Ultimate strength of aluminium plates under biaxial loading

Ultimate Strength of Aluminium Plates Under Biaxial

Loading

Odd Halvdan Holt Kristensen’ Torgeir Moan, (M)’

ABSTIL4CT

This paper addresses the collapse strength of rectangular aluminium plates under uniaxial and biaxial compressive stress. Both non-welded and welded plates with residual stresses and soft zones have been

considered. Numerical simulations using the non-linear jinite element program ABA QUS have been

carried out. The plates have been loaded longitudinally, transversally and biaxially until final collapse

strength has been reached. Several dt~erent aluminium alloys and welded conditions have been

investigated. The results are compared with

improvement of current approaches are given.

NOMENCLATURE a b c E k n RX RY t w Wi x Y D E q G Length of plate Breadth of plate

Proportionality parameter for biaxial design curve

Young’s modulus

Proportionality constant between transverse and longitudinal stresses

Knee-factor in the Ramberg-Osgood law Longitudinal strength ratio of plate in biaxial compression

Transverse strength ratio of plate in biaxial compression

Plate thickness

Plate out-of-plane deflection

Normalizing constant for plate out-of-plane deflection

Longitudinal co-ordinate of plate Transverse co-ordinate of plate Plate slenderness ratio

Strain

Curve parameter in the biaxial interaction formula

Stress

~o.2 0.2 Y. tensile proof stress

1_{NorwegianUnivemity}

of Science andTechnology, Norway

existing design codes and recommendations jor

ax Longitudinal mean stress

aXu Mean uniaxial ultimate stress in longitudinal

compression

~Y Transverse mean stress

~yu Mean uniaxial ultimate stress in transverse

compression

INTRODUCTION

There has been a significant research on the design of steel structures used in the marine industry. Using aluminium instead of steel has made it possible to produce lighter and faster vessels. Design rules for aluminium structures, however, have often been established by changing the material parameters in design codes for steel structures. This approach sometimes give far from optimal design and in other cases the requirements are non-conservative.

In Norway there is a concerted effort to improve procedures for calculating global load effects as well as strength formulations for fatigue and ultimate limit states.

One issue of concern in developing new design codes is the ultimate strength of the hull plating. This plating is normally located between longitudinal

stiffeners and transverse girders. The stiffeners are either extruded or welded to the plates, and the transverse girders are welded to the plates.

During the process of welding, zones with

reduced yield strength and residual stresses are created. Close to the welds there will be a zone with reduced yield strength and residual stresses in tension. Further away from the welds, there is a corresponding stress field in compression to maintain equilibrium. In a whole structure, that consists of many welded parts, a single component can be exposed to various residual stress fields, but it will only have a reduced yield strength in a zone close to a weld. The residual stress fields are strongly dependent on the sequential order in which the structure has been assembled. Welding also introduces post welding distortions that has to be kept within specified tolerance limits.

The extent of the residual stresses and reduction of the yield strength close to the welds will depend on the different aluminium alloys used. In this paper three different alloys are considered. They have a high, moderate and a low yield strength, respectively.

Normally plates have been considered as the first line of defence in design. They transfer the loads into the stiffeners, and when interframe strength is calculated, one uses a beam-column approach which includes an effective width of the plating. For this reason the aim here is to estimate this ultimate strength of the plate.

The non-linear finite element program ABAQUS, which has been validated against test results (Aalberg Table 1: Material properties taken fromBS8118

et al, 1998; Hopperstad et ai, 1994; Kristensen, 1997) is used to determine the stiffness and ultimate strength capacity of plates with typical dimensions for catamarans and other fast going vessels. The plates all had the same length and breadth, but their thickness was altered to cover all ranges of slenderness of practical interest for the marine industry. Their boundary conditions were chosen to represent the boundary conditions that the plates will have in a real marine structure. The plates were strained axially and transversally beyond ultimate load. For one of the aluminium alloys some of the plates were also analysed for proportional biaxial loading.

NUMERICAL MODELLING

Three different aluminium alloys have been

investigated. The first alloy is 6082-T6 (Al

SiMgMn). This is a heat treatable alloy that is solution heat-treated and then artificially aged. The second alloy is 5083-0 (Al Mg4,5Mn0,7). This is a non-heat-treatable alloy in it’s annealed state. Alloy number three is alloy 5083-F, with the same alloying elements as 5083-0, but the material is as fabricated,

with no formal heat treatment. The material

properties of the different alloys according to British Standard 8118 for the structural use of aluminium (British Standards Institution, 1991) are given in Table 1. These are the values for plates.

-Alloy Minimum 0.2 $ZOMinimum tensile Minimum Modulus of Loss of strength

tensile proof strength elongation elasticity due to welding

stress [N/mm2] [N/mm2] at failure [70] [N/mm2] [%]

6082-T6 240 295 8 70000 50

5083-0 125 275 12 70000 0

5083-F 175 320 18 70000 0

The material parameters used in the analyses are not the exact values taken from Table 1. They were taken from experimental values in the Ph. D. thesis by David Shane Mofflin (Mofflin, 1983). The material properties were found from compression rather than tension tests since failure due to buckling is essentially a compression failure. The values

presented are true values and not minimum

guaranteed values. They represent typical values from a lot of tests performed.

Determining the buckling capacity of aluminium structures in an adequate way does require knowledge about the whole stress strain relationship for the alloy analysed, not only the 0.270 tensile proof stress of the

material. One of the most common approaches is to use the Ramberg-Osgood law (Mazzolani, 1995).

[1

E++o.

ow

Cr0.2

The value of the exponent, n, E and CY0,2 in the

Ramberg-Osgood law has been experimentally

determined in the work done by Mofflin. The

material parameters for the different alloys used in the numerical analyses are given in Table 2. Because the variation between the different elastic moduli was small, the modulus of elasticity was kept constant.

