Modified analytical analysis for calculating the stress of anti-slide structures with continuous ladders basing on geological environmental protection

Abstract

In anti-slide structures with continuous ladders (hereinafter referred to as ASCLs), horizontal and vertical reinforced concrete anti-slide members are connected head-to-tail in a ladder-like configuration, positioned along the sliding surfaces of slopes. The anti-slide members are interconnected and anchored from the sliding mass through the sliding zone into the underlying intact bedrock, so as to resist the landslide thrust and replace the weak materials within the sliding zone. By improving the load arrangement, the structural stress of the horizontal anti-slide members, and landslide thrust calculation of ASCLs, numerical extreme values of the axial force, shear force and bending moment were verified in a reasonable numerical range via the extreme values of the analytical and modified analytical results. The extreme value of the axial force of the modified analytical calculation result was between the analytical and numerical results. The maximum modified analytical calculation result of the shear force was between the analytical and numerical results, and the minimum modified analytical calculation result of the shear force was relatively close to the analytical and numerical results. The extreme modified analytical calculation result of the bending moment of the ASCLs was between the analytical and numerical results. The correctness of the numerical calculation results was well verified via the analytical and modified analytical results.

1. Introduction

1.1. Literature review

Fattah, M. Y. et al. ¹ investigated the response and behavior of machine foundations resting on dry and saturated sand experimentally. It was found that the rate of increase in excess pore water pressure ratio decreased remarkably at a depth of 0.5 B–1.5 B (B is the footing width) for medium and loose dense sand, respectively. Moreover, excess pore water pressure ratio increased with increasing the eccentricity of dynamic load. The generated pore water pressure was always greater under the point of load application. Its value reduced with a certain percentages at any point away from the point of load application. Aqoub et al. ² 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. ³ 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 ⁴ 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.

Zhu et al. ⁵ took the Monkey Stone landslide in Fengjie County, Chongqing as an example, and used FLAC3D to analysis the slope stability reinforced by continuous anti-slide key structures. The shear deformation, displacement, safety and stability coefficient of the sliding mass, as well as the mechanical and deformation characteristics of the anti-slide structure under different water level conditions had been researched. Wang et al. ⁶ calculated and analyzed the waste slope reinforced by continuous anti-slide keys via the Geo-Studio numerical simulation software. Based on the multi-objective optimization theory, the optimal design scheme for continuous anti-slide keys was obtained, thereby achieving the best balance between safety and economy. Gong Chen et al. ⁷ analyzed the analytical and numerical calculations of the mechanical behaviour of the ASCLs, the scientific research and academic value of the analytical calculation formulas for the stress on the ASCLs had been derived and deduced, and the analytical and numerical simulation calculation results had been compared and analysed for verification. However, the essential reasons for the significant differences between the analytical calculation and numerical simulation results had not been discussed. Furthermore, the reasonability of the structural stress analysis calculation model and formula should be revised and improved.

1.2. Summary of the structures

Anti-slide structures with continuous ladders (ASCLs), also known as stepped displacement anti-slide keys, stepped-key structures or continuous pile-key structures, differ significantly from anti-slide piles and anti-slide keys in terms of their structural form, structural layout and structural stress. The horizontal and vertical reinforced concrete anti-slide members connected head-to-tail in a ladder-like configuration are positioned along the slip surface of a slope. ASCLs are complex and hyperstatic mechanical systems. ASCLs are applicable to landslides where the rock masses above and below slip surfaces are integrated, the structures in the sliding bodies are intact, the positions of the slip surfaces are clear and deep, and the landslide thrust is large. The global stability of 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 The Monkey Stone landslide in Fengjie County of Chongqing city, representing the most complex form of geohazard governance in the Three Gorges Reservoir Area in terms of sequential bedrock landslides. ⁷ The longitudinal profile of one ASCL along the slip surface of a slope is shown in Figure 1. Many ASCLs are connected via horizontal binding beams that are similar to the binding beams in the frame structures to increase the global stability and anti-slide capacity. The plane layout of one ASCL is shown in Figure 2.

Figure 1.

The longitudinal profile of one ASCL along the slip surface of a slope. ⁷

Figure 2.

The plane layout of one ASCL. ⁷

1.3. Research significance

1.3.1. Research limitations

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, the modified analytical analysis methods is required to verify them. The deficiencies of the work by Gong Chen et al. ⁷ are as follows: 1. It is inappropriate for the slope surface to be covered with building loads, and the influence of water level fluctuations, such as the water level and heavy rain, should also be considered. 2. The effect of the structural load may decrease, ignoring the self-weight effect of the horizontal anti-slide members. 3. The thrust value of the landslide calculated by the transfer coefficient method is too large because of the transfer coefficient method, which is rooted in limit equilibrium theory. Furthermore, the function of the actual multistage retaining structure has been not considered when the single transfer coefficient method is used to calculate the landslide thrust of ASCLs.

1.3.2. Research significance

The research significance is as follows: 1. The accuracy of the stress analysis results is improved by modifying the analytical calculation method for the stress of ASCLs. 2. Scientific references and a basis for the formulation of relevant norms and the stress analysis of ASCLs are provided.

2. Improvement measures

On the basis of the above research limitations, this section describes the modified measures proposed in this study. Improvement measures are proposed with respect to three aspects: the layout of building load, the stress calculation of horizontal anti-slide members, and the landslide thrust calculation.

2.1. Improvement measures for building load layout

Considering the influence of landslides, which suffer from impact, such as water level changes and heavy rain, the building load at a position above the highest water level of the slope surface is arranged, and the building load (40 kN/m²) is applied to the landslide mass above the elevation of 193 m.

2.2. Improvement measures for stress calculation of horizontal anti-slide members

Because of the anti-slide member (elastic member) EF, which is arranged horizontally, the foundation coefficient, which is K_V along the vertical direction (at the same ground depth), can be approximately regarded as equal. The initial load effect and stress diagram of the elastic member EF are shown in Figure 3. The differential relation equation between the displacement function of the elastic member EF and the load function q(x) is shown in Formula (1). Formula (1) is a fourth-order nonhomogeneous differential equation with constant coefficients, and the fourth-order homogeneous differential equation with constant coefficients is shown in Formula (2). The analytic solution to Formula (2) is obtained as follows: the characteristic root of λ is shown in Formula (3) by substituting the formula y=e^λx into Formula (2); the particular solution of formula (2) is shown in Formula (4). The Euler formula and its transformations are subsequently shown in Formula (5).

$\frac{d^{4}y}{d{x}^4}+4{\beta }_{EF}^{4}y=\frac{q\left(x\right)}{E_{EF}{I}_{EF}}$

(1)

$\frac{d^{4}y}{{dx}^4}+4{\beta }_{EF}^{4}y=0$

(2)

$\{\begin{array}{l}{\lambda }^4+4{\beta }_{EF}^4=0\\ {\lambda }^2=\pm \sqrt{-4{\beta }_{EF}^4}=\pm \sqrt{i^{2}4{\beta }_{EF}^4}=\pm 2i{\beta }_{EF}^2=\{\begin{array}{l}2i{\beta }_{EF}^2\\ -2i{\beta }_{EF}^2\end{array}\\ \lambda =\{\begin{array}{l}\pm \sqrt{2i{\beta }_{EF}^2}=\{\begin{array}{l}{\beta }_{EF}\sqrt{2i}\\ -{\beta }_{EF}\sqrt{2i}\end{array}\\ \pm \sqrt{-2i{\beta }_{EF}^2}=\{\begin{array}{l}{\beta }_{EF}i\sqrt{2i}\\ -{\beta }_{EF}i\sqrt{2i}\end{array}\end{array}\\ \{\begin{array}{l}{\left(1+i\right)}^2=1+2i+i^2=2i,i^2=-1\\ \sqrt{2i}=\sqrt{{\left(1+i\right)}^2}=1+i\end{array}\\ \lambda =\{\begin{array}{l}{\lambda }_1={\beta }_{EF}\left(1+i\right)\\ {\lambda }_2=-{\beta }_{EF}\left(1+i\right)\\ {\lambda }_3={\beta }_{EF}i\left(1+i\right)={\beta }_{EF}i+{\beta }_{EF}{i}^2={\beta }_{EF}i-{\beta }_{EF}\\ {\lambda }_4=-{\beta }_{EF}i\left(1+i\right)=-{\beta }_{EF}i-{\beta }_{EF}{i}^2=-{\beta }_{EF}i+{\beta }_{EF}\end{array}\end{array}$

(3)

$\{\begin{array}{l}{y}_1=e^{{\beta }_{EF}x}{e}^{{\beta }_{EF}ix}\\ y_2=e^{{-\beta }_{EF}x}{e}^{{-\beta }_{EF}ix}\\ y_3=e^{-{\beta }_{EF}x}{e}^{{\beta }_{EF}ix}\\ y_4=e^{{\beta }_{EF}x}{e}^{{-\beta }_{EF}ix}\end{array}$

(4)

$\{\begin{array}{l}\hspace{0.17em}{e}^{\left(a+ib\right)x}=e^{ax}\left(cosx+isinbx\right)\\ e^{ibx}=cosbx+isinbx\\ e^{-ibx}=cosbx-isinbx\\ \mathit{cos}\hspace{0.17em}{\beta }_{EF}x=\frac{1}{2}\left(e^{i{\beta }_{EF}x}+e^{-i{\beta }_{EF}x}\right)\\ \mathit{sin}\hspace{0.17em}{\beta }_{EF}x=\frac{1}{2i}\left(e^{i{\beta }_{EF}x}-e^{-i{\beta }_{EF}x}\right)\end{array}$

(5)

Figure 3.

