How to Measure Residual Stress in a Multi-layered Vessel?

first of all Rajasekhar from Sri Lanka (I guess) STOP SHOUTING!!!!

You must make use of Lame's Equation to evaluate the radial pressure acting at the junction of the cylinderiacal vessels.

apply Lame's equation to the inner and outer cylinder each. Applying the boundary conditions.
Aslo take into account the equation used to evaluate the value of the difference in junction diameter before they were shrunk

Using the two Lame's equations and equation to evaluate change in junction diameter, you will get the radial pressure acting at the junction.

Since you have the values of the constants, apply Lame's equation at the junction to get the value of Hoop stress

For further clarification, on this i suggest that you lay your hands on a good book about Strength of Materials and go through the chapters on Thick Cylinderical Shells

You got an input for computing the stress but one detail was not mentioned. I do not know how were the contact areas machined but you should also consider for the computations the fact that when pressed surfaces change their roughnesses and thus a part of geometrical press fit gets lost. So in fact the pressure is less than assumed by the method you were indicated.

But as far as i understand your question you mentioned "how to measure.." and not "how to compute...". So that has to be analysed in more details. When an assembly as you mention is made the tangential stress at the outer diameter will reach a level depending on the geometry and amplitude of the press fit. If before the shrinking is done you could bond on the outer diameter strain gauges in axial and tangential directions to build up a bridge. You measure after the operation the values and using the output you can determine the strains and thus the actual pressure at in between surfaces. If the vessels are in use and if the outer temperature does not increase over the allowed limits for the bonded gauges you can get an information first for the effect generated by internal pressure and second if a press-fit loss occurs (due to further surface deformations or material creep) over time making a comparison between the values obtained after shrinking (without internal pressure) and outputs at any later time under same loading conditions( no internal pressure and temperature as near as possible to the one when 1st measurement was done).

I hope now you have all the input as well for computing (correctly) as for measuring when you need it.

I remember there is a ultrasonic stress tester can test this residual streess and an instrument mde of russia according to gravitive magnetic method to get the stress value.

Of course there are very sophisticated possibilities but they are related to cost. The approach i suggested has the advantage to be accessible to almost every body and to a very acceptable cost. Strain gauges are quite cheep and a conditioner is also available as a module to be connected to a PC or laptop saving the complex lab dedicated devices.

THANK U FOR U R INSTANT REPLY