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Limit States of Materials and Structures Dieter Weichert Institute - - PowerPoint PPT Presentation

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Limit States of Materials and Structures Dieter Weichert Institute of General Mechanics RWTH-Aachen University JUBILEE SCIENTIFIC CONFERENCE PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH Methodology:

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JUBILEE SCIENTIFIC CONFERENCE

“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

Limit States of Materials and Structures

Dieter Weichert

Institute of General Mechanics RWTH-Aachen University

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“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

Methodology: Direct Methods

Access to vital information on structural behaviour without path-dependent analysis of the process Here: Limit states due to inelastic effects

  • Instantaneous collapse
  • Low/High Cycle Fatigue
  • Ratchetting

INSTANTANEOUS COLLAPSE  LIMIT ANALYSIS FAILURE UNDER VARIABLE LOADS  SHAKEDOWN ANALYSIS

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“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

Practical interest:

  • Design and assessment of structures and structural

elements operating beyond elasticity

  • No need for step-by-step calculation
  • Reduced set of data required
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“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

CONTENT THEORETICAL DEVELOPMENTS (LOWER-BOUND METHODS)

CLASSICAL FORMULATION EXTENSIONS

NUMERICAL METHODS

INTERIOR POINT OPTIMISATION SELECTIVE ALGORITHM

EXAMPLES

PIPES AND PRESSURE VESSELS COMPOSITES

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“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

ε σ σmax ο ο σ ε ο σ ε ο σ ε σmax σmax σmax σmin σmin σmin σmin

Purely elastic 

p (x, t) = 0

Shakedown lim

t → ∞

. p (x, t) = 0 Low-cycle fatigue ∆

p = ∫ 0

T 

. p (x, t) = 0 Ratchetting ∆

p = ∫ 0 T

 . p (x, t) ≠ 0

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“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

Acknowledgements

  • J. Groß-Weege

J.B. Tritsch

  • M. Hachemi
  • A. Belouchrani
  • M. Boulbibane

Hamadouche

  • B. Nayroles
  • L. Raad
  • F. Schwabe

A.D. Nguyen

  • S. Mouhtamid
  • M. Chen
  • J. Simon
  • G. Chen
  • Ch. Broeckmann
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“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

  • Additive decomposition of total strains into elastic and plastic part:

 = e + p

  • Convexity of the yield surface and validity of normality rule:

〈(x) − (s)(x), 

. p(x)〉 ≥ 0

  • Linear elastic-perfectly plastic or linear elastic-unlimited linear hardening

material:

σY ε σ

Perfectly plastic Linear kinematical hardening

s2

p

e . s1 s(s)

F

s

GENERAL ASSUMPTIONS

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MELAN‘S THEOREM (1938)

An elastic-perfectly body will shake down if there exists a positive real number α > 1 and a time-independent field of residual stresses ρ

° (x) such

that: P(s)(x) = { (s) / F(x, α (s)) < σF}, ∀x ∈ V, ∀t > 0 where (s)(x, t) = (E)(x, t) + ρ

°(x)

 The field of purely elastic stresses satisfies

Div σ(E)= − f * in V n.σ(E) = p*

  • n Sp

 The field of residual stresses satisfies

Div ρ

°

= 0 in V n.ρ

° = 0

  • n Sp
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“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

ILLUSTRATION OF MELAN´S THEOREM

s2 s1 F (s) = 0

Fictitious elastic solution domain

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“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

s2 s1 F (s) = 0

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“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

s2 s1 F (s) = 0

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“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

s2 s1 F (s) = 0

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“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

s2 s1 F (s) = 0

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“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

s2 s1 F (s) = 0

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“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

s2 s1 F (s) = 0

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“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

s2 s1 F (s) = 0

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“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

s2 s1 F (s) = 0

Time-independent residual stress field

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“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

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“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

Optimization: max α

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“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

Grüning (1926), Bleich (1932), Melan (1936), Symonds (1951), Koiter (1956), Gvozdev (1938), Drucker, Prager & Greenberg (1951), Hodge (1959)

  • Leading ideas: Positiveness of dissipation, boundedness of

free energy, failure if rate of external work exceeds rate of dissipation.

