ELASTIC SHAKEDOWN IN PRESSURE VESSEL COMPONENTS UNDER PROPORTIONAL AND NON-PROPORTIONAL LOADING

ELASTIC SHAKEDOWN IN PRESSURE VESSEL COMPONENTS UNDER PROPORTIONAL AND NON-PROPORTIONAL LOADING

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## ELASTIC SHAKEDOWN IN PRESSURE VESSEL COMPONENTS UNDER PROPORTIONAL AND NON-PROPORTIONAL LOADING

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**1. **ELASTIC SHAKEDOWN INPRESSURE VESSEL COMPONENTSUNDER PROPORTIONAL AND NON-PROPORTIONALLOADING Dr Martin Muscat
Department of Mechanical Engineering
University of Malta
Dr D.Mackenzie, Dr R.Hamilton, P. Makulsawatudom
Department of Mechanical Engineering
University of Strathclyde

**2. **SUMMARY OF PRESENTATION Introduction – What is shakedown ?
Achieving shakedown using EN13445/3 PV code
A proposed method for Elastic shakedown loads - Cases of proportional & non-proportional loading.
Discussion

**3. **What is shakedown ?
A structure made of elastic-perfectly plastic material subjected to cyclic loading exhibits an initial short-term transient response followed by one of three types of steady state response:
(1) Elastic shakedown
(2) Plastic shakedown
(3) Ratcheting

**4. **Elastic shakedown - Response is wholly elastic after the initial transient response.Plastic shakedown - Reverse plasticity occurs leading to low cycle fatigue.Ratcheting - The plastic strain increases incrementally with every load cycle until incremental plastic collapse eventually occurs.

**5. ** Ratcheting should be avoided in structural design.
Plastic shakedown is acceptable but produces plastic strain which must be addressed within a fatigue analysis.
A way to eliminate both problems is to design in order to achieve elastic shakedown.

**6. **Design methods inBS EN13445/3 Design by rule
Follow a set of rules to calculate a thickness
Design by analysis
Elastic route (Annex C) uses a stress categorisation procedure & appropriate design stress limits
Direct route (Annex B) is based on inelastic analysis & circumvents the stress categorisation procedure

**7. **The Direct routeof Design by Analysis EN code design checks against different failure modes:
Excessive deformation
Progressive plastic deformation
Instability
Fatigue
Loss of static equilibrium

**8. **
Elastic perfectly plastic material model
Small deformation theory
The check for preventing ratchetting requires the von Mises yield criterion Some analysis requirements

**9. **
BS EN13445/3 gives a set of :
Principles - which are general definitions and requirements which must be satisfied in a design check
Application rules - are generally recognised rules which follow the principles and satisfy their requirements Principles & Application rules

**10. **Principle
The Principle for preventing progressive plastic deformation is ‘For all relevant load cases, on repeated application of specified action cycles progressive plastic deformation shall not occur’

**11. **Application rules
The shakedown rule, the principle is fulfilled if it can be shown that the equivalent stress concentration free model shakes down to elastic behaviour under the action cycles considered.
The technical adaptation rule, the principle is fulfilled if it can be shown that the maximum absolute value of the principal strain does not exceed 5% under the action cycles considered.

**12. **The technical adaptation rule
The simplest and most accurate is to apply conventional cyclic elastic-plastic analysis and examine the plastic strain accumulation after each cycle:
A trial and error basis
Time consuming
Requires large computer power and storage
Useful for complex load cycles involving more than one load frequency

**13. **The shakedown rule
In EN13445/3 Zeman’s and Preiss’s deviatoric mapping of stress state technique is given as an application tool for the shakedown rule.
The deviatoric map is based on Melan’s lower bound shakedown theorem and may be used to evaluate shakedown loads for structures subject to proportional loading.
A major disadvantage of the deviatoric map is that it is somewhat tedious to use.

**14. **Preventingprogressive plastic deformationNew methods for calculatingelastic shakedown loads Based on:
Melan’s lower bound elastic shakedown theorem for cases of proportional loading.
Polizzotto’s lower bound elastic shakedown theorem for cases of non-proportional loading.
Non-linear finite element analysis.

**17. **Iterations between the calculated lower bound and the limit load are then used to systematically converge to a self equilibrating residual stress field where the maximum residual stress is slightly less than or equal to ?Y.

**18. **Advantages of the proposed method (proportional loading)
Accurate for calculating elastic shakedown loads when compared to full elasto plastic cyclic analysis
Relatively easy to apply
Automatic - most of the analysis is done by the computer
Eliminates the need for low cycle fatigue analysis

**20. **The proposed method (Non-proportional loading)
Inelastic analyses are performed to obtain a number of time independent stress fields, s corresponding to a number of cyclic load levels.

**21. **The time independent stress field A stable time independent stress field is not always obtained after the first cycle of loading.
This depends on the geometry and on the loading cycle.
It is recommended that a check is made to determine whether the time independent stress field used for the analysis has stabilised or not.
A stable time independent stress field was always obtained in less than 10 load/unload cycles

**23. **A typical graph showing the normalised post transient stress field for each load level is shown below.

**24. **Advantages of the proposed method (non-proportional loading)
Can be used for non-proportional loading
Accurate for calculating elastic shakedown loads
Relatively easy to apply
Automatic - most of the analysis done by the computer
Eliminates the need for low cycle fatigue analysis

**25. **Thick cylinder with offset cross-holes

**26. **Elastic plastic response of plain cylinder and cylinder with offset cross-hole The applied pressure PA is normalised w.r.t. the yield pressure of a corresponding plain cylinder.
The figures show the boundary between the elastic shakedown region and cyclic plasticity.
The lower bound shakedown loads calculated by the proposed method were verified by performing full cyclic plastic analysis.

**27. **Nozzle/Cylinder intersection

**28. **Result 10 load/unload cycles used to obtain the time independent stress field.
Elastic shakedown pressure is calculated to be 19MPa.
Full elastic-plastic cyclic analysis gives a result of 19.2MPa.
The deviatoric mapping of stress state technique gives a result of 17.65MPa.
Elastic compensation gives a result of 13.75MPa.

**29. **Some Comments
The full nonlinear analysis (100 cycles) took 7 hours on a Pentium III dual 1GHz Xeon processor having 1GB RAM.
The non-linear superposition method took 1 hour to finish.
Therefore the proposed method can reduce the design time considerably.

**30. **CONCLUSIONS The proposed methods can be fully automated with little intervention from the side of the analyst.
The methods proposed here are well suited to be used as elastic shakedown load calculation tools in the new BS EN 13445/3 Annex B to satisfy the principle which prevents ratchetting.