A historical view on the development of shakedown · PDF fileInstitute of General Mechanics RWTH Aachen University A historical view on the development of shakedown theory D. Weichert

Institute of General MechanicsRWTH Aachen University

A historical view on the development of

shakedown theory

D. Weichert

GAMM-2011, Graz, April

shakedown theory

Institute of General MechanicsRWTH Aachen University

A historical view on the development of

shakedown theory

Weichert

2011, Graz, April 19, 2011

shakedown theory

Subject:

Determination of limit states without calculating the

mechanical

Direct M

INTRODUCTIONINTRODUCTION

INSTANTANEOUS COLLAPSE

FAILURE UNDER VARIABLE LOADS

without calculating the evolution of

mechanical field quantities

Direct Methods

LIMIT ANALYSIS

FAILURE UNDER VARIABLE LOADS SHAKEDOWN ANALYSIS

♦ Objectives

Direct access to vital information on structural behaviour

Here in particular: Limit states of structures

o Instantaneous collapse

o Low/High Cycle Fatigue


o Low/High Cycle Fatigue

o Ratchetting

Practical importance:

o Design and assessment of structures and structural elements

beyond elasticity

o No need for step-by-step calculation

o Reduced set of data required

Direct access to vital information on structural behaviour

Here in particular: Limit states of structures

Design and assessment of structures and structural elements operating

step calculation

CONTENTSCONTENTS

•STARTING POINT: ERNST MELAN’s THEOREM

•PORTRAIT OF ERNST MELAN

•LOWER-BOUND SHAKEDOWN THEOREM

•UPPER-BOUND THEOREM

•MODERN DEVELOPMENTS

•EXAMPLES

•THE FUTURE

STARTING POINT: ERNST MELAN’s THEOREM

PORTRAIT OF ERNST MELAN

BOUND SHAKEDOWN THEOREM

BOUND THEOREM

MODERN DEVELOPMENTS


ε

σ

σmax

ο ο

σ

ε ο

σmax σmax

σmin

σmin σmin

Purely elastic

p (x, t) = 0

Shakedown

limt → ∞

. p

(x, t) = 0

Low

ο

σ

ε ο

σ

ε

max σmax

min σmin

Low-cycle fatigue

∆p = ∫ 0

T

. p

(x, t) = 0

Ratchetting

∆p = ∫ 0

T

. p

(x, t) ≠ 0

STARTING POINT: MELAN‘S THEOREM (1936/1938)STARTING POINT: MELAN‘S THEOREM (1936/1938)

*16.11.1890

TH Prag:

Promotion 1917:

Professional Engineer 1916

Statthalterei Graz

Habilitation 1922 TH

ao. Prof. 1923 TH Prag:

o. Prof. 1925 TH Wien:

STARTING POINT: MELAN‘S THEOREM (1936/1938)STARTING POINT: MELAN‘S THEOREM (1936/1938)

*16.11.1890 Brünn – †10.12.1963 Wien

Civil Engineering

1917: Torsion von Umdrehungskörpern

Professional Engineer 1916-1920:

Graz / Wagner-Biró / TH Berlin / Wagner-Biró

1922 TH Wien: Theory of Elasticity

1923 TH Prag: Mechanics in Civil Engng./

Resistance of Materials

1925 TH Wien: Theory of Elasticity

Emeritierung 1962

MELAN‘S WORKMELAN‘S WORK

Die gewöhnl. u. partiellen Differenzengleichungen d. Baustatikgenaue Berechnung v. Trägerrosten, 1942 (mit R. Schindler); Einführung in d. Baustatik, 1950; Wärmespannungen, 1953 (mit H. Parkusin: Beton u. Eisen 18, 1919, S. 83-85; Ein Btr(Teplitz-Schönau) 1, 1920, S. 417-19, 427-29; Die Verteilung d. Kraft in e. Streifen v. Breite, in: Zs. f. angew. Math. u. Mechanik 5, 1925, S. 314durch e. Einzelkraft im Innern beanspruchten Halbscheibe, ebd. 12, 1932, S. 343Application of Theories of Elasticity and PlasticityBoston Society of Civil Engineers 23, 1936, S. 317 ff.; Boston Society of Civil Engineers 23, 1936, S. 317 ff.; Integralgleichungen auf Probleme d. Statik, in: Plastizität d. räuml. Kontinuums, in: IngenieurWärmespannungen in e. Scheibe infolge e. wandernden Wärmequelle, ebd. 20, 1952, S. 4648; Ein Btr. z. Auflösung linearer Gleichungssysteme mit positiv definiter Matrix mittels Iteration, in: Sitzungsberr. d. Österr. Ak. d. Wiss. 151, 1942, S. 249Spannungs- u. Verzerrungszustand e. gelochten Scheibe b. nichtlinearem SpannungsDehnungsgesetz, in: Österr. Ingenieur-Archiv 1, 1946, S. 14stationärer Wärmefelder, ebd. 9, 1955, S. 171 ff.; Kugel, in: Acta Physica Austriaca 10, 1956, S. 81ihrer Höhe abgespannter Maste, in: Der Bauing

Bd. 2; Massivbrücken (Fritsche), Bd. 3; Stahlbrücken (Hartmann), 1948/50. [NDB 16 (1990]

. u. partiellen Differenzengleichungen d. Baustatik, 1927 (mit F. Bleich); Die , 1942 (mit R. Schindler); Einführung in d. Baustatik,

Parkus); Die Druckverteilung durch e. elast. Schicht, Btr. z. Torsion v. Rotationskörpern, in: Techn. Bll. 29; Die Verteilung d. Kraft in e. Streifen v. endl.

. Math. u. Mechanik 5, 1925, S. 314-18; Der Spannungszustand d. durch e. Einzelkraft im Innern beanspruchten Halbscheibe, ebd. 12, 1932, S. 343-46; The

Plasticity to Foundation Problems, in: Journal of theEngineers 23, 1936, S. 317 ff.; Anwendung linearer Engineers 23, 1936, S. 317 ff.; Anwendung linearer

, in: Annali di Matematica 16, 1937, S. 263-73; Zur

, in: Ingenieur-Archiv 9, 1938, S. 116-26; Wärmespannungen in e. Scheibe infolge e. wandernden Wärmequelle, ebd. 20, 1952, S. 46-

. z. Auflösung linearer Gleichungssysteme mit positiv definiter Matrix mittels . d. Wiss. 151, 1942, S. 249-54; Ein rotationssymmetr.

u. Verzerrungszustand e. gelochten Scheibe b. nichtlinearem Spannungs-Archiv 1, 1946, S. 14-21; Spannungen infolge nicht

stationärer Wärmefelder, ebd. 9, 1955, S. 171 ff.; Wärmespannungen b. d. Abkühlung e. 10, 1956, S. 81-86; Die genaue Berechnung mehrfach in

Bauing. 35, 1960, S. 416 ff. – Hrsg.: Der Brückenbau,

Bd. 2; Massivbrücken (Fritsche), Bd. 3; Stahlbrücken (Hartmann), 1948/50. [NDB 16 (1990]

MELAN‘S WORKMELAN‘S WORK

Melan used modern and advanced mathematical methods to solve general, but

practically motivated problems practically motivated problems

used modern and advanced mathematical methods to solve general, but

practically motivated problems practically motivated problems

MELAN‘S MELAN‘S LIFELIFE

Brilliant student

Father famous engineer and professor

Highly estimated by students

Member of Academy of SciencesMember of Academy of Sciences

Married, no Children, wife passed away 1948

Father famous engineer and professor

Married, no Children, wife passed away 1948

LOWERLOWER--BOUND DIRECT METHODS BOUND DIRECT METHODS

♦♦♦♦ Statement *

The structure shakes down if there exist

field and a sanctuary of elasticity C

with

Ω∈∀∈ xx ),(C

with

where

∀≤= xxx , ) ,( )( Yσ f C

Eα

* Melan, E.: Ingenieur-Archiv I X, 116-126, (1938 )

** Nayroles, B.; Weichert, D.: C.R. Acad. Sci. 316, 1493-1498 (1993)

BOUND DIRECT METHODS BOUND DIRECT METHODS

The structure shakes down if there exist α > 1, a time-independent residual stress

such that **:

Γ

C

εεεε .

p σσσσ

Γ

σσσσ

−−−−

Ω∈

1498 (1993)

