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摘要:[目的] 基于极限平衡理论，提出了一种求解一般条件下条形浅基础极限承载力的精确方法。[方法] 地基土被视为服从Mohr-Coulomb屈服准则的理想弹塑性材料，并且假定它是各向同性的、均匀的以及不可压缩或不可膨胀的理想连续介质。通过分析基础与土之间的相对运动和相互作用，将条形浅基础极限承载力问题分为两类问题。建立一个以总竖向极限承载力为目标函数的最小值模型，进而采用滑移线法求解极限承载力而无需事先对塑性区和非塑性楔作任何假定，还提出一种工程上方便实用的简化方法。此外，重点研究了基础两侧均布荷载相同的第一类问题，推导出Terzaghi极限承载力方程的适用条件以及其三个承载力系数的理论精确解，提出一个新承载力方程替代Terzaghi方程，并且通过无量纲分析提出几何力学相似原理。[结果] 研究结果表明：当基础完全光滑时，研究得到的总竖向极限承载力与现有方法得到的结果相当吻合；然而当基础完全粗糙时，现有方法低估了极限承载力。[结论] 经典的Prandtl机构并不是无重土地基上完全光滑基础极限承载力问题的塑性破坏机构。Abstract:[Introduction] Based on the limit equilibrium theory, an accurate approach is proposed to solve the ultimate bearing capacity of shallow strip footings under general conditions.[Method] The foundation soil is considered to be an ideal elastic-plastic material, which obeys the Mohr-Coulomb yield criterion, and is assumed to be an ideal continuous medium that is isotropic, homogeneous and incompressible or non-expansive. Through analyzing the relative motion and interaction between the footing and soil, the problem of the ultimate bearing capacity of shallow strip footings is divided into two categories. A minimum model with the total vertical ultimate bearing capacity as its objective function is established to solve the ultimate bearing capacity using the slip-line method with no need to make any assumptions on the plastic zone and non-plastic wedge in advance. A convenient and practical simplified method is also proposed for practical engineering purposes. Furthermore, the first category of the problem in the case of the same uniform surcharges on both sides of footing is the focus of the study: the applicable conditions of Terzaghi′s ultimate bearing capacity equation as well as the theoretical exact solutions to its three bearing capacity factors are derived, and a new bearing capacity equation is put forward as a replacement for Terzaghi′s equation. The geometric and mechanical similarity principle is proposed by a dimensionless analysis.[Result] The results show that for perfectly smooth footings, the total vertical ultimate bearing capacity obtained by the present method is in good agreement with those by existing methods; whereas for perfectly rough footings, the existing methods underestimate the ultimate bearing capacity.[Conclusion] The classic Prandtl mechanism is not the plastic failure mechanism of the ultimate bearing capacity problem of perfectly smooth footings on weightless soil.
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表 1 竖向承载力系数Nv0和Nv1
Table 1
Vertical bearing capacity factors Nv0 and Nv1 工况 φ＝25°，δ＝-10° φ＝25°，δ＝10° η2/η1 -1 -0.5 0 0 0.5 1 η1 Nv0 Nv1 Nv0 Nv1 Nv0 Nv1 Nv0 Nv1 Nv0 Nv1 Nv0 Nv1 0.0 0 3.422 9 0 3.422 9 0 3.422 9 0 11.330 0 11.330 0 11.330 0.2 2.551 7 7.442 1 2.798 1 7.945 1 3.013 3 8.407 9 5.225 5 21.239 5.327 4 21.515 5.422 3 21.784 0.4 5.103 5 10.365 5.5962 11.187 6.026 6 11.936 10.451 27.988 10.655 28.426 10.845 28.851 0.6 7.655 2 13.117 8.3943 14.228 9.0399 15.234 15.676 34.131 15.982 34.711 16.267 35.270 0.8 10.207 15.797 11.192 17.183 12.053 18.433 20.902 39.992 21.309 