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the 5th Asia Pacific congress on Computational Mechanics and 4th International Symposium on Computational Mechanics CEPAO AO shakedow kedown, n, new w co comput putational ational asp spects ects Hoang Van-Long, Nguyen Dang-Hung

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CEPAO AO shakedow kedown, n, new w co comput putational ational asp spects ects

the 5th Asia Pacific congress on Computational Mechanics and 4th International Symposium on Computational Mechanics

Singapore, 11-14 Dec. 2013

Hoang Van-Long, Nguyen Dang-Hung

University of Liège, Belgium 1

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Introduction CEPAO package (Calcul Elasto-Plastique, Analyse-Optimisation)

Built-up in 1980’s at University of Liège (Belgium) for 2D framed structures New development: 3D framed structures

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Input data Output

Global organization

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4

  • The one/two/three-linear behaviors of the mild steel are

considered.

  • The frames are submitted to fixed or repeated load.
  • The second-order effect is taken into account.
  • The beam-to-column joints could be rigid or semi-rigid.
  • The compact or slender cross-sections are examined.
  • The investigation is carried out using direct or step-by-

step methods.

  • Both

analysis and

  • ptimization

methodologies are applied.

General features

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Element features

3D plastic hinges 3D beam-column element

Yield surface Constitutive law

Orbison -1982 (Single, smooth, convex: step-by-step method) 16 facet-polyherdron (direct method)

  • r

C p

λN e 

Normality rule:

  • Formulation by using the elastic Bernoulli beam theory
  • P- effect by using stability fountions (in step-by-step analaysis)
  • Imperfection effect by using European buckling curves (in step-by-step analaysis)

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Rigid-plastic formulations

λ d f Bd λ N λ s      ξ Min

T C T

p T c T p

Min n s N f s B s) , n (   Z

b Wx x c  

T

Min 

λ λ N s Bd Nλ λ s      ξ Min

C T E T

p e T C T T p p

Z Min n ρ) (s N ρ B l n ρ) , n (    

Limit analysis by kinematical approach Limit design by statical approach Shakedown analysis by kinematical approach Shakedown design by statical approach Linear programming formulation

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b Wx x c  

T

Min 

Very large dimension

20 story frame Problem W dimension Analysis (921)x(18 961) Optimization (16 380)x(19 220)

Several tachniques have been proposed and adopted in order to reduce the proplem sizes

Rigid-plastic formulations

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                           

p C C R CC RC RC RR C R

e e e D D D D s s

T

λ N e   

C p C

λ

C C T C

     N F s N H

                          

C R 2 C CC CC 1 C CC T RC 2 C RC RC 1 C RC RR C R

e e R N D D R N D D R N D D R N D D s s

Normality rule Plastic constitutive Elastic constitutive relation

Elastic Plastic

e D s Δ Δ 

Elastic-plastic constitutive relation

Elastic-plastic formulations

T

K B D B 

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P = 250 Mpa

E = 206000 Mpa Floor pressure: 4.8 kN/m2 Wind load (Y direction): 26.7 kN/node

Frame I.1 – Six-story space frame

  • I. 3-D rigid frames analysis

Numerical examples

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p = 345 Mpa

E = 200000Mpa Floor pressure: 4.8 kN/m2 Wind load (Y direction): 0.96 kN/m2

Frame I.2 – Twenty-story space frame

Numerical examples

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Auther Model Frame I.1 Frame I.2 Liew JYR - 2000 Plastic-hinge 2.010 - Kim SE - 2001 Plastic-hinge 2.066 - Cuong NH - 2006 Fiber-plastic-hinge 2.066 1.003 Liew JYR - 2001 Plastic-hinge - 1.031 Jiang XM - 2002 Fibre-element - 1.000 Chorean C.G.- 2005 Distributed plasticity, n=300 1.998 1.005 n=30 2.124 1.062 CEPAO Plastic-hinge, hardening ignored 2.033 1.024 hardening considered 2.149 1.051

Load multipliers given by second-order analysis

Numerical examples

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Load multipliers given CEPAO

Method, model Frame 1 Frame 2

Elastic-plastic, first-order 2.489 1.689 (instantanenous) Elastic-plastic, second-order 2.033 1.024 (unstableness) Limit analysis 2.412 1.698 (instantanenous) Shakedown analysis, load a 2.311 1.614 (incremental) Shakedown analysis, load b 1.670 0.987 (Alternating) load a: 0 floor pressure  4.8  (kN/m2); 0  wind load  0.96  (kN/m2) load b:0  floor pressure  4.8  (kN/m2);-0.96 wind load  0.96 (kN/m2)

