# Unstiffened cylinders

## Values for calculation

$T$ $\mathrm{°C}$
$T_{test}$ $\mathrm{°C}$
$P$ $\mathrm{MPa}$
$P_{test}$ $\mathrm{MPa}$
$e_a$ $\mathrm{mm}$
$L$ $\mathrm{mm}$
$R$ $\mathrm{mm}$
$ν$
$R_{p0.2/T}$ $\mathrm{MPa}$
$R_{p0.2/T_{test}}$ $\mathrm{MPa}$
$R_{p1.0/T}$ $\mathrm{MPa}$
$R_{p1.0/T_{test}}$ $\mathrm{MPa}$
$E_T$ $\mathrm{MPa}$
$E_{T_{test}}$ $\mathrm{MPa}$

## Calculation

### 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}$
$$σ_{e_{test}}=\cfrac{R_{p0.2/T_{test}}}{1.25}$$
$\text{else}$
$$σ_{e_{test}}=R_{p0.2/T_{test}}$$

### Pressure at which mean circumferential stress in cylindrical or conical shell midway between stiffeners, or in a spherical shell, reaches yield point for normal operating load cases

$$P_y=\cfrac{σ_e\cdot e_a}{R}$$

### Pressure at which mean circumferential stress in cylindrical or conical shell midway between stiffeners, or in a spherical shell, reaches yield point for testing load cases

$$P_{y_{test}}=\cfrac{σ_{e_{test}}\cdot e_a}{R}$$

### Parameter $Z$

$$Z=\cfrac{π\cdot R}{L}$$

### Number of circumferential waves for an unstiffened part of a cylinder

$$n_{cyl}=2\div 20$$

### Mean elastic circumferential strain at collapse

$$ε=\cfrac{1}{n_{cyl}^2-1+\cfrac{Z^2}{2}}\cdot\left\{\cfrac{1}{\left(\cfrac{n_{cyl}^2}{Z^2}+1\right)^2}+\cfrac{e_a^2}{12\cdot R^2\cdot\left(1-ν^2\right)}\cdot\left(n_{cyl}^2-1+Z^2\right)^2\right\}$$

### Theoretical elastic instability pressure for collapse of a perfect cylindrical, conical or spherical shell for normal operating load cases

$$P_m=\cfrac{E_T\cdot e_a\cdot ε}{R}$$

### Theoretical elastic instability pressure for collapse of a perfect cylindrical, conical or spherical shell for testing load cases