40.703 21.689 41.385 1.0 12.759 18.439 13.991 20.093 15.067 21.579 26.127 45.692 26.637 46.527 27.111 47.327 1.2 15.310 21.059 16.789 22.975 18.080 24.693 31.353 51.288 31.964 52.243 32.534 53.156 1.4 17.862 23.663 19.587 25.839 21.093 27.785 36.578 56.813 37.292 57.884 37.956 58.907 1.6 20.414 26.256 22.385 28.689 24.106 30.861 41.804 62.286 42.619 63.471 43.378 64.602 1.8 22.966 28.843 25.183 31.531 27.120 33.927 47.029 67.720 47.946 69.018 48.800 70.254 2.0 25.517 31.423 27.981 34.365 30.133 36.984 52.255 73.123 53.274 74.533 54.223 75.873 2.2 28.069 33.999 30.779 37.194 33.146 40.034 57.480 78.503 58.601 80.023 59.645 81.466 2.4 30.621 36.572 33.577 40.019 36.160 43.080 62.706 83.864 63.928 85.493 65.067 87.039 2.6 33.173 39.142 36.375 42.840 39.173 46.121 67.931 89.208 69.256 90.946 70.489 92.593 
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表 2 完全光滑基础结果比较
Table 2
Comparison of results for perfectly smooth footings φ/(°) 本文方法 Martin(2004)[22] Bolton和Lau(1993)[16] Chen(1975)[24] Sokolovskii(1965)[14] Nv Nγ Quv/kN Nγ Quv/kN Error/ % Nγ Quv/kN Error/ % Nγ Quv/kN Error/ % Nγ Quv/kN Error/ % q＝0 5 0.085 0.085 0.85 0.084 0.84 -0.05 0.09 0.90 6.51 0.131 1.31 55.03 0.17 1.70 101.18 10 0.281 0.281 2.81 0.281 2.81 0.07 0.29 2.90 3.28 0.461 4.61 64.17 0.56 5.60 99.43 15 0.699 0.699 6.99 0.699 6.99 0.03 0.71 7.10 1.60 1.16 11.60 66.00 1.40 14.00 100.34 20 1.578 1.578 15.78 1.579 15.79 0.06 1.60 16.00 1.39 2.68 26.80 69.83 3.16 31.60 100.25 25 3.461 3.461 34.61 3.461 34.61 0.01 3.51 35.10 1.43 5.90 59.00 70.49 6.92 69.20 99.97 30 7.655 7.655 76.55 7.653 76.53 -0.03 7.74 77.40 1.10 12.70 127.00 65.89 15.30 153.00 99.86 35 17.599 17.599 175.99 17.577 175.77 -0.13 17.80 178.00 1.14 28.60 286.00 62.51 35.20 352.00 100.01 40 43.293 43.293 432.93 43.187 431.87 -0.24 44.00 440.00 1.63 71.60 716.00 65.38 86.50 865.00 99.80 q＝20 kPa 5 1.346 0.210 33.46 0.210* 33.46 0.00 0.09 32.25 -3.60 0.131 32.66 -2.38 0.17 33.05 -1.21 10 3.562 0.619 55.62 0.619* 55.62 0.00 0.29 52.33 -5.92 0.461 54.04 -2.84 0.56 55.03 -1.06 15 7.293 1.411 92.93 1.411* 92.93 0.00 0.71 85.92 -7.54 1.16 90.42 -2.70 1.40 92.82 -0.12 20 13.767 2.968 157.67 2.968* 157.67 0.00 1.60 143.99 -8.68 2.68 154.79 -1.83 3.16 159.59 1.22 25 25.462 6.138 274.62 6.138* 274.62 0.00 3.51 248.34 -9.57 5.90 272.24 -0.87 6.92 282.44 2.85 30 47.723 12.921 497.23 12.921* 497.23 0.00 7.74 445.42 -10.42 12.70 495.02 -0.44 15.30 521.02 4.78 35 93.059 28.467 950.59 28.467* 950.59 -0.00 17.80 843.92 -11.22 28.60 951.92 0.14 35.20 1017.92 7.08 40 193.890 67.499 1 958.90 67.496* 1 958.86 -0.00 44.00 1 723.90 -12.00 71.60 1 999.90 2.09 86.50 2 148.90 9.70 q＝40 kPa 5 2.496 0.225 64.96 0.225* 64.96 0.00 0.09 63.61 -2.08 0.131 64.02 -1.45 0.17 64.41 -0.85 10 6.545 0.660 105.45 0.660* 105.45 0.00 0.29 101.76 -3.50 0.461 103.47 -1.88 0.56 104.46 -0.94 15 13.264 1.499 172.64 1.499* 172.64 0.00 0.71 164.75 -4.57 1.16 169.25 -1.97 1.40 171.65 -0.58 20 24.748 3.151 287.48 3.151* 287.48 0.00 1.60 271.98 -5.39 2.68 282.78 -1.64 3.16 287.58 0.03 25 45.164 6.515 491.64 6.515* 491.64 -0.00 3.51 461.58 -6.11 5.90 485.49 -1.25 6.92 495.68 0.82 30 83.336 13.732 873.36 13.732* 873.36 0.00 7.74 813.44 -6.86 12.70 863.04 -1.18 15.30 889.04 1.80 35 159.508 30.324 1635.08 30.322* 1 635.06 -0.00 17.80 1 509.84 -7.66 28.60 1 617.84 -1.05 35.20 1 683.84 2.98 40 324.929 72.148 3289.29 72.142* 3 289.23 -0.00 44.00 3 007.81 -8.56 71.60 3 283.81 -0.17 86.50 3 432.81 4.36 注：*表示该Nγ值是基于Terzaghi方程由软件ABC计算的Quv值进行换算得到的。
