Formula for permeability coefficient of coarse-grained soil based on parameters of two-dimensional fractal gradation model
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摘要: 粗粒土在自然界中分布广泛且在工程中应用普遍。渗透性是粗粒土的重要性质，对于同种粗粒土，因颗粒级配与孔隙特征几乎决定其渗透性，原则上其渗透系数应该可基于颗粒级配和孔隙特征参量计算得到。采用基于分形理论建立的双分形级配模型对粗粒土的连续级配、间断级配进行定量描述，以确定该级配模型的适用性并获得级配模型参数。在此基础上，基于双分形级配模型参数和Kozeny-Carman公式构造出包含级配参量及孔隙率的渗透系数计算公式。在论述了计算公式中各项物理意义后，基于现有文献的实测数据验证该计算公式的有效性。结果表明，双分形级配模型能准确唯一地定量描述粗粒土的颗粒级配；建立的含颗粒级配参量和孔隙率的粗粒土渗透系数计算公式是合理的，该公式适用于计算连续和间断级配粗粒土的渗透系数。Abstract: The coarse-grained soil is widely distributed in nature and used in engineering. The permeability is one of its key properties, and for the same type of coarse-grained soil, its permeability is almost determined by the grain-size distribution and the pore characteristics. In principle, its permeability coefficient can be calculated based on the grain-size distribution and the pore characteristic parameters. The two-dimensional fractal gradation model based on the fractal theory is used to quantitatively describe the continuous gradation and gap gradation of the coarse-grained soil so as to determine the applicability of the gradation model and obtain its parameters. Then, based on the parameters of the two-dimensional fractal gradation model and the Kozeny-Carman formula, a formula for calculating the permeability coefficient including grain-size distribution parameters and the porosity is established. After discussing the physical meaning of each item in the formula, the validation of the formula is verified by using the measured data in the existing researches. The results show that the two-dimensional gradation model can accurately and uniquely describe the grain size distribution of the coarse-grained soil. The established formula for calculating the permeability coefficient of the coarse-grained soil including grain-size distribution parameters and the porosity is reasonable, and it is suitable for calculating the permeability coefficient of continuously graded and gap-graded coarse-grained soil.
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Keywords:
- coarse-grained soil /
- permeability coefficient /
- fractal model /
- grain-size distribution /
- porosity
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0. 引言
随着中国交通行业的不断发展,中国桥梁建设水平得到大幅提升,对桥梁跨越能力的要求也不断增长,悬索桥作为所有桥型中跨越能力最大的桥型,越来越成为跨越大江、大河的主要解决方案。但是随着悬索桥跨度的不断增加,锚碇规模急剧扩大,造成锚碇建设成本过高。因此研究锚碇沉井基础的受力变形特性对于悬索桥的锚碇优化设计显得尤为重要。
Alampalli[1]在1994年研究了沉井在承受竖向和水平向荷载时的结构响应；李永盛[2]和李家平等[3]分别在1995年和2005年通过模型试验探讨了沉井基础的变形机制和破坏失稳形式；穆保岗等[4]在2017年通过模型试验研究了水平荷载长期作用下沉井变位的特性；Liu等[5]在2019年通过模型试验结合数值模拟分析研究了重力式锚碇的稳定性。
本文首先进行了在分级水平荷载下的沉井在砂箱中的模型试验,然后基于PLAXIS 3D软件建立了有限元模型,并分析了沉井的位移及沉井前侧和沉井底部的土压力,研究了水平荷载条件下沉井的受力变形规律。
1. 依托工程及模型试验
本文依托南京仙新路大桥北锚碇沉井工程,沉井长度为70 m,宽度为50 m,高度为49.5 m。该工程地基土以粉砂和中砂为主。
本试验采用的模型槽平面尺寸为4.0 m×2.0 m,高1.0 m。地基土采用中砂,其相对密度为2.68,最大孔隙比0.881,最小孔隙比0.463,不均匀系数3.89,曲率系数0.92。模型试验分层填筑地基土,控制每层填土的厚度为0.1 m,最终得到地基土的干密度为1.55 g/cm3,含水率0.63%,相对密实度为59.61%,内摩擦角为34.5°（快剪）。
沉井模型平面尺寸为0.7 m×0.5 m,高0.495 m,由厚度为22 mm的钢板焊接而成,试验过程将沉井看成刚体,不考虑沉井自身的变形,为模拟沉井与土体相互作用的界面,通过在沉井表面黏2～3 mm的砂粒实现[6],如图1所示。
