1EFyFE+φ2EFφFEβEF+φ3EFMFEEEFIEFβEF2+φ4EFFQFEEEFIEFβEF3|x=lEF=0,where xED is the horizontal displacement at Node E along the ED direction, xDE is the initial horizontal displacement at Node D, φDE is the initial angular displacement at Node D, MDE is the initial bending moment at Node D, FQDE is the initial shear force at Node D, EDE is the elastic coefficient of DE, IDE is the section inertial moment of DE, βDE is the deformation coefficient of DE, lDE is the length of DE, yEF is the vertical displacement at Node E along the EF direction, φFE is the initial angular displacement at Node F, MFE is the initial bending moment at Node F, FQFE is the initial shear force at Node F, EEF is the elastic coefficient of EF, IEF is the section inertial moment of EF, βEF is the deformation coefficient of EF, and lEF is the length of EF.(14)φED=φEF,φED=−4βDEφ4DExDE+φ1DEφDE+φ2DEMDEEDEIDEβDE+φ3DEFQDEEDEIDEβDE2|y=lDE,φEF=−4βEFφ4EFyFE+φ1EFφFE+φ2EFMFEEEFIEFβEF+φ3EFFQFEEEFIEFβEF2|x=lEF,where φED is the angular displacement at Node E along the ED direction and φEF is the angular displacement at Node E along the EF direction.(15)∑ME=0,MED+MEF=0,MED=EDEIDE−4βDE2φ3DExDE+−4βDEφ4DEφDE+φ1DEMDEEDEIDE+φ2DEFQDEEDEIDEβDE|y=lDE,MEF=EEFIEF−4βEF2φ3EFyFE+−4βEFφ4EFφFE+φ1EFMFEEEFIEF+φ2EFFQFEEEFIEFβEF|x=lEF,where ∑ME is the resultant moment at Node E, MED is the bending moment at Node E along the ED direction, and MEF is the bending moment at Node E along the EF direction.(16)∑Fx=0,FQED+FNEF=0,∑Fy=0,FNED−FQEF=0,FQED=EDEIDE−4βDE3φ2DExDE+−4βDE2φ3DEφDE+−4βDEφ4DEMDEEDEIDE+φ1DEFQDEEDEIDE|y=lDE,FQEF=EEFIEF−4βEF3φ2EFyFE+−4βEF2φ3EFφFE+−4βEFφ4EFMFEEEFIEF+φ1EFFQFEEEFIEF|x=lEF,where ∑Fx is the resultant force at Node E along the x-direction, ∑Fy is the resultant force at Node E along the y-direction, FQED is the shear force at Node E along the ED direction, FQEF is the shear force at Node E along the EF direction, FNED is the axial force of DE, FNEF is the axial force of EF, MED is the bending moment at Node E along the ED direction, and MEF is the bending moment at Node E along the EF direction.


## 3.2.1. Analytical Calculation of the Structural Stress of AB (Elastic Member)

The initial effects and stress diagram ofAB (elastic member) are shown in Figure 10. Node B on the slip surfaces is taken as the starting point, and the analytical calculation of the displacement and internal stress of the section, which is x m distance from Node B along the BA direction, is shown in formulas (9) and (10). The analytical calculation of the structural stress of LM and OP is the same as that of AB and is not repeated here.whereMJK is the bending moment at Node J along the JK direction, MKJ is the bending moment at Node K along the KJ direction, MLK is the bending moment at Node L along the LK direction, MKL is the bending moment at Node K along the KL direction, MKN is the bending moment at Node K along the KN direction, MNK is the bending moment at Node N along the NK direction, MON is the bending moment at Node O along the ON direction, MNO is the bending moment at Node N along the NO direction, FQJK is the shear force at Node J along the JK direction, FQKJ is the shear force at Node K along the KJ direction, FQLK is the shear force at Node L along the LK direction, FQKL is the shear force at Node K along the KL direction, FQKN is the shear force at Node K along the KN direction, FQNK is the shear force at Node N along the NK direction, FQON is the shear force at Node O along ON direction, FQNO is the shear force at Node N along the NO direction, FNKJ is the axial force of JK, FNKL is the axial force of KL, FNKN is the axial force of KN, FNNO is the axial force of NO, θK1 is the angular displacement of K1, θK2 is the angular displacement of K2, θN1 is the angular displacement of N1, θN2 is the angular displacement of N2, qJK is the rectangular load of the load decomposition of JK, qJK′ is the maximum of the triangular load of the load decomposition of JK, qKN is the rectangular load of the load decomposition of KN, qKN′ is the maximum of the triangular load of the load decomposition of KN, qKL is the rectangular load of the load decomposition of KL, qKL′ is the maximum of the triangular load of the load decomposition of KL, qNO is the rectangular load of the load decomposition of NO, qNO′ is the maximum of the triangular load of the load decomposition of NO, iJK is the linear stiffness of JK, iKL is the linear stiffness of KL, iKN is the linear stiffness of KN, iNO is the linear stiffness of NO, lJK is the length of JK, lKL is the length of KL, lKN is the length of KN, and lNO is the length of NO.Figure 10
The initial effects and stress diagram ofAB (elastic member): (a) the initial effects of AB (elastic member); (b) the stress diagram of AB (elastic member).(9)yx=φ1AByBA+φ2ABφBAβAB+φ3ABMBAEABIABβAB2+φ4ABFQBAEABIABβAB2,φx=−4βABφ4AByAB+φ1ABφBA+φ2ABMBAEABIABβAB+φ3ABFQBAEABIABβAB2,Mx=EABIAB−4βAB2φ3AByBA+−4βABφ4ABφBA+φ1ABMBAEABIAB+φ2ABFQBAEABIABβAB,Qx=EABIAB−4βAB2φ2AByBA+−4βAB2φ3ABφBA+−4βABφ4ABMBAEABIAB+φ1ABFQBAEABIABβAB,wherey (x) is the vertical displacement of the section that is x m distance from Node B along the BA direction, φ (x) is the angular displacement of the section that is x m distance from Node B along the BA direction, M (x) is the bending moment of the section that is x m distance from Node B along the BA direction, Q (x) is the shear force of the section that is x m distance from Node B along the BA direction, yBA is the initial vertical displacement of Node B, φBA is the initial angular displacement of Node B, MBA is the initial bending moment of Node B, FQBA is the initial shear force of Node B, EAB is the elastic coefficient of AB, IAB is the section inertial moment of AB, and βAB is the deformation coefficient of AB.As mentioned before, the boundary condition of NodeA is shown in formula (11) considering the bottom of AB (elastic member), which is a hinge support. By calculating formula (11), formula (12) is obtained. Furthermore, formula (12) is plugged into formula (9) to obtain the structural effects of AB (elastic member).(10)φ1AB=cosβABxchβABx,φ2AB=sinβABxchβABx+cosβABxshβABx2,φ3AB=sinβABxshβABx2,φ4AB=sinβABxchβABx−cosβABxshβABx4,shβABx=eβABx−e−βABx2,chβABx=eβABx+e−βABx2,βAB=KVBPAB4EABIAB1/4,where KV is the vertical foundation coefficient of the slip bed, BP(AB) is the calculation width of AB, and e is the natural exponent that is equal to 2.718281…(11)MAB=EABIAB−4βAB2φ3AByBA+−4βABφ4ABφBA+φ1ABMBAEABIAB+φ2ABFQBAEABIABβAB|x=lAB=0,yA=φ1AByBA+φ2ABφBAβAB+φ3ABMBAEABIABβAB2+φ4ABFQBAEABIABβAB3|x=lAB=0,(12)yBA=MBAEABIABβAB24φ3ABφ4AB+φ1ABφ2AB4φ2ABφ3AB−4φ1ABφ4AB+FQBAEABIABβAB34φ4AB2+φ2AB24φ2ABφ3AB−4φ1ABφ4AB|x=lAB,φBA=−MBAEABIABβABφ1AB2+4φ3AB24φ2ABφ3AB−4φ1ABφ4AB−FQBAEABIABβAB24φ3ABφ4AB+φ1ABφ2AB4φ2ABφ3AB−4φ1ABφ4AB|x=lAB,where lAB is the length of AB, MAB is the bending moment at Node A along the AB direction, and yA is the vertical displacement of Node A.