$$P_{m_{test}}=\cfrac{E_{T_{test}}\cdot e_a\cdot ε}{R}$$

### Radio $P_m/P_y$

$$P_m/P_y=\cfrac{P_m}{P_y}$$

### Radio $P_{m_{test}}/P_{y_{test}}$

$$P_{m_{test}}/P_{y_{test}}=\cfrac{P_{m_{test}}}{P_{y_{test}}}$$

### Radio $P_r/P_y$

$\text{if }\ P_m/P_y<0.25$
$$P_r/P_y=0+\cfrac{0.125-0}{0.25-0}\cdot\left(P_m/P_y-0\right)$$
$\text{else if }\ P_m/P_y<0.5$
$$P_r/P_y=0.125+\cfrac{0.251-0.125}{0.5-0.25}\cdot\left(P_m/P_y-0.25\right)$$
$\text{else if }\ P_m/P_y<0.75$
$$P_r/P_y=0.251+\cfrac{0.375-0.251}{0.75-0.5}\cdot\left(P_m/P_y-0.5\right)$$
$\text{else if }\ P_m/P_y<1$
$$P_r/P_y=0.375+\cfrac{0.5-0.375}{1-0.75}\cdot\left(P_m/P_y-0.75\right)$$
$\text{else if }\ P_m/P_y<1.25$
$$P_r/P_y=0.5+\cfrac{0.605-0.5}{1.25-1}\cdot\left(P_m/P_y-1\right)$$
$\text{else if }\ P_m/P_y<1.5$
$$P_r/P_y=0.605+\cfrac{0.68-0.605}{1.5-1.25}\cdot\left(P_m/P_y-1.25\right)$$
$\text{else if }\ P_m/P_y<1.75$
$$P_r/P_y=0.68+\cfrac{0.72-0.68}{1.75-1.5}\cdot\left(P_m/P_y-1.5\right)$$
$\text{else if }\ P_m/P_y<2$
$$P_r/P_y=0.72+\cfrac{0.755-0.72}{2-1.75}\cdot\left(P_m/P_y-1.75\right)$$
$\text{else if }\ P_m/P_y<2.25$
$$P_r/P_y=0.755+\cfrac{0.78-0.755}{2.25-2}\cdot\left(P_m/P_y-2\right)$$
$\text{else if }\ P_m/P_y<2.5$
$$P_r/P_y=0.78+\cfrac{0.803-0.78}{2.5-2.25}\cdot\left(P_m/P_y-2.25\right)$$
$\text{else if }\ P_m/P_y<2.75$
$$P_r/P_y=0.803+\cfrac{0.822-0.803}{2.75-2.5}\cdot\left(P_m/P_y-2.5\right)$$
$\text{else if }\ P_m/P_y<3$
$$P_r/P_y=0.822+\cfrac{0.836-0.822}{3-2.75}\cdot\left(P_m/P_y-2.75\right)$$
$\text{else if }\ P_m/P_y<3.25$
$$P_r/P_y=0.836+\cfrac{0.849-0.836}{3.25-3}\cdot\left(P_m/P_y-3\right)$$
$\text{else if }\ P_m/P_y<3.5$
$$P_r/P_y=0.849+\cfrac{0.861-0.849}{3.5-3.25}\cdot\left(P_m/P_y-3.25\right)$$
$\text{else if }\ P_m/P_y<3.75$
$$P_r/P_y=0.861+\cfrac{0.87-0.861}{3.75-3.5}\cdot\left(P_m/P_y-3.5\right)$$
$\text{else if }\ P_m/P_y<4$
$$P_r/P_y=0.87+\cfrac{0.879-0.87}{4-3.75}\cdot\left(P_m/P_y-3.75\right)$$
$\text{else if }\ P_m/P_y<4.25$
$$P_r/P_y=0.879+\cfrac{0.887-0.879}{4.25-4}\cdot\left(P_m/P_y-4\right)$$
$\text{else if }\ P_m/P_y<4.5$
$$P_r/P_y=0.887+\cfrac{0.896-0.887}{4.5-4.25}\cdot\left(P_m/P_y-4.25\right)$$
$\text{else if }\ P_m/P_y<4.75$
$$P_r/P_y=0.896+\cfrac{0.905-0.896}{4.75-4.5}\cdot\left(P_m/P_y-4.5\right)$$
$\text{else if }\ P_m/P_y<5$
$$P_r/P_y=0.905+\cfrac{0.914-0.905}{5-4.75}\cdot\left(P_m/P_y-4.75\right)$$
$\text{else if }\ P_m/P_y<5.25$
$$P_r/P_y=0.914+\cfrac{0.917-0.914}{5.25-5}\cdot\left(P_m/P_y-5\right)$$
$\text{else if }\ P_m/P_y<5.5$
$$P_r/P_y=0.917+\cfrac{0.923-0.917}{5.5-5.25}\cdot\left(P_m/P_y-5.25\right)$$
$\text{else if }\ P_m/P_y<5.75$
$$P_r/P_y=0.923+\cfrac{0.929-0.923}{5.75-5.5}\cdot\left(P_m/P_y-5.5\right)$$
$\text{else if }\ P_m/P_y<6$
$$P_r/P_y=0.929+\cfrac{0.935-0.929}{6-5.75}\cdot\left(P_m/P_y-5.75\right)$$
$\text{else if }\ P_m/P_y<6.25$
$$P_r/P_y=0.935+\cfrac{0.941-0.935}{6.25-6}\cdot\left(P_m/P_y-6\right)$$
$\text{else if }\ P_m/P_y<6.5$
$$P_r/P_y=0.941+\cfrac{0.947-0.941}{6.5-6.25}\cdot\left(P_m/P_y-6.25\right)$$
$\text{else if }\ P_m/P_y<6.75$
$$P_r/P_y=0.947+\cfrac{0.953-0.947}{6.75-6.5}\cdot\left(P_m/P_y-6.5\right)$$
$\text{else if }\ P_m/P_y<7$
$$P_r/P_y=0.953+\cfrac{0.959-0.953}{7-6.75}\cdot\left(P_m/P_y-6.75\right)$$
$\text{else}$
$$P_r/P_y=0.959$$

