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Flat ends welded directly to the shell

Circular flat ends welded directly to the shell l cyl D i e s e
Circular flat ends welded directly to the shell

Values for calculation

$ \{T\in\mathbb{R}\mid T\geq0\} $
This set includes all real numbers that are greater than or equal to zero. Therefore, it contains all positive numbers and zero, but no negative numbers.
$ T $ $ \mathrm{°C} $
$ \{T_{test}\in\mathbb{R}\mid T_{test}\geq0\} $
This set includes all real numbers that are greater than or equal to zero. Therefore, it contains all positive numbers and zero, but no negative numbers.
$ T_{test} $ $ \mathrm{°C} $
$ \{P\in\mathbb{R}\mid P>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ P $ $ \mathrm{MPa} $
$ \{P_{test}\in\mathbb{R}\mid P_{test}>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ P_{test} $ $ \mathrm{MPa} $
$ \{ν\in\mathbb{R}\mid ν>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ ν $
$ \{D_i\in\mathbb{R}\mid D_i>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ D_i $ $ \mathrm{mm} $
$ \{e_s\in\mathbb{R}\mid e_s>0\} $
This set includes all real numbers that are greater than zero. Therefore, it contains only positive numbers. Zero and negative numbers are not included in this set.
$ e_s $ $ \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} $
$ R_{p0.2/T/s} $ $ \mathrm{MPa} $
$ R_{p0.2/T_{test}/s} $ $ \mathrm{MPa} $
$ R_{p1.0/T/s} $ $ \mathrm{MPa} $
$ R_{p1.0/T_{test}/s} $ $ \mathrm{MPa} $
$ R_{m/20/s} $ $ \mathrm{MPa} $
$ R_{m/T/s} $ $ \mathrm{MPa} $
$ R_{m/T_{test}/s} $ $ \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}

Nominal design stress at calculation temperature of the shell for normal operating load cases

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

Nominal design stress at calculation temperature of the shell for testing load cases

$\text{if }\ \text{type }$$\text{of }$$\text{material }$$\text{of }$$\text{the }$$\text{shell}= \text{Cast steels}$
$\text{if }\ \text{type }$$\text{of }$$\text{material }$$\text{of }$$\text{the }$$\text{shell}= \text{Cast steels}$
Equations in LaTeX code
\text{if }\ \text{type }$$\text{of }$$\text{material }$$\text{of }$$\text{the }$$\text{shell}= \text{Cast steels}
$$f_{s_{test}}=\cfrac{R_{p0.2/T_{test}/s}}{1.33}$$
$f_{s_{test}}=\cfrac{R_{p0.2/T_{test}/s}}{1.33}$
Equations in LaTeX code
f_{s_{test}}=\cfrac{R_{p0.2/T_{test}/s}}{1.33}
$\text{else if }\ \text{type }$$\text{of }$$\text{material }$$\text{of }$$\text{the }$$\text{shell}= \text{Austenitic steels}\wedge\text{min. }$$\text{elongation }$$\text{after }$$\text{fracture }$$\text{of }$$\text{the }$$\text{shell}\geq 35$
$\text{else if }\ \text{type }$$\text{of }$$\text{material }$$\text{of }$$\text{the }$$\text{shell}= \text{Austenitic steels}\wedge\text{min. }$$\text{elongation }$$\text{after }$$\text{fracture }$$\text{of }$$\text{the }$$\text{shell}\geq 35$
Equations in LaTeX code
\text{else if }\ \text{type }$$\text{of }$$\text{material }$$\text{of }$$\text{the }$$\text{shell}= \text{Austenitic steels}\wedge\text{min. }$$\text{elongation }$$\text{after }$$\text{fracture }$$\text{of }$$\text{the }$$\text{shell}\geq 35
$$f_{s_{test}}=\max\left(\cfrac{R_{p1.0/T_{test}/s}}{1.05}, \cfrac{R_{m/T_{test}/s}}{2}\right)$$
$f_{s_{test}}=\max\left(\cfrac{R_{p1.0/T_{test}/s}}{1.05}, \cfrac{R_{m/T_{test}/s}}{2}\right)$
Equations in LaTeX code
f_{s_{test}}=\max\left(\cfrac{R_{p1.0/T_{test}/s}}{1.05}, \cfrac{R_{m/T_{test}/s}}{2}\right)
$\text{else if }\ \text{type }$$\text{of }$$\text{material }$$\text{of }$$\text{the }$$\text{shell}= \text{Austenitic steels}\wedge 30\le \text{min. }$$\text{elongation }$$\text{after }$$\text{fracture }$$\text{of }$$\text{the }$$\text{shell}< 35$
$\text{else if }\ \text{type }$$\text{of }$$\text{material }$$\text{of }$$\text{the }$$\text{shell}= \text{Austenitic steels}\wedge 30\le \text{min. }$$\text{elongation }$$\text{after }$$\text{fracture }$$\text{of }$$\text{the }$$\text{shell}< 35$
Equations in LaTeX code
\text{else if }\ \text{type }$$\text{of }$$\text{material }$$\text{of }$$\text{the }$$\text{shell}= \text{Austenitic steels}\wedge 30\le \text{min. }$$\text{elongation }$$\text{after }$$\text{fracture }$$\text{of }$$\text{the }$$\text{shell}< 35
$$f_{s_{test}}=\cfrac{R_{p1.0/T_{test}/s}}{1.05}$$
$f_{s_{test}}=\cfrac{R_{p1.0/T_{test}/s}}{1.05}$
Equations in LaTeX code
f_{s_{test}}=\cfrac{R_{p1.0/T_{test}/s}}{1.05}
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$f_{s_{test}}=\cfrac{R_{p0.2/T_{test}/s}}{1.05}$$
$f_{s_{test}}=\cfrac{R_{p0.2/T_{test}/s}}{1.05}$
Equations in LaTeX code
f_{s_{test}}=\cfrac{R_{p0.2/T_{test}/s}}{1.05}

