buckling ultimate strength aluminum

ABS TECHNICAL PAPERS 2005

BUCKLING AND ULTIMATE STRENGTH OF ALUMINUM PLATES AND STIFFENED PANELS IN

MARINE STRUCTURES Xiaozhi WANG, Haihong SUN

American Bureau of Shipping, Houston, USA

Akira AKIYAMA American Bureau of Shipping, Yokohama, Japan

Aiping DU American Bureau of Shipping, Singapore

Presented at Fifth International Forum on Aluminum Ships, Tokyo, Japan, October 11-13, 2005

ABSTRACT Although aluminum alloys are well suited for some applications in marine structures, their unique material characteristics make the structural response different than steel. When structural elements are subjected to compressive loads, the buckling and collapse capacity is one of the most crucial factors governing the design. Ultimate strength of longitudinally stiffened panels is very important because it governs the structural capacity. Such panels are subjected to longitudinal compression, transverse compression, shear, and local bending. As the mechanical properties of aluminum alloy typically vary more significantly between the parent metal, weld metal and HAZ (Heat Affected Zone), as compared to those of steel, it would be anticipated that the existing formulations for steel structures may not be accurate when applied to aluminum alloy panels. It is the objective of this paper to develop the criteria for buckling and ultimate strength of plate panels and stiffened panels under longitudinal compression, and validate the criteria by non-linear FEM (Finite Element Method). Typical “marine” grade aluminum alloys will be investigated. Recommendations will be made based on the study and validation. KEY WORDS: aluminum material, buckling, HAZ, ship structures, stiffened panel, ultimate strength INTRODUCTION Aluminum is an attractive material as it is light, strong, clean, normally ductile, easily formed and fabricated, and readily available. It is recyclable and thus environmentally friendly. The accumulated experience over many years shows that aluminum alloys of “marine” grade offer both safety and reliability for use in marine structures. Although aluminum alloys are well suited for some applications in marine structures, their unique material

characteristics make the structural response different than steel. When structural elements are subjected to compressive loads, the buckling and collapse capacity is one of the most crucial factors governing the design. Ultimate strength of longitudinally stiffened panels is very important because it governs the structural capacity. Such panels are subjected to longitudinal and transverse compression arising from hull girder bending, and local bending arising from lateral pressure. As the mechanical properties of aluminum alloy typically vary more significantly between the parent metal, weld metal and HAZ, as compared to those of steel, it would be anticipated that the existing formulations for steel structures may not be accurate when applied to aluminum alloy panels. In this paper, the unique material characteristics of aluminum alloys will be addressed first. Recommended formula for buckling and ultimate strength of aluminum plate and stiffened panel under longitudinal compression will be presented, supported by finite element analysis. Finally, conclusions will be made based on the study. MATERIAL CHARACTERISTICS OF ALUMINIUM ALLOYS Compared to steel, the material characteristics of aluminum alloys can be focused on two issues: stress-strain relationship and heat-affected-zone. Stress-Strain Law A generalized law was proposed by Ramberg and Osgood (1943) for aluminum alloy:

)(σεε =

n

yE)(002.0

σσσε +=

where n = knee factor, which can be determined by Mazzolani

Buckling and Ultimate Strength of Aluminum Plates and Stiffened Panels in Marine Structures 119


(1995) as )1ln(

2lnχk

n+

=

y

t

t

ytkσσ

εσσ

10−

=

tσ = ultimate tensile strength

yσ = minimum yield strength (0.2% strain)

tε = fracture strain Table 1 lists the typical mechanical properties used in non-heat-treatable sheet and plate aluminum alloy based on ABS (2003). Table 1 Mechanical property of aluminum alloys, ref.

