The Stress Calculation Methods of Antislide Structures with Continuous Ladders

Chen, Gong; Xu, Qiang

doi:https://doi.org/10.1155/2020/8829205

Advances in Civil Engineering

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Research Article | Open Access

Volume 2020 | Article ID 8829205 | https://doi.org/10.1155/2020/8829205

The Stress Calculation Methods of Antislide Structures with Continuous Ladders

Gong Chen¹and Qiang Xu¹

Academic Editor: Hui Yao

Received08 Sept 2020

Revised09 Nov 2020

Accepted07 Dec 2020

Published21 Dec 2020

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.

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 Figure 1. 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].

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 Figure 3.

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 Figure 4.

(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 by DEF 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.

Otherwise, if DEF 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.

The internal stress at Node E 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).where x_E is the horizontal displacement of Node E, y_E 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.where ∑F_x is the resultant force at Node E along the x-direction, ∑F_y is the resultant force at Node E along the y-direction, ∑M_E is the resultant moment at Node E, F_QED is the shear force at Node E along the ED direction, F_QEF is the shear force at Node E along the EF direction, F_NED is the axial force of DE, F_NEF is the axial force of EF, M_ED is the resultant moment at Node E along the ED direction, and M_EF is the resultant moment at Node E along the EF direction.

3. Analytical Calculations

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

3.1.1. Analytical Calculation of the Structural Stress of BCD

The load decomposition of BCD 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.

where M_BC is the bending moment at Node B along the BC direction, M_CB is the bending moment at Node C along the CB direction, M_DC is the bending moment at Node D along the DC direction, M_CD is the bending moment at Node C along the CD direction, F_QBC is the shear force at Node B along the BC direction, F_QCB is the shear force at Node C along the CB direction, F_QDC is the shear force at Node D along the DC direction, F_QCD is the shear force at Node C along the CD direction, F_NCB is the axial force of BC, F_NCD is the axial force of CD, θ_C1 is the angular displacement of C₁, θ_C2 is the angular displacement of C₂, q_BC is the rectangular load of the load decomposition of BC, is the maximum of the triangular load of the load decomposition of BC, q_CD is the rectangular load of the load decomposition of CD, is the maximum of the triangular load of the load decomposition of CD, i_BC is the linear stiffness of BC, i_CD is the linear stiffness of CD, l_BC is the length of BC, and l_CD is the length of CD.

3.1.2. Analytical Calculation of the Structural Stress of JKLNO

The load decomposition of JKLNO is shown in Figure 9, and the analytical calculation of the structural stress of JKLNO is shown in formulas (5)∼(8).

(a)

(b)

(c)

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 of AB (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.

where M_JK is the bending moment at Node J along the JK direction, M_KJ is the bending moment at Node K along the KJ direction, M_LK is the bending moment at Node L along the LK direction, M_KL is the bending moment at Node K along the KL direction, M_KN is the bending moment at Node K along the KN direction, M_NK is the bending moment at Node N along the NK direction, M_ON is the bending moment at Node O along the ON direction, M_NO is the bending moment at Node N along the NO direction, F_QJK is the shear force at Node J along the JK direction, F_QKJ is the shear force at Node K along the KJ direction, F_QLK is the shear force at Node L along the LK direction, F_QKL is the shear force at Node K along the KL direction, F_QKN is the shear force at Node K along the KN direction, F_QNK is the shear force at Node N along the NK direction, F_QON is the shear force at Node O along ON direction, F_QNO is the shear force at Node N along the NO direction, F_NKJ is the axial force of JK, F_NKL is the axial force of KL, F_NKN is the axial force of KN, F_NNO is the axial force of NO, θ_K1 is the angular displacement of K₁, θ_K2 is the angular displacement of K₂, θ_N1 is the angular displacement of N₁, θ_N2 is the angular displacement of N₂, q_JK is the rectangular load of the load decomposition of JK, is the maximum of the triangular load of the load decomposition of JK, q_KN is the rectangular load of the load decomposition of KN, is the maximum of the triangular load of the load decomposition of KN, q_KL is the rectangular load of the load decomposition of KL, is the maximum of the triangular load of the load decomposition of KL, q_NO is the rectangular load of the load decomposition of NO, is the maximum of the triangular load of the load decomposition of NO, i_JK is the linear stiffness of JK, i_KL is the linear stiffness of KL, i_KN is the linear stiffness of KN, i_NO is the linear stiffness of NO, l_JK is the length of JK, l_KL is the length of KL, l_KN is the length of KN, and l_NO is the length of NO.

where y (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, y_BA is the initial vertical displacement of Node B, φ_BA is the initial angular displacement of Node B, M_BA is the initial bending moment of Node B, F_QBA is the initial shear force of Node B, E_AB is the elastic coefficient of AB, I_AB is the section inertial moment of AB, and β_AB is the deformation coefficient of AB.

As mentioned before, the boundary condition of Node A 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).where is the vertical foundation coefficient of the slip bed, B_P(AB) is the calculation width of AB, and e is the natural exponent that is equal to 2.718281…where l_AB is the length of AB, M_AB is the bending moment at Node A along the AB direction, and y_A is the vertical displacement of Node A.

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

The boundary conditions at Node E 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:where x_ED is the horizontal displacement at Node E along the ED direction, x_DE is the initial horizontal displacement at Node D, φ_DE is the initial angular displacement at Node D, M_DE is the initial bending moment at Node D, F_QDE is the initial shear force at Node D, E_DE is the elastic coefficient of DE, I_DE is the section inertial moment of DE, β_DE is the deformation coefficient of DE, l_DE is the length of DE, y_EF is the vertical displacement at Node E along the EF direction, φ_FE is the initial angular displacement at Node F, M_FE is the initial bending moment at Node F, F_QFE is the initial shear force at Node F, E_EF is the elastic coefficient of EF, I_EF is the section inertial moment of EF, β_EF is the deformation coefficient of EF, and l_EF is the length of EF.where φ_ED is the angular displacement at Node E along the ED direction and φ_EF is the angular displacement at Node E along the EF direction.where ∑M_E is the resultant moment at Node E, M_ED is the bending moment at Node E along the ED direction, and M_EF is the bending moment at Node E along the EF direction.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, F_QED is the shear force at Node E along the ED direction, F_QEF is the shear force at Node E along the EF direction, F_NED is the axial force of DE, F_NEF is the axial force of EF, M_ED is the bending moment at Node E along the ED direction, and M_EF 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 Figure 11.

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 () is established to be 875000 kN/m³, and the horizontal foundation coefficient of the slip beds (K_H) is established to be 612500 kN/m³. The mechanical parameters of the geotechnical landslides are shown in Table 1.

As 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 Table 2. 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, is the vertical foundation coefficient of the slip bed, K_H 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 B_P is the calculation width of the member.

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 (F_n) is equal to zero. If F_n is not equal to zero, F_s, which is the actual stability factor, should be adjusted until F_n 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.where F_i is the landslide thrust of the ith slice, F_i−1 is the landslide thrust of the (i − 1)th slice, is the gravity of the ith slice, α_i is the horizontal angle of the bottom of the ith slice, α_i−1 is the horizontal angle of the bottom of the (i − 1)th slice, φ_i is the angle of internal friction of the bottom of the ith slice, c_i is the cohesion of the bottom of the ith slice, l_i is the length of the bottom of the ith slice, ψ_i−1 is the transfer coefficient, is the reduced transfer coefficient, T_i is the sliding force of the ith slice, R_i is the sliding resistance of the ith slice, and F_s 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].where E_P 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 Table 3.

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):where σ_x is the analytical calculation of the horizontal elastic resistance below the slip surfaces, σ_y is the analytical calculation of the vertical elastic resistance below the slip surfaces, q_x is the numerical calculation of the horizontal interaction between the antislide members and geotechnical materials, q_y 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 Figure 12. 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.

A comparative table of the structural displacements of the antislide members is given in Table 4. 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.

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 Figure 13, 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.

A comparative table of the axial force of the antislide members is given in Table 5. 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.

4.4.3. Comparative Analysis of the Shear Force

A comparative figure of the shear force of the ASCL is shown in Figure 15. 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.

A comparative table of the shear force of the antislide members is given in Table 6. 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.

4.4.4. Comparative Analysis of the Bending Moment

The comparative figure of the bending moment of the ASCL is shown in Figure 16. 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.

A comparative table of the bending moments of the antislide members is shown in Table 7. 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.

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 Figure 17. 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 Table 8. 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.

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.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This research was supported by the Natural Science Foundation of Education Department of Sichuan Province (Grant no. 17ZB0013) and the Natural Science Foundation of Chengdu Textile College (Grant no. 2015fzlkc06).

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Copyright © 2020 Gong Chen and Qiang Xu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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