Slope Calculation Analysis Based on Arbitrary Polygonal Hybrid Stress Elements Considering Gravity

Abstract

This article proposes an arbitrary polygonal hybrid stress element considering gravity. It derives an arbitrary polygonal hybrid stress element considering gravity alone for slope stability related engineering analysis. In the stability analysis of slopes, slope disasters caused by gravity erosion have recently become an urgent problem to be solved through engineering. The traditional finite element analysis of slope stability faces problems such as a large number of divided elements and slow calculation efficiency. By introducing high-order stress fields through stress hybridization elements, accurate results can be simulated using a small number of elements. When dividing the mesh, most of the cell shapes are asymmetric, and the shape of the cell can be any polygon, which can simulate the geometric shape of complex slopes well and more accurately calculate the stress distribution in different parts, thus accurately simulating the stability situation in engineering. By comparing with the corresponding commercial software MARC 2020, the effectiveness and efficiency of the element were verified. It has been proven that any polygonal hybrid stress element has the advantage of flexible mesh division, which can obtain high-order stress fields and stress concentration phenomena with fewer elements. Applying this element to practical problems of slopes in engineering has also achieved good calculation results.

1. Introduction

China is the country with the largest scale of hydropower energy development and construction in the world. At the same time, its geological conditions are complex. In the last century, many slope disasters occurred, and the country established a dedicated department to study and prevent them. In recent years, with global warming, the intensity and frequency of rainstorms caused by extreme weather have increased, and slope disasters caused by gravity erosion have made large-scale hydropower stations face unprecedented slope stability problems [1,2]. Gravity erosion is a mass failure on steep slopes caused by its own weight. The forms of gravity erosion include avalanches, landslides, mudslides, and sinkhole formation, but the mechanisms and dynamics of each erosion type are different. In the construction of large hydropower stations, if the left bank accumulation body experiences a complete landslide, it will directly endanger the normal operation of the power station inlet and dam. For example, in the Baihetan hydropower station, the excavation work of the underground power plant cavern on the left bank is carried out in a deep rock mass with discontinuity. This rock mass has the characteristics of high plateau stress and has encountered rock failure such as peeling during actual excavation work. Through discontinuous discrete element modeling, the evolution of the displacement field and stress field is reproduced. Through numerical simulation and on-site monitoring data, discontinuity will affect stress redistribution and cause arch and sidewall effects. Therefore, studying the influence of gravity on the stability of the left bank accumulation body slope is of great significance for ensuring the stable operation of the project and safeguarding the safety of people’s lives and property.

The traditional quantitative analysis methods for slope stability are mainly divided into uncertainty analysis methods and deterministic analysis methods [3,4,5]. Uncertainty mainly includes reliability evaluation methods and artificial neural network analysis methods, while mainstream deterministic analysis methods mainly include limit equilibrium method and numerical analysis method. In slope stability technology, finite element method in numerical analysis is a commonly used method. The reliability evaluation method [6,7,8] often cannot truly reflect the stability of actual slopes when establishing limit state equations, and is often used as an auxiliary means in practical engineering. Artificial Neural Network Analysis (ANN) is a neural network constructed by humans with the ability to learn, generalize, and process information in biological neural networks, which can achieve certain functions. Although artificial neural network analysis has a high degree of nonlinear representation ability, it has not yet formed a complete system [9,10,11,12]. The limit equilibrium method is the earliest and most commonly used quantitative analysis method in engineering practice. It is currently a relatively mature and widely used method for slope stability analysis, but it cannot analyze the process of slope failure and has certain limitations [13,14]. The finite element method (FEM) is currently the best method, and its calculation results have been widely recognized. It can effectively solve problems with large deformation and displacement discontinuity. However, there are some convergence issues with the strength reduction method based on the finite element method. Under sufficient and reasonable grid density, the finite element reduction method is effective in analyzing slope stability [15,16,17,18]. The boundary element method (BEM) often only divides the dangerous parts of the boundary region into stability analysis, and is not suitable for analyzing heterogeneous and nonlinear slopes. In terms of slope stability, traditional finite element analysis needs to be linked to traditional safety factors, and the basis for determining slope stability is not yet mature [19,20]. Discrete Element Method (DEM) is mainly used to calculate how a large number of particles move under given conditions, which can well reflect the slip of the contact surface between rock blocks. While solving for large displacements, it can also calculate the deformation and stress distribution inside the rock blocks. However, in complex engineering problems, the determination of some parameters is highly arbitrary [21,22,23,24].

