概要

均等な内水圧および外水圧を受けるRC円形圧力トンネルを、トンネル軸方向に均一な厚肉円筒として、平面ひずみ状態でモデル化する。モデル化においては、以下の考え方を採用した。

圧力水路の構造は鉄筋コンクリート構造とする。
内水圧を受ける場合のモデル化範囲は、圧力水路の覆工（鉄筋含む）および周辺岩盤とする。
内水圧を受ける覆工コンクリートは、ひび割れが発生しているものとし、周方向の応力は分担しない。鉄筋は完全弾性体として扱う。
外水圧は覆工コンクリートの外面に作用するものとし、岩盤はモデル化しない。
外水圧を受ける覆工には半径方向・周方向ともに圧縮応力が作用するため、コンクリート・鉄筋とも完全弾性体として扱う。
温度変化量は、簡略化のため、覆工内で均一とし、岩盤内での温度変化は考慮しない。
水路における鉄筋のかぶりは、断面寸法に対し比較的大きいので、これを考慮する。

f:id:damyarou:20190527132711j:plain

Design of RC Circular Pressure Tunnel

Basic Equations for Elastic Cylinder Model in Plane Strain State

In this discussion, a long circular cylinder including surrounding area is considered as a pressure tunnel model using polar coordinates. Symbols used are shown below in this discussion.

$\sigma_r$	normal stress in the radial direction
$\epsilon_r$	normal strain in the radial direction
$\sigma_{\theta}$	normal stress in the circumferential direction
$\epsilon_{\theta}$	normal strain in the circumferential direction
$\sigma_z$	normal stress in the axial direction
$\epsilon_z$	normal strain in the axial direction
$u$	displacement in the radial direction
$r$	distance to a point in a cylinder from origin of polar coordinates in the radial direction
$a$	distance to inner boundary of a cylinder from origin of polar coordinates in the radial direction
$E$	elastic modulus of material
$\nu$	Poisson's ratio of material
$\alpha$	thermal expansion coefficient of material
$T$	temperature change of material

Basic Equations for Isotropic Elastic Material in Plane Strain State

Stress - Strain Relationship

Stress - strain relationships for an elastic material in polar coordinates are shown below.

$\begin{equation} \begin{cases} &\epsilon_{r} -\alpha T = \cfrac{1}{E}\{\sigma_{r} -\nu (\sigma_{\theta}+\sigma_{z})\} \\ &\epsilon_{\theta}-\alpha T = \cfrac{1}{E}\{\sigma_{\theta}-\nu (\sigma_{z} +\sigma_{r})\} \\ &\epsilon_{z} -\alpha T = \cfrac{1}{E}\{\sigma_{z} -\nu (\sigma_{r} +\sigma_{\theta})\} \end{cases} \end{equation}$

Considering a condition os $\epsilon_z=0$ , followings can be obtained from above.

$\begin{equation} \begin{cases} &\epsilon_{r} -\alpha T = \cfrac{1}{E}\{(1-\nu^2) \sigma_{r} -\nu (1+\nu) \sigma_{\theta}\} \\ &\epsilon_{\theta}-\alpha T = \cfrac{1}{E}\{(1-\nu^2) \sigma_{\theta}-\nu (1+\nu) \sigma_{r}\} \\ &\sigma_z=\nu (\sigma_r+\sigma_{\theta}) - E \alpha T \end{cases} \end{equation}$

$\begin{equation} \begin{cases} \sigma_r =\cfrac{E}{(1+\nu)(1-2\nu)}\{(1-\nu) \epsilon_r + \nu\epsilon_{\theta} - (1+\nu)\alpha T\} \\ \sigma_{\theta}=\cfrac{E}{(1+\nu)(1-2\nu)}\{\nu \epsilon_r + (1-\nu) \epsilon_{\theta} - (1+\nu)\alpha T\} \end{cases} \end{equation}$

Strain - Displacement Relationship

Relationships between strain and displacement can be expressed as follows.

$\begin{equation} \epsilon_r=\cfrac{du}{dr} \qquad \epsilon_{\theta}=\cfrac{u}{r} \end{equation}$

Equilibrium Equation

An equilibrium equation of stress can be expressed as follow.

$\begin{equation} \cfrac{d\sigma_r}{dr}+\cfrac{\sigma_r-\sigma_{\theta}}{r}=0 \end{equation}$

General Expressions of Displacements and Stresses

$\begin{equation} \cfrac{d\sigma_r}{dr}+\cfrac{\sigma_r-\sigma_{\theta}}{r}=\cfrac{E(1-\nu)}{(1+\nu)(1-2\nu)} \left\{\cfrac{d^2u}{dr^2}+\cfrac{1}{r}\cfrac{du}{dr}-\cfrac{u}{r^2}-\cfrac{1+\nu}{1-\nu} \alpha \cfrac{dT}{dr}\right\}=0 \end{equation}$

$\begin{equation} \cfrac{d^2u}{dr^2}+\cfrac{1}{r}\cfrac{du}{dr}-\cfrac{u}{r^2}=\cfrac{1+\nu}{1-\nu} \alpha \cfrac{dT}{dr} \quad \rightarrow \quad \cfrac{d}{dr}\left[\cfrac{1}{r}\cfrac{d(ru)}{dr}\right]=\cfrac{1+\nu}{1-\nu} \alpha \cfrac{dT}{dr} \end{equation}$

$\begin{align} &\cfrac{d}{dr}\left[\cfrac{1}{r}\cfrac{d(ru)}{dr}\right]=0 & \rightarrow & & &u=C_1\cdot r + \cfrac{C_2}{r} & &\text{(general solution)}\\ &\cfrac{d}{dr}\left[\cfrac{1}{r}\cfrac{d(ru)}{dr}\right]=\cfrac{1+\nu}{1-\nu} \alpha \cfrac{dT}{dr} & \rightarrow & & &u=\cfrac{1+\nu}{1-\nu} \alpha \cfrac{1}{r} \int_a^r T r dr & &\text{(paticular solution)} \end{align}$

From above, following general expressions of displacement and stresses can be obtained. (These equations are shown in 'Theory of Elasticity, S. Timoshenko and J. N. Goodier, Chapter 14. Thermal Stress, 135. The Long Circular Cylinder.')

$\begin{align} &u =\ \ \cfrac{1+\nu}{1-\nu} \alpha \cfrac{1}{r} \int_a^r T r dr & &+ C_1\cdot r + \cfrac{C_2}{r} \\ &\sigma_r = -\cfrac{\alpha E}{1-\nu} \cfrac{1}{r^2} \int_a^r T r dr & &+ \cfrac{E}{(1+\nu)(1-2\nu)}\cdot C_1-\cfrac{E}{(1+\nu)}\cfrac{C_2}{r^2} \\ &\sigma_{\theta}=\ \ \cfrac{\alpha E}{1-\nu} \cfrac{1}{r^2} \int_a^r T r dr - \cfrac{\alpha E T}{1-\nu} & &+ \cfrac{E}{(1+\nu)(1-2\nu)}\cdot C_1+\cfrac{E}{(1+\nu)}\cfrac{C_2}{r^2} \\ &\sigma_z = -\cfrac{\alpha E T}{1-\nu} & &+ \cfrac{2 \nu E}{(1+\nu)(1-2\nu)}\cdot C_1 \end{align}$

In case of uniform distribution of temperature in a material, thermal items can be expressed as results of integrations.

$\begin{align} &\cfrac{1+\nu}{1-\nu} \alpha \cfrac{1}{r} \int_a^r T r dr & &=\ \ \cfrac{1+\nu}{1-\nu} \alpha T \cfrac{r^2-a^2}{2 r} \\ &\cfrac{\alpha E}{1-\nu} \cfrac{1}{r^2} \int_a^r T r dr & &=-\cfrac{E \alpha T}{1-\nu} \cfrac{r^2-a^2}{2 r^2} \\ &\cfrac{\alpha E}{1-\nu} \cfrac{1}{r^2} \int_a^r T r dr - \cfrac{\alpha E T}{1-\nu} & &=-\cfrac{E \alpha T}{1-\nu} \cfrac{r^2+a^2}{2 r^2} \end{align}$

Therefore, displacement and stresses can be expressed as follows.

