# Flat ends with a relief groove for a pair of openings

## Values for calculation

$T$ $\mathrm{°C}$
$T_{test}$ $\mathrm{°C}$
$P$ $\mathrm{MPa}$
$P_{test}$ $\mathrm{MPa}$
$ν$
$D_i$ $\mathrm{mm}$
$e_{af}$ $\mathrm{mm}$
$e_s$ $\mathrm{mm}$
$r_d$ $\mathrm{mm}$
$h_w$ $\mathrm{mm}$
$d$ $\mathrm{mm}$
$k$ $\mathrm{mm}$
$R_{p0.2/T}$ $\mathrm{MPa}$
$R_{p0.2/T_{test}}$ $\mathrm{MPa}$
$R_{p1.0/T}$ $\mathrm{MPa}$
$R_{p1.0/T_{test}}$ $\mathrm{MPa}$
$R_{m/20}$ $\mathrm{MPa}$
$R_{m/T}$ $\mathrm{MPa}$
$R_{m/T_{test}}$ $\mathrm{MPa}$
$R_{p0.2/T/s}$ $\mathrm{MPa}$
$R_{p0.2/T_{test}/s}$ $\mathrm{MPa}$
$R_{p1.0/T/s}$ $\mathrm{MPa}$
$R_{p1.0/T_{test}/s}$ $\mathrm{MPa}$
$R_{m/20/s}$ $\mathrm{MPa}$
$R_{m/T/s}$ $\mathrm{MPa}$
$R_{m/T_{test}/s}$ $\mathrm{MPa}$

## Calculation

### Maximum allowed value of the nominal design stress for normal operating load cases

$\text{if }\ \text{type }$$\text{of }$$\text{material}= \text{Cast steels}$
$$f_d=\min\left(\cfrac{R_{p0.2/T}}{1.9}, \cfrac{R_{m/20}}{3}\right)$$
$\text{else if }\ \text{type }$$\text{of }$$\text{material}= \text{Austenitic steels}\wedge\text{min. }$$\text{elongation }$$\text{after }$$\text{fracture}\geq 35$$f_d=\max\left[\cfrac{R_{p1.0/T}}{1.5}, \min\left(\cfrac{R_{p1.0/T}}{1.2}, \cfrac{R_{m/T}}{3}\right)\right]$$\text{else if }\ \text{type }$$\text{of }$$\text{material}= \text{Austenitic steels}\wedge 30\le \text{min. }$$\text{elongation }$$\text{after }$$\text{fracture}< 35$
$$f_d=\cfrac{R_{p1.0/T}}{1.5}$$
$\text{else}$
$$f_d=\min\left(\cfrac{R_{p0.2/T}}{1.5}, \cfrac{R_{m/20}}{2.4}\right)$$

### Maximum allowed value of the nominal design stress for testing load cases

$\text{if }\ \text{type }$$\text{of }$$\text{material}= \text{Cast steels}$
$$f_{test}=\cfrac{R_{p0.2/T_{test}}}{1.33}$$
$\text{else if }\ \text{type }$$\text{of }$$\text{material}= \text{Austenitic steels}\wedge\text{min. }$$\text{elongation }$$\text{after }$$\text{fracture}\geq 35$$f_{test}=\max\left(\cfrac{R_{p1.0/T_{test}}}{1.05}, \cfrac{R_{m/T_{test}}}{2}\right)$$\text{else if }\ \text{type }$$\text{of }$$\text{material}= \text{Austenitic steels}\wedge 30\le \text{min. }$$\text{elongation }$$\text{after }$$\text{fracture}< 35$
$$f_{test}=\cfrac{R_{p1.0/T_{test}}}{1.05}$$
$\text{else}$
$$f_{test}=\cfrac{R_{p0.2/T_{test}}}{1.05}$$