The initial load effect and stress diagram of the elastic member EF.

By being transformed with the Euler formula, the particular solution displayed in Formula (2), which is a real number solution, is shown in Formula (6), and the transformation process is shown in the superposition principle of advanced mathematics (Volume 1), which is not elaborated here. Because the general solution of Formula (2) is a linear combination of its four particular solutions, which are linearly independent, the general solution of Formula (2) is shown in Formula (7).

The boundary conditions of the elastic member EF are shown in Formula (8). Formula (9) is deduced by substituting Formula (7) into Formula (8); then, Formula (10) is deduced from Formula (9). Moreover, Formula (11) is deduced from uniting Formula (10), and the expression of y(x), which is shown in Formula (12), is deduced by substituting Formula (11) into Formula (7).

$\{\begin{array}{l}{y}_1=e^{{\beta }_{EF}x}\hspace{0.17em}\mathit{cos}{\beta }_{EF}x\\ y_2=e^{{\beta }_{EF}x}\hspace{0.17em}\mathit{sin}{\beta }_{EF}x\\ y_3=e^{-{\beta }_{EF}x}\hspace{0.17em}\mathit{cos}{\beta }_{EF}x\\ y_4=e^{-{\beta }_{EF}x}\hspace{0.17em}\mathit{sin}{\beta }_{EF}x\end{array}$

(6)

${y=e}^{{\beta }_{EF}x}\left(\mathit{Acos}{\beta }_{EF}x+Bsin{\beta }_{EF}x\right)+e^{-{\beta }_{EF}x}\left(Ccos{\beta }_{EF}x+Dsin{\beta }_{EF}x\right)$

(7)

$\{\begin{array}{l}y=f\left(x\right),y{│}_{x=0}=y_{FE}\\ \frac{dy}{dx}=\phi \left(x\right),\frac{dy}{dx}{│}_{x=0}={\phi }_{FE}\\ \frac{d^{2}y}{d{x}^2}=\frac{M\left(x\right)}{E_{EF}{I}_{EF}},\frac{d^{2}y}{d{x}^2}{│}_{x=0}=\frac{M_{FE}}{E_{EF}{I}_{EF}}\\ \frac{d^{3}y}{d{x}^3}=\frac{Q\left(x\right)}{E_{EF}{I}_{EF}},\frac{d^{3}y}{d{x}^3}{│}_{x=0}=\frac{F_{QFE}}{E_{EF}{I}_{EF}}\end{array}$

(8)

$\{\begin{array}{l}{y=e}^{{\beta }_{EF}x}\left(\mathit{Acos}{\beta }_{EF}x+Bsin{\beta }_{EF}x\right)+e^{-{\beta }_{EF}x}\left(Ccos{\beta }_{EF}x+Dsin{\beta }_{EF}x\right)\\ {y│}_{x=0}=A+C=y_{FE}\\ \frac{dy}{dx}=A\left({\beta }_{EF}{e}^{{\beta }_{EF}x}\mathit{cos}{\beta }_{EF}x-{\beta }_{EF}{e}^{{\beta }_{EF}x}\mathit{sin}{\beta }_{EF}x\right)+B\left({\beta }_{EF}{e}^{{\beta }_{EF}x}\mathit{sin}{\beta }_{EF}x+{\beta }_{EF}{e}^{{\beta }_{EF}x}\mathit{cos}{\beta }_{EF}x\right)+\\ \hspace{1em}C\left(-{\beta }_{EF}{e}^{-{\beta }_{EF}x}\mathit{cos}{\beta }_{EF}x-{\beta }_{EF}{e}^{-{\beta }_{EF}x}\mathit{sin}{\beta }_{EF}x\right)+D\left(-{\beta }_{EF}{e}^{-{\beta }_{EF}x}\mathit{sin}{\beta }_{EF}x+{\beta }_{EF}{e}^{-{\beta }_{EF}x}\mathit{cos}{\beta }_{EF}x\right)\\ \frac{dy}{dx}{│}_{x=0}=A{\beta }_{EF}+B{\beta }_{EF}-C{\beta }_{EF}+D{\beta }_{EF}={\phi }_{FE}\\ \frac{d^{2}y}{d{x}^2}=A\left({\beta }_{EF}^2{e}^{{\beta }_{EF}x}\hspace{0.17em}\mathit{cos}{\beta }_{EF}x-{\beta }_{EF}^2{e}^{{\beta }_{EF}x}\mathit{sin}{\beta }_{EF}x-{\beta }_{EF}^2{e}^{{\beta }_{EF}x}\hspace{0.17em}\mathit{sin}{\beta }_{EF}x-{\beta }_{EF}^2{e}^{{\beta }_{EF}x}\mathit{cos}{\beta }_{EF}x\right)+\\ \hspace{1em}B\left({\beta }_{EF}^2{e}^{{\beta }_{EF}x}\mathit{sin}{\beta }_{EF}x+{\beta }_{EF}^2{e}^{{\beta }_{EF}x}\mathit{cos}{\beta }_{EF}x+{\beta }_{EF}^2{e}^{{\beta }_{EF}x}\mathit{cos}{\beta }_{EF}x-{\beta }_{EF}^2{e}^{{\beta }_{EF}x}\mathit{sin}{\beta }_{EF}x\right)+\\ \hspace{1em}C\left({\beta }_{EF}^2{e}^{-{\beta }_{EF}x}\mathit{cos}{\beta }_{EF}x+{\beta }_{EF}^2{e}^{-{\beta }_{EF}x}\mathit{sin}{\beta }_{EF}x+{\beta }_{EF}^2{e}^{-{\beta }_{EF}x}\mathit{sin}{\beta }_{EF}x-{\beta }_{EF}^2{e}^{-{\beta }_{EF}x}\mathit{cos}{\beta }_{EF}x\right)+\\ \hspace{1em}D\left({\beta }_{EF}^2{e}^{-{\beta }_{EF}x}\mathit{sin}{\beta }_{EF}x-{\beta }_{EF}^2{e}^{-{\beta }_{EF}x}\mathit{cos}{\beta }_{EF}x-{\beta }_{EF}^2{e}^{-{\beta }_{EF}x}\mathit{cos}{\beta }_{EF}x-{\beta }_{EF}^2{e}^{-{\beta }_{EF}x}\mathit{sin}{\beta }_{EF}x\right)\\ {\frac{d^{2}y}{d{x}^2}│}_{x=0}=2B{\beta }_{EF}^2-2D{\beta }_{EF}^2=\frac{M_{FE}}{E_{EF}{I}_{EF}}\\ \frac{d^{3}y}{d{x}^3}=-2A{\beta }_{EF}^2\left({\beta }_{EF}{e}^{{\beta }_{EF}x}\mathit{sin}{\beta }_{EF}x+{\beta }_{EF}{e}^{{\beta }_{EF}x}\mathit{cos}{\beta }_{EF}x\right)+2B{\beta }_{EF}^2\left({{\beta }_{EF}e}^{{\beta }_{EF}x}\mathit{cos}{\beta }_{EF}x-{\beta }_{EF}{e}^{{\beta }_{EF}x}\mathit{sin}{\beta }_{EF}x\right)+\\ 2C{\beta }_{EF}^2\left(-{\beta }_{EF}{e}^{-{\beta }_{EF}x}\mathit{sin}{\beta }_{EF}x+{\beta }_{EF}{e}^{-{\beta }_{EF}x}\mathit{cos}{\beta }_{EF}x\right)-2D{\beta }_{EF}^2\left(-{\beta }_{EF}{e}^{-{\beta }_{EF}x}\mathit{cos}{\beta }_{EF}x-{\beta }_{EF}{e}^{-{\beta }_{EF}x}\mathit{sin}{\beta }_{EF}x\right)\\ {\frac{d^{3}y}{d{x}^3}│}_{x=0}=-2A{\beta }_{EF}^3+2B{\beta }_{EF}^3+2C{\beta }_{EF}^3+2D{\beta }_{EF}^3=\frac{F_{QFE}}{E_{EF}{I}_{EF}}\end{array}$

