1. ELASTIC SHAKEDOWN IN PRESSURE VESSEL COMPONENTS UNDER PROPORTIONAL AND NON-PROPORTIONAL LOADING
Dr Martin Muscat
Department of Mechanical Engineering
University of Malta
Dr D.Mackenzie, Dr R.Hamilton, P. Makulsawatudom
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
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 in BS 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 route of Design by Analysis
EN code design checks against different failure modes:
Excessive deformation
Progressive plastic deformation
Instability
Fatigue
Loss of static equilibrium

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

9. Principles & Application rules
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

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. Preventing progressive plastic deformation New methods for calculating elastic 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.

15. Melan’s theorem
Melan’s theorem states that for cases of proportional loading elastic shakedown is achieved if:
|r |  y (1)
|r + e |  y or |s |  y (2)
where y is the yield stress.
e is the elastic stress field.
r is the residual stress field.
s is the shakedown stress field.

16. The proposed method (proportional loading)
Inelastic analyses are performed to obtain a number of shakedown stress fields, si corresponding to a number of cyclic load levels.
Corresponding elastic stress fields, sei are found by performing a single elastic analysis and invoking proportionality.
A self-equilibrating residual stress field, ri, is obtained for each load level by using the superposition equation ri = si - ei.
A lower bound to the elastic shakedown load is established by examining the residual stress fields ri for each load level to establish the highest load at which the calculated residual stress field satisfies the yield condition.

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.
Lower bound
Yield stress
Upper bound

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

19. Polizzotto’s theorem states that for a steady/cyclic load given by
P(t) = Pc(t) + Po
elastic shakedown is achieved if:
|pt|  y where pt = |c(t) + s| (1)
y is the yield stress
pt is the post transient stress field
c(t) is the elastic stress response to Pc(t)
s is a time independent stress field in equilibrium with Po

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

22. The proposed method …
Corresponding elastic stress fields sc, are found by performing a single elastic analysis and invoking proportionality.
The post transient stress fields pt, are obtained for each load level by using the superposition equation pt = s + sc.
A lower bound to the elastic shakedown load is established by examining the post transient stress fields pt for each load level to establish the highest load at which pt satisfies the yield condition.

23. A typical graph showing the normalised post transient stress field for each load level is shown below.
Lower bound
Yield stress
Upper bound

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
Cylinder Geometry
Finite element mesh

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
In plane steady moment acting on nozzle = 711.1Nm
Cyclic internal pressure
Young’s Modulus = GPa
Yield stress (Shell & weld) = 234MPa
Yield stress (Nozzle) = 343MPa

28. Maximum post transient stress v.s.Cyclic pressure
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.
Maximum post transient stress v.s.Cyclic pressure

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.