A Combined Analytical and Numerical Approach for the Evaluation of Radial Loads on the Lining of Vertical Shafts

The evaluation of the load acting on a shaft support is of fundamental importance for the correct dimensioning of the structure. The load acting on the support can appear somewhat complex. One approach to define the load on the lining may be to use the convergence-confinement method (CCM) normally used in the tunneling design. This process involves intersecting the convergence-confinement (CC) curve with the support reaction line. However, in order to be able to adopt this technique, it is necessary to know the radial displacement of the shaft wall at the point in which the support is to be installed. Using the equations of Vlachopoulos and Diederichs (Rock Mech Rock Eng 42:131–146, 2009) the reaction line of the support can be calculated. Numerical models developed with Flac 2D v.6.0 considering the Mohr–Coulomb criterion and an ideal elasto-plastic behavior simulating stepwise excavation and support installation were developed. The relation between applied internal stress and radial displacement of the wall shaft, obtained by the numerical simulation was compared with the CC curve obtained by the CCM and it showed a good match between the two methods. However, an iterative procedure has also been used to insert the reaction line in the CC graph. The result shows lower initial displacements (and therefore greater radial stress) when compared with the values obtained by numerical calculation with the axisymmetric model. It is therefore recommended the combined use of the CCM (analytical method) and the axisymmetric numerical model (step by step simulation) to obtain the values of the final load on the lining and the final plastic radius, necessary for the correct design of supporting structures on the shaft wall.