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Bolted non-circular flat end with narrow-face gasket

Values for calculation

$T$ $\mathrm{°C}$
$T_{test}$ $\mathrm{°C}$
$T_{assembly}$ $\mathrm{°C}$
$P$ $\mathrm{MPa}$
$P_{test}$ $\mathrm{MPa}$
$a'$ $\mathrm{mm}$
$b'$ $\mathrm{mm}$
$W$ $\mathrm{N}$
$c$ $\mathrm{mm}$
$n$
$t_B$ $\mathrm{mm}$
$R_{p0.2/T}$ $\mathrm{MPa}$
$R_{p0.2/T_{test}}$ $\mathrm{MPa}$
$R_{p0.2/T_{assembly}}$ $\mathrm{MPa}$
$R_{p1.0/T}$ $\mathrm{MPa}$
$R_{p1.0/T_{test}}$ $\mathrm{MPa}$
$R_{p1.0/T_{assembly}}$ $\mathrm{MPa}$
$R_{m/20}$ $\mathrm{MPa}$
$R_{m/T}$ $\mathrm{MPa}$
$R_{m/T_{test}}$ $\mathrm{MPa}$
$R_{m/T_{assembly}}$ $\mathrm{MPa}$

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}

Maximum allowed value of the nominal design stress for assembly 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}
$$f_A=\min\left(\cfrac{R_{p0.2/T_{assembly}}}{1.9}, \cfrac{R_{m/20}}{3}\right)$$
$f_A=\min\left(\cfrac{R_{p0.2/T_{assembly}}}{1.9}, \cfrac{R_{m/20}}{3}\right)$
Equations in LaTeX code
f_A=\min\left(\cfrac{R_{p0.2/T_{assembly}}}{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_A=\max\left[\cfrac{R_{p1.0/T_{assembly}}}{1.5}, \min\left(\cfrac{R_{p1.0/T_{assembly}}}{1.2}, \cfrac{R_{m/T_{assembly}}}{3}\right)\right]$$
$f_A=\max\left[\cfrac{R_{p1.0/T_{assembly}}}{1.5}, \min\left(\cfrac{R_{p1.0/T_{assembly}}}{1.2}, \cfrac{R_{m/T_{assembly}}}{3}\right)\right]$
Equations in LaTeX code
f_A=\max\left[\cfrac{R_{p1.0/T_{assembly}}}{1.5}, \min\left(\cfrac{R_{p1.0/T_{assembly}}}{1.2}, \cfrac{R_{m/T_{assembly}}}{3}\right)\right]
$\text{else if }\ \text{type }$$\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
$$f_A=\cfrac{R_{p1.0/T_{assembly}}}{1.5}$$
$f_A=\cfrac{R_{p1.0/T_{assembly}}}{1.5}$
Equations in LaTeX code
f_A=\cfrac{R_{p1.0/T_{assembly}}}{1.5}
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$f_A=\min\left(\cfrac{R_{p0.2/T_{assembly}}}{1.5}, \cfrac{R_{m/20}}{2.4}\right)$$
$f_A=\min\left(\cfrac{R_{p0.2/T_{assembly}}}{1.5}, \cfrac{R_{m/20}}{2.4}\right)$
Equations in LaTeX code
f_A=\min\left(\cfrac{R_{p0.2/T_{assembly}}}{1.5}, \cfrac{R_{m/20}}{2.4}\right)

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

$\text{if }\ \cfrac{a'}{b'}\le 0.16$
$\text{if }\ \cfrac{a'}{b'}\le 0.16$
Equations in LaTeX code
\text{if }\ \cfrac{a'}{b'}\le 0.16
$$C_4=0.75$$
$C_4=0.75$
Equations in LaTeX code
C_4=0.75
$\text{else if }\ \cfrac{a'}{b'}< 0.363$
$\text{else if }\ \cfrac{a'}{b'}< 0.363$
Equations in LaTeX code
\text{else if }\ \cfrac{a'}{b'}< 0.363
$$C_4=0.75-0.44+\sqrt{0.44^2-\left(\cfrac{a'}{b'}-0.16\right)^2}$$
$C_4=0.75-0.44+\sqrt{0.44^2-\left(\cfrac{a'}{b'}-0.16\right)^2}$
Equations in LaTeX code
C_4=0.75-0.44+\sqrt{0.44^2-\left(\cfrac{a'}{b'}-0.16\right)^2}
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$C_4=-0.6117\cdot \cfrac{a'}{b'}+0.9224$$
$C_4=-0.6117\cdot \cfrac{a'}{b'}+0.9224$
Equations in LaTeX code
C_4=-0.6117\cdot \cfrac{a'}{b'}+0.9224

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

$$C_3=\sqrt{C_4+\cfrac{6\cdot W\cdot c}{P\cdot n\cdot t_B\cdot a'^2}}$$
$C_3=\sqrt{C_4+\cfrac{6\cdot W\cdot c}{P\cdot n\cdot t_B\cdot a'^2}}$
Equations in LaTeX code
C_3=\sqrt{C_4+\cfrac{6\cdot W\cdot c}{P\cdot n\cdot t_B\cdot a'^2}}

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)

Minimum thickness for the flanged extension

$$e_1=\sqrt{\cfrac{6\cdot W\cdot c}{n\cdot t_B\cdot\min\left(f_d, f_{test}, f_A\right)}}$$
$e_1=\sqrt{\cfrac{6\cdot W\cdot c}{n\cdot t_B\cdot\min\left(f_d, f_{test}, f_A\right)}}$
Equations in LaTeX code
e_1=\sqrt{\cfrac{6\cdot W\cdot c}{n\cdot t_B\cdot\min\left(f_d, f_{test}, f_A\right)}}
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