(9)

$\{\begin{array}{l}A+C=y_{FE}\\ A+B-C+D=\frac{{\phi }_{FE}}{{\beta }_{EF}}\\ B-D=\frac{M_{FE}}{2E_{EF}{I}_{EF}{\beta }_{EF}^2}\\ A-B-C-D=-\frac{F_{QFE}}{2E_{EF}{I}_{EF}{\beta }_{EF}^3}\end{array}$

(10)

$\{\begin{array}{l}A=\frac{4E_{EF}{I}_{EF}{\beta }_{EF}^3{y}_{FE}+2E_{EF}{I}_{EF}{\beta }_{EF}^2{\phi }_{FE}-F_{QFE}}{8E_{EF}{I}_{EF}{\beta }_{EF}^3}\\ B=\frac{2E_{EF}{I}_{EF}{\beta }_{EF}^2{\phi }_{FE}+F_{QFE}+2M_{FE}{\beta }_{EF}}{8E_{EF}{I}_{EF}{\beta }_{EF}^3}\\ C=\frac{4E_{EF}{I}_{EF}{\beta }_{EF}^3{y}_{FE}-2E_{EF}{I}_{EF}{\beta }_{EF}^2{\phi }_{FE}+F_{QFE}}{8E_{EF}{I}_{EF}{\beta }_{EF}^3}\\ D=\frac{2E_{EF}{I}_{EF}{\beta }_{EF}^2{\phi }_{FE}+F_{QFE}-2M_{FE}{\beta }_{EF}}{8E_{EF}{I}_{EF}{\beta }_{EF}^3}\end{array}$

(11)

$\begin{array}{l}y=e^{{\beta }_{EF}x}\left(\frac{4E_{EF}{I}_{EF}{\beta }_{EF}^3{y}_{FE}+2E_{EF}{I}_{EF}{\beta }_{EF}^2{\phi }_{FE}-F_{QFE}}{8E_{EF}{I}_{EF}{\beta }_{EF}^3}\mathit{cos}{\beta }_{EF}x+\frac{2E_{EF}{I}_{EF}{\beta }_{EF}^2{\phi }_{FE}+F_{QFE}+2M_{FE}{\beta }_{EF}}{8E_{EF}{I}_{EF}{\beta }_{EF}^3}\mathit{sin}{\beta }_{EF}x\right)+\\ {\hspace{1em}e}^{{-\beta }_{EF}x}\left(\frac{4E_{EF}{I}_{EF}{\beta }_{EF}^3{y}_{FE}-2E_{EF}{I}_{EF}{\beta }_{EF}^2{\phi }_{FE}+F_{QFE}}{8E_{EF}{I}_{EF}{\beta }_{EF}^3}\mathit{cos}{\beta }_{EF}x+\frac{2E_{EF}{I}_{EF}{\beta }_{EF}^2{\phi }_{FE}+F_{QFE}-2M_{FE}{\beta }_{EF}}{8E_{EF}{I}_{EF}{\beta }_{EF}^3}\mathit{sin}{\beta }_{EF}x\right)\end{array}$

(12)

The derivatives of the trigonometric and hyperbolic functions are shown in Formula (13). Formula (14) is deduced by uniting Formula (8), Formula (12) and Formula (13). The first to fourth derivatives of φ₁, φ₂, φ₃ and φ₄ in sequence are shown in Formula (15). Moreover, Formula (16) is deduced by uniting Formula (14) and Formula (15).

$\{\begin{array}{l}shx=\frac{e^x-e^{-x}}{2}\\ chx=\frac{e^x+e^{-x}}{2}\\ {\left[sinx\right]}^{\prime }=cosx\\ {\left[cosx\right]}^{\prime }=-sinx\\ {\left[shx\right]}^{\prime }=chx\\ {\left[chx\right]}^{\prime }=shx\end{array}$

(13)

$\{\begin{array}{l}y\left(x\right)={\phi }_1{y}_{FE}+{\phi }_2\frac{{\phi }_{FE}}{{\beta }_{EF}}+{\phi }_3\frac{M_{FE}}{E_{EF}{I}_{EF}{\beta }_{EF}^2}+{\phi }_4\frac{F_{QFE}}{E_{EF}{I}_{EF}{\beta }_{EF}^3}\\ \frac{dy}{dx}=\phi \left(x\right)=\frac{d{\phi }_1}{dx}{y}_{FE}+\frac{d{\phi }_2}{dx}\frac{{\phi }_{FE}}{{\beta }_{EF}}+\frac{d{\phi }_3}{dx}\frac{M_{FE}}{E_{EF}{I}_{EF}{\beta }_{EF}^2}+\frac{d{\phi }_4}{dx}\frac{F_{QFE}}{E_{EF}{I}_{EF}{\beta }_{EF}^3}\\ \frac{d^{2}y}{d{x}^2}=\frac{M\left(x\right)}{E_{EF}{I}_{EF}}={\frac{d^2{\phi }_1}{d{x}^2}y}_{FE}+\frac{d^2{\phi }_2}{d{x}^2}\frac{{\phi }_{FE}}{{\beta }_{EF}}+\frac{d^2{\phi }_3}{d{x}^2}\frac{M_{FE}}{E_{EF}{I}_{EF}{\beta }_{EF}^2}+\frac{d^2{\phi }_4}{d{x}^2}\frac{F_{QFE}}{E_{EF}{I}_{EF}{\beta }_{EF}^3}\\ \frac{d^{3}y}{d{x}^3}=\frac{Q\left(x\right)}{E_{EF}{I}_{EF}}=\frac{d^3{\phi }_1}{d{x}^3}{y}_{FE}+\frac{d^3{\phi }_2}{d{x}^3}\frac{{\phi }_{FE}}{{\beta }_{EF}}+\frac{d^3{\phi }_3}{d{x}^3}\frac{M_{FE}}{E_{EF}{I}_{EF}{\beta }_{EF}^2}+\frac{d^3{\phi }_4}{d{x}^3}\frac{F_{QFE}}{E_{EF}{I}_{EF}{\beta }_{EF}^3}\\ {\phi }_1=\mathit{cos}\hspace{0.17em}{\beta }_{EF}xch{\beta }_{EF}x\\ {\phi }_2=\frac{\mathit{sin}\hspace{0.17em}{\beta }_{EF}xch{\beta }_{EF}x+\mathit{cos}{\beta }_{EF}xsh{\beta }_{EF}x}{2}\\ {\phi }_3=\frac{\mathit{sin}\hspace{0.17em}{\beta }_{EF}xsh{\beta }_{EF}x}{2}\\ {\phi }_4=\frac{\mathit{sin}\hspace{0.17em}{\beta }_{EF}xch{\beta }_{EF}x-\mathit{cos}{\beta }_{EF}xsh{\beta }_{EF}x}{4}\end{array}$

(14)

$\{\begin{array}{l}{\phi }_1=\mathit{cos}{\beta }_{EF}xch{\beta }_{EF}x\\ \frac{d{\phi }_1}{dx}={\beta }_{EF}\left(\mathit{cos}{\beta }_{EF}xsh{\beta }_{EF}x-\mathit{sin}{\beta }_{EF}xch{\beta }_{EF}x\right)=-4{\beta }_{EF}{\phi }_4\\ \frac{d^2{\phi }_1}{d{x}^2}=-2{\beta }_{EF}^2\mathit{sin}{\beta }_{EF}xsh{\beta }_{EF}x=-4{\beta }_{EF}^2{\phi }_3\\ \frac{d^3{\phi }_1}{d{x}^3}=-2{\beta }_{EF}^3\left(\mathit{sin}{\beta }_{EF}xch{\beta }_{EF}x+\mathit{cos}{\beta }_{EF}xsh{\beta }_{EF}x\right)=-4{\beta }_{EF}^3{\phi }_2\\ \frac{d^4{\phi }_1}{d{x}^4}=-4{\beta }_{EF}^4{\phi }_1\\ {\phi }_2=\frac{\mathit{sin}{\beta }_{EF}xch{\beta }_{EF}x+\mathit{cos}{\beta }_{EF}xsh{\beta }_{EF}x}{2}\\ \frac{d{\phi }_2}{dx}={\beta }_{EF}\mathit{cos}{\beta }_{EF}xch{\beta }_{EF}x={\beta }_{EF}{\phi }_1\\ \frac{d^2{\phi }_2}{d{x}^2}={\beta }_{EF}\frac{d{\phi }_1}{dx}=-4{\beta }_{EF}^2{\phi }_4\\ \frac{d^3{\phi }_2}{d{x}^3}={\beta }_{EF}\frac{d^2{\phi }_1}{d{x}^2}=-4{\beta }_{EF}^3{\phi }_3\\ \frac{d^4{\phi }_2}{d{x}^4}=-4{\beta }_{EF}^4{\phi }_2\\ {\phi }_3=\frac{\mathit{sin}{\beta }_{EF}xsh{\beta }_{EF}x}{2}\\ \frac{d{\phi }_3}{dx}=\frac{{\beta }_{EF}}{2}\left(\mathit{sin}{\beta }_{EF}xch{\beta }_{EF}x+\mathit{cos}{\beta }_{EF}xsh{\beta }_{EF}x\right)={\beta }_{EF}{\phi }_2\\ \frac{d^2{\phi }_3}{d{x}^2}={\beta }_{EF}\frac{d{\phi }_2}{dx}={\beta }_{EF}^2{\phi }_1\\ \frac{d^3{\phi }_3}{d{x}^3}={\beta }_{EF}\frac{d^2{\phi }_2}{d{x}^2}=-4{\beta }_{EF}^3{\phi }_4\\ \frac{d^4{\phi }_3}{d{x}^4}=-4{\beta }_{EF}^4{\phi }_3\\ {\phi }_4=\frac{\mathit{sin}{\beta }_{EF}xch{\beta }_{EF}x-\mathit{cos}{\beta }_{EF}xsh{\beta }_{EF}x}{4}\\ \frac{d{\phi }_4}{dx}=\frac{{\beta }_{EF}\mathit{sin}{\beta }_{EF}xsh{\beta }_{EF}x}{2}={\beta }_{EF}{\phi }_3\\ \frac{d^2{\phi }_4}{d{x}^2}={\beta }_{EF}\frac{d{\phi }_3}{dx}={\beta }_{EF}^2{\phi }_2\\ \frac{d^3{\phi }_4}{d{x}^3}={\beta }_{EF}\frac{d^2{\phi }_3}{d{x}^2}={\beta }_{EF}^3{\phi }_1\\ \frac{d^4{\phi }_4}{d{x}^4}=-4{\beta }_{EF}^4{\phi }_4\end{array}$