Table 2: Material parameters used in the numerical analyses

,

Alloy 0.270 tensile roof

1’ Knee-factor, n, Modulus of elasticity Loss of yield

stress [N/mm ] in the Ramberg- [N/mm2] strength due to

Osgood law welding [7.]

6082-T6 292 26 70000 50

5083-0 91 10 70000 10

5083-F 190 15 70000 10

All plates had a length of 900 mm and a breadth of from 1-3, except for some plates made of alloy 6082-300 mm. The thickness of the plates was varied to T6. Numerical analyses were then performed for ~ in

cover slenderness ratios the range of 1-5. Table 3 states the thickness of the

plates analysed. (2)

Table 3: The thickness of the plates analysed

Plate slenderness [-] Thickness 6082-T6 [mm] Thickness 5083-0 [mm] Thickness 5083-F [mm]

1.0 19.37 10.82 15.63 1.5 12.92 7.21 10.42 2.0 9.69 5.41 7.81 2.5 7.75 4.33 6.25 3.0 6.46 3.61 5.21 3.5 5.54 4.0 4.84 4.5 4.31 5.0 3.88

The plates were given an initial imperfection. The imperfection was introduced as a sine-wave out of plane deflection of the nodes. When the plates were axially loaded, three half waves in the longitudinal direction and one half wave in the transverse direction were used. In the case of transverse loading, one half wave as well in longitudinal and transverse direction was used. These are the linear eigenmodes when the plate is subjected to axial and transverse loading respectively (Hughes, 1988; Mazzolani, 1995).

For biaxial loading a combination of several

modes was used. From initial deflection

measurements of steel and aluminium plates the

sinusoidal mode with one half wave in both

Figure 1: HAZ along long edges

longitudinal and transverse direction seem to dominate, with the higher modes having smaller amplitudes (Antoniou, 1984; Clarke et al 1985). A deflection pattern given as

‘=w[si’t}025si@

‘3)

was chosen. The deflection amplitude was adjusted to give maximum out-of-flatness equal to 0.005b or 0.01 b. Biaxial analyses were only carried out for a maximum out-of-flatness amplitude equal to 0.005b. Five different patterns of soft zones due to welding

Figure 3: HAZ along all edges Figure 4: HAZ in the middle - longitudinal direction

Figure 5: HAZ in the middle - transverse direction

have been investigated. They are shown in Fimres The breadth of the soft zones according to BS

1-5. The width ~f the soft ~ones was general~ set 8118 is given in Table 4. The materials h&e been equal to 25 mm for the plates in the numerical assumed to be at a sufficiently low temperature at the analyses. For plates of material 6082-T6 soft zones start of the deposition of every weld and the materials

with a width of 12.5 mm was also investigated. have been assumed to have at least two valid heat

paths. Table 4: Extent of heat-affected-zone according to British Standard8118

Plate slenderness [-] Extent of HAZ Extent of HAZ Extent of HAZ

6082-T6 [mm] 5083-0 [mm] 5083-F [mm] 1.0 26.46 23.61 25.21 1.5 24.31 21.63 23.47 2.0 23.23 16.23 22.60 2.5 22.58 12.99 18.75 3.0 19.38 10.83 15.63 3.5 16.62 4.0 14.52 4.5 12.93 5.0 11.64

The width of the parts of the plate with residual stresses in tension was set identical to the width of the soft zones. The residual stresses were assumed to be uniformly distributed over the breadth of the soft zones, and the value of the residual stresses equal to 75 % of the reduced yield stress. Further away from the welds there will be a corresponding residual stress field in compression. This corresponding stress field was also assumed to be uniformly distributed over the parts of the plates that were not having a reduced yield strength. The value of the residual stress in compression was adjusted to give zero resulting residual forces. The calculation of the total residual stress field for the plates with soft zones along all edges was carried out by adding two residual stress fields; one residual stress field containing soft zones

along long edges only and one residual stress field containing soft zones along short edges only.

Plates with no heat affected zones, as well as plates with residual stresses only or soft zones only were investigated.

Due to the location of the plates, between longitudinal stiffeners and transversal girders, a reasonable assumption will be to assume the edges to be free to move in their own plane, but with the edges forced to remain straight. In the real structure there will be some degree of fixation of the plate, but in the numerical analyses, the edges were all assumed to be

simply supported. This is believed to be a

conservative assumption.

Due to symmetry one quarter of the plate was modelled for all the uniaxial cases. The element

model consisted of 432 shell elements. They all were quadratic with characteristic element length 12.5 mm. The element type used was a four noded thick shell element (type S4R). An incremental method with true Newton-Raphson iteration was used in the analyses. When the biaxial analyses were carried out, the whole plate was modelled with the same elements and equal element dimensions.

NUMERICAL RESULTS

A significant sensitivity study has been carried out. Only the most relevant cases for establishing design rules will be shown herein.

Axial Compression

Axial compression means that the plates are strained by giving all the nodes at one of the short edges the same axial displacement. The total force was then recorded by summing the node reaction forces at the other end of the plate. To calculate compressive stresses, the total force was divided by the cross section area of the plate. Normalised stresses were found by dividing the stresses by the 0.2 % tensile proof stress.

Normalisation is not a straight forward process

when dealing with aluminium alloys because

aluminium alloys, unlike for instance mild steel, do not have a well defined yielding plateau. Another stress value that could have been used for making the curves non-dimensional is the stress that corresponds to equal plastic and elastic deformations.

The deflection of one of the nodes at the loaded edge was divided by the length of the plate to give the strain, and the strain was divided by the yield strain to give the normalised strain.

Axial Compression, Alloy 6082-T6

Figure 6 shows the maximum value of the stresