a historical view on the development of shakedown · pdf fileinstitute of general mechanics...
TRANSCRIPT
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
INTRODUCTIONINTRODUCTION
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
INTRODUCTIONINTRODUCTION
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
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
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
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
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
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
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
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
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
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
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
EXAMPLE FOR ADVANCED FORMULATIONEXAMPLE FOR ADVANCED FORMULATION
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]
EXAMPLE FOR ADVANCED FORMULATIONEXAMPLE FOR ADVANCED FORMULATION
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
EXAMPLE FOR ADVANCED FORMULATIONEXAMPLE FOR ADVANCED FORMULATION
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.
EXAMPLE FOR ADVANCED FORMULATIONEXAMPLE FOR ADVANCED FORMULATION
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
EXAMPLE FOR ADVANCED FORMULATIONEXAMPLE FOR ADVANCED FORMULATION
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
EXAMPLE FOR ADVANCED FORMULATIONEXAMPLE FOR ADVANCED FORMULATION
[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
EXAMPLE FOR ADVANCED FORMULATIONEXAMPLE FOR ADVANCED FORMULATION
EXAMPLE FOR ADVANCED FORMULATIONEXAMPLE FOR ADVANCED FORMULATION
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.
EXAMPLE FOR ADVANCED FORMULATIONEXAMPLE FOR ADVANCED FORMULATION
) ∀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
APPLICATION OF THE INTERIOR POINT METHODAPPLICATION OF THE INTERIOR POINT METHOD
( )
( )( )( )
+−==−+
−===
=
−=
mNRESNKSisPF
NRESNKSiCc
s
f
rYiijEir
jij
i
rj
),13(,0,
)3(,1,0
,,
)(
σα
ρ
ρα
α
x
x
x
o
o
o
APPLICATION OF THE INTERIOR POINT METHODAPPLICATION OF THE INTERIOR POINT METHOD
* Nguyen, A.D.: PhD., RWTH Aachen (2007)
* Mouhtamid, S.: PhD., RWTH Aachen (2007)
APPLICATION OF THE INTERIOR POINT METHODAPPLICATION OF THE INTERIOR POINT METHOD
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
PIPEPIPE--JUNCTION WITH 60JUNCTION WITH 60OO ANGLEANGLE
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:
PIPEPIPE--JUNCTION WITH 60JUNCTION WITH 60OO ANGLEANGLE
Equivalent von Mises stresses due to internal pressure:
ANGLEANGLE
Equivalent von Mises stresses due to temperature:
PIPEPIPE--JUNCTION WITH 60JUNCTION WITH 60OO ANGLEANGLE
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
THERMOTHERMO--MECHANICAL EXAMPLES MECHANICAL EXAMPLES
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)
MECHANICAL EXAMPLES MECHANICAL EXAMPLES
Mechanism (B)
(Lokal)
0,5 0,8 1,0
Mechanism (A)
(Global)
Weege
Present results
Q
Q0
889 (1985)
(1988)
THERMOTHERMO--MECHANICAL EXAMPLES MECHANICAL EXAMPLES
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
MECHANICAL EXAMPLES MECHANICAL EXAMPLES
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
THERMOTHERMO--MECHANICAL EXAMPLES MECHANICAL EXAMPLES
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)
MECHANICAL EXAMPLES MECHANICAL EXAMPLES
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
SELECTIVE ALGORITHM SELECTIVE ALGORITHM -- EXAMPLEEXAMPLEEXAMPLEEXAMPLE
SELECTIVE ALGORITHM SELECTIVE ALGORITHM -- EXAMPLEEXAMPLEEXAMPLEEXAMPLE
SELECTIVE ALGORITHM SELECTIVE ALGORITHM -- EXAMPLEEXAMPLEEXAMPLEEXAMPLE
SELECTIVE ALGORITHM SELECTIVE ALGORITHM -- EXAMPLEEXAMPLEEXAMPLEEXAMPLE
SELECTIVE ALGORITHM SELECTIVE ALGORITHM -- EXAMPLEEXAMPLEEXAMPLEEXAMPLE
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
DESIGN METHODS : Plastic propertiesDESIGN METHODS : Plastic properties
Comparation
Comparison between 2D and 3D elements
Matrix Fiber
E(GPa)
υ
σy(MPa)
210
0.3
280
2.1
0.2
140
DESIGN METHODS : Plastic propertiesDESIGN METHODS : Plastic properties
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
DESIGN METHODS : Plastic propertiesDESIGN METHODS : Plastic properties
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
DESIGN METHODS : Plastic propertiesDESIGN METHODS : Plastic properties
,
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
DESIGN METHODS : Plastic propertiesDESIGN METHODS : Plastic properties
,
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
DESIGN METHODS : Plastic propertiesDESIGN METHODS : Plastic properties
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
DESIGN METHODS : Plastic propertiesDESIGN METHODS : Plastic properties
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