(15)

$\{\begin{array}{l}y\left(x\right)={\phi }_1{y}_{FE}+{\phi }_2\frac{{\phi }_{FE}}{{\beta }_{EF}}+{\phi }_3\frac{M_{FE}}{E_{EF}{I}_{EF}{\beta }_{EF}^2}+{\phi }_4\frac{F_{QFE}}{E_{EF}{I}_{EF}{\beta }_{EF}^3}\\ \phi \left(x\right)=\left(-4{\beta }_{EF}{\phi }_4\right)y_{FE}+{\phi }_1{\phi }_{FE}+{\phi }_2\frac{M_{FE}}{E_{EF}{I}_{EF}{\beta }_{EF}}+{\phi }_3\frac{F_{QFE}}{E_{EF}{I}_{EF}{\beta }_{EF}^2}\\ M\left(x\right)=E_{EF}{I}_{EF}\left[{\left(-4{\beta }_{EF}^2{\phi }_3\right)y}_{FE}+\left(-4{\beta }_{EF}{\phi }_4\right){\phi }_{FE}+{\phi }_1\frac{M_{FE}}{E_{EF}{I}_{EF}}+{\phi }_2\frac{F_{QFE}}{E_{EF}{I}_{EF}{\beta }_{EF}}\right]\\ Q\left(x\right)=E_{EF}{I}_{EF}\left[\left(-4{\beta }_{EF}^3{\phi }_2\right)y_{FE}+{\left(-4{\beta }_{EF}^2{\phi }_3\right)\phi }_{FE}+\left(-4{\beta }_{EF}{\phi }_4\right)\frac{M_{FE}}{E_{EF}{I}_{EF}}+{\phi }_1\frac{F_{QFE}}{E_{EF}{I}_{EF}}\right]\end{array}$

(16)

The real solution of Formula (2) is deduced by Formula (3)–Formula (16). Furthermore, according to Formula (16), Formula (17) is the general solution of Formula (1), which should conform to Formula (1) and the boundary conditions of the elastic member EF.

$\{\begin{array}{l}y\left(x\right)={\phi }_1\left(y_{FE}-\frac{q\left(x\right)}{K_V{B}_{P\left(EF\right)}}\right)+{\phi }_2\frac{{\phi }_{FE}}{{\beta }_{EF}}+{\phi }_3\frac{M_{FE}}{E_{EF}{I}_{EF}{\beta }_{EF}^2}+{\phi }_4\frac{F_{QFE}}{E_{EF}{I}_{EF}{\beta }_{EF}^3}+\frac{q\left(x\right)}{K_V{B}_{P\left(EF\right)}}\\ \phi \left(x\right)=\left(-4{\beta }_{EF}{\phi }_4\right)\left(y_{FE}-\frac{q\left(x\right)}{K_V{B}_{P\left(EF\right)}}\right)+{\phi }_1{\phi }_{FE}+{\phi }_2\frac{M_{FE}}{E_{EF}{I}_{EF}{\beta }_{EF}}+{\phi }_3\frac{F_{QFE}}{E_{EF}{I}_{EF}{\beta }_{EF}^2}\\ M\left(x\right)=E_{EF}{I}_{EF}\left[\left(-4{\beta }_{EF}^2{\phi }_3\right)\left(y_{FE}-\frac{q\left(x\right)}{K_V{B}_{P\left(EF\right)}}\right)+\left(-4{\beta }_{EF}{\phi }_4\right){\phi }_{FE}+{\phi }_1\frac{M_{FE}}{E_{EF}{I}_{EF}}+{\phi }_2\frac{F_{QFE}}{E_{EF}{I}_{EF}{\beta }_{EF}}\right]\\ Q\left(x\right)=E_{EF}{I}_{EF}\left[\left(-4{\beta }_{EF}^3{\phi }_2\right)\left(y_{FE}-\frac{q\left(x\right)}{K_V{B}_{P\left(EF\right)}}\right)+{\left(-4{\beta }_{EF}^2{\phi }_3\right)\phi }_{FE}+\left(-4{\beta }_{EF}{\phi }_4\right)\frac{M_{FE}}{E_{EF}{I}_{EF}}+{\phi }_1\frac{F_{QFE}}{E_{EF}{I}_{EF}}\right]\\ {\phi }_1=\mathit{cos}{\beta }_{EF}xch{\beta }_{EF}x\\ {\phi }_2=\frac{\mathit{sin}{\beta }_{EF}xch{\beta }_{EF}x+\mathit{cos}{\beta }_{EF}xsh{\beta }_{EF}x}{2}\\ {\phi }_3=\frac{\mathit{sin}{\beta }_{EF}xsh{\beta }_{EF}x}{2}\\ {\phi }_4=\frac{\mathit{sin}{\beta }_{EF}xch{\beta }_{EF}x-\mathit{cos}{\beta }_{EF}xsh{\beta }_{EF}x}{4}\\ {\beta }_{EF}={\left[\frac{K_V{B}_{P\left(EF\right)}}{4E_{EF}{I}_{EF}}\right]}^{\frac{1}{4}}\end{array}$

(17)

Here, f(x) is the vertical displacement function of the elastic member EF, x is the horizontal distance from any point of the elastic member EF to point F, β_EF is the deformation coefficient of the elastic member EF, E_EF is the material elastic modulus of the elastic member EF, I_EF is the inertia moment of cross-section of the elastic member EF, φ(x) is the rotation angle function of the elastic member EF, M(x) is the moment function of the elastic member EF, Q(x) is the shear force function of the elastic member EF, y_FE is the vertical displacement of point F on the elastic member EF, φ_FE is the angular displacement of point F on the elastic member EF, M_FE is the bending moment of point F on the elastic member EF, F_QFE is the shear force of point F on the elastic member EF, and B_P(EF) is the calculated width of the elastic member EF.

2.3. Improvement measures for landslide thrust calculation

The applicable conditions of various landslide thrust calculation methods are summarized as follows. Based on the geological characteristics of the Monkey Stone landslide, the transfer coefficient method is adopted to calculate its landslide thrust.

1. When the sliding surface is a single plane, the landslide thrust is calculated by using the method specifically designed for single planar sliding surfaces. ⁸

2. When the sliding surface is a circular arc or approximately regarded as a circular arc, the simplified Bishop method is adopted for calculating the landslide thrust, considering that the overall moment balance plays a dominant role in the stability analysis and the requirement for computational simplicity. ⁸

3. When the sliding surface is a continuous curved surface or composed of irregular (steep) broken line segments, the Janbu method is adopted to calculate the landslide thrust. ⁸

4. When the sliding surface is composed of broken line segments with gentle dip angles and slight variations between adjacent segments, the transfer coefficient method is adopted to calculate the landslide thrust. ⁸

5. When the sliding surface has a steep dip angle and the sliding mass exhibits obvious block dislocation during sliding, the block limit equilibrium method is adopted to calculate the landslide thrust. ⁸

The results of the landslide thrust and anti-slide force on the structure above the sliding surface directly affect the stress calculation and analysis results of the structure below the sliding surface and even the entire structure. On the basis of the unbalanced thrust method (transfer coefficient method), the landslide thrust acting on the structure above the sliding surface is calculated and analysed, and the method assumes that only thrust is transmitted between the blocks; however, tension is not transmitted, and the cracking between the blocks is not considered. Moreover, the landslide thrust at the top of the sliding block is 0. The assumption is feasible for the stress calculation in a single-stage retaining structure. However, the calculated value of the landslide thrust may be too large when the assumption is applied to the stress calculation in multistage retaining structures, such as multistage anti-slide piles, multistage anti-slide keys, multistage anti-shear holes, and ASCLs. The reasons underlying this phenomenon can be analyzed as follows.