## 3.2.2. Analytical Calculation of the Structural Stress ofDEF (Elastic Combination)

The boundary conditions at NodeE of DEF (elastic combination) are shown in formulas (13)∼(15), and the equilibrium condition of the internal stress at Node E of DEF (elastic combination) is shown in formula (16). According to the boundary conditions, the initial effects of DE and EF are solved. The effects of the members are obtained considering their initial effects, which are plugged into the displacement and internal stress calculations of the members. The analytical calculation of the structural stress of the HIJ is the same as that of the DEF and is not repeated here:(13)xED=φ1DExDE+φ2DEφDEβDE+φ3DEMDEEDEIDEβDE2+φ4DEFQDEEDEIDEβDE3|y=lDE=0,yEF=φ1EFyFE+φ2EFφFEβEF+φ3EFMFEEEFIEFβEF2+φ4EFFQFEEEFIEFβEF3|x=lEF=0,where xED is the horizontal displacement at Node E along the ED direction, xDE is the initial horizontal displacement at Node D, φDE is the initial angular displacement at Node D, MDE is the initial bending moment at Node D, FQDE is the initial shear force at Node D, EDE is the elastic coefficient of DE, IDE is the section inertial moment of DE, βDE is the deformation coefficient of DE, lDE is the length of DE, yEF is the vertical displacement at Node E along the EF direction, φFE is the initial angular displacement at Node F, MFE is the initial bending moment at Node F, FQFE is the initial shear force at Node F, EEF is the elastic coefficient of EF, IEF is the section inertial moment of EF, βEF is the deformation coefficient of EF, and lEF is the length of EF.(14)φED=φEF,φED=−4βDEφ4DExDE+φ1DEφDE+φ2DEMDEEDEIDEβDE+φ3DEFQDEEDEIDEβDE2|y=lDE,φEF=−4βEFφ4EFyFE+φ1EFφFE+φ2EFMFEEEFIEFβEF+φ3EFFQFEEEFIEFβEF2|x=lEF,where φED is the angular displacement at Node E along the ED direction and φEF is the angular displacement at Node E along the EF direction.(15)∑ME=0,MED+MEF=0,MED=EDEIDE−4βDE2φ3DExDE+−4βDEφ4DEφDE+φ1DEMDEEDEIDE+φ2DEFQDEEDEIDEβDE|y=lDE,MEF=EEFIEF−4βEF2φ3EFyFE+−4βEFφ4EFφFE+φ1EFMFEEEFIEF+φ2EFFQFEEEFIEFβEF|x=lEF,where ∑ME is the resultant moment at Node E, MED is the bending moment at Node E along the ED direction, and MEF is the bending moment at Node E along the EF direction.(16)∑Fx=0,FQED+FNEF=0,∑Fy=0,FNED−FQEF=0,FQED=EDEIDE−4βDE3φ2DExDE+−4βDE2φ3DEφDE+−4βDEφ4DEMDEEDEIDE+φ1DEFQDEEDEIDE|y=lDE,FQEF=EEFIEF−4βEF3φ2EFyFE+−4βEF2φ3EFφFE+−4βEFφ4EFMFEEEFIEF+φ1EFFQFEEEFIEF|x=lEF,where ∑Fx is the resultant force at Node E along the x-direction, ∑Fy is the resultant force at Node E along the y-direction, FQED is the shear force at Node E along the ED direction, FQEF is the shear force at Node E along the EF direction, FNED is the axial force of DE, FNEF is the axial force of EF, MED is the bending moment at Node E along the ED direction, and MEF is the bending moment at Node E along the EF direction.


## 4. Engineering Case

### 4.1. Engineering Overview

Taking the ASCL of the Houzishi landslides as an example, the stability and internal stress of the antislide structures are calculated and analysed by taking 1 m (unit width) of the sliding bodies along the transverse direction of the landslides. The layout of the ASCL in the Houzishi landslides is shown in Figure11.Figure 11
The layout of the ASCL in the Houzishi landslide. According to the unbalanced thrust method, the sliding body is divided into 21 sticks to calculate the landslide thrust and resistance along the vertical direction. The horizontal axis represents the horizontal length of the landslide, while the vertical axis represents the Yellow Sea elevation of the landslide.Landslides are caused by the increasing sliding force of the sliding bodies and the decreasing mechanical properties of the geotechnical materials. The total load of a landslide is assumed to be the dead weight of the sliding bodies plus the ground surface load, which is equal to 100 kN/m.The sliding bodies are simplified to be massive marls, the sliding zones are simplified to be plastic clays, and the slip beds are simplified to be massive carbonaceous sandstones. The vertical foundation coefficient of the slip beds (KV) is established to be 875000 kN/m3, and the horizontal foundation coefficient of the slip beds (KH) is established to be 612500 kN/m3. The mechanical parameters of the geotechnical landslides are shown in Table 1.Table 1
The mechanical parameters of the geotechnical landslides.
Landslide locationMaterialDensity (kN/m3)Elastic modulus (MPa)Compressive strength (MPa)Tensile strength (MPa)Cohesion (MPa)Angle of internal friction (°)Poisson’s ratioSliding bodiesMarls233800201.40.5350.3Sliding zonePlastic clays18.516——0.03200.35Slip bedsCarbonaceous sandstones251100083.752.4751.1440.26As mentioned before, simplifying the horizontal and vertical antislide members below the slip surfaces to be beams under the foundation and embedded in the slip bed, the analytical calculation of the structural stress is carried out. The mechanical parameters of the antislide members are shown in Table2. The judgement method of the mechanical properties of the members embedded in the slip bed is shown in the following formula:where β is the deformation coefficient of the member, K is the foundation coefficient of the slip bed, KV is the vertical foundation coefficient of the slip bed, KH is the horizontal foundation coefficient of the slip bed, h2 is the anchorage depth of the member that is embedded in the slip bed, and BP is the calculation width of the member.Table 2
The mechanical parameters of the antislide members.
Sequence numberMemberMember length (m)Partition lengthSection size (length × width, m); mechanical properties of the membersStrength of concrete; Poisson’s ratioStrength of rebarElastic modulus (MPa)Partition memberLength (m)1AC9.434AB4.7173 × 3; (elastic member)C40; 0.2HRB500; HRB4002.6 × 104BC4.7173 × 3; (−)2CE10.000CD5.0023 × 3; (−)DE4.9983 × 3; (elastic member)3EG15.752EF7.8763 × 3; (elastic member)FG7.8763 × 3; (−)4GI10.000GH4.4793 × 3; (−)HI5.5213 × 3; (elastic member)5IN30.624IJ9.7083 × 3; (elastic member)JK9.7083 × 3; (−)KN11.2083 × 3; (−)6KM12.624KL5.5623 × 3; (−)LM7.0623 × 3; (elastic member)7NP22.414NO10.4573 × 3; (−)OP11.9573 × 3; (elastic member)(17)β=KBP4EI1/4,rigid members:βh2≤1.0,elastic members:βh2>1.0,


### 4.2. Landslide Thrust and Resistance in front of the Structures

#### 4.2.1. Landslide Thrust

The implicit solution of the unbalanced thrust method (transfer coefficient method) is an iterative solution. The landslide body is balanced to solve the stability factor (Fs) by steadily reducing the strength of the geotechnical material at the bottom of sliding bodies. The solution is determined as follows: first, the sliding force of the left side of the first stick on the top of a sliding body (F0) is assumed to be zero; second, the landslide thrust of the other stick is calculated according to formulas (18) and (19) until the landslide thrust of the nth stick (Fn) is equal to zero. If Fn is not equal to zero, Fs, which is the actual stability factor, should be adjusted until Fn is equal to zero [32].According to the implicit solution of the unbalanced thrust method (transfer coefficient method), the landslide thrust and the sliding resistance of each stick in the slip bodies can be calculated. The landslide thrust of the antislide members is calculated by the sliding force, which is projected onto the horizontal and vertical antislide members.(18)Fi=Wisinαi−Wicosαitanφi+cili+Ψi−1Fi−1,Ψi−1=cosαi−1−αi−sinαi−1−αitanφi,(19)Fi=Ti−RiFs+Ψi−1′Fi−1,Ti=Wisinαi,Ri=Wicosαitanφi+cili,Ψi−1′=cosαi−1−αi−sinαi−1−αitanφiFs,where Fi is the landslide thrust of the ith slice, Fi−1 is the landslide thrust of the (i − 1)th slice, Wi is the gravity of the ith slice, αi is the horizontal angle of the bottom of the ith slice, αi−1 is the horizontal angle of the bottom of the (i − 1)th slice, φi is the angle of internal friction of the bottom of the ith slice, ci is the cohesion of the bottom of the ith slice, li is the length of the bottom of the ith slice, ψi−1 is the transfer coefficient, Ψi−1′ is the reduced transfer coefficient, Ti is the sliding force of the ith slice, Ri is the sliding resistance of the ith slice, and Fs is the stability factor.


#### 4.2.2. Resistance in front of the Structures

The passive earth pressure in front of the structures is described in formula (20) [33]. An infinite passive earth pressure is not realistic, even if the ground above the structures is close to horizontal and the friction between the antislide structures and the geotechnical is ignored. Therefore, the resistance of the horizontal antislide members is set to be the horizontal projection of the excess sliding force in the case of ultimate status [34].(20)EP=12γ1h12λ0,λ0=cos2φ1+αcos2α⋅cosα−δ1+sinφ1+δ⋅sinφ1+i/cosα−δ⋅cosα−i2,where EP is the passive earth pressure in front of the structure, γ1 is the unit weight of the geotechnical material in front of the structure, h1 is the stress height of the structure in the sliding body, λ0 is the coefficient of passive earth pressure of the structure, φ1 is the angle of internal friction in front of the structure, δ is the friction angle of the structural back, α is the intersection angle between the back and vertical directions of the structure, and i is the ground angle of the structure.


