EN 1999 - Eurocode 9: Design of aluminium structures Part ...eurocodes.jrc.ec.europa.eu/doc/WS2008/EN1999_9_Mandara.pdf · EN 1999 - Eurocode 9: Design of aluminium structures

Brussels, 18-20 February 2008 – Dissemination of information workshop 1

EUROCODESBackground and Applications

EN 1999 - Eurocode 9: Design of aluminium structuresPart 1.5 - Shell structures (A. Mandara)

EN 1999 - Eurocode 9: Design of aluminiumstructures Part 1.5 - Shell structures

A. Mandara

Department of Civil EngineeringSecond University of Naples – School of Engineering

Real Casa dell’Annunziata – Via Roma, 29, Aversa (CE)

Eurocodes - Background and Applications“Dissemination of information for training” workshop

Brussels 18-20 February 2008



Aluminium shells – applications




The EN1999-1-5General part




The EN1999-1-5Annexes




The prEN1993-1-6




w θ, v

x, u

t

r

l

t

r1

hL

r2

r

β β

w

φ

θ

r

t

Shell configurations allowed for in EN1999-1-5





Types of shell analysis in EN1999-1-5



Specific issues for aluminium alloy shells in EN1999-1-5




• shell plastic buckling• imperfection sensitivity analysis of aluminium cylinders;• set-up of buckling curves for aluminium shells;• definition of imperfection classes for plastic buckling; • interaction between load cases;• introduction of additional shell configurations;

• stiffened shells• imperfection sensitivity analysis of stiffened cylinders;• validation of EN1993-1-6 procedures and harmonization with

EN1999 rules;• effect of welding effect (HAZ zones)

• imperfection sensitivity analysis of welded cylinders;• definition of simplified design procedures;

Background activity - Main investigated aspects




Parametric analysis:Shell geometrical data and material

features

The ABAQUS model




Imperfection model

⎥⎦⎤

⎢⎣⎡ −

⎥⎦⎤

⎢⎣⎡ −

= −−−−∑ Ryyke

Lxxkeww o

yyyko

xxxk oyox

)(cos)(cos 2)(

2)(

0

21

21 ππ




Buckling response of axially loaded cylinders

0

1000

2000

3000

4000

5000

6000

0 2 4 6 8 10 12

Abbassamenti Assiali [mm]

Car

ichi

Ass

iali

[KN

]

R/t = 200f02 = 200 N/mmqTipo IncastratoImperfezione Asimmetrica

ABAQUS

Pcr,th = 5293.38 KN

W0/t =0.01

W0/t =3

0

1000

2000

3000

4000

5000

6000

0 10 20 30 40 50 60 70 80

Abbassamenti Assiali [mm]

Car

ichi

Ass

iali

[KN

]

R/t = 200f02 = 200 N/mmqTipo IncastratoImperfezione Assial-Simmetrica

ABAQUS

Pcr,th = 5293.38 KN

W0/t =0.01

W0/t =3




Deflected shapes at buckling (cylinders under uniform external pressure)

Diagramma Carichi-Spostamenti Radiali

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 5 10 15 20 25 30 35 40

Spoatamenti Radiali [mm]

Car

ichi

Sup

erfic

iali

[N/m

mq]

R/t = 200 L/R=2f02 = 100 N/mmq

W 0 = 0.1 mmTipo Appoggiato

ABAQUS

Pcr,th = 0.058 N/mmq

Diagramma Carichi-Spostamenti Radiali

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 10 20 30 40 50 60

Spoatamenti Radiali [mm]

Car

ichi

Sup

erfic

iali

[N/m

mq]

R/t = 100 L/R=2f02 = 100 N/mmqW 0 = 0,75 mm

Tipo Appoggiato

ABAQUS

Pcr,th = 0.058 N/mmq




Imperfection sensitivity curves (axially loaded cylinders)

Cylinders under axial load Imperfection Sensitivity Curves

0,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

0,0 0,2 0,4 0,6 0,8 1,0 1,2

W0/t

Pu /Pcr,th

R/t = 200 R = 1000 mm t = 5 mmf02 = 200 N/mm2

P cr,th = 5293.38 KNσ cr,th = 168.49 N/mm 2

CLASS A CLASS B CLASS C




Imperfection sensitivity curves (cylinders under external pressure)

