Comparison of traditional and whole-space stereographic projections in block stability analysis
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摘要: 工程地质分析中大量采用的极射赤平投影是半球投影：将三维空间的平面和直线投影到赤道圆内的一段圆弧或一个点。通过分析多组结构面在赤道圆内的投影弧交点与临空面投影弧的相对位置关系，判断能否形成不稳定块体。然而，这类分析判断缺乏严格的理论依据。块体理论中的全空间赤平投影，将整个平面及面的上半空间和下半空间均投影到赤道面上，通过分析多组结构面投影形成的节理锥和临空面投影圆的相对位置关系，判断块体的可动性和失稳模式，其理论依据是完备的、准确的。通过构造算例，指出“工程地质手册”和“水利水电工程边坡设计规范”中采用传统赤平投影进行边坡稳定分析时可能存在的不足，并揭示了其原因。并且，通过全空间赤平投影，展示了节理锥的连续变化及其意义，分析了扁平或尖长块体的投影区特征。最后讨论了全空间赤平投影的优势，及其简单易用、适合推广的特点。Abstract: The stereographic projection used in engineering geological analysis is a hemispherical projection: a plane or a straight line in 3-D space is projected to be an arc or a point inside the equatorial circle. This kind of projection is widely used to judge the unstable blocks cut by discontinuities by analyzing the relationship of locations between the intersection of arcs projected by multiple sets of discontinuities and the arc projected by the free surface. Nevertheless, this kind of judgement lacks a rigous theoretical basis. The whole-space stereographic projection proposed in the block theory projects a whole plane and both its upper-half and lower-half space onto the equatorial plane. The removability and failure mode of blocks can be determined by analyzing the location relationship between the joint pyramid projected by multiple sets of discontinuities and the circle projected by the free surface. This approach is complete and accurate in theory. In this study, the shortcomings of the traditional stereographic projection used for slope stability analysis in "Geological engineering handbook" and "Design code for engineering slopes of water resources and hydropower projects" are pointed out by analyzing some specially designed examples, and the reasons for these shortcomings are released. Through the whole-space stereographic projection, the continuous variation of morphology of joint pyramid is exposed, the characteristics of the shape of projection zone projected by flat or sharp blocks are analyzed, and the significance of these issues is explored. Finally, the advantages of the whole-space stereographic projection are summarized, and the simplicity in using and the feasibility for popularization are pointed out.
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图 1 棱锥的表示及不同临空面上出现的可动节理锥
Figure 1. Diagram of pyramids and removable joint pyramids exposed on various free surfaces
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图 3 不同节理锥与坡面形成的块体形态
Figure 3. Morphologies of blocks formed by intersection of different joint pyramids and slope surfaces
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图 4 改变面2的倾向后的全空间赤平投影
Figure 4. Different whole-space stereographic projections by changing dip direction of plane 2
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图 5 边坡失稳判别的赤平投影
Figure 5. Stereographic projection used in judgment of slope stability
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图 6 第3组结构面倾向变化时的全空间赤平投影
Figure 6. Variation of whole-space stereographic projection by changing dip direction of discontinuity set 3
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图 7 第3组结构面倾向变化时得到的块体形态
Figure 7. Different morphologies of blocks by changing dip direction of discontinuity set 3
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图 8 面P3通过面P2与临空面交棱时的投影
Figure 8. Whole-space stereographic projection when face 3 passing through intersection line between face 2 and free surface
表 1 第3组结构面P3倾向变化时可动节理锥及失稳模式变化
Table 1 Variation of removable joint pyramids and failure modes by changing dip direction of discontinuity set 3
情形 面P3倾向 可动JP编号 失稳模式 稳定系数 1 40° 001 12 2.645 2 105° 001 12 2.645 3 190° 101 23 5.727 4 250° 000 12 2.645 5 285° 100 23 0.645 6 335° 100 23 1.494 
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