Study on calculation method of bending moment and deflection of underlying layer under soil pressure
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摘要: 面层是基坑支护系统中重要的组成部分，为了对面层进行合理设计，利用薄板小挠度弯曲理论建立了面层的薄板受力模型，对喷射混凝土基坑支护型式的面层进行了受力分析。为简化计算，假设混凝土面层为弹性体，将面层简化为作用在基坑边坡上的四边简支矩形薄板，推导了主动土压力作用下面层挠度和弯矩的一般表达式，并用有限差分软件FLAC3D模拟了某基坑开挖后面层的弯矩和挠度情况。结果表明：挠度和弯矩的理论计算值与数值模拟值吻合度较好，其中最大挠度和最大弯矩的计算值与模拟值之间的误差分别为6 % 和13 %，模拟得到的挠度和弯矩云图与公式计算得到的分布规律一致，均是从中心向四周逐渐减小，验证了该方法的有效性。推导的公式可为面层的设计提供参考。Abstract: Surface layer is an important part of foundation pit supporting system. For the purpose of designing the surface layer reasonably, this paper established a forced model of the surface layer by using the small deflection bending theory of thin plate, and the forced analysis was carried out on the surface layer of shotcrete foundation pit support type. In order to simplify the calculation, the concrete surface layer is assumed to be an elastic body, and the surface layer is simplified as a rectangular thin plate with four sides simply supported on the foundation pit slope, which derives in the general expressions of the deflection and bending moment of the bottom layer under active earth pressure. And the bending moment and deflection of the bottom layer after the foundation pit excavation are simulated by the finite difference software FLAC3D. The results show that the theoretical values of the deflection and bending moment are in agreement with those of numerical simulation, the margin of error between the calculation value and the simulation value of the maximum deflection and maximum bending moment are 6 % and 13 % respectively, the distribution laws which simulation cloud pictures of the deflection and bending moment show are consistent with those calculated by the formulas. The deflection and bending moment both reduce gradually from the surface center to boundary, which verifies the effectiveness of the proposed method. The formulas can provide reference for the design of surface layer.
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Key words:
- surface layer /
- small deflection bending of thin plat /
- deflection /
- bending moment /
- validity
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表 1 材料物理、力学参数
Table 1. Physical and mechanical parameters of the materials
类型 密度ρ/kg·m-3) 弹性模量E/MPa 泊松比μ 黏聚力c/kPa 内摩擦角φ/(°) 砂土 3 000 10 0.2 0 30 混凝土 2 300 3×104 0.2 - - 
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表 2 面层各点挠度的计算值和模拟值
Table 2. Calculation and simulation values of the surface layer deflection
监测点 计算值/mm 模拟值/mm 误差/mm O 7.76 7.26 0.50 1 5.49 4.42 1.07 2 5.49 4.96 0.53 3 5.49 5.47 0.02 4 5.49 4.96 0.53 5 3.88 3.03 0.86 6 3.88 3.81 0.07 7 3.88 3.81 0.07 8 3.88 3.02 0.86
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表 3 面层各点弯矩的计算值和模拟值
Table 3. Calculation and simulation values of the surface layer bending moment
监测点 计算值/(N·m) 模拟值/(N·m) 误差 O 53 218.85 46 138.80 13% 1 37 631.41 29 839.31 20% 2 37 631.41 33 942.20 10% 3 37 631.41 37 614.50 0% 4 37 631.41 37 107.40 1% 5 26 609.43 24 261.50 9% 6 26 609.43 33 050.80 20% 7 26 609.43 31 360.60 18% 8 26 609.43 21 895.20 18%
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