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Junction between the small end of a cone and a cylinder

Geometry of cone/cylinder intersection small end e 1 e 2 α l 1 l 2 D c e cyl e con
Geometry of cone/cylinder intersection small end

Values for calculation

$ T $ $ \mathrm{°C} $
$ T_{test} $ $ \mathrm{°C} $
$ P $ $ \mathrm{MPa} $
$ P_{test} $ $ \mathrm{MPa} $
$ α $ $ \mathrm{°} $
$ D_c $ $ \mathrm{mm} $
$ e_1 $ $ \mathrm{mm} $
$ e_2 $ $ \mathrm{mm} $
$ e_{cyl} $ $ \mathrm{mm} $
$ e_{con} $ $ \mathrm{mm} $
$ z $
$ 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$
$$f_{test}=\cfrac{R_{p1.0/T_{test}}}{1.05}$$
$\text{else}$
$$f_{test}=\cfrac{R_{p0.2/T_{test}}}{1.05}$$

Factor $ s $

$$s=\cfrac{e_2}{e_1}$$

Factor $ τ $

$\text{if }\ s< 1$
$$τ=s\cdot\sqrt{\cfrac{s}{\cos{α}}}+\sqrt{\cfrac{1+s^2}{2}}$$
$\text{else}$
$$τ=1+\sqrt{s\cdot\left\{\cfrac{1+s^2}{2\cdot\cos{α}}\right\}}$$

Factor $ β_H $

$$β_H=0.4\cdot\sqrt{\cfrac{D_c}{e_1}}\cdot\cfrac{\tan{α}}{τ}+0.5$$

Required or analysis thickness at a junction at the small end of a cone

$$e_j=\max\left(\cfrac{P\cdot D_c\cdot β_H}{2\cdot f_d\cdot z}, \cfrac{P_{test}\cdot D_c\cdot β_H}{2\cdot f_{test}}\right)$$

Length along cylinder

$$l_1=\sqrt{D_c\cdot e_1}$$

Length along cone at large or small end

$$l_2=\sqrt{\cfrac{D_c\cdot e_2}{\cos{α}}}$$

Requirements

$$ e_1\geq \max\left(e_{cyl}, e_j\right) $$ $$ e_2\geq \max\left(e_{con}, e_j\right) $$