In response to the above issues, this article proposes an arbitrary polygonal hybrid stress element that considers gravity. The hybrid stress element introduces a high-order stress field that considers gravity, and accurate results can be simulated using a small number of elements. When dividing the mesh, the shape of the elements can be any polygon with any number of sides, which is also very effective for asymmetric elements. Therefore, it can simulate the complex geometric shape of slopes well, calculate the stress distribution in different parts more accurately, and accurately predict the stability state in engineering problems. Hybrid stress elements have the advantages of easy grid division, fast calculation speed, and high calculation accuracy in the analysis of large deformations and multiple materials. They can respond promptly and provide guidance when facing practical engineering problems.

2. Materials and Methods

In 2019, Yang et al. [25] proposed a schematic diagram of the element based on the polygonal hybrid stress element method (PHSEM), as shown in Figure 1. The mesh division of the analysis model is composed of polygons with an indefinite number of edges. In real structures, there are also irregular polygons that can be efficiently and conveniently meshed based on the spatial distribution of materials and the characteristics of the structure, effectively simulating complex and irregular models. For a typical polygonal hybrid stress element, the element composition is shown in Figure 2.

A typical hybrid stress element is shown in Figure 2, where all outer boundaries of the element are composed of a given displacement boundary $\mathsf{\partial }{\Omega }_u$, a given force boundary $\mathsf{\partial }{\Omega }_t$, a common boundary $\mathsf{\partial }{\Omega }_e^{\prime }$ between elements, and a free boundary $\mathsf{\partial }{\Omega }_f$, i.e., $\mathsf{\partial }{\Omega }_e=\mathsf{\partial }{\Omega }_e^{\prime }\cup \mathsf{\partial }{\Omega }_u\cup \mathsf{\partial }{\Omega }_t\cup \mathsf{\partial }{\Omega }_f$. Based on the principle of minimum surplus energy in the virtual work principle, the element is subjected to stress. The finite element model of function $\mathit{\sigma }$ as a field variable, according to the modified residual functional expression:

$$\begin{array}{c}{\prod }_{mc}={\displaystyle \sum _e}\left\{{\int }_{{\Omega }_e}{\displaystyle \frac{1}{2}}\sigma :S:\sigma d\Omega -{\int }_{\mathsf{\partial }{\Omega }_e}n\cdot \sigma \cdot ud\mathsf{\partial }\Omega +{\int }_{\mathsf{\partial }{\Omega }_t}\overline{T}\cdot ud\mathsf{\partial }\Omega \right\}.\end{array}$$

(1)

Under static conditions without external force, the stress field satisfies the equilibrium equation when considering gravity:

$$\begin{array}{c}\mathsf{\nabla }\cdot \sigma +f=0.\end{array}$$

(2)

The solution to its equilibrium equation is:

$$\begin{array}{c}\sigma ={\sigma }_e+{\sigma }_e^{*}.\end{array}$$

(3)

In the two-dimensional problem of hybrid stress elements, Airy stress function polynomials with different terms can be used to satisfy polynomial completeness.

$$\begin{array}{c}{\sigma }_e=\mathrm{P}\mathsf{\beta }.\end{array}$$

(4)

$$\begin{array}{c}{\sigma }_e^{*}=\left\{\begin{array}{c}{\sigma }_x^{*}\\ {\sigma }_y^{*}\\ {\tau }_{xy}^{*}\end{array}\right\}=\left\{\begin{array}{c}{f}_x\cdot x\\ f_y\cdot y\\ 0\end{array}\right\}.\end{array}$$

(5)

Among them, ${\sigma }_e$ is a column vector containing three stress components, and ${\sigma }_e^{\mathrm{*}}$ is a column vector containing three gravity stress components. $\beta$ is a column vector containing $m$ unknown coefficients ${\beta }_1,{\beta }_2\cdots {\beta }_m$. $\beta$ is the stress coefficient of the assumed independent stress field inside the element, $P$ is a $3\times m$ parameter matrix obtained from the Airy stress function, and $P$ is the stress interpolation matrix.