$\begin{align} &u =\ \ \cfrac{1+\nu}{1-\nu} \alpha T \cfrac{r^2-a^2}{2 r} & &+ C_1\cdot r + \cfrac{C_2}{r} \\ &\sigma_r =-\cfrac{E \alpha T}{1-\nu} \cfrac{r^2-a^2}{2 r^2} & &+ \cfrac{E}{(1+\nu)(1-2\nu)}\cdot C_1-\cfrac{E}{(1+\nu)}\cfrac{C_2}{r^2} \\ &\sigma_{\theta}=-\cfrac{E \alpha T}{1-\nu} \cfrac{r^2+a^2}{2 r^2} & &+ \cfrac{E}{(1+\nu)(1-2\nu)}\cdot C_1+\cfrac{E}{(1+\nu)}\cfrac{C_2}{r^2} \\ &\sigma_z = -\cfrac{\alpha E T}{1-\nu} & &+ \cfrac{2 \nu E}{(1+\nu)(1-2\nu)}\cdot C_1 \end{align}$

Basic Equations for No-tension Material in Circumferential Direction

For no-tension material in circumferential direction such as concrete under the internal pressure, the equilibrium equation becomes shown below considering a condition of $\sigma_{\theta}=0$ .

$\begin{equation} \cfrac{d \sigma_r}{dr} + \cfrac{\sigma_r}{r} = 0 \end{equation}$

Using an assumption of Poisson's ratio of zero ( $\nu=0$ ), the stress in the radial direction can be expressed as follow.

$\begin{equation} \sigma_r=E\epsilon_r-E \alpha T \quad \rightarrow \quad \sigma_r=E\cfrac{du}{dr}-E \alpha T \quad \left(\epsilon_r=\cfrac{du}{dr}\right) \end{equation}$

$\begin{equation} \cfrac{d \sigma_r}{dr} + \cfrac{\sigma_r}{r} = E \left\{ \cfrac{d^2 u}{dr^2} + \cfrac{1}{r} \cfrac{du}{dr} - \alpha \left( \cfrac{dT}{dr} + \cfrac{T}{r} \right) \right\}=0 \end{equation}$

$\begin{align} &\cfrac{d^2 u}{dr^2} + \cfrac{1}{r}\cfrac{du}{dr}=0 & \rightarrow & & &u=C_1 + C_2\cdot \ln(r) & &\text{(general solution)}\\ &\cfrac{d^2 u}{dr^2} + \cfrac{1}{r}\cfrac{du}{dr}=\alpha \left(\cfrac{dT}{dr}+\cfrac{T}{r}\right) & \rightarrow & & &u=\alpha \int_a^r T dr & &\text{(paticular solution)} \end{align}$

From above, following basic equations for no-tension material in circumferential direction can be obtained.

$\begin{align} &u=C_1+C_2\cdot \ln(r) + \alpha \int_a^r T dr \\ &\sigma_r=E \cfrac{C_2}{r} \end{align}$

Using an assumption of uniform distribution of temperature, a integration of thermal item becoms $\alpha T (r-a)$ . Therefore, displacement and stresses can be expressed as follows.

$\begin{align} &u=C_1+C_2\cdot \ln(r) + \alpha T (r-a) \\ &\sigma_r=E \cfrac{C_2}{r} \\ &\sigma_{\theta}=0 \end{align}$

Model of RC Circular Tunnel under Internal Pressure

Double Reinforcement Section

Components of Model

Components of Model
Bedrock	Elastic material. Thermal stress is ignored.
Concrete (outer cover)	No-tension material. Thermal stress is considered.
Outer Reinforcement	Elastic material. Thermal stress is considered.
Concrete (middle location)	No-tension material. Thermal stress is considered.
Inner Reinforcement	Elastic material. Thermal stress is considered.
Concrete (Inner cover)	No-tension material. Thermal stress is considered.

Coordinate of Boundary in Radial Direction
$r_7$	Outer boundary of bedrock
$r_6$	External surface of concrete lining
$r_5$	Boundary of outer cover concrete and outer reinforcement
$r_4$	Boundary of outer reinforcement and middle concrete
$r_3$	Boundary of middle concrete and inner reinforcement
$r_2$	Boundary of inner reinforcement and inner cover concrete
$r_1$	Inner surface of concrete lining

Basic Equations for Each Material

Bedrock ( $r_6 \leqq r \leqq r_7$ )

$\begin{align} % displacement of bedrock &u=C_{g1}\cdot r + \cfrac{C_{g2}}{r} \\ % stress of bedrock &\sigma_r=\cfrac{E_g}{(1+\nu_g)(1-2\nu_g)}\cdot C_{g1}-\cfrac{E_g}{(1+\nu_g)}\cfrac{C_{g2}}{r^2} \\ &\sigma_{\theta}=\cfrac{E_g}{(1+\nu_g)(1-2\nu_g)}\cdot C_{g1}+\cfrac{E_g}{(1+\nu_g)}\cfrac{C_{g2}}{r^2} \\ \end{align}$

Outer Cover Concrete ( $r_5 \leqq r \leqq r_6$ )

$\begin{align} % displacement of outer cover &u=C_{co1}+C_{co2}\cdot \ln(r) + \alpha_c T (r-r_5) \\ % stress of outer cover &\sigma_r=E_c \cfrac{C_{co2}}{r} \\ &\sigma_{\theta}=0 \end{align}$

Outer Reinforcement ( $r_4 \leqq r \leqq r_5$ )

$\begin{align} % displacement of outer rebar &u =\ \ \cfrac{1+\nu_s}{1-\nu_s} \alpha_s T \cfrac{r^2-r_4{}^2}{2 r} & &+ C_{so1}\cdot r + \cfrac{C_{so2}}{r} \\ % stress of outer rebar &\sigma_r =-\cfrac{E_s \alpha_s T}{1-\nu_s} \cfrac{r^2-r_4{}^2}{2 r^2} & &+ \cfrac{E_s}{(1+\nu_s)(1-2\nu_s)}\cdot C_{so1}-\cfrac{E_s}{(1+\nu_s)}\cfrac{C_{so2}}{r^2} \\ &\sigma_{\theta}=-\cfrac{E_s \alpha_s T}{1-\nu_s} \cfrac{r^2+r_4{}^2}{2 r^2} & &+ \cfrac{E_s}{(1+\nu_s)(1-2\nu_s)}\cdot C_{so1}+\cfrac{E_s}{(1+\nu_s)}\cfrac{C_{so2}}{r^2} \end{align}$

Middle Concrete ( $r_3 \leqq r \leqq r_4$ )

$\begin{align} % displacement of middle concrete &u=C_{cm1}+C_{cm2}\cdot \ln(r) + \alpha_c T (r-r_3) \\ % stress of middle rebar &\sigma_r=E_c \cfrac{C_{cm2}}{r} \\ &\sigma_{\theta}=0 \end{align}$

Inner Reinforcement ( $r_2 \leqq r \leqq r_3$ )

$\begin{align} % displacement of inner rebar &u =\ \ \cfrac{1+\nu_s}{1-\nu_s} \alpha_s T \cfrac{r^2-r_2{}^2}{2 r} & &+ C_{si1}\cdot r + \cfrac{C_{si2}}{r} \\ % stress of inner rebar &\sigma_r =-\cfrac{E_s \alpha_s T}{1-\nu_s} \cfrac{r^2-r_2{}^2}{2 r^2} & &+ \cfrac{E_s}{(1+\nu_s)(1-2\nu_s)}\cdot C_{si1}-\cfrac{E_s}{(1+\nu_s)}\cfrac{C_{si2}}{r^2} \\ &\sigma_{\theta}=-\cfrac{E_s \alpha_s T}{1-\nu_s} \cfrac{r^2+r_2{}^2}{2 r^2} & &+ \cfrac{E_s}{(1+\nu_s)(1-2\nu_s)}\cdot C_{si1}+\cfrac{E_s}{(1+\nu_s)}\cfrac{C_{si2}}{r^2} \end{align}$

Inner Cover Concrete ( $r_1 \leqq r \leqq r_2$ )

$\begin{align} % displacement of inner cover &u=C_{ci1}+C_{ci2}\cdot \ln(r) + \alpha_c T (r-r_1) \\ % stress of inner cover &\sigma_r= E_c \cfrac{C_{ci2}}{r} \\ &\sigma_{\theta}=0 \end{align}$

Simultaneous linear equations

To fix un-known parameters, simultaneous linear equations will be created considering boundary conditions shown below.

