Use of Zarka‘s Method at FHL

Theory

Implementation in ANSYS

Experience

Hartwig HübelAndreas Kretzschmar

(FH Lausitz)

Cracow March 11, 2003

Background, History

in nuclear industry: elastic-plastic response required for life assessment under cyclic loading:

strain range ∆ε cyclic accumulated (ratchet-) strain εacc

displacements at end of life

20 years ago:only rough estimates available as „simplified“ elastic-plastic methods (knock-down factors), without capturing effects of:

individual geometry of structure kind of loading (thermal, displacement-, force-controlled, ...)

hardening of material

3

Background, History (ctd.)

publication of some examples obtained by Zarka with Zarka‘s method revealed:

however: method remained largely obscure:

superior quality of results (approximations almost exact) obtained with little numerical effort (few lin. elastic analyses)

description of theory appeared to be not complete some assumptions seemed to be heuristically motivated

by end of seventies Zarka‘s method came up, claiming it was able to account for these effects

why that ambiguous?

when advantageous?question:

some applications outside Zarka‘s team showed poor results

4

Outline of Zarka‘s methodmaterial model:

- yield surface must exist- hardening required:

» purely kinematic

σ´ - space

simplified method:

⇒“infinite” ratcheting ruled out:limit state = ES or PS, associated with ”finite” ratcheting

only partial information: - no evolution of σ, ε with cycles - only post-shakedown quantities - loss in accuracy (Vp, σ, ε)

only reduced effort:- direct method- few elastic analyses sufficient

+ some “local” calculations

- temperature dependence:only size of yield surface, i.e. σy (not E, ν, Et)

» linear or multilinear

σ

ε

σyEt

5

Specialisation, Modification

Zarka-method

specialisation,

modifications

simplified theory of plastic zones (VFZT)

- moderate levels of strain (e.g. no deep drawing)- isotropic material - no strength differential effects (tensile = compressive)- linear and bilinear kinematic hardening (bilin., trilin. σ−ε)- Mises yield surface- temperature-dependent yield stress- mainly cyclic loading between 2 extreme states of loading - logic of iterative improvement:

identifying Vp, estimating TIV Y

6

Reformulation of “Exact” Plastic Theory

residual state can be obtained by 1 mea with modif. elastic materialdata (E*) and modif. loading (ε0), if Vp and Y are known

idea of Zarka-Method: estimation of Vp and Y

yesεel-pl = σel-pl/Eεel-pl = σel-pl/E + ξ/Cexact(el-pl)

loadappliedin el. zone Ve (=V-Vp)in plastic zone Vpkind of

analysis

yesεf.el = σf.el/Eεf.el = σf.el/Efictitious elastic (f.el)

initial strainε0 inVp

=> ε*= ρ•(1/E)=> ε*= ρ•(1/E*) + ε0with 1/E*=1/E+1/Cand ε0=Y/C

modif. elast.(mea)

noεel-pl-εf.el =(σel-pl-σf.el)/Eε* ρ

εel-pl-εf.el =(σel-pl-σf.el)/E + ξ/Cε* ρ ξ = ρ+Y

differencedef

here: - uniaxial stress state- bilinear σ−ε-diagram (kinem. hardening)- monotonic loading

σ

εEEtσy

σ

εpl

C

7

Mon. Loading: Estimation of Yhere: bilinear σ−ε-diagram (lin. kinem. hardening)

σ´ is unknown; estimation not possible, since center ξ (i.e. εpl) also unknown

Y is unknown; but estimation possible, since center σ´f.el is known

Mises yield condition:f(σ´,ξ,σy) f(Y,σ´f.el,σy)

Y = ξ−(σ´−σ´f.el)ρ´

def

projection

multiaxial stress: Vp and Y unknown; local estimation of Y sufficient, since field effects of Y are relieved compared to σ (=heuristic)

σ´ - space:

σ´

ξ=C·εpl*.. .

Y – space:

Y

*σ´f.el. ..

8

uniaxial stress: Y exactly known by projection; only Vp unknown

numerical effort required: few linear analyses (fictitious elastic and modified elastic) local calculations

iterative improvement of Vp and Y

Mon. Loading: Algorithmfictitious elastic analysis (f.el) of max. loading σf.el

estimate extension of plastic zone Vp Vp

modify elastic parameters:Ε, ν → E* , ν*

modify loading:real loading → initial strain ε0

estimate TIV Y Y

modified elastic analysis (mea) ρ

elastic-plastic solution by superposition, e.g. σel-pl=σf.el + ρ σel-pl,εel-pl,

displace.etc.

