Minimum plate thickness
\begin{equation}
t_0=\cfrac{D_0+800}{400}
\end{equation}
| $t_0$ | : Minimum plate thickness including corrosion allowance ($\geqq$6mm) |
| $D_0$ | : Internal diameter of pipe |
Pressure sharing design by bedrock
\begin{equation}
\sigma=\cfrac{P D}{2 t}(1-\lambda)
\end{equation}
\begin{equation}
\lambda=\cfrac{1-\cfrac{E_s}{P} \cdot \alpha_s \cdot \Delta T \cdot \cfrac{2t}{D}}
{1+(1+\beta_c)\cfrac{E_s}{E_c} \cdot \cfrac{2t}{D} \cdot \ln{\cfrac{D_R}{D}}+(1+\beta_g)\cfrac{E_s}{E_g} \cdot \cfrac{m_g+1}{m_g} \cdot \cfrac{2t}{D}}
\end{equation}
| $\sigma$ | : Tensile stress of pipe |
| $D$ | : Internal diameter for stress calculation ($=D_0+\epsilon$) |
| $t$ | : Plate thickness of pipe excluding corrosion allowance ($=t_0-\epsilon$) |
| $\epsilon$ | : Allowance thickness for corrosion and wear (=1.5mm) |
| $\lambda$ | : Sharing ratio of internal pressure by bedrock |
| $E_s$ | : Elastic modulus of steel (=206,000 MPa) |
| $\alpha_s$ | : Thermal expansion coefficient of steel ($=1.2 \times 10^{-5} /^\circ C$) |
| $\Delta T$ | : Temperature change of steel penstock (temperature drop is positive) |
| $\beta_c$ | : Plastic deformation coefficient of concrete (=0) |
| $E_c$ | : Elastic modulus of concrete (=20,600 MPa) |
| $D_R$ | : Excavation diameter of tunnel |
| $\beta_g$ | : Plastic deformation coefficient of bedrock |
| $E_g$ | : Elastic modulus of bedrock |
| $m_g$ | : Poisson's number of bedrock |
Buckling pressure of Steel pipe without stiffener
(1) Stress for calculating critical buckling pressure (Amstutz's formula)
\begin{equation}
\left(\frac{k_0}{r_m}+\frac{\sigma_N}{E_s^*}\right)\left(1+12\frac{{r_m}^2}{t^2}\frac{\sigma_N}{E_s^*}\right)^{1.5}
=3.36\frac{r_m}{t}\frac{\sigma_F^*-\sigma_N}{E_s^*}\left(1-\frac{1}{2}\frac{r_m}{t}\frac{\sigma_F^*-\sigma_N}{E_s^*}\right) \qquad\qquad (1)
\end{equation}
(2) Critical buckling pressure
\begin{equation}
p_k=\cfrac{\sigma_N}{\cfrac{r_m}{t}\left(1+0.35\cfrac{r_m}{t}\cfrac{\sigma_F^*-\sigma_N}{E_s^*}\right)} \qquad\qquad (2)
\end{equation}
(3) Properties of steel material
\begin{equation}
E_s^*=\frac{E_s}{1-{\nu_s}^2} \qquad
\sigma_F^*=\mu\frac{\sigma_F}{\sqrt{1-\nu_s+{\nu_s}^2}} \qquad
\mu=1.5-0.5\cfrac{1}{(1+0.002\cdot E_s/\sigma_F)^2}
\end{equation}
(4) Gap between outer surface of steel pipe and inner surface of backfill concrete
\begin{equation}
k_0=\cfrac{(\alpha_s\Delta T+\beta_g\sigma_a\eta/E_s) r_0'}{1+\beta_g}
\end{equation}
If compressive stress $\sigma_v$ is applied in the steel pipe, $k_0/r_m$ shall be changed to $-\sigma_v/E_s^*$ .
(5) Definitions of symbols
| $p_k$ | : Critical buckling pressure |
| $D_0$ | : Internal diameter of the pipe |
| $t_0$ | : Plate thickness of the pipe |
| $t$ | : plate thickness of the pipe excluding margin thickness ($=t_0-\epsilon$) |
| $\epsilon$ | : Margin thickness for corrosion and wear |
| $E_s$ | : Elastic modulus of steel |
| $\nu_s$ | : Poison's ratio of steel |
| $r_m$ | : Radius to the center of shell thickness ($=(D_0+t_0)/2$) |
| $r_0'$ | : Radius to the external surface of shell ($=(D_0+2 t_0)/2$) |
| $\sigma_N$ | : Critical axial stress at the moment of buckling |
| $\sigma_F$ | : Yield stress of steel shell |
| $E_s^*$ | : Modified elastic modulus including Poisson's effect |
| $\sigma_F^*$ | : Modified yield stress including Poisson's effect |
| $\mu$ | : Coefficient for supporting effect |
| $k_0$ | : Gap between concrete and external surface of steel pipe |
| $\alpha_s$ | : Coefficient of thermal expansion of steel |
| $\Delta T$ | : Temperature change of steel pipe (positive is decrease of temperature) |
| $\beta_g$ | : Plastic deformation modulus of be |
| $\eta$ | : Joint efficiency |
| $\sigma_a$ | : Allowable stress of steel pipe |
