Ground Displacements and Stresses around Single and Twin Tunnels using Analytical and Numerical Approaches

Abstract

Significant emphasis is given nowadays on exploring underground space- in the form of tunnels- for (i) fast commuting in urban areas, (ii) connectivity to remote areas under different weather conditions, (iii) military and strategical advantages, (iv) housing missiles, nuclear facilities, oil storage facilities and research observatories, (v) water supply and sewage drains. To determine the response of tunnels, geotechnical engineers often utilize several analytical and numerical based approaches. Extensive work seems to have been carried out in literature by using different numerical tools, especially, the finite elements (FE), to study the response of ground to shallow tunnelling. Similarly, numerous analytical works have been reported on tunnels in predicting the stress and displacement response. The ground response, especially the ground surface vertical settlement, becomes critically important to geotechnical, structural and tunnelling engineers. In the present thesis, it has been attempted to apply analytical and finite element (FE) based approaches with an objective of predicting ground displacements and stresses due to laying of single and twin tunnels in soft ground. The objective of this research is to solve single and twin tunnel problems incorporating the presence of pseudo-static earthquake body forces. It is aimed to (i) develop new analytical solutions enabling to incorporate ground volume loss (GVL) and buoyant force problems for tunnels in weightless and ponderable (weighty) media, and to (ii) use the FE method to develop design charts for ground surface settlement for single and twin lined (supported) and unlined (unsupported) tunnels. The plane strain formulation, considering the tunnel as very long as compared to its cross-section, helps to reduce the difficulties of the modelling of the problem since the three dimensional analysis often becomes quite complicated. The study deals with planar tunnelling problems in linearly elastic and linearly elastic perfect plastic medium. For developing the analytical based solutions, computer codes have been written in MATLAB, while for the numerical FE based models, the commercially available codes OPTUM G2 and PLAXIS 2D have been employed. In the initial section of the work, the analytical solutions based on the Airy’s stress function method (ASFM) and complex variable method (CVM) have been determined. The ASFM has been used to obtain solution for a non-uniformly deforming lined circular tunnel in an elastic medium. The non-uniform radial displacement around the tunnel periphery for a given GVL was modelled by using the Gaussian distribution function. To simplify the analytical formulation, the soil-liner interface was assumed to be frictionless. The horizontal and vertical ground displacements (u, v), the radial & circumferential stresses (σr and σθ ) in soil media, and the axial forces and the bending moments in the liner were computed. The results were thoroughly compared and validated with the FE based solutions and with field measurements from various recorded tunnelling case-studies. The CVM has been applied to obtain analytical solutions for an unlined single circular tunnel deforming non-uniformly (i) in a weightless medium, and (ii) in a ponderable medium without and with horizontal earthquake body forces. The effects of various parameters, namely, pseudo-static earthquake acceleration (αh) in horizontal direction, ground volume loss (���), cover to diameter ratio, the ratio (wt) of weight of liner to weight of soil excavated on ground displacements and stresses have been examined. For these analytical solutions, necessary validation has been provided by comparing the results with that obtained on the basis of the FE based analyses. It was observed that (i) the shallow tunnels generally cause more ground settlements as compared to deeper tunnels, (ii) a higher magnitude of GVL leads to greater ground surface settlements, (iii) heave gets generated on unloading (excavation) of the tunnel cavity for lower magnitude of wt, and (iv) the pseudo-static earthquake horizontal body forces cause the deformations and stresses to become asymmetric about the tunnel’s centreline and it leads to significant changes in horizontal displacement patterns. The CVM has also been employed for the case of non-uniformly deforming twin circular tunnels in the presence of pseudo-static earthquake body forces by using a stepwise iterative procedure. The effects of tunnel centre to centre spacing, cover depths, orientation with respect to each other, ��� and �h on stress and displacement patterns have been thoroughly examined. It was observed that the shallower tunnel mainly dictates the overall ground settlement response - both vertical and horizontal displacements- of twin tunnels. Similar to a single tunnel, the presence of �h changes the horizontal displacement patterns around twin tunnels significantly. For twin tunnels with dissimilar values of ���, a tunnel with greater value of ��� leads to greater ground surface settlements on the top of its crown. In the latter part, single and twin circular tunnels have been modelled in FE based code OPTUM G2. In addition to the vertical ground surface displacements (�!), the non-dimensional stability numbers ( γD c + σi ) corresponding to ultimate collapse have also been computed by performing lower and upper bound two-dimensional finite element limit analysis (FELA); here γ and c refer to unit weight and cohesion of soil media, D is the dimeter of the tunnel and σi is the supporting pressure on the tunnel periphery. By using the Mohr-Coulomb yield criterion, an associated flow rule and adaptive mesh pattern, a comprehensive set of analyses have been performed. Apart from the effect of different soil parameters E, c, �, and γ, the effects of spacing (�) in case of twin tunnels and cover (�) to diameter (D) ratio of twin tunnels, has also been carefully examined on the results; here E and � refer to Young’s modulus and friction angle of media. It was noted that the magnitude of ! &! ' " increases with (i) an increase in the value of ( γD c + σi ), (ii) decrease in the value of �/�, (iii) an increase in the value of �/�, (iv) decrease in the values of �, � and �. For all the problems studied in this thesis, stress and displacement contour plots have been drawn comprehensively by varying the various input material and geometric parameters. The analytical and numerical solutions presented in this thesis will be useful for predicting ground displacements and stresses for single and twin tunnel problems