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

Geometry of cone/cylinder intersection with knuckle - Large end D c ≥ e cyl ≥ e con ≥ e 2 α ≥ 1.4l 1 ≥ 0.7 l 2 ≥ e 1 0.5 l 1 r ≥ 1.4 l 2
Geometry of cone/cylinder intersection with knuckle - Large end

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}$
$\text{if }\ \text{type }$$\text{of }$$\text{material}= \text{Cast steels}$
Equations in LaTeX code
\text{if }\ \text{type }$$\text{of }$$\text{material}= \text{Cast steels}
$$f_d=\min\left(\cfrac{R_{p0.2/T}}{1.9}, \cfrac{R_{m/20}}{3}\right)$$
$f_d=\min\left(\cfrac{R_{p0.2/T}}{1.9}, \cfrac{R_{m/20}}{3}\right)$
Equations in LaTeX code
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 }$$\text{fracture}\geq 35$
$\text{else if }\ \text{type }$$\text{of }$$\text{material}= \text{Austenitic steels}\wedge\text{min. }$$\text{elongation }$$\text{after }$$\text{fracture}\geq 35$
Equations in LaTeX code
\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]$$
$f_d=\max\left[\cfrac{R_{p1.0/T}}{1.5}, \min\left(\cfrac{R_{p1.0/T}}{1.2}, \cfrac{R_{m/T}}{3}\right)\right]$
Equations in LaTeX code
f_d=\max\left[\cfrac{R_{p1.0/T}}{1.5}, \min\left(\cfrac{R_{p1.0/T}}{1.2}, \cfrac{R_{m/T}}{3}\right)\right]
$\text{else if }\ \text{type }$$\text{of }$$\text{material}= \text{Austenitic steels}\wedge 30\le \text{min. }$$\text{elongation }$$\text{after }$$\text{fracture}< 35$
$\text{else if }\ \text{type }$$\text{of }$$\text{material}= \text{Austenitic steels}\wedge 30\le \text{min. }$$\text{elongation }$$\text{after }$$\text{fracture}< 35$
Equations in LaTeX code
\text{else if }\ \text{type }$$\text{of }$$\text{material}= \text{Austenitic steels}\wedge 30\le \text{min. }$$\text{elongation }$$\text{after }$$\text{fracture}< 35
$$f_d=\cfrac{R_{p1.0/T}}{1.5}$$
$f_d=\cfrac{R_{p1.0/T}}{1.5}$
Equations in LaTeX code
f_d=\cfrac{R_{p1.0/T}}{1.5}
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$f_d=\min\left(\cfrac{R_{p0.2/T}}{1.5}, \cfrac{R_{m/20}}{2.4}\right)$$
$f_d=\min\left(\cfrac{R_{p0.2/T}}{1.5}, \cfrac{R_{m/20}}{2.4}\right)$
Equations in LaTeX code
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}$
$\text{if }\ \text{type }$$\text{of }$$\text{material}= \text{Cast steels}$
Equations in LaTeX code
\text{if }\ \text{type }$$\text{of }$$\text{material}= \text{Cast steels}
$$f_{test}=\cfrac{R_{p0.2/T_{test}}}{1.33}$$
$f_{test}=\cfrac{R_{p0.2/T_{test}}}{1.33}$
Equations in LaTeX code
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 }$$\text{fracture}\geq 35$
$\text{else if }\ \text{type }$$\text{of }$$\text{material}= \text{Austenitic steels}\wedge\text{min. }$$\text{elongation }$$\text{after }$$\text{fracture}\geq 35$
Equations in LaTeX code
\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)$$
$f_{test}=\max\left(\cfrac{R_{p1.0/T_{test}}}{1.05}, \cfrac{R_{m/T_{test}}}{2}\right)$
Equations in LaTeX code
f_{test}=\max\left(\cfrac{R_{p1.0/T_{test}}}{1.05}, \cfrac{R_{m/T_{test}}}{2}\right)
$\text{else if }\ \text{type }$$\text{of }$$\text{material}= \text{Austenitic steels}\wedge 30\le \text{min. }$$\text{elongation }$$\text{after }$$\text{fracture}< 35$
$\text{else if }\ \text{type }$$\text{of }$$\text{material}= \text{Austenitic steels}\wedge 30\le \text{min. }$$\text{elongation }$$\text{after }$$\text{fracture}< 35$
Equations in LaTeX code
\text{else if }\ \text{type }$$\text{of }$$\text{material}= \text{Austenitic steels}\wedge 30\le \text{min. }$$\text{elongation }$$\text{after }$$\text{fracture}< 35
$$f_{test}=\cfrac{R_{p1.0/T_{test}}}{1.05}$$
$f_{test}=\cfrac{R_{p1.0/T_{test}}}{1.05}$
Equations in LaTeX code
f_{test}=\cfrac{R_{p1.0/T_{test}}}{1.05}
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$f_{test}=\cfrac{R_{p0.2/T_{test}}}{1.05}$$
$f_{test}=\cfrac{R_{p0.2/T_{test}}}{1.05}$
Equations in LaTeX code
f_{test}=\cfrac{R_{p0.2/T_{test}}}{1.05}

Factor $ β $

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

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

Factor $ γ $

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

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

$$l_1=\sqrt{D_c\cdot e_1}$$
$l_1=\sqrt{D_c\cdot e_1}$
Equations in LaTeX code
l_1=\sqrt{D_c\cdot e_1}

Length along cone at large or small end

$$l_2=\sqrt{\cfrac{D_c\cdot e_2}{\cos{α}}}$$
$l_2=\sqrt{\cfrac{D_c\cdot e_2}{\cos{α}}}$
Equations in LaTeX code
l_2=\sqrt{\cfrac{D_c\cdot e_2}{\cos{α}}}

Requirements

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