#### 4.2.3. Calculation Results of the Sliding Force and Resistance

By using the implicit solution of the unbalanced thrust method (transfer coefficient method), the distributed line load on the antislide members is determined and shown in Table3.Table 3
The distributed line load on the antislide members.
Sequence numberStructureMemberMember length (m)Sliding force (kN)Sliding resistance (kN)Distributed line load (kN/m)1BCDBC4.71721259.82819479.445377.440CD5.00220049.66659.5773996.4192FGHFG7.87616255.55414355.040241.304GH4.47928583.57860.6376368.1493JKLJK9.70820640.66917914.429280.824KL5.56235912.62764.4576445.1944KNOKN11.20817176.50714598.502230.015NO10.45739324.825256.2123736.121


### 4.3. Elastic Resistance below the Slip Surfaces

The analytical solution of the elastic resistance below the slip surfaces is shown in formula (21), and the numerical solution of the interaction between the antislide members and slip bed is shown in formula (22):(21)σx=−KVBPyx,σy=−KHBPxy,(22)qx=dQydy,qy=dQxdx,where σx is the analytical calculation of the horizontal elastic resistance below the slip surfaces, σy is the analytical calculation of the vertical elastic resistance below the slip surfaces, qx is the numerical calculation of the horizontal interaction between the antislide members and geotechnical materials, qy is the numerical calculation of the vertical interaction between the antislide members and geotechnical materials, dQ (x) is the horizontal shear force difference of the microelement on the antislide member, dQ (y) is the vertical shear force difference of the microelement on the antislide member, dx is the horizontal length of the microelement that is equal to 0.5 m on the antislide member by using numerical simulation software (MIDAS GTS NX), and dy is the vertical length of the microelement that is equal to 0.5 m on the antislide member by using numerical simulation software (MIDAS GTS NX).


### 4.4. Comparative Analysis of the Structural Effects

#### 4.4.1. Comparative Analysis of the Displacement

Due to the small structural displacements, the initial location and displacements of the structures cannot be distinguished with the numerical simulation at the original scaling. Taking the shape of the ASCL as a benchmark, the analytical calculation and the numerical simulation of the displacements were enlarged at different scales. A comparative figure of the enlarged displacements is given in Figure12. In Figure 12, the global displacements caused deformation toward downwards to the right, and the results of the analytical calculation of the structural displacements were similar to the distribution of the numerical results, showing that the ASCL was bearing the landslide thrust and the resistance of the geotechnical materials.Figure 12
A comparative figure of the enlarged displacements. The black lines represent the initial shape of the ASCL, and the green lines represent the slip surface. Taking the initial shape of the ASCL as a standard, the red lines represent the displacements of the analytical calculations, while the pink lines represent the displacements of the numerical calculations.A comparative table of the structural displacements of the antislide members is given in Table4. In Table 4 and the following tables, E was equal to 10. In Table 4, the structural displacement results of the analytical calculation and numerical simulation were different due to the different calculation methods and assumptions. When the structural mechanics and elastic foundation beam method were adopted to analytically calculate the structural displacements, the minor loads that included the gravity of the antislide members, friction, and end-bearing force between the antislide members and geotechnical materials were ignored, and the resulting structural displacements of the horizontal antislide members presented only vertical displacements, without horizontal displacements, while that of the vertical antislide members presented only horizontal displacements, without vertical displacements. Otherwise, due to the minor loads that included the gravity of antislide members, the interaction between the antislide members and geotechnical materials, and the effects that were simulated as the intercoupling between the antislide members and geotechnical materials, the maximum numerically determined structural displacements were greater than the maximum analytically calculated structural displacements. The corresponding ratios of the maximum displacements of the global structures were equal to 0.820 for the horizontal displacement, 0.031 for the vertical displacement, and 4.342 for the angular displacement. Thus, the numerical simulation of the structural displacements was proposed to be adopted.Table 4
A comparative table of the structural displacements of the antislide members.
MemberMaximum analytical calculation of horizontal displacement Dx (m)Maximum numerical simulation of horizontal displacement Dx′ (m)Dx/Dx′Maximum analytical calculation of vertical displacement Dy (m)Maximum numerical simulation of vertical displacement Dy′ (m)Dy/Dy′Maximum analytical calculation of angular displacement Dr (rad)Maximum numerical simulation of angular displacement Dr′ (rad)Dr/Dr′AC06.626E − 030−6.047E − 04−1.747E − 020.0351.795E − 043.235E − 040.555CE2.270E − 036.623E − 030.3430−1.746E − 0207.741E − 041.155E − 046.702EG05.914E − 030−4.737E − 04−1.758E − 020.0272.215E − 041.607E − 041.378GI3.408E − 036.026E − 030.5660−1.679E − 0201.104E − 031.194E − 049.246IN05.271E − 030−4.720E − 04−1.941E − 020.0242.564E − 044.473E − 040.573KM4.162E − 035.249E − 030.7930−1.858E − 0201.216E − 032.537E − 044.793NP5.431E − 035.459E − 030.9950−1.652E − 0201.942E − 032.965E − 046.550Total5.431E − 036.626E − 030.820−6.047E − 04−1.941E − 020.0311.942E − 034.473E − 044.342


#### 4.4.2. Comparative Analysis of the Axial Force

As mentioned before, the global displacements presented a deformation downwards to the right, resulting in an apparent domino effect. This domino effect is shown in Figure13, and a comparison of the axial forces of the ASCL is shown in Figure 14. According to Figures 13 and 14, the trend of the analytical calculation of the structural axial force was close to the trend of the numerical simulation of the structural axial force. The analytical calculation and numerical simulation of the axial force of the horizontal antislide member indicated mostly axial tension, while the analytical calculation and numerical simulation of the axial force of the vertical antislide members indicated mostly axial pressure, confirming that the global displacements reflected deformation downwards to the right. The horizontal antislide members acted as tie rods, while the vertical antislide members bearing the resultant force between the thrust and the sliding resistance presented deformation downwards to the right. Notably, the horizontal antislide members that were not strictly acting as tie rods bore the axial force, the shear force, and the bending moment.Figure 13
The domino effect. The domino effect occurs when several dominoes are arranged in rows according to certain spacing; when the first domino is pushed over, the other dominoes behind it will be pushed over in a chain reaction.Figure 14
A comparison of the axial forces of the ASCL. The black lines represent the initial shape of the ASCL, and the green lines represent the slip surface. Taking the initial shape of the ASCL as a standard, the red lines represent the axial force of the analytical calculations, while the pink lines represent the axial force of the numerical calculations.A comparative table of the axial force of the antislide members is given in Table5. In Table 5, the analytical calculation result of the axial force was different from the numerical simulation result of axial force due to the different calculation methods and assumptions. The ratio of the maximum axial forces of the global structures was equal to 10.755, while the ratio of the minimum axial forces of the global structures was equal to 0.176. Based on the principle of internal stress envelope diagrams, the maximum result from the analytical calculation of the axial force, which was equal to 33800.857 kN, and the minimum result from the numerical simulation of the axial force, which was equal to −39304.948 kN, were proposed to be adopted.Table 5
A comparative table of the axial force of the antislide members.
MemberMaximum analytical calculation of axial forceFN (kN)Maximum numerical simulation of axial forceFN′ (kN)FN/FN′Minimum analytical calculation of axial forceFn (kN)Minimum numerical simulation of axial forceFn′ (kN)Fn/Fn′AC8680.4443142.7642.7628680.444−1160.419−7.480CE588.064−11339.797−0.052−6338.100−17527.280.362EG11722.3411284.8289.1211859.380−463.996−4.007GI−129.082−10593.5920.012−6931.905−22643.8620.306IN33800.8572133.85915.8404134.072−1777.112−2.326KM−4862.311−9423.5720.516−4862.311−27649.1740.176NP1256.725−6426.624−0.1961256.725−39304.948−0.032Total33800.8573142.76410.755−6931.905−39304.9480.176


#### 4.4.3. Comparative Analysis of the Shear Force

A comparative figure of the shear force of the ASCL is shown in Figure15. In Figure 15, the trend of the analytical calculation results of the structural shear force was close to the trend of the numerical simulation results of the structural shear force. The horizontal antislide members bore a positive shear force due to the vertical loads, while the top of the vertical antislide members bore a negative shear force due to the horizontal antislide members acting as tie rods. However, the negative shear force of the vertical antislide members decreased gradually and became a positive shear force because of the resultant force due to the thrust and the sliding resistance. Assuming that a slip surface is a boundary in the ASCL, the shear force of the members close to the slip surfaces was very high, and the shear force direction of the members above the slip surfaces and below the slip surfaces was opposite due to the interaction between the antislide members and geotechnical materials below the slip surfaces, which can decrease the shear force of the members above the slip surfaces.Figure 15
A comparative figure of the shear force of the ASCL. The black lines represent the initial shape of the ASCL, and the green lines represent the slip surface. Taking the initial shape of the ASCL as a standard, the red lines represent the shear force of the analytical calculations, while the pink lines represent the shear force of the numerical calculations.A comparative table of the shear force of the antislide members is given in Table6. The analytical calculation result of the shear force was different from the numerical simulation result of the shear force due to the different calculation methods and assumptions. In Table 6, the ratio of the maximum shear forces of the global structures was equal to 2.061, while the ratio of the minimum shear forces of the global structures was equal to 1.035. The analytical calculation result of the structural shear force exhibited a hysteretic effect, unlike the numerical simulation result of the structural shear force, as the maximum result of the analytical calculation of the shear force was observed behind the members and the maximum result of the numerical simulation of the shear force was observed in the middle of the members. Based on the principle of internal stress envelope diagrams, the maximum shear force from the analytical calculation, which was equal to 22169.628 kN, and the minimum shear force from the analytical calculation, which was equal to −16901.873 kN, were proposed to be adopted.Table 6
A comparative table of the shear force of the antislide members.
MemberMaximum analytical calculation of shear forceFQ (kN)Maximum numerical simulation of shear forceFQ′ (kN)FQ/FQ′Minimum analytical calculation of shear forceFq (kN)Minimum numerical simulation of shear forceFq′ (kN)Fq/Fq′AC2368.4477591.1860.312−2010.671−9953.1840.202CE11309.6452135.7935.295−8680.444−255.00334.041EG1771.4324952.2780.358−6338.100−8730.5150.726GI16800.6013912.3114.294−11722.341391.741−29.924IN3834.73010757.6620.356−6931.905−16337.6650.424KM18946.2971770.99610.698−16901.873−809.66320.875NP22169.6283380.7616.558−16898.984−2215.3487.628Total22169.62810757.6622.061−16901.873−16337.6651.035