Stress in the radial direction at outer boundary of bedrock is equal to zero.
Displacement and stress in the radial direction have continuity at the boundaries of each material.
Stress in the radial direction at inner surface of concrete lining is equal to the internal pressure.
Positive sign of the internal pressure $P_a$ is toward outer direction.

$\begin{align} &1.&r=r_7 \;\text{(stress)}\quad &\left\{ % stress of bedrock \cfrac{E_g}{(1+\nu_g)(1-2\nu_g)}\cdot C_{g1}-\cfrac{E_g}{(1+\nu_g)}\cfrac{C_{g2}}{r_7{}^2} \right\} & &=0 \\ % &2.&r=r_6 \;\text{(disp.)}\quad &\left\{ % displacement of bedrock C_{g1}\cdot r_6 + \cfrac{C_{g2}}{r_6} \right\}-\left\{ % displacement of outer cover C_{co1}+C_{co2}\cdot \ln(r_6) \right\} & &=\alpha_c T (r_6-r_5) \\ % &3.&r=r_6 \;\text{(stress)}\quad &\left\{ % stress of bedrock \cfrac{E_g}{(1+\nu_g)(1-2\nu_g)}\cdot C_{g1}-\cfrac{E_g}{(1+\nu_g)}\cfrac{C_{g2}}{r_6{}^2} \right\}-\left\{ % stress of outer cover E_c \cfrac{C_{co2}}{r_6} \right\} & &=0 \\ % &4.&r=r_5 \;\text{(disp.)}\quad &\left\{ % displacement of outer cover C_{co1}+C_{co2}\cdot \ln(r_5) \right\}-\left\{ % displacement of outer rebar C_{so1}\cdot r_5 + \cfrac{C_{so2}}{r_5} \right\} & &=\cfrac{1+\nu_s}{1-\nu_s} \alpha_s T \cfrac{r_5{}^2-r_4{}^2}{2 r_5} \\ % &5.&r=r_5 \;\text{(stress)}\quad &\left\{ % stress of outer cover E_c \cfrac{C_{co2}}{r_5} \right\}-\left\{ % stress of outer rebar \cfrac{E_s}{(1+\nu_s)(1-2\nu_s)}\cdot C_{so1}-\cfrac{E_s}{(1+\nu_s)}\cfrac{C_{so2}}{r_5{}^2} \right\} & &=-\cfrac{E_s \alpha_s T}{1-\nu_s} \cfrac{r_5{}^2-r_4{}^2}{2 r_5{}^2} \\ % &6.&r=r_4 \;\text{(disp.)}\quad &\left\{ % displacement of outer rebar C_{so1}\cdot r_4 + \cfrac{C_{so2}}{r_4} \right\}-\left\{ % displacement of middle concrete C_{cm1}+C_{cm2}\cdot \ln(r_4) \right\} & &=\alpha_c T (r_4-r_3) \\ &7.&r=r_4 \;\text{(stress)}\quad &\left\{ % stress of outer rebar \cfrac{E_s}{(1+\nu_s)(1-2\nu_s)}\cdot C_{so1}-\cfrac{E_s}{(1+\nu_s)}\cfrac{C_{so2}}{r_4{}^2} \right\}-\left\{ % stress of middle rebar E_c \cfrac{C_{cm2}}{r_4} \right\} & &=0 \\ % &8.&r=r_3 \;\text{(disp.)}\quad &\left\{ % displacement of middle concrete C_{cm1}+C_{cm2}\cdot \ln(r_3) \right\}-\left\{ % displacement of inner rebar C_{si1}\cdot r_3 + \cfrac{C_{si2}}{r_3} \right\} & &=\cfrac{1+\nu_s}{1-\nu_s} \alpha_s T \cfrac{r_3{}^2-r_2{}^2}{2 r_3} \\ &9.&r=r_3 \;\text{(stress)}\quad &\left\{ % stress of middle rebar E_c \cfrac{C_{cm2}}{r_3} \right\}-\left\{ % stress of inner rebar \cfrac{E_s}{(1+\nu_s)(1-2\nu_s)}\cdot C_{si1}-\cfrac{E_s}{(1+\nu_s)}\cfrac{C_{si2}}{r_3{}^2} \right\} & &=-\cfrac{E_s \alpha_s T}{1-\nu_s} \cfrac{r_3{}^2-r_2{}^2}{2 r_3{}^2} \\ % &10.&r=r_2 \;\text{(disp.)}\quad &\left\{ % displacement of inner rebar C_{si1}\cdot r_2 + \cfrac{C_{si2}}{r_2} \right\}-\left\{ % displacement of inner cover C_{ci1}+C_{ci2}\cdot \ln(r_2) \right\} & &=\alpha_c T (r_2-r_1) \\ &11.&r=r_2 \;\text{(stress)}\quad &\left\{ % stress of inner rebar \cfrac{E_s}{(1+\nu_s)(1-2\nu_s)}\cdot C_{si1}-\cfrac{E_s}{(1+\nu_s)}\cfrac{C_{si2}}{r_2{}^2} \right\}-\left\{ % stress of inner cover E_c \cfrac{C_{ci2}}{r_2} \right\} & &=0 \\ % &12.&r=r_1 \;\text{(stress)}\quad &\left\{ E_c \cfrac{C_{ci2}}{r_1} \right\} & &=-P_a \end{align}$

Matrix Expression of Simultaneous linear equations for Computer Programing

$\begin{equation} \begin{bmatrix} a_{1,1} & a_{1,2} & 0 & 0 & \dots & 0 & 0 & 0 & 0 \\ a_{2,1} & a_{2,2} & a_{2,3} & a_{2,4} & \dots & 0 & 0 & 0 & 0 \\ a_{3,1} & a_{3,2} & a_{3,3} & a_{3,4} & \dots & 0 & 0 & 0 & 0 \\ 0 & 0 & a_{4,3} & a_{4,4} & \dots & 0 & 0 & 0 & 0 \\ 0 & 0 & a_{5,3} & a_{5,4} & \dots & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \dots & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \dots & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & \dots &a_{8,9} & a_{8,10} & 0 & 0 \\ 0 & 0 & 0 & 0 & \dots &a_{9,9} & a_{9,10} & 0 & 0 \\ 0 & 0 & 0 & 0 & \dots &a_{10,9} & a_{10,10} & a_{10,11} & a_{10,12} \\ 0 & 0 & 0 & 0 & \dots &a_{11,9} & a_{11,10} & a_{11,11} & a_{11,12} \\ 0 & 0 & 0 & 0 & \dots & 0 & 0 & a_{12,11} & a_{12,12} \end{bmatrix} \begin{Bmatrix} C_{g1} \\ C_{g2} \\ C_{co1} \\ C_{co2} \\ C_{so1} \\ C_{so2} \\ C_{cm1} \\ C_{cm2} \\ C_{si1} \\ C_{si2} \\ C_{ci1} \\ C_{ci2} \\ \end{Bmatrix} = \begin{Bmatrix} 0 \\ \alpha_c T (r_6-r_5) \\ 0 \\ \frac{1+\nu_s}{1-\nu_s} \alpha_s T \frac{r_5{}^2-r_4{}^2}{2 r_5} \\ -\frac{E_s \alpha_s T}{1-\nu_s} \frac{r_5{}^2-r_4{}^2}{2 r_5{}^2} \\ \alpha_c T (r_4-r_3) \\ 0 \\ \frac{1+\nu_s}{1-\nu_s} \alpha_s T \frac{r_3{}^2-r_2{}^2}{2 r_3} \\ -\frac{E_s \alpha_s T}{1-\nu_s} \frac{r_3{}^2-r_2{}^2}{2 r_3{}^2} \\ \alpha_c T (r_2-r_1) \\ 0 \\ -P_a \end{Bmatrix} \end{equation}$