ABS (2003) Minimum Yield Strength

(at 0.2% Offset) in N/mm2

Ramberg-Osgood Model

Base Metal

Weld Metal

Knee Factor

Alloy and Temper

σ0b σ0w n 5083 - H111 165.7 145 17 5083 - H116 213.8 165 24 5083 - H323 234.4 165 20 5083 - H343 268.7 165 15 5086 - H111 145.1 124 24 5086 - H32, H34, H116 193.2 131 26

5383 - H116, H321 215.8 145 15

5383 - H34 264.8 145 22 5456 - H116, H321 227.5 179 15

5456 - H323 248.1 179 24 5456 - H343 282.4 179 18

Figure 1 represents typical stress-strain curves reproduced from the Ramberg-Osgood material model for 5083-H116 in the base and welded material conditions, based on the parameters in Table 1.

0

50

100

150

200

250

300

350

0.0% 0.5% 1.0% 1.5% 2.0%

Base metal

Weld metal

ε0.2 Strain, ε

Stre

ss, σ

Fig.1 Stress-strain curves of base metal and weld

metal of 5083-H116

Heat Affected Zone In welded profiles, the heat input removes some of beneficial effects from heat treatment or strain hardening and leads to a decrease in the elastic limit, which results in the strength redistribution along the cross section profile with the minimum at the welds. The first experimental analyses carried out by Hill, Clark and Brungraber (1962) were on the plate joints with longitudinal welds at the center, using 6000 series alloy. As an upper bound of HAZ extent, as denoted by z, they found a value of 0.74 in. On this basis, the French specifications give z values equal to or smaller than 25 mm (1 in). They also found the reductions of yield strength compared with the parent metal were between 33 and 50 percent. Mazzolani (1971) studied this problem. As expected, the distributions depend greatly on the heat treatment. Whereas the decrease of yield strength at welds is about 10% for the non-heat-treat alloys, the decrease reaches 40-50% for heat-treated alloys. As specified in the ship rules and guides, due consideration should be made for critical or extensive weld zones for yield strength because the localized reduction in strength properties to the parent alloy occurs near welds, as shown in Figure 1. The strength of the butt-welded aluminum alloys is dependent on the use of compatible strength welding wires. The extent of the HAZ is commonly taken from the “1-in” rule, i.e. the softening effect extends through 1-in in all directions from the weld, ref. Kontoleon et al. (2000) and Rigo et al (2004). Another rule of thumb is to take the extent of HAZ as approximately three times the average thickness of the welded components, based on Paik and Duran (2004). The relationship between HAZ extent and plate thickness for fillet weld and butt weld in BS 8118 (1991) can be expressed as:

{ }3/20,3min AA ttz += , mm for fillet weld, excluding 7xxx series alloys

{ }3/20,/3min 2AAB tttz += , mm

for butt weld, excluding 7xxx series alloys where tA is the lesser of 0.5(tB + tC) and 1.5tB, tB and tC are the thickness of the thinnest and thickest elements connected by welding, respectively. The modified factors, α and η in BS 8118, are ignored in the analysis by Kristensen and Moan (1999). Figure 2 shows the relationship between the extent of HAZ and plate thickness according to BS 8118, ref. Kristensen and Moan (1999).

120 Buckling and Ultimate Strength of Aluminum Plates and Stiffened Panels in Marine Structures


0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 5 10 15 20 25

Plate Thickness, mm

Rat

io =

Ext

ent o

f HA

Z/P

late

Thi

ckne

ss

6082-T6

5083-O5083-F

Fig. 2 Extent of HAZ according to BS 8118, ref.

Kristensen and Moan (1999) If all of the elements connected by welding have the same thickness, the above formula can be rewritten as:

⎩⎨⎧

>+≤

=mm5.7if3/20mm5.7if3

tttt

z , mm

It can be seen that the “1-inch” rule and “three times the average thickness” rule for the HAZ extent are the two special cases from the formula. Figure 3 shows a referenced Sealium@ (5383 alloy) sample for the hardness measurement of the butt-welded metal, ref. Paik and Duran (2004). The extent of HAZ in BS 8118 (1999) is fairly close to the hardness measurement.