#### 4.4.4. Comparative Analysis of the Bending Moment

The comparative figure of the bending moment of the ASCL is shown in Figure16. In Figure 16, the direction of the analytical calculation result of the structural bending moment was contrary to the direction of the numerical simulation result of the structural bending moment at the top junction between the horizontal antislide members and the vertical antislide members due to the difference between the analytical calculation, which was a piecewise calculation, and the numerical simulation, which was a global calculation. The trend of the analytical calculation of the structural bending moment approached the trend of the numerical simulation of the structural bending moment, while the location of the member section gradually moved to the bottom joints of the vertical antislide members from the top joints between the horizontal antislide members and the vertical antislide members. Assuming that a slip surface is a boundary in the ASCL, the bending moments near the slip surfaces were larger. The top outer edges and the bottom outer edges of the horizontal antislide members and the vertical antislide members easily exhibited tensile bending moments due to the global displacements that resulted in the apparent domino effect; however, the bending moment reversed due to the interaction between the antislide members and geotechnical materials.Figure 16
The comparative figure of the bending moment of the ASCL. The black lines represent the initial shape of the ASCL, and the green lines represent the slip surface. Taking the initial shape of the ASCL as a standard, the red lines represent the bending moment of the analytical calculations, while the pink lines represent the bending moment of the numerical calculations.A comparative table of the bending moments of the antislide members is shown in Table7. In Table 7, the ratio of the maximum bending moments of the global structures was equal to 0.964, while the ratio of the minimum bending moments of the global structures was equal to 2.821. For the same reason, the analytical calculation result of the structural bending moment exhibited a hysteretic effect, unlike the numerical simulation result of the structural bending moment. Based on the principle of internal stress envelope diagrams, the maximum bending moment from the numerical simulation, which was 23385.630 kN·m, and the minimum bending moment from the analytical calculation, which was appropriately decreased to −36901.790 kN·m, were proposed to be adopted.Table 7
A comparative table of the bending moments of the antislide members.
MemberMaximum analytical calculation of bending momentM (kN·m)Maximum numerical simulation of bending momentM′ (kN·m)M/M′Minimum analytical calculation of bending momentm (kN·m)Minimum numerical simulation of bending momentm′ (kN·m)m/m′AC4521.4715658.9110.799−3948.781−14354.3600.275CE5419.2692605.4322.080−23265.751−6865.6373.389EG4388.1889634.8920.455−20468.400−12945.6021.581GI7644.0378805.7040.868−32908.479−8560.6543.844IN12859.18423385.6300.550−24614.286−22101.5421.114KM9288.1865922.7151.568−36901.7900+∞NP22543.87723385.6300.964−62354.262−4332.75614.391Total22543.87723385.6300.964−62354.262−22101.5422.821


#### 4.4.5. Comparative Analysis of the Interaction between the Members and Geotechnical Materials

A comparative figure of the interaction between the antislide members and geotechnical materials is shown in Figure17. In Figure 17, the trend of analytical calculation result of the interaction between the antislide members and geotechnical materials approached the trend of the numerical simulation result of the interaction between the antislide members and geotechnical materials. Assuming that a slip surface is a boundary in the ASCL, the interaction between the antislide members and geotechnical materials on the slip surfaces was greater, the horizontal antislide members above the slip surfaces bare the downward load, while the vertical antislide members above the slip surfaces bare the rightward load. However, the interaction below the slip surfaces was reversed and decreased gradually due to the anchoring effects between the antislide members below the slip surfaces and the geotechnical materials on the slip bed.Figure 17
A comparative figure of the interaction between the antislide members and geotechnical materials. The black lines represent the initial shape of the ASCL, and the green lines represent the slip surface. Taking the initial shape of the ASCL as a standard, the red lines represent the interaction between the antislide members and geotechnical materials of the analytical calculations, while the pink lines represent the interaction between the antislide members and geotechnical materials of the numerical calculations.A comparative table of the interaction between the antislide members and geotechnical materials is shown in Table8. In Table 8, the maximum numerical simulation of the interaction between the antislide members and geotechnical materials was equal to 44410.500 kN/m, so larger values were unreliable and should be eliminated if there was no approximate numerical simulation result provided nearby. Based on the principle of internal stress envelope diagrams, the maximum interaction between the antislide members and geotechnical materials from the numerical simulation, which was equal to 19906.4 kN/m, and the minimum interaction between the antislide members and geotechnical materials from the numerical simulation, which was equal to −19826.88 kN/m, were proposed to be adopted.Table 8
A comparative table of the interaction between the antislide members and geotechnical materials.
MemberMaximum analytical calculation of interactionQ (kN/m)Maximum numerical simulation of interactionQ′ (kN/m)Q/Q′Minimum analytical calculation of interactionq (kN/m)Minimum numerical simulation of interactionq′ (kN/m)q/q′AC2116.45019906.4000.106−377.440−19826.8800.019CE3996.4194144.9500.964−5561.500−2786.0021.996EG1657.9506617.8000.251−241.304−17461.0300.014GI6368.1496823.8900.933−8349.600−5572.6401.498IN1185.10044410.5000.027−280.824−18480.4100.015KM6445.1942584.8502.493−10196.900−1224.2888.329NP3736.1215745.2900.650−13305.950−5764.5002.308Total6445.19444410.5000.145−13305.950−19826.8800.671


## 4.1. Engineering Overview

Taking the ASCL of the Houzishi landslides as an example, the stability and internal stress of the antislide structures are calculated and analysed by taking 1 m (unit width) of the sliding bodies along the transverse direction of the landslides. The layout of the ASCL in the Houzishi landslides is shown in Figure11.Figure 11
The layout of the ASCL in the Houzishi landslide. According to the unbalanced thrust method, the sliding body is divided into 21 sticks to calculate the landslide thrust and resistance along the vertical direction. The horizontal axis represents the horizontal length of the landslide, while the vertical axis represents the Yellow Sea elevation of the landslide.Landslides are caused by the increasing sliding force of the sliding bodies and the decreasing mechanical properties of the geotechnical materials. The total load of a landslide is assumed to be the dead weight of the sliding bodies plus the ground surface load, which is equal to 100 kN/m.The sliding bodies are simplified to be massive marls, the sliding zones are simplified to be plastic clays, and the slip beds are simplified to be massive carbonaceous sandstones. The vertical foundation coefficient of the slip beds (KV) is established to be 875000 kN/m3, and the horizontal foundation coefficient of the slip beds (KH) is established to be 612500 kN/m3. The mechanical parameters of the geotechnical landslides are shown in Table 1.Table 1
The mechanical parameters of the geotechnical landslides.
Landslide locationMaterialDensity (kN/m3)Elastic modulus (MPa)Compressive strength (MPa)Tensile strength (MPa)Cohesion (MPa)Angle of internal friction (°)Poisson’s ratioSliding bodiesMarls233800201.40.5350.3Sliding zonePlastic clays18.516——0.03200.35Slip bedsCarbonaceous sandstones251100083.752.4751.1440.26As mentioned before, simplifying the horizontal and vertical antislide members below the slip surfaces to be beams under the foundation and embedded in the slip bed, the analytical calculation of the structural stress is carried out. The mechanical parameters of the antislide members are shown in Table2. The judgement method of the mechanical properties of the members embedded in the slip bed is shown in the following formula:where β is the deformation coefficient of the member, K is the foundation coefficient of the slip bed, KV is the vertical foundation coefficient of the slip bed, KH is the horizontal foundation coefficient of the slip bed, h2 is the anchorage depth of the member that is embedded in the slip bed, and BP is the calculation width of the member.Table 2
The mechanical parameters of the antislide members.
Sequence numberMemberMember length (m)Partition lengthSection size (length × width, m); mechanical properties of the membersStrength of concrete; Poisson’s ratioStrength of rebarElastic modulus (MPa)Partition memberLength (m)1AC9.434AB4.7173 × 3; (elastic member)C40; 0.2HRB500; HRB4002.6 × 104BC4.7173 × 3; (−)2CE10.000CD5.0023 × 3; (−)DE4.9983 × 3; (elastic member)3EG15.752EF7.8763 × 3; (elastic member)FG7.8763 × 3; (−)4GI10.000GH4.4793 × 3; (−)HI5.5213 × 3; (elastic member)5IN30.624IJ9.7083 × 3; (elastic member)JK9.7083 × 3; (−)KN11.2083 × 3; (−)6KM12.624KL5.5623 × 3; (−)LM7.0623 × 3; (elastic member)7NP22.414NO10.4573 × 3; (−)OP11.9573 × 3; (elastic member)(17)β=KBP4EI1/4,rigid members:βh2≤1.0,elastic members:βh2>1.0,


