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

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: 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  LIMIT ANALYSIS INSTANTANEOUS COLLAPSE FAILURE UNDER VARIABLE LOADS  SHAKEDOWN ANALYSIS “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 “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 “PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

σ σ σ σ σ max σ max σ max σ max ε ε ε ε ο ο ο ο σ min σ min σ min σ min Purely elastic Shakedown Low-cycle fatigue Ratchetting p = ∫ 0 T . p ( x , t ) = 0 . p ( x , t ) = 0 p = ∫ 0 T . p ( x , t ) ≠ 0 ∆  p ( x , t ) = 0 ∆   t → ∞    lim “PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

Acknowledgements J. Groß-Weege F. Schwabe J.B. Tritsch A.D. Nguyen M. Hachemi S. Mouhtamid A. Belouchrani M. Chen M. Boulbibane J. Simon Hamadouche G. Chen B. Nayroles Ch. Broeckmann L. Raad “PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

GENERAL ASSUMPTIONS  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: s 2 . σ p e F Linear kinematical hardening s σ Y s (s) s 1 Perfectly plastic ε “PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

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) ) < σ F }, ∀ 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 * on S p  The field of residual stresses satisfies Div ρ = 0 ° in V n. ρ ° = 0 on S p

ILLUSTRATION OF MELAN´S THEOREM s 2 Fi c titious elastic solution domain s 1 F ( s ) = 0 “PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

s 2 s 1 F ( s ) = 0 “PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

s 2 s 1 F ( s ) = 0 “PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

s 2 s 1 F ( s ) = 0 “PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

s 2 s 1 F ( s ) = 0 “PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

s 2 s 1 F ( s ) = 0 “PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

s 2 s 1 F ( s ) = 0 “PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

s 2 s 1 F ( s ) = 0 “PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

s 2 Time-independent residual stress field s 1 F ( s ) = 0 “PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

Optimization: max α “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 “PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

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

NUMERICAL METHODS “PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

PIPES AND PRESSURE VESSELS

 Tube under moving thermal loading  Geometry and initial loading h R Q 0 = h σ Y L / R = 0.733 ∆ T 0 = 2 σ Y /( E α ϑ ) h / R = 1/400 Τ i = Τ 0 + ∆ T ∆ L / L = 0.06 Q Τ 0 Q ∆ L L  Mechanical characteristics 2.1 × 10 + 5 Young’s modulus (MPa) Poisson’s ratio 0.3 Yield stress (MPa) 360 1.2 × 10 − 5 Thermal expansion coefficient ( K − 1 )

 Shakedown domains 1,0 Mechanism ( B) (Lokal) 0,8 ∆ϑ ∆ϑ 0 0,5 Mechanism ( A) Ponter 0,3 (Global) Gross-Weege Present results 0,0 0,0 0,3 0,5 0,8 1,0 Q Q 0 * Ponter, A.R.S.; Karadeniz, S.: J. Appl. Mech. 52 , 883-889 (1985) * Gross-Weege, J.: , Doctor thesis, Ruhr-Universität, Bochum (1988)

Flange under internal pressure and axial load “PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

Admissible domains Q Q 0 “PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

Limited kinematic hardening “PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

Nozzle under thermo-mechanical loading Dr.-thesis Jaan Simon “PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

Numerical data Loads Corners NC Variables n Equality Inequality constraints Constraints All three loads P x ,P y and T m E m I vary independently: 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 Elements NE 6 376 All material parameters are considered Gaussian points NG 51 008 as temperature-independent. Nodes NK 9 645 “PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

Elastic solution “PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

Shakedown Domain “PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

Shakedown Domain Influence of hardening “PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

PERIODIC AND NON- PERIODIC COMPOSITES

Periodic composite with influence of interface 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 σ Y (MPa)  137 68.5 Perfect bond. 1.0 0.8 Σ y 2 0.6 Σ ( ) ϕ ϕ σ Y l 0.4 L σ max = 150 MPa y 0.2 σ max = 50 MPa 1 δ δ t 0.0 0 30 60 90 Σ ϕ F. Schwabe, Min Chen, A. Hachemi “PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

Fibre-reinforced composite, 2D/3D E 22 0,8 (a) Limit analysis (b) Shakedown 0,6 E 11 ErreurŹ! 0,4 (a) (b) 3D-analysis 0,2 D 2D-analysis L 0 0 0,2 0,4 0,6 0,8 Matrix Fibre Young’s Modulus E (GPa) 210 2.1 ErreurŹ! Poisson’s ratio ν 0.3 0.2 Yield stress σ Y (MPa) 280 140 “PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

Non-periodic composites 2D/2.5D and mesh sensitivity test _______________________________________ Geng Chen, Christoph Broeckmann (IWM) “PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

“PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

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Ultimate strength and grain size “PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”

Endurance limit and grain size “PRACTICAL APPLICATIONS OF INNOVATIVE SOLUTIONS RESULTING FROM SCIENTIFIC RESEARCH”