模型试验中设计荷载为62kg,本文中水平荷载分级施加,每级荷载为设计荷载的～0.5倍,试验过程中每级荷载施加持续15 min直至土体破坏（土体破坏表现为沉井盖板处的位移急剧增大）,沉井加载示意图如图2所示。
2. 有限元分析
本研究建立的有限元模型完全基于模型试验,土体及沉井的单元形状均为四面体十节点实体单元,数值模型的网格如图3所示。
砂土的本构模型采用土体硬化（HS）模型,土层参数[7]取值见表1。
表 1 土层参数Table 1. Soil parameters土层 γ /(kN·m-3)e Es /MPaErefoed /MPaEref50 /MPaErefur /MPac′ φ′ /(°)ψ /(°)m 砂土 15.6 0.723 10.2 10.2 10.2 30.6 0 34.5 0 0.5 注： γ 为砂土的重度；e 为砂土的孔隙比；Es 为砂土的压缩模量；Erefoed 为砂土的主固结加载切线刚度；Eref50 为砂土的标准三轴排水试验割线刚度；Erefur 为砂土的卸载重加载刚度；c′ 为砂土的有效黏聚力；φ′ 为砂土的有效摩擦角；ψ 为砂土的膨胀角；m 为砂土的刚度应力水平相关幂值。3. 试验结果与有限元结果对比
3.1 沉井顶部位移
在沉井盖板顶部设置3个位移测量点A、B、C。在位移测量点上放置位移靶标,采用TH-ISM-ST机器视觉测量仪对靶标位移进行测量,分辨率为0.01 mm,靶标布置如图4。
模型试验和数值模拟的位移对比如图5所示,由图易知,模型试验和数值模拟的靶标位移较为一致,本文中取水平位移随设计荷载增加而不断增加的线弹性阶段为水平承载力极限值[4],即安全系数取值为4。
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图 5 模试验和数值模拟的位移对比Figure 5. Comparison of displacements between model tests and numerical simulations3.2 沉井前侧壁土压力随施加荷载变化
在沉井前侧设置8个土压力盒,布置如图6所示,由于土压力盒对称分布,且沉井左右侧完全对称,因此取沉井左右两侧土压力盒平均值作为最终结果,结果如图7所示,其中模型试验中3号及3＇号土压力盒数据较差,本文中已舍弃,余下的沉井前侧土压力盒数据和数值模拟结果较吻合。
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图 7 模试验和数值模拟的沉井前侧土压力对比Figure 7. Comparison of soil pressures on front side of caisson between model tests and numerical simulations3.3 沉井前侧壁土压力随施加荷载变化
在沉井底部设置12个土压力盒,布置如图8所示,同理,取沉井左右两侧土压力盒平均值作为最终结果,结果如图9所示,其中8号及8＇号土压力盒数据较差,本文中已舍弃,余下的沉井底部土压力盒数据和数值模拟结果对比,发现当施加荷载/设计荷载的值小于等于4时较一致,当其值大于4之后,二者的结果相差较大。
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图 9 模型试验和数值模拟的沉井前侧土压力对比Figure 9. Comparison of soil pressures on bottom of the caisson between model tests and numerical simulations4. 结论
本文在已有的研究基础上,通过开展模型试验和数值模拟计算,得到水平荷载下沉井的受力变位特性,主要得出以下结论：
（1）对锚碇沉井基础在砂土中的受力变位特性进行了试验研究和有限元分析,结果显示,水平荷载下锚碇沉井基础在砂土中的破坏模式为倾覆破坏,且安全系数远大于2,说明现阶段规范[8]中锚碇设计较为保守,有进一步的优化空间。
（2）通过PLAXIS 3D软件建立了锚碇沉井基础的有限元模型,采用应变硬化的本构模型,结果表明模型试验的结果和有限元模型计算的结果较为一致,说明数值建模过程中的土体本构模型及参数取值可靠,表明PLAXIS 3D软件能够较好的模拟锚碇沉井在砂土中的受力变形行为。
上述模型试验和有限元分析,只是针对水平荷载条件下锚碇沉井基础在砂土中的受力特性开展的研究,只考虑了单层干砂的地基土层,尚需更近一步探索。
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图 1 花岗岩级配碎石试验级配曲线及双分形模型拟合曲线
Figure 1. Fitting curves of granite-graded macadam by tests and two-dimensional fractal model
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图 2 渭河粗粒土试验级配曲线及双分形模型拟合曲线
Figure 2. Fitting curves of coarse-grained soil from Weihe River by tests and two-dimensional fractal model
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图 3 砂岩粗粒料试验级配曲线及双分形模型拟合曲线
Figure 3. Fitting curves of sandstone coarse-grained materials by tests and two-dimensional fractal model
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图 4 花岗岩级配碎石渗透系数实测值及计算值
Figure 4. Measured and calculated values of permeability coefficient of granite-graded macadam
表 1 粗粒土双分形级配模型参数
Table 1 Parameters of two-dimensional fractal gradation model for coarse-grained soil