## 4.2. Landslide Thrust and Resistance in front of the Structures

### 4.2.1. Landslide Thrust

The implicit solution of the unbalanced thrust method (transfer coefficient method) is an iterative solution. The landslide body is balanced to solve the stability factor (Fs) by steadily reducing the strength of the geotechnical material at the bottom of sliding bodies. The solution is determined as follows: first, the sliding force of the left side of the first stick on the top of a sliding body (F0) is assumed to be zero; second, the landslide thrust of the other stick is calculated according to formulas (18) and (19) until the landslide thrust of the nth stick (Fn) is equal to zero. If Fn is not equal to zero, Fs, which is the actual stability factor, should be adjusted until Fn is equal to zero [32].According to the implicit solution of the unbalanced thrust method (transfer coefficient method), the landslide thrust and the sliding resistance of each stick in the slip bodies can be calculated. The landslide thrust of the antislide members is calculated by the sliding force, which is projected onto the horizontal and vertical antislide members.(18)Fi=Wisinαi−Wicosαitanφi+cili+Ψi−1Fi−1,Ψi−1=cosαi−1−αi−sinαi−1−αitanφi,(19)Fi=Ti−RiFs+Ψi−1′Fi−1,Ti=Wisinαi,Ri=Wicosαitanφi+cili,Ψi−1′=cosαi−1−αi−sinαi−1−αitanφiFs,where Fi is the landslide thrust of the ith slice, Fi−1 is the landslide thrust of the (i − 1)th slice, Wi is the gravity of the ith slice, αi is the horizontal angle of the bottom of the ith slice, αi−1 is the horizontal angle of the bottom of the (i − 1)th slice, φi is the angle of internal friction of the bottom of the ith slice, ci is the cohesion of the bottom of the ith slice, li is the length of the bottom of the ith slice, ψi−1 is the transfer coefficient, Ψi−1′ is the reduced transfer coefficient, Ti is the sliding force of the ith slice, Ri is the sliding resistance of the ith slice, and Fs is the stability factor.


### 4.2.2. Resistance in front of the Structures

The passive earth pressure in front of the structures is described in formula (20) [33]. An infinite passive earth pressure is not realistic, even if the ground above the structures is close to horizontal and the friction between the antislide structures and the geotechnical is ignored. Therefore, the resistance of the horizontal antislide members is set to be the horizontal projection of the excess sliding force in the case of ultimate status [34].(20)EP=12γ1h12λ0,λ0=cos2φ1+αcos2α⋅cosα−δ1+sinφ1+δ⋅sinφ1+i/cosα−δ⋅cosα−i2,where EP is the passive earth pressure in front of the structure, γ1 is the unit weight of the geotechnical material in front of the structure, h1 is the stress height of the structure in the sliding body, λ0 is the coefficient of passive earth pressure of the structure, φ1 is the angle of internal friction in front of the structure, δ is the friction angle of the structural back, α is the intersection angle between the back and vertical directions of the structure, and i is the ground angle of the structure.


### 4.2.3. Calculation Results of the Sliding Force and Resistance

By using the implicit solution of the unbalanced thrust method (transfer coefficient method), the distributed line load on the antislide members is determined and shown in Table3.Table 3
The distributed line load on the antislide members.
Sequence numberStructureMemberMember length (m)Sliding force (kN)Sliding resistance (kN)Distributed line load (kN/m)1BCDBC4.71721259.82819479.445377.440CD5.00220049.66659.5773996.4192FGHFG7.87616255.55414355.040241.304GH4.47928583.57860.6376368.1493JKLJK9.70820640.66917914.429280.824KL5.56235912.62764.4576445.1944KNOKN11.20817176.50714598.502230.015NO10.45739324.825256.2123736.121


## 4.2.1. Landslide Thrust

The implicit solution of the unbalanced thrust method (transfer coefficient method) is an iterative solution. The landslide body is balanced to solve the stability factor (Fs) by steadily reducing the strength of the geotechnical material at the bottom of sliding bodies. The solution is determined as follows: first, the sliding force of the left side of the first stick on the top of a sliding body (F0) is assumed to be zero; second, the landslide thrust of the other stick is calculated according to formulas (18) and (19) until the landslide thrust of the nth stick (Fn) is equal to zero. If Fn is not equal to zero, Fs, which is the actual stability factor, should be adjusted until Fn is equal to zero [32].According to the implicit solution of the unbalanced thrust method (transfer coefficient method), the landslide thrust and the sliding resistance of each stick in the slip bodies can be calculated. The landslide thrust of the antislide members is calculated by the sliding force, which is projected onto the horizontal and vertical antislide members.(18)Fi=Wisinαi−Wicosαitanφi+cili+Ψi−1Fi−1,Ψi−1=cosαi−1−αi−sinαi−1−αitanφi,(19)Fi=Ti−RiFs+Ψi−1′Fi−1,Ti=Wisinαi,Ri=Wicosαitanφi+cili,Ψi−1′=cosαi−1−αi−sinαi−1−αitanφiFs,where Fi is the landslide thrust of the ith slice, Fi−1 is the landslide thrust of the (i − 1)th slice, Wi is the gravity of the ith slice, αi is the horizontal angle of the bottom of the ith slice, αi−1 is the horizontal angle of the bottom of the (i − 1)th slice, φi is the angle of internal friction of the bottom of the ith slice, ci is the cohesion of the bottom of the ith slice, li is the length of the bottom of the ith slice, ψi−1 is the transfer coefficient, Ψi−1′ is the reduced transfer coefficient, Ti is the sliding force of the ith slice, Ri is the sliding resistance of the ith slice, and Fs is the stability factor.


## 4.2.2. Resistance in front of the Structures

The passive earth pressure in front of the structures is described in formula (20) [33]. An infinite passive earth pressure is not realistic, even if the ground above the structures is close to horizontal and the friction between the antislide structures and the geotechnical is ignored. Therefore, the resistance of the horizontal antislide members is set to be the horizontal projection of the excess sliding force in the case of ultimate status [34].(20)EP=12γ1h12λ0,λ0=cos2φ1+αcos2α⋅cosα−δ1+sinφ1+δ⋅sinφ1+i/cosα−δ⋅cosα−i2,where EP is the passive earth pressure in front of the structure, γ1 is the unit weight of the geotechnical material in front of the structure, h1 is the stress height of the structure in the sliding body, λ0 is the coefficient of passive earth pressure of the structure, φ1 is the angle of internal friction in front of the structure, δ is the friction angle of the structural back, α is the intersection angle between the back and vertical directions of the structure, and i is the ground angle of the structure.


## 4.2.3. Calculation Results of the Sliding Force and Resistance

By using the implicit solution of the unbalanced thrust method (transfer coefficient method), the distributed line load on the antislide members is determined and shown in Table3.Table 3
The distributed line load on the antislide members.
Sequence numberStructureMemberMember length (m)Sliding force (kN)Sliding resistance (kN)Distributed line load (kN/m)1BCDBC4.71721259.82819479.445377.440CD5.00220049.66659.5773996.4192FGHFG7.87616255.55414355.040241.304GH4.47928583.57860.6376368.1493JKLJK9.70820640.66917914.429280.824KL5.56235912.62764.4576445.1944KNOKN11.20817176.50714598.502230.015NO10.45739324.825256.2123736.121


## 4.3. Elastic Resistance below the Slip Surfaces

The analytical solution of the elastic resistance below the slip surfaces is shown in formula (21), and the numerical solution of the interaction between the antislide members and slip bed is shown in formula (22):(21)σx=−KVBPyx,σy=−KHBPxy,(22)qx=dQydy,qy=dQxdx,where σx is the analytical calculation of the horizontal elastic resistance below the slip surfaces, σy is the analytical calculation of the vertical elastic resistance below the slip surfaces, qx is the numerical calculation of the horizontal interaction between the antislide members and geotechnical materials, qy is the numerical calculation of the vertical interaction between the antislide members and geotechnical materials, dQ (x) is the horizontal shear force difference of the microelement on the antislide member, dQ (y) is the vertical shear force difference of the microelement on the antislide member, dx is the horizontal length of the microelement that is equal to 0.5 m on the antislide member by using numerical simulation software (MIDAS GTS NX), and dy is the vertical length of the microelement that is equal to 0.5 m on the antislide member by using numerical simulation software (MIDAS GTS NX).


