Semianalytical Analysis of Overexcavation and Critical Support Pressure for Support Design in TBM Tunneling through Squeezing Rock Condition

Acknowledgments

Notation

E₁

elastic modulus of the shield;

E₂

elastic modulus of the concrete lining;

E_r

elastic modulus of the rock mass;

f

failure criterion;

f₀

displacement reduction ratio;

g

flow potential;

K₁

stiffness of the shield;

K₂

stiffness of the concrete lining;

K_ψ

dilatancy coefficient;

m_b, s, a

strength constants in the Hoek-Brown failure criterion;

n

number of annuli in the plastic zone by the finite difference method;

p_i

support pressure;

p_i,1

contacting pressure between the shield and the rock mass;

p_i,cri1

required critical support pressure corresponding to ɛ_i,cri1 = 1%;

p_i,cri2

required critical support pressure corresponding to ɛ_i,cri2 = 2.5%;

p_i,cri3

required critical support pressure corresponding to ɛ_i,cri3 = 5.0%;

p_i,cri4

required critical support pressure corresponding to ɛ_i,cri4 = 10%;

p_i,fic

fictitious support pressure added on the tunnel periphery;

p_i,fin

contacting pressure between the concrete lining and the rock mass;

p_i,max

maximum capacity of the support;

R

location of the rock mass to the center of the circular opening;

R*

normalized plastic radius;

R₀

radius of the circular opening;

R_p

radius of the plastic zone;

$R_p^{max}$

maximum plastic radius;

r_(i)

radial distance to the center of a circular opening at the inner boundary of the ith annulus;

t₁

thickness of the shield;

t₂

thickness of the shotcrete;

u

radial displacement;

u₀

radial displacement at the tunnel periphery;

u_0,1

rock displacement at the tunnel periphery when the rock mass and the shield reach an equilibrium;

u_0,fin

rock displacement at the tunnel periphery when the rock mass and the concrete lining reach an equilibrium;

u_0,fin1

u_0,fin solved by the FDM analysis;

u_0,fin2

u_0,fin solved by the proposed solution;

u_0,ini

initial radial displacement at the tunnel face;

u_0,max

maximum radial displacement;

u_r(i)

radial displacement at r = r_(i);

Δr

radial increment in the plastic zone;

ΔR₀

size of the overexcavation;

Δt₁

deformation of the shield;

Δt₂

deformation of the lining;

ɛ_r

radial strain;

${\epsilon }_r^e$

elastic radial strain;

${\epsilon }_r^p$

plastic radial strain;

ɛ_θ

tangential strain;

ɛ_θ(i)

tangential strain at r = r_(i);

ɛ_θ,cri1

critical strain level of 1%;

ɛ_θ,cri2

critical strain level of 2.5%;

ɛ_θ,cri3

critical strain level of 5%;

ɛ_θ,cri4

critical strain level of 10%;

ɛ_θ,fin

tunnel strain corresponding to u_0,fin, ɛ_θ,fin = u_0,fin/R₀;

ɛ_θ,ini

tunnel strain corresponding to u_0,ini, ɛ_θ,ini = u_0,ini/R₀;

${\epsilon }_{\theta }^e$

elastic tangential strain;

${\epsilon }_{\theta }^p$

plastic tangential strain;

μ₁

Poisson's ratio of the shield;

μ₂

Poisson's ratio of the concrete lining;

σ₀

in situ stress field;

σ₁

major principal stress;

σ₃

minor principal stress;

σ_c,max

compressive strength of the concrete lining;

σ_ci

compressive strength of the intact rock;

σ_cm

compressive strength of the rock mass;

σ_r

radial stress of the rock mass;

σ_r2

radial stress at the elastoplastic boundary;

σ_θ

tangential stress of the rock mass; and

ψ

dilatancy angle.

References