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ANALYTIC SOLUTIONS OF ELASTIC TUNNELING PROBLEMS Phần 5

Sự dịch chuyển theo hướng xuyên tâm, uθ là việc di chuyển theo hướng tiếp tuyến, và ug là số tiền mà các đường hầm hợp đồng theo hướng xuyên tâm do mất đất không ổn định. | Section 5.3 Comparison to Yu s Solution 37 Figure 5.2 Displacements scaled by Yji for v 0 and r h 0.7. the horizontal constraint is imposed along the vertical axis of symmetry . For now it is assumed that the stresses along the surface more than five times the tunnel depth away from the origin will be small enough to result in a higher so-called small strain stiffness of the ground see for example 1 which would severely limit the deformations beyond these points. It should be emphasized that the stresses are not affected by the choice of rigid body motion 23 . The computer program which has generated Figure 5.1 and Figure 5.2 can also be used to calculate the maximum errors in the normalized stresses along the surface which are on the order of 1 X 10-12. The maximum errors in the normalized displacements along the tunnel are on the order of 1 X 10-7. These values correspond to the problem shown in Figure 5.1 and Figure 5.2. In addition a numerical integration of the tractions along the tunnel can be performed by the program. This integration has been performed after the superposition of the gravitational stresses 5.7 - 5.9 . This results in a buoyant force in terms of yh2 after deformation which is on the order of 1 X 10-17 indicating that the complete solution is indeed in equilibrium within computational accuracy. 5.3 Comparison to Yu s Solution The results of the calculation have also been compared to the solution by Yu 45 for a rigid buoyant tunnel in an infinite plane. Comparisons of the isotropic stress 11 ơxx ơyy and deviatoric stress ựị ơyy - ơxx 2 ơ2y are shown in Figure 5.3 and Figure 5.4 respectively. It is clear from these plots that the stresses in the ground and around the tunnel are heavily influenced by the presence of a stress-free surface. The stress free surface is included in the exact solution presented in this chapter but is absent from Yu s solution. It is also apparent 38 Buoyancy of a Rigid Tunnel Chapter 5 x h Figure 5.3 Comparison between