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Flat ends with a hub for tapered shell for a pair of openings

Tapered shell l cyl D eq D i e s e af r D F
Tapered shell

Values for calculation

$ T $ $ \mathrm{°C} $
$ T_{test} $ $ \mathrm{°C} $
$ P $ $ \mathrm{MPa} $
$ P_{test} $ $ \mathrm{MPa} $
$ D_i $ $ \mathrm{mm} $
$ D_F $ $ \mathrm{mm} $
$ e_s $ $ \mathrm{mm} $
$ e_{af} $ $ \mathrm{mm} $
$ r $ $ \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} $

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$
$$f_{test}=\cfrac{R_{p1.0/T_{test}}}{1.05}$$
$\text{else}$
$$f_{test}=\cfrac{R_{p0.2/T_{test}}}{1.05}$$

Length of cylindrical shell

$$l_{cyl}=0.5\cdot \sqrt{\left(D_i+e_s\right)\cdot e_s}$$

Equivalent diameter of an end with a hub

$$D_{eq}=\cfrac{D_i+D_F}{2}$$

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

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

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

$$e=\max\left\{\left(Y_1\cdot\left(C_1\cdot D_{eq}\cdot\sqrt{\cfrac{P}{f_d}}\right)\right), C_1\cdot Y_2\cdot D_i\cdot\sqrt{\cfrac{P}{f_d}}\right\}$$

$$e_{af}\geq e$$

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

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

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

Requirements

$$ r\geq e_s $$ $$ r\geq 1.3\cdot e_{af} $$ $$ d\le 0.5\cdot D_i $$
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