Fig. 3 Hardness measurement for a 6 mm Sealium@

butt-welded sample, ref. Paik and Duran (2004) In EuroCode 9 (1999), the extent of HAZ are specified for a MIG weld laid on unheated material and with interpass cooling to 600C or less when multi-pass welds are laid, 0 < t ≤ 6 mm z = 20 mm 6 < t ≤ 12 mm z = 30 mm 12 < t ≤ 25 mm z = 35 mm t > 25mm z = 40 mm

The extent of HAZ for a TIG weld is greater because the heat input is greater than for a MIG weld, which is: 0 < t ≤ 6 mm z = 30 mm The above figures apply to in-line butt welds or to fillet welds at T junctions in 6xxx, 7xxx or work-hardened series 5xxx series alloys. In BS8118 (1991) and EuroCode 9 (1999), the influence of temperature is taken into account. The extent of HAZ is increased when temperature rises above 600C. In this paper, the HAZ size is defined according to BS8118 (1991). BUCKLING AND ULTIMATE STRENGTH OF ALUMINUM PLATE PANELS Buckling Strength of Plate Panels in Axial Compression The critical buckling strength may be obtained from the following equation: σc = σE when σE ≤ 0.5σyb

= σyb ⎟⎟⎠

⎞⎜⎜⎝

⎛−

E

yb

σσ4

1 when σE > 0.5σyb

where σE = elastic buckling stress

= 2

2191.0 ⎟

⎠⎞

⎜⎝⎛

stEm ψ N/mm2 (kgf/mm2, psi)

ψ = reduction factor considering HAZ softening effect due to welding

m1 = buckling coefficient, ref. Table 3 in Part 3, Chapter 2, Section 3 of ABS (2003)

E = 6.9 × 104 N/mm2 (7000 kgf/mm2, 10 × 106 psi) t = thickness of plating, in mm (in.) s = shorter side of plate panel, in mm (in.) l = longer side of plate panel, in mm (in.) σyb = minimum yield strength of base material, in

N/mm2 (kgf/mm2, psi) Ultimate Strength of Plate Panels in Axial Compression The ultimate strength may be calculated from the following modified Faulkner’s formula:

⎪⎩

⎪⎨⎧

>−

≤= 112

11

2 βββ

β

σσ

for

for

yb

u

where

Ets ybσ

ψβ 1

=

Ets ybσ

β =



In the above two equations, the reduction factor may be defined as ψ = 1 for 5.01.01 βη −≥

= )1(42.1142.15.0

ηβ

−− for 5.01.01 βη −<

where η = σyw/σyb

σyw = minimum yield strength of material weld metal, in N/mm2 (kgf/mm2, psi)

The above formula is valid for β ≥ 1. This condition is satisfied in the design of aluminum crew and supply boats, in which the typical range of β is between 1.5 and 3.0. Finite Element Analysis Figure 4 shows aluminum plating between stiffeners subject to axial compression. In a typically continuous aluminum stiffened plate structure, the stiffeners should be designed in such a way that the stiffener does not fail prior to plate buckling. Often, the plate is assumed to be simply supported at all four edges, which are kept straight in plane during the deformation. This assumption is widely accepted in the buckling and ultimate strength assessment of ship structures, ref. Sun and Wang (2005).

σa Simply supported

Longitudinal

Transverse Fig. 4 A typical aluminum plating

Due to the double symmetry, one quarter of the plate is modeled in the following analysis. The FE model is shown in Figure 5 and HAZ is represented by fine mesh with reduced yielding stress.

Fig. 5 Finite element model

Comparison Parametric analyses are performed for different slenderness ratio and different strength ratio for weld and base material, which are given below: l = 600 ~ 2000 mm s = 200 ~ 500 mm t = 4 ~ 20 mm β = 1 ~ 4 η = ybyw σσ / = 0.4 ~1.0 In the finite element analysis, the initial imperfection is assumed to be of the following form, ref. Sun and Wang (2005):

)sin()]sin()sin([ 03010

sy

lx

lx

tW p παπδπδ +=

where

sl

=α

tWp

32 0

01 =δ ,t

Wp

30

03 =δ , sW p 09.00 =

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

η = 0.4 η = 0.5 η = 0.6 η = 0.7

η = 0.8 η = 0.9 η = 1.0

Axial strain/initial yield strain

Axi

al st

ress

/yie

ld

stre

ss

Fig. 6 Load – end shortening curves of a plate panel with varying strength ratio for weld and base material