### Nominal design stress at calculation temperature of the shell for normal operating load cases

$\text{if }\ \text{type }$$\text{of }$$\text{material }$$\text{of }$$\text{the }$$\text{shell}= \text{Cast steels}$$f_s=\min\left(\cfrac{R_{p0.2/T/s}}{1.9}, \cfrac{R_{m/20/s}}{3}\right)$$\text{else if }\ \text{type }$$\text{of }$$\text{material }$$\text{of }$$\text{the }$$\text{shell}= \text{Austenitic steels}\wedge\text{min. }$$\text{elongation }$$\text{after }$$\text{fracture }$$\text{of }$$\text{the }$$\text{shell}\geq 35$
$$f_s=\max\left[\cfrac{R_{p1.0/T/s}}{1.5}, \min\left(\cfrac{R_{p1.0/T/s}}{1.2}, \cfrac{R_{m/T/s}}{3}\right)\right]$$
$\text{else if }\ \text{type }$$\text{of }$$\text{material }$$\text{of }$$\text{the }$$\text{shell}= \text{Austenitic steels}\wedge 30\le \text{min. }$$\text{elongation }$$\text{after }$$\text{fracture }$$\text{of }$$\text{the }$$\text{shell}< 35$$f_s=\cfrac{R_{p1.0/T/s}}{1.5}$$\text{else}$$f_s=\min\left(\cfrac{R_{p0.2/T/s}}{1.5}, \cfrac{R_{m/20/s}}{2.4}\right)$$### Nominal design stress at calculation temperature of the shell for testing load cases \text{if }\ \text{type }$$\text{of }$$\text{material }$$\text{of }$$\text{the }$$\text{shell}= \text{Cast steels}$
$$f_{s_{test}}=\cfrac{R_{p0.2/T_{test}/s}}{1.33}$$
$\text{else if }\ \text{type }$$\text{of }$$\text{material }$$\text{of }$$\text{the }$$\text{shell}= \text{Austenitic steels}\wedge\text{min. }$$\text{elongation }$$\text{after }$$\text{fracture }$$\text{of }$$\text{the }$$\text{shell}\geq 35$$f_{s_{test}}=\max\left(\cfrac{R_{p1.0/T_{test}/s}}{1.05}, \cfrac{R_{m/T_{test}/s}}{2}\right)$$\text{else if }\ \text{type }$$\text{of }$$\text{material }$$\text{of }$$\text{the }$$\text{shell}= \text{Austenitic steels}\wedge 30\le \text{min. }$$\text{elongation }$$\text{after }$$\text{fracture }$$\text{of }$$\text{the }$$\text{shell}< 35$
$$f_{s_{test}}=\cfrac{R_{p1.0/T_{test}/s}}{1.05}$$
$\text{else}$
$$f_{s_{test}}=\cfrac{R_{p0.2/T_{test}/s}}{1.05}$$

### Lower of the nominal design stresses $f_d$ of the end and $f_s$ of the shell for normal operating load cases

$$f_{min}=\min\left\{f_d, f_s\right\}$$

### Lower of the nominal design stresses $f_{test}$ of the end and $f_{s_{test}}$ of the shell for testing load cases

$$f_{min_{test}}=\min\left\{f_{test}, f_{s_{test}}\right\}$$

### Length of cylindrical shell

$$l_{cyl}=\sqrt{\left(D_i+e_s\right)\cdot e_s}$$

### Factor $B_1$

$$B_1=1-\cfrac{3\cdot f_d}{P}\cdot\left(\cfrac{e_s}{D_i+e_s}\right)^2+\cfrac{3}{16}\cdot\left(\cfrac{D_i}{D_i+e_s}\right)^4\cdot\cfrac{P}{f_d}-\cfrac{3}{4}\cdot\cfrac{\left(2\cdot D_i+e_s\right)\cdot e_s^2}{\left(D_i+e_s\right)^3}$$

### Factor $B_{1_{test}}$

$$B_{1_{test}}=1-\cfrac{3\cdot f_{test}}{P_{test}}\cdot\left(\cfrac{e_s}{D_i+e_s}\right)^2+\cfrac{3}{16}\cdot\left(\cfrac{D_i}{D_i+e_s}\right)^4\cdot\cfrac{P_{test}}{f_{test}}-\cfrac{3}{4}\cdot\cfrac{\left(2\cdot D_i+e_s\right)\cdot e_s^2}{\left(D_i+e_s\right)^3}$$

### Factor $A_1$

$$A_1=B_1\cdot\left[1-B_1\cdot\cfrac{e_s}{2\cdot\left(D_i+e_s\right)}\right]$$

### Factor $A_{1_{test}}$

$$A_{1_{test}}=B_{1_{test}}\cdot\left[1-B_{1_{test}}\cdot\cfrac{e_s}{2\cdot\left(D_i+e_s\right)}\right]$$

### Shape factor for calculation of circular flat ends

$$C_1=\max\left\{\left[0.40825\cdot A_1\cdot\cfrac{D_i+e_s}{D_i}\right], \left[0.299\cdot \left(1+1.7\cdot\cfrac{e_s}{D_i}\right)\right]\right\}$$

### Shape factor for calculation of circular flat ends for testing load cases

$$C_{1_{test}}=\max\left\{\left[0.40825\cdot A_{1_{test}}\cdot\cfrac{D_i+e_s}{D_i}\right], \left[0.299\cdot \left(1+1.7\cdot\cfrac{e_s}{D_i}\right)\right]\right\}$$

### Factor $g$

$$g=\cfrac{D_i}{D_i+e_s}$$

### Factor $H$

$$H=\sqrt[4]{12\cdot\left(1-ν^2\right)}\cdot\sqrt{\cfrac{e_s}{D_i+e_s}}$$

### Factor $J$

$$J=\cfrac{3\cdot f_{min}}{P}-\cfrac{D_i^2}{4\cdot\left(D_i+e_s\right)\cdot e_s}-1$$