9

Mon. Loading: Iterative Improvement

iterative improvement of Vp:

→

≤>−

VeVp

yplel

v σσ

iterative improvement of Y:

with from previous iterationplelij

−σ

Y – space:

Y

*σ´f.el.

∗Y*=-ρ’

projectionwith from previous iteration

ijρ

10

iteration of Vp partly necessary exact results obtained after maximum 1 iteration

σ

ε

EEtσy

u

• hand calculation ⇒ analytical solution• e.g. for specific configuration (A1/A2, l1/l2; Et/E):

Example 1: Mon. Loading: 2 Bars in Series

1

1.5

2

2.5

3

1 2 3σf.el / σy

withoutiteration

1 iterationexact

εel-pl

εf.el

11

σ

εE

Etσy

• hand calculation ⇒ analytical solution; for specific Et/E:

after 1 iteration close to exact results

∆w0

Example 2a: Mon. Loading: Beam of ideal I-Profile

withoutiteration

1

1.2

1.4

1.6

1.8

2.0

2.2

2.4

1 1.5 2 2.5 3 3.5σf.el / σy

1 iteration

exact

εel-pl

εf.el

12

∆w0 ε

σ

EEt=0σy

hand calculation possible for Et=0 (via initial stress):

exact

1 iteration

withoutiteration

top surface

middle surface

mid

leng

th

clam

ped

plastic zone

1

1.5

2

2.5

3

3.5

1 1.25 1.5 1.75 2 2.25

σf.el / σy

exact 1 iteration

withoutiteration

εel-pl

εf.el

Example 2b: Mon. Loading: Beam of rect. Cross Section

Implementation in FE-codes

extension for specific structures (trusses, beams):- trilinear σ – ε- increm. projection in ES, if > 2 states of loading

implementation in any commercial FE code (ANSYS, ABAQUS, ADINA, ...) usually not easy:

- access to elastic material properties and application of initial strain usually inconsistent:E element, ε0 Gauss or nodal point

- modification of elastic material properties and of loading (initial strain) required during iterative solution

* ** *

Ve

Vp- exact definition of Vp somewhat arbitrary

(Vp, if σv>σy in > 50% of Gauss-points )

implementation at FHL in ANSYS :for general structures (1D, 2D, 3D):

- bilinear σ – ε- T-dependent σy- mon. + cyclic loading (2 states of loading)

ε

σ

E

EtT1

σy,2Et

T2

σy,1

13

Ways of Implementationin ANSYS

1. VEFZOT: APDL-Macro (=ANSYS parameter language; slow)initial strain simulated by:* thermal strain at nodal points* orthotropic coefficient of thermal expansion ( deviatoric)* orthotropic directions changing from el. to el. sometimes poor qual.

2. Userpl.f: FORTRAN user subroutine (compiling and linking required)initial strain simulated by:* inelastic strain at Gauss points* formally equilibrium iterations performed, although not necessary

3. IS-Macros (APDL-Macro) or Ustress.f (FORTRAN-subroutine):initial strain simulated by initial stress at Gauss points

4. Particular implementation for trusses and bending beams:initial strain simulated by ts and ∆t across section

14

Example 3: Mon. Loading: Hollowed Plate

axial strain

σ

εE

Etσy

displace-ment

+ Miseseffective stress

15

exact VFZTplastic zone

plastic zone

exact VFZT

Example 4: Mon. Loading: Crack in a Plate

σ

ε

EEt

monotonic loading

crack

16

Mises effect. stress:

fictitious elastic

VFZT, 3. mea evolutive

plastic zone

Cyclic Loading: ES: Estimation of Y

elastic shakedown (ES): if Ω exists at each point of structure

Y

*σ´f.el

min

*σ´f.el

max

Ω.∗Y*=-ρ’

Y

*σ´f.elmin

*σ´f.elmax

Ω

.∗Y*=-ρ’

Y

*σ´f.el

min

*σ´f.elmax

Ω∗Y*=-ρ’.

