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Shell and Head Blank Plate

When manufacturing a pressure vessel the engineer must know the specification (dimensions, material and quantity) of each of its parts. Such specification must be in the BOM (Bill of Materials) which is often in the technical drawing. For materials made of plates, the engineer must provide the blank (or flat) plate dimensions (length, width and thickness) so the manufacturer knows which plate to buy and how to cut it before forming.

Nowadays many CAD softwares have an option to flatten the 3D model but it can get tricky when you need more than one plate to create the shell or head (it often happens with spherical shells).

1. Types of Shell and Head

There are 2 types of shell and 6 types of heads:

  • Cylindrical Shell
  • Spherical Shell
  • Conical Head
  • Toriconical Head
  • Torispherical Head
  • Hemispherical Head
  • Ellipsoidal Head
  • Flat Head

The calculation when using only one plate is quite simples, but can get pretty complex when you need more plates, specially because in such case you also need to choose where the weldings will be placed.

2. Cylindrical Shell Blank Plate

Of all the shell and head types, the cylindrical shell is the easiest to calculate the blank plate. You basically need to find the perimeter, which will be the plate width, because the plate length is the shell height.

Fig. 1.1 basically show the transformation of the cylindrical shell into a flat plate.

head-003 Fig. 2.1 - Shell transformation into flat plate
$$\begin{align}bpw_{shell} = {\pi \cdot (D_i + t)}\tag{1}\end{align}$$ $$\begin{align}bpl_{shell} = {h}\tag{2}\end{align}$$


  • bpwshell = shell blank plate width
  • bplshell = shell blank plate length
  • Di = inside diameter
  • t = plate thickness
  • h = height
$$\begin{align}e_{cylinder} = {\left(\frac{50 \cdot \ t}{R_f}\right) \cdot \left(1-\frac{R_f}{R_o} \right)}\tag{3}\end{align}$$


  • e: calculated forming strain or extreme fiber elongation
  • Rf: final mean radius
  • Ro: original mean radius, equal to infinity for flat plate
  • t: nominal thickness of the plate

Example: if a cylinder have inside diameter of 1500 mm and plate thickness of 25 mm:

$$\begin{align}e_{cylinder} = {\left(\frac{50 \cdot \ 25}{762.5}\right) \cdot \left(1-\frac{762.5}{\infty} \right) = 1.639\%}\tag{4}\end{align}$$

Which means that the plate will reduce its thickness in 1.6%, from 25 to 24.59 mm.

3. Spherical Shell Blank Plate

Due to the complexity of spherical shells and the impossibility to create it with a single plate and/or a single weld, I've decided to create an article just for it: Spherical Shell and Hemispherical Head Layout

4. Conical Head Blank Plate

The conical head can be a little tricky because it doesn't form a square or rectangular plate but a rounded one with a cross section missing, in the case where it have a minor diameter, as shown in Fig. 4.1.

head-003 Fig. 4.1 - Conical head transformation into flat plate

First, find the blank plate major radius (R) using equation (5), then find the minor diameter (r) using equation (6) and, finally, the cut angle (θ) using equation (7).

$$\begin{align}R = {\frac{D/2}{sin(\beta)}}\tag{5}\end{align}$$ $$\begin{align}r = {R - \frac{h}{cos(\beta)}}\tag{6}\end{align}$$ $$\begin{align}\theta = {\frac{180 \cdot (2 \cdot R - D)}{R}}\tag{7}\end{align}$$

The blank plate width (bpw) will always be the same as R, but the plate length (bpl) will vary according to θ. If it's less than 180, use equation (8), if it's bigger than 180 and have a minor diameter, use equation (9) and, if it is greater than 180 but doesn't have a minor diameter, use equation (10).

$$\begin{align}bpl_{\theta < 180} = {R + R \cdot sin\left(\frac{180 - \theta}{2}\right)}\tag{8}\end{align}$$ $$\begin{align}bpl_{\theta > 180,r} = {R - r \cdot sin\left(\frac{\theta - 180}{2}\right)}\tag{9}\end{align}$$ $$\begin{align}bpl_{\theta > 180} = {R}\tag{10}\end{align}$$


  • D: mean major diameter
  • d: mean minor diameter
  • h: height
  • β: half angle
  • R: blank plate major radius
  • r: blank plate minor radius
  • bpl: blank plate length
  • bpw: blank plate width
  • θ: blank plate cut angle

5. Toriconical Head Blank Plate

6. Torispherical Head Blank Plate

There's a lot of debate about the blank plate calculation for a torispherical head, if you ask a designer engineer he'll probably say that you should ask your Vendor, because each Vendor have his preferred method to calculate it.

I'll use the middle perimeter (dashed line in the Fig. 6.1) to calculate the circular blank plate used to manufacture the torispherical head. It's easy this way because I know, for sure, that the blank plate will be circular and that the middle surface will have way less deformation than the outer and inner surfaces.

head-003 Fig. 6.1 - Torispherical head transformation into flat plate

According to Fig. 6.1:

  • θ: knuckle radius angle
  • β: crown radius angle
  • Φ: crown radius half angle
  • R: middle crown radius
  • r: middle knuckle radius
  • s: straight flange length
  • D: middle diameter
  • bpd: blank plate diameter

From the variables above, we only have to find the angles. For that, we'll use the equations (11), (12) and (13).

$$\begin{align}\phi = {asin \left(\frac{D - 2 \cdot r}{2 \cdot R - 2 \cdot r}\right)}\tag{11}\end{align}$$ $$\begin{align}\beta = {2 \cdot \phi}\tag{12}\end{align}$$ $$\begin{align}\theta = {90 - \phi}\tag{13}\end{align}$$

Now that we have the angles, we'll find the crown and knuckle arc lengths using equations (14) and (15).

$$\begin{align}r_{arc} = {\frac{2 \cdot r \cdot \pi \cdot \phi}{360}}\tag{14}\end{align}$$ $$\begin{align}R_{arc} = {\frac{4 \cdot R \cdot \pi \cdot \beta}{360}}\tag{15}\end{align}$$

Finally to find the blank plate diameter (bpd) we use the equation (16).

$$\begin{align}bpd = {2 \cdot r_{arc} + R_{arc} + 2 \cdot s}\tag{16} \end{align}$$ $$\begin{align}e = {\left(\frac{75 \cdot \ t}{R_f}\right) \cdot \left(1-\frac{R_f}{R_o} \right)}\tag{17}\end{align}$$

7. Ellipsoidal Head Blank Plate

As the cylindrical shell, ellipsoidal head blank plate is pretty easy to be calculated as there isn't many variables involved as you can see in the Fig. 7.1 below.

head-003 Fig. 7.1 - Ellipsoidal head transformation into flat plate

To find the perimeter (which will be the blank plate diameter) of the semi-surface section of the head, use the equation (18) shown below. The reason we use the semi-surface is because it have way less deformation compared to the outer and inner surfaces.

$$\begin{align}bpd = {2 \cdot \pi \cdot \sqrt{\frac{h^2 + (D/2)^2}{2}} + 2 \cdot s}\tag{18}\end{align}$$
  • bpd: head blank plate diameter
  • D: middle diameter
  • h: inner height or major axis of the ellipse, in a 2:1 ellipsoidal head[ 2 ], this variable is approximately D/4
  • s: straight flange length

8. Hemispherical Head Blank Plate

Due to the complexity of hemispherical heads, I've decided to create an article just for it: Spherical Shell and Hemispherical Head Layout

9. Flat Head Blank Plate

I believe that flat heads doesn't need drawing as it's just a rounded flat plate. The width and height will be equal to the diameter.


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