Advanced stability analysis of regular stiffened plates and complex plated elements

Abstract

Further info

This file contains the slides of the conference presentation.

If you are interested in the full paper, please email the author directly with the request:

geri@vbt.bme.hu

complex plated elements

Proc. of SDSS' Rio 2010 International Colloquium on Stability and Ductility of Steel Structures. Rio de Janeiro, 2010. pp. 81-100. (ISBN: 978-85-285-0137-7).

Vigh L G, Dunai L, Advanced stability analysis of regular stiffened plates and complex plated elements, Proc. of SDSS’ Rio 2010 International In the paper research on advanced stability analysis of stiffened steel plates are presented. Regular panels are studied by laboratory and virtual tests and the buckling resistances are calculated and evaluated by conventional Eurocode method, by numerical buckling analyses of the plate elements under the actual stress conditions, and by material and geometrical nonlinear finite element analyses imperfections included, following the FEM based design recommendations. The stability analyses of complex plated elements are related to the design of a new Danube tied arch bridge. The non-conventional constructional solutions and complex loading conditions are studied by refined multi- level finite element models. The relative safeties of the different methods of the critical plated elements are determined.

The paper highlights the practical problems of the advanced stability analysis: definition of critical point of the element, handling stress concentrations and definition of imperfections.

Keywords: stiffened plate, buckling, imperfection, finite element model, FEM based design

CEN, Brussels, 2005.

[3] Vigh, L.G., Virtual and real test based analysis and design of non-conventional thin-walled metal structures, PhD dissertation, Budapest University of Technology and Economics, 2006.

[4] Dunai, L., Jakab, G., Néző, J., Topping, B.H.V., “Experiments on welded plate girders: fabrication, imperfection and behaviour”, Proc. 1st International Conference on Advances in Experimental Structural Engineering (AESE ’05), Vol. 1, pp. 51-58, Nagoya, Japan, 19-21 July, 2005.

[5] Vigh, L.G., “On the Eurocode buckling formulas of multi-stiffened metal plates”, Proc. of International Colloquium on Stability and Ductility of Steel Structures (SDSS 2006), Vol. 1, pp. 545-552, Lisbon, Portugal, 6-8 September, 2006.

[6] ANSYS Structural Analysis Guide, Online Documentation ANSYS Inc., 2005.

[7] Jakab, G., Szabó, G., Dunai, L., “Imperfection sensitivity of welded beams: experiment and simulation”,

International Conference in Metal Structures: Steel – A New and Traditional Material For Building (ICMS 2006), pp.

173-182, Brasov, Romania, 20-22 September, 2006.

[8] Horváth, A., Dunai, L., Domanovszky, S., Pesti, Gy., “The Pentele Danube bridge in Dunaújváros, Hungary; design, research and construction”, Bauingenieur, pp. 419-438, 2008.

[9] Dunai, L., Farkas, Gy., Szatmári, I., Joó, A., Tóth, A., Vigh, L.G., Dunaújváros Danube Bridge – River Bridge – Additional static analyses 2 (in Hungarian), 7th research report, BME Department of Structural Engineering, Budapest, August 2006.

Advanced stability analysis of regular stiffened plates and

complex plated elements

László Gergely VIGH László DUNAI

Associate Professor Professor

Budapest University of Technology and Economics

Department of Structural Engineering

Contents

• Introduction

• Plate stability ↔ FEM

• Experimental and numerical study

• Design method evaluation

• Stiffened plate analysis – example

• Complex stiffened plate analysis – example

• Conclusions

Introduction – plated elements

Introduction – strategy

Regular multi-stiffened plates and complex plated elements

Standard formulae and FEM based design

Experimental and numerical research programme

Stability checking of a new Danube bridge

Eurocode 3 Part 1-5

(1)basic procedure for stiffened plates in complex stress fields (no use of numerical models)

(2) partial use of FEM: plate slenderness from GNB (3) reduced stress method

(4) FEM based design – full numerical simulation

Plate stability ↔ FEM

0 , ,

M y eff c Rd

c

f N A

= γ

Experimental study

Test program

Geometric imperfections

Residual stresses

Ultimate behaviour

Failure modes

Design method evaluation

a) plate-type: b) column-like:

Rd c

Ed N

N ≤ _,

Check:

Consideration of both plate-type and column-like buckling

p cr

y c A p

f

, ,

σ λ = β

ρ

c cr

y c A c

f

, ,

σ λ = β

p

σ

cr ,

critical stress for overall buckling, e.g. from orthotropic plate theory

2 1 ,

1 , 2

,

A a

EI

sl sl c

cr

σ = π

χ

c

c) interpolation: ^ρ

^c

^{= ( ρ − χ}

^c

^{) ξ ( 2 − ξ ) + χ}

^c

^{1 ( 0 1 )}

,

,

− ≤ ≤

= ξ

σ ξ σ

c cr

p cr

0 , ,

M y eff c Rd

c

f N A

= γ

Cross-section resistance – with effective area

Eurocode 3 – axial load resistance – basic procedure

Effective cross-section

Eurocode 3 – shear load resistance – basic procedure

( )

E red red

cr E

cr E

B bend cr

p A

E A comp cr

k k

k

b E t

k k

σ σ

σ τ

σ σ

ν σ π

σ

τ

=

=

=

 





 





= −

=

, ,

2 2

2

,

12 1

= ∑ ∑

= = M m

N n mn

b y n a

x A m

y x w

1 1

sin sin

) ,

( π π

LBA analysis LBA analysis LBA analysis LBA analysis (energy method):

(energy method):

(energy method):

(energy method):

b

s

t

p

b

_p

t

s

y

x

i^th stiffener

(n

_s

+1)b

_p

= b

a

Parameter Range

a

600 ~ 4200 mm

α 1 ~ 7

t_p

2 ~ 10 mm

n_s

1 ~ 5

b_s

20 ~ 100 mm

t_s

2 ~ 10 mm

ψ -1 ~ 1

τ ^/ σ ^{0 ~ 3}

γ 0.1 ~ 2700

δ 0.006 ~ 1

Buckling coefficients – parametric study

Overall buckling Local buckling

N ( )

^_







 



 

 +  +



 



 + +

= +

2 2

)

(

( 1 )

1 1

1

γ α α

α δ

n m m

m

k n _s

s ov

A ^{( )}

( ¹ )

^{2 2}



 



 + +

= m n m

k_A^loc _s

α

α M

N+M δ

γ α α

α α

) 1 ( 1

) 1 ( 1 1 1

' '

2 2

lim 1 lim

) (

+ +

+ +

 +







 +

=

s ov s

AB n

n C

m m C

k k⁽_AB^loc) = A₁n_s² + A₂n_s + A₃

4 20

' 12

3 12

' 6

2 6

' 2

1 2

' 0

lim lim

lim lim

lim lim

lim

=

≤ ⇒

<

=

≤ ⇒

<

=

≤ ⇒

<

=

≤ ⇒

<

m m m m

α α α

α α α

α α α

α α

4 20

) 1 ( 12

3 12

) 1 ( 6

2 6

) 1 ( 2

1 2

) 1 ( 0

=

≤ ⇒ +

<

=

≤ ⇒ +

<

=

≤ ⇒ +

<

=

≤ ⇒ +

<

m n

m n

m n

m n

s s

s s

α α

α α

a) short panel: 0<

φ

≤1 b) long panel:

φ

≥1 c) local – overall interaction: if

γ

₀^*<

γ

<

γ

₁^*

[ ]

^3/4

2

4 3

2 ) (

) 1 ' (

'

α γ α

τ τ τ

+

=

=

=

s p

x ov y

k n

D B k B

k

4

4 3

) (

) 1 ( ' '

τ

γ

τ τ

+

=

=

=

s p

y ov x

n k

D B B k k

φ φ

'=

φ

^'=

φ

¹

( ^{φ φ} ) ^η

φ

τ

φ

2 2

' 75 . 2 ' 1 . 0 95 . 1 ' 92 . 1 ' 567 . 0 25 . 3

'= − + + + +

k

( )

*

0

* 1

0 , , 1 ,

* , 0 0

, , ) (

γ γ γ

γ

^{τ τ}

τ

τ −

− − +

= _loc ^{loc loc}

ov k k

k k

( ) ^α

α

_m,i = n_s +i ; i=0,1















≥

 +











 +

<

 +











 +

=

1 if

) 4 (

34 . 5

1 if

) 34 (

. 4 5

, 2

2 ,

, 2

2 , ,

,

i m s

i m

i m s

i m i

loc

i n

i n k

α α α α

τ

0 5 10 15 20 25 30 35 40

0 1 2 3 4 5

α αα

α(B_y/B_x)^1/4 [-]

kττττ(ov) [-]

EC9-1-1 (Bulson) original

EC9-1-1 (Bulson) modified

EC3-1-5 and EC9/2

energy sol. (long plate)

γ = 5 γ = 500

0 20 40 60 80 100 120 140 160 180 200

0 100 200 300 400 500

γγγγ [-]

kττττ [-]

EC9 (Bulson) energy sol.