Lower of the nominal design stresses $ f_d $ of the end and $ f_s $ of the shell for normal operating load cases

$$f_{min}=\min\left\{f_d, f_s\right\}$$
$f_{min}=\min\left\{f_d, f_s\right\}$
Equations in LaTeX code
f_{min}=\min\left\{f_d, f_s\right\}

Lower of the nominal design stresses $ f_{test} $ of the end and $ f_{s_{test}} $ of the shell for testing load cases

$$f_{min_{test}}=\min\left\{f_{test}, f_{s_{test}}\right\}$$
$f_{min_{test}}=\min\left\{f_{test}, f_{s_{test}}\right\}$
Equations in LaTeX code
f_{min_{test}}=\min\left\{f_{test}, f_{s_{test}}\right\}

Length of cylindrical shell

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

Factor $ B_1 $

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

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

Factor $ g $

$$g=\cfrac{D_i}{D_i+e_s}$$
$g=\cfrac{D_i}{D_i+e_s}$
Equations in LaTeX code
g=\cfrac{D_i}{D_i+e_s}

Factor $ H $

$$H=\sqrt[4]{12\cdot\left(1-ν^2\right)}\cdot\sqrt{\cfrac{e_s}{D_i+e_s}}$$
$H=\sqrt[4]{12\cdot\left(1-ν^2\right)}\cdot\sqrt{\cfrac{e_s}{D_i+e_s}}$
Equations in LaTeX code
H=\sqrt[4]{12\cdot\left(1-ν^2\right)}\cdot\sqrt{\cfrac{e_s}{D_i+e_s}}

Factor $ J $

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

Factor $ U $

$$U=\cfrac{2\cdot\left(2-ν\cdot g\right)}{\sqrt{3\cdot\left(1-ν^2\right)}}$$
$U=\cfrac{2\cdot\left(2-ν\cdot g\right)}{\sqrt{3\cdot\left(1-ν^2\right)}}$
Equations in LaTeX code
U=\cfrac{2\cdot\left(2-ν\cdot g\right)}{\sqrt{3\cdot\left(1-ν^2\right)}}

Factor $ f_1 $

$$f_1=2\cdot g^2-g^4$$
$f_1=2\cdot g^2-g^4$
Equations in LaTeX code
f_1=2\cdot g^2-g^4

Factor $ A $

$$A=\left(\cfrac{3}{4}\cdot\cfrac{U\cdot D_i}{e_s}-2\cdot J\right)\cdot\left(1+ν\right)\cdot\left[1+\left(1-ν\right)\cdot\cfrac{e_s}{D_i+e_s}\right]$$
$A=\left(\cfrac{3}{4}\cdot\cfrac{U\cdot D_i}{e_s}-2\cdot J\right)\cdot\left(1+ν\right)\cdot\left[1+\left(1-ν\right)\cdot\cfrac{e_s}{D_i+e_s}\right]$
Equations in LaTeX code
A=\left(\cfrac{3}{4}\cdot\cfrac{U\cdot D_i}{e_s}-2\cdot J\right)\cdot\left(1+ν\right)\cdot\left[1+\left(1-ν\right)\cdot\cfrac{e_s}{D_i+e_s}\right]

Factor $ B $

$$B=\left[\left(\cfrac{3}{8}\cdot\cfrac{U\cdot D_i}{e_s}-J\right)\cdot H^2-\cfrac{3}{2}\cdot\left(2-ν\cdot g\right)\cdot g\right]\cdot H$$
$B=\left[\left(\cfrac{3}{8}\cdot\cfrac{U\cdot D_i}{e_s}-J\right)\cdot H^2-\cfrac{3}{2}\cdot\left(2-ν\cdot g\right)\cdot g\right]\cdot H$
Equations in LaTeX code
B=\left[\left(\cfrac{3}{8}\cdot\cfrac{U\cdot D_i}{e_s}-J\right)\cdot H^2-\cfrac{3}{2}\cdot\left(2-ν\cdot g\right)\cdot g\right]\cdot H

Factor $ F $

$$F=\left(\cfrac{3}{8}\cdot U\cdot g+\cfrac{3}{16}\cdot f_1\cdot\cfrac{D_i+e_s}{e_s}-2\cdot J\cdot\cfrac{e_s}{D_i+e_s}\right)\cdot H^2-3\cdot\left(2-ν\cdot g\right)\cdot g\cdot\cfrac{e_s}{D_i+e_s}$$
$F=\left(\cfrac{3}{8}\cdot U\cdot g+\cfrac{3}{16}\cdot f_1\cdot\cfrac{D_i+e_s}{e_s}-2\cdot J\cdot\cfrac{e_s}{D_i+e_s}\right)\cdot H^2-3\cdot\left(2-ν\cdot g\right)\cdot g\cdot\cfrac{e_s}{D_i+e_s}$
Equations in LaTeX code
F=\left(\cfrac{3}{8}\cdot U\cdot g+\cfrac{3}{16}\cdot f_1\cdot\cfrac{D_i+e_s}{e_s}-2\cdot J\cdot\cfrac{e_s}{D_i+e_s}\right)\cdot H^2-3\cdot\left(2-ν\cdot g\right)\cdot g\cdot\cfrac{e_s}{D_i+e_s}

Factor $ G $

$$G=\left[\cfrac{3}{8}\cdot f_1-2\cdot J\cdot\left(\cfrac{e_s}{D_i+e_s}\right)^2\right]\cdot H$$
$G=\left[\cfrac{3}{8}\cdot f_1-2\cdot J\cdot\left(\cfrac{e_s}{D_i+e_s}\right)^2\right]\cdot H$
Equations in LaTeX code
G=\left[\cfrac{3}{8}\cdot f_1-2\cdot J\cdot\left(\cfrac{e_s}{D_i+e_s}\right)^2\right]\cdot H

Factor $ a $

$$a=\cfrac{B}{A}$$
$a=\cfrac{B}{A}$
Equations in LaTeX code
a=\cfrac{B}{A}

Factor $ b $

$$b=\cfrac{F}{A}$$
$b=\cfrac{F}{A}$
Equations in LaTeX code
b=\cfrac{F}{A}

Factor $ c $

$$c=\cfrac{G}{A}$$
$c=\cfrac{G}{A}$
Equations in LaTeX code
c=\cfrac{G}{A}

Factor $ N $

$$N=\cfrac{b}{3}-\cfrac{a^2}{9}$$
$N=\cfrac{b}{3}-\cfrac{a^2}{9}$
Equations in LaTeX code
N=\cfrac{b}{3}-\cfrac{a^2}{9}

Factor $ Q $

$$Q=\cfrac{c}{2}-\cfrac{a\cdot b}{6}+\cfrac{a^3}{27}$$
$Q=\cfrac{c}{2}-\cfrac{a\cdot b}{6}+\cfrac{a^3}{27}$
Equations in LaTeX code
Q=\cfrac{c}{2}-\cfrac{a\cdot b}{6}+\cfrac{a^3}{27}

Factor $ K $

$$K=\cfrac{N^3}{Q^2}$$
$K=\cfrac{N^3}{Q^2}$
Equations in LaTeX code
K=\cfrac{N^3}{Q^2}

Factor $ S $

$\text{if }\ Q\geq 0$
$\text{if }\ Q\geq 0$
Equations in LaTeX code
\text{if }\ Q\geq 0
$$S=\sqrt[3]{Q\cdot\left[1+\left(1+K\right)^{1/2}\right]}$$
$S=\sqrt[3]{Q\cdot\left[1+\left(1+K\right)^{1/2}\right]}$
Equations in LaTeX code
S=\sqrt[3]{Q\cdot\left[1+\left(1+K\right)^{1/2}\right]}
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$S=-\sqrt[3]{\left|Q\right|\cdot\left[1+\left(1+K\right)^{1/2}\right]}$$
$S=-\sqrt[3]{\left|Q\right|\cdot\left[1+\left(1+K\right)^{1/2}\right]}$
Equations in LaTeX code
S=-\sqrt[3]{\left|Q\right|\cdot\left[1+\left(1+K\right)^{1/2}\right]}

Shape factor for calculation of circular flat ends

$$C_2=\cfrac{\left(D_i+e_s\right)\cdot\left(\cfrac{N}{S}-S-\cfrac{a}{3}\right)}{D_i\cdot\sqrt{\cfrac{P}{f_{min}}}}$$
$C_2=\cfrac{\left(D_i+e_s\right)\cdot\left(\cfrac{N}{S}-S-\cfrac{a}{3}\right)}{D_i\cdot\sqrt{\cfrac{P}{f_{min}}}}$
Equations in LaTeX code
C_2=\cfrac{\left(D_i+e_s\right)\cdot\left(\cfrac{N}{S}-S-\cfrac{a}{3}\right)}{D_i\cdot\sqrt{\cfrac{P}{f_{min}}}}

Minimum required thickness for a flat end with a hub

$\text{if }\ C_2\geq 0.3$
$\text{if }\ C_2\geq 0.3$
Equations in LaTeX code
\text{if }\ C_2\geq 0.3
$$e=\max\left\{\left(C_1\cdot D_i\cdot\sqrt{\cfrac{P}{f_d}}\right), \left(C_2\cdot D_i\cdot\sqrt{\cfrac{P}{f_{min}}}\right)\right\}$$
$e=\max\left\{\left(C_1\cdot D_i\cdot\sqrt{\cfrac{P}{f_d}}\right), \left(C_2\cdot D_i\cdot\sqrt{\cfrac{P}{f_{min}}}\right)\right\}$
Equations in LaTeX code
e=\max\left\{\left(C_1\cdot D_i\cdot\sqrt{\cfrac{P}{f_d}}\right), \left(C_2\cdot D_i\cdot\sqrt{\cfrac{P}{f_{min}}}\right)\right\}
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$e=C_2\cdot D_i\cdot\sqrt{\cfrac{P}{f_{min}}}$$
$e=C_2\cdot D_i\cdot\sqrt{\cfrac{P}{f_{min}}}$
Equations in LaTeX code
e=C_2\cdot D_i\cdot\sqrt{\cfrac{P}{f_{min}}}

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

$$e_{test}=C_{1_{test}}\cdot D_i\cdot\sqrt{\cfrac{P_{test}}{f_{test}}}$$
$e_{test}=C_{1_{test}}\cdot D_i\cdot\sqrt{\cfrac{P_{test}}{f_{test}}}$
Equations in LaTeX code
e_{test}=C_{1_{test}}\cdot D_i\cdot\sqrt{\cfrac{P_{test}}{f_{test}}}

Requirements

$$ \cfrac{P}{f_{min}}\le 0.1 $$
$\cfrac{P}{f_{min}}\le 0.1$
Equations in LaTeX code
\cfrac{P}{f_{min}}\le 0.1
$$ \cfrac{P_{test}}{f_{min_{test}}}\le 0.1 $$
$\cfrac{P_{test}}{f_{min_{test}}}\le 0.1$
Equations in LaTeX code
\cfrac{P_{test}}{f_{min_{test}}}\le 0.1
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