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表 3 完全粗糙基础的结果比较
Table 3
Comparison of results for perfectly rough footings φ/(°) 本文方法 Martin(2004)[22] Bolton和Lau(1993)[16] Chen(1975)[24] Terzaghi(1943)[3] Nv Quv/ kN Nγ Quv/kN Error/ % Nγ Quv/kN Error/ % Nγ Quv/kN Error/% Nγ Quv/ kN Error/ % q＝0 5 0.114 1.14 0.113 1.13 -0.81 0.62 6.20 442.43 0.382 3.82 234.21 0.50 5.00 337.45 10 0.447 4.47 0.433 4.33 -3.16 1.71 17.10 282.29 1.16 11.60 159.33 1.20 12.00 168.28 15 1.267 12.67 1.181 11.81 -6.76 3.17 31.70 150.28 2.73 27.30 115.54 2.50 25.00 97.38 20 3.183 31.83 2.839 28.39 -10.80 5.97 59.70 87.58 5.87 58.70 84.44 5.00 50.00 57.10 25 7.618 76.18 6.491 64.91 -14.79 11.60 116.00 52.28 12.40 124.00 62.78 9.70 97.00 27.34 30 18.083 180.83 14.754 147.54 -18.41 23.60 236.00 30.51 26.70 267.00 47.66 19.70 197.00 8.95 35 43.677 436.77 34.476 344.76 -21.07 51.00 510.00 16.77 60.20 602.00 37.83 42.40 424.00 -2.92 40 111.435 1 114.35 85.566 855.66 -23.21 121.00 1 210.00 8.58 147.00 1 470.00 31.92 100.40 1 004.00 -9.90 q＝20 kPa 5 1.621 36.21 0.376* 35.11 -3.03 0.62 37.55 3.71 0.382 35.17 -2.86 0.50 37.84 4.49 10 4.516 65.16 1.122* 60.65 -6.92 1.71 66.53 2.10 1.16 61.03 -6.34 1.20 65.87 1.09 15 9.722 117.22 2.579* 104.61 -10.76 3.17 110.52 -5.71 2.73 106.12 -9.47 2.50 113.92 -2.81 20 19.187 211.87 5.457* 182.56 -13.83 5.97 187.69 -11.41 5.87 186.69 -11.89 5.00 198.77 -6.18 25 36.951 389.51 11.326* 326.50 -16.18 11.60 329.24 -15.47 12.40 337.24 -13.42 9.70 351.41 -9.78 30 72.205 742.05 23.887* 606.89 -18.21 23.60 604.02 -18.60 26.70 635.02 -14.42 19.70 646.11 -12.93 35 147.066 1 490.66 52.651* 1 192.43 -20.01 51.00 1 175.92 -21.11 60.20 1 267.92 -14.94 42.40 1 252.79 -15.96 40 320.843 3 228.43 124.75* 2 531.45 -21.59 121.00 2 493.90 -22.75 147.00 2 753.90 -14.70 100.40 2 629.42 -18.55 q＝40 kPa 5 2.951 69.51 0.418* 66.89 -3.77 0.62 68.91 -0.87 0.382 66.53 -4.29 0.50 70.67 1.68 10 8.061 120.61 1.233* 111.19 -7.81 1.71 115.96 -3.86 1.16 110.46 -8.42 1.20 119.74 -0.72 15 16.902 209.02 2.814* 185.78 -11.12 3.17 189.35 -9.41 2.73 184.95 -11.52 2.50 202.85 -2.95 20 32.560 365.60 5.926* 315.23 -13.78 5.97 315.68 -13.66 5.87 314.68 -13.93 5.00 347.55 -4.94 25 61.424 654.24 12.262* 549.10 -16.07 11.60 542.49 -17.08 12.40 550.49 -15.86 9.70 605.82 -7.40 30 117.337 1 213.37 25.824* 994.29 -18.06 23.60 972.04 -19.89 26.70 1 003.04 -17.33 19.70 1 095.23 -9.74 35 233.012 2 370.12 56.911* 1 900.95 -19.80 51.00 1 841.84 -22.29 60.20 1 933.84 -18.41 42.40 2 081.59 -12.17 40 493.947 4 979.47 134.96* 3 917.45 -21.33 121.00 3 777.81 -24.13 147.00 4 037.81 -18.91 100.40 4 254.83 -14.55 注：*表示该Nγ值是基于Terzaghi方程由软件ABC计算的Quv值进行换算得到的。
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