Cylinders under external pressureImperfection sensitivity curve

0,0

0,2

0,4

0,6

0,8

1,0

0,0 0,2 0,4 0,6 0,8 1,0

W0/t

Pu /Pcr,th

L = 1000 mm R= 1000 mm t= 10 mmR/t = 100 L/R=1f02 = 100 N/mm2 n = 9 Pcr,th = 0.620 N/mm2

L = 1000 mm R= 1000 mm t= 10 mmR/t = 100 L/R=1f02 = 100 N/mm2 n=8Pcr,th = 0.610 N/mm2

n=8

Class A Class CClass B

n = 9




Imperfection sensitivity curves (cylinders under torsion)

Cylinders under torsionImperfection sensitivity curve

0,0

0,2

0,4

0,6

0,8

1,0

0,0 0,2 0,4 0,6 0,8 1,0

W0/t

Pu /Pcr,th

Longitudinal ImperfectionHelical Imperfection

R/t = 200 L/R = 2 t = 5 mmf02=200 N/mm2

τcr,th = 49.358 N/mm2

Class CClass BClass A




( )βα−−= xexP 1)(

( ) ( ) ( ) αβ−−αααβ

==/1/1/1

/11)( xex

dxxdPxp

Cylinders under axial loadWeak hardening alloys

- R/t = 100-

0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.20 0.40 0.60 0.80 1.00

x

P(x)

Class AClass BClass CWeibull Curve AWeibull Curve BWeibull Curve C5% Percentile Value

A

B

C

f02 = 200 N/mm2

Semi-probabilistic interpretation of buckling data(axially loaded cylinders) - Weibull’s law





- R/t = 200-

0.00

0.20

0.40

0.60

0.80

1.00

0.00 0.20 0.40 0.60 0.80 1.00

x

P(x)

Class AClass BClass CWeibull Curve AWeibull Curve BWeibull Curve C5% Percentile Value

A

B

C

f02 = 200 N/mm2

( ) ( ) ( ) αβ−−αααβ

==/1/1/1

/11)( xex

dxxdPxp

Semi-probabilistic interpretation of buckling data(axially loaded cylinders) - Weibull’s law












Shell buckling – EC3 formulation

λλλαχ

λλλλλ

λλβχ

λλχη

≤⇔=

≤<⇔⎟⎟⎠

⎞⎜⎜⎝

⎛

−−

−=

≤⇔=

p

pp

2

00

0

0

1

1

βαλ

σλ

−=

=

1p

xRc

ykf




Shell buckling – fabrication tolerance classes in EC3

0 max00 max

gxo

gx

Uw wUt t

≤ ⇔ ≤


Fabrication tolerance quality

class

Description

Class A Excellent Class B High Class C Normal

r t

Δwox

inward

gx

Δwox

gx

t

Gauge lengthm a x

4o

g x

U

R t=

Dimple tolerance parameter



Expressions of buckling factors according to EC3

Axial (meridional) load External pressure

and torsion (shear) λ 0 0.20 0.40

β 0.60 0.60

η 1.00 1.00

Axial (meridional) load

External pres-sure (αθ) and

torsion (shear) (ατ)

Fabrication tolerance

quality class

Description

Q αx αθ or ατ Class A Excellent 40 0,75 Class B High 25 0,65 Class C Normal 16 ( ) 44.1

//191.11

62.0

trQx

+=α

0,50




Cylinders under axial load Weak hardening alloys

Quality Class C0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.50 1.00 1.50 2.00

λ

χ Minimum ValueMedium ValueMaximum Value5%Percentile ValueExperimental Value

EC3 Curve

Modified EC3 Curve

Comparison of EC3 buckling curves with simulation data




Cylinders under external pressureStrong hardening alloys

Quality class A

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 1.0 2.0 3.0 4.0 5.0

λ

χ L/R=2 R/T=200L/R=2 R/T=100L/R=2 R/T=50L/R=1 R/T=200L/R=1 R/T=100L/R=1 R/T=50L/R=4 R/T=200L/R=4 R/T=100L/R=4 R/T=50

EC3 Curve

Modified EC3 Curve





Cylinders under torsionWeak hardening alloys

Quality class C

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

λ

χ L/R=2 R/T=200 Imp. 1

L/R=2 R/T=200 Imp. 2

L/R=2 R/T=100 Imp. 1

L/R=2 R/T=100 Imp. 2

L/R=2 R/T=50 Imp. 1

L/R=2 R/T=50 Imp. 2

L/R=4 R/T=200 Imp. 2

L/R=4 R/T=100 Imp. 2

L/R=4 R/T=50 Imp. 2

EC3 Curve





Shell buckling - proposal for pr1999-1-5

( )[ ]200

22

15.0

1

λλλαφ

λφφχ

αχχ

+−+=

−+=

=

perf

perfx

xRc

ykfσ

λ =

Axial (meridional) load External pressure (αθ) and torsion (ατ) Fabrication tolerance

quality class

Description Q αx α ref αθ or ατ

Class A Excellent 40 0,75 Class B High 25 0,65 Class C Normal 16 ( ) 44.1

//191.11

62.0

trQx

+=α

0,50 ( )( ), 20

11 0, 2 1 /ref ref

θ τα =+ − α λ − λ α

Alloy Axial (meridional) load External pressure Shear (torsion) λ0 α0 λ0 α0 λ0 α0

Weak hardening alloys 0.2 0.35 0.3 0.55 0.5 0.3 Strong hardening alloys 0.1 0.2 0.2 0.7 0.4 0.4





Quality Class A

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.50 1.00 1.50 2.00

λ

χ Minimum Value

Medium Value

Maximum Value

5% Percentile Value

Buckling curves - proposal for EC9




Cylinders under external pressureWeak hardening alloys

Quality class A

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 1.00 2.00 3.00 4.00 5.00 6.00

λ






Cylinders under torsion Weak hardening alloys

Quality class B

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 1.00 2.00 3.00 4.00 5.00

λ














0,0

0,2

0,4

0,6

0,8

1,0

1,2

0,0 0,2 0,4 0,6 0,8 1,0

w0/t

Pu /Pcr,th

R/t = 50 f 0.2 = 200 N/mm2

P cr,th = 24806.01 kNσcr,th = 197.40 N/mm2

clamped ends hinged ends

Imperfection 1

2

3

4

w0*/t

A B C

Exploitation of plastic buckling features(axially loaded cylinders)




0,0

0,2

0,4

0,6

0,8

1,0

1,2

0,0 0,2 0,4 0,6 0,8 1,0

w0/t

Pu /Pcr,th

R/t = 200 f 0.2 = 100 N/mm2

P cr,th = 2694.32 kNσcr,th = 85.76 N/mm 2

clamped ends hinged ends

Imperfection 1

234

w0*/t

A B C

Exploitation of plastic buckling features(axially loaded cylinders)




HINGED ENDS

0.00

0.03

0.06

0.09

0.12

0.15

0.18

0.21

0.24

0.27

0.30

0 50 100 150 200 250 300

R/t

w0*/t

f 0.2 = 100 N/mm2

f 0.2 = 200 N/mm2

f 0.2 = 300 N/mm2

Exploitation of plastic buckling features (axially loaded cylinders) – Imperfection limit w0*/t




CLAMPED ENDS

0.00

0.02

0.04

0.06

0.08

0.10

0 50 100 150 200 250 300

R/t

w0*/t

f 0.2 = 100 N/mm2f 0.2 = 200 N/mm2

f 0.2 = 300 N/mm2

Exploitation of plastic buckling features (axially loaded cylinders) – Imperfection limit w0*/t




Exploitation of plastic buckling features (axially loaded cylinders) Definition of quality Class A-plus in prEN-1999-1-5

Fabrication tolerance quality class Description Value of U0,.max (f0.2 in N/mm2)

Clamped ends Hinged ends

Class A-plus Excellent 0.2

1 2.25 0.01t Rf R t

⎛ ⎞+⎜ ⎟⎜ ⎟

⎝ ⎠

0.2

1 5 0.02t Rf R t

⎛ ⎞+⎜ ⎟⎜ ⎟

⎝ ⎠

Class A Very high 0,006 Class B High 0,01 Class C Normal 0,016

Q Fabrication tolerance quality class Description Clamped

ends Hinged

ends αx

Class A-plus Excellent 60 50 Class A Very high 40 Class B High 25 Class C Normal 16