("exact")

interaction region

n

_s

= 5

n

_s

= 4

n

_s

= 3

n

_s

= 2

n

_s

= 1 γ

1*

γ

0*

k

_τ,loc,0

k

_τ,loc,1

Stability analysis of stiffened

plates of a new Danube bridge

Submodel of Joint #1

Case t

_p

b a stiffener Nr. [mm] [m] [m]

___________________________________________

1 40 2 4.56 2 x 280-22

2 30 3.8 4.56 5 x 280-22

3 50 2 2.125 2 x T270-150-22

4 20 3.8 3.9 5 x 280-22

5 16 3.8 3.86 5 x 280-22

6 20 2 3.86 2 x 280-22

___________________________________________

t

_p

– plate thickness; b – plate width; a – plate length between transverse stiffeners or diaphragms

stiffeners

b

t

_p

stiffeners

b

t

_p

stiffeners

b

t

_p

stiffeners

b

t

_p

FE mesh and material model

E_t= E/10000 = 21 N/mm²

a) global imperfection of stiffener

b) imperfection of subpanel

c) local imperfection of stiffener

( ^{/ 400 , / 400} )

0

min a b

e

_w

= e

0_w

= ^min ( a ^{/ 400 ,} b ^{/ 400} ) ^φ

⁰

^{= 1 / 50}

~ alternatively, relevant buckling shapes, i.e.

a) overall buckling,

b) local buckling of subpanels,

c) torsion mode of the stiffener

combination of the imperfections:

leading (100%) + others (70%)

PROBLEM when using

buckling shapes as imperfections:

overall/local plate buckling usually accompanied by the torsion of stiffener

the requirements for the imperfection amplitudes

are difficult to satisfy

σ

cr

= 930,9 MPa

0 50 100 150 200 250 300 350 400 450 500

0 10 20 30 40 50

Lateral deflection of panel center [mm]

Load [N/mm2 ]

FEM EC3-1-5

Hungarian Standard (allowable stress) 1,47 x Hungarian Standard

normal imperfection

large imperfection

( )

mm 6 , 50 5 280 1

mm 5 400 / 2000 400

/ , 400 / min

0 2

, 0

1 , 0

=

⋅

=

=

=

=

=

b

φ

x w

h e

b a

e

10 8472 661

, 0

6 , 5

10 1932 588

, 2

00 , 5

2 3 , 0

1 3 , 0

⋅ =

=

⋅ =

=

−

−

α

α

( )

mm 4 , 50 5 270 1

mm 5 400 / 2000 400

/ , 400 / min

0 2

, 0

1 , 0

=

⋅

=

=

=

=

=

b

φ

x w

h e

b a

e

12135 10

445 , 0

4 , 5

10 6614 756

, 0

00 , 5

2 3 , 0

1 3 , 0

⋅ =

=

⋅ =

=

−

−

α α

σ

cr

= 3560 MPa

0 50 100 150 200 250 300 350 400 450 500

0 1 2 3 4 5 6

Lateral deflection of panel center [mm]

Load [N/mm2 ]

FEM EC3-1-5

Hungarian Standard (allowable s tress) 1,47 x Hungarian Standard

normal imperfection

large imperfection

σ

cr

= 881 MPa σ

cr

= 745 MPa

( )

mm 625 , 1 400 / 650

mm 6 , 50 5 280 1

mm 5 , 9 400 / 3800 400

/ , 400 / min

3 , 0

0 2

, 0

1 , 0

=

=

=

⋅

=

=

=

=

=

w b x

w

e

h e

b a

e

φ

10 1999 813

, 0

625 , 1

10 9705 577

, 0

6 , 5

10 4170 278

, 2

5 , 9

3 3 , 0

2 3 , 0

1 3 , 0

⋅ =

=

⋅ =

=

⋅ =

=

−

−

−

α α α

0 50 100 150 200 250 300 350 400

0 10 20 30 40 50

Lateral deflection of panel center [mm]

Load [N/mm2 ]

FEM EC3-1-5

1,47 x Hungarian Standard

Hungarian Standard (allowable stress) normal imperfection

large imperfection

Algorithm of Method 3

• Definition of critical point – peak stresses

• Choice of buckling curve – failure mode, post-buckling behaviour

• Imperfections - combination

• Conservative results

• Role of experiments

• Numerical modelling

• Parallel application of methods

Buckling checking of regular stiffened plates:

• modified buckling coefficients for multi-stiffened plates;

• accuracy, failure mode prediction.

FEM based buckling checking:

• GMNI analyis – actual imperfections – virtual experiments;

• experiments – model verification – accuracy.

Application for irregular plated elements:

• Parralel application of different methods;

• benefits of using FE based checking;

• practical shortcomings of the application.