Y – space

*σ´f.el

min

*σ´f.el

maxΩ

load histogram:

timemax

mincyclic load

const. loadtime

Vp, if Y*=-ρ‘ is outside Ω:

17

Example 5: ES: 2 Bars Parallel

F

T

timeF

T

020406080

100120140

0 500 1000 1500 2000 2500no. of cycles

εel-pl/εy

numerical effort required to achieve ES: exact: 1000 cy. * 10 loadsteps * 5 iterations

~ ca. 50 000 linear elast. anal. VFZT: 2 f.el. + 2 mea = 4 linear elast. anal.

02468

101214

0 5 10 15 20 25no. of cycles

εel-pl/εy

exact:

σ

ε

EEtσy

εy

elastic shakedown

εacc

identical results

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Example 6: ES: Thermal Stratification

σ

ε

EEt

εy

p

∆T

time

≈ elastic shakedown

0 100 200no. of cycles

10

5

0

hoop εpεy

VFZT:good approxim. with 1 mea (i.e. 3 linear elastic analyses)

shakedown

evolutive

hoop εacc

fictitious elastic analysis

evolutiveanalysis

1 2 3 4 5 6 7 meaVFZT

∆Tcoldhot*

p

19

Cyclic Loading: PS: Estimation of ∆Y

range values: Vp, if ∆σv > 2σy:

load histogram:

timemax

mincyclic load

const. loadtime

Y – space

*σ´f.el

min

*σ´f.el

max

*σ´f.el

min

*σ´f.el

max

∆Y *σ´f.el

min

*σ´f.elmax

∆Y

plastic shakedown (PS): if there is nointersection at least at 1 point of structure

σ

ε

εmean

∆ε

*εacc

20

Cyclic Loading: PS: Estimation of Ymean

mean values: Vp, if σv,min > σy or σv,max > σy:

in Ve,∆:projection of Y*± ½ ∆Y on max (or on min, if closer):

*σ´f.el

min

*σ´f.el

max

Ymean

in Vp,∆:

*σ´f.el

min

*σ´f.elmax

½ ∆Y½ ∆Y

Ymean

∗Y*=-ρ’mean

21

Example 7: PS: Beam under N, ∆w0

∆w0

N

N

w0

time

≈

plastic shakedown

σ

εE

Etσy

∆ε

εacc

fictitious elasticanalysis

evolutive analysis

1 2 3 4 5 6 7 meaVFZT

22

Example 7 (ctd.): PS: Beam: ∆ε

23

VFZT:

∆w0

N

N

w0

time≈

plastic shake-down

fictitious elastic

evolutive: 15 cycles

1 mea

2 mea

3 mea

Example 7 (ctd.): PS: Beam: εacc

24

∆w0

N

N

*w0

time≈

plasticshakedown

fictitious elastic

evolutive: 15 cycles

2 mea

3 mea

5 mea

1 mea

VFZT:

4 mea

Example 8: PS: Cylindrical Shell under Axial

Force and TemperatureCL

T

N

radius

axial ∆ε

axial εacc

fictitious elastic analysis

evolutiveanalysis

1 2 3 4 5 6 7 meaVFZT

σ

εE

Et

Et/E=0.03ν=0.3

N

T

time

≈

σNf.el/σy=0.8

σTf.el/σy=2.2 (PS)

25

fictitious elastic Zarka’s method

evolutive (100 cycles)

1 mea 2 mea 3 mea

Example 8 (ctd.): PS: Cylindrical Shell: axial ∆ε

CL

T

N

radius

Application: Ratcheting-Interaction Diagrams

which combination of load parameters yields ES, PS, R for specific configuration of structure and loading? by shakedown theorems no post-shakedown quantities

load 1

load

2

E

ES

PS R

Zarka‘s method: ES, PS known a priori R not possible: subregions ES2, PS2

post-shakedown quantities (∆ε, εacc)

load 1

load

2

E

ES1

PS1

ES2

PS2

26

εel-pl

εy

without iteration

exact after max.1 iteration

0

5

0 1 2 3σt/σy0

2

1

0 1

E

Sa

SbSc

P

exact with-out iteration

exact with1 iteration

σtσy

σpσy

hand calculation for any configuration of loading (T, F) and hard. param. (Et/E):

exact after maximum 1 iteration

Example 9: Ratch-Inter: 2 Bars Parallel

F

T F

T

time

σ

ε

EEtσy

εy

σp/σy = 0.2

σp/σy = 0.5

27

0

1

2

3

4

5

0 1 2 3 4

εy1εel-pl

σ σt y1

0,05

0,2

0,5

1,0

σp σy1

σ

ε

σy1

σy2

g

e

b f h

a

c

d

j

k

l

σpσy1

σtσy1

0 0,5 1

2

1

0

E

S

P

Example 10: Ratch-Inter: 2 Bars Parallel (Trilinear)