ILLUSTRATION OF THE BASIC IDEAS OF SHAKEDOWNILLUSTRATION OF THE BASIC IDEAS OF SHAKEDOWN

s2

F (s) = 0

ILLUSTRATION OF THE BASIC IDEAS OF SHAKEDOWNILLUSTRATION OF THE BASIC IDEAS OF SHAKEDOWN--THEORYTHEORY

s1


s2

F (s) = 0


s1


s2

F (s) = 0


s1


s2

F (s) = 0


s1


s2

F (s) = 0


s1


s2

F (s) = 0


s1


s2

F (s) = 0


s1


s2

F (s) = 0


s1


s2

F (s) = 0


s1


s2

F (s) = 0


Unfeasable

s1

Additive decomposition of total strains into an elastic and a plastic part:

=

Convexity of the yield surface and validity of normality rule:

⟨(x) − (s)

CLASSICAL BASIC ASSUMPTIONSCLASSICAL BASIC ASSUMPTIONS

Linear elastic-perfectly plastic or linear elastic

σY

ε

σ

Perfectly plastic

Linear kinematical hardening

Additive decomposition of total strains into an elastic and a plastic part:

= e +

p

Convexity of the yield surface and validity of normality rule:

(s)(x), . p

(x)⟩ ≥ 0

CLASSICAL BASIC ASSUMPTIONSCLASSICAL BASIC ASSUMPTIONS

perfectly plastic or linear elastic-unlimited linear hardening material:

s2

pe.

s1s(s)

Fs

Grüning (1926), Bleich (1932), Melan (1936

Koiter (1956, 1960), Gvozdev (1938), Drucker, Prager & Greenberg (1951),

Hodge (1959)

Leading ideas: Positiveness of dissipation, boundedness of free

energy, failure if rate of external work exceeds dissipation.

FOUNDATIONSFOUNDATIONS

Shakedown analysis (SDA) and Limit analysis (LA) separately

developed.

“Incomplete” formulations, leading to a “static” and a “kinematic” approach. Bounding methods are naturally introduced

926), Bleich (1932), Melan (1936), Symonds (1951),

), Gvozdev (1938), Drucker, Prager & Greenberg (1951),

Leading ideas: Positiveness of dissipation, boundedness of free energy, failure if rate of external work exceeds rate of

Shakedown analysis (SDA) and Limit analysis (LA) separately

“Incomplete” formulations, leading to a “static” and a “kinematic” approach. Bounding methods are naturally introduced

LINES OF DEVELOPMENTLINES OF DEVELOPMENT

ApplicationsApplications toto specificspecific

TheoreticalTheoretical extensionsextensions

• geometrical effects

• larger classes of material

NumericalNumerical SDA SDA andand LA LA

LifetimeLifetime predictionprediction

specificspecific structuresstructures

material behavior

Thinwalled structures, Beams, Frames, Plates, Shells

Theories of plastic hinges andyield-lines, interaction diagramms, upper bounds for collapse loads

PragerPrager, , OlszakOlszak, , HodgeHodge, , SawczukSawczuk, ,

FoundationsRoads, DamsContact

Slip-line and slipsimplified lowerassociated flow

APPLICATIONS TO SPECIFIC STRUCTURESAPPLICATIONS TO SPECIFIC STRUCTURES

PragerPrager, , OlszakOlszak, , HodgeHodge, , SawczukSawczuk, , SokólSokól--SupelSupel, , DuszekDuszek, , ZyczkowskiZyczkowski, , König, König, MahrenholtzMahrenholtz, Leers, Stein, , Leers, Stein, Borkowski, Borkowski, HaythorthwaiteHaythorthwaite, , HeymanHeyman, , Horne, Horne, BreeBree, Neal, , Neal, GokhfeldGokhfeld, , CherniavskiCherniavski, Jones, Maier, , Jones, Maier, CorradiCorradi, , ControContro, Groß, Groß--WeegeWeege, Weichert, , Weichert, MassonnetMassonnet, Save, , Save, GrundyGrundy, Tin, Tin--LoiLoi, , Dang Hung, Dang Hung, KaliszkiKaliszki

JohnsonJohnson, Sharp, Booker, , Sharp, Booker, Weichert, Collins, Weichert, Collins, BoulbibaneBoulbibane, Maier, Hai, Maier, HaiShiauShiau, Pastor, Anderson, Wong, Kapoor, Pastor, Anderson, Wong, Kapoor