文献来源 级配编号 级配类型 D1 D2 RT1/mm RT2/mm MT1 MT2 R2 文献[4] JP1 连续 2.592 1.912 45 3.0623 64.00 36.00 0.9971 JP2 连续 2.588 2.528 45 9.1935 55.00 45.00 0.9972 JP3 连续 2.564 2.580 45 10.3868 49.95 50.05 0.9980 JP4 连续 2.500 2.000 45 3.9093 69.00 31.00 0.9983 JP5 连续 2.512 2.472 45 8.1350 67.25 32.75 0.9966 JP6 连续 2.616 2.204 45 7.1977 83.15 16.85 0.9969 JP7 连续 2.400 2.024 45 4.9916 75.00 25.00 0.9996 JP8 连续 2.400 2.224 45 4.4177 75.00 25.00 0.9991 JP9 连续 2.396 2.416 45 3.9097 74.35 25.65 0.9981 文献[5] TYU1 连续 1.904 2.328 20 1.4164 77.00 23.00 0.9962 TYU2 连续 2.292 0.104 20 0.3199 90.00 10.00 0.9979 TYU3 连续 2.404 1.284 20 0.6747 92.05 7.95 0.9969 TYU4 连续 2.308 2.380 20 3.6713 71.15 28.85 0.9960 TYU5 连续 2.300 2.380 20 6.2355 47.10 52.90 0.9964 TYU6 连续 2.324 2.296 20 5.0453 41.85 58.15 0.9970 TYU7 连续 2.404 1.924 20 2.6729 39.95 60.05 0.9981 TYU8 连续 1.840 1.980 20 2.4032 20.00 80.00 0.9984 TYU9 连续 2.544 0.848 20 1.1443 28.35 71.65 0.9979 TYU10 连续 2.692 0.400 20 1.0331 17.00 83.00 0.9974 文献[7] 1-3# 间断 2.600 1.584 60 24.5036 58.00 42.00 0.9927 1-4# 间断 2.376 2.172 60 24.5033 43.50 56.50 0.9959 1-7# 间断 2.616 1.704 60 18.9698 66.05 33.95 0.9915 2-1# 间断 2.620 2.004 60 21.5590 65.00 35.00 0.9928 2-2# 间断 1.500 2.608 60 27.8484 16.05 83.95 0.9824 2-3# 间断 2.604 1.204 60 0.5987 76.00 24.00 0.9807 
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表 2 渭河粗粒土试样渗透系数实测值及计算值
Table 2 Measured and calculated values of permeability coefficient of coarse-grained soil samples from Weihe River
级配
编号渗透系数k/
(10-2cm·s-1)M(r<dc)MT/% 相对误差δ/% 试验
值计算
值[5]式（10）计算值 δ[5] δ式（10） TYU1 6.70 9.50 6.64 31.54 41.79 0.90 TYU2 4.50 7.90 4.71 31.74 75.56 4.67 TYU3 2.80 — 2.66 35.79 — 5.00 TYU4 2.40 5.20 2.48 41.54 116.67 3.33 TYU5 2.30 4.10 2.31 42.97 78.26 0.43 TYU6 2.10 3.00 1.86 48.13 42.86 11.43 TYU7 1.90 2.20 1.74 72.13 15.79 8.42 TYU8 1.00 1.50 1.35 81.95 50.00 35.00 TYU9 0.96 — 1.07 83.01 — 11.46 TYU10 0.88 — 0.77 92.16 — 12.50 注：相对误差δ=|k计算值-k实测值|/k实测值×100%。
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表 3 砂岩粗粒料的渗透系数实测值与计算值
Table 3 Measured and calculated values of permeability coefficient of sandstone coarse-grained materials
级配
编号实测值 M(r<dc)MT/% 渗透系数计算值/(10-2 cm·s-1) 相对误差/% ρd/
(g·cm-3)k/
(10-2 cm·s-1)k式（11） kT kC k水科院 k杨志浩 δ式（11） δT δC δ水科院 δ杨志浩 1-3# 1.89 84.10 32.27 84.14 4.15 37.12 149.84 8.78 0.05 95.07 55.86 78.17 89.56 1-4# 1.91 60.71 32.10 60.39 20.45 143.42 138.75 15.62 0.53 66.32 136.24 128.55 74.27 1-7# 1.89 6.17 38.95 5.32 2.51 25.65 31.71 6.09 13.78 59.32 315.72 413.94 1.30 2-1#① 1.91 7.14 40.80 8.56 1.67 18.21 22.20 4.90 19.89 76.61 155.04 210.92 31.37 2-1#② 1.82 8.76 40.80 9.22 2.54 27.93 30.93 6.73 5.25 71.00 218.84 253.08 23.17 2-1#③ 1.74 10.64 40.80 10.04 3.50 39.91 40.39 8.79 5.64 67.11 275.09 279.61 17.39 2-2#① 1.91 0.79 50.08 0.75 0.90 8.80 0.89 2.32 5.06 13.92 1013.92 12.66 193.67 2-2#② 2.00 0.39 50.08 0.29 0.54 5.53 0.61 1.65 25.64 38.46 1317.95 56.41 323.08 2-2#③ 1.79 1.17 50.08 1.69 1.55 15.47 1.37 3.53 44.44 32.48 1222.22 17.09 201.71 2-3#① 1.91 0.59 56.83 0.58 0.67 5.17 0.50 1.32 1.69 13.56 776.27 15.25 123.73 2-3#② 2.00 0.11 56.83 0.11 0.40 3.25 0.34 0.94 0 263.64 2854.55 209.09 754.55 2-3#③ 1.81 0.90 56.83 1.33 1.06 8.30 0.72 1.87 47.78 17.78 822.22 20.00 107.78
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