Analytical solution of steady seepage field of foundation pit considering thickness of retaining wall
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摘要: 对悬挂式考虑厚度挡墙支护下的各向同性土层中基坑的二维稳态渗流场进行了解析研究。根据对称性取基坑半截面，将周围土层划分为5个规则的区域，利用叠加法和分离变量法将5个区域内的水头分布表示为级数解的形式，结合区域间的连续条件得出基坑周围渗流场的显式解析解。将本文解析解得到的水头计算结果和挡墙上水压力计算结果与有限元软件进行对比，结果吻合较好，验证了本文解析解的正确性。将一维渗流、二维渗流中流网法以及不考虑挡墙厚度情况下和考虑挡墙厚度情况下本文解析解计算得到的挡墙上的水压力与有限元解进行对比分析，发现考虑挡墙厚度情况下本文解析解的计算结果最为精确，指出了其他几种解法的误差。最后对基坑渗流场影响因素进行了参数分析，基坑内侧宽度、不透水层顶面至挡墙底部距离、坑内水位、挡墙厚度等因素对基坑水头分布有较大影响，随着挡墙厚度增大，最大水力梯度降到挡墙底部，基坑外侧渗流场远场水头增大，挡墙底部水头明显减小。Abstract: The two-dimensional steady-state seepage field of a foundation pit in an isotropic soil layer supported by a suspended retaining wall considering thickness is analytically studied. The distribution of water head in the five regions is expressed in the form of a graded solution by using the superposition method and the method of separation of variables, and the explicit analytical solution of the seepage field around the pit is derived by combining the continuity conditions between the regions. Compared with the angle-preserving transform solution, this analytical solution can directly solve the values of water head at any point inside and outside the pit without generating singularities. The correctness of the analytical solution is verified by comparing with the calculated results of the water head obtained from the finite element software. The distribution of total water head obtained from the analytical solution of this paper is curved. The factors such as the inner width of the pit, the distance from the top of the impermeable layer to the bottom of the retaining wall, the water level in the pit and the thickness of the retaining wall have a great influence on the distribution of water head of the pit. As the thickness of the retaining wall increases, the water heads in the far field of the seepage field of the pit and the bottom of the retaining wall change more, and the maximum hydraulic gradient drops to the bottom of the retaining wall. Compared with the flow network methods in one-dimensional seepage and two-dimensional seepage and the proposed analytical solution without considering the thickness of the retaining wall, it is more accurate to calculate the water pressure in the design by using the analytical solution of two-dimensional seepage considering the thickness of the retaining wall.
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图 2 本文解与有限元软件水头计算结果对比
Figure 2. Comparison between solution in this paper and calculated results of water head by finite element software
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图 3 不同计算方法下水压力计算结果比较
Figure 3. Comparison of calculation results of water pressure among different calculation methods
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图 4 c对基坑周围总水头分布的影响
Figure 4. Effects of c on distribution of total water head around foundation pit
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图 5 a对基坑周围总水头分布的影响
Figure 5. Effects of a on distribution of total water head around foundation pit
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图 6 h2对基坑周围总水头分布的影响
Figure 6. Effects of h2 on distribution of total water head around foundation pit
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图 7 d对基坑周围总水头分布的影响
Figure 7. Effects of d on distribution of total water head around foundation pit
表 1 基坑渗流模型的工程参数
Table 1 Parameters of seepage model for foundation pit (单位: m)
b c h1 h2 a d 23 7 15 9 6 2 
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表 2 基坑外侧挡墙上各点的水压力分布
Table 2 Distribution of water pressure at each point on retaining wall outside foundation pit
计算点号 深度zi /m 水压力/kPa 一维渗流 本文解(考虑挡墙厚度) 有限元法(考虑挡墙厚度) 流网法 有限元法(不考虑挡墙厚度) 本文解(不考虑挡墙厚度) a 0 0 0 0 0 0 0 b 5 49.00 40.49 39.76 36.00 37.71 38.45 c 9 88.20 71.97 70.62 62.10 66.73 68.04 d 12 117.60 94.23 92.54 78.50 86.76 88.46 e 15 147.00 113.67 111.63 94.90 102.85 104.91 f 17 166.60 121.41 118.16 101.30 105.85 107.12
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表 3 基坑外侧挡墙上各点水压力不同计算方法误差对比
Table 3 Comparison of errors among different methods for water pressure at each point on retaining wall outside foundation pit
计算点号 深度zi /m 本文解(不考虑挡墙厚度)较有限元法(不考虑挡墙厚度)误差/% 流网法较有限元法(不考虑挡墙厚度)误差/% 一维渗流较本文解(不考虑挡墙厚度)误差/% 本文解(考虑挡墙厚度)较有限元法(考虑挡墙厚度)误差/% 本文解(不考虑挡墙厚度)较本文解(考虑挡墙厚度)误差/% a 0 0 0 0 0 0 b 5 1.96 -4.53 27.44 1.84 -5.04 c 9 1.96 -6.94 29.63 1.91 -5.46 d 12 1.96 -9.52 32.94 1.83 -6.12 e 15 2.00 -7.73 40.12 1.83 -7.71 f 17 1.20 -4.30 55.53 2.75 -11.77
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表 4 基坑内侧挡墙上各点的水压力分布
Table 4 Distribution of water pressure at each point on retaining wall inside foundation pit
计算点号 深度zi /m 水压力/kPa 一维渗流 本文解(考虑挡墙厚度) 有限元法(考虑挡墙厚度) 流网法 有限元法(不考虑挡墙厚度) 本文解(不考虑挡墙厚度) d 0 0 0 0 0 0 0 e 3 29.40 46.23 45.90 42.50 50.22 50.68 f 5 49.00 82.34 83.90 75.13 95.36 94.13
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表 5 基坑内侧挡墙上各点水压力不同计算方法误差对比
Table 5 Comparison of errors among different calculation methods of water pressure at each point on retaining wall inside foundation pit
计算点号 深度zi /m 本文解(不考虑挡墙厚度)较有限元法(不考虑挡墙厚度)误差/% 流网法较有限元法(不考虑挡墙厚度)误差/% 一维渗流较本文解(不考虑挡墙厚度)误差/% 本文解(考虑挡墙厚度)较有限元法(考虑挡墙厚度)误差/% 本文解(不考虑挡墙厚度)较本文解(考虑挡墙厚度)误差/% d 0 0 0 0 0 0 e 3 0.92 -15.37 -41.99 0.72 9.63 f 5 -1.29 -23.38 -47.94 -1.86 14.32
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