Foundations, Pavements, Dams, Soils, Rolling

slip surface theories, lower bound methods, flow rules

Composites, PorousMaterials

Local failure, homogenisationtechniques, Numerical analysis on level of RVE, Design of composites, multi-physics modeling

APPLICATIONS TO SPECIFIC STRUCTURESAPPLICATIONS TO SPECIFIC STRUCTURES

, Sharp, Booker, , Sharp, Booker, RaadRaad, , Weichert, Collins, Weichert, Collins, CliffeCliffe, Sloan, , Sloan, PonterPonter, ,

, Maier, Hai, Maier, Hai--Su, Hossain, Su, Hossain, , Pastor, Anderson, Wong, Kapoor, Pastor, Anderson, Wong, Kapoor

Tarn Tarn & Dvorak,& Dvorak, PonterPonter, , SuquetSuquet, Maier, , Maier, Weichert, Weichert, CocchettiCocchetti, , CarvelliCarvelli, Schwabe, , Schwabe, HachemiHachemi, , DébordesDébordes, , MagoriecMagoriec, Li, , Li, ChenChen

Mathematical foundations

Mathematical setting (convexanalysis), initial b.v.p., dynamicshakedown, bi-potential approach, optimisation

Prager, Prager, CeradiniCeradini, , NayrolesNayroles, , DébordesDébordes, , SuquetSuquet, , KamenjarzhKamenjarzh, Weichert, , Weichert,

Geometrical

Second order effectsstability problems

MaierMaier, Weichert, Groß, Weichert, GroßPolizzottoPolizzotto, Stumpf, , Stumpf,

THEORETICAL EXTENSIONSTHEORETICAL EXTENSIONS

SuquetSuquet, , KamenjarzhKamenjarzh, Weichert, , Weichert, RafalskiRafalski, Pham, , Pham, KaliszkiKaliszki, Stein, , Stein, WiechmannWiechmann, De , De SaxcéSaxcé, , BousshineBousshine

PolizzottoPolizzotto, Stumpf, , Stumpf, QuocQuoc Son, Son, ZyczkowskiZyczkowskiSiemazskoSiemazsko, , SawczukSawczuk

Geometrical effects

effects, large strains, problems

, Weichert, Groß, Weichert, Groß--WeegeWeege, , , Stumpf, , Stumpf, SaczukSaczuk, Gary, , Gary,

Material laws

Hardening, Damage, thermal effects, Non-associated flow rules, Visco-plasticity, cracked bodies, interface failure

MelanMelan, Maier, König, , Maier, König, GokhfeldGokhfeld, , CherniavskiCherniavski, Mandel, Weichert, Groß, Mandel, Weichert, Groß--

THEORETICAL EXTENSIONSTHEORETICAL EXTENSIONS

, Stumpf, , Stumpf, SaczukSaczuk, Gary, , Gary, ZyczkowskiZyczkowski, König, , König, TritschTritsch, , SawczukSawczuk, , DuszekDuszek

CherniavskiCherniavski, Mandel, Weichert, Groß, Mandel, Weichert, Groß--WeegeWeege, , HachemiHachemi, , PonterPonter, , DoroszDorosz, , PolizzottoPolizzotto, Stein, Pham, , Stein, Pham, BelouchraniBelouchrani

Efficient optimisation is one key toshakedown and limit analysis

Non-linear optimisation (convex)

For technical problems, the number of

NUMERICAL METHODSNUMERICAL METHODS

For technical problems, the number ofoptimisation variables is of order 104 –106

of

Reduced base approach

Stein, Zhang, Staat, Heitzer, Vu D.K.

Interior point approach

Pastor, Krabbenhoft, Akoa, Weichert, Simon

Selective algorithm

Weichert, Mouhtamid, Hachemi, Simonof–

Weichert, Mouhtamid, Hachemi, Simon

Simplified Methods

Zarka, Inglebert

Elastic compensation

Ponter, Engelhardt

♦♦♦♦ Local thermodynamic potentials

ΨF (e,T) = ΨeF (e,T)

ΨM (e,, T, D) = ΨeM

(e,T, D) + ΨpM

ΨC ([[[[ue]]]] d) = Ψe

C ([[[[u

e]]]], d)

with

EXAMPLE FOR ADVANCED FORMULATIONEXAMPLE FOR ADVANCED FORMULATION

Ω = Ω M ∪ Ω F ∪Ω C

where

ΩF : elastic domain (fibre phase)

ΩM : elastic-plastic domain (matrix phase)

ΩC : band-shaped domain (cohesive zone)

in Ω M pM

(, D) in Ω F

in Ω C


Ω C δ

Γ

Ω F

Ω M

ΨeF =

1

2ρ (e − α ϑ ϑ I):L: (e − α ϑ

ΨeM

= 1

2ρ (1 − D) (e − α ϑ ϑ I):L: (

ΨpM

= 1

2ρ (1 − D)

T.Z.