1. When the multistage retaining structure effect is considered in the process of calculating the landslide thrust acting on multistage retaining structures and according to extreme design thinking, if any level of retaining structure has stabilized the landslide thrust of the block it is in, the theoretical landslide thrust exerted by this block on the subsequent block should be 0. Theoretically, the block where any retaining structure is located no longer transmits the landslide thrust downwards, the next block has no thrust acting on the previous block, and the subsequent piece may slip away. Therefore, the theoretical anti-slide force of any block where the retaining structure is located is 0.

2. Under the same conditions, an envelope phenomenon occurs in the landslide thrust curves when considering the multistage retaining structure effect and when neglecting it. That is, there is a landslide thrust envelope effect. The landslide thrust envelope curve of multistage anti-slide piles is shown in Figure 4, where the red curve encloses the yellow curve, the yellow curve encloses the green curve, the green curve encloses the purple curve, and the purple curve encloses the blue curve.

3. In Figure 4, when the stress on multistage retaining structures is calculated via the unbalanced thrust method (transfer coefficient method) without considering the effect of multistage retaining structures, relatively small stress occurs in the two ends of the multistage retaining structure, whereas greater stress is found in the middle and front parts of the structure due to the trend where the thrust curve of the landslide is small at both ends and large in the middle and front parts. Therefore, the analytical calculation result produces a certain delay–lag effect. When the stress on multistage retaining structures is calculated by the unbalanced thrust method (transfer coefficient method) considering the effect of multistage retaining structures, because the thrust curve of the landslide is a step-by-step envelope from the back to the front, it may cause larger stress in the rear structure of the multistage retaining structures and less stress in the front structure. Furthermore, five levels of anti-slide piles, namely, anti-slide pile 1, anti-slide pile 2, anti-slide pile 3, anti-slide pile 4 and anti-slide pile 5, are set from the back to the front (from left to right). When the stress on anti-slide pile 1 is calculated, the structural design of anti-slide pile 1 can be carried out on the basis of the landslide thrust curve corresponding to the red line. When the stress on anti-slide pile 2 is calculated, it is assumed that anti-slide pile 1 has balanced the unbalanced sliding force of the block (i.e., block 1). That is, the landslide thrust transmitted from block 1 to block 2 is 0, the structural design of anti-slide pile 2 can be carried out on the basis of the landslide thrust curve corresponding to the yellow line. The structural design of anti-slide pile 3 can be carried out on the basis of the landslide thrust curve corresponding to the green line, the structural design of anti-slide pile 4 can be carried out on the basis of the landslide thrust curve corresponding to the purple line, and the structural design of anti-slide pile 5 can be carried out on the basis of the landslide thrust curve corresponding to the blue line.

The hypothesis of “zero-block recursion” is innovatively proposed, which means that the calculation of the landslide thrust for the retaining structure at this level is designed under extreme conditions when the landslide thrust of the block where the upper-level retaining structure is located is 0. The hypothesis of “zero-block recursion” is applicable only to the stress calculation of multilevel retaining structures. Under the same conditions, the landslide thrust value calculated when considering the multilevel retaining structure effect is smaller than that calculated when the multilevel retaining structure effect is not considered. Notably, the landslide thrust value calculated ignoring the effect of multilevel retaining structures is relatively large; the anti-slide effect of the actual multilevel retaining structure, whose value tends to be conservative and safe, is not considered, and the thrust curve of the landslide tends to be small at first, then large, and then small again. In contrast, the landslide thrust value calculated considering the effect of multilevel retaining structures is relatively small, the anti-slide effect of the actual multilevel retaining structure, whose value tends to be overly theoretical, is considered, and the thrust curve of the landslide tends to increase at the rear and gradually decreases at the front. However, in actual multistage retaining structure landslides, interaction forces exist between each block of the sliding body; that is, the force in the block where the retaining structure of this stage is located is transmitted to the next block and relatively small with respect to the situation in which the multilevel retaining structure effect is ignored. According to Figures 3 and 11 in Gong Chen et al., ⁷ the block diagram of the sliding body of ASCLs is shown in Figure 5, to which the numbers of anti-slide members in subsequent figures and tables shall conform. The table of sliding body corresponding to anti-slide structures above the sliding surface is shown in Table 1. Notably, the bars at the back are those with smaller numerical numbers among bars 1 to 6 in Figure 4, such as block 1 which is at the end, and block 6 which is at the front. The components ranked later refer to those with capital letters among the AC, CE, EG, GI, IN, KM and NP components in Figure 5, such as member NP which is at the end, and member AC which is at the front. The original analytical calculation method proposed by Chen Gong et al. ⁷ does not consider the continuous support effect of the actual multilevel retaining structure on the sliding body blocks. However, the modified analytical calculation method accounts for the continuous support effect of the actual multilevel retaining structure on the sliding body blocks, theoretically assuming that the landslide thrust of the block where this stage of the retaining structure is located is 0 and that the next block may slide away. Therefore, there is no landslide thrust or anti-slide force acting on the block where this stage of the retaining structure is located.

Table 1.

The table of sliding body corresponding to anti-slide structures above the sliding surface.

Sequence number	Anti-slide structure above the sliding surface	Sliding block section of the structure	Sliding block of 0 sliding thrust	Sliding block of sliding thrust calculation	Resistance of sliding block section of the structure (kN)
1	BCD	4	0	0 - 21	0
2	FGH	6	4	4 - 21	0
3	JKL	8	6	6 - 21	0
4	KNO	9	8	8 - 21	0

Figure 4.

The landslide thrust envelope curve of multistage anti-slide piles.

Figure 5.

The block diagram of the sliding body of ASCLs.

3. Project overview and numerical calculation results

3.1. Project overview

The Monkey Stone landslide is located on the left bank of the Yangtze River in the Three Gorges Reservoir Area, at the front of the Sanmashan Community in the new county seat of Fengjie, Chongqing city, and along the river. The Monkey Stone landslide starts from Baiyangping Valley in the west, ends at Shuijinggou in the east, extends to Lianjie Avenue in the north and reaches the riverbed of the Yangtze River in the south. The landslide plane is fan shaped, and the sliding bodies below the riverside road spread out to the east and west. In the Monkey Stone landslide, which is a bedrock layer cutting landslide, the leading edge elevation is 100 m, the elevation at the rear edge is 250 m, the length is 370 m from north to south, the width is 320 m to 420 m from east to west, the area is 12.19×10⁴ m², the average stacking thickness is 45 m to 60 m, the thickest part is 66 m, and the volume is approximately 450×10⁴ m³. There are many buildings located on the sliding body, including the People’s Court of Fengjie County, the Outpatient Department of the Health School, the Maternal and Child Health Care Station, Buyun Street and Dengjiawu Market, the County Passenger Transport Center, the County Bus Transport Company, the County Port and Shipping Transport Center and Wharf, and the farmers’ market. The overall stability of the Monkey Stone landslide is closely related to the safety of two roads in Sanmashan Community, commercial pedestrian streets, water and land passenger transport, and many buildings and people’s lives and property in Fengjie County. Once a landslide becomes unstable after the Three Gorges Reservoir is filled with water, it directly threatens the lives and property of residents, damages the integrity of urban functions, and severely affects local social stability and economic development. The continuation and treatment project of the Monkey Stone landslide adopted a comprehensive treatment plan featuring the combination of the following measures: setting up stepped reinforced concrete replacement anti-slide keys (referred to as ladder keys) along the slide belt, underwater stone thrown at the front edge to press the foot, underground drainage, bank slope protection in water level fluctuation areas, and safety monitoring. ⁷

Taking the type of ASCL in the Monkey Stone landslide as an example, a landslide mass with a unit width of 1 m was extracted along the transverse direction of the landslide to calculate and analyse its stability and the stress on the anti-slide structure. According to the implicit solution method in the transfer coefficient method, the safety and stability coefficient (Fs) of the landslide is calculated. By using this coefficient, the cohesion force and internal friction angle of the sliding strip are continuously reduced until the remaining sliding thrust of the last sliding block approaches 0. Then, the safety and stability coefficient (Fs) of the landslide can be calculated. The safety stability coefficient (Fs) of the landslide is set to meet its strength requirements. The combination of the load conditions for landslides is as follows: self-weight of the sliding body + building load: 40 kN/m² (above 193 m elevation of the sliding body) + 175 m normal water storage level in front of the dam (stable static water level) + a rainstorm that occurs only once every five years. The suggested design values of the physical and mechanical parameters of the rock and soil mass of the Monkey Stone landslide are shown in Table 2, ⁹ and the member design parameters of the ASCLs are shown in Table 3. ⁷ In accordance with the content described in Table 2, ⁹ the coefficients of the sliding bed foundation in the Monkey Stone landslide are as follows: K_V=400000 kN/m³, K_H=280000 kN/m³.

