PRE-STRESSING IN THICK CYLINDRICAL SHELL

 

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PRE-STRESSING IN THICK CYLINDRICAL

SHELL

 

Pre-stressing means stresses are created during

manufacturing. Thick shells are made from thin shells by

hoop shrinking. Due to hoop shrinking, outer cylinder is put

into tension. The inner cylinder is put into compression.

These pre-stresses make the stress distribution uniform

under fluid pressure.

 

Refer to Fig. below:

Fig. Pre-stresses              Fig. resultant stress variation

PRE-STRESSES IN CYLINDER (due to hoop shrinking)

 AB stress = –2psr22/ (r22 –r12)  compressive

 CD stress = ps(r32+r22)/ (r32 –r22)    Tensile

STRESSES DUE TO FLUID PRESSURE

 AA’ stress= p((r32+r12)/ (r32 –r12)

 CC’ stress= pr12((r32+r22)/[r22(r32 –r12)]

FINAL RESULTANT STRESSES

Final stress at A = AA’ – AB=p((r32+r12)/ (r32 –r12) — 2psr22/ (r22 –r12)

Resultant stress at C = CC’ + CD =pr12((r32+r22)/[r22(r32 –r12)]+ ps(r32+r22)/ (r32 –r22)

Stresses due to fabrication (pre-stresses ) are coupled with stresses due to fluid pressure to get the final distribution of stresses. These are found to be more uniform as compared to stresses in a single walled thick vessel.

ADVANTAGES OF PRE-STRESSING

  1. Stresses are more uniform.

  2. Vessel of less thickness can withstand higher fluid pressures.

  3. For a certain fluid pressure, vessel of lesser thickness will be required.

  4. Less stress will be produced for a certain fluid pressure.

  5. Further, it is applicable  when the thick vessel is made from number of thin cylinders.

MOST EFFECTIVE PROPORTIONS OF THE CYLINDER AND THE JACKET

In order to find the best proportions, final stress at A and at C are equated equal to the allowable stress. Therefore, the equations become

Final stress at A = AA’ – AB=p((r32+r12)/ (r32 –r12) — 2psr22/ (r22 –r12) = σallow

Resultant stress at C = CC’ + CD

=pr12((r32+r22)/[r22(r32 –r12)]+ p s (r32+r22)/ (r32 –r22)=σallow

Since proportions are to be found, known are p, r1 and σallow. Unknown are r2, r3 and ps. Here an assumption is to be made about radial deflection such that pre-stresses are equal to the elastic strength of the material. Heating temperatures are such that the material of the vessel does not melt. Now if we take r2 =Cr1 and r3=  C2r1 in which C is a constant for a set of values for p and σallow.  Further analysis will show that the best stress distribution will exist if the radii are made in a GP series i.e. r2/r1=r3/r2.