Constitutive Models and Determining Methods Effects on Application of Convergence–Confinement Method in Underground Excavation

Abstract

Stress reduction factor (λ) is an important component in the two-dimensional (2D) analysis of tunnel using convergence–confinement method (CCM). So far, however, there has been little discussion about effective parameters on λ. The aim of this study was to evaluate what the effect of constitutive model and determining method of λ is on CCM application in circular tunnel in shallow depth. For this purpose, at first, a series of parametric studies were carried out in different circular tunnel radius and depth. Parameters calibration of each constitutive model was carried out by triaxial simulation using finite difference method (FLAC). The results showed significant impact of constitutive model on λ in tunnel walls. The effects of constitutive model on λ were reduced by increasing depth, radius and distance from tunnel face. Then a comparison of the 2D analysis using CCM to in situ monitoring data of Lyon–Vaise tunnel was carried out. To this aim, two different constitutive models (CJS2 and Mohr–Coulomb) and various determining method of λ (direct and indirect method) were used. The results represented that λ obtained via indirect method cause more precise results. CJS2 constitutive model enable the convergence–confinement method to more accurate prediction of tunnel behavior than Mohr–Coulomb.

Appendix: Brief Representation of CJS2 Constitutive Model

1.1 Elastic Mechanism

Elastic law of this model has been given as follow:

$$\dot{\varepsilon }_{ij}^{e} = \frac{{\dot{s}_{ij} }}{2G} + \frac{{\dot{I}_{1} }}{9K}\delta_{ij}$$

(1)

where $\dot{s}_{ij}$ is second invariant of stress deviator tensor, $\dot{I}_{1}$ is first invariant stress tensor, K and G are elastic shear and bulk modulus of material. K and G are a function of confine pressure as:

$$K = K_{0}^{e} \left( {\frac{{I_{1} }}{{3P_{a} }}} \right)^{n}$$

(2)

$$G = G_{0}^{e} \left( {\frac{{I_{1} }}{{3P_{a} }}} \right)^{n}$$

(3)

P _a is reference pressure which equal to 100 kPa and $K_{e}^{0}$,  $G_{e}^{0}$ and n are elastic parameters of model that can be determine by laboratory results.

1.2 Isotropic Plastic Mechanism

Yield surface in this mechanism is as follow:

$$f^{i} \left( {I_{1} ,Q} \right) = \frac{{I_{1} }}{3} - \left( {Q + \frac{{I_{1c} }}{3}} \right) \le 0$$

(4)

Flow rule is associated in this mechanism.

$$\varepsilon_{ij}^{pi} = \lambda^{i} \frac{{\partial f^{i} }}{{\partial \sigma_{ij} }} = \frac{{\lambda^{i} }}{3}\delta_{ij}$$

(5)

where λ ⁱ is plastic modulus.

1.3 Deviatoric Plastic Mechanism

The failure surface can be expressed as follow:

$$s_{II} h(\theta ) - R_{m} \left( {I_{1} + I_{1c} } \right) = 0$$

(6)

where $s_{II} = \sqrt {s_{ij} s_{ij} }$, R _m is equivalent radius of failure surface, I _1c is cohesion parameter and h(θ) = (1 − γ cos3θ)1/6, θ is Lode angle and γ is constant model parameter.

Figure 19 shows plastic, characteristic and failure mechanism of CJS2 constitutive model in deviatoric space.

Fig. 19

Reproduced with permission from Maleki and Mousivand (2014)

Presentation of different mechanism of CJS2 constitutive model; a and b deviatoric stress space.

Yield surface in this constitutive model is similar to failure surface as follow:

$$f\left( {\sigma_{ij} ,R} \right) = s_{II} h(\theta ) - R\left( {I_{1} + I_{1c} } \right) = 0$$

(7)

where $R = \frac{{AR_{m} P}}{{R_{m} + AP}}$ is associated hardening of model, P and A are hardening variable and constant model parameter respectively. Failure mode occurs when the parameter P approach infinity. The characteristic surface separates the contractancy and dilatancy states in the stress space can be define as below:

$$s_{II} h(\theta_{s} ) - R_{c} \left( {I_{1} + I_{1c} } \right) = 0$$

(8)

where R _c is equivalent radius of characteristic surface.