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Flat ends with a hub for uniform thickness shell

Uniform thickness shell l cyl D eq D i e s e af r
Uniform thickness shell

Values for calculation

$T$ $\mathrm{°C}$
$T_{test}$ $\mathrm{°C}$
$P$ $\mathrm{MPa}$
$P_{test}$ $\mathrm{MPa}$
$D_i$ $\mathrm{mm}$
$e_s$ $\mathrm{mm}$
$e_{af}$ $\mathrm{mm}$
$r$ $\mathrm{mm}$
$R_{p0.2/T}$ $\mathrm{MPa}$
$R_{p0.2/T_{test}}$ $\mathrm{MPa}$
$R_{p1.0/T}$ $\mathrm{MPa}$
$R_{p1.0/T_{test}}$ $\mathrm{MPa}$
$R_{m/20}$ $\mathrm{MPa}$
$R_{m/T}$ $\mathrm{MPa}$
$R_{m/T_{test}}$ $\mathrm{MPa}$

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}

Length of cylindrical shell

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

Equivalent diameter of an end with a hub

$$D_{eq}=D_i-r$$
$D_{eq}=D_i-r$
Equations in LaTeX code
D_{eq}=D_i-r

Factor $ B_1 $

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

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

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

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

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

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

Minimum required thickness for a flat end with a hub

$$e=C_1\cdot D_{eq}\cdot\sqrt{\cfrac{P}{f_d}}$$
$e=C_1\cdot D_{eq}\cdot\sqrt{\cfrac{P}{f_d}}$
Equations in LaTeX code
e=C_1\cdot D_{eq}\cdot\sqrt{\cfrac{P}{f_d}}
$$e_{af}\geq e$$
$e_{af}\geq e$
Equations in LaTeX code
e_{af}\geq e

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

$$e_{test}=C_{1_{test}}\cdot D_{eq}\cdot\sqrt{\cfrac{P_{test}}{f_{test}}}$$
$e_{test}=C_{1_{test}}\cdot D_{eq}\cdot\sqrt{\cfrac{P_{test}}{f_{test}}}$
Equations in LaTeX code
e_{test}=C_{1_{test}}\cdot D_{eq}\cdot\sqrt{\cfrac{P_{test}}{f_{test}}}
$$e_{af}\geq e_{test}$$
$e_{af}\geq e_{test}$
Equations in LaTeX code
e_{af}\geq e_{test}

Requirements

$$r\geq e_s$$
$r\geq e_s$
Equations in LaTeX code
r\geq e_s
$$r\geq 1.3\cdot e_{af}$$
$r\geq 1.3\cdot e_{af}$
Equations in LaTeX code
r\geq 1.3\cdot e_{af}
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