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Welded non-circular flat ends

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} $
$ \{a'\in\mathbb{R}\mid a'>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.
$ a' $ $ \mathrm{mm} $
$ \{b'\in\mathbb{R}\mid b'>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.
$ b' $ $ \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}

Shape factors for calculation of flat ends of non-circular shape

$\text{if }\ \cfrac{a'}{b'}\le 0.4588$
$\text{if }\ \cfrac{a'}{b'}\le 0.4588$
Equations in LaTeX code
\text{if }\ \cfrac{a'}{b'}\le 0.4588
$$C_3=0.706256$$
$C_3=0.706256$
Equations in LaTeX code
C_3=0.706256
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$C_3=-3.128\cdot\left(\cfrac{a'}{b'}\right)^6+8.6107\cdot\left(\cfrac{a'}{b'}\right)^5-7.9244\cdot\left(\cfrac{a'}{b'}\right)^4+2.4595\cdot\left(\cfrac{a'}{b'}\right)^3-0.4326\cdot\left(\cfrac{a'}{b'}\right)^2+0.3789\cdot\cfrac{a'}{b'}+0.5912$$
$C_3=-3.128\cdot\left(\cfrac{a'}{b'}\right)^6+8.6107\cdot\left(\cfrac{a'}{b'}\right)^5-7.9244\cdot\left(\cfrac{a'}{b'}\right)^4+2.4595\cdot\left(\cfrac{a'}{b'}\right)^3-0.4326\cdot\left(\cfrac{a'}{b'}\right)^2+0.3789\cdot\cfrac{a'}{b'}+0.5912$
Equations in LaTeX code
C_3=-3.128\cdot\left(\cfrac{a'}{b'}\right)^6+8.6107\cdot\left(\cfrac{a'}{b'}\right)^5-7.9244\cdot\left(\cfrac{a'}{b'}\right)^4+2.4595\cdot\left(\cfrac{a'}{b'}\right)^3-0.4326\cdot\left(\cfrac{a'}{b'}\right)^2+0.3789\cdot\cfrac{a'}{b'}+0.5912

Thickness of the flat end

$$e=\max\left(C_3\cdot a'\cdot\sqrt{\cfrac{P}{f_d}}, C_3\cdot a'\cdot\sqrt{\cfrac{P_{test}}{f_{test}}}\right)$$
$e=\max\left(C_3\cdot a'\cdot\sqrt{\cfrac{P}{f_d}}, C_3\cdot a'\cdot\sqrt{\cfrac{P_{test}}{f_{test}}}\right)$
Equations in LaTeX code
e=\max\left(C_3\cdot a'\cdot\sqrt{\cfrac{P}{f_d}}, C_3\cdot a'\cdot\sqrt{\cfrac{P_{test}}{f_{test}}}\right)
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