LESSON 31 DESIGN OF THIN CYLINDRICAL & SPHERICAL SHELLS

31.1          Pressure Vessels

Pressure vessels are the vessels used to store or supply fluids under pressure. Stored fluid may undergo a change of state inside the pressure vessel e.g. in steam boiler or may undergo some chemical reaction. In nuclear / thermal power plants, chemical industries and various other industries, pressure vessels are used for storage and supply of different fluids, like water, gas, steam, air etc. Pressure vessels are generally made of steel plates by bending them to desired shapes and joining the ends by welding. Pressure vessels have to be designed very carefully as their failure may cause dangerous accident due to explosion or leakage of fluid.

31.2         Classification of Pressure Vessels

31.2.1    According to Dimensions

Thin Shell Pressure Vessels

Pressure vessels with inner diameter to wall thickness ratio greater than 20 are called thin shell pressure vessels. These are used in boilers, tanks, pipes etc.

Thick Shell Pressure Vessels

Pressure vessels with inner diameter to wall thickness ratio less than 20 are called thick shell pressure vessels. These are used in high pressure cylinders, tanks and gun barrels etc.

31.2.2    According to Shape

Depending upon the geometric shape, the pressure vessels can be classified into Cylindrical, Spherical or Conical pressure vessels.

31.2.3    According to End Construction

Open Ended Pressure Vessels

e.g. cylinder with a piston.

Due to internal fluid pressure, circumferential hoop stress is induced.

 lose Ended Pressure Vessels

e.g. a tank.

Due to internal pressure, circumferential hoop stress and longitudinal stress is induced.

Design of pressure vessels with inner diameter to wall thickness ratio greater than 20 (i.e. thin shell) and having cylindrical and spherical shape, are discussed in the following articles.

31.3         Design of Thin Cylindrical Shells

31.3.1    Circumferential Hoop Stress

Due to internal pressure, cylindrical shell may fail along the longitudinal section as shown in figure 31.1. To avoid this kind of failure, the circumferential hoop stress should not exceed the yield strength of the material. Consider free body diagram of half portion of the cylinder as shown in figure 31.2.

Figure 31.1 Failure of Thin Cylindrical Shell along the Longitudinal Axis

Figure 31.2 Hoop Stress in Thin Cylindrical Shell

 

 

Let       p          = Internal Pressure

            d          = Internal Diameter of Cylinder

            t           = Wall Thickness of Cylinder

            l           = Length of Cylinder

Considering the equilibrium of forces,

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31.3.2    Longitudinal Stress

In addition to hoop stress, longitudinal stress is also induced in close ended pressure vessels. Thin cylindrical shells may fail along the transverse section (as figure 31.3), if this longitudinal stress reaches the yield strength of the material. Consider free body diagram of the upper half of the cylinder as shown in figure 31.4. Considering the equilibrium of forces,

Figure 31.3 Failure of Thin Cylindrical Shell along the Transverse Section

Figure 31.4 Longitudinal Stress in Thin Cylindrical Shell

31.4 Considering the equilibrium of forces,

312.png

It can be observed from the expressions of transverse hoop stress and longitudinal stress that

313.png

Therefore hoop stress should be the design criteria for thin cylindrical shells subjected to internal pressure and desired minimum wall thickness can be estimated from the design equation based on allowable value of hoop stress.

Figure 31.5 Hoop Stress in Thin Spherical Shell

31.5          Design of Thin Spherical Shells

Consider free body diagram of half portion of a sphere as shown in figure 31.5.

Let       p          = Internal Pressure

            d          = Internal Diameter of Sphere

            t           = Wall Thickness of Sphere

 314.png

Desired minimum wall thickness can be estimated from the design equation based on allowable value of hoop stress.

 

 

References

  1. Design of Machine Elements by VB Bhandari

  2. Analysis and Design of Machine Elements by V.K. Jadon

  3. Design of Machine Elements by C.S. Sharma & K. Purohit

  4. Machine Design by P.C. Sharma & D.K. Aggarwal

  5. Machine Design by R.S. Khurmi

 

Last modified: Tuesday, 25 March 2014, 4:28 AM