HONGJUN LI Department of Mechanical Engineering ... THEMATIC NETWORK COMPETITIVE AND SUSTAINABLE GROWTH (GROWTH) PROGRAMME Introduction FEA is an established analysis method used in

FENET THEMATIC NETWORKCOMPETITIVE AND SUSTAINABLE GROWTH(GROWTH) PROGRAMME

HONGJUN LIDepartment of Mechanical Engineering

University of Strathclyde Glasgow, Scotland, UK


Introduction

FEA is an established analysis method used in Pressure Vessel Design • Most DBA is still based on elastic analysis• Inelastic analysis is becoming more widely used

This presentation will describe new approaches to inelastic design developed at the University of Strathclyde by

Hongjun LiMartin Muscat

Bobby HamiltonDonald Mackenzie

Specifically • A new criterion of plastic collapse for strain hardening plastic

analysis• A bounding theorem method for calculating shakedown loads


INELASTIC FAILURE MODES

Gross plastic deformation under static load• In elastic DBA, prevented by limiting the elastic primary

stress in the vessel.• In inelastic DBA calculate permissible load through

inelastic analysisIncremental plastic collapse (ratchetting)• In elastic DBA, shakedown is “assured” by limiting the

elastic primary plus secondary stress to twice yield• In inelastic DBA, calculate permissible load through

inelastic shakedown analysis


Inelastic Material Models

ε

σ

ε

E

σ

ε

E

EP

σ

ε

E

σ

E

(a) Non-linear stress-strain (b) Multi-linear isotropic hardening

(c) Bilinear kinematic hardening (d) Perfect plasticity

ELASTIC

PLASTIC

σ YYIELD σ Y

YIELD

σ YYIELDσ Y

YIELD

ELASTIC

PLASTIC

ELASTIC ELASTIC

PLASTICPLASTIC


Gross Plastic Deformationunder Static Load


Types of inelastic analysis

Limit analysis• Assumes an elastic-perfectly plastic material model and small

deformation theory • The limit load is the highest load satisfying equilibrium between

external and internal forces• May be assumed to be the ductile collapse load in DBA.

• The allowable load is two thirds of limit load“Plastic” analysis • Strain hardening and large deformation effects may be included• A criterion of plastic collapse is applied to determine the allowable

load• The “plastic load” characterises gross plastic deformation

• The allowable load is two thirds of plastic load


Limit AnalysisAdvantages

No inelastic stress strain design data needed Load-path independent – don’t need to know load historyConservative if non-linear geometric weakening is not significantEasier than elastic analysis

Disadvantages× Not conservative if non-linear geometric weakening is

significant× Does not account for enhanced strength due to strain

hardening× Can be difficult to evaluate limit load if solution converges

for unrealistic deformations


Example: Multilinear Hardening Plastic Analysis

σ

ε


ASME Criterion of Plastic Collapse

The twice elastic slope criterion (TES)

Applied to characteristic load-deformation curve obtained by plastic analysis

L o ad

D efo rm atio n

P P

kk /2


Deformation Parameter

Little guidance on nature and location of deformation parameter given

Choice can significantly affect the calculated plastic load.

• Gerdeen proposed the product of the load and deformation parameters should have units of work (Nm) if possible

A. CANTILEVERBENDING

B. MOMENTLOADING

C. PRESSURISEDCYLINDER

D. PRESSURISEDCLOSED VESSEL

E. PRESSURE INHEAD

F. PRESSURE INNOZZLE

F

M

d

θ

w

dw

P

∆V

P

P

P


Problems with the ASME Procedure

The method is heuristic or arbitrary• Based on experimental experience with specific

configurationsChoice of suitable load and deformation parameters

• Especially for non-proportional combined loading• In some cases no intersection occurs between the load-

deformation curve and collapse limit lineThe plastic load is influenced by the elastic response

• Value of plastic load is dependent on FE modellingassumptions

Does not adequately represent the effect material strain hardening has on plastic deformation


A New Plastic Criterion

L oad

P las ticW ork

S lo p e

Based on plastic work conceptsThe evolution of the gross plastic deformation mechanism is characterised by

The rate of change of slope of the Wp-Q curve, orThe curvature at a point on the curve

23

2

2

2

1

1

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+

=

dQdW

dQWd

p

p

ρ


Curvature and Gross Plastic Collapse

Example: bilinear hardening beam in pure bending

Post-yield curvature indicates post-yield stress redistribution

Plastic deformation The maximum rate of stress redistribution

corresponds with maximum curvature Subsequent decrease in curvature

indicates decreasing rate of stress redistribution

Minimum or zero curvature indicates little or no redistribution

Gross plastic deformation

curvature

HardeningAnalysis

WP (Nm)

M(Nm)

Discontinuity


Curvature and Gross Plastic Collapse

Plastic Work Curvature or PWC criterion of plastic load • The load corresponding to

constant or zero curvature after stress redistribution

curvature

HardeningAnalysis

WP (Nm)

M(Nm)

Discontinuity


Pipe Bend Example: ASME Approach

0

100

200

300

400

500

600

700

800

900

1000

0 5 10 15 20 25 30

Rotation -Degree

Mom

ent-2

*kN

mClosing & perfectly plasticClosing & 5% bilinearOpening & perfectly plasticOpening & 2% bilinear


Pipe Bend Example: Closing Moment

637.5

Moment kNm

500

Plastic work

720

Moment kNm

Plastic work

600

Bilinear Hardening MTES=700 kNm

MPWC=720 kNm

Perfectly Plastic MTES= NA

MPWC=637.5 kNm


Pipe Bend Example: Opening Moment

1180

Moment kNm

Plastic WorkPlastic Work

Moment kNm

1120

MTES=980 kNm

MPWC=1180 kNm

Bilinear Hardening MTES=915 kNm

MPWC=1120 kNm

Perfectly Plastic


Shakedown & Ratchettingunder Cyclic Load


Ductile Failure under Cyclic Loads

• Maximum load between first yield and plastic collapse• Elastic shakedown • Plastic shakedown (alternating plasticity)• Ratchetting

• DBA must ensure shakedown of the vessel


P

t

PS

PY

A

B

C

DA B

DC

Elastic Shakedown of Thick Cylinder



Preventing Cyclic Failure in PVD

High cycle & low cycle fatigue • Perform a fatigue analysis

• Establish the design life of the vessel

Ratchetting• Ensuring that the structure shakes-down to elastic

action


Shakedown Analysis: Plastic DBA

Incremental elastic-plastic Finite Element Analysis• Simulate the structural response for a given load history• Monitor the resulting plastic strain accumulation

• No plastic strain accumulation & No alternating plasticity: elastic shakedown

• Alternating plasticity: plastic shakedown• Plastic strain increases with each cycle: ratchetting


Elastic Shakedown of thick Vessel

Incremental elastic-plastic analysis

0

ε1p

t

t

P

Py


Plastic Shakedown of Thick Vessel


0

ε1p

tP

Py


Ratchetting of Thick Vessel


0t

t

P

Py

ε1p


Incremental FEA

Does not predict the specific value of the shakedown load• Demonstrates the structural response for a given load

level• A number of simulations at different loads needed to

identify shakedown load• PV design codes simply require that shakedown is

demonstrated for a specific load • Can be difficult to distinguish between plastic

shakedown and ratchetting• Large number of cycles may be needed• Expensive in computer requirements


Shakedown Bounding Theorems

Similar approach to (lower bound) limit load analysis• Establishes a lower bound on the shakedown load for

design purposes• Does not calculate realistic stress and strain values• Independent of load history

Melan’s (lower bound) theorem:• ‘For a given time dependent cyclic load set, a structure

made of elastic-perfectly plastic material will exhibit shakedown if a constant residual stress field can be found such that the yield condition is not violated for any combination of time dependent cyclic elastic stress and residual stress’.


Muscat’s Lower Bound Method

The lower bound limit load theorem can be coupled with FEA to evaluate elastic shakedown loadsMuscat’s method uses static FEA to calculate “elastic” and “elastic plus residual” stress fields for Melan’s theorem• Limit analysis is performed with results stored for n load

levels up to the limit load • “Elastic plus residual” stress is assumed to be the

limit load solution for each load Pn

– Satisfies Melan’s theorem• “Elastic” stress is the conventional elastic stress at

load Pn


Applying Melan’s Theorem

Residual stress calculated by superposition of “Elastic plus residual” and “Elastic” stress fields• Residual stress = (Elastic Plus Residual)- (Elastic)

Load Pn is a lower bound on the shakedown load if• Maximum residual stress is less than yield• Maximum “Elastic plus residual” stress is less than

yield


Shakedown of a Thick Cylinder

Calculate stress fields by limit analysis for

load Pn

Elastic Stress:From scaled initial elastic response

Elastic plus Residual Stress:from limit analysis


Calculate residual stress field for load Pn -

(Elastic plus residual) – (Elastic) Stress

= Residual Stress


(Elastic plus residual) ≤σy: Yes

Pn is a lower bound shakedown load

-

Apply Melan’s Theorem

(Residual) ≤σy: Yes


Conclusion: Inelastic Analysis in PV DBA

Inelastic analysis allows allowable loads to be calculated by simulating inelastic response• Limit analysis, Limit load• Plastic analysis, Plastic load• A criterion for plastic collapse is required for gross

plastic deformation design in plastic analysis• Problems with TES• The Plastic Work Curvature criterion is a consistent

and robust criterion which accounts for the effects of geometric non-linearity and strain hardening


Shakedown can be assessed through• Performing incremental elastic plastic analysis and

monitoring plastic strain accumulation over a number of cycles

• Applying a lower bound theorem based approach such as Muscat’s Method to the results of a (static) limit analysis

– Requires only a single limit analysis– Clearly establishes the elastic shakedown load


I would like thank Dr Mackenzie for his help during the preparation of this presentation!

QUESTIONS?

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HONGJUN LI Department of Mechanical Engineering ... THEMATIC NETWORK COMPETITIVE AND SUSTAINABLE GROWTH (GROWTH) PROGRAMME Introduction FEA is an established analysis method used in