Analytical Method for Segmental Tunnel Linings Reinforced by Secondary Lining Considering Interfacial Slippage and Detachment

Acknowledgments

Notation

A₁, A₂

area of the section of segmental and secondary linings, respectively;

$\overline{\mathbf{A}},{\overline{\mathbf{A}}}^{\mathrm{\prime }}$

normalized system matrice beyond and within the range of the track bed, respectively;

a

associated parameter of segmental lining;

b, b′, b₁, b₂

associated parameters of secondary lining and the part within the range of track bed, width of the section of segmental and secondary linings, respectively;

c

associated parameter of segmental lining;

d, d′

associated parameters of secondary lining and the part within the range of track bed, respectively;

E_s, E₁, E₂, $E_1^{\mathrm{\prime }}$, $E_2^{\mathrm{\prime }}$

Young's moduli of the soil, two linings beyond and within the range of the track bed, respectively;

$\overline{\mathbf{f}}$

integral load vector;

$\overline{\mathbf{H}}$

transfer matrix for the whole lining;

h₁, h₂, $h_2^{\mathrm{\prime }}$

cross-sectional heights of segmental lining, secondary lining, and the part of secondary lining within the range of the track bed, respectively;

I₁, I₂, ${\overline{I}}_1$, ${\overline{I}}_2$

moments of inertia of two linings and their normalizations, respectively;

$\overline{\mathbf{J}}$, ${\overline{\mathbf{J}}}^{\mathrm{\prime }}$

normalized transfer matrice of the joint and bolt, respectively;

K_sz

stiffness of a single radial spring element;

k_gs, k_gz, ${\overline{k}}_{gs}$, ${\overline{k}}_{gz}$

tangential and radial stiffnesses of the soil spring and their normalizations, respectively;

k_r, k_t, ${\overline{k}}_r$, ${\overline{k}}_t$

radial and tangential stiffnesses of the bolt and their normalizations, respectively;

k_s, k_z, ${\overline{k}}_s$, ${\overline{k}}_z$

tangential and radial stiffnesses at the interface and their normalizations, respectively;

k_u, k_w, k_φ, ${\overline{k}}_w$, ${\overline{k}}_u$, ${\overline{k}}_{\phi }$

tangential, radial, and rotational stiffnesses of the joint and their normalizations, respectively;

M₁, M₂

bending moments of segmental and secondary linings, respectively;

N_z, N₁, N₂

radial force at the interface, axial forces of segmental and secondary linings, respectively;

$\overline{\mathbf{p}}$

integral load vector for the whole lining;

p₁, p₂, p₃, p₄, p₅

vertical overburden soil pressure, reaction pressure at the bottom of lining, total lateral earth pressure developed at the crown of the tunnel lining, additional horizontal soil pressure, dead load of lining, respectively;

Q_s, Q₁, Q₂

tangential force at the interface, shear forces of segmental and secondary linings, respectively;

$\overline{\mathbf{q}}$

normalized load vector;

q_s, q_z, ${\overline{q}}_s$, ${\overline{q}}_z$

distributed loads along the circumferential and radial directions and their normalizations, respectively;

R₀, R₁, R₂, $R_2^{\mathrm{\prime }}$

radii of the centerline of the interface, segmental lining, secondary lining, and the part of secondary lining within the range of the track bed, respectively;

$\overline{\mathbf{T}}$

transfer matrix;

u_B, u_C, u₁, u₂

tangential displacements at the interface and the curvilinear middle plane of two linings, respectively;

w₁, w₂

radial displacements at the curvilinear middle plane of two linings, respectively;

$\overline{\mathbf{x}}$, ${\overline{\mathbf{x}}}_0$, ${\overline{\mathbf{x}}}_1$

normalized state vectors at any cross section and the front and back ends of the curved beams, respectively;

Δu, Δw, $\mathrm{\Delta }\overline{u}$, $\mathrm{\Delta }\overline{w}$

tangential slippage, radial detachment, and their normalizations, respectively;

θ₀, θ₁

central angles at the front and back ends of the curved beams, respectively;

υ

Poisson's ratio of the soil; and

φ₁, φ₂

rotation angles of the cross section at the centerline of two linings, respectively.

References