ΨeC =

12 [ue]

T.Q~

δ.[ue]


C 2 δ

with

Q~

δ = (I − d)Q δ =

(1 − d

Θ

Θ

and Qδ = nT.C.n/δ

Coupling terms Θi represent the directional interaction of damage.

ϑ I) + Cε ϑ 2 in Ω F

(e − α ϑ ϑ I) + Cε ϑ 2 in Ω M

in Ω M

in Ω C


C

dn)Qδn Θ

2 Θ

3

Θ2

(1 − dt)Qδt Θ4

Θ3

Θ4

(1 − ds)Qδs

represent the directional interaction of damage.

♦♦♦♦ Forces and fluxes

F = ρ ∂ ΨF

∂ e = L:(e − α ϑ ϑ I)

M = ρ ∂ ΨM

∂ e = (1 − D) L:(e − α ϑ ϑ

= − ρ ∂ ΨM

∂ e = − (1 − D) Z.


Y = − ρ ∂ ΨM

∂ D =

12

(e − α ϑ ϑ I):L:(

= − ∂ ΨC

∂ [ue] = Q

~ δ.[ue]

y = − ∂ ΨC

∂ d =

12 [ue]

T.Qδ.[ue]

g = − grad T

T

in ΩF

ϑ I) in ΩM

in ΩM


e − α ϑ ϑ I) + 12

T

.Z. in ΩM

in ΩC

in ΩC

in Ω

♦♦♦♦ Clausius-Duhem

M:.

p + :

.[u.

Assumption: each term fulfils respectively the dissipation inequality

♦♦♦♦ Dissipative potentials

ADVANCED FORMULATIONADVANCED FORMULATION

♦♦♦♦ Dissipative potentials

F(M, ,D) ≤ 0 in Ω M

G(,d) ≤ 0 in Ω C

♦♦♦♦ Specific forms

F = 32 (

M

1 − D −

1 − D) : (

M

1 − D

G = (τn

1 − dn)

2

+ 1

β 2(

τt

1 − dt)

2

+ 1

γ

:. + Y D

. + g.q ≥ 0

u. in]+ y.d

. ≥ 0

Assumption: each term fulfils respectively the dissipation inequality

D −

1 − D)

1/2

− σY ≤ 0 in Ω M

1

γ 2(

τs

1 − ds )

2

1/2

− σmax ≤ 0 in Ω C

♦♦♦♦ Maximum dissipation principle

(M− (s)M

):.

p + (− (s)).. ≥ 0

(− (s)).[u. in] ≥ 0

where

.

p ∈ δ F; . ∈ δ F in Ω M

. in ∈ δ Ω


[u. in] ∈ δ G in Ω C

♦♦♦♦ Unified presentation

= ∂ P

∂ ; ⟨ − (s)⟩.

. ≥ 0

∀(s)M

; (s) ∈ F−−−−

⊂ F in Ω M

∀ (s) ∈ G−−−−

⊂ G in Ω C



Find αSD = max°(r), °(r), °

α

subjected to

(s)(y) ∈ G

−−−−(y)

(s)(y) ∈ F−−−−

(y

<(s)> =

such that (s) = α (c) + °(r) ,

(s)

where τ

(s)

n = n.(s).n , τ

(s)

t = n.

Important: Material damage has to be controled by the use of appropriate damage mode.g. linking damage variables to the plastic deformation.


) ∀y ∈ Ω C

y) ∀y ∈ Ω M∪Ω F

∀y ∈ Ω

= α (c) + °(r)

.(s).t , n.t = 0 , n.n = 1, t.t = 1

Material damage has to be controled by the use of appropriate damage models e.g. linking damage variables to the plastic deformation.