Table 2.

The suggested design values of the physical and mechanical parameters of the rock and soil mass of the Monkey Stone landslide. ⁹ .

Landslide location	Material	Density (kN/m3) natural/Saturation	Cohesion (kPa) natural/Saturation	Angle of internal friction (°) natural/Saturation	Elastic modulus (MPa)	Tensile strength (MPa)	Poisson’s ratio
Sliding body	Broken stone and soil	23/23.5	60/50	28/25	1960	0.01	0.29
Sliding zone	Slip soil	23/23.5	35/23	23.4/20.7	26	0.005	0.34
Sliding bed	Argillaceous siltstone, Silty sandstone	25.3/25.6	1700/1400	47/42	6500	0.05	0.25

Table 3.

The member design parameters of the ASCLs. ⁷ .

Sequence number	Member	Member length (m)	Partition length		Section size (length × width, m); mechanical properties of the members	Strength of concrete; Poisson’s ratio	Strength of rebar	Elastic modulus(Mpa)
			Partition member	Length (m)
1	AC	9.434	AB	4.717	2.5 × 2.5; (elastic member)	C40; 0.2	HRB400; HRB335	2.6×104
			BC	4.717	2.5 × 2.5; (-)
2	CE	10.000	CD	5.002	2.5 × 2.5; (-)
			DE	4.998	2.5 × 2.5; (elastic member)
3	EG	15.752	EF	7.876	2.5 × 2.5; (elastic member)
			FG	7.876	2.5 × 2.5; (-)
4	GI	10.000	GH	4.479	2.5 × 2.5; (-)
			HI	5.521	2.5 × 2.5; (elastic member)
5	IN	30.624	IJ	9.708	2.5 × 2.5; (elastic member)
			JK	9.708	2.5 × 2.5; (-)
			KN	11.208	2.5 × 2.5; (-)
6	KM	12.624	KL	5.562	2.5 × 2.5; (-)
			LM	7.062	2.5 × 2.5; (elastic member)
7	NP	22.414	NO	10.457	2.5 × 2.5; (-)
			OP	11.957	2.5 × 2.5; (elastic member)

3.2. Numerical simulation calculation and analysis results of the structure

The stress of the ASCLs in the Monkey Stone landslide has been numerically simulated, calculated and analysed via MIDAS GTS NX. In the numerical simulation, horizontal constraints are applied to the vertical edges of the sliding bed, while both horizontal and vertical constraints are imposed on its horizontal edges. The mesh sizes of the sliding mass, sliding zone and sliding bed are all 10 meters. Materials above the maximum water level are in a natural state, whereas those below are fully saturated. Geotechnical material characteristic parameters are assigned in accordance with Table 2, and material characteristic parameters of anti-slide structures are assigned in accordance with Table 3. Slope stability analysis is performed using the SAM method for the investigated cases.

The numerical simulation modelling diagram and displacement cloud diagram of the Monkey Stone landslide are shown in Figures 6 and 7, respectively. The axial force, shear force and bending moment diagram of ASCLs obtained from the numerical calculations are shown in Figures 8–10, respectively. In Figure 7, the displacement of the sliding body above the ASCLs is small; conversely, it is larger elsewhere. This phenomenon confirms that the anti-slide structures play a significant supporting role for the sliding body above it, but its supporting role for the sliding body below it is relatively small. Therefore, the continuation and treatment project of the Monkey Stone landslide adopted a comprehensive treatment plan featuring the following measures: setting up stepped reinforced concrete anti-slide keys (referred to as ladder keys) along the slide belt, underwater stone thrown at the front edge to press the foot, underground drainage, bank slope protection in water level fluctuation areas, and safety monitoring.

Figure 6.

The numerical simulation modelling diagram.

Figure 7.

The displacement cloud diagram of the Monkey Stone landslide.

Figure 8.

The axial force diagram of ASCLs obtained from the numerical calculations.

Figure 9.

The shear force diagram of ASCLs obtained from the numerical calculations.

Figure 10.

The bending moment diagram of ASCLs obtained from the numerical calculations.

4. Comparative analysis of structural stress

Firstly, the reaction force of the structure support above the sliding surface is calculated via the displacement method of structural mechanics. Then, the internal stress of the structure below the sliding surface is calculated via the elastic foundation beam method, the cantilever pile method and the bottom boundary condition. Finally, the internal stress of the specified section in the structure is calculated via the section method. In the overall coordinate system, from left to right is the positive X-axis, from bottom to top is the positive Y-axis, and the positive Z-axis follows the right-hand rule. The axial force is positively related to the tensile force. The shear force is positive when it causes the isolation body to rotate clockwise; conversely, it is negative. The bending moment is drawn on one side of the tensile edge. The internal force diagram of the axial force, shear force and bending moment are positive when plotted on the outside of the overall structure; conversely, they are negative. Notably, in the subsequent tables, the calculation results containing the letter “E” are all represented using scientific notation, such as 5.431E-03 m=5.431×10^-3 m.

4.1. Structural load

The landslide thrust above the sliding surface and the anti-slide force in front of the pile are calculated via the transfer coefficient method (implicit solution method) and the passive earth pressure method, and the above acting force is simplified. The load action table (analytical solution) for the members above the sliding surface in the Monkey Stone landslide is shown in Table 4, and the load action table (modified analytical solution) for the members above the sliding surface in the Monkey Stone landslide is shown in Table 5.

Table 4.

The load action table (analytical solution) for the members above the sliding surface in the Monkey Stone landslide.

Sequence number	Structure	Horizontal inclination angle of sliding surface (°)	Sliding thrust of sliding block (kN)	Resistance of sliding block (kN)	Member	Member length (m)	Sliding force (kN)	Resistance (kN)	Distributed line load (kN/m)
1	BCD	45.794	25986.217	23415.992	BC	4.717	18627.897	16785.463	546.844
					CD	5.002	18118.634	87.890	3604.707
2	FGH	29.627	29844.718	25877.317	FG	7.876	14753.783	12792.492	405.271
					GH	4.479	25942.881	92.197	5771.530
3	JKL	29.669	38058.741	32394.968	JK	9.708	18838.644	16035.141	445.033
					KL	5.562	33069.219	94.322	5928.604
4	KNO	22.141	38811.697	32159.708	KN	11.208	14627.631	12120.582	379.934
					NO	10.457	35949.690	377.513	3401.757

Table 5.

The load action table (modified analytical solution) for the members above the sliding surface in the Monkey Stone landslide.

Sequence number	Structure	Horizontal inclination angle of sliding surface (°)	Sliding thrust of sliding block (kN)	Resistance of sliding block (kN)	Member	Member length (m)	Sliding force (kN)	Resistance (kN)	Distributed line load (kN/m)
1	BCD	45.794	25986.217	0	BC	4.717	18627.897	0	4105.348
					CD	5.002	18118.634	0	3622.278
2	FGH	29.627	7631.444	0	FG	7.876	3772.616	0	635.252
					GH	4.479	6633.725	0	1481.073
3	JKL	29.669	10668.517	0	JK	9.708	5280.795	0	700.213
					KL	5.562	9269.869	0	1666.643
4	KNO	22.141	4036.087	0	KN	11.208	1521.149	0	291.970
					NO	10.457	3738.463	0	357.508

4.2. Comparison and analysis on structural displacement

The comparison analysis table of the maximum displacement of ASCLs is shown in Table 6, the comparison analysis table of the minimum displacement of ASCLs is shown in Table 7.

Table 6.

The comparison analysis table of the maximum displacement of ASCLs.

