# Junction between the large end of a cone and a cylinder with a knuckle

## Values for calculation

$T$ $\mathrm{°C}$
$T_{test}$ $\mathrm{°C}$
$P$ $\mathrm{MPa}$
$P_{test}$ $\mathrm{MPa}$
$α$ $\mathrm{°}$
$D_c$ $\mathrm{mm}$
$r$ $\mathrm{mm}$
$e_1$ $\mathrm{mm}$
$e_2$ $\mathrm{mm}$
$e_{cyl}$ $\mathrm{mm}$
$e_{con}$ $\mathrm{mm}$
$R_{p0.2/T}$ $\mathrm{MPa}$
$R_{p0.2/T_{test}}$ $\mathrm{MPa}$
$R_{p1.0/T}$ $\mathrm{MPa}$
$R_{p1.0/T_{test}}$ $\mathrm{MPa}$
$R_{m/20}$ $\mathrm{MPa}$
$R_{m/T}$ $\mathrm{MPa}$
$R_{m/T_{test}}$ $\mathrm{MPa}$

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

### Factor $β$

$$β=\cfrac{1}{3}\cdot\sqrt{\cfrac{D_c}{e_j}}\cdot\cfrac{\tan{α}}{1+\cfrac{1}{\sqrt{\cos{α}}}}-0.15$$

### Factor $ρ$

$$ρ=\cfrac{0.028\cdot r}{\sqrt{D_c\cdot e_j}}\cdot\cfrac{α}{1+\cfrac{1}{\sqrt{\cos{α}}}}$$

### Factor $γ$

$$γ=1+\cfrac{ρ}{1.2\cdot\left(1+\cfrac{0.2}{ρ}\right)}$$

### Required or analysis thickness at a junction at the large end of a cone

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

### 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

$$e_1\geq \max\left(e_{cyl}, e_j\right)$$$$e_2\geq \max\left(e_{con}, e_j\right)$$$$r< 0.3\cdot D_c$$