Complementary to

Easy to adapt to

STRATEGY FOR NUMERICAL APPLICATIONSSTRATEGY FOR NUMERICAL APPLICATIONS

Robust

Fast

to step-by-step methods:

to commercial FE-codes

STRATEGY FOR NUMERICAL APPLICATIONSSTRATEGY FOR NUMERICAL APPLICATIONS

♦♦♦♦ Direct method as an optimisation problem

( )( )

>

≤+

=

0

0,

0

..

max

α

σα

ρ

α

YiijEir

jij

PF

C

ts

APPLICATION OF THE INTERIOR POINT METHODAPPLICATION OF THE INTERIOR POINT METHOD

( )( ) ( )

++−=

++=

==

≤≤

=

ℜ∈

where

.NRESNRESNKSm

.NGSNGSn

nkmi

xxx

c

ts

f

n

n

kLkkL

i

l

l

)2)112(()3(

)2)12(()16(

,1;,1

0

..

)(min

x

xn

x

* Akoa, F. et al.: J. Global. Optim. 37: 609-630 (2007)

* Nguyen, A.D.: PhD., RWTH Aachen (2007)

* Mouhtamid, S.: PhD., RWTH Aachen (2007)

* Simon, J.: .: PhD., RWTH Aachen (2011), in process

Direct method as an optimisation problem *

( )( )

<<∞−<<−∞

=−+

=

−

0,0

0,

0

..

min

r

rYiijEir

jij

s

sPF

C

ts

α

σα

ρ

α


( )

( )( )( )

+−==−+

−===

=

−=

mNRESNKSisPF

NRESNKSiCc

s

f

rYiijEir

jij

i

rj

),13(,0,

)3(,1,0

,,

)(

σα

ρ

ρα

α

x

x

x

o

o

o


* Nguyen, A.D.: PhD., RWTH Aachen (2007)

* Mouhtamid, S.: PhD., RWTH Aachen (2007)


PIPEPIPE--JUNCTION WITH 60JUNCTION WITH 60OO ANGLEANGLE

Pipe Nozzle

Length [mm] 600.00 157.15

Thickness [mm] 3.6 2.6

Inner radius [mm] 53.55 18.60

Material: steel, all material parameters

as temperature-independent

ANGLEANGLE

parameters are considered

independent


Element-type: square, 8 nodes per

solid45 (structural), solid70 (thermal) in ANSYS

Number of elements: 510

Number of nodes: 1136

Boundary conditions:

Left end of pipe is clamped

Right end of pipe is fixed in longitudinal direction

Nozzle is assumed closed without restrictions on displacements

ANGLEANGLE

per element

solid45 (structural), solid70 (thermal) in ANSYS

Right end of pipe is fixed in longitudinal direction

Nozzle is assumed closed without restrictions on displacements

Equivalent von Mises stresses due to internal pressure:


Equivalent von Mises stresses due to internal pressure:

ANGLEANGLE

Equivalent von Mises stresses due to temperature:


Equivalent von Mises stresses due to temperature:

ANGLEANGLE

Results of the shakedown analysis:

PIPEPIPE--JUNCTION WITH 60JUNCTION WITH 60OO ANGLEANGLEANGLEANGLE

THERMOTHERMO--MECHANICAL EXAMPLES MECHANICAL EXAMPLES

Tube under moving thermal loading

Τ0

Τi = Τ0 + ∆T Q

h

R

Mechanical characteristics

∆L

L

Q

Young’s modulus (MPa)

Poisson’s ratio

Yield stress (MPa)

Thermal expansion coefficient

MECHANICAL EXAMPLES MECHANICAL EXAMPLES

Geometries and initial loading

L/R = 0.733 Q0 = h σY

h/R = 1/400 ∆T0 = 2 σY/(E αϑ)

∆L/L = 0.06

Thermal expansion coefficient ( K−1)

2.1×10+5

0.3

360

1.2×10−5


Shakedown domains

0,5

0,8

1,0

∆ϑ∆ϑ 0

0,0

0,3

0,0 0,3

Ponter

Gross-Weege

Present results

∆ϑ 0

* Ponter, A.R.S.; Karadeniz, S.: J. Appl. Mech. 52, 883-889 (1985)

* Gross-Weege, J.: , Doctor thesis, Ruhr-Universität, Bochum (1988)


Mechanism (B)

(Lokal)

0,5 0,8 1,0

Mechanism (A)

(Global)

Weege

Present results

Q

Q0

889 (1985)

(1988)


