# Ellipsoidal ends

## Values for calculation

$T$ $\mathrm{°C}$
$T_{test}$ $\mathrm{°C}$
$P$ $\mathrm{MPa}$
$P_{test}$ $\mathrm{MPa}$
$D_i$ $\mathrm{mm}$
$D_e$ $\mathrm{mm}$
$h_i$ $\mathrm{mm}$
$z$
$R_{p0.2/T}$ $\mathrm{MPa}$
$R_{p0.2/T_{test}}$ $\mathrm{MPa}$
$R_{p1.0/T}$ $\mathrm{MPa}$
$R_{p1.0/T_{test}}$ $\mathrm{MPa}$
$R_{m/20}$ $\mathrm{MPa}$
$R_{m/T}$ $\mathrm{MPa}$
$R_{m/T_{test}}$ $\mathrm{MPa}$

## 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}$
$$f_d=\min\left(\cfrac{R_{p0.2/T}}{1.9}, \cfrac{R_{m/20}}{3}\right)$$
$\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$
$$f_{test}=\cfrac{R_{p1.0/T_{test}}}{1.05}$$
$\text{else}$
$$f_{test}=\cfrac{R_{p0.2/T_{test}}}{1.05}$$

### Design stress for buckling formula for normal operating load cases

$\text{if }\ \text{type }$$\text{of }$$\text{material}= \text{Austenitic steels}\wedge\text{production }$$\text{according }$$\text{to }$$\text{temperature}=\text{cold}\wedge\text{seamless }$$\text{product}=\text{yes}$
$$f_b=\cfrac{1.6\cdot R_{p1.0/T}}{1.5}$$
$\text{else}$
$$f_b=\cfrac{R_{p0.2/T}}{1.5}$$

### Design stress for buckling formula for testing load cases

$\text{if }\ \text{type }$$\text{of }$$\text{material}= \text{Austenitic steels}\wedge\text{production }$$\text{according }$$\text{to }$$\text{temperature}=\text{cold}\wedge\text{seamless }$$\text{product}=\text{yes}$
$$f_{b_{test}}=\cfrac{1.6\cdot R_{p1.0/T}}{1.05}$$
$\text{else}$
$$f_{b_{test}}=\cfrac{R_{p0.2/T}}{1.05}$$

### Shape factor for an ellipsoidal end

$$K=D_i/\left(2\cdot h_i\right)$$

### Inside radius of curvature of a knuckle

$$r=D_i\cdot\left(\left(0.5/K\right)-0.08\right)$$

### Inside spherical radius of central part of torispherical end

$$R=D_i\cdot\left(0.44\cdot K+0.02\right)$$

### Required thickness of end to limit membrane stress in central part

$$e_s=\cfrac{P\cdot R}{2\cdot f_d\cdot z-0.5\cdot P}$$

### Required thickness of end to limit membrane stress in central part for testing load cases

$$e_{s_{test}}=\cfrac{P_{test}\cdot R}{2\cdot f_{test}-0.5\cdot P_{test}}$$

### Required thickness of knuckle to avoid plastic buckling

$$e_b=\left(0.75\cdot R+0.2\cdot D_i\right)\cdot\left[\cfrac{P}{111\cdot f_b}\cdot\left(\cfrac{D_i}{r}\right)^{0.825}\right]^\left(\cfrac{1}{1.5}\right)$$

### Required thickness of knuckle to avoid plastic buckling for testing load cases

$$e_{b_{test}}=\left(0.75\cdot R+0.2\cdot D_i\right)\cdot\left[\cfrac{P_{test}}{111\cdot f_{b_{test}}}\cdot\left(\cfrac{D_i}{r}\right)^{0.825}\right]^\left(\cfrac{1}{1.5}\right)$$

### Parameter $Y$

$$Y=\min\left(e/R, 0.04\right)$$

### Parameter $Z$

$$Z=\log_{10}\left(1/Y\right)$$

### Ratio of knuckle inside radius to shell inside diameter

$$X=r/D_i$$

### Parameter $N$

$$N=1.006-\cfrac{1}{6.2+\left(90\cdot Y\right)^4}$$

### Factor $β_{0.06}$

$$β_{0.06}=N\cdot\left(-0.3635\cdot Z^3+2.2124\cdot Z^2-3.2937\cdot Z+1.8873\right)$$

### Factor $β_{0.1}$

$$β_{0.1}=N\cdot\left(-0.1833\cdot Z^3+1.0383\cdot Z^2-1.2943\cdot Z+0.837\right)$$

### Factor $β_{0.2}$

$$β_{0.2}=\max\left\{0.95\cdot\left(0.56-1.94\cdot Y-82.5\cdot Y^2\right), 0.5\right\}$$

### Factor $β$

$\text{if }\ X= 0.06$
$$β=β_{0.06}$$
$\text{else if }\ 0.06 < X < 0.1$
$$β=25\cdot\left\{\left(0.1-X\right)\cdot β_{0.06}+\left(X-0.06\right)\cdot β_{0.1}\right\}$$
$\text{else if }\ X= 0.1$
$$β=β_{0.1}$$
$\text{else if }\ 0.1 < X < 0.2$
$$β=10\cdot\left\{\left(0.2-X\right)\cdot β_{0.1}+\left(X-0.1\right)\cdot β_{0.2}\right\}$$
$\text{else}$
$$β=β_{0.2}$$

### Required thickness of knuckle to avoid axisymmetric yielding

$$e_y=\cfrac{β\cdot P\cdot\left(0.75\cdot R+0.2\cdot D_i\right)}{f_d}$$

### Required thickness of knuckle to avoid axisymmetric yielding for testing load cases

$$e_{y_{test}}=\cfrac{β\cdot P_{test}\cdot\left(0.75\cdot R+0.2\cdot D_i\right)}{f_{test}}$$

### Required thickness

$$e=\max\left(e_s, e_{s_{test}}, e_y, e_{y_{test}}, e_b, e_{b_{test}}\right)$$

## Requirements

$$r \le 0.2\cdot D_i$$$$r \geq 0.06\cdot D_i$$$$r \geq 2\cdot e$$$$e \le 0.08\cdot D_e$$$$D_i/2 < R \le D_e$$$$1.7 < K < 2.2$$