Numerical examples

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Frame I.1 Frame I.2

Top-story sways (m) Load Load Top-story sways (m)

Numerical examples

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Instantaneous mechanism (first-order) unstableness (second-order) Instantaneous mechanism (limit analysis) Incremental mechanism (shakedown analysis) Altenating plasticity (shakedown analysis)

Numerical examples

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Frame I.1 Frame I.2

Top-story sways (m) Load Load Har. No har. No har. Har. Top-story sways (m)

Hardening effect Numerical examples

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Plastic-hinge occurs

section 7

Local buckling (2.08) (2.13) (2.254)

Local buckling check

Numerical examples

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  • II. 2-D bending frames analysis (Casciaro -2002)

Load domain: 9  p1 10; 0  p2  5; - 500  P3  500

Mechanical properties E I MP Column 300000 540000 1800000 Beam 300000 67500 450000

4 frames: 1: 3 spans, 4 stories 2: 4 spans, 6 stories 3: 5 spans, 9 stories 4: 6 spans, 10 stories

Numerical examples

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Frame (ns x nb) Limit analysis Shakedown analysis Casciaro CEPAO Dif. Casciaro CEPAO Dif. 34 2.4612 2.4612 0.0% 2.0134 2.0102 0.0% 46 1.8610 1.8610 0.0% 1.3993 1.2655

  • 10.5%

59 1.2000 1.2000 0.0% 0.7533 0.7076

  • 6.4%

610 1.1532 1.1532 0.0% 0.7209 0.6771

  • 6.5%

Alternating plasticity at Load Multiplier Section A (1) 1.1846 Section B (2) 0.6816 Section C (3) 0.6533 Upper bounds Load Upper bound CEPAO Casciaro Dis.

Numerical examples

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Frame III.1

  • III. 3-D rigid frames optimization

(shakedown design)

Numerical examples

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Frame III.1(American sections)

Numerical examples

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Initial member-size (295kN)

  • ptimal

member-size (169kN)

(Second-order analysis) Load Initial Optimal Top-story sways (m)

Optimal?

Numerical examples

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Frame III.2

Numerical examples

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Frame III.2 (American sections)

(Limit design)

Convergence!

Numerical examples

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Frame III.3 (Shakedown design)

Numerical examples

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Frame III.3 (American sections) Frame III.3 (European sections)

Numerical examples

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Direct method (2.412) Step-by-step method (2.489) Direct method (1.698) Step-by-step method (1.689)

  • IV. Convergence of SBS and Direct methods

Numerical examples

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Direct method (2.126) Step-by-step method (2.175)

Numerical examples

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Direct method (2.469) Step-by-step method (2.402)

Numerical examples

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Direct method (2.226) Step-by-step method (2.264)

Numerical examples

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(Bjorhovde-1990) Moment-rotation relationship for connexions

  • V. 2-D semi-rigid frames

Numerical examples

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Frame V.1a: E = 2.1E7; I = 118.5E-6; Mp = 20; h = 0.3; Frame V.1b: column: E = 2.1E7; I = 85.2E-6; Mp = 10; beam: E = 2.1E7; = 118.5E-6; Mp = 20; h = 0.3.

  • V. 2-D semi-rigid frames, Tin Loi 1993

Load domain:

01 02 Limit and shakedown analysis

Numerical examples

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Frame V.1a Frame V.1b

Connexion strength Connexion strength Load Load

Numerical examples

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Loading Geometry

  • V. 2-D semi-rigid frames, Jaspart – 1991 (using FINELG)

Type 1

Numerical examples

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Type 2 Geometry Loading

Numerical examples

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Type 3 Geometry Loading

Numerical examples

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Type 1

Numerical examples

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Type 2

Numerical examples

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Type 3

Numerical examples

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Load domain

  • Shakedown:

a) 0 1  1, 0 2  1 b) -1 1 1, 0  2  1.

  • Limit: 1=2=;

Groups of elements:

  • Optimal: 40 different

groups of elements, load factor  = 0.25

  • Analysis: 8 different

groups of elements:

  • V. 2-D semi-rigid frames

E=2E8, σp=2E5

Numerical examples

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(First order) (Second order) Top-story sways (m) Top-story sways (m) Rotation moment Load Load

Numerical examples

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Joint strength Load factors

  • Numerical examples

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Variation of weight according to connexion strengths Theoretic weight Real weight Connexion strength Connexion strength

Numerical examples

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Concluding remarks

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Thank you for your attention!

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