( )1.44

,00,2

1

1 0.61 2.60x

x xE

Q f

α

λ λ

=⎛ ⎞

+ −⎜ ⎟⎜ ⎟⎝ ⎠




Cylinders under axial loadStrong hardening alloys

Quality Class A-plus

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0.00 0.50 1.00 1.50 2.00

λ

χ Minimum ValueMedium ValueMaximum Value5% Percentile Value

EC3

Proposed EC9 Curve

Hinged ends

Exploitation of plastic buckling features (axially loaded cylinders) - Class A-plus buckling curves




Shell buckling – summary of EC9 formulation

Unstiffened shells

Stiffened shells

Load cases • axial compression• external pressure• torsion




Material buckling class Axial (meridional) load External pressure Shear (torsion)

λx,0 μx λθ,0 μθ λτ,0 μτ A (Weak hardening alloys) 0.2 0.35 0.3 0.55 0.5 0.3 B (Strong hardening alloys) 0.1 0.2 0.2 0.7 0.4 0.4


Shell buckling – summary of EC9 formulation



Shell buckling – fabrication tolerance classes according to EC9

Class 1Class 2Class 3Class 4

Axial (meridional) load External pressure (αθ) and torsion (ατ) Fabrication

tolerance quality

class Q αx αref αθ or ατ

Class 1 16 0,50 Class 2 25 0,65 Class 3 40 0,75 Class 4 50-60

( ) 44.1//191.11

62.0

trQx

+=α

- ( )( ), 2

0

11 0,2 1 /ref ref

θ τα =+ − α λ − λ α












Interaction between load cases

• interaction between load cases• axial compression – external pressure• axial compression – torsion• external pressure – torsion

• validation of EN1993-1-6 procedures• proposal for an alternative formulation




Interaction between load cases ENV1993-1-6 formulation




σcr/σu

Pcr/Pu

Axial compression and External pressure

Interaction between load case ENV 1993-1-6 – Interaction domains




σcr/σu

τcr/τu

Axial compression and Torsion





Pcr/Pu

τcr/τu

External pressure and Torsion





Interaction between load casesprEN1993-1-6 formulation and proposal for prEN1999-1-5

prEN1993-1-6 prEN1999-1-5




σcr/σu

Pcr/PuEC3Class AClass BClass CClass AClass BClass CProposal Class AProposal Class BProposal Class C

Axial compression and External pressure

Interaction between load casesEN 1999-1-5 – Interaction domains




σcr/σu

τcr/τuEC3Class AClass BClass CClass AClass BClass CProposal Class AProposal Class BProposal Class C

Axial compression and Torsion





Pcr/Pu

τcr/τu

EC3Class AClass BClass CProposal Class AProposal Class BProposal Class C

External pressure and Torsion





Interaction buckling check according to EC9












Parametric analysis – Stiffened shells

Stiffenersection

Shellgeometry

Stiffenersize [mm] [mm] [mm] [mm]

Circular radius 5 10 25 50

Square side 5 10 25 50

5x20 10x20 25x20 50x20

5x10 10x10 25x10 50x10

5x5 10x5 25x5 50x5

Rectangular

R/t=50

R/t=100

R/t=200 sides












Stiffened shells – Proposal for EN19991-5Axial load

s 2xRc 12

θ s 3

EA A1.2 1n 1 A5 C d Aω1

0.80

x

s

x

EAC d

withφ

⎛ ⎞⎛ ⎞= α + +⎜ ⎟⎜ ⎟

⎝ ⎠⎝ ⎠+

α =

External pressure

2nRc 12

3

A1p AArj

0.50with

θ

θ

⎛ ⎞= α +⎜ ⎟

⎝ ⎠

α =

where

[ ]

i

2θφ45

2ssφ44

sss14

θφ12

ssφ11

jl

πrω

/rDDνC

/rdEIDC

)/(rdEAeC

CCνC

/dEACC

=

=

+=

=

=

+=

[ ][ ] 2

rtrstsφθ66

2rrθ55

rrr25

φθ33

rrθ22

/r/dGI/d0,5(GIDC

/rdEIDC

)/(rdEAeC

CC

/dEACC

++=

+=

−=

=

+=

[ ]