Figure 6 illustrates the load-end shortening curves of a plate panel (l = 900 mm, s = 300 mm, t = 6.68 mm, E = 69000 N/mm2, σyb = 213.8 N/mm2), in which the HAZ softening effect to the ultimate strength of a plate panel is demonstrated. Figure 7 shows the comparison results of prediction by the modified Faulkner’s formula with the FE analysis, in which a total of 132 FEA results with different values of aspect ratio and slenderness ratio are used.



Fig. 7 Comparison of the modified Faulkner’s formula

with the FE analysis It can be seen that the modified Faulkner’s formula predicts reasonably conservative ultimate strength of plate panels. BUCKLING AND ULTIMATE STRENGTH OF ALUMINUM STIFFENED PANELS Critical Buckling Stress The critical buckling stress of longitudinals in compression may be obtained from the following equation σc = σE when σE ≤ 0.5σy

= ⎟⎟⎠

⎞⎜⎜⎝

⎛−

E

yy σ

σσ

41 when σE > 0.5σy

where σy = minimum yield strength of stiffener under consideration, in N/mm2 (kgf/mm2, psi). If there is a large difference between the minimum yield strength of a stiffener and the plating, the minimum yield strength from the weighting of areas is to be used σE = elastic buckling stress, in N/mm2 (kgf/mm2, psi) Column Elastic Buckling Stress The elastic buckling stress σE of a stiffener with the associated effective plating, with respect to axial compression may be obtained from the following equation:

21 lACEI

e

eE =σ N/mm2 (kgf/mm2, psi)

where Ie = moment of inertia, cm4 (in4), of stiffener, including effective plate flange, se

C1 = 1000 (1000, 144) Ae = cross-sectional area, in cm2 (in2), of stiffener,

including effective plate flange, sel = span of stiffener, in m (ft) se = effective width of plating = C s

⎪⎩

⎪⎨⎧

>−

≤= 112

11

2 βββ

β

for

forC

Web and Flange Buckling Local buckling is considered satisfactory provided the following proportions are not exceeded. - Flat bars, Outstanding Face Bars and Flanges dw/tw ≤ 0.5(E/σyb)1/2

- Built-up Sections, Angle Bars and Tee Bars dw/tw ≤ 1.5(E/σyb)1/2

- Bulb Plates dw/tw ≤ 0.85(E/σyb)1/2

When σa < 0.80σyb, the values of dw/tw may be increased by the factor (0.80σyb /σa), where σa is the working stress and is to be taken not less than 0.55σyb. dw and tw are the thickness and height of the stiffener web, as shown in Figure 8.

Centroid of Stiffener

b1 bf

b2

t

d w

tw

t f

⊕

z 0

y0

se

y

z

Fig. 8 Sectional dimensions of a stiffened panel

Comparison To verify the column buckling strength of stiffened panels, the series of nonlinear analyses is performed using ANSYS (2004) program. The engineering model to be used in the finite element analysis is shown in Figure 9, ref. Sun and Wang (2005). The FE model and boundary condition is shown in Figure 10. The situation considered is that all stiffeners are welded to the plate and the stiffener-plate conjunction region is HAZ. The initial deflection mode is taken as the first buckling mode, which deteriorates the ultimate strength most significantly. Based on recommendation in Sun and Wang (2005), the initial deflection amplitude is taken



into account, as given by: LW0 = 0.15%l

L

M

N

P O

A

B

D E

G

C

H

I

J F

AC,BD,DE: u = 0, θx = θz = 0 LN,MO,OP: u = S1, θx = θz = 0 FH, IJ: w = 0, θz = 0 GI: w = 0, v= S2, θy = 0 CN: v = 0, θx = 0, θz = 0 AL: v = S3, θx = 0, θz = 0 S1, S2 and S3 are the parameters to keep the boundaries straight or in-plane.

