ELASTIC SHAKEDOWN INPRESSURE VESSEL COMPONENTS

UNDER PROPORTIONAL AND NON-PROPORTIONAL

LOADING

Dr Martin MuscatDepartment of Mechanical Engineering

University of Malta

Dr D.Mackenzie, Dr R.Hamilton, P. MakulsawatudomDepartment of Mechanical Engineering

University of Strathclyde

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

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

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.

• 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.

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

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

• Elastic perfectly plastic material model• Small deformation theory• The check for preventing ratchetting requires the

von Mises yield criterion

Some analysis requirements

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

• 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’

• 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.

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 basisTime consumingRequires large computer power and storage

• Useful for complex load cycles involving more than one load frequency

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.

Preventingprogressive plastic deformation

New 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.

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.

Melan’s theorem

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, ei 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.

•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.

Residual stress (obtained by superposition) v.s. Applied load

0

100

200

300

400

500

600

700

800

900

0 1 2 3 4 5 6

Applied load

Res

idu

al S

tres

s

Yield stress

Upper bound

Lower bound

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

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

Polizzotto’s theorem

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.

P2

P1

tttt t1 2 3 4 5

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

The proposed method …

•Corresponding elastic stress fields c, 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 + c.

•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.

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

Yield stress

Upper bound

Lower bound

Normalised post transient stress field (obtained by superposition) v.s. Applied load

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

0 1 2 3 4 5 6

Applied load

Normalised post transient stress

field

pt/Y

Lower bound

Pi

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

Thick cylinder with offset cross-holes

Finite element mesh

Rc

11.2mm

L

Rb

Blend Detail

ab

Cylinder Geometry

Elastic plastic response of plain cylinder and

cylinder with offset cross-hole

PA

PY,Cyl

b/a

0

0.5

1

1.5

2

2.5

3

1 1.5 2 2.5 3 3.5 4

Circular crosshole

Cyclic Plasticity

Max. design pressure

Elastic shakedown

0

0.5

1

1.5

2`

2.5

3

1 1.5 2 2.5 3 3.5 4

Cyclic Plasticity

Elastic shakedown

Max. design pressure

PA

PY,Cyl

Plain Cylinder • 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.

Nozzle/Cylinder intersection

In plane steady moment acting on nozzle = 711.1NmCyclic internal pressure

Young’s Modulus = 210.125GPaYield stress (Shell & weld) = 234MPaYield stress (Nozzle) = 343MPa

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.

180

200

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240

260

280

300

320

340

360

380

0 5 10 15 20 25 30

Applied cyclic pressure

Max

. p

ost

tra

nsi

ent

stre

ss

Max. post transient stress - nozzle Max. post transient stress - shell/weld

Maximum post transient stress v.s.Cyclic pressure

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.

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.