For two-dimensional plane problems, the stress field containing gravity can be specifically expressed as:

$$\begin{array}{c}\left\{\begin{array}{c}{\sigma }_x\\ {\sigma }_y\\ {\tau }_{xy}\end{array}\right\}=\left[\begin{array}{ccccccc}1& y& 0& 0& 0& x& 0\\ 0& 0& 1& x& 0& 1& y\\ 0& 0& 0& 0& 1& -y& -x\end{array}\begin{array}{c}\dots \\ \dots \\ \dots \end{array}\right]\left\{\begin{array}{c}{\beta }_1\\ ⋮\\ {\beta }_m\end{array}\right\}+\left\{\begin{array}{c}{\sigma }_x^{*}\\ {\sigma }_y^{*}\\ {\tau }_{xy}^{*}\end{array}\right\}=\mathrm{p}\mathsf{\beta }+\left\{\begin{array}{c}{\sigma }_x^{*}\\ {\sigma }_y^{*}\\ {\tau }_{xy}^{*}\end{array}\right\}.\end{array}$$

(6)

Interpolation of the generalized displacement $d$ of the node can obtain the boundary displacement $u$:

$$\begin{array}{c}u=\mathrm{L}\mathrm{d}.\end{array}$$

(7)

By substituting Equations (3)–(7) into Equation (1), we can obtain the modified residual energy functional expression with gravity after finite element discretization:

$$\begin{array}{c}{\Pi }_{mc}={\displaystyle \sum _e}\left({\displaystyle \frac{1}{2}}{\beta }^{T}H\beta +{\beta }^{T}B-{\beta }^{T}Gd-{f_{\Gamma }}^{T}d+f^{T}d+{\displaystyle \frac{1}{2}}{\int }_{\Omega }{{\sigma }_e^{*}}^{T}S{\sigma }_e^{*}d\Omega \right),\end{array}$$

(8)

of which

$$\begin{array}{c}H={\int }_{\Omega }{P}^{T}SPd\Omega ,\end{array}$$

(9)

$$\begin{array}{c}G={\int }_{\mathsf{\partial }\Omega }{P}^T{n}^{T}Ld\mathsf{\partial }\Omega ,\end{array}$$

(10)

$$\begin{array}{c}F={\int }_{\mathsf{\partial }{\Omega }_t}\overline{T}Ld\mathsf{\partial }\Omega ,\end{array}$$

(11)

$$\begin{array}{c}{F}_{\Gamma }={\int }_{\mathsf{\partial }\Omega }{L}^{T}n{\sigma }_e^{*}d\mathsf{\partial }\Omega ,\end{array}$$

(12)

$$\begin{array}{c}B={\int }_{\Omega }{P}^{T}S{\sigma }_e^{*}d\Omega .\end{array}$$

(13)

The unknown variables in the equation are the coefficient matrix $\beta$ and the node displacement $d$.

According to the revised residual energy stationary condition:

$$\begin{array}{c}{\displaystyle \frac{\mathsf{\partial }{\prod }_{mc}}{\mathsf{\partial }\beta }}=0,{\displaystyle \frac{\mathsf{\partial }{\prod }_{mc}}{\mathsf{\partial }d}}=0.\end{array}$$

(14)

The expression of stress parameters within each element can be obtained:

$$\begin{array}{c}{\rm B}=H^{-1}\left(Gd-B\right).\end{array}$$

(15)

The expression for solving generalized displacement can be obtained:

$$\begin{array}{c}F=G^T{H}^{-1}Gd-G^T{H}^{-1}B+f_{\Gamma },\end{array}$$

(16)

where the stiffness matrix is expressed as:

$$\begin{array}{c}{K}^e=G^T{H}^{-1}G,\end{array}$$

(17)

of which

$$\begin{array}{c}{K}^{e}d=f+G^T{H}^{-1}B-f_{\Gamma }.\end{array}$$

(18)

Equation (18) is the fundamental equation of any polygon considering gravity in two dimensions, with the unknown variable being the linear equation system of node displacement $d$.

3. Results

This section establishes three numerical examples. The first example is a classic concrete gravity dam model, which uses a hybrid stress element to simulate the concrete gravity dam model. The model material is selected as the dam body concrete, mainly used to verify the effectiveness of the PHSEM finite element model considering gravity. The second example is a typical concrete gravity dam model containing two materials, used to analyze the feasibility of hybrid stress elements in complex material research and practical engineering applications. The third example is an actual model of a large hydropower station slope, used to verify the effectiveness and efficiency of the PHSEM method in practical engineering applications.

3.1. Numerical Example 1

The structure of the example model is shown in Figure 3, which is a typical concrete gravity dam with a dam slope of 1:2 and a dam height of 10 m [26]. The PHSEM model and Marc model use the same boundary conditions, with complete constraint in the lower left corner and vertical displacement constraint in the lower right corner. When considering gravity, the simulation takes into account a gravity acceleration of $9.8 \mathrm{m}/{\mathrm{s}}^2$.

Figure 4 shows the specific mesh division of two models. The PHSEM model mesh is shown in Figure 4a. The PHSEM model has four nodes and one arbitrary polygon hybrid stress element, using a 250 term, 21st order Airy stress function. The Marc model mesh is shown in Figure 4b, with a total of 6956 nodes and 6713 planar fully strain integrated quadrilateral elements. The specific material parameters of the two models are the same, both are dam concrete, and their material mechanics parameters are shown in

Table 1. Physical and Mechanical Parameters of Dam Body.

Parameter	Elastic Modulus (MPa)	Poisson’s Ratio	Unit Weight (kN/m³)
Dam concrete	2.0 × 104	0.167	23.52

[26].

In order to verify the effectiveness of the PHSEM finite element model considering gravity, PHSEM models and Marc models with the same boundary conditions and structure were established. Figure 5a,b show a comparison of the stress maps ${\sigma }_x$ in the horizontal direction, considering gravity, between the PHSEM model and the Marc model. Figure 5c,d show a comparison of the stress maps ${\sigma }_y$ in the vertical direction, and Figure 5e,f show a comparison of the equivalent Mises stress maps $\overline{\sigma }$. From the distribution of the stress map, it can be seen that the stress cloud map distribution of the PHSEM model is consistent with the results of the Marc model, verifying the effectiveness of any polygonal hybrid stress element considering gravity. On the right side of the model, there is no constraint in the horizontal direction. In the vertical stress cloud map, it can be seen from the figure that due to the effect of gravity, the stress at the top of the model is the highest, and the stress at the bottom of the model is the lowest, and the transition of the stress cloud map band is gentle.

In order to more accurately evaluate the accuracy of PHSEM model calculations, a path curve can be selected for detailed comparison of its stress values. Figure 6 shows the comparison of stress values between the PHSEM model and the Marc model on the selected path. Select the path shown in Figure 6a and compare the values of the horizontal stress ${\sigma }_x$, vertical stress ${\sigma }_y$, and equivalent Mises stress $\overline{\sigma }$. It can be seen from the graph that the three stress trends are consistent, and the values are basically the same. From this, it can be seen that the results of the PHSEM model are in good agreement with those of the Marc model, and the stress field distribution calculated by PHSEM considering gravity is correct.

