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Cylinder under internal pressure (P>0)

Values for calculation

$ \{e_a\in\mathbb{R}\mid e_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.
$ e_a $ $ \mathrm{mm} $
$ \{D\in\mathbb{R}\mid D>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 $ $ \mathrm{mm} $
$ \{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} $
$ \{F\in\mathbb{R}\} $
This set includes all real numbers.
$ F $ $ \mathrm{N} $
$ \{F_{test}\in\mathbb{R}\} $
This set includes all real numbers.
$ F_{test} $ $ \mathrm{N} $
$ \{M\in\mathbb{R}\mid M\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.
$ M $ $ \mathrm{Nm} $
$ \{M_{test}\in\mathbb{R}\mid M_{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.
$ M_{test} $ $ \mathrm{Nm} $
$ \{w/l\in\mathbb{R}\mid w/l\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.
$ w/l $
$ 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} $
$ E_T $ $ \mathrm{MPa} $
$ E_{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}

Stress calculated from the pressure

$$σ_P=\cfrac{P\cdot D}{2\cdot e_a}$$
$σ_P=\cfrac{P\cdot D}{2\cdot e_a}$
Equations in LaTeX code
σ_P=\cfrac{P\cdot D}{2\cdot e_a}

Stress calculated from the pressure for testing load cases

$$σ_{P_{test}}=\cfrac{P_{test}\cdot D}{2\cdot e_a}$$
$σ_{P_{test}}=\cfrac{P_{test}\cdot D}{2\cdot e_a}$
Equations in LaTeX code
σ_{P_{test}}=\cfrac{P_{test}\cdot D}{2\cdot e_a}

Maximum longitudinal stress (positive if tensile), taking account of all loads

$$σ_{max}=\cfrac{F\cdot D+4\cdot M}{π\cdot D^2\cdot e_a}$$
$σ_{max}=\cfrac{F\cdot D+4\cdot M}{π\cdot D^2\cdot e_a}$
Equations in LaTeX code
σ_{max}=\cfrac{F\cdot D+4\cdot M}{π\cdot D^2\cdot e_a}
$$f_d\geq σ_{max}$$
$f_d\geq σ_{max}$
Equations in LaTeX code
f_d\geq σ_{max}

Maximum longitudinal stress (positive if tensile), taking account of all loads for testing load cases

$$σ_{max_{test}}=\cfrac{F_{test}\cdot D+4\cdot M_{test}}{π\cdot D^2\cdot e_a}$$
$σ_{max_{test}}=\cfrac{F_{test}\cdot D+4\cdot M_{test}}{π\cdot D^2\cdot e_a}$
Equations in LaTeX code
σ_{max_{test}}=\cfrac{F_{test}\cdot D+4\cdot M_{test}}{π\cdot D^2\cdot e_a}
$$f_{test}\geq σ_{max_{test}}$$
$f_{test}\geq σ_{max_{test}}$
Equations in LaTeX code
f_{test}\geq σ_{max_{test}}

Minimum longitudinal stress (positive if tensile), taking account of all loads

$$σ_{min}=\cfrac{F\cdot D-4\cdot M}{π\cdot D^2\cdot e_a}$$
$σ_{min}=\cfrac{F\cdot D-4\cdot M}{π\cdot D^2\cdot e_a}$
Equations in LaTeX code
σ_{min}=\cfrac{F\cdot D-4\cdot M}{π\cdot D^2\cdot e_a}

Minimum longitudinal stress (positive if tensile), taking account of all loads for testing load cases

$$σ_{min_{test}}=\cfrac{F_{test}\cdot D-4\cdot M_{test}}{π\cdot D^2\cdot e_a}$$
$σ_{min_{test}}=\cfrac{F_{test}\cdot D-4\cdot M_{test}}{π\cdot D^2\cdot e_a}$
Equations in LaTeX code
σ_{min_{test}}=\cfrac{F_{test}\cdot D-4\cdot M_{test}}{π\cdot D^2\cdot e_a}

Maximum longitudinal compressive stress

$\text{if }\ σ_{min} < 0$
$\text{if }\ σ_{min} < 0$
Equations in LaTeX code
\text{if }\ σ_{min} < 0
$$σ_c=-σ_{min}$$
$σ_c=-σ_{min}$
Equations in LaTeX code
σ_c=-σ_{min}
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$σ_c=σ_{min}$$
$σ_c=σ_{min}$
Equations in LaTeX code
σ_c=σ_{min}