Circular plate under pressure and temperature

p

R

h

ϑ

Mechanical characteristics

Young’s modulus (MPa)

Poisson’s ratio

Yield stress (MPa)

Thermal expansion coefficient


Circular plate under pressure and temperature

Geometries and initial loading

h/R = 1/25

p0 = 4σYh2/[(1 + ν) R2]

ϑ0 = 6(1 − ν)σY/(E αϑ)

Thermal expansion coefficient ( K−1)

1.6×10+5

0.3

360

2.0×10−5


Loading domains

0,6

0,8

1

c

a

0/ϑϑ

a. Alter. Plasticity

b. Alter. Plasticity

c. Accum. Plasticity

p = const.)(

* Gokhfeld, D.A; Cherniavsky, O.F.: Sijthoff-Noordhoff, Leyden

0

0,2

0,4

0 0,2 0,4 0,6 0,8 1

Present solution

b

p/p0

Gokhfeld &

Cherniavsky (1980)


0,6

0,8

1

Dc = 0.24

εR = 0.37

εD = 0.02

0/ϑϑ

Noordhoff, Leyden (1980)

0

0,2

0,4

0 0,2 0,4 0,6 0,8 1

p/p0

Without damage

Model of Lemaitre (1985)

Model of Shichun & Hua (1990)

Thin pipe under internal pressure and temperature:

Pipe Pipe underunder thermomechanicalthermomechanical loadingloading

Thin pipe under internal pressure and temperature:

∆T and p vary independently

all parameters are assumed as

temperature-independent

loadingloading

R/h = 10

FE-mesh and relevant numbers of optimization problem:

Elements NE

Gaussian points

Pipe Pipe underunder thermomechanicalthermomechanical loadingloading

Element-type:

Nodes NK

Corners NC

Variables

Equality constraints

Inequality constraints

mesh and relevant numbers of optimization problem:

NE 600

Gaussian points NG 4 800

loadingloading

type: solid, 8 nodes per element

solid45, solid70 in ANSYS

984

4

196 801

Equality constraints mE 146 952

Inequality constraints mI 38 400

Results of shakedown analysis:

Pipe Pipe underunder thermomechanicalthermomechanical loadingloadingloadingloading

Nbr of iterations: 7645

Running time: 2144 s

working station:

Sun W 1100z

CPU 2,4GHz

RAM 5120 MB

SELECTIVE ALGORITHM*SELECTIVE ALGORITHM*

IDEA: Concentration on “active” zones

Strategy:

Gauß-Points are “active” if

If an element contains an active GP, then all GP in this element and in all

surrounding elements are set active.

0.8eq Yσ σ≥

SD/LA factor are calculated on the active zone

Evolution of active zone has to be monitored

____________________________________________* J. Simon, M. Kreimeier (ongoing research)

IDEA: Concentration on “active” zones

If an element contains an active GP, then all GP in this element and in all

surrounding elements are set active.

0.8eq Y GP is activeσ σ ⇒

SD/LA factor are calculated on the active zone

Evolution of active zone has to be monitored

____________________________________________

SELECTIVE ALGORITHM SELECTIVE ALGORITHM -- EXAMPLEEXAMPLE

Element-type: square, 8 nodes per element

Number of elements: 400

Number of nodes: 882

Loading: surface tractions px and py with angle 30

EXAMPLEEXAMPLE

element solid45 in ANSYS

with angle 30o in loading space

SELECTIVE ALGORITHM SELECTIVE ALGORITHM -- EXAMPLEEXAMPLEEXAMPLEEXAMPLE







Direct Methods target without detour limit states of structures.

They can be used for failure prediction and safe design of structures.

They are limited to certain classes of material laws.

They are complementary to incremental simulation methods.

Calculation efficiency has significantly improved.

CONCLUSIONSCONCLUSIONS

Calculation efficiency has significantly improved.

Perspectives: Further reduction of CPU

material behaviour, lifetime prediction

EN 1994-2)

target without detour limit states of structures.

They can be used for failure prediction and safe design of structures.

They are limited to certain classes of material laws.

hey are complementary to incremental simulation methods.

significantly improved. significantly improved.