$\begin{align} &a_{1,1}=\cfrac{E_g}{(1+\nu_g)(1-2\nu_g)} & &a_{1,2}=-\cfrac{E_g}{(1+\nu_g)r_7{}^2} & &a_{1,3}=0 & &a_{1,4}=0 \\ % &a_{2,1}=r_6 & &a_{2,2}=\cfrac{1}{r_6} & &a_{2,3}=-1 & &a_{2,4}=-\ln(r_6) \\ &a_{3,1}=\cfrac{E_g}{(1+\nu_g)(1-2\nu_g)} & &a_{3,2}=-\cfrac{E_g}{(1+\nu_g)r_6{}^2} & &a_{3,3}=0 & &a_{3,4}=-\cfrac{E_c}{r_6} \\ % &a_{4,3}=1 & &a_{4,4}=\ln(r_5) & &a_{4,5}=-r_5 & &a_{4,6}=-\cfrac{1}{r_5} \\ &a_{5,3}=0 & &a_{5,4}=\cfrac{E_c}{r_5} & &a_{5,5}=-\cfrac{E_s}{(1+\nu_s)(1-2\nu_s)} & &a_{5,6}=\cfrac{E_s}{(1+\nu_s)r_5{}^2} \\ % &a_{6,5}=r_4 & &a_{6,6}=\cfrac{1}{r_4} & &a_{6,7}=-1 & &a_{6,8}=-\ln(r_4) \\ &a_{7,5}=\cfrac{E_s}{(1+\nu_s)(1-2\nu_s)} & &a_{7,6}=-\cfrac{E_s}{(1+\nu_s)r_4{}^2} & &a_{7,7}=0 & &a_{7,8}=-\cfrac{E_c}{r_4} \\ % &a_{8,7}=1 & &a_{8,8}=\ln(r_3) & &a_{8,9}=-r_3 & &a_{8,10}=-\cfrac{1}{r_3} \\ &a_{9,7}=0 & &a_{9,8}=\cfrac{E_c}{r_3} & &a_{9,9}=-\cfrac{E_s}{(1+\nu_s)(1-2\nu_s)} & &a_{9,10}=\cfrac{E_s}{(1+\nu_s)r_3{}^2} \\ % &a_{10,9}=r_2 & &a_{10,10}=\cfrac{1}{r_2} & &a_{10,11}=-1 & &a_{10,12}=-\ln(r_2) \\ &a_{11,9}=\cfrac{E_s}{(1+\nu_s)(1-2\nu_s)} & &a_{11,10}=-\cfrac{E_s}{(1+\nu_s)r_2{}^2} & &a_{11,11}=0 & &a_{11,12}=-\cfrac{E_c}{r_2} \\ % &a_{12,9}=0 & &a_{12,10}=0 & &a_{12,11}=0 & &a_{12,12}=\cfrac{E_c}{r_1} \end{align}$

Single Reinforcement Section

Components of Model

Components of Model
Bedrock	Elastic material. Thermal stress is ignored.
Concrete (outer cover)	No-tension material. Thermal stress is considered.
Inner Reinforcement	Elastic material. Thermal stress is considered.
Concrete (Inner cover)	No-tension material. Thermal stress is considered.

Coordinate of Boundary in Radial Direction
$r_5$	Outer boundary of bedrock
$r_4$	External surface of concrete lining
$r_3$	Boundary of outer cover concrete and reinforcement
$r_2$	Boundary of reinforcement and inner cover concrete
$r_1$	Inner surface of concrete lining

Basic Equations for Each Material

Bedrock ( $r_4 \leqq r \leqq r_5$ )

Outer Cover Concrete ( $r_3 \leqq r \leqq r_4$ )

$\begin{align} % displacement of outer cover &u=C_{co1}+C_{co2}\cdot \ln(r) + \alpha_c T (r-r_3) \\ % stress of outer cover &\sigma_r=E_c \cfrac{C_{co2}}{r} \\ &\sigma_{\theta}=0 \end{align}$

Reinforcement ( $r_2 \leqq r \leqq r_3$ )

$\begin{align} % displacement of rebar &u =\ \ \cfrac{1+\nu_s}{1-\nu_s} \alpha_s T \cfrac{r^2-r_2{}^2}{2 r} & &+ C_{si1}\cdot r + \cfrac{C_{si2}}{r} \\ % stress of rebar &\sigma_r =-\cfrac{E_s \alpha_s T}{1-\nu_s} \cfrac{r^2-r_2{}^2}{2 r^2} & &+ \cfrac{E_s}{(1+\nu_s)(1-2\nu_s)}\cdot C_{si1}-\cfrac{E_s}{(1+\nu_s)}\cfrac{C_{si2}}{r^2} \\ &\sigma_{\theta}=-\cfrac{E_s \alpha_s T}{1-\nu_s} \cfrac{r^2+r_2{}^2}{2 r^2} & &+ \cfrac{E_s}{(1+\nu_s)(1-2\nu_s)}\cdot C_{si1}+\cfrac{E_s}{(1+\nu_s)}\cfrac{C_{si2}}{r^2} \end{align}$

Inner Cover Concrete ( $r_1 \leqq r \leqq r_2$ )

$\begin{align} % displacement of inner cover &u=C_{ci1}+C_{ci2}\cdot \ln(r) + \alpha_c T (r-r_1) \\ % stress of inner cover &\sigma_r= E_c \cfrac{C_{ci2}}{r} \\ &\sigma_{\theta}=0 \end{align}$

Simultaneous linear equations

To fix un-known parameters, simultaneous linear equations will be created considering boundary conditions shown below.

Stress in the radial direction at outer boundary of bedrock is equal to zero.
Displacement and stress in the radial direction have continuity at the boundaries of each material.
Stress in the radial direction at inner surface of concrete lining is equal to the internal pressure.
Positive sign of the internal pressure $P_a$ is toward outer direction.

$\begin{align} &1.&r=r_5 \;\text{(stress)}\quad &\left\{ % stress of bedrock \cfrac{E_g}{(1+\nu_g)(1-2\nu_g)}\cdot C_{g1}-\cfrac{E_g}{(1+\nu_g)}\cfrac{C_{g2}}{r_5{}^2} \right\} & &=0 \\ % &2.&r=r_4 \;\text{(disp.)}\quad &\left\{ % displacement of bedrock C_{g1}\cdot r_4 + \cfrac{C_{g2}}{r_4} \right\}-\left\{ % displacement of outer cover C_{co1}+C_{co2}\cdot \ln(r_4) \right\} & &=\alpha_c T (r_4-r_3) \\ % &3.&r=r_4 \;\text{(stress)}\quad &\left\{ % stress of bedrock \cfrac{E_g}{(1+\nu_g)(1-2\nu_g)}\cdot C_{g1}-\cfrac{E_g}{(1+\nu_g)}\cfrac{C_{g2}}{r_4{}^2} \right\}-\left\{ % stress of outer cover E_c \cfrac{C_{co2}}{r_4} \right\} & &=0 \\ % &4.&r=r_3 \;\text{(disp.)}\quad &\left\{ % displacement of middle concrete C_{co1}+C_{co2}\cdot \ln(r_3) \right\}-\left\{ % displacement of inner rebar C_{si1}\cdot r_3 + \cfrac{C_{si2}}{r_3} \right\} & &=\cfrac{1+\nu_s}{1-\nu_s} \alpha_s T \cfrac{r_3{}^2-r_2{}^2}{2 r_3} \\ % &5.&r=r_3 \;\text{(stress)}\quad &\left\{ % stress of middle rebar E_c \cfrac{C_{co2}}{r_3} \right\}-\left\{ % stress of inner rebar \cfrac{E_s}{(1+\nu_s)(1-2\nu_s)}\cdot C_{si1}-\cfrac{E_s}{(1+\nu_s)}\cfrac{C_{si2}}{r_3{}^2} \right\} & &=-\cfrac{E_s \alpha_s T}{1-\nu_s} \cfrac{r_3{}^2-r_2{}^2}{2 r_3{}^2} \\ % &6.&r=r_2 \;\text{(disp.)}\quad &\left\{ % displacement of inner rebar C_{si1}\cdot r_2 + \cfrac{C_{si2}}{r_2} \right\}-\left\{ % displacement of inner cover C_{ci1}+C_{ci2}\cdot \ln(r_2) \right\} & &=\alpha_c T (r_2-r_1) \\ % &7.&r=r_2 \;\text{(stress)}\quad &\left\{ % stress of inner rebar \cfrac{E_s}{(1+\nu_s)(1-2\nu_s)}\cdot C_{si1}-\cfrac{E_s}{(1+\nu_s)}\cfrac{C_{si2}}{r_2{}^2} \right\}-\left\{ % stress of inner cover E_c \cfrac{C_{ci2}}{r_2} \right\} & &=0 \\ % &8.&r=r_1 \;\text{(stress)}\quad &\left\{ E_c \cfrac{C_{ci2}}{r_1} \right\} & &=-P_a \end{align}$