Maximum longitudinal compressive stress for testing load cases

$\text{if }\ σ_{min_{test}} < 0$
$\text{if }\ σ_{min_{test}} < 0$
Equations in LaTeX code
\text{if }\ σ_{min_{test}} < 0
$$σ_{c_{test}}=-σ_{min_{test}}$$
$σ_{c_{test}}=-σ_{min_{test}}$
Equations in LaTeX code
σ_{c_{test}}=-σ_{min_{test}}
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$σ_{c_{test}}=σ_{min_{test}}$$
$σ_{c_{test}}=σ_{min_{test}}$
Equations in LaTeX code
σ_{c_{test}}=σ_{min_{test}}

Nominal elastic limit for shell for normal operating load cases

$\text{if }\ \text{type }$$\text{of }$$\text{material}= \text{Austenitic steels}$
$\text{if }\ \text{type }$$\text{of }$$\text{material}= \text{Austenitic steels}$
Equations in LaTeX code
\text{if }\ \text{type }$$\text{of }$$\text{material}= \text{Austenitic steels}
$$σ_e=\cfrac{R_{p0.2/T}}{1.25}$$
$σ_e=\cfrac{R_{p0.2/T}}{1.25}$
Equations in LaTeX code
σ_e=\cfrac{R_{p0.2/T}}{1.25}
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$σ_e=R_{p0.2/T}$$
$σ_e=R_{p0.2/T}$
Equations in LaTeX code
σ_e=R_{p0.2/T}

Nominal elastic limit for shell for testing load cases

$\text{if }\ \text{type }$$\text{of }$$\text{material}= \text{Austenitic steels}$
$\text{if }\ \text{type }$$\text{of }$$\text{material}= \text{Austenitic steels}$
Equations in LaTeX code
\text{if }\ \text{type }$$\text{of }$$\text{material}= \text{Austenitic steels}
$$σ_{e_{test}}=\cfrac{R_{p0.2/T_{test}}}{1.25}$$
$σ_{e_{test}}=\cfrac{R_{p0.2/T_{test}}}{1.25}$
Equations in LaTeX code
σ_{e_{test}}=\cfrac{R_{p0.2/T_{test}}}{1.25}
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$σ_{e_{test}}=R_{p0.2/T_{test}}$$
$σ_{e_{test}}=R_{p0.2/T_{test}}$
Equations in LaTeX code
σ_{e_{test}}=R_{p0.2/T_{test}}

Factor $ K $

$$K=\cfrac{1.21\cdot E_T\cdot e_a}{σ_e\cdot D}$$
$K=\cfrac{1.21\cdot E_T\cdot e_a}{σ_e\cdot D}$
Equations in LaTeX code
K=\cfrac{1.21\cdot E_T\cdot e_a}{σ_e\cdot D}

Factor $ K_{test} $

$$K_{test}=\cfrac{1.21\cdot E_{T_{test}}\cdot e_a}{σ_{e_{test}}\cdot D}$$
$K_{test}=\cfrac{1.21\cdot E_{T_{test}}\cdot e_a}{σ_{e_{test}}\cdot D}$
Equations in LaTeX code
K_{test}=\cfrac{1.21\cdot E_{T_{test}}\cdot e_a}{σ_{e_{test}}\cdot D}

Elastic imperfection reduction factor

$\text{if }\ w/l\geq 0.01\wedge D/e_a\le 424$
$\text{if }\ w/l\geq 0.01\wedge D/e_a\le 424$
Equations in LaTeX code
\text{if }\ w/l\geq 0.01\wedge D/e_a\le 424
$$α=\cfrac{0.83}{\sqrt{1+0.005\cdot D/e_a}}\cdot\left(1.5-50\cdot w/l\right)$$
$α=\cfrac{0.83}{\sqrt{1+0.005\cdot D/e_a}}\cdot\left(1.5-50\cdot w/l\right)$
Equations in LaTeX code
α=\cfrac{0.83}{\sqrt{1+0.005\cdot D/e_a}}\cdot\left(1.5-50\cdot w/l\right)
$\text{else if }\ w/l\geq 0.01\wedge D/e_a> 424$
$\text{else if }\ w/l\geq 0.01\wedge D/e_a> 424$
Equations in LaTeX code
\text{else if }\ w/l\geq 0.01\wedge D/e_a> 424
$$α=\cfrac{0.7}{\sqrt{1+0.005\cdot D/e_a}}\cdot\left(1.5-50\cdot w/l\right)$$
$α=\cfrac{0.7}{\sqrt{1+0.005\cdot D/e_a}}\cdot\left(1.5-50\cdot w/l\right)$
Equations in LaTeX code
α=\cfrac{0.7}{\sqrt{1+0.005\cdot D/e_a}}\cdot\left(1.5-50\cdot w/l\right)
$\text{else if }\ D/e_a\le 424$
$\text{else if }\ D/e_a\le 424$
Equations in LaTeX code
\text{else if }\ D/e_a\le 424
$$α=\cfrac{0.83}{\sqrt{1+0.005\cdot D/e_a}}$$
$α=\cfrac{0.83}{\sqrt{1+0.005\cdot D/e_a}}$
Equations in LaTeX code
α=\cfrac{0.83}{\sqrt{1+0.005\cdot D/e_a}}
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$α=\cfrac{0.7}{\sqrt{1+0.005\cdot D/e_a}}$$
$α=\cfrac{0.7}{\sqrt{1+0.005\cdot D/e_a}}$
Equations in LaTeX code
α=\cfrac{0.7}{\sqrt{1+0.005\cdot D/e_a}}