3.2. Numerical Example 2

The structure of the example model is shown in Figure 7. The model is a typical bi material dam, divided into an upper part of the dam body and a lower part of the dam foundation. The dam body is a typical concrete gravity dam with a slope of 1:2 and a height of 10 m, similar to the previous example [26]. The dam foundation is a rectangular shape with a length of 130 m and a width of 38 m. The PHSEM model and Marc model use the same boundary conditions, with complete constraints on the lower left and lower right corners of the dam foundation, and only horizontal displacement constraints on the upper left and upper right corners. Simulate the situation when considering gravity, with a gravity acceleration of $9.8 \mathrm{m}/{\mathrm{s}}^2$.

Figure 8 shows the specific mesh division of two models. The PHSEM model mesh is shown in Figure 8a. The PHSEM model has four nodes and two arbitrary polygon hybrid stress elements, using a 250 term, 21st order Airy stress function. The Marc model grid is shown in Figure 4b, with a total of 131,069 nodes and 130,216 planar fully strain integrated quadrilateral elements. The specific material parameters of the two models are the same, both being the concrete of the dam body and the rock mass of the dam foundation. Their material mechanics parameters are shown in

Table 2. Physical and Mechanical Parameters of Dams.

Parameter	Elastic Modulus (MPa)	Poisson’s Ratio	Unit Weight (kN/m³)
Dam concrete	2.0 × 104	0.167	23.52
Dam foundation rock mass	2.0 × 104	0.2	21.56

[26].

Figure 9 shows the comparison of horizontal stress ${\sigma }_x$ cloud maps, vertical stress ${\sigma }_y$ cloud maps, and equivalent Mises stress $\overline{\sigma }$ 773 cloud maps between the PHSEM model and the Marc model. From the figure, it can be seen that the horizontal stress cloud maps of the PHSEM model and the Marc model are slightly different at the interface between the two materials. The vertical stress cloud map and the equivalent Mises stress cloud map are basically consistent, verifying the effectiveness of any polygonal hybrid stress element. In the vertical stress cloud map, it can be seen from the figure that due to the effect of gravity, the stress at the top and left and right sides of the model is relatively high, while the stress at the bottom of the model is the smallest. The stress cloud map at the junction of the two materials has a uniform transition of stress bands.

Figure 10 shows the comparison of stress values along the selected paths for the PHSEM model and Marc model. Select the path shown in Figure 10a and compare the values of the horizontal stress ${\sigma }_x$, vertical stress ${\sigma }_y$, and equivalent Mises stress $\overline{\sigma }$ 773. As shown in the figure, the numerical fitting trends of the horizontal stress ${\sigma }_x$, vertical stress ${\sigma }_y$, and equivalent Mises stress $\overline{\sigma }$ are consistent. Except for a slight error in the horizontal stress ${\sigma }_x$, the other two terms are basically the same in terms of numerical values. From this, it can be seen that the results of the PHSEM model are in good agreement with those of the Marc model, and the stress field distribution calculated by PHSEM is correct.

3.3. Numerical Example 3

This example is a profile model of the left bank accumulation slope of a large hydropower station (Figure 11), which calculates the stress of the model under the consideration of gravity. The bottom of the model is 257.15 m long, with a left side height of 156.5 m and a right side height of 34.5 m. There are four types of materials from top to bottom, namely alluvial deposits ($pl{Q}_3$), ancient landslide remnants (${delQ}_2$), sandstone shale ($P_{2}X$), and basalt ($P_2\beta$). The specific properties of the elastic modulus, Poisson’s ratio, and density of the materials are shown in

Table 3. Material profile of left bank accumulation slope of a large hydropower station.

Parameter	Elastic Modulus (MPa)	Poisson’s Ratio	Unit Weight (kN/m³)
Namely alluvial deposits $(pl{Q}_3$)	25	0.35	19.6
Ancient landslide remnants $({delQ}_2$)	35	0.3	22.05
Sandstone shale $(P_{2}X$)	750	0.35	24.5
Basalt $(P_2\beta$)	7500	0.25	27.44

. The model constrains vertical displacement at the bottom and horizontal displacement at the left, with a gravitational acceleration of $9.8 \mathrm{m}/{\mathrm{s}}^2$. In practical engineering, parameters such as elastic modulus and Poisson’s ratio are functions of confining pressure. In numerical calculations, for the convenience of calculation and comparison, we have taken a relatively suitable value.