## 4.4. Comparative Analysis of the Structural Effects

### 4.4.1. Comparative Analysis of the Displacement

Due to the small structural displacements, the initial location and displacements of the structures cannot be distinguished with the numerical simulation at the original scaling. Taking the shape of the ASCL as a benchmark, the analytical calculation and the numerical simulation of the displacements were enlarged at different scales. A comparative figure of the enlarged displacements is given in Figure12. In Figure 12, the global displacements caused deformation toward downwards to the right, and the results of the analytical calculation of the structural displacements were similar to the distribution of the numerical results, showing that the ASCL was bearing the landslide thrust and the resistance of the geotechnical materials.Figure 12
A comparative figure of the enlarged displacements. The black lines represent the initial shape of the ASCL, and the green lines represent the slip surface. Taking the initial shape of the ASCL as a standard, the red lines represent the displacements of the analytical calculations, while the pink lines represent the displacements of the numerical calculations.A comparative table of the structural displacements of the antislide members is given in Table4. In Table 4 and the following tables, E was equal to 10. In Table 4, the structural displacement results of the analytical calculation and numerical simulation were different due to the different calculation methods and assumptions. When the structural mechanics and elastic foundation beam method were adopted to analytically calculate the structural displacements, the minor loads that included the gravity of the antislide members, friction, and end-bearing force between the antislide members and geotechnical materials were ignored, and the resulting structural displacements of the horizontal antislide members presented only vertical displacements, without horizontal displacements, while that of the vertical antislide members presented only horizontal displacements, without vertical displacements. Otherwise, due to the minor loads that included the gravity of antislide members, the interaction between the antislide members and geotechnical materials, and the effects that were simulated as the intercoupling between the antislide members and geotechnical materials, the maximum numerically determined structural displacements were greater than the maximum analytically calculated structural displacements. The corresponding ratios of the maximum displacements of the global structures were equal to 0.820 for the horizontal displacement, 0.031 for the vertical displacement, and 4.342 for the angular displacement. Thus, the numerical simulation of the structural displacements was proposed to be adopted.Table 4
A comparative table of the structural displacements of the antislide members.
MemberMaximum analytical calculation of horizontal displacement Dx (m)Maximum numerical simulation of horizontal displacement Dx′ (m)Dx/Dx′Maximum analytical calculation of vertical displacement Dy (m)Maximum numerical simulation of vertical displacement Dy′ (m)Dy/Dy′Maximum analytical calculation of angular displacement Dr (rad)Maximum numerical simulation of angular displacement Dr′ (rad)Dr/Dr′AC06.626E − 030−6.047E − 04−1.747E − 020.0351.795E − 043.235E − 040.555CE2.270E − 036.623E − 030.3430−1.746E − 0207.741E − 041.155E − 046.702EG05.914E − 030−4.737E − 04−1.758E − 020.0272.215E − 041.607E − 041.378GI3.408E − 036.026E − 030.5660−1.679E − 0201.104E − 031.194E − 049.246IN05.271E − 030−4.720E − 04−1.941E − 020.0242.564E − 044.473E − 040.573KM4.162E − 035.249E − 030.7930−1.858E − 0201.216E − 032.537E − 044.793NP5.431E − 035.459E − 030.9950−1.652E − 0201.942E − 032.965E − 046.550Total5.431E − 036.626E − 030.820−6.047E − 04−1.941E − 020.0311.942E − 034.473E − 044.342


### 4.4.2. Comparative Analysis of the Axial Force

As mentioned before, the global displacements presented a deformation downwards to the right, resulting in an apparent domino effect. This domino effect is shown in Figure13, and a comparison of the axial forces of the ASCL is shown in Figure 14. According to Figures 13 and 14, the trend of the analytical calculation of the structural axial force was close to the trend of the numerical simulation of the structural axial force. The analytical calculation and numerical simulation of the axial force of the horizontal antislide member indicated mostly axial tension, while the analytical calculation and numerical simulation of the axial force of the vertical antislide members indicated mostly axial pressure, confirming that the global displacements reflected deformation downwards to the right. The horizontal antislide members acted as tie rods, while the vertical antislide members bearing the resultant force between the thrust and the sliding resistance presented deformation downwards to the right. Notably, the horizontal antislide members that were not strictly acting as tie rods bore the axial force, the shear force, and the bending moment.Figure 13
The domino effect. The domino effect occurs when several dominoes are arranged in rows according to certain spacing; when the first domino is pushed over, the other dominoes behind it will be pushed over in a chain reaction.Figure 14
A comparison of the axial forces of the ASCL. The black lines represent the initial shape of the ASCL, and the green lines represent the slip surface. Taking the initial shape of the ASCL as a standard, the red lines represent the axial force of the analytical calculations, while the pink lines represent the axial force of the numerical calculations.A comparative table of the axial force of the antislide members is given in Table5. In Table 5, the analytical calculation result of the axial force was different from the numerical simulation result of axial force due to the different calculation methods and assumptions. The ratio of the maximum axial forces of the global structures was equal to 10.755, while the ratio of the minimum axial forces of the global structures was equal to 0.176. Based on the principle of internal stress envelope diagrams, the maximum result from the analytical calculation of the axial force, which was equal to 33800.857 kN, and the minimum result from the numerical simulation of the axial force, which was equal to −39304.948 kN, were proposed to be adopted.Table 5
A comparative table of the axial force of the antislide members.
MemberMaximum analytical calculation of axial forceFN (kN)Maximum numerical simulation of axial forceFN′ (kN)FN/FN′Minimum analytical calculation of axial forceFn (kN)Minimum numerical simulation of axial forceFn′ (kN)Fn/Fn′AC8680.4443142.7642.7628680.444−1160.419−7.480CE588.064−11339.797−0.052−6338.100−17527.280.362EG11722.3411284.8289.1211859.380−463.996−4.007GI−129.082−10593.5920.012−6931.905−22643.8620.306IN33800.8572133.85915.8404134.072−1777.112−2.326KM−4862.311−9423.5720.516−4862.311−27649.1740.176NP1256.725−6426.624−0.1961256.725−39304.948−0.032Total33800.8573142.76410.755−6931.905−39304.9480.176


### 4.4.3. Comparative Analysis of the Shear Force

A comparative figure of the shear force of the ASCL is shown in Figure15. In Figure 15, the trend of the analytical calculation results of the structural shear force was close to the trend of the numerical simulation results of the structural shear force. The horizontal antislide members bore a positive shear force due to the vertical loads, while the top of the vertical antislide members bore a negative shear force due to the horizontal antislide members acting as tie rods. However, the negative shear force of the vertical antislide members decreased gradually and became a positive shear force because of the resultant force due to the thrust and the sliding resistance. Assuming that a slip surface is a boundary in the ASCL, the shear force of the members close to the slip surfaces was very high, and the shear force direction of the members above the slip surfaces and below the slip surfaces was opposite due to the interaction between the antislide members and geotechnical materials below the slip surfaces, which can decrease the shear force of the members above the slip surfaces.Figure 15
A comparative figure of the shear force of the ASCL. The black lines represent the initial shape of the ASCL, and the green lines represent the slip surface. Taking the initial shape of the ASCL as a standard, the red lines represent the shear force of the analytical calculations, while the pink lines represent the shear force of the numerical calculations.A comparative table of the shear force of the antislide members is given in Table6. The analytical calculation result of the shear force was different from the numerical simulation result of the shear force due to the different calculation methods and assumptions. In Table 6, the ratio of the maximum shear forces of the global structures was equal to 2.061, while the ratio of the minimum shear forces of the global structures was equal to 1.035. The analytical calculation result of the structural shear force exhibited a hysteretic effect, unlike the numerical simulation result of the structural shear force, as the maximum result of the analytical calculation of the shear force was observed behind the members and the maximum result of the numerical simulation of the shear force was observed in the middle of the members. Based on the principle of internal stress envelope diagrams, the maximum shear force from the analytical calculation, which was equal to 22169.628 kN, and the minimum shear force from the analytical calculation, which was equal to −16901.873 kN, were proposed to be adopted.Table 6
A comparative table of the shear force of the antislide members.
MemberMaximum analytical calculation of shear forceFQ (kN)Maximum numerical simulation of shear forceFQ′ (kN)FQ/FQ′Minimum analytical calculation of shear forceFq (kN)Minimum numerical simulation of shear forceFq′ (kN)Fq/Fq′AC2368.4477591.1860.312−2010.671−9953.1840.202CE11309.6452135.7935.295−8680.444−255.00334.041EG1771.4324952.2780.358−6338.100−8730.5150.726GI16800.6013912.3114.294−11722.341391.741−29.924IN3834.73010757.6620.356−6931.905−16337.6650.424KM18946.2971770.99610.698−16901.873−809.66320.875NP22169.6283380.7616.558−16898.984−2215.3487.628Total22169.62810757.6622.061−16901.873−16337.6651.035


### 4.4.4. Comparative Analysis of the Bending Moment

The comparative figure of the bending moment of the ASCL is shown in Figure16. In Figure 16, the direction of the analytical calculation result of the structural bending moment was contrary to the direction of the numerical simulation result of the structural bending moment at the top junction between the horizontal antislide members and the vertical antislide members due to the difference between the analytical calculation, which was a piecewise calculation, and the numerical simulation, which was a global calculation. The trend of the analytical calculation of the structural bending moment approached the trend of the numerical simulation of the structural bending moment, while the location of the member section gradually moved to the bottom joints of the vertical antislide members from the top joints between the horizontal antislide members and the vertical antislide members. Assuming that a slip surface is a boundary in the ASCL, the bending moments near the slip surfaces were larger. The top outer edges and the bottom outer edges of the horizontal antislide members and the vertical antislide members easily exhibited tensile bending moments due to the global displacements that resulted in the apparent domino effect; however, the bending moment reversed due to the interaction between the antislide members and geotechnical materials.Figure 16
The comparative figure of the bending moment of the ASCL. The black lines represent the initial shape of the ASCL, and the green lines represent the slip surface. Taking the initial shape of the ASCL as a standard, the red lines represent the bending moment of the analytical calculations, while the pink lines represent the bending moment of the numerical calculations.A comparative table of the bending moments of the antislide members is shown in Table7. In Table 7, the ratio of the maximum bending moments of the global structures was equal to 0.964, while the ratio of the minimum bending moments of the global structures was equal to 2.821. For the same reason, the analytical calculation result of the structural bending moment exhibited a hysteretic effect, unlike the numerical simulation result of the structural bending moment. Based on the principle of internal stress envelope diagrams, the maximum bending moment from the numerical simulation, which was 23385.630 kN·m, and the minimum bending moment from the analytical calculation, which was appropriately decreased to −36901.790 kN·m, were proposed to be adopted.Table 7
A comparative table of the bending moments of the antislide members.
MemberMaximum analytical calculation of bending momentM (kN·m)Maximum numerical simulation of bending momentM′ (kN·m)M/M′Minimum analytical calculation of bending momentm (kN·m)Minimum numerical simulation of bending momentm′ (kN·m)m/m′AC4521.4715658.9110.799−3948.781−14354.3600.275CE5419.2692605.4322.080−23265.751−6865.6373.389EG4388.1889634.8920.455−20468.400−12945.6021.581GI7644.0378805.7040.868−32908.479−8560.6543.844IN12859.18423385.6300.550−24614.286−22101.5421.114KM9288.1865922.7151.568−36901.7900+∞NP22543.87723385.6300.964−62354.262−4332.75614.391Total22543.87723385.6300.964−62354.262−22101.5422.821


