关于圆形隧道卡斯特纳公式的讨论

隧道围岩的应力及变形是隧道力学中的基本问题,它直接影响到隧道的安全评价及施工支护措施。20世纪50年代提出的卡斯特纳公式,描述了圆形隧道的塑性区半径和塑性区的应力分布规律,公式已成为隧道力学分析的理论基础。以满足莫尔-库伦准则或Mises准则的理想弹塑性材料为例,按照塑性力学加卸载准则,证明了圆形隧道的开挖过程是加载过程,支护过程是卸载过程;对圆形隧道开挖和支护过程进行了弹塑性FLAC有限差分数值计算,同时针对外压和地应力均为0、内压值为支护应力的隧道进行弹性分析。结果表明,围岩的应力分布在开挖后与卡氏公式相符,但在支护后则与卡氏公式不符。弹塑性分析中,支护引起的围岩应力增量及位移增量分别与弹性分析中的应力和位移相同。因此在涉及塑性变形的隧道力学分析中,卡氏公式只适用于开挖阶段,不适用于支护阶段。

Discussion on Kastner Formulae of Circular Tunnel

Stress and deformation of surrounding rock which directly affecting the safety evaluation and the support or lining measures to a tunnel are basic issues in tunnel engineering. Kastner formulae proposed in the 1950s describe the range of plastic zone and the stress distribution in a circular tunnel. These formulae have acted as the theoretical basis of tunnel analysis for more than half century. The ideal elastoplastic material satisfying Mohr-Coulomb criterion or Mises criterion was adopted as a tunnel material. According to the loading and unloading criteria of plastic mechanics, it was proved that the excavation of circular tunnel means loading in terms of plastic mechanics, while support means unloading. The elastoplastic FLAC finite difference analyses were carried out for excavation and support process of a circular tunnel. At the same time, the elastic analysis was carried out for the tunnel whose external pressure and ground stress were 0 and the internal pressure was support stress. The results show that the stress distribution of surrounding rock is consistent with Kastner formulae for stage after excavation, but not for stage after support. In elastoplastic analysis, the stress increment and displacement increment of surrounding rock caused by support are the same as those in elastic analysis. Therefore, in the tunnel mechanical analysis involving plastic deformation, Kastner formulae are only applicable to the excavation stage, not to the support stage.