A simple equation for obtaining finite strain solutions from small strain analyses of tunnels with very large convergences

Abstract

This paper presents a novel, very simple, accurate, theoretically well-founded and widely applicable relationship expressing the tunnel convergences obtained from large strain elasto-plastic analyses as a hyperbolic function solely of the corresponding small strain convergences. It can thus be used for 'selfcorrecting' small strain solutions, removing the need for large strain elasto-plastic analyses at least at the preliminary design stage and quantifying a hitherto unknown error caused by disregarding the geometric non-linearity. The proposed relationship can be proved rigorously for the plane strain rotationally symmetric ground response problem with a general elasto-plastic constitutive law with or without dilatancy and hardening. Numerical analyses of characteristic two-and three-dimensional excavation problems show that this relationship is generally applicable, irrespective of the in situ stress state and the tunnel geometry. It is therefore very useful from an engineering point of view for the design of tunnels crossing heavily squeezing ground, where the convergences may be so large (sometimes well in excess of 10% of the tunnel radius) that the usual small strain elasto-plastic analyses are deficient.

Topics of Analytical and Computational Methods in Tunnel Engineering

“…Closed-form solutions can be found for every criterion of soil and rock literature in the case of isotropic or hydrostatic stress field (k = 1), because of the axisymmetric type of this case. Recently, Vrakas and Anagnostou presented an extension of the small strain analyzes to obtain finite strain solutions [47]. For the general case of stress fields (k 6 ¼ 1), closed-form solutions have been obtained so far for the Tresca criterion [48] and for the Mohr-Coulomb criterion [49].…”

Fluid injection‐induced cavity expansion in dry porous medium

“…The fluid injection problem is analyzed for two cases, see Table 1 for the values of the parameters n , M , Y , Γ, α, and β used for these simulations. The duration of the simulation $$t_f$$ is selected so that the cavity radius has experienced a maximum increase of about 10%, which is a practical limit for the validity of the small deformation theory underlying the model 40 . The cases have been selected so that $$c < s$$ in Case 1 and $$s < c$$ in Case 2 at the end of the simulation.…”

“…The evolution of the cavity radius 𝑎, permeation front 𝑠, and permeation coefficient 𝜁 are determined by solving the two ODEs in Equation (38) together with Equation (40). These three equations are solved numerically considering the initial conditions for both 𝑎 and 𝑠 to be 1.…”

“…Equations ( 48)-( 50) and the kinematic boundary condition for Phase I, Equation (40), are numerically solved to determine 𝑡 𝑒 directly. A first-order estimate of 𝑡 𝑒 is obtained from Equations ( 48) and ( 50) by neglecting the change in cavity radius during phase I (i.e., 𝑎 = 1) and assuming that the permeation coefficient to be equal to its initial value, 𝜁 0 (Equation B.1),…”

“…The duration of the simulation 𝑡 𝑓 is selected so that the cavity radius has experienced a maximum increase of about 10%, which is a practical limit for the validity of the small deformation theory underlying the model. 40 The cases have been selected so that 𝑐 < 𝑠 in Case 1 and 𝑠 < 𝑐 in Case 2 at the end of the simulation. The ODEs (38), which are coupled with nonlinear algebraic Equation (40) for Phase I, Equations ( 44) and (45) for Phase II and Equations ( 46) and (47) for Phase III are solved numerically using an adaptive Runge-Kutta method.…”