28

TP

0 0.2 10.4 0.6 0.8

1

2

3

4

0

σt/σy

σp/σy

0 iteration

1 iteration2 iterations(≈exact) Pc

Sc

Pb

ESa

Sb

close to exact after 2 iterations

0

0.5

1

1.5

0 1 2 3 4

σpσy

= 0.4

0 iteration

εel-pl

εy

σt/σy

1 iteration

2 iterations

≈ exact withoutiteration

exact

p

∆Ttime

p ∆T

Example 11: Ratch-Inter:Bree Problem

hand calculation for any configuration of loading (T, P) and hard. param. (Et/E):

29

Example 12: Ratch-Inter:Beam under N, ∆w0

accumulated strain:

0 0,5 1,00,00

0,50

1,00

1,50

2,00

2,50

3,00

3,50

10,0-12,0

8,0-10,0

6,0-8,0

4,0-6,0

2,0-4,0

0,0-2,0

εacc/εy:

σw0σy

f.el

σN /σyf.el

∆w0

N w0

N

*time≈

max. load level

30

1.Benchmark CUT-FHL:Thick-walled Cylinder

thick walled cylinder under internal pressure

p

material: lin. kin. hardeningσ

εE

Etσy

εy

mon. loading, load level σv

f.el/σy=1.665 timep

combination ν Et/EA 0.49 0.001B 0.49 0.1C 0.3 0.001D 0.3 0.1

31

1.Benchmark CUT-FHL (ctd.):Thick-walled Cylinder

results:combinations A, B, D: all residual stress- and strain-components after 6 mea

close to exact values (difference <1%);combination C (Et/E=0.001, ν=0.3):

res.

stra

in

radius

exactVFZT

inner outer

hoopaxialradial

res.

stre

ss [M

pa]

radiusinner outer

hoop

axialradial

no improvement with more mea loss in accuracy: why?

32

2.Benchmark CUT-FHL: Hertz Contact

lin. kin. hardeningEt/E = 0.1

σ

εEEtσy

Hertz contact (plane stress)

q (parabolic, regularized)

stationary cycl. loading (ES);load level σv

f.el/σy=1.388 timeq

33

2.Benchmark CUT-FHL (ctd.): Hertz Contact

results: residual stresses along vertical section:

sect

ion

top

x

y

1 mea 7 mea 30 mea

top

ρx

ρy

evolutive

slow approxi-mation to exact solution:why ?

34

3.Benchmark CUT-FHL: Bree Problem

cycl. loading (ES, PS)

p

∆Ttime

thin-walled cylinder under internal pressure and radial temperature gradient

p ∆T

lin. kin. hardening:Et/E = 0.1,T-dependent yield stress

σ

εE

Etσy1

εy1

σy2

40.2Pb

41.91.91.2σt /σy

0.80.80.20.7σp /σy

PcScSbSacombi-nation

f.el

f.el

35

3.Benchmark CUT-FHL (ctd.): Bree Problem

results for combination Pc:

0

0.1

0.2

0.3

0.4

0.5

1 2 3 4 5 6 7 mea

[%]εacc,hoop

εacc,axial

evolutive analysis

VFZT

fictitious elastic analysis

- accum. state at minimum load level:- range values = exact after 2 mea

σy= 200 at inner and outer surface for min. load level (no temperature),σy= 161.1 at inner surface and 238.9 at outer surface for max. load level

(fully developed T-gradient)

εaccreasonable, but not superior: why?

36

Nozzle:T.

σ´ - space

path of fictitious elastic stress

coolantmedium

Further Development: Non-radial Loading

3-Bar Problem: T1 T2 T3T1

time

T2T3

Moving Load:

37

SummaryZarka’s Method (and it’s modifications by FHL) is a simplified method to provide post-shakedown quantities in ES and PS by direct means(∆ε for fatigue, εacc for ratcheting-assessment, displacements)

examples show in case of radial loading:- good approximation to exact solution- at small numerical effort (few linear elastic analyses)- in particular if directional redistribution is not strongly developed- range-values (PS) usually better than mean values

is implemented in commercial FE-program ANSYS for monotonic and cyclic loading

benchmarks between CUT and FHL raised some questions

38