Matrix Expression of Simultaneous linear equations for Computer Programing

$\begin{equation} \begin{bmatrix} a_{1,1} & a_{1,2} & 0 & 0 & 0 & 0 & 0 & 0 \\ a_{2,1} & a_{2,2} & a_{2,3} & a_{2,4} & 0 & 0 & 0 & 0 \\ a_{3,1} & a_{3,2} & a_{3,3} & a_{3,4} & 0 & 0 & 0 & 0 \\ 0 & 0 & a_{4,3} & a_{4,4} &a_{4,5} & a_{4,6} & 0 & 0 \\ 0 & 0 & a_{5,3} & a_{5,4} &a_{5,5} & a_{5,6} & 0 & 0 \\ 0 & 0 & 0 & 0 &a_{6,5} & a_{6,6} & a_{6,7} & a_{6,8} \\ 0 & 0 & 0 & 0 &a_{7,5} & a_{7,6} & a_{7,7} & a_{7,8} \\ 0 & 0 & 0 & 0 & 0 & 0 & a_{8,7} & a_{8,8} \end{bmatrix} \begin{Bmatrix} C_{g1} \\ C_{g2} \\ C_{co1} \\ C_{co2} \\ C_{si1} \\ C_{si2} \\ C_{ci1} \\ C_{ci2} \\ \end{Bmatrix} = \begin{Bmatrix} 0 \\ \alpha_c T (r_4-r_3) \\ 0 \\ \frac{1+\nu_s}{1-\nu_s} \alpha_s T \frac{r_3{}^2-r_2{}^2}{2 r_3} \\ -\frac{E_s \alpha_s T}{1-\nu_s} \frac{r_3{}^2-r_2{}^2}{2 r_3{}^2} \\ \alpha_c T (r_2-r_1) \\ 0 \\ -P_a \end{Bmatrix} \end{equation}$

$\begin{align} &a_{1,1}=\cfrac{E_g}{(1+\nu_g)(1-2\nu_g)} & &a_{1,2}=-\cfrac{E_g}{(1+\nu_g)r_5{}^2} & &a_{1,3}=0 & &a_{1,4}=0 \\ % &a_{2,1}=r_4 & &a_{2,2}=\cfrac{1}{r_4} & &a_{2,3}=-1 & &a_{2,4}=-\ln(r_4) \\ &a_{3,1}=\cfrac{E_g}{(1+\nu_g)(1-2\nu_g)} & &a_{3,2}=-\cfrac{E_g}{(1+\nu_g)r_4{}^2} & &a_{3,3}=0 & &a_{3,4}=-\cfrac{E_c}{r_4} \\ % &a_{4,3}=1 & &a_{4,4}=\ln(r_3) & &a_{4,5}=-r_3 & &a_{4,6}=-\cfrac{1}{r_3} \\ &a_{5,3}=0 & &a_{5,4}=\cfrac{E_c}{r_3} & &a_{5,5}=-\cfrac{E_s}{(1+\nu_s)(1-2\nu_s)} & &a_{5,6}=\cfrac{E_s}{(1+\nu_s)r_3{}^2} \\ % &a_{6,5}=r_2 & &a_{6,6}=\cfrac{1}{r_2} & &a_{6,7}=-1 & &a_{6,8}=-\ln(r_2) \\ &a_{7,5}=\cfrac{E_s}{(1+\nu_s)(1-2\nu_s)} & &a_{7,6}=-\cfrac{E_s}{(1+\nu_s)r_2{}^2} & &a_{7,7}=0 & &a_{7,8}=-\cfrac{E_c}{r_2} \\ % &a_{8,5}=0 & &a_{8,6}=0 & &a_{8,7}=0 & &a_{8,8}=\cfrac{E_c}{r_1} \end{align}$

Model of RC Circular Tunnel under External Pressure

Double Reinforcement Section

Components of Model

Components of Model
Concrete (outer cover)	Elastic material. Thermal stress is considered.
Outer Reinforcement	Elastic material. Thermal stress is considered.
Concrete (middle location)	Elastic material. Thermal stress is considered.
Inner Reinforcement	Elastic material. Thermal stress is considered.
Concrete (Inner cover)	Elastic material. Thermal stress is considered.

Coordinate of Boundary in Radial Direction
$r_6$	External surface of concrete lining
$r_5$	Boundary of outer cover concrete and outer reinforcement
$r_4$	Boundary of outer reinforcement and middle concrete
$r_3$	Boundary of middle concrete and inner reinforcement
$r_2$	Boundary of inner reinforcement and inner cover concrete
$r_1$	Inner surface of concrete lining

Basic Equations for Each Material

Outer Cover Concrete ( $r_5 \leqq r \leqq r_6$ )

$\begin{align} % displacement of outer concrete &u =\ \ \cfrac{1+\nu_c}{1-\nu_c} \alpha_c T \cfrac{r^2-r_5{}^2}{2 r} & &+ C_{co1}\cdot r + \cfrac{C_{co2}}{r} \\ % stress of outer concrete &\sigma_r =-\cfrac{E_c \alpha_c T}{1-\nu_c} \cfrac{r^2-r_5{}^2}{2 r^2} & &+ \cfrac{E_c}{(1+\nu_c)(1-2\nu_c)}\cdot C_{co1}-\cfrac{E_c}{(1+\nu_c)}\cfrac{C_{co2}}{r^2} \\ &\sigma_{\theta}=-\cfrac{E_c \alpha_c T}{1-\nu_c} \cfrac{r^2+r_5{}^2}{2 r^2} & &+ \cfrac{E_c}{(1+\nu_c)(1-2\nu_c)}\cdot C_{co1}+\cfrac{E_c}{(1+\nu_c)}\cfrac{C_{co2}}{r^2} \end{align}$

Outer Reinforcement ( $r_4 \leqq r \leqq r_5$ )

Middle Concrete ( $r_3 \leqq r \leqq r_4$ )

$\begin{align} % displacement of middle concrete &u =\ \ \cfrac{1+\nu_c}{1-\nu_c} \alpha_c T \cfrac{r^2-r_3{}^2}{2 r} & &+ C_{cm1}\cdot r + \cfrac{C_{cm2}}{r} \\ % stress of middle concrete &\sigma_r =-\cfrac{E_c \alpha_c T}{1-\nu_c} \cfrac{r^2-r_3{}^2}{2 r^2} & &+ \cfrac{E_c}{(1+\nu_c)(1-2\nu_c)}\cdot C_{cm1}-\cfrac{E_c}{(1+\nu_c)}\cfrac{C_{cm2}}{r^2} \\ &\sigma_{\theta}=-\cfrac{E_c \alpha_c T}{1-\nu_c} \cfrac{r^2+r_3{}^2}{2 r^2} & &+ \cfrac{E_c}{(1+\nu_c)(1-2\nu_c)}\cdot C_{cm1}+\cfrac{E_c}{(1+\nu_c)}\cfrac{C_{cm2}}{r^2} \end{align}$

Inner Reinforcement ( $r_2 \leqq r \leqq r_3$ )

Inner Cover Concrete ( $r_1 \leqq r \leqq r_2$ )

$\begin{align} % displacement of inner concrete &u =\ \ \cfrac{1+\nu_c}{1-\nu_c} \alpha_c T \cfrac{r^2-r_1{}^2}{2 r} & &+ C_{ci1}\cdot r + \cfrac{C_{ci2}}{r} \\ % stress of inner concrete &\sigma_r =-\cfrac{E_c \alpha_c T}{1-\nu_c} \cfrac{r^2-r_1{}^2}{2 r^2} & &+ \cfrac{E_c}{(1+\nu_c)(1-2\nu_c)}\cdot C_{ci1}-\cfrac{E_c}{(1+\nu_c)}\cfrac{C_{ci2}}{r^2} \\ &\sigma_{\theta}=-\cfrac{E_c \alpha_c T}{1-\nu_c} \cfrac{r^2+r_1{}^2}{2 r^2} & &+ \cfrac{E_c}{(1+\nu_c)(1-2\nu_c)}\cdot C_{ci1}+\cfrac{E_c}{(1+\nu_c)}\cfrac{C_{ci2}}{r^2} \end{align}$

Simultaneous linear equations

To fix un-known parameters, simultaneous linear equations will be created considering boundary conditions shown below.

