THIN SPHERICAL SHELL

 

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THIN SPHERICAL SHELL

 

A vessel is thin if its wall thickness is small. It will be due

inside pressure is small. It will also due to high value of

allowable stress for the material of the shell. Thin spherical

shells are used for the storage.

 

 Stresses in a thin spherical shell

  • Hoop stress or Tangential stress or Circumferential stress(σh)———-It acts in the tangential direction at the point of consideration (say in a horizontal circle)

  • Hoop stress or Tangential stress or Circumferential stress(σh)———-It acts in the tangential direction at the point of consideration (say in a vertical circle)

  • Radial stress(σr)———-It acts in the radial direction at the point of consideration

NOTE: All the three stresses are tensile or compression and are at right angles to each other. Therefore, these are principal stresses.

Hoop stress is tensile.

Hoop stress is tensile.

Radial stress is compression stress.

Derivation of formula of various stresses

  • (a) σh = hoop stress

It will break the sphere at the circumference as shown in Fig.

Refer to Fig. and equating the resistance of the material to the breaking force.

Material resistance = σh πD t

Breaking force due to internal fluid pressure = p x projected area

= p (π/4)Di2

Therefore  σh πD t = p (π/4)Di2

σh = pD/4t

If the efficiency of the circumferential joint is considered then  σh = pD/2t ηcircum

(b) Radial stress = σr = inside fluid pressure

σr = pi

From the equations it can be easily concluded that the radial stress is negligible as compared to hoop stresses because of d/t ratio value is minimum 20.

Built-up spherical shells

Shells with joints are called built-up spherical shells.

Change in the dimensions of the shell

(i) Change in diameter of the shell, δD

δD/D = Change in circumference / Original circumference= π δD/πD

Therefore δD/D= σh/E -μ σh/E = pDi/4tE – μ pD/4tE

δD/D = (pDi/4tE) (1-μ)

δD = (pDiDi /4tE) (2-μ)= (pDi2 /4tE) (2-μ)

δD = (pDi2 /4tE) (1-μ)

(ii)Change in volume, δV

δV/V = Volumetric strain = 3 δD/D

δV/V = 3 (pDi/4tE) (1-μ)

δV/V =  (¾)(pDi/tE) (1–μ)

V = (4π/3) R3 = (π/6) D3

δV = (pDi/4tE) (1–μ) (π/6) D3

δV =(π/6)(pD4/tE) (1–μ)=(πpD4/6tE) (1–μ)

Advantages of thin spherical shells over thin cylindrical shells

(i) Thickness requirement is less for the same pressure, same diameter and same material.

(ii) Higher pressure can be with the same thickness of a cylindrical shell of same material and same diameter

(iii) It can store or process more volume in the same space i.e. it is compact.

Disadvantages of thin spherical shells over thin cylindrical shells

(i) It is difficult to manufacture and a costly affair.

(ii) It is difficult to transport from the industry to the point of use.

(iii) It is difficult to support at the site of use.

(iv)It is difficult to repair and maintain.

Thus invariably cylindrical thin shells are in use. Only sparingly a spherical shell is used for storage in a refinery.