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

developed.

  • “Incomplete” formulations, leading to a “static” and a

“kinematic” approach. Bounding methods are naturally introduced

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“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

Mathematical foundations Mathematical setting (convex analysis), initial b.v.p., dynamic shakedown, bi- potential approach Geometrical effects Second order effects, large strains, stability problems Material laws Hardening, damage, thermal effects, Non- associated flow rules, Visco-plasticity, cracked bodies, interface failure Specific Applications Thinwalled structures, Beams, Frames, Plates, Shells Foundations, Pavements, Roads, Dams, Soils, Rolling Contact Composites, Porous Materials

EXTENSIONS OF CLASSICAL THEOREMS

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“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

NUMERICAL METHODS

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PIPES AND PRESSURE VESSELS

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 Tube under moving thermal loading  Mechanical characteristics  Geometry and initial loading

Τ0 Τi = Τ0 + ∆T ∆L L Q Q h R

L/R = 0.733 Q0 = h σY h/R = 1/400 ∆T0 = 2 σY/(E αϑ) ∆L/L = 0.06 Young’s modulus (MPa) Poisson’s ratio Yield stress (MPa) Thermal expansion coefficient ( K−1) 2.1×10+5 0.3 360 1.2×10−5

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 Shakedown domains

0,0 0,3 0,5 0,8 1,0 0,0 0,3 0,5 0,8 1,0 Mechanism (B) (Lokal) Mechanism (A) (Global) Ponter Gross-Weege Present results

Q Q0 ∆ϑ ∆ϑ 0

* Ponter, A.R.S.; Karadeniz, S.: J. Appl. Mech. 52, 883-889 (1985) * Gross-Weege, J.: , Doctor thesis, Ruhr-Universität, Bochum (1988)

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“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

Flange under internal pressure and axial load

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“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

Q Q

Admissible domains

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“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

Limited kinematic hardening

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“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

Nozzle under thermo-mechanical loading

Dr.-thesis Jaan Simon

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“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

All three loads Px ,Py and T vary independently: All material parameters are considered as temperature-independent.

Numerical data

Elements NE 6 376 Gaussian points NG 51 008 Nodes NK 9 645 Loads Corners NC Variables n Equality constraints mE Inequality Constraints mI 1 2 561 089 283 975 102 016 2 4 1 071 169 794 055 204 032 3 8 2 091 329 1 813 582 408 064

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“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

Elastic solution

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“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

Shakedown Domain

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“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

Shakedown Domain

Influence of hardening

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PERIODIC AND NON- PERIODIC COMPOSITES

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“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

y

1

y

2

ϕ

Σ Σ

t l L

δ δ

ϕ

0.0 0.2 0.4 0.6 0.8 1.0 30 60 90

σmax = 50 MPa σmax = 150 MPa

Perfect bond.

Σ ( ) σY ϕ

  • F. Schwabe, Min Chen, A. Hachemi

Matrix Fiber Interface

Young’s modulus E (GPa) 67.2 370 0.67 Poisson’s ratio ν 0.318 0.318 0.0 Yield stress σY (MPa) 137  68.5

Periodic composite with influence of interface

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“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

Fibre-reinforced composite, 2D/3D

E22 E11

D L Matrix Fibre

Young’s Modulus E (GPa) 210 2.1 Poisson’s ratio ν 0.3 0.2 Yield stress σY (MPa) 280 140

0,2 0,4 0,6 0,8 0,2 0,4 0,6 0,8 (b) (a) 3D-analysis 2D-analysis Limit analysis Shakedown (a) (b)

ErreurŹ! ErreurŹ!

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“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

Non-periodic composites

2D/2.5D and mesh sensitivity test

Geng Chen, Christoph Broeckmann (IWM)

_______________________________________

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“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

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“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

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“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

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“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

Ultimate strength and grain size

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“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

Endurance limit and grain size

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“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

CORRELATION MATRIX Group 3

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“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

  • 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.
  • Perspectives: Further reduction of CPU-time, extension to

larger classes of material behaviour, lifetime prediction material design.

CONCLUSIONS