Stress in the radial direction at outer surface of concrete lining is equal to the external pressure.
Displacement and stress in the radial direction have continuity at the boundaries of each material.
Stress in the radial direction at inner surface of concrete lining is equal to zero.
Positive sign of the external pressure $P_b$ is toward inner direction.

$\begin{align} &1.\quad r=r_6 \;\text{(stress)}& &\quad \\ &\left\{ % stress of outer concrete \cfrac{E_c}{(1+\nu_c)(1-2\nu_c)}\cdot C_{co1}-\cfrac{E_c}{(1+\nu_c)}\cfrac{C_{co2}}{r_6{}^2} \right\} & &=-P_b+\cfrac{E_c \alpha_c T}{1-\nu_c} \cfrac{r_6{}^2-r_5{}^2}{2 r_6{}^2} \\ % &2.\quad r=r_5 \;\text{(disp.)}& &\quad \\ &\left\{ % displacement of outer concrete C_{co1}\cdot r_5 + \cfrac{C_{co2}}{r_5} \right\}-\left\{ % displacement of outer rebar C_{so1}\cdot r_5 + \cfrac{C_{so2}}{r_5} \right\} & &=\cfrac{1+\nu_s}{1-\nu_s} \alpha_s T \cfrac{r_5{}^2-r_4{}^2}{2 r_5} \\ % &3.\quad r=r_5 \;\text{(stress)}& &\quad \\ &\left\{ % stress of outer concrete \cfrac{E_c}{(1+\nu_c)(1-2\nu_c)}\cdot C_{co1}-\cfrac{E_c}{(1+\nu_c)}\cfrac{C_{co2}}{r_5{}^2} \right\}-\left\{ % stress of outer rebar \cfrac{E_s}{(1+\nu_s)(1-2\nu_s)}\cdot C_{so1}-\cfrac{E_s}{(1+\nu_s)}\cfrac{C_{so2}}{r_5{}^2} \right\} & &=-\cfrac{E_s \alpha_s T}{1-\nu_s} \cfrac{r_5{}^2-r_4{}^2}{2 r_5{}^2} \\ % &4.\quad r=r_4 \;\text{(disp.)}& &\quad \\ &\left\{ % displacement of outer rebar C_{so1}\cdot r_4 + \cfrac{C_{so2}}{r_4} \right\}-\left\{ % displacement of middle concrete C_{cm1}\cdot r_4 + \cfrac{C_{cm2}}{r_4} \right\} & &=\cfrac{1+\nu_c}{1-\nu_c} \alpha_c T \cfrac{r_4{}^2-r_3{}^2}{2 r_4} \\ % &5.\quad r=r_4 \;\text{(stress)}& &\quad \\ &\left\{ % stress of outer rebar \cfrac{E_s}{(1+\nu_s)(1-2\nu_s)}\cdot C_{so1}-\cfrac{E_s}{(1+\nu_s)}\cfrac{C_{so2}}{r_4{}^2} \right\}-\left\{ % stress of middle concrete \cfrac{E_c}{(1+\nu_c)(1-2\nu_c)}\cdot C_{cm1}-\cfrac{E_c}{(1+\nu_c)}\cfrac{C_{cm2}}{r_4{}^2} \right\} & &=-\cfrac{E_c \alpha_c T}{1-\nu_c} \cfrac{r_4{}^2-r_3{}^2}{2 r_4{}^2} \\ % &6.\quad r=r_3 \;\text{(disp.)}& &\quad \\ &\left\{ % displacement of middle concrete C_{cm1}\cdot r_3 + \cfrac{C_{cm2}}{r_3} \right\}-\left\{ % displacement of inner rebar C_{si1}\cdot r_3 + \cfrac{C_{si2}}{r_3} \right\} & &=\cfrac{1+\nu_s}{1-\nu_s} \alpha_s T \cfrac{r_3{}^2-r_2{}^2}{2 r_3} \\ % &7.\quad r=r_3 \;\text{(stress)}& &\quad \\ &\left\{ % stress of middle concrete \cfrac{E_c}{(1+\nu_c)(1-2\nu_c)}\cdot C_{cm1}-\cfrac{E_c}{(1+\nu_c)}\cfrac{C_{cm2}}{r_3{}^2} \right\}-\left\{ % stress of inner rebar \cfrac{E_s}{(1+\nu_s)(1-2\nu_s)}\cdot C_{si1}-\cfrac{E_s}{(1+\nu_s)}\cfrac{C_{si2}}{r_3{}^2} \right\} & &=-\cfrac{E_s \alpha_s T}{1-\nu_s} \cfrac{r_3{}^2-r_2{}^2}{2 r_3{}^2} \\ % &8.\quad r=r_2 \;\text{(disp.)}& &\quad \\ &\left\{ % displacement of inner rebar C_{si1}\cdot r_2 + \cfrac{C_{si2}}{r_2} \right\}-\left\{ % displacement of inner concrete C_{ci1}\cdot r_2 + \cfrac{C_{ci2}}{r_2} \right\} & &=\cfrac{1+\nu_c}{1-\nu_c} \alpha_c T \cfrac{r_2{}^2-r_1{}^2}{2 r_2} \\ % &9.\quad r=r_2 \;\text{(stress)}& &\quad \\ &\left\{ % stress of inner rebar \cfrac{E_s}{(1+\nu_s)(1-2\nu_s)}\cdot C_{si1}-\cfrac{E_s}{(1+\nu_s)}\cfrac{C_{si2}}{r_2{}^2} \right\}-\left\{ % stress of inner concrete \cfrac{E_c}{(1+\nu_c)(1-2\nu_c)}\cdot C_{ci1}-\cfrac{E_c}{(1+\nu_c)}\cfrac{C_{ci2}}{r_2{}^2} \right\} & &=-\cfrac{E_c \alpha_c T}{1-\nu_c} \cfrac{r_2{}^2-r_1{}^2}{2 r_2{}^2} \\ % &10.\quad r=r_1 \;\text{(stress)}& &\quad \\ &\left\{ % stress of inner concrete \cfrac{E_c}{(1+\nu_c)(1-2\nu_c)}\cdot C_{ci1}-\cfrac{E_c}{(1+\nu_c)}\cfrac{C_{ci2}}{r_1^2} \right\} & &=0 \end{align}$

Matrix Expression of Simultaneous linear equations for Computer Programing

$\begin{equation} \begin{bmatrix} a_{1,1} & a_{1,2} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ a_{2,1} & a_{2,2} & a_{2,3} & a_{2,4} & 0 & 0 & 0 & 0 & 0 & 0 \\ a_{3,1} & a_{3,2} & a_{3,3} & a_{3,4} & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & a_{4,3} & a_{4,4} & a_{4,5} & a_{4,6} & 0 & 0 & 0 & 0 \\ 0 & 0 & a_{5,3} & a_{5,4} & a_{5,5} & a_{5,6} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & a_{6,5} & a_{6,6} & a_{6,7} & a_{6,8} & 0 & 0 \\ 0 & 0 & 0 & 0 & a_{7,5} & a_{7,6} & a_{7,7} & a_{7,8} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & a_{8,7} & a_{8,8} & a_{ 8,9} & a_{ 8,10} \\ 0 & 0 & 0 & 0 & 0 & 0 & a_{9,7} & a_{9,8} & a_{ 9,9} & a_{ 9,10} \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & a_{10,9} & a_{10,10} \end{bmatrix} \begin{Bmatrix} C_{co1} \\ C_{co2} \\ C_{so1} \\ C_{so2} \\ C_{cm1} \\ C_{cm2} \\ C_{si1} \\ C_{si2} \\ C_{ci1} \\ C_{ci2} \\ \end{Bmatrix} = \begin{Bmatrix} -P_b+\frac{E_c \alpha_c T}{1-\nu_c} \frac{r_6{}^2-r_5{}^2}{2 r_6{}^2} \\ \frac{1+\nu_s}{1-\nu_s} \alpha_s T \frac{r_5{}^2-r_4{}^2}{2 r_5} \\ -\frac{E_s \alpha_s T}{1-\nu_s} \frac{r_5{}^2-r_4{}^2}{2 r_5{}^2} \\ \frac{1+\nu_c}{1-\nu_c} \alpha_c T \frac{r_4{}^2-r_3{}^2}{2 r_4} \\ -\frac{E_c \alpha_c T}{1-\nu_c} \frac{r_4{}^2-r_3{}^2}{2 r_4{}^2} \\ \frac{1+\nu_s}{1-\nu_s} \alpha_s T \frac{r_3{}^2-r_2{}^2}{2 r_3} \\ -\frac{E_s \alpha_s T}{1-\nu_s} \frac{r_3{}^2-r_2{}^2}{2 r_3{}^2} \\ \frac{1+\nu_c}{1-\nu_c} \alpha_c T \frac{r_2{}^2-r_1{}^2}{2 r_2} \\ -\frac{E_c \alpha_c T}{1-\nu_c} \frac{r_2{}^2-r_1{}^2}{2 r_2{}^2} \\ 0 \end{Bmatrix} \end{equation}$

