The Hyperstatic Reaction Method for the Analysis of the Sprayed Concrete Linings Behavior in Tunneling

Hyperstatic Reaction Method considers the interaction between the shotcrete lining and the rock mass modelling half tunnel section by beam elements connected by nodes, which can develop bending moments, axial forces and shear forces. The interaction between ground and support is represented by Winkler type springs in the nodes of the numerical model, in the normal and tangential direction. 1944 analyses were developed considering three different tunnel radius values (2.0, 4.5 and 7.0 m), three different rock types (RMR = 40, RMR = 60 and RMR = 80), three different stress ratios acting on the linings (\(K = 1, \,K = 0.5\) and \(K = 0\)), three different lining thickness values (0.1, 0.2 and 0.3 m) and three values of the average elastic modulus of the shotcrete (6000, 9000 and 12,000 MPa). Besides, also two different stiffness values of the tangential springs at nodes were considered. Results show that the lining thickness and the elastic modulus of the shotcrete have a strong influence on the bending moment values which develop inside. The maximum axial force is not practically influenced by the lining elastic modulus and thickness. Both maximum bending moments and maximum axial forces increase in a linear way with the vertical load value; tunnel radius, \(K\) and RMR values of the rock mass have also a great influence on the maximum bending moment and axial force. Besides, it is possible to note how the absence of shear interaction between the lining and the rock mass causes bending moment values in the lining higher with respect to the case where the stiffness of the tangential springs at nodes equals to half the stiffness of the normal springs. For this reason, the neglectance of the shear interaction between the lining and the tunnel wall leads to an overdesign of the SC lining.