### 4.4.5. Comparative Analysis of the Interaction between the Members and Geotechnical Materials

A comparative figure of the interaction between the antislide members and geotechnical materials is shown in Figure17. In Figure 17, the trend of analytical calculation result of the interaction between the antislide members and geotechnical materials approached the trend of the numerical simulation result of the interaction between the antislide members and geotechnical materials. Assuming that a slip surface is a boundary in the ASCL, the interaction between the antislide members and geotechnical materials on the slip surfaces was greater, the horizontal antislide members above the slip surfaces bare the downward load, while the vertical antislide members above the slip surfaces bare the rightward load. However, the interaction below the slip surfaces was reversed and decreased gradually due to the anchoring effects between the antislide members below the slip surfaces and the geotechnical materials on the slip bed.Figure 17
A comparative figure of the interaction between the antislide members and geotechnical materials. The black lines represent the initial shape of the ASCL, and the green lines represent the slip surface. Taking the initial shape of the ASCL as a standard, the red lines represent the interaction between the antislide members and geotechnical materials of the analytical calculations, while the pink lines represent the interaction between the antislide members and geotechnical materials of the numerical calculations.A comparative table of the interaction between the antislide members and geotechnical materials is shown in Table8. In Table 8, the maximum numerical simulation of the interaction between the antislide members and geotechnical materials was equal to 44410.500 kN/m, so larger values were unreliable and should be eliminated if there was no approximate numerical simulation result provided nearby. Based on the principle of internal stress envelope diagrams, the maximum interaction between the antislide members and geotechnical materials from the numerical simulation, which was equal to 19906.4 kN/m, and the minimum interaction between the antislide members and geotechnical materials from the numerical simulation, which was equal to −19826.88 kN/m, were proposed to be adopted.Table 8
A comparative table of the interaction between the antislide members and geotechnical materials.
MemberMaximum analytical calculation of interactionQ (kN/m)Maximum numerical simulation of interactionQ′ (kN/m)Q/Q′Minimum analytical calculation of interactionq (kN/m)Minimum numerical simulation of interactionq′ (kN/m)q/q′AC2116.45019906.4000.106−377.440−19826.8800.019CE3996.4194144.9500.964−5561.500−2786.0021.996EG1657.9506617.8000.251−241.304−17461.0300.014GI6368.1496823.8900.933−8349.600−5572.6401.498IN1185.10044410.5000.027−280.824−18480.4100.015KM6445.1942584.8502.493−10196.900−1224.2888.329NP3736.1215745.2900.650−13305.950−5764.5002.308Total6445.19444410.5000.145−13305.950−19826.8800.671


## 4.4.1. Comparative Analysis of the Displacement

Due to the small structural displacements, the initial location and displacements of the structures cannot be distinguished with the numerical simulation at the original scaling. Taking the shape of the ASCL as a benchmark, the analytical calculation and the numerical simulation of the displacements were enlarged at different scales. A comparative figure of the enlarged displacements is given in Figure12. In Figure 12, the global displacements caused deformation toward downwards to the right, and the results of the analytical calculation of the structural displacements were similar to the distribution of the numerical results, showing that the ASCL was bearing the landslide thrust and the resistance of the geotechnical materials.Figure 12
A comparative figure of the enlarged displacements. The black lines represent the initial shape of the ASCL, and the green lines represent the slip surface. Taking the initial shape of the ASCL as a standard, the red lines represent the displacements of the analytical calculations, while the pink lines represent the displacements of the numerical calculations.A comparative table of the structural displacements of the antislide members is given in Table4. In Table 4 and the following tables, E was equal to 10. In Table 4, the structural displacement results of the analytical calculation and numerical simulation were different due to the different calculation methods and assumptions. When the structural mechanics and elastic foundation beam method were adopted to analytically calculate the structural displacements, the minor loads that included the gravity of the antislide members, friction, and end-bearing force between the antislide members and geotechnical materials were ignored, and the resulting structural displacements of the horizontal antislide members presented only vertical displacements, without horizontal displacements, while that of the vertical antislide members presented only horizontal displacements, without vertical displacements. Otherwise, due to the minor loads that included the gravity of antislide members, the interaction between the antislide members and geotechnical materials, and the effects that were simulated as the intercoupling between the antislide members and geotechnical materials, the maximum numerically determined structural displacements were greater than the maximum analytically calculated structural displacements. The corresponding ratios of the maximum displacements of the global structures were equal to 0.820 for the horizontal displacement, 0.031 for the vertical displacement, and 4.342 for the angular displacement. Thus, the numerical simulation of the structural displacements was proposed to be adopted.Table 4
A comparative table of the structural displacements of the antislide members.
MemberMaximum analytical calculation of horizontal displacement Dx (m)Maximum numerical simulation of horizontal displacement Dx′ (m)Dx/Dx′Maximum analytical calculation of vertical displacement Dy (m)Maximum numerical simulation of vertical displacement Dy′ (m)Dy/Dy′Maximum analytical calculation of angular displacement Dr (rad)Maximum numerical simulation of angular displacement Dr′ (rad)Dr/Dr′AC06.626E − 030−6.047E − 04−1.747E − 020.0351.795E − 043.235E − 040.555CE2.270E − 036.623E − 030.3430−1.746E − 0207.741E − 041.155E − 046.702EG05.914E − 030−4.737E − 04−1.758E − 020.0272.215E − 041.607E − 041.378GI3.408E − 036.026E − 030.5660−1.679E − 0201.104E − 031.194E − 049.246IN05.271E − 030−4.720E − 04−1.941E − 020.0242.564E − 044.473E − 040.573KM4.162E − 035.249E − 030.7930−1.858E − 0201.216E − 032.537E − 044.793NP5.431E − 035.459E − 030.9950−1.652E − 0201.942E − 032.965E − 046.550Total5.431E − 036.626E − 030.820−6.047E − 04−1.941E − 020.0311.942E − 034.473E − 044.342


## 4.4.2. Comparative Analysis of the Axial Force

As mentioned before, the global displacements presented a deformation downwards to the right, resulting in an apparent domino effect. This domino effect is shown in Figure13, and a comparison of the axial forces of the ASCL is shown in Figure 14. According to Figures 13 and 14, the trend of the analytical calculation of the structural axial force was close to the trend of the numerical simulation of the structural axial force. The analytical calculation and numerical simulation of the axial force of the horizontal antislide member indicated mostly axial tension, while the analytical calculation and numerical simulation of the axial force of the vertical antislide members indicated mostly axial pressure, confirming that the global displacements reflected deformation downwards to the right. The horizontal antislide members acted as tie rods, while the vertical antislide members bearing the resultant force between the thrust and the sliding resistance presented deformation downwards to the right. Notably, the horizontal antislide members that were not strictly acting as tie rods bore the axial force, the shear force, and the bending moment.Figure 13
The domino effect. The domino effect occurs when several dominoes are arranged in rows according to certain spacing; when the first domino is pushed over, the other dominoes behind it will be pushed over in a chain reaction.Figure 14
A comparison of the axial forces of the ASCL. The black lines represent the initial shape of the ASCL, and the green lines represent the slip surface. Taking the initial shape of the ASCL as a standard, the red lines represent the axial force of the analytical calculations, while the pink lines represent the axial force of the numerical calculations.A comparative table of the axial force of the antislide members is given in Table5. In Table 5, the analytical calculation result of the axial force was different from the numerical simulation result of axial force due to the different calculation methods and assumptions. The ratio of the maximum axial forces of the global structures was equal to 10.755, while the ratio of the minimum axial forces of the global structures was equal to 0.176. Based on the principle of internal stress envelope diagrams, the maximum result from the analytical calculation of the axial force, which was equal to 33800.857 kN, and the minimum result from the numerical simulation of the axial force, which was equal to −39304.948 kN, were proposed to be adopted.Table 5
A comparative table of the axial force of the antislide members.
MemberMaximum analytical calculation of axial forceFN (kN)Maximum numerical simulation of axial forceFN′ (kN)FN/FN′Minimum analytical calculation of axial forceFn (kN)Minimum numerical simulation of axial forceFn′ (kN)Fn/Fn′AC8680.4443142.7642.7628680.444−1160.419−7.480CE588.064−11339.797−0.052−6338.100−17527.280.362EG11722.3411284.8289.1211859.380−463.996−4.007GI−129.082−10593.5920.012−6931.905−22643.8620.306IN33800.8572133.85915.8404134.072−1777.112−2.326KM−4862.311−9423.5720.516−4862.311−27649.1740.176NP1256.725−6426.624−0.1961256.725−39304.948−0.032Total33800.8573142.76410.755−6931.905−39304.9480.176