$\begin{align} &a_{1,1}=\cfrac{E_c}{(1+\nu_c)(1-2\nu_c)} & &a_{1,2}=-\cfrac{E_c}{(1+\nu_c)r_6{}^2} & &a_{1,3}=0 & &a_{1,4}=0 \\ % &a_{2,1}=r_5 & &a_{2,2}=\cfrac{1}{r_5} & &a_{2,3}=-r_5 & &a_{2,4}=-\cfrac{1}{r_5} \\ &a_{3,1}=\cfrac{E_c}{(1+\nu_c)(1-2\nu_c)} & &a_{3,2}=-\cfrac{E_c}{(1+\nu_c)r_5{}^2} & &a_{3,3}=-\cfrac{E_s}{(1+\nu_s)(1-2\nu_s)} & &a_{3,4}=\cfrac{E_s}{(1+\nu_s)r_5{}^2} \\ % &a_{4,3}=r_4 & &a_{4,4}=\cfrac{1}{r_4} & &a_{4,5}=-r_4 & &a_{4,6}=-\cfrac{1}{r_4} \\ &a_{5,3}=\cfrac{E_s}{(1+\nu_s)(1-2\nu_s)} & &a_{5,4}=-\cfrac{E_s}{(1+\nu_s)r_4{}^2} & &a_{5,5}=-\cfrac{E_c}{(1+\nu_c)(1-2\nu_c)} & &a_{5,6}=\cfrac{E_c}{(1+\nu_c)r_4{}^2} \\ % &a_{6,5}=r_3 & &a_{6,6}=\cfrac{1}{r_3} & &a_{6,7}=-r_3 & &a_{6,8}=-\cfrac{1}{r_3} \\ &a_{7,5}=\cfrac{E_c}{(1+\nu_c)(1-2\nu_c)} & &a_{7,6}=-\cfrac{E_c}{(1+\nu_c)r_3{}^2} & &a_{7,7}=-\cfrac{E_s}{(1+\nu_s)(1-2\nu_s)} & &a_{7,8}=\cfrac{E_s}{(1+\nu_c)r_3{}^2} \\ % &a_{8,7}=r_2 & &a_{8,8}=\cfrac{1}{r_2} & &a_{8,9}=-r_2 & &a_{8,10}=-\cfrac{1}{r_2} \\ &a_{9,7}=\cfrac{E_s}{(1+\nu_s)(1-2\nu_s)} & &a_{9,8}=-\cfrac{E_s}{(1+\nu_s)r_2{}^2} & &a_{9,9}=-\cfrac{E_c}{(1+\nu_c)(1-2\nu_c)} & &a_{9,10}=\cfrac{E_c}{(1+\nu_c)r_2{}^2} \\ % &a_{10,7}=0 & &a_{10,8}=0 & &a_{10,9}=\cfrac{E_c}{(1+\nu_c)(1-2\nu_c)} & &a_{10,10}=-\cfrac{E_c}{(1+\nu_c)r_1{}^2} \end{align}$

Single Reinforcement Section

Components of Model

Components of Model
Concrete (outer cover)	Elastic material. Thermal stress is considered.
Inner Reinforcement	Elastic material. Thermal stress is considered.
Concrete (Inner cover)	Elastic material. Thermal stress is considered.

Coordinate of Boundary in Radial Direction
$r_4$	External surface of concrete lining
$r_3$	Boundary of reinforcement and outer cover concrete
$r_2$	Boundary of reinforcement and inner cover concrete
$r_1$	Inner surface of concrete lining

Basic Equations for Each Material

Outer Cover Concrete ( $r_3 \leqq r \leqq r_4$ )

$\begin{align} % displacement of outer concrete &u =\ \ \cfrac{1+\nu_c}{1-\nu_c} \alpha_c T \cfrac{r^2-r_3{}^2}{2 r} & &+ C_{co1}\cdot r + \cfrac{C_{co2}}{r} \\ % stress of outer concrete &\sigma_r =-\cfrac{E_c \alpha_c T}{1-\nu_c} \cfrac{r^2-r_3{}^2}{2 r^2} & &+ \cfrac{E_c}{(1+\nu_c)(1-2\nu_c)}\cdot C_{co1}-\cfrac{E_c}{(1+\nu_c)}\cfrac{C_{co2}}{r^2} \\ &\sigma_{\theta}=-\cfrac{E_c \alpha_c T}{1-\nu_c} \cfrac{r^2+r_3{}^2}{2 r^2} & &+ \cfrac{E_c}{(1+\nu_c)(1-2\nu_c)}\cdot C_{co1}+\cfrac{E_c}{(1+\nu_c)}\cfrac{C_{co2}}{r^2} \end{align}$

Reinforcement ( $r_2 \leqq r \leqq r_3$ )

Inner Cover Concrete ( $r_1 \leqq r \leqq r_2$ )

Simultaneous linear equations

To fix un-known parameters, simultaneous linear equations will be created considering boundary conditions shown below.

Stress in the radial direction at outer surface of concrete lining is equal to the external pressure.
Displacement and stress in the radial direction have continuity at the boundaries of each material.
Stress in the radial direction at inner surface of concrete lining is equal to zero.
Positive sign of the external pressure $P_b$ is toward inner direction.

$\begin{align} &1.\quad r=r_4 \;\text{(stress)}& &\quad \\ &\left\{ % stress of outer concrete \cfrac{E_c}{(1+\nu_c)(1-2\nu_c)}\cdot C_{co1}-\cfrac{E_c}{(1+\nu_c)}\cfrac{C_{co2}}{r_4{}^2} \right\} & &=-P_b+\cfrac{E_c \alpha_c T}{1-\nu_c} \cfrac{r_4{}^2-r_3{}^2}{2 r_4{}^2} \\ % &2.\quad r=r_3 \;\text{(disp.)}& &\quad\\ &\left\{ % displacement of outer concrete C_{co1}\cdot r_3 + \cfrac{C_{co2}}{r_3} \right\}-\left\{ % displacement of inner rebar C_{si1}\cdot r_3 + \cfrac{C_{si2}}{r_3} \right\} & &=\cfrac{1+\nu_s}{1-\nu_s} \alpha_s T \cfrac{r_3{}^2-r_2{}^2}{2 r_3} \\ % &3.\quad r=r_3 \;\text{(stress)}& &\quad \\ &\left\{ % stress of outer concrete \cfrac{E_c}{(1+\nu_c)(1-2\nu_c)}\cdot C_{co1}-\cfrac{E_c}{(1+\nu_c)}\cfrac{C_{co2}}{r_3{}^2} \right\}-\left\{ % stress of inner rebar \cfrac{E_s}{(1+\nu_s)(1-2\nu_s)}\cdot C_{si1}-\cfrac{E_s}{(1+\nu_s)}\cfrac{C_{si2}}{r_3{}^2} \right\} & &=-\cfrac{E_s \alpha_s T}{1-\nu_s} \cfrac{r_3{}^2-r_2{}^2}{2 r_3{}^2} \\ % &4.\quad r=r_2 \;\text{(disp.)}& &\quad \\ &\left\{ % displacement of inner rebar C_{si1}\cdot r_2 + \cfrac{C_{si2}}{r_2} \right\}-\left\{ % displacement of inner concrete C_{ci1}\cdot r_2 + \cfrac{C_{ci2}}{r_2} \right\} & &=\cfrac{1+\nu_c}{1-\nu_c} \alpha_c T \cfrac{r_2{}^2-r_1{}^2}{2 r_2} \\ % &5.\quad r=r_2 \;\text{(stress)}& &\quad \\ &\left\{ % stress of inner rebar \cfrac{E_s}{(1+\nu_s)(1-2\nu_s)}\cdot C_{si1}-\cfrac{E_s}{(1+\nu_s)}\cfrac{C_{si2}}{r_2{}^2} \right\}-\left\{ % stress of inner concrete \cfrac{E_c}{(1+\nu_c)(1-2\nu_c)}\cdot C_{ci1}-\cfrac{E_c}{(1+\nu_c)}\cfrac{C_{ci2}}{r_2{}^2} \right\} & &=-\cfrac{E_c \alpha_c T}{1-\nu_c} \cfrac{r_2{}^2-r_1{}^2}{2 r_2{}^2} \\ % &6.\quad r=r_1 \;\text{(stress)}& &\quad \\ &\left\{ % stress of inner concrete \cfrac{E_c}{(1+\nu_c)(1-2\nu_c)}\cdot C_{ci1}-\cfrac{E_c}{(1+\nu_c)}\cfrac{C_{ci2}}{r_1^2} \right\} & &=0 \end{align}$

