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Spherical Shell and Hemispherical Head Layout

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While writing the article Shell & Head Blank Plate I had to research how to calculate the blank plate for every shell and head type. For me, the most difficult for both finding documentation and to calculate was the spherical shell. You can't create it with no welding (as it happens with some heads) or with only one welding (as it happens with the cylindrical shell). You have to break it in parts which is called crown and petals.

I really don't know why there isn't much documentation about spherical shell layout. I've found a Youtube video that helped me a little (it isn't very clear) and an app called Petal Layout: Sphere, Dish End (which I even didn't try to use as I don't have an Android phone)

Here I'll explain to you, step by step, with drawings, how creating the spherical shell layout can be pretty easy. By the way, I advice you to use AutoCAD for that, but you can do it in any other CAD software or even on paper, it'll just be a little more difficult.

Note (1):

It's not the intention of this article to explain about the fabrication process.

1. Spherical Shell and Hemispherical Head

This type of shell and head are mostly used in vessels with high pressure, as it gives the lowest thickness[ 1 ] (because there's a better stress distribution) compared to other types. It is basically composed by the crown (one for head and two for shell) and petals as shown in Fig. 1.1.

Fig. 1.1 - Hemispherical Head and Spherical Shell

It's up for the design engineer to decide the amount of petals that it'll have and the maximum crown size. It must be taken into consideration the available size of plates, equipment and the fabrication process.

The plate sizes available can be easily found in the internet[ 2 ] but the equipment and fabrication process must be discussed with the fabrication shop. Most Vendors are happy to help you with that and they usually have much more knowledge of the process than a design engineer.

2. Drawing Process

The drawing from Fig. 2.1 below will be used to explain the step by step process. As I explain what you should do, you can come back to this figure to have a better understanding.

Fig. 2.1 - Hemispherical Head Drawing

The drawing can be done using any CAD software or even a paper, it's up to you to decide what's best for you. I personally like AutoCAD because I'm very familiar with it as I use if since 2007.

Information you'll need before start:

D: mean diameter
C: mean crown diameter
Number of petals

Note (2):

mean diameter = inside diameter + plate thickness

Note (3):

the step by step below is for the hemispherical head, if you want to draw the spherical shell, just mirror the head and remove the bottom horizontal line.

Note (4):

many steps can be done without calculation, just using CAD software. It's up to you to choose what's best for you.

2.1. Step #1

Draw the semi-circle using the mean diameter (D).

2.2. Step #2

Draw the crown line using the mean crown diameter (C)

2.3. Step #3

Find the α angle by connecting a line from the intersection between C and the semi-circle to the center point.

2.4. Step #4

Decide how many parts you'll divide the α angle. The quantity will affect the precision of the petal drawing (Fig. 7.1). I would say that having an arc (b) with around 100 mm is enough. You can use the equation (1) to find the number of parts for a given arc size. Use equation (2) to find the β angle.

$$\begin{align}n = {\frac{\pi \cdot D \cdot \alpha}{360 \cdot b}}\tag{1}\end{align}$$ $$\begin{align}\beta = {\frac{\alpha}{n}}\tag{2}\end{align}$$

Now that you have the number of parts and the β angle, draw it in the semi-circle.

Note (5):

you can adjust the mean crown diameter to have rounded α and β angles.

2.5. Step #5

Draw a circle with the mean diameter (D) just below the semi-circle. Also draw the mean crown diameter (C) as well.

2.6. Step #6

Draw vertical lines (shown in red in the Fig. 2.1) connecting the intersection of part lines and semi-circle to the bottom circle horizontal center line.

Place numbers or letters to identify each point in the bottom circle. I'll help to draw the petals later on.

2.7. Step #7

Decide the number of petals you'll have. Take into consideration the available plate size, equipment and fabrication process. Don't feel under pressure to get it right in the first try. You can change it later if you want as it's quite simple to re-draw it.

The angle γ can be found using equation (3) where m is the number of petals. .

$$\begin{align}\gamma = {\frac{360}{m}}\tag{3}\end{align}$$

2.8. Step #8

Now we'll gonna draw the petal as shown in Fig. 8.2. The total height will be the arc b multiplied by the number of parts, as per equation (4).

$$\begin{align}ph = {b \cdot n}\tag{4}\end{align}$$

The numbers shown in the Fig. 8.1 is related to the numbers drawn in the bottom circle in the Fig. 2.1. The distance between each point is perimeter of the arc created between each number in the bottom circle.

The perimeters can be easily found using CAD software but we'll show the analytical way in here. You'll have to find the variable k from Fig. 8.1 below for each part, use equation (6) for that.

Fig. 8.1 - Finding variable k

First, find the angle θ using equation (5) where u is the number of the part. For example, in the Fig. 2.1, to find θ for points 1 and 6, the value of u would be 1.

$$\begin{align}\theta = {\alpha - u \cdot \beta}\tag{5}\end{align}$$

Now find the variable k using equation (6) where R is the head mean radius (basically D divided by 2). The subscript p and q is used for the points, for example: 1 and 6, 2 and 7, and so on.

$$\begin{align}k_{p \cdot q} = {cos(\theta) \cdot R}\tag{6}\end{align}$$

k is the radius of the part, as shown in Fig. 2.1. The perimeter them can be found using equation (7).

$$\begin{align}arc_{p \cdot q} = {\frac{\pi \cdot 2 \cdot k_{p \cdot q}}{360} \cdot \gamma}\tag{7}\end{align}$$

Now that you have all the perimeters or arc you can draw the petal as shown in Fig. 8.2. If you're drawing the spherical shell, just mirror it as shown in Fig. 8.3.

Fig. 8.2 - Hemispherical Head Petal

Fig. 8.3 - Spherical Shell Petal

2.9. Step #9

Finally, after creating all the points, you just need to connect each one with a straight line. That's why you need to use many points, because if you use just a few, you will not create the arc of each petal side. It's possible to create an equation for generate such curvature, but I think there's no need for such precision.

3. Conclusion

I know, it seems pretty complex because of the amount of text I wrote but if you use a CAD software, it get much easier and you'll take just a few minutes to draw it.

Keep in mind that the dimension of the petal shown in Fig. 8.2 and 8.3 is the blank plate (flattened) dimension. It'll later be bend to create the petal, which will them be welded to one another and to the crown.