$$P< \cfrac{P_r/P_y\cdot P_y}{1.5}$$

### Radio $P_{r_{test}}/P_{y_{test}}$

$\text{if }\ P_{m_{test}}/P_{y_{test}}<0.25$
$$P_{r_{test}}/P_{y_{test}}=0+\cfrac{0.125-0}{0.25-0}\cdot\left(P_{m_{test}}/P_{y_{test}}-0\right)$$
$\text{else if }\ P_{m_{test}}/P_{y_{test}}<0.5$
$$P_{r_{test}}/P_{y_{test}}=0.125+\cfrac{0.251-0.125}{0.5-0.25}\cdot\left(P_{m_{test}}/P_{y_{test}}-0.25\right)$$
$\text{else if }\ P_{m_{test}}/P_{y_{test}}<0.75$
$$P_{r_{test}}/P_{y_{test}}=0.251+\cfrac{0.375-0.251}{0.75-0.5}\cdot\left(P_{m_{test}}/P_{y_{test}}-0.5\right)$$
$\text{else if }\ P_{m_{test}}/P_{y_{test}}<1$
$$P_{r_{test}}/P_{y_{test}}=0.375+\cfrac{0.5-0.375}{1-0.75}\cdot\left(P_{m_{test}}/P_{y_{test}}-0.75\right)$$
$\text{else if }\ P_{m_{test}}/P_{y_{test}}<1.25$
$$P_{r_{test}}/P_{y_{test}}=0.5+\cfrac{0.605-0.5}{1.25-1}\cdot\left(P_{m_{test}}/P_{y_{test}}-1\right)$$
$\text{else if }\ P_{m_{test}}/P_{y_{test}}<1.5$
$$P_{r_{test}}/P_{y_{test}}=0.605+\cfrac{0.68-0.605}{1.5-1.25}\cdot\left(P_{m_{test}}/P_{y_{test}}-1.25\right)$$
$\text{else if }\ P_{m_{test}}/P_{y_{test}}<1.75$
$$P_{r_{test}}/P_{y_{test}}=0.68+\cfrac{0.72-0.68}{1.75-1.5}\cdot\left(P_{m_{test}}/P_{y_{test}}-1.5\right)$$
$\text{else if }\ P_{m_{test}}/P_{y_{test}}<2$
$$P_{r_{test}}/P_{y_{test}}=0.72+\cfrac{0.755-0.72}{2-1.75}\cdot\left(P_{m_{test}}/P_{y_{test}}-1.75\right)$$
$\text{else if }\ P_{m_{test}}/P_{y_{test}}<2.25$
$$P_{r_{test}}/P_{y_{test}}=0.755+\cfrac{0.78-0.755}{2.25-2}\cdot\left(P_{m_{test}}/P_{y_{test}}-2\right)$$
$\text{else if }\ P_{m_{test}}/P_{y_{test}}<2.5$
$$P_{r_{test}}/P_{y_{test}}=0.78+\cfrac{0.803-0.78}{2.5-2.25}\cdot\left(P_{m_{test}}/P_{y_{test}}-2.25\right)$$
$\text{else if }\ P_{m_{test}}/P_{y_{test}}<2.75$
$$P_{r_{test}}/P_{y_{test}}=0.803+\cfrac{0.822-0.803}{2.75-2.5}\cdot\left(P_{m_{test}}/P_{y_{test}}-2.5\right)$$
$\text{else if }\ P_{m_{test}}/P_{y_{test}}<3$
$$P_{r_{test}}/P_{y_{test}}=0.822+\cfrac{0.836-0.822}{3-2.75}\cdot\left(P_{m_{test}}/P_{y_{test}}-2.75\right)$$
$\text{else if }\ P_{m_{test}}/P_{y_{test}}<3.25$
$$P_{r_{test}}/P_{y_{test}}=0.836+\cfrac{0.849-0.836}{3.25-3}\cdot\left(P_{m_{test}}/P_{y_{test}}-3\right)$$
$\text{else if }\ P_{m_{test}}/P_{y_{test}}<3.5$
$$P_{r_{test}}/P_{y_{test}}=0.849+\cfrac{0.861-0.849}{3.5-3.25}\cdot\left(P_{m_{test}}/P_{y_{test}}-3.25\right)$$
$\text{else if }\ P_{m_{test}}/P_{y_{test}}<3.75$
