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 r2 and r3) is put on to the inner cylinder (with radii as r1 and r2). Then it is allowed to cool. A pressure will be developed both on the jacket and the cylinder. Let ps is the shrinkage pressure developed after hoop shrinking  at the common radius achieved. This pressure ps will be  only external pressure for the cylinder and only internal pressure for the jacket.

Let the radii are as under;

r1 is the inner radius of the cylinder

r2 is the common radius =outer radius of cylinder

r2 is the common radius = inner radius of the jacket

r3 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 dr2 be the change in radius at radius r2.

Change in circumference = 2 π dr2

Original circumference =  2π r2

Circumferential strain= 2 π dr2/2π r2 = dr2/r2

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)

 

dr2/r2 = (σh/E –μσr/E) at the radius r2    (7)

σh at r2 due to only external ps =   (–ps r22 –psr12) /((r22 –r12)

= –ps ((r22+r12]/ (r22 –r12)  Compressive

σh = -ps

σr at r2 due to only ps = –ps                        Compressive

Substituting the values in eq (7), we get

dr2/r2 = (-ps r22/E)[(r22+r12]/ (r22 –r12) –μ]

(dr2)Cyl = (–ps r2/E) [((r22+r12)/((r22 –r12) –μ)]               (8)

 

CIRCUMFERENTIAL STRAIN FOR THE JACKET (ONLY INTERNAL PRESSURE)

 

Radii are r2 and r3

dr2/r2 =( σh/E –μσr/E) at the radius r2 in the jacket   (9)

σh at r2 due to only internal pressure ps will be

σh = ps(r32+r22)/ (r32 –r22)    Tensile

σr at r2 due to only internal ps = –ps  Compression          

Substituting the values in eq (9), we get

dr2/r2 = [ps(r32+r22/ (r32 –r22)] /E  — μ(-ps)/E

(dr2)Jacket = (ps r2/E) [(r32 +r22)/((r32 –r22) +μ)]                (10)

Jacket internal radius will increase by dr2.

 

FINAL RADIAL DEFLECTION

 

Final change will be the sum of the two changes in dr2. While adding take only with positive sign

Total dr2 = Initial difference in the radii

=(dr2)Jacket +(dr2)Cylinder

=(ps r2/E) [(r32+r22)/((r32 –r22)–μ)]+ (ps r2/E) [(r22 +r12)/((r22 –r12) +μ)]

Rad Def=   (ps r2/E) [(r32+r22)/((r32 –r22)+ (r22 +r12)/((r22 –r12) ]

 

CONCLUSION

 

Thus initial stresses (pre-stressing of the vessel) will be developed in the vessel only during fabrication. These pre-stresses are such that these will stress at the innermost radius and increase as we go outwards. This will result in uniform stresses in entire wall of the thick shell. Thus these pre-stresses have become useful.