Buckling pressure of Steel pipe with stiffeners
Steel pipe
(1) Critical backling pressure (Timoshenko's formula)
\begin{equation}
\cfrac{(1-\nu_s^2) r_0' p_k}{E_s t}=
\cfrac{1-\nu_s^2}{(n^2-1)\left(1+\cfrac{n^2 L^2}{\pi^2 r_0'^2}\right)^2}+
\cfrac{t^2}{12 r_0'^2}\left\{(n^2-1)+\cfrac{2 n^2-1-\nu_s}{1+\cfrac{n^2 L^2}{\pi^2 r_0'^2}}\right\}
\end{equation}
(2) Modified interval of stiffeners
\begin{equation}
L'=(L+1.56 \sqrt{r_m t} \cos^{-1}\lambda)
\left(1+0.037\cdot \cfrac{\sqrt{r_m t}}{L+1.56 \sqrt{r_m t}\cos^{-1}\lambda} \cdot \cfrac{t^3}{I_s}\right)
\end{equation}
\begin{equation}
\lambda=1-(1+T)\cdot \cfrac{1+\cfrac{t_r}{1.56 \sqrt{r_m t}}}{1+\cfrac{S_0}{1.56 t \sqrt{r_m t}}}
\end{equation}
\begin{equation}
T=\cfrac{2}{t_r+1.56\sqrt{r_m t}} \cdot
\cfrac{\cfrac{r_0'^2}{t}-\cfrac{(t_r+1.56\sqrt{r_m t}) r_0'^2}{S_0+1.56 t \sqrt{r_m t}}}
{\cfrac{3}{[3(1-\nu_s^2)]^{0.75}}\left(\cfrac{r_0'}{t}\right)^{1.5} \cfrac{\sinh{\beta L}+\sin{\beta L}}{\cosh{\beta L}-\cos{\beta L}}
+\cfrac{2 r_0'^2}{S_0+1.56 t \sqrt{r_m t}}}
\end{equation}
\begin{equation}
\beta=\cfrac{[3(1-\nu_s^2)]^{0.25}}{\sqrt{r_m t}} \qquad
S_0=t_r (t+h_r) \qquad
I_s=\cfrac{t_r}{12}(h_r+t)^3
\end{equation}
Stiffener
(1) Critical buckling stress
\begin{equation}
\sigma_{cr}=\sigma_N \left\{1-\cfrac{r_0'}{e}\cfrac{\sigma_F-\sigma_N}{(1+3 \pi/2) E_s}\right\}
\end{equation}
(2) Stress for calculating critical buckling stress (Amstutz's formula)
\begin{equation}
\left(\frac{k_0}{r_m}+\frac{\sigma_N}{E_s}\right)\left(1+\frac{{r_m}^2}{i^2}\frac{\sigma_N}{E_s}\right)^{1.5}
=1.68\frac{r_m}{e}\frac{\sigma_F-\sigma_N}{E_s}\left(1-\frac{1}{4}\frac{r_m}{e}\frac{\sigma_F-\sigma_N}{E_s}\right)
\end{equation}
\begin{equation}
e=\cfrac{b t^2/2 + t_r h_r (t+h_r/2)}{A} \qquad b=t_r+1.56\sqrt{r_m t} \qquad A=b t + t_r h_r
\end{equation}
\begin{equation}
i=\sqrt{\cfrac{I}{A}} \qquad I=\cfrac{b}{3}\left\{(t-e)^3-e^3\right\}+\cfrac{t_r}{3}\left\{(h_r+t-e)^3-(t-e)^3\right\}
\end{equation}
(3) Average compressive stress
\begin{equation}
\sigma_c=\cfrac{p' r_0' (t_r+1.56\sqrt{r_m t})}{S_0+1.56 t \sqrt{r_m t}}
\end{equation}
(4) Modified external pressure
\begin{equation}
p'=\cfrac{p}{t_r+1.56\sqrt{r_m t}}
\left\{
(t_r+1.56\sqrt{r_m t})+2 \cdot \cfrac{\cfrac{r_0'^2}{t}-\cfrac{(t_r+1.56\sqrt{r_m t}) r_0'^2}{S_0+1.56 t \sqrt{r_m t}}}
{\cfrac{3}{[3(1-\nu_s^2)]^{0.75}}\left(\cfrac{r_0'}{t}\right)^{1.5} \cfrac{\sinh{\beta L}+\sin{\beta L}}{\cosh{\beta L}-\cos{\beta L}}
+\cfrac{2 r_0'^2}{S_0+1.56 t \sqrt{r_m t}}}
\right\}
\end{equation}
(5) Safety factor against buckling of stiffener
\begin{equation}
S_f=\cfrac{\sigma_{cr}}{\sigma_c}
\end{equation}
Definitions of symbols
| $p_k$ | : Critical buckling pressure |
| $n$ | : Number of winkles |
| $L$ | : Interval of stiffeners |
| $L'$ | : Modified interval of stiffeners |
| $t_r$ | : plate thickness of stiffeners excluding margin thickness |
| $h_r$ | : Height of stiffeners |
| $\sigma_{cr}$ | : Critical buckling stress of stiffeners |
| $\sigma_{c}$ | : Average compressive stress of stiffeners |
| $e$ | : Distance from centroid of combined section to internal surface of pile |
| $i$ | : Radius of gyration of combined section |
| $p$ | : External pressure |
| $p'$ | : Modified external pressure |
| $S_f$ | : Safety factor against stiffener buckling |
Programs
| Filename | Description |
|---|---|
| a_d.txt | shell scropt for execution of 'f90_ams_d' |
| a_dd.txt | shell script for execution of 'f90_ams_dd' |
| a_stiff.txt | shell script for execution of 'f90_stiff' |
| f90_ams_d.f90 | calculation of buckling pressure without stiffeners |
| f90_ams_dd.f90 | calculation of required plate thickness |
| f90_stiff.f90 | calculation of buckling pressure with stiffeners |