Fig. 9 Engineering model

Fig. 10 FE model of a stiffened panel Figure 11 illustrates some representative results based on the nonlinear FE analysis.

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

η = 0.4

η = 0.5

η = 0.6

η = 0.7

η = 0.8

η = 0.9

η = 1.0

l = 1219mm, s = 304.8mmt = 7.9375mmdw = 101.6 mm, bf = 50.8mmtw = tf = 6.35mmsyb = 213.8 MPa

Axial strain/initial yield strain

Axi

al st

ress

/yie

ld st

ress

Fig. 11 Load-end shortening curves of a stiffened panel

with varying strength ratio for weld and base material

Figure 12 shows the comparison results of prediction by the proposed formula with the FE analysis, in which a total of 56 FEA results are used.

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

Bias Mean = 0.8923COV = 4.64%

FEA: σu/σyb Pr

opos

ed F

orm

ula:

σcr/σ

yb

Fig. 12 Comparison of proposed formula with the FE

analysis It can be seen that the predictions of the proposed formula are on the conservative side for the stiffened panels selected from typical aluminum crew and supply boats. CONCLUSIONS This paper proposed buckling and ultimate strength formulas for plate panels and stiffened panels in aluminum ship structures. Based on the previous experiences from steel structures, a modified Faulkner’s formula was put forward for ultimate strength of plate under longitudinal compression and a reduction factor is introduced to take into account the HAZ softening effect. The column buckling formula used for steel ship structures was also extended for stiffened aluminum panel under axial compression.

It was demonstrated that the proposed formulas are of reasonable accuracy compared with the nonlinear FE analysis. Incorporating these formulas into the existing ship rules, the buckling and ultimate strength of aluminum plate and stiffened panels can be assessed in a more rational way.

Please note that the conclusion of this paper is the opinion of authors.



ACKNOWLEDGEMENTS The authors would like to thank Mr. Jim Speed of American Bureau of Shipping for proof reading this manuscript.

Kristensen, O. H. H. and Moan, T. (1999), “Ultimate Strength of Aluminum Plates under Biaxial Loading”, PRADS’1999.

Paik, J. K. and Duran, A. (2004), “Ultimate Strength of

Aluminum Plates and Stiffened Panels for Marine Structures”, Marine Technology, vol. 41, no. 3, 108-121.

REFERENCES ABS (2003), “Rules for Building and Classing High-

Speed Naval Craft”. Mazzolani, F. M. (1971), Inelastic Behavior of Welded

Aluminum Prifiles, Costruzioni Metalliche, 1971, No.5.

ABS (2004), “Guide for Buckling and Ultimate Strength Assessment for Offshore Structures”.

ANSYS (2004), Release 8.1 Documentation, ANSYS,

INC. Mazzoloni, F. M. (1995), “Aluminum Alloy Structures”,

E&FN Spon. BS 8118 (1991), “Structural Use of Aluminum, Part 1:

Code of Practice for Design”. Ramberg, W. and Osgood, W. R. (1943), “Description of

stress-strain curves by three parameters”, NACA Techn. No. 902.

EuroCode 9 (1999), “Design of Aluminum Structures- Part 1-1: General Rules”, DD ENV, 1999-1-1:2000.

Rigo, P., et. al. (2004), “Ultimate Strength of Aluminium

Stiffened Panels: Sensitivity Analysis”, 9th Symposium on Practical Design of Ships and Other Floating Structures (PRADS), Luebeck-Travemuende, Germany.

Hill, H. N., Clark, J. W., and Brungraber, R. J. (1962),

“Design of Welded Aluminum Structures”, Trans. ASCE, 127, 1962.

Kontoleon, M. J., et. al. (2000), “Butt-welded Aluminum

joints: A Numerical Study of The HAZ effect on the Ultimate Tension Strength”, in Baniotopoulos, C. C., and Wald, F., Editors, The Paramount Role of Joints in the Reliability Response of Structures, 337-346.

Sun, H. and Wang, X. (2005), “Buckling and Ultimate Strength Assessment of FPSO Structures”, Trans. SNAME, Houston, USA.


buckling ultimate strength aluminum

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