Buckling reduction factor

$\text{if }\ α\cdot K< 0.5$
$\text{if }\ α\cdot K< 0.5$
Equations in LaTeX code
\text{if }\ α\cdot K< 0.5
$$Δ=\cfrac{0.75\cdot α\cdot K}{1.5}$$
$Δ=\cfrac{0.75\cdot α\cdot K}{1.5}$
Equations in LaTeX code
Δ=\cfrac{0.75\cdot α\cdot K}{1.5}
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$Δ=\cfrac{1-\cfrac{0.4123}{\left(α\cdot K\right)^{0.6}}}{1.5}$$
$Δ=\cfrac{1-\cfrac{0.4123}{\left(α\cdot K\right)^{0.6}}}{1.5}$
Equations in LaTeX code
Δ=\cfrac{1-\cfrac{0.4123}{\left(α\cdot K\right)^{0.6}}}{1.5}

Buckling reduction factor for testing load cases

$\text{if }\ α\cdot K_{test}< 0.5$
$\text{if }\ α\cdot K_{test}< 0.5$
Equations in LaTeX code
\text{if }\ α\cdot K_{test}< 0.5
$$Δ_{test}=\cfrac{0.75\cdot α\cdot K_{test}}{1.05}$$
$Δ_{test}=\cfrac{0.75\cdot α\cdot K_{test}}{1.05}$
Equations in LaTeX code
Δ_{test}=\cfrac{0.75\cdot α\cdot K_{test}}{1.05}
$\text{else}$
$\text{else}$
Equations in LaTeX code
\text{else}
$$Δ_{test}=\cfrac{1-\cfrac{0.4123}{\left(α\cdot K_{test}\right)^{0.6}}}{1.05}$$
$Δ_{test}=\cfrac{1-\cfrac{0.4123}{\left(α\cdot K_{test}\right)^{0.6}}}{1.05}$
Equations in LaTeX code
Δ_{test}=\cfrac{1-\cfrac{0.4123}{\left(α\cdot K_{test}\right)^{0.6}}}{1.05}

Maximum permitted compressive longitudinal stress

$$σ_{c,all}=σ_e\cdot Δ$$
$σ_{c,all}=σ_e\cdot Δ$
Equations in LaTeX code
σ_{c,all}=σ_e\cdot Δ
$$σ_c\le σ_{c,all}$$
$σ_c\le σ_{c,all}$
Equations in LaTeX code
σ_c\le σ_{c,all}

Maximum permitted compressive longitudinal stress for testing load cases

$$σ_{c,all_{test}}=σ_{e_{test}}\cdot Δ_{test}$$
$σ_{c,all_{test}}=σ_{e_{test}}\cdot Δ_{test}$
Equations in LaTeX code
σ_{c,all_{test}}=σ_{e_{test}}\cdot Δ_{test}
$$σ_{c_{test}}\le σ_{c,all_{test}}$$
$σ_{c_{test}}\le σ_{c,all_{test}}$
Equations in LaTeX code
σ_{c_{test}}\le σ_{c,all_{test}}

Requirements

$$ σ_P+σ_c\le f_d $$
$σ_P+σ_c\le f_d$
Equations in LaTeX code
σ_P+σ_c\le f_d
$$ σ_{P_{test}}+σ_{c_{test}}\le f_{test} $$
$σ_{P_{test}}+σ_{c_{test}}\le f_{test}$
Equations in LaTeX code
σ_{P_{test}}+σ_{c_{test}}\le f_{test}
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