The mesh division of the PHSEM model is shown in Figure 12a: The PHSEM model considering gravity has a total of 137 nodes and 31 arbitrary polygonal hybrid stress elements. Using a 250 term, 21st order Airy stress function. The Marc model mesh is shown in Figure 12b, with a total of 175,132 nodes and 172,445 planar fully strain integrated quadrilateral elements.

Figure 13 shows the comparison of horizontal stress ${\sigma }_x$ cloud maps, vertical stress ${\sigma }_y$ cloud maps, and equivalent Mises stress $\overline{\sigma }$ cloud maps between the PHSEM model and Marc model of the left bank accumulation slope profile of a large hydropower station considering gravity. From the figure, it can be seen that the different layered materials also have a certain influence on the distribution of stress, and the stress concentration phenomenon is also captured in the PHSEM model. The stress cloud distribution of the PHSEM model is consistent with the stress trend of the Marc model, and the values are basically the same, which verifies the effectiveness of the arbitrary polygonal hybrid stress element in practical engineering applications. Under the same computational conditions, the PHSEM method takes 7 s to calculate, while the Marc model takes 30 s, resulting in a 76.6% increase in computational speed.

Through the calculations of the above three examples, it can be seen that in the Marc model, ordinary finite elements are low order elements, and a small number of elements cannot reflect the stress concentration phenomenon of actual problems. It is necessary to divide a large number of elements to achieve good calculation accuracy. However, the PHSEM method uses high-order stress elements for calculation, and a small number of elements can also capture the stress concentration phenomenon. The arbitrary polygonal hybrid stress element considering physical strength can obtain results similar to commercial finite element software Marc 2020 with only a small number of elements, and has higher computational efficiency, demonstrating the effectiveness and efficiency of the PHSEM method. By verifying the stress field, we can further predict its failure and analyze the consequences of structural safety, which is our next step of work.

The effectiveness of the PHSEM method was verified in the presented examples, and in fact, good computational results were obtained using the PHSEM method for the vast majority of grids, arbitrary boundary conditions, and material properties. The calculated results can simulate actual working conditions very well.

4. Conclusions

This article proposes a new element for arbitrary polygon hybrid stress considering physical strength. The element is validated through the first and second examples, and the calculation results are compared with those of the commercial software Marc 2020 to verify the effectiveness and efficiency of the arbitrary polygon hybrid stress element. Through the third example, it is demonstrated that the arbitrary polygon hybrid stress element has the advantages of flexible mesh division and the ability to simulate more accurate results using a small number of elements, compared to the ordinary finite element method in analyzing practical engineering problems of slopes.

• Compared with other calculation methods, the PHSEM method uses a high-order stress field and has a relatively small number of arbitrary polygonal hybrid stress elements. Ordinary displacement finite elements require fine mesh division at complex structures, while arbitrary polygonal hybrid stress elements can be simulated with only a few or even one element.
• When calculating multiple materials, traditional finite element methods perform stress smoothing in post-processing to ensure stress continuity, while the arbitrary polygon hybrid stress element method treats stress as a field variable, with different high-order stress fields at the interface, where surface stress is continuous, but stress is discontinuous. Stress discontinuity phenomenon that can better reflect practical engineering problems.
• The arbitrary polygonal hybrid stress element considering physical strength can accurately reflect the stress field and has high computational efficiency when analyzing practical engineering problems, providing a new method for the calculation of practical engineering problems.

In this study, the arbitrary polygon hybrid stress element has the advantage of flexible mesh division and the ability to obtain high-order stress fields and stress concentration phenomena with fewer elements compared to the ordinary finite element method in analyzing practical engineering problems of dams. At the same time, it is more convenient and efficient when dealing with complex models, and can select elements with any number of edges, which makes up for some shortcomings of traditional finite element methods in engineering applications and provides a convenient and efficient new element for simulating practical engineering problems.