Perspectives: Further reduction of CPU-time, extension to larger classes of

lifetime prediction, introduction into standards (BS 5500,

DESIGN METHODS FOR COMPOSITES*DESIGN METHODS FOR COMPOSITES*

Local design: exclusively RVE-level

Global design: interaction between structure and RVE

Heterogeneous Material

Localisation

Strain Method

___________________________

* M. Chen (ongoing research)

LA in global

Micro results

Macro results

Homogenised

Parameters

Homogenisation

Globalisation

Elastic Properties

DESIGN METHODS FOR COMPOSITES*DESIGN METHODS FOR COMPOSITES*

level

Global design: interaction between structure and RVE

Heterogeneous Material

RVE

Localisation

LM

LA in global

design

Limit domain

Yield surface

Approximationof von Mises Yield criterion

Plastic Properties

DESIGN METHODS : Plastic propertiesDESIGN METHODS : Plastic properties

Validation test


Comparation

Comparison between 2D and 3D elements

Matrix Fiber

E(GPa)

υ

σy(MPa)

210

0.3

280

2.1

0.2

140


Comparation

0

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8

U1/U0

U2/U0

SD_3D

LM_3D

EL_3D

SD_2D*

LM_2D*

EL_2D*

* F. Schwabe. RWTH-Aachen. Phd. Thesis (2000)

Limit Load

Matrix(A1) Fiber(A12O3)

E(GPa)

υ

σy(MPa)

70

0.3

80

370

0.3

2000

Comparison with incremental method


0

20

40

60

80

100

0 20 40 60 80 100

PX/MPa

PY/MPa

Inc.Meth.

LANCELOT

IPDCA

,

22E

11E

Influence of geometrical parameters


,

Normalized Shakedown Domain

0

0.2

0.4

0.6

0.8

0 0.2 0.4

(U1/U0)/K

(U2/U0)*K

Normalized Elastic Domain

0

0.1

0.2

0.3

0.4

0 0.1 0.2 0.3 0.4

(U1/U0)/K

(U1/U0)*K 45

35

30

20

M / Ma K b K= = ⋅

tan

M = constant value

K

a b

ϕ=

⋅ =

,

Influence of geometrical parameters


,

Normalized Shakedown Domain

0.4 0.6 0.8

(U1/U0)/K

45

35

30

20

Normalized Limit Domain

0

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8 1

(U1/U0)/K

(U2/U0)*K 45

35

30

20


Square pattern Rotated pattern

Radius of fiber R=15 R=15

A 50 70.7

B 50 70.7

Volume fraction 7.0686

Shakedown Domain

Influence of pattern

0

1

2

3

4

5

0 1 2 3 4U1/U0

U2/U0

Square

Rotated Square

Hexagonal


Rotated pattern Hexagonal pattern

R=15

93.0

53.7 Square pattern

Rotated pattern

Hexagonal pattern

5 6

Rotated Square

Hexagonal

DESIGN METHODS : Elastic propertiesDESIGN METHODS : Elastic properties

Homogenisation Theory

060

80

100

120

140

160

Youngs M

odulu

s/M

pa

00.24

0.26

0.28

0.3

0.32

0.34P

ois

son R

atio

Homogenisation Theory

DESIGN METHODS : Elastic propertiesDESIGN METHODS : Elastic properties

20.0635 - 0.8389 72.3360R RE +=

Material Al Al2O3

E (MPa) 70000 370000

υ 0.3 0.3

σy (MPa) 80 2000

5 10 15 20 25 30 35 40

Radius of Fiber/mm

homogenized Youngs Modulus

5 10 15 20 25 30 35 40

Radius of Fiber/mm

homogenized Poisson Ratio

20.000047 0.000871 0.298097R Ru += − +

Limit Domain

100

DESIGN METHODSDESIGN METHODS

State 1: Onset of plasticity

State 2: Debonding

State 3: Overall plastic flow

0

20

40

60

80

0 20 40 60 80 100

PX/MPa

PY/M

Pa

Limit_R25

Limit_R15

Py

Px

Limit Domain

100

150

200

250

PY/MPa

R5

R10

R15

R20

R25

R30

R35

R40

DESIGN METHODDESIGN METHOD

0

50

100

0 50 100 150 200

PX/MPa

R10

R15

R20

R25

R30

R35

R40

> ca.30%η

Limit domain increased quickly

250

Documents

A historical view on the development of shakedown · PDF fileInstitute of General Mechanics RWTH Aachen University A historical view on the development of shakedown theory D. Weichert