An Analytical Method for Vertical Additional Stresses and Displacements in 3-Layer Ground Under Rectangular Uniform Load
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摘要: 对于实践中常见的三层地基的竖向附加应力与位移计算问题，现有理论方法尚不能进行合理且可便捷操作的解析. 为了在理论计算方面解决此问题，基于弹性理论建立了三层地基的层状弹性半空间分析模型，通过中间变量转换，运用变量状态空间理论与Hankel积分变换方法，推导出了矩形均布垂直荷载下三层地基竖向附加应力与位移的紧凑解析解，并提出了避免数值溢出的高效率数值计算实现方法，包括地表沉降和地基内部竖向附加应力与位移的高精度数值积分边界处理方法. 实例分析结果表明：该文方法与FLAC3D数值模拟结果吻合良好，与规范建议的有限深度地基模型的计算误差约为8%；对于各层差异明显的三层地基，当地表下一层土体厚度大于荷载宽度时，现行规范的均质地基方法计算的地基中附加应力误差较大；对由上而下依次为中-软-硬土层的地基，在土体深度与荷载宽度之比≤0.75时，三层地基理论的计算值小于传统均质地基方法的结果，反之，传统方法可能低估了地基中的附加应力. 该文方法揭示出上两层土体不同厚度时，地基的中上部区域的附加应力差异较为显著，而增大最上面一层土体厚度，可明显提高地基中沿深度的应力扩散效率.Abstract: The existing theoretical methods cannot reasonably and practicably analyze vertical additional stresses and displacements in a 3-layer ground, which is common in actual engineering. To solve this theoretical calculation problem, a semi-infinite layered elastic model for the 3-layer ground was established based on the elastic theory. According to the proposed transformation of intermedium variables, the state space theory and the Hankel integral transformation, the closed-form analytical solutions to the vertical additional stresses and displacements under a rectangular uniform load on the 3-layer ground, were deduced. Also, an effective numerical calculation tactic was provided to carry out the proposed method to avoid the possible numerical overflow. Meanwhile, an approach for the integral upper bound to obtain the high-accuracy numerical results of the settlements at the ground surface and the vertical stresses and displacements in the ground was put forward. Analytical results of some examples show that, the proposed solutions agree well with the numerical results via FLAC3D, and the error between the proposed model and the finitely deep foundation model based on China codes is about 8%. If the 1st layer thickness of the multi-layer ground with greatly different 3 layers is larger than the load width, the error of the additional stresses calculated according to the current codes will be very high. As for the ground sequentially including medium, soft and hard layers from the surface, the proposed solutions are obviously less than those obtained with the traditional method for homogeneous ground within the range where the ratio of the soil depth to the load width is not more than 0.75. If the soil depth is over the range, the traditional method will underestimate the additional stress in the ground. Moreover, the proposed method reveals that the thicknesses of the upper 2 layers have a great influence on the additional stresses in the upper and middle areas of the ground, and the stress dispersion efficiency along the depth evidently increase with the thickness of the 1st layer under the surface.
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图 1 矩形面积荷载作用下三层地基分析模型
Figure 1. The analysis model for the 3-layer foundation under a rectangular uniform load
图 2 实际矩形面积荷载分割示意图
Figure 2. Schematic diagram of the division of the actual rectangular uniform load
图 3 本方法计算求解的技术框架图
Figure 3. The computation technology infrastructure diagram of the proposed method
图 4 均布矩形面积荷载作用下均质地基的竖向附加应力与竖向位移
Figure 4. Vertical additional stresses and displacements of a homogeneous ground under a uniform rectangular uniform load
图 5 均布矩形面积荷载作用下双层地基的竖向附加应力与地面沉降
Figure 5. Vertical additional stresses and surface settlements of a double-layer ground under a uniform rectangular uniform load
图 6 均布矩形面积荷载作用下三层地基的竖向附加应力与竖向位移
Figure 6. Vertical additional stresses and displacements of a three-layer ground under a uniform rectangular uniform load
图 7 实例三层地基的竖向附加应力和竖向位移沿深度分布曲线
Figure 7. Profiles of vertical additional stresses and displacements along the depth of a 3-layer ground example
图 8 上部两土层不同软硬关系下三层地基中的竖向附加应力随深度变化曲线
Figure 8. Profiles of vertical additional stresses in the 3-layer example along the depth under different properties of the upper 2 soil layers
图 9 两种典型软硬土层组合地基的竖向附加应力随深度变化曲线
Figure 9. Distribution curves of vertical additional stresses along the depth of the 3-layer example under 2 typical combinations of soft and hard layers
图 10 中-软-硬土层组合地基的上两层土体厚度对竖向附加应力的影响
Figure 10. Influence of the thickness of the upper two layers in the medium-soft-hard soil stratum on vertical additional stress
图 11 不同X取值下的计算耗时和计算精度
Figure 11. Computational time costs and accuracy under different values of X
表 1 中-软-硬土层组合地基上两层不同厚度时的附加应力系数
Table 1. Coefficients of vertical additional stresses in a medium-soft-hard soil stratum under different thicknesses of the upper 2 layers
z/B homoge nization solution h1=20 m, h2=10 m h1=10 m, h2=20 m h1=h2=15 m h1=h2=20 m h1=h2=10 m k relative deviation/% k relative deviation/% k relative deviation/% k relative deviation/% k relative deviation/% 0.5 0.818 0.807 -1.4 0.769 -6.0 0.773 -5.5 0.783 -4.3 0.833 1.8 1.0 0.549 0.574 4.6 0.615 12.0 0.591 7.7 0.518 -5.6 0.675 22.9 1.5 0.395 0.485 22.9 0.496 25.6 0.488 23.5 0.444 12.3 0.480 21.6 2.0 0.303 0.378 24.7 0.375 23.9 0.376 24.0 0.378 24.7 0.359 18.4 2.5 0.244 0.301 23.2 0.295 20.9 0.298 22.1 0.307 25.7 0.280 14.9 3.0 0.202 0.245 21.3 0.239 18.4 0.243 20.2 0.253 25.4 0.227 12.4 
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