Study on the Mechanical Behavior of a Secondary Tunnel Lining with a Yielding Layer in Transversely Isotropic Rock Stratum

Abstract

For tunnels excavated in soft layered rock strata, the secondary lining is susceptible to asymmetrical pressure and substantial concrete cracking. To solve this problem, a supporting system that combines a secondary lining with a highly deformable layer is proposed. First, field investigations were conducted to reveal the mechanical behaviors of the secondary lining of two typical tunnels situated in phyllite formation in China. Then, similarity model tests were used to analyze the mechanical responses of the secondary lining in layered rock stratum with different inclination angles. Finally, numerical simulation was adopted to systemically investigate the yielding mechanism of yielding layer in different cases. The results show that: (1) The distribution of inner force and deformation of lining is non-uniform due to the anisotropy of rock mass and geo-stress field, with the positive bending moment and larger axial force appearing mainly in the region where the tangent of the tunnel contour is parallel or vertical to the weak planes, respectively. (2) The existence of yielding layer has no influence on the distributed feature of inner force and displacement around tunnel perimeter, while it can significantly lower the magnitude of inner force and displacement of the lining. (3) The distinctly compressive zones of yielding layer coincide with the zones of positive bending moment, and the line connecting the central points of the evidently compressive zones on both sides will deflect towards the direction of minor principal stress to a certain extent. (4) The inner force and deformation of lining increase with the increase in the strength or decrease in the thickness of yielding layer, and this changing trend is related to the different supporting curves for linings with yielding layer of different strength and thickness.

Appendix

As it is difficult to quantify the support characteristic curve of yielding supporting system, we presented the schematic convergence–confinement curves in this article.

For rock bolts, its supporting stiffness, k_bol, is (Oreste 2003):

$$k_{{{\text{bol}}}} = \frac{1}{{S_{{\text{t}}} S_{{\text{r}}} (4L/\pi \varphi^{2} E_{{{\text{st}}}} + Q)}},$$

(3)

where E_st, L and φ are the elastic modulus, length and diameter of bolts; S_t and S_r are the circumferential spacing and longitudinal spacing; Q is the load–deformation constant for the anchor and head.

For shotcrete, its supporting stiffness, k_shot, is (Oreste 2003):

$$k_{{{\text{shot}}}} = \frac{{E_{{{\text{shot}}}} }}{{1 + \nu_{{{\text{shot}}}} }} \cdot \frac{{R^{2} - (R - t_{{{\text{shot}}}} )^{2} }}{{(1 - 2\nu_{{{\text{shot}}}} )R^{2} + (R - t_{{{\text{shot}}}} )^{2} }} \cdot \frac{1}{R},$$

(4)

where E_shot and ν_shot are the elastic modulus and Poisson’s ratio of shotcrete, respectively; R is the equivalent radius of the tunnel; t_shot is the thickness of shotcrete.

For steel arch, its supporting stiffness, k_st, is (Oreste 2003):

$$k_{{{\text{st}}}} = \frac{{E_{{{\text{st}}}} A_{{{\text{st}}}} }}{{d(R - \frac{{h_{st} }}{2})^{2} }},$$

(5)

where E_st, A_st, h_st and d are the elastic modulus, area, height and spacing of steel arch.

For secondary lining, its supporting stiffness, k_sec, is (Oreste 2003):

$$k_{{\sec}} = \frac{{E_{{{\text{con}}}} }}{{1 + \nu_{{{\text{con}}}} }} \cdot \frac{{R^{2} - (R - t_{{\sec}} )^{2} }}{{(1 - 2\nu_{{{\text{con}}}} )R^{2} + (R - t_{{\sec}} )^{2} }} \cdot \frac{1}{R},$$

(6)

where E_sec and ν_sec are the elastic modulus and Poisson’s ratio of secondary lining, respectively; R is the equivalent radius of the tunnel; t_sec is the thickness of secondary lining.

For yielding layer, there is no explicit expression describing its supporting characteristic curve. Based on its stress–strain curve, the schematic image of its support characteristic curve is shown in Fig. 

Fig. 20

Support characteristic curves: a yielding layer; b the preliminary supports; c the secondary supports

20a.

In each installation stage of supports, the stiffness of the system is simply given by the sum of the stiffness of each single element in the system.

Thus, for preliminary supports, the stiffness of the system is (Fig. 20b):

$$k_{{{\text{tot1}}}} = k_{{{\text{bol}}}} + k_{{{\text{shot}}}} + k_{{{\text{st}}}} .$$

(7)

For secondary supports (secondary lining and yielding layer), the stiffness of the system is (Fig. 20c):

Stage AB: the yielding layer is in its elastic stage and the total stiffness:

$$k_{{{\text{tot2}}}} = k_{{{\text{yield1}}}} + k_{{\sec}} .$$

(8)

Stage BC: the yielding layer is in its yielding stage and its stiffness changes to zero, and due to the yielding effect of yielding layer, the total stiffness is:

$$k_{{{\text{tot2}}}} = \alpha_{{\text{(u)}}} k_{{\sec}} ,$$

(9)

where α_(u) is the effective ratio of secondary lining, and 0 < α_(u) ≤ 1.

Point C: the yielding limit is reached, and the total stiffness is:

$$k_{{{\text{tot2}}}} = k_{{\sec}} .$$

(10)

Stage CD: the yielding layer is in its plastic stage, and the total stiffness is:

$$k_{{{\text{tot2}}}} = k_{{{\text{yield2}}}} + k_{{\sec}} .$$

(11)