## 4.4.3. Comparative Analysis of the Shear Force

A comparative figure of the shear force of the ASCL is shown in Figure15. In Figure 15, the trend of the analytical calculation results of the structural shear force was close to the trend of the numerical simulation results of the structural shear force. The horizontal antislide members bore a positive shear force due to the vertical loads, while the top of the vertical antislide members bore a negative shear force due to the horizontal antislide members acting as tie rods. However, the negative shear force of the vertical antislide members decreased gradually and became a positive shear force because of the resultant force due to the thrust and the sliding resistance. Assuming that a slip surface is a boundary in the ASCL, the shear force of the members close to the slip surfaces was very high, and the shear force direction of the members above the slip surfaces and below the slip surfaces was opposite due to the interaction between the antislide members and geotechnical materials below the slip surfaces, which can decrease the shear force of the members above the slip surfaces.Figure 15
A comparative figure of the shear force of the ASCL. The black lines represent the initial shape of the ASCL, and the green lines represent the slip surface. Taking the initial shape of the ASCL as a standard, the red lines represent the shear force of the analytical calculations, while the pink lines represent the shear force of the numerical calculations.A comparative table of the shear force of the antislide members is given in Table6. The analytical calculation result of the shear force was different from the numerical simulation result of the shear force due to the different calculation methods and assumptions. In Table 6, the ratio of the maximum shear forces of the global structures was equal to 2.061, while the ratio of the minimum shear forces of the global structures was equal to 1.035. The analytical calculation result of the structural shear force exhibited a hysteretic effect, unlike the numerical simulation result of the structural shear force, as the maximum result of the analytical calculation of the shear force was observed behind the members and the maximum result of the numerical simulation of the shear force was observed in the middle of the members. Based on the principle of internal stress envelope diagrams, the maximum shear force from the analytical calculation, which was equal to 22169.628 kN, and the minimum shear force from the analytical calculation, which was equal to −16901.873 kN, were proposed to be adopted.Table 6
A comparative table of the shear force of the antislide members.
MemberMaximum analytical calculation of shear forceFQ (kN)Maximum numerical simulation of shear forceFQ′ (kN)FQ/FQ′Minimum analytical calculation of shear forceFq (kN)Minimum numerical simulation of shear forceFq′ (kN)Fq/Fq′AC2368.4477591.1860.312−2010.671−9953.1840.202CE11309.6452135.7935.295−8680.444−255.00334.041EG1771.4324952.2780.358−6338.100−8730.5150.726GI16800.6013912.3114.294−11722.341391.741−29.924IN3834.73010757.6620.356−6931.905−16337.6650.424KM18946.2971770.99610.698−16901.873−809.66320.875NP22169.6283380.7616.558−16898.984−2215.3487.628Total22169.62810757.6622.061−16901.873−16337.6651.035


## 4.4.4. Comparative Analysis of the Bending Moment

The comparative figure of the bending moment of the ASCL is shown in Figure16. In Figure 16, the direction of the analytical calculation result of the structural bending moment was contrary to the direction of the numerical simulation result of the structural bending moment at the top junction between the horizontal antislide members and the vertical antislide members due to the difference between the analytical calculation, which was a piecewise calculation, and the numerical simulation, which was a global calculation. The trend of the analytical calculation of the structural bending moment approached the trend of the numerical simulation of the structural bending moment, while the location of the member section gradually moved to the bottom joints of the vertical antislide members from the top joints between the horizontal antislide members and the vertical antislide members. Assuming that a slip surface is a boundary in the ASCL, the bending moments near the slip surfaces were larger. The top outer edges and the bottom outer edges of the horizontal antislide members and the vertical antislide members easily exhibited tensile bending moments due to the global displacements that resulted in the apparent domino effect; however, the bending moment reversed due to the interaction between the antislide members and geotechnical materials.Figure 16
The comparative figure of the bending moment of the ASCL. The black lines represent the initial shape of the ASCL, and the green lines represent the slip surface. Taking the initial shape of the ASCL as a standard, the red lines represent the bending moment of the analytical calculations, while the pink lines represent the bending moment of the numerical calculations.A comparative table of the bending moments of the antislide members is shown in Table7. In Table 7, the ratio of the maximum bending moments of the global structures was equal to 0.964, while the ratio of the minimum bending moments of the global structures was equal to 2.821. For the same reason, the analytical calculation result of the structural bending moment exhibited a hysteretic effect, unlike the numerical simulation result of the structural bending moment. Based on the principle of internal stress envelope diagrams, the maximum bending moment from the numerical simulation, which was 23385.630 kN·m, and the minimum bending moment from the analytical calculation, which was appropriately decreased to −36901.790 kN·m, were proposed to be adopted.Table 7
A comparative table of the bending moments of the antislide members.
MemberMaximum analytical calculation of bending momentM (kN·m)Maximum numerical simulation of bending momentM′ (kN·m)M/M′Minimum analytical calculation of bending momentm (kN·m)Minimum numerical simulation of bending momentm′ (kN·m)m/m′AC4521.4715658.9110.799−3948.781−14354.3600.275CE5419.2692605.4322.080−23265.751−6865.6373.389EG4388.1889634.8920.455−20468.400−12945.6021.581GI7644.0378805.7040.868−32908.479−8560.6543.844IN12859.18423385.6300.550−24614.286−22101.5421.114KM9288.1865922.7151.568−36901.7900+∞NP22543.87723385.6300.964−62354.262−4332.75614.391Total22543.87723385.6300.964−62354.262−22101.5422.821


## 4.4.5. Comparative Analysis of the Interaction between the Members and Geotechnical Materials

A comparative figure of the interaction between the antislide members and geotechnical materials is shown in Figure17. In Figure 17, the trend of analytical calculation result of the interaction between the antislide members and geotechnical materials approached the trend of the numerical simulation result of the interaction between the antislide members and geotechnical materials. Assuming that a slip surface is a boundary in the ASCL, the interaction between the antislide members and geotechnical materials on the slip surfaces was greater, the horizontal antislide members above the slip surfaces bare the downward load, while the vertical antislide members above the slip surfaces bare the rightward load. However, the interaction below the slip surfaces was reversed and decreased gradually due to the anchoring effects between the antislide members below the slip surfaces and the geotechnical materials on the slip bed.Figure 17
A comparative figure of the interaction between the antislide members and geotechnical materials. The black lines represent the initial shape of the ASCL, and the green lines represent the slip surface. Taking the initial shape of the ASCL as a standard, the red lines represent the interaction between the antislide members and geotechnical materials of the analytical calculations, while the pink lines represent the interaction between the antislide members and geotechnical materials of the numerical calculations.A comparative table of the interaction between the antislide members and geotechnical materials is shown in Table8. In Table 8, the maximum numerical simulation of the interaction between the antislide members and geotechnical materials was equal to 44410.500 kN/m, so larger values were unreliable and should be eliminated if there was no approximate numerical simulation result provided nearby. Based on the principle of internal stress envelope diagrams, the maximum interaction between the antislide members and geotechnical materials from the numerical simulation, which was equal to 19906.4 kN/m, and the minimum interaction between the antislide members and geotechnical materials from the numerical simulation, which was equal to −19826.88 kN/m, were proposed to be adopted.Table 8
A comparative table of the interaction between the antislide members and geotechnical materials.
MemberMaximum analytical calculation of interactionQ (kN/m)Maximum numerical simulation of interactionQ′ (kN/m)Q/Q′Minimum analytical calculation of interactionq (kN/m)Minimum numerical simulation of interactionq′ (kN/m)q/q′AC2116.45019906.4000.106−377.440−19826.8800.019CE3996.4194144.9500.964−5561.500−2786.0021.996EG1657.9506617.8000.251−241.304−17461.0300.014GI6368.1496823.8900.933−8349.600−5572.6401.498IN1185.10044410.5000.027−280.824−18480.4100.015KM6445.1942584.8502.493−10196.900−1224.2888.329NP3736.1215745.2900.650−13305.950−5764.5002.308Total6445.19444410.5000.145−13305.950−19826.8800.671


## 5. Conclusions and Future Work

To verify the effect of antislide structures with continuous ladders on the slope stability, a new analytical method (the new stress calculation methods of antislide structures with continuous ladders that are based on the displacement method of structural mechanics and elastic foundation beam method) has been proposed. By researching the results of analytical methods and numerical simulation methods, the conclusions and future work are concluded as follows.The trends of the displacements, axial force, shear force, bending moment, and interaction between the antislide members and geotechnical materials from the analytical calculations approached the corresponding trends from the numerical simulations. The results were different between the analytical calculations and the numerical simulations due to the different calculation methods and assumptions. However, by eliminating some unreliable values, the ultimate values from the analytical calculation results (shear force, bending moment, and interaction between the antislide members and geotechnical materials) were close to the ultimate values from the numerical simulation results, and these effects were greater on the slip surfaces than on the surrounding areas. The ultimate analytical result exhibited a hysteretic effect, unlike the ultimate numerical simulation result because the analytical calculation of structural stress was a piecewise calculation while the numerical simulation of structural stress considered a global calculation; this phenomenon was understood as the hysteretic effect of the analytical calculation due to a larger domino spacing relative to the numerical simulation. Based on the principle of internal stress envelope diagrams, the maximum absolute values of the analytical calculation results and numerical simulation results were proposed to be adopted, and the other values were proposed to a reference for the structural stress.Research on the internal materials of the ASCLs (reinforced concrete, prestressed concrete, steel reinforced concrete, compound materials, etc.), seismic calculation of the ASCLs, physical simulation test of the ASCLs (jack loading test, large shake table test, centrifugal model test, etc.), structural stress calculation of the ASCLs under the multiple loads, structural reliability, etc., should be performed. Furthermore, the research findings of ASCLs, as they are further developed and improved, should be applied in practical engineering applications.



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*Source: 8829205-2020-12-21.xml*