Numerical simulation of round robin numerical test on tunnels using a simplified kinematic hardening model

Abstract

The paper summarizes the numerical simulation of the round robin numerical test on tunnels performed in Aristotle University of Thessaloniki. The main issues of the numerical simulation are presented along with representative comparisons of the computed response with the recorded data. For the simulation, the finite element method is implemented, using ABAQUS. The analyses are performed on prototype-scale models under plane strain conditions. While the tunnel behavior is assumed to be elastic, the soil nonlinear behavior during shaking is modeled using a simplified kinematic hardening model combined with a von Mises failure criterion and an associated plastic flow rule. The model parameters are adequately calibrated using available laboratory test results for the specific fraction of sand. The soil–tunnel interface is also accounted and simulated adequately. The effect of the interface friction on the tunnel response is investigated for one test case, as this parameter seems to affect significantly the tunnel lining axial forces. Finally, the internal forces of the tunnel lining are also evaluated with available closed-form solutions, usually used in the preliminary stages of design and compared with the experimental data and the numerical predictions. The numerical analyses can generally reproduce reasonably well the recorded response. Any differences between the experimental data and the numerical results are mainly attributed to the simplification of the used model and to differences between the assumed and the actual mechanical properties of the soil and the tunnel during the test.

Appendix

According to Wang [26] and assuming the full-slip conditions, the following formulations are proposed for the computation of the maximum axial force (N _max) and bending moment (M _max) of the lining:

$$ M_{\hbox{max} } = \pm \frac{1}{6}K_{1} \frac{{E_{\text{m}} }}{{(1 + \nu_{\text{m}} )}}R^{2} \gamma_{\hbox{max} } $$

(14)

$$ N_{\hbox{max} } = \pm \frac{1}{6}K_{1} \frac{{E_{m} }}{{(1 + \nu_{m} )}}R\gamma_{\hbox{max} } $$

(15)

where

$$ K_{1} = \frac{{12(1 - \nu_{\text{m}} )}}{{2F + 5 - 6\nu_{\text{m}} }} $$

(16)

and

$$ F = \frac{{E_{\text{m}} \left( {1 - \nu_{\text{l}}^{2} } \right)R^{3} }}{{6E_{\text{l}} I_{\text{l}} (1 + \nu_{\text{m}} )}} $$

(17)

the flexibility ratio. E _m is the soil elastic modulus, v _m is the soil Poisson ratio, E _l is the lining elastic modulus, v _l is the lining Poisson ratio, I _l is the moment of inertia of the tunnel lining (per unit width), R is the tunnel radius, and γ _max the maximum shear strain at tunnel depth.

According to Penzien [18], the lining internal forces can be computed, assuming the full-slip conditions, using the following expressions:

$$ N(\theta ) = - \frac{{12E_{l} I_{l} \varDelta d_{\text{stru}}^{n} }}{{D^{3} \left( {1 - v_{\text{l}}^{2} } \right)}}\cos 2\left( {\theta + \frac{\pi }{4}} \right) $$

(18)

$$ M(\theta ) = - \frac{{6E_{l} I_{l} \varDelta d_{\text{stru}}^{n} }}{{D^{2} \left( {1 - v_{\text{l}}^{2} } \right)}}\cos 2\left( {\theta + \frac{\pi }{4}} \right) $$

(19)

where N(θ) and M(θ) are the axial force and the bending moment, respectively, and D is the tunnel diameter and:

$$ \pm \varDelta d_{\text{stru}}^{n} = \pm R^{n} \varDelta d_{\text{ff}} $$

(20)

$$ R^{n} = \frac{{\varDelta d_{\text{stru}}^{n} }}{{\varDelta d_{\text{ff}} }} = \pm \frac{{4(1 - v_{\text{m}} )}}{{(1 + a^{n} )}} $$

(21)

$$ a^{n} = \frac{{12(5 - 6v_{\text{m}} )E_{\text{l}} I_{\text{l}} }}{{D^{3} G_{\text{m}} \left( {1 - v_{\text{l}}^{2} } \right)}} $$

(22)

with G _m being the soil shear modulus and $\varDelta d_{\text{ff}}$ the free-field ovaling distortion correlated with the shear strain γ _max.