An improved analytical approach for analyzing a circular opening excavated in a strain-softening rock mass

Abstract

A broken zone usually exists in the deeply buried tunnels after excavation, and the rock mass in the broken zone is in the state of unloading failure and exhibiting a nonlinear dilatancy effect. In order to study the deformation of isotropic rock mass in an initial hydrostatic stress field, a method is proposed to calculate the radius of the broken zone of the surrounding rock in a circular tunnel. Based on the unloading experiment of rock samples, a layer-wise summation method is established to calculate the displacement of the circular tunnel after excavation. The results reflect the impact of the nonlinear dilatancy effect on the displacement of the tunnel wall along the radial direction. Moreover, the method does not include complex integral calculation. Comparing with the Kastner’s (Houska 1981) method and Y K Lee’s (Lee and Pietruszczak 2008) method. This article reveals that the residual strength parameters c_r, φ_r and dilation angle ψ, that are essential in Y K Lee’s method, are highly sensitive to displacement calculation, which makes the Y K Lee’s method difficult to apply. In contrast, the method proposed in this study provides accurate prediction and has access to obtain its parameters more easily, making it easy to popularize.

Appendix

The Y, N, B, and C in this paper are defined respectively,

$$ Y=2c\cos \varphi /\left(1-\sin \varphi \right); $$

$$ N=\left(1+\sin \varphi \right)/\left(1-\sin \varphi \right); $$

$$ B=\left[2-\frac{1+\nu }{E}\sin \varphi \left({p}_0+c\cot \varphi \right)\right]\frac{1+\nu }{E}\sin \varphi \cdotp \left({p}_0+c\cot \varphi \right){\left[\frac{\left({p}_0+c\cot \varphi \right)\left(1-\sin \varphi \right)}{p_i+c\cot \varphi}\right]}^{\frac{1-\sin \varphi }{\sin \varphi }}; $$

$$ \mathrm{C}={\left({R}_c-{u}_C\right)}^2-\sum \limits_{i=1}^n\left(1+{k}_i\right)\left({r}_i^2-{r}_{i+1}^2\right). $$

List of symbols

bradius of a tunnel

p₀ initial stress in a rock mass before the excavation is made

c, φrock mass cohesion, angle of internal friction of the rock mass

E, νYoung’s modulus, Poisson’s ratio

G shear modulus of rock mass

γ (kN/m³)weight of rock mass

Hdepth of tunnel

σ₁axial pressure in triaxial compression test

σ₂confining pressure in triaxial compression test

σ_θhoop stress

σ_rradial stress

ε_θtangential strain

ε_rradial strain

ε_vvolume strain

p_isupport pressure

p_icsupport pressure for the occurrence of broken zone

r_iith layer’s radius

h_ithe thickness of the ith layer

k_icoefficient of dilation in ith layer

R_pradius of the plastic zone

R_cradius of the broken zone

c_r residual cohesion

φ_rresidual friction angle

ψdilation angle

u_iradial displacement at r = r_i

u_B radial displacement of the elasto-plastic interface

u_C radial displacement of the plastic-broken interface