Three-dimensional slope stability based on the finite element limit equilibrium method
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摘要: 提出了一種基于有限元彈塑性應力場和極限平衡狀態的三維邊坡穩定分析方法——三維有限元極限平衡法。首先，考慮三維空間中滑動方向，提出滑動面上一點在滑動方向上的極限平衡條件，并證明滑動面上土體整體達到極限平衡狀態與滑動面上土體各處在滑動方向上處于極限平衡狀態等價。再通過剛體極限平衡假定計算主滑方向和滑動面上各點滑動方向。最后，定義局部安全系數為抗剪強度與滑動方向上剪應力投影的比值，基于三維邊坡整體極限平衡條件將局部安全系數通過積分中值定理轉變為整體安全系數。該方法計算簡單，消除了剪應力比形式定義安全系數滑動面形狀限制，具備合理性與有效性。算例驗證結果表明，該方法滑動方向假設合理，安全系數與嚴格三維極限平衡法結果一致。Abstract: A finite element limit equilibrium method was proposed based on finite element stress analysis combined with a limit equilibrium condition to analyze the slope stability. The local safety factor defined in the form of shear strength and shear stress ratio in a three-dimensional (3D) space does not consider the sliding direction influence on the calculation results. In this paper, a 3D finite element limit equilibrium method that considers the sliding direction was proposed. This method was different from the limit equilibrium and strength reduction methods and analyzed slope stability through the “true” stress state without reducing the material strength parameters. First, considering the sliding direction in the 3D space, the limit equilibrium condition of a point was proposed on the slip surface in the sliding direction. An equivalent relationship was proved of the slip surface was in the limit equilibrium state, and each point of the slip surface was in the limit equilibrium state in the sliding direction. Then, the main sliding direction and the sliding direction of each point on the slip surface were calculated assuming the rigid body limit equilibrium. Finally, the local safety factor was defined as the ratio of the shear strength to the shear stress projection in the sliding direction. Based on the equivalent relationship of the limit equilibrium state of the 3D slope, the local safety factor was transformed into a global safety factor by applying the integral median theorem. The method is simple to calculate, eliminates the limitation of the slip surface shape of the safety factor defined by the shear stress ratio form, and is reasonable and effective. The verification result of the calculation example shows that the sliding direction assumption of the method is reasonable, and the safety factor is consistent with the result of the strict 3D limit equilibrium method.
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圖 1 滑動面應力分析。(a)二維；(b)三維
Figure 1. Stress on a slip surface: (a) two-dimensional; (b) three-dimensional
圖 2 滑動面構造步驟. (a)三角形網格；(b)網格節點調整；(c)滑動面與坡面相交
Figure 2. Construction steps of the slip surface: (a) triangle mesh; (b) mesh node adjustment; (c) slip surface intersects the slope surface
圖 4 算例1. (a)材料參數和斷面；(b)工況1；(c)工況2
Figure 4. Example 1: (a) material parameters and cross-section; (b) case 1; (c) case 2
圖 5 單元數對安全系數的影響. (a)滑動面；(b)有限元
Figure 5. Influence of the number of units on the safety factor: (a) slip surface; (b) FEM
圖 6 工況1的滑動方向與主滑方向. (a)等軸測圖；(b)俯視圖
Figure 6. Sliding direction and the main sliding direction of case 1: (a) isometric view; (b) top view
圖 7 工況1的剪應力矢量方向與主滑方向. (a)等軸測圖；(b)俯視圖
Figure 7. Shear stress vector direction and the main sliding direction of case 1: (a) isometric view; (b) top view
圖 8 算例2楔形體破壞. (a)對稱楔形體；(b)非對稱楔形體
Figure 8. Example 2: wedge failure: (a) symmetric wedge; (b) asymmetric wedge
圖 9 非對稱楔形體的滑動方向與主滑方向. (a)等軸測圖；(b)俯視圖
Figure 9. Sliding direction and the main sliding direction of an asymmetrical wedge: (a) isometric view; (b) top view
圖 10 非對稱楔形體的剪應力矢量方向與主滑方向. (a)等軸測圖；(b)俯視圖
Figure 10. Shear stress vector direction and the main sliding direction of an asymmetrical wedge: (a) isometric view; (b) top view
表 2 算例2的材料參數
Table 2. Mechanical parameters used in example 2
Parameters Symmetric wedge Asymmetric wedge Rock Structural surfaces Rock Structural surfaces Weight, γ/
(kN·m?3)26 20 26 20 Poisson ratio, ν 0.49 0.49 0.49 0.49 Friction angle,
φ/ (°)45 20 45 30 Cohesion,
c/kPa1000 20 1000 50 
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表 3 算例2的幾何信息
Table 3. Geometric information used in example 2
(°) Surface Symmetric wedge Asymmetric wedge Dip direction Dip Dip direction Dip Left structural surface 115 45 120 40 Right structural surface 245 45 240 60 Upper surface 180 10 180 0 Slope surface 180 60 180 60
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