### Factor $U$

$$U=\cfrac{2\cdot\left(2-ν\cdot g\right)}{\sqrt{3\cdot\left(1-ν^2\right)}}$$

### Factor $f_1$

$$f_1=2\cdot g^2-g^4$$

### Factor $A$

$$A=\left(\cfrac{3}{4}\cdot\cfrac{U\cdot D_i}{e_s}-2\cdot J\right)\cdot\left(1+ν\right)\cdot\left[1+\left(1-ν\right)\cdot\cfrac{e_s}{D_i+e_s}\right]$$

### Factor $B$

$$B=\left[\left(\cfrac{3}{8}\cdot\cfrac{U\cdot D_i}{e_s}-J\right)\cdot H^2-\cfrac{3}{2}\cdot\left(2-ν\cdot g\right)\cdot g\right]\cdot H$$

### Factor $F$

$$F=\left(\cfrac{3}{8}\cdot U\cdot g+\cfrac{3}{16}\cdot f_1\cdot\cfrac{D_i+e_s}{e_s}-2\cdot J\cdot\cfrac{e_s}{D_i+e_s}\right)\cdot H^2-3\cdot\left(2-ν\cdot g\right)\cdot g\cdot\cfrac{e_s}{D_i+e_s}$$

### Factor $G$

$$G=\left[\cfrac{3}{8}\cdot f_1-2\cdot J\cdot\left(\cfrac{e_s}{D_i+e_s}\right)^2\right]\cdot H$$

### Factor $a$

$$a=\cfrac{B}{A}$$

### Factor $b$

$$b=\cfrac{F}{A}$$

### Factor $c$

$$c=\cfrac{G}{A}$$

### Factor $N$

$$N=\cfrac{b}{3}-\cfrac{a^2}{9}$$

### Factor $Q$

$$Q=\cfrac{c}{2}-\cfrac{a\cdot b}{6}+\cfrac{a^3}{27}$$

### Factor $K$

$$K=\cfrac{N^3}{Q^2}$$

### Factor $S$

$\text{if }\ Q\geq 0$
$$S=\sqrt[3]{Q\cdot\left[1+\left(1+K\right)^{1/2}\right]}$$
$\text{else}$
$$S=-\sqrt[3]{\left|Q\right|\cdot\left[1+\left(1+K\right)^{1/2}\right]}$$

### Shape factor for calculation of circular flat ends

$$C_2=\cfrac{\left(D_i+e_s\right)\cdot\left(\cfrac{N}{S}-S-\cfrac{a}{3}\right)}{D_i\cdot\sqrt{\cfrac{P}{f_{min}}}}$$

### Calculation coefficient $Y_1$ for opening reinforcement

$$Y_1=\min\left\{2, \sqrt[3]{\cfrac{k}{k-d}}\right\}$$

### Calculation coefficient $Y_2$ for opening reinforcement

$$Y_2=\sqrt{\cfrac{k}{k-d}}$$

### Minimum required thickness for a flat end with a hub

$\text{if }\ C_2\geq 0.3$
$$e=\max\left\{\left(Y_1\cdot\left(\max\left\{\left(C_1\cdot D_i\cdot\sqrt{\cfrac{P}{f_d}}\right), \left(C_2\cdot D_i\cdot\sqrt{\cfrac{P}{f_{min}}}\right)\right\}\right)\right), C_1\cdot Y_2\cdot D_i\cdot\sqrt{\cfrac{P}{f_d}}\right\}$$
$\text{else}$
$$e=\max\left\{\left(Y_1\cdot\left(C_2\cdot D_i\cdot\sqrt{\cfrac{P}{f_{min}}}\right)\right), C_1\cdot Y_2\cdot D_i\cdot\sqrt{\cfrac{P}{f_d}}\right\}$$

$$e_{af}\geq e$$

### Minimum required thickness for a flat end with a hub for testing load cases

$$e_{test}=\max\left\{\left(Y_1\cdot\left(C_{1_{test}}\cdot D_i\cdot\sqrt{\cfrac{P_{test}}{f_{test}}}\right)\right), C_1\cdot Y_2\cdot D_i\cdot\sqrt{\cfrac{P_{test}}{f_{test}}}\right\}$$

$$e_{af}\geq e_{test}$$

### Minimum required thickness under a relief groove

$$e_r=\max\left\{e_s, e_s\cdot\left(\cfrac{f_s}{f_d}\right)\right\}$$

### Minimum required thickness under a relief groove for testing load cases

$$e_{r_{test}}=\max\left\{e_s, e_s\cdot\left(\cfrac{f_{s_{test}}}{f_{test}}\right)\right\}$$

## Requirements

$$\cfrac{P}{f_{min}}\le 0.1$$$$\cfrac{P_{test}}{f_{min_{test}}}\le 0.1$$$$d\le 0.5\cdot D_i$$$$r_d \geq \max\left(0.25\cdot e_s, 5\right)$$$$h_w \geq e_{af}-2$$