$$P_{r_{test}}/P_{y_{test}}=0.861+\cfrac{0.87-0.861}{3.75-3.5}\cdot\left(P_{m_{test}}/P_{y_{test}}-3.5\right)$$
$\text{else if }\ P_{m_{test}}/P_{y_{test}}<4$
$$P_{r_{test}}/P_{y_{test}}=0.87+\cfrac{0.879-0.87}{4-3.75}\cdot\left(P_{m_{test}}/P_{y_{test}}-3.75\right)$$
$\text{else if }\ P_{m_{test}}/P_{y_{test}}<4.25$
$$P_{r_{test}}/P_{y_{test}}=0.879+\cfrac{0.887-0.879}{4.25-4}\cdot\left(P_{m_{test}}/P_{y_{test}}-4\right)$$
$\text{else if }\ P_{m_{test}}/P_{y_{test}}<4.5$
$$P_{r_{test}}/P_{y_{test}}=0.887+\cfrac{0.896-0.887}{4.5-4.25}\cdot\left(P_{m_{test}}/P_{y_{test}}-4.25\right)$$
$\text{else if }\ P_{m_{test}}/P_{y_{test}}<4.75$
$$P_{r_{test}}/P_{y_{test}}=0.896+\cfrac{0.905-0.896}{4.75-4.5}\cdot\left(P_{m_{test}}/P_{y_{test}}-4.5\right)$$
$\text{else if }\ P_{m_{test}}/P_{y_{test}}<5$
$$P_{r_{test}}/P_{y_{test}}=0.905+\cfrac{0.914-0.905}{5-4.75}\cdot\left(P_{m_{test}}/P_{y_{test}}-4.75\right)$$
$\text{else if }\ P_{m_{test}}/P_{y_{test}}<5.25$
$$P_{r_{test}}/P_{y_{test}}=0.914+\cfrac{0.917-0.914}{5.25-5}\cdot\left(P_{m_{test}}/P_{y_{test}}-5\right)$$
$\text{else if }\ P_{m_{test}}/P_{y_{test}}<5.5$
$$P_{r_{test}}/P_{y_{test}}=0.917+\cfrac{0.923-0.917}{5.5-5.25}\cdot\left(P_{m_{test}}/P_{y_{test}}-5.25\right)$$
$\text{else if }\ P_{m_{test}}/P_{y_{test}}<5.75$
$$P_{r_{test}}/P_{y_{test}}=0.923+\cfrac{0.929-0.923}{5.75-5.5}\cdot\left(P_{m_{test}}/P_{y_{test}}-5.5\right)$$
$\text{else if }\ P_{m_{test}}/P_{y_{test}}<6$
$$P_{r_{test}}/P_{y_{test}}=0.929+\cfrac{0.935-0.929}{6-5.75}\cdot\left(P_{m_{test}}/P_{y_{test}}-5.75\right)$$
$\text{else if }\ P_{m_{test}}/P_{y_{test}}<6.25$
$$P_{r_{test}}/P_{y_{test}}=0.935+\cfrac{0.941-0.935}{6.25-6}\cdot\left(P_{m_{test}}/P_{y_{test}}-6\right)$$
$\text{else if }\ P_{m_{test}}/P_{y_{test}}<6.5$
$$P_{r_{test}}/P_{y_{test}}=0.941+\cfrac{0.947-0.941}{6.5-6.25}\cdot\left(P_{m_{test}}/P_{y_{test}}-6.25\right)$$
$\text{else if }\ P_{m_{test}}/P_{y_{test}}<6.75$
$$P_{r_{test}}/P_{y_{test}}=0.947+\cfrac{0.953-0.947}{6.75-6.5}\cdot\left(P_{m_{test}}/P_{y_{test}}-6.5\right)$$
$\text{else if }\ P_{m_{test}}/P_{y_{test}}<7$
$$P_{r_{test}}/P_{y_{test}}=0.953+\cfrac{0.959-0.953}{7-6.75}\cdot\left(P_{m_{test}}/P_{y_{test}}-6.75\right)$$
$\text{else}$
$$P_{r_{test}}/P_{y_{test}}=0.959$$

$$P_{test}< \cfrac{P_{r_{test}}/P_{y_{test}}\cdot P_{y_{test}}}{1.1}$$