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Cylinder under internal pressure (P>0)

Values for calculation

$ e_a $ $ \mathrm{mm} $
$ D $ $ \mathrm{mm} $
$ T $ $ \mathrm{°C} $
$ T_{test} $ $ \mathrm{°C} $
$ P $ $ \mathrm{MPa} $
$ P_{test} $ $ \mathrm{MPa} $
$ F $ $ \mathrm{N} $
$ F_{test} $ $ \mathrm{N} $
$ M $ $ \mathrm{Nm} $
$ M_{test} $ $ \mathrm{Nm} $
$ 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}$
$$f_d=\min\left(\cfrac{R_{p0.2/T}}{1.9}, \cfrac{R_{m/20}}{3}\right)$$
$\text{else if }\ \text{type }$$\text{of }$$\text{material}= \text{Austenitic steels}\wedge\text{min. }$$\text{elongation }$$\text{after }$$\text{fracture}\geq 35$
$$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$
$$f_d=\cfrac{R_{p1.0/T}}{1.5}$$
$\text{else}$
$$f_d=\min\left(\cfrac{R_{p0.2/T}}{1.5}, \cfrac{R_{m/20}}{2.4}\right)$$

Maximum allowed value of the nominal design stress for testing load cases

$\text{if }\ \text{type }$$\text{of }$$\text{material}= \text{Cast steels}$
$$f_{test}=\cfrac{R_{p0.2/T_{test}}}{1.33}$$
$\text{else if }\ \text{type }$$\text{of }$$\text{material}= \text{Austenitic steels}\wedge\text{min. }$$\text{elongation }$$\text{after }$$\text{fracture}\geq 35$
$$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$
$$f_{test}=\cfrac{R_{p1.0/T_{test}}}{1.05}$$
$\text{else}$
$$f_{test}=\cfrac{R_{p0.2/T_{test}}}{1.05}$$

Stress calculated from the pressure

$$σ_P=\cfrac{P\cdot D}{2\cdot e_a}$$

Stress calculated from the pressure for testing load cases

$$σ_{P_{test}}=\cfrac{P_{test}\cdot D}{2\cdot e_a}$$

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

$$σ_{max}=\cfrac{F\cdot D+4\cdot M}{π\cdot D^2\cdot e_a}$$

$$f_d\geq σ_{max}$$

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

$$σ_{max_{test}}=\cfrac{F_{test}\cdot D+4\cdot M_{test}}{π\cdot D^2\cdot e_a}$$

$$f_{test}\geq σ_{max_{test}}$$

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

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

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

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

Maximum longitudinal compressive stress

$\text{if }\ σ_{min} < 0$
$$σ_c=-σ_{min}$$
$\text{else}$
$$σ_c=σ_{min}$$

Maximum longitudinal compressive stress for testing load cases

$\text{if }\ σ_{min_{test}} < 0$
$$σ_{c_{test}}=-σ_{min_{test}}$$
$\text{else}$
$$σ_{c_{test}}=σ_{min_{test}}$$

Nominal elastic limit for shell for normal operating load cases

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

Nominal elastic limit for shell for testing load cases

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

Factor $ K $

$$K=\cfrac{1.21\cdot E_T\cdot e_a}{σ_e\cdot D}$$

Factor $ K_{test} $

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

Elastic imperfection reduction factor

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

Buckling reduction factor

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

Buckling reduction factor for testing load cases

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

Maximum permitted compressive longitudinal stress

$$σ_{c,all}=σ_e\cdot Δ$$

$$σ_c\le σ_{c,all}$$

Maximum permitted compressive longitudinal stress for testing load cases

$$σ_{c,all_{test}}=σ_{e_{test}}\cdot Δ_{test}$$

$$σ_{c_{test}}\le σ_{c,all_{test}}$$

Requirements

$$ σ_P+σ_c\le f_d $$ $$ σ_{P_{test}}+σ_{c_{test}}\le f_{test} $$
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