Mean deformation tensor and mean deformation ellipse of an excavated tunnel section

Terzaghi, based on graphical methods, introduced the concept of the deformation ellipse as a tool to understand and visualize the deformation of tunnel sections after their excavation, originally assumed as perfect circles. The deformation ellipse gave the opportunity to pass from a set of displacement data expressing the convergence of a tunnel section in a complicated and occasionally mutually inconsistent way, to the concept of deformation of a circle to an ellipse, and thereby to understand the pattern and the changes of the pattern of deformation of a tunnel during its excavation, and search for the causes of this change. Nevertheless, because of its graphic character and limitation in the available data, this approach seems to have rarely been used.In this paper we present analytical solutions for the mean tensor of deformation and of the mean deformation ellipse of a tunnel section, regarded as a deforming, pure mathematical curve. This transformation of a circle to an ellipse corresponds to the strain due to re-arrangement of stresses just after the section excavation, and hence describes the deformation of the surrounding rockmass and of support, independently of any stress-focusing models and measurements. This technique can be used for all types of geodetic and geotechnical monitoring data (tape, extensometer, total station, laser scanner etc. measurements) in tunnels and other underground excavations and its only requirements are (a) uniform, plane strain conditions in each control section during and shortly after the underground excavation, and (b) a nearly uniform distribution of observations along the tunnel section. Uncertainties in estimates permit to confirm whether the adopted model of a mean deformation ellipse is suitable to describe the deformed excavation section. The robustness and value of this approach are highlighted in two case-studies.