Member	Maximum analytical calculated horizontal displacement (m)	Maximum numerical calculated horizontal displacement (m)	Maximum modified analytical calculated horizontal displacement (m)	Maximum analytical calculated vertical displacement (m)	Maximum numerical calculated vertical displacement (m)	Maximum modified analytical calculated vertical displacement (m)	Maximum analytical calculated angular displacement (rad)	Maximum numerical calculated angular displacement (rad)	Maximum modified analytical calculated angular displacement (rad)
AC	0	5.168E-03	0	0	-2.815E-02	0	2.481E-05	5.543E-05	7.227E-05
CE	5.705E-03	5.163E-03	6.352E-03	0	-2.901E-02	0	5.979E-05	2.146E-04	7.335E-05
EG	0	5.033E-03	0	0	-2.679E-02	0	4.932E-05	4.278E-04	4.377E-05
GI	8.609E-03	5.702E-03	2.553E-03	0	-2.596E-02	0	7.819E-05	3.608E-04	2.687E-05
IN	0	5.341E-03	0	6.733E-06	-2.400E-02	1.341E-05	2.216E-04	7.005E-04	7.281E-05
KM	1.057E-02	5.401E-03	3.297E-03	0	-2.591E-02	0	1.101E-04	8.260E-05	3.886E-05
NP	1.384E-02	5.934E-03	1.586E-03	0	-2.142E-02	0	5.312E-04	5.388E-04	6.592E-05
Total	1.384E-02	5.934E-03	6.352E-03	6.733E-06	-2.142E-02	1.341E-05	5.312E-04	7.005E-04	7.335E-05

Table 7.

The comparison analysis table of the minimum displacement of ASCLs.

Member	Minimum analytical calculated horizontal displacement (m)	Minimum numerical calculated horizontal displacement (m)	Minimum modified analytical calculated horizontal displacement (m)	Minimum analytical calculated vertical displacement (m)	Minimum numerical calculated vertical displacement (m)	Minimum modified analytical calculated vertical displacement (m)	Minimum analytical calculated angular displacement (rad)	Minimum numerical calculated angular displacement (rad)	Minimum modified analytical calculated angular displacement (rad)
AC	0	5.072E-03	0	-1.826E-03	-3.020E-02	-7.331E-03	-5.018E-04	-3.419E-04	-2.013E-03
CE	0	4.768E-03	0	0	-2.985E-02	0	-1.678E-03	-6.646E-05	-1.877E-03
EG	0	5.001E-03	0	-1.645E-03	-2.901E-02	-1.860E-03	-5.220E-04	-1.736E-04	-5.792E-04
GI	0	5.025E-03	0	0	-2.679E-02	0	-2.419E-03	-6.807E-05	-7.272E-04
IN	0	5.260E-03	0	-1.345E-03	-2.905E-02	-1.971E-03	-6.101E-04	-6.243E-04	-5.686E-04
KM	0	3.509E-03	0	0	-2.734E-02	0	-2.655E-03	-3.028E-04	-8.358E-04
NP	-1.025E-03	2.435E-03	-1.202E-04	0	-2.400E-02	0	-4.239E-03	-2.760E-04	-4.896E-04
Total	-1.025E-03	2.435E-03	-1.202E-04	-1.826E-03	-3.020E-02	-7.331E-03	-4.239E-03	-6.243E-04	-2.013E-03

According to Table 6, the maximum analytical calculated horizontal displacement value of 1.384E-02 m was generated at the 1/2.143 section of member NP, the maximum numerical calculated horizontal displacement of 5.934E-03 m was generated at section 1/7.471 of member NP, and the maximum modified analytical calculated horizontal displacement value of 6.352E-03 m was generated at section 1/1.999 of member CE. The maximum analytical calculated vertical displacement of 6.733E-06 m was generated at the 1/1.538 section of member IN, the maximum numerical calculated vertical displacement of -2.142E-02 m was generated at the end section of member NP, and the maximum modified analytical calculated vertical displacement value of 1.341E-05 m was generated at the 1/1.538 section of member IN. The maximum analytical calculated angular displacement value of 5.312E-04 rad was generated at the 1/2.802 section of member NP, and the maximum numerical calculated angular displacement value of 7.005E-04 rad was generated at the 1/1.097 section of member IN; the maximum modified analytical calculation angular displacement value of 7.335E-05 rad was generated at the 1/2.857 section of member CE. Furthermore, according to Table 7, the minimum analytical calculated horizontal displacement value of -1.025E-03 m was generated at the 1/1.214 section of member NP, the minimum numerical calculated horizontal displacement value of 2.435E-03 m was generated at the end section of member NP, and the minimum modified analytical calculated horizontal displacement value of -1.202E-04 m was generated at the 1/1.214 section of member NP. The minimum analytical calculated vertical displacement value of -1.826E-03 m was generated at the 1/2 section of member AC, the minimum numerical calculated vertical displacement value of -3.020E-02 m was generated at the 1/1.307 section of member AC, and the minimum modified analytical calculated vertical displacement value of -7.331E-03 m was generated at the 1/2 section of member AC. The minimum analytical calculated angular displacement value of -4.239E-03 rad was generated at section 1/2.143 of member NP, the minimum numerical calculated angular displacement value of -6.243E-04 rad was generated at section 1/3.224 of member IN, and the minimum modified analytical calculation angular displacement value of -2.013E-03 rad was generated at the 1/2 section of member AC.

Owing to the different displacement solution methods and assumed conditions that are adopted, differences exist among the analytical calculated, numerical and modified analytical calculated displacement values of members. The analytical and modified analytical calculation displacement of the members are solved by using the displacement method of structural mechanics, the elastic foundation beam method, the cantilever pile method and the bottom boundary condition, ignoring secondary loads such as the self-weight of the vertical anti-slide members. The calculated displacements of the horizontal anti-slide members include only vertical and angular displacements, without horizontal displacements. The calculated displacements of the vertical anti-slide members include only horizontal and angular displacements, without vertical displacements. The numerical calculation displacement of the members accounts for the self-weight of the members and the interaction stress between the members and the rock or soil mass, simulating the mutual coupling effect between the structure and the rock or soil mass. When the ASCL is subjected to landslide thrust and the anti-slide forces of the rock and soil mass, the overall displacement of the structure leans forward to the right and below. The shape trends of the analytical, numerical and modified analytical calculated displacement of the structure are similar. The absolute values of the analytical horizontal and angular displacement of the members arranged at the back are greater, whereas the absolute values of the numerical calculation of the vertical displacement of the members arranged at the front are greater.

4.3. Comparison and analysis on the structural axial force

The comparison analysis table of the axial forces of ASCLs is shown in Table 8, and the comparison analysis figure of the axial forces of ASCLs is shown in Figure 11.

Table 8.

The comparison analysis table of the axial forces of ASCLs.

Member	Maximum analytical calculated axial force (kN)	Maximum numerical calculated axial force (kN)	Maximum modified analytical calculated axial force (kN)	Minimum analytical calculated axial force (kN)	Minimum numerical calculated axial force (kN)	Minimum modified analytical calculated axial force (kN)
AC	7773.924	5054.577	6852.231	7773.924	-1038.541	6852.231
CE	106.265	-10555.04	-6970.539	-6238.026	-16358.361	-7200.616
EG	10418.198	876.094	2087.208	1248.381	-799.570	1522.632
GI	-785.127	-10258.966	-2103.942	-6976.413	-20694.130	-2887.235
IN	30882.577	858.438	5383.353	3097.368	-2095.662	900.140
KM	-6427.909	-6764.634	-5343.914	-6427.909	-23724.116	-5343.914
NP	289.668	-5189.091	-1097.532	289.668	-31657.138	-1097.532
Total	30882.577	5054.577	6852.231	-6976.413	-31657.138	-7200.616

Figure 11.

The comparison analysis figure of the axial forces of ASCLs.

Combining the contents described in Table 8 and Figure 11, there are differences in the analytical, numerical and modified analytical calculation of the axial force value of the members because of different solutions and assumed conditions of the axial force. At the section of 1/3 to 1/1.577 of member IN, the maximum analytical calculated axial force value of 30882.577 kN was generated. At the 1/3.145 section of member AC, the maximum numerical calculated axial force value of 5054.577 kN was generated. At the section of 0 to 1 of member AC (i.e., the entire section of the component), the maximum modified analytical calculated axial force value of 6852.231 kN was generated. At the section of 1/2.233 to 1 of member GI, the minimum analytical calculated axial force value of -6976.413 kN was generated. At the section of 1/1.956 of member NP, the minimum numerical calculated axial force value of -31657.138 kN was generated. At the section of 0 to 1/2 member CE, the minimum modified analytical calculated axial force value of -7200.616 kN was generated. As mentioned before, the overall displacement of the structure leans forward to the right and below. Furthermore, the graphical trend of the member analytical and modified analytical calculation of the axial force is close to the numerical calculation. In the horizontal anti-slide members, the axial forces are mostly in a positive tensile state. In the vertical anti-slide members, the axial forces are all in a negative compression state. When the vertical anti-slide member is subjected to the combined force of the landslide thrust and the anti-slide force in front of the pile, it undergoes a right-downwards and forwards displacement. The horizontal anti-slide member plays the role of a “pull rod” for the vertical anti-slide member. Notably, the horizontal anti-slide members are not strictly tension rods (pure tensile stress), but their actual force is in a combined force state of simultaneously bearing axial force, shear force and bending moment. The extreme modified analytical calculated axial force of ASCLs lies between the extreme analytical and numerical calculation value of the axial force.

