Analytical Approach to Evaluating the Influence of the Compressible Layer on the Time-Dependent Response of Deep Soft-Rock Tunnels

Data Availability Statement

Notation

∗

convolution symbol;

A₁–A₅

constant;

B(s)

function of flexibility coefficient;

c

cohesion of rock (MPa);

E

elastic modulus of surrounding rock (MPa);

E₁

initial elastic of compressible layer (MPa);

E₂

ultimate elastic modulus of compressible layer (MPa);

E₃

elastic modulus of rigid lining (MPa);

G

shear modulus of surrounding rock (MPa);

G₀

shear modulus of Maxwell and Kelvin (MPa);

G₁

shear modulus of Kelvin (MPa);

G_f1

initial shear modulus of compressible layer (MPa);

G_f2

shear modulus of compressible layer (MPa);

G_r

initial shear modulus of concrete lining (MPa);

h

dilatancy coefficient;

J(s)

expression of J(t) in Laplace Space;

j(t)

flexibility modulus (1/MPa);

K

bulk modulus (MPa);

k_P

constant coefficient related to rock properties;

m

initial stress release coefficient;

P_a

fictitious supporting force (MPa);

P_l1(s)

expression of P_l1(Δt) in Laplace Space;

P_l1(t − t₂)

interaction force in the interface between compressible layer and surrounding rocks (MPa);

P_l2 (s)

expression of P_l2 (t − t₃) in Laplace Space;

P_l2 (t − t₃)

interaction force in the interface between compressible layer and concrete lining. (MPa);

$\begin{array}{rl}& \overline{P^{\mathrm{\prime }}(s)},\overline{P^{\mathrm{″}}(s)},\overline{Q^{\mathrm{\prime }}(s)},\overline{Q^{\mathrm{″}}(s)}\end{array}$

Laplace operation of Burgers model;

R_L

influence radius of tunnel face (m);

R_P

plastic radius (m);

r₀

tunnel radius (m);

t₁

time of surrounding rock entering plastic stage (days);

t₂

installation time of compressible layer and lining (days);

t₃

yielding time of compressible layer (days);

t₄

initial time of strengthening stage for compressible layer (days);

u_r(t)

radial displacement of surrounding rock (mm);

$u_r^l$(t)

radial displacement of lining (mm);

$u_r^P$(t)

radial displacement of surrounding rock in the plastic zone (mm);

$u_r^s$(t)

radial displacement increment of surrounding rock after support installation (mm);

$u_r^{ve}$(t)

radial displacement of surrounding rock in the viscoelastic zone (mm);

v

tunnel excavation rate (m/days);

X

distance from tunnel face to studied section (m);

X₁

entering plastic stage with the excavation distance (m);

ΔP_l1(s)

expression of ΔP_l1(t − t₄) in Laplace Space;

ΔP_l1(t − t₄)

stress increase in the interface between surrounding rocks and compressible layer (MPa);

δ(Δt)

Dirac delta function;

${\epsilon }_{\theta }^{ve}$(R_P, t)

tangential strain in viscoelastic–plastic boundary;

${\epsilon }_r^{ve}$(R_P, t)

radial strain in viscoelastic–plastic boundary;

η₁

viscosity coefficient of Maxwell (MPa × days);

η₂

viscosity coefficient of Kelvin (MPa × days);

λ

stress release coefficient;

μ

Poisson’s ratio of surrounding rock;

μ′

Poisson’s ratio of compressible layer;

μ″

Poisson’s ratio of concrete lining;

σ₁

maximum principal stress (MPa);

σ₃

minimum principal stress (MPa);

σ_c

uniaxial compressive strength of rocks (MPa);

σ_r

radial stress (MPa);

σ_θ

tangential stress (MPa); and

φ

internal friction angle of rocks (°)

References