# RADIAL DEFLECTION THICK SHELL

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**RADIAL DEFLECTION THICK SHELL**

## Meaning of radial deflection is change in radius due to fluid

## pressure. It can also be due to shrinkage pressure developed

## due to hoop shrinking. It help in the design and fabrication

## of thick vessels made of thin shells.

**Hoop shrinking**

#### Hot Outer cylinder (called the jacket with radii as r_{2} and r_{3}) is put on to the inner cylinder (with radii as r_{1} and r_{2}). Then it is allowed to cool. A pressure will be developed both on the jacket and the cylinder. Let p_{s} is the shrinkage pressure developed after hoop shrinking at the common radius achieved. This pressure p_{s} will be only external pressure for the cylinder and only internal pressure for the jacket.

#### Let the radii are as under;

#### r_{1} is the inner radius of the cylinder

#### r_{2} is the common radius =outer radius of cylinder

#### r_{2} is the common radius = inner radius of the jacket

#### r_{3} is the outer radius of the jacket

#### The radial displacement will be found from the change in circumference at the point of consideration (at the common radius). Let dr_{2} be the change in radius at radius r_{2}.

#### Change in circumference = 2 π dr_{2}

#### Original circumference = 2π r_{2}

#### Circumferential strain= 2 π dr_{2}/2π r_{2} = dr_{2}/r_{2}

**Circumferential strain**

#### It will be considered in two steps:

#### (i) first for the cylinder

#### (ii) Secondly for the jacket

**CIRCUMFERENTIAL STRAIN FOR THE CYLINDER (ONLY EXTERNAL PRESSURE)**

#### dr_{2}/r_{2} = (σ_{h}/E –μσ_{r}/E) at the radius r_{2} (7)

#### σ_{h} at r_{2} due to only external p_{s }= (–p_{s} r_{2}^{2} –p_{s}r_{1}^{2}) /((r_{2}^{2} –r_{1}^{2})

#### = –p_{s }((r_{2}^{2}+r_{1}^{2}]/ (r_{2}^{2} –r_{1}^{2}) Compressive

#### σ_{h} = -p_{s}

#### σ_{r} at r_{2} due to only p_{s }= –p_{s } Compressive

#### Substituting the values in eq (7), we get

#### dr_{2}/r_{2} = (-p_{s} r_{2}^{2}/E)[(r_{2}^{2}+r_{1}^{2}]/ (r_{2}^{2} –r_{1}^{2}) –μ]

#### (dr_{2})_{Cyl} = (–p_{s} r_{2}/E) [((r_{2}^{2}+r_{1}^{2})/((r_{2}^{2} –r_{1}^{2}) –μ)] (8)

**CIRCUMFERENTIAL STRAIN FOR THE JACKET (ONLY INTERNAL PRESSURE)**

**Radii are r**_{2} and r_{3}

_{2}and r

_{3}

#### dr_{2}/r_{2} =( σ_{h}/E –μσ_{r}/E) at the radius r_{2} in the jacket (9)

#### σ_{h} at r_{2} due to only internal pressure p_{s} will be

#### σ_{h} = p_{s}(r3^{2}+r2^{2})/ (r_{3}^{2} –r_{2}^{2}) Tensile

#### σ_{r} at r_{2} due to only internal p_{s }= –p_{s } Compression_{ }

#### Substituting the values in eq (9), we get

#### dr_{2}/r_{2} = [p_{s}(r3^{2}+r2^{2}/ (r_{3}^{2} –r_{2}^{2})] /E — μ(-p_{s})/E

#### (dr_{2})_{Jacket} = (p_{s} r_{2}/E) [(r_{3}^{2} +r_{2}^{2})/((r_{3}^{2} –r_{2}^{2}) +μ)] (10)

#### Jacket internal radius will increase by dr_{2}.

**FINAL RADIAL DEFLECTION**

#### Final change will be the sum of the two changes in dr_{2}. While adding take only with positive sign

#### Total dr_{2} = Initial difference in the radii

#### =(dr_{2})Jacket +(dr_{2})Cylinder

#### =(p_{s} r_{2}/E) [(r_{3}^{2}+r_{2}^{2})/((r_{3}^{2} –r_{2}^{2})–μ)]+ (p_{s} r_{2}/E) [(r_{2}^{2} +r_{1}^{2})/((r_{2}^{2} –r_{1}^{2}) +μ)]

#### Rad Def= (p_{s} r_{2}/E) [(r_{3}^{2}+r_{2}^{2})/((r_{3}^{2} –r_{2}^{2})+ (r_{2}^{2} +r_{1}^{2})/((r_{2}^{2} –r_{1}^{2}) ]

**CONCLUSION**