Matrix Expression of Simultaneous linear equations for Computer Programing

$\begin{equation} \begin{bmatrix} a_{1,1} & a_{1,2} & 0 & 0 & 0 & 0 \\ a_{2,1} & a_{2,2} & a_{2,3} & a_{2,4} & 0 & 0 \\ a_{3,1} & a_{3,2} & a_{3,3} & a_{3,4} & 0 & 0 \\ 0 & 0 & a_{4,3} & a_{4,4} & a_{4,5} & a_{4,6} \\ 0 & 0 & a_{5,3} & a_{5,4} & a_{5,5} & a_{5,6} \\ 0 & 0 & 0 & 0 & a_{6,5} & a_{6,6} \end{bmatrix} \begin{Bmatrix} C_{co1} \\ C_{co2} \\ C_{si1} \\ C_{si2} \\ C_{ci1} \\ C_{ci2} \\ \end{Bmatrix} = \begin{Bmatrix} -P_b+\frac{E_c \alpha_c T}{1-\nu_c} \frac{r_4{}^2-r_3{}^2}{2 r_4{}^2} \\ \frac{1+\nu_s}{1-\nu_s} \alpha_s T \frac{r_3{}^2-r_2{}^2}{2 r_3} \\ -\frac{E_s \alpha_s T}{1-\nu_s} \frac{r_3{}^2-r_2{}^2}{2 r_3{}^2} \\ \frac{1+\nu_c}{1-\nu_c} \alpha_c T \frac{r_2{}^2-r_1{}^2}{2 r_2} \\ -\frac{E_c \alpha_c T}{1-\nu_c} \frac{r_2{}^2-r_1{}^2}{2 r_2{}^2} \\ 0 \end{Bmatrix} \end{equation}$

$\begin{align} &a_{1,1}=\cfrac{E_c}{(1+\nu_c)(1-2\nu_c)} & &a_{1,2}=-\cfrac{E_c}{(1+\nu_c)r_4{}^2} & &a_{1,3}=0 & &a_{1,4}=0 \\ % &a_{2,1}=r_3 & &a_{2,2}=\cfrac{1}{r_3} & &a_{2,3}=-r_3 & &a_{2,4}=-\cfrac{1}{r_3} \\ &a_{3,1}=\cfrac{E_c}{(1+\nu_c)(1-2\nu_c)} & &a_{3,2}=-\cfrac{E_c}{(1+\nu_c)r_3{}^2} & &a_{3,3}=-\cfrac{E_s}{(1+\nu_s)(1-2\nu_s)} & &a_{3,4}=\cfrac{E_s}{(1+\nu_c)r_3{}^2} \\ % &a_{4,3}=r_2 & &a_{4,4}=\cfrac{1}{r_2} & &a_{4,5}=-r_2 & &a_{4,6}=-\cfrac{1}{r_2} \\ &a_{5,3}=\cfrac{E_s}{(1+\nu_s)(1-2\nu_s)} & &a_{5,4}=-\cfrac{E_s}{(1+\nu_s)r_2{}^2} & &a_{5,5}=-\cfrac{E_c}{(1+\nu_c)(1-2\nu_c)} & &a_{5,6}=\cfrac{E_c}{(1+\nu_c)r_2{}^2} \\ % &a_{6,3}=0 & &a_{6,4}=0 & &a_{6,5}=\cfrac{E_c}{(1+\nu_c)(1-2\nu_c)} & &a_{6,6}=-\cfrac{E_c}{(1+\nu_c)r_1{}^2} \end{align}$

Thank you.

記事の先頭に行く

概要

Design of RC Circular Pressure Tunnel

Basic Equations for Elastic Cylinder Model in Plane Strain State

Basic Equations for Isotropic Elastic Material in Plane Strain State

Stress - Strain Relationship

Strain - Displacement Relationship

Equilibrium Equation

General Expressions of Displacements and Stresses

Basic Equations for No-tension Material in Circumferential Direction

Model of RC Circular Tunnel under Internal Pressure

Double Reinforcement Section

Components of Model

Basic Equations for Each Material

Bedrock ()

Outer Cover Concrete ()

Outer Reinforcement ()

Middle Concrete ()

Inner Reinforcement ()

Inner Cover Concrete ()

Simultaneous linear equations

Matrix Expression of Simultaneous linear equations for Computer Programing

Single Reinforcement Section

Components of Model

Basic Equations for Each Material

Bedrock ()

Outer Cover Concrete ()

Reinforcement ()

Inner Cover Concrete ()

Simultaneous linear equations

Matrix Expression of Simultaneous linear equations for Computer Programing

Model of RC Circular Tunnel under External Pressure

Double Reinforcement Section

Components of Model

Basic Equations for Each Material

Outer Cover Concrete ()

Outer Reinforcement ()

Middle Concrete ()

Inner Reinforcement ()

Inner Cover Concrete ()

Simultaneous linear equations

Matrix Expression of Simultaneous linear equations for Computer Programing

Single Reinforcement Section

Components of Model

Basic Equations for Each Material

Outer Cover Concrete ()

Reinforcement ()

Inner Cover Concrete ()

Simultaneous linear equations

Matrix Expression of Simultaneous linear equations for Computer Programing

Thank you.

Bedrock ( $r_6 \leqq r \leqq r_7$ )

Outer Cover Concrete ( $r_5 \leqq r \leqq r_6$ )

Outer Reinforcement ( $r_4 \leqq r \leqq r_5$ )

Middle Concrete ( $r_3 \leqq r \leqq r_4$ )

Inner Reinforcement ( $r_2 \leqq r \leqq r_3$ )

Inner Cover Concrete ( $r_1 \leqq r \leqq r_2$ )

Bedrock ( $r_4 \leqq r \leqq r_5$ )

Outer Cover Concrete ( $r_3 \leqq r \leqq r_4$ )

Reinforcement ( $r_2 \leqq r \leqq r_3$ )

Inner Cover Concrete ( $r_1 \leqq r \leqq r_2$ )

Outer Cover Concrete ( $r_5 \leqq r \leqq r_6$ )

Outer Reinforcement ( $r_4 \leqq r \leqq r_5$ )

Middle Concrete ( $r_3 \leqq r \leqq r_4$ )

Inner Reinforcement ( $r_2 \leqq r \leqq r_3$ )

Inner Cover Concrete ( $r_1 \leqq r \leqq r_2$ )

Outer Cover Concrete ( $r_3 \leqq r \leqq r_4$ )

Reinforcement ( $r_2 \leqq r \leqq r_3$ )

Inner Cover Concrete ( $r_1 \leqq r \leqq r_2$ )