23312

233

225223311

23

214

221233

22522

2225

2223311

2

1422

12252

2233122

2

252

225566452

4444

1

)C(Cω)CωC)(CCC(ωA

)Cωj)(CCωC(Cω)Cj)(CCC(ω

)Cωj)(CCj)(CC(C2ωA

C2jCC)C(C2ωCωjA

+−+++=

+++−++−

+++=

+++++=




Equivalent orthotropic properties of corrugated sheeting (from prEN1999-1-6)

( )

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⋅=⋅=

=⋅=

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=⋅=

⎟⎟⎠

⎞⎜⎜⎝

⎛+

⋅=⋅=

⎟⎟⎠

⎞⎜⎜⎝

⎛+⋅=⋅=

⋅=⋅=

ϕ

ϕ

ϕ

2

223

xyθ

2y

2

222

3

xφ

2

22xyθ

2

22

yθ

2

3

xφ

4l

dπ1

12

tGIGD

0,13EtdIED

4l

dπ1

1

ν-112

EtIED

4l

dπ1

tGtGC

4l

dπ1tEtEC

3d

2tEtEC

Stiffened shells – Proposal for EN19991-5




0,0

0,2

0,4

0,6

0,8

1,0

1,2

0,0 0,2 0,4 0,6 0,8 1,0 1,2 1,4

λ

χ

L/R=2; R/t=50

L/R=2; R/t=100

L/R=2; R/t=200

Strong hardening alloysQuality class A

Q* = 1.3Q

Q* = Q

0, /x xRk xRcn nλ =

Axial load

Stiffened shells – Proposal for prEN19991-5


( )[ ]200

22

15.0

1

λλλαφ

λφφχ

αχχ

+−+=

−+=

=

perf

perfx



0,0

0,2

0,4

0,6

0,8

1,0

1,2

0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0

λ

χ

L/R=2; R/t=50

L/R=2; R/t=100

L/R=2; R/t=200

Strong hardening alloysQuality class C

0, /nRk nRcp pθλ =

External pressure

Stiffened shells – Proposal for prEN19991-5


( )[ ]200

22

15.0

1

λλλαφ

λφφχ

αχχ

+−+=

−+=

=

perf

perfx



General buckling curve formulation

( )[ ]200

22

15.0

1

λλλαφ

λφφχ

αχχ

+−+=

−+=

=

perf

perfx

Stiffened shells – EC9 formulation












Rolling Welding

MIGMIGbhaz = 20 mmbhaz = 30 mmbhaz = 35 mmbhaz = 40 mm

TIGbhaz = 30 mm

0 t 6mm< ≤6 t 12mm< ≤

12 t 25mm< ≤t 25mm>

0 t 6mm< ≤


Effect of welding (HAZ zones):definition of simplified design procedures




Effect of welding – Parametric analysis



0,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

0,0 0,1 0,2 0,3 0,4 0,5 0,6

w0/t

Pu /Pcr,th

α = 0.86α = 0.80

α = 0.72

α = 0.72 α = 0.71α = 0.67

Welded

Unwelded

R/t = 50; f 0.2 = 200 N/mm2; ρo,haz=0,53

Class 3 Class 2 Class 1

Effect of welding – Imperfection sensitivity curves, axial compression




0,0

0,2

0,4

0,6

0,8

1,0

1,2

1,4

0,0 0,1 0,2 0,3 0,4 0,5 0,6

w0/t

Pu /Pcr,th

α = 0.61α = 0.67

α = 0.54

α = 0.56α = 0.68

α = 0.76

Welded

Unwelded

Class 3 Class 2 Class 1

R/t = 100; f 0.2 = 200 N/mm2; ρ0,haz = 0,53

Effect of welding – Imperfection sensitivity curves, axial compression












λ i,wλ i,0

ρo,haz

1,0

00

ρi,w

λi,w,0 λ i

,0,w o o

,w ,0(1 ) i i

ii i

λ − λρ = ω + − ω

λ − λ

ω0 = ω (ρ0,haz)

λi,0 = λi,0 (λi,w,0)


Effect of welding according to EC9

Documents

EN 1999 - Eurocode 9: Design of aluminium structures Part ...eurocodes.jrc.ec.europa.eu/doc/WS2008/EN1999_9_Mandara.pdf · EN 1999 - Eurocode 9: Design of aluminium structures