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

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 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 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

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

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

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

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 basis Time consuming Requires 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.

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.

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.

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.

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

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

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.

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 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.

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

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 Cylinder Geometry Finite element mesh

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.

Nozzle/Cylinder intersection In plane steady moment acting on nozzle = 711.1Nm Cyclic internal pressure Young’s Modulus = 210.125GPa Yield stress (Shell & weld) = 234MPa Yield stress (Nozzle) = 343MPa

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

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.