4.4. Comparison and analysis on the structural shear force

The comparison analysis table of the shear forces of ASCLs is shown in Table 9, and the comparison analysis figure of the shear forces of ASCLs is shown in Figure 12.

Table 9.

The comparison analysis table of the shear forces of ASCLs.

Member	Maximum analytical calculated shear force (kN)	Maximum numerical calculated shear force (kN)	Maximum modified analytical calculated shear force (kN)	Minimum analytical calculated shear force (kN)	Minimum numerical calculated shear force (kN)	Minimum modified analytical calculated shear force (kN)
AC	2685.728	6525.172	12164.310	-2082.203	-9996.498	-9058.726
CE	10256.821	2245.374	11266.404	-7773.924	318.078	-6852.231
EG	2406.787	4281.904	2899.302	-6238.026	-8170.158	-6970.539
GI	15432.484	3446.075	4546.518	-10418.198	504.021	-2087.208
IN	4547.969	9451.661	3628.621	-6976.413	-12405.207	-3169.047
KM	17341.223	1504.376	5334.976	-15633.673	-528.463	-3934.893
NP	20323.269	2444.238	2290	-15248.904	-2059.853	-1448.461
Total	20323.269	9451.661	12164.310	-15633.673	-12405.207	-9058.726

Figure 12.

The comparison analysis figure of the shear forces of ASCLs.

Combining the contents described in Table 9 and Figure 12, there are differences in the analytical, numerical, and modified analytical calculation of the shear force value of the members, because of different solutions and assumed conditions of the shear force. At the 1/2.143 section of member NP, the maximum analytical calculated shear force value of 20323.269 kN was generated; at the 1/1.594 section of member IN, the maximum numerical calculated shear force value of 9451.661 kN was generated; and at the 1/2 section of member AC, the maximum modified analytical calculated shear force value of 12164.310 kN was generated. At the initial section of member KM, the minimum analytical calculated shear force value of -15633.673 kN was generated; at the end section of member IN, the minimum numerical calculated shear force value of -12405.207 kN was generated; and at the initial section of member AC, the minimum modified analytical calculated shear force value of -9058.726 kN was generated. The maximum value of the numerical calculation of the shear force in the members arranged relatively in the middle was produced, and the maximum value of the analytical calculation of the shear force in the members arranged relatively in the back was produced. The graphical trend of the shear force in the member analytical and modified analytical calculation is close to that in the numerical calculation. Owing to the vertical loads, the horizontal anti-slide members above the sliding surface generate positive shear forces, and owing to the “tension rod” effect of the horizontal anti-slide member on the top of the vertical anti-slide member, a negative shear force is generated near the top of the vertical anti-slide member. However, as the vertical anti-slide member is subjected to the combined force of the landslide thrust and the anti-slide force in front of the pile, the negative shear force value of the vertical anti-slide member gradually decreases and increases in the direction of the positive shear force value (from top to bottom). Taking the sliding surface as the boundary, the sliding body undergoes shear slip failure along the sliding surface. The shear force of the member is relatively large near the sliding surface, and the shear force of the member above the sliding surface is opposite to that below the sliding surface because the structure‒rock mass interaction below the sliding surface counteracts the shear force of the member above the sliding surface. The maximum modified analytical calculated shear force value of ASCLs lies between the maximum analytical and numerical calculated shear force value, whereas the minimum modified analytical calculated shear force value is relatively close to the minimum analytical and numerical calculated shear force value.

4.5. Comparison and analysis on the structural bending moment

The comparison analysis table of the bending moments of ASCLs is shown in Table 10, and the comparison analysis figure of the bending moments of ASCLs is shown in Figure 13.

Table 10.

The comparison analysis table of the bending moments of ASCLs.

Member	Maximum analytical calculated bending moment (kN·m)	Maximum numerical calculated bending moment (kN·m)	Maximum modified analytical calculated bending moment (kN·m)	Minimum analytical calculated bending moment (kN·m)	Minimum numerical calculated bending moment (kN·m)	Minimum modified analytical calculated bending moment (kN·m)
AC	4905.508	4853.661	19790.572	-3375.999	-12729.971	-6504.544
CE	4962.435	5046.539	6267.353	-22039.210	-7519.382	-24656.603
EG	5675.610	8880.791	6194.902	-20385.401	-12780.054	-22546.718
GI	7130.899	8557.917	2660.316	-31595.389	-8704.033	-9652.303
IN	13799.476	17640.948	9262.995	-25917.973	-18215.506	-7995.689
KM	8439.228	4399.002	2928.142	-34169.436	-114.484	-10838.582
NP	20866.638	17683.278	2609.348	-58534.409	-2766.007	-6794.462
Total	20866.638	17683.278	19790.572	-58534.409	-18215.506	-24656.603

Figure 13.

The comparison analysis figure of the bending moments of ASCLs.

Combining the contents described in Table 10 and Figure 13, at the 1/4.981 section of member NP, the maximum analytical calculated bending moment value of 20866.638 kN·m was generated; at the initial section of member NP, the maximum numerical calculated bending moment value of 17683.278 kN·m was generated; at the 1/3.145 section of member AC, the maximum modified analytical calculated bending moment value of 19790.572 kN·m was generated. At the section of 1/1.799 of member NP, the minimum analytical calculated bending moment value of -58534.409 kN·m was generated; at the 1/2.509 section of member IN, the minimum numerical calculated bending moment value of -18215.506 kN·m was generated; and at the 1/1.250 section of member CE, the minimum modified analytical calculated bending moment value of -24656.603 kN·m was generated. The trend of the analytical and modified analytical calculation of the bending moment graph are close to the trend of the numerical calculation. However, at the junction of the top of the horizontal and vertical anti-slide members, the direction of the analytical calculation of the bending moment value is opposite to that of the numerical calculation, because the analytical calculation of the bending moment adopts the block calculation of the force on the anti-slide structure, whereas the numerical calculation of the bending moment adopts the overall calculation of the force on the anti-slide structure. Moreover, while the position of the member section gradually moves from the junction of the top of the horizontal and vertical anti-slide members to the junction of the bottom of the vertical anti-slide members, the analytical calculation of the bending moment value gradually approaches that of the numerical calculation of the bending moment value. Taking the sliding surface as the boundary, the sliding body undergoes shear sliding failure along the sliding surface, and the bending moment value of the member is relatively large near the sliding surface. Owing to the overall displacement of the structure, the outer edges of the top of the horizontal and vertical anti-slide members are prone to be in a tensile state. However, owing to the interaction between the structure and the rock and soil mass, the bending moment at the bottom of the component gradually reverses. The extreme modified analytical calculation result of the bending moment of the ASCLs was between the extreme analytical and numerical calculation result of the bending moment.

5. Conclusion

The graphical trends of the analytical and modified analytical calculation solution for the stress of the ASCLs are similar. In the analytical calculation solution, the stress on the structure ranked later generates the maximum value; in the modified analytical calculation solution, the stress on the structure ranked higher generates the maximum value; and in the numerical calculation solution, the structures placed in the middle and later parts of the sequence are subjected to the maximum stress. However, the axial force values of different algorithms differ. The reason for this is that the analytical and modified analytical calculation solution ignore the influences of friction forces, bonding forces, mechanical bite forces and end-bearing forces between the member and the rock and soil mass. In the process of calculating the stress on the structure using the unbalanced thrust method (transfer coefficient method), the analytical calculation solution ignores the anti-slide effect of the multilevel retaining structure on the sliding body blocks, as a result, the analytical calculated stress value of the structure is relatively large, and the stress on the structure ranked later is greater. By using the “zero block recurrence” assumption, the modified analytical calculation solution accounts for the anti-slide effect of the multilevel retaining structure on the sliding block while ignoring the influence of the sliding force between the sliding blocks. As a result, the modified analytical calculated stress value of the structure is relatively small, and the stress on the structure with a higher sequence is greater. The above two methods yield the two extreme force states. The extreme values of the axial force, shear force and bending moment of the numerical calculation solution were verified to be within a reasonable range by combining the analytical and modified analytical calculation. The extreme modified analytical calculated axial force of ASCLs lies between the extreme analytical and numerical calculation value of the axial force. The maximum modified analytical calculated shear force value of ASCLs lies between the maximum analytical and numerical calculated shear force value, whereas the minimum modified analytical calculated shear force value is relatively close to the minimum analytical and numerical calculated shear force value. The extreme modified analytical calculation result of the bending moment of the ASCLs was between the extreme analytical and numerical calculation result of the bending moment. The correctness of numerical calculation solutions can be well verified by using analytical and modified analytical calculation solutions. Laboratory physical model tests will be implemented to further verify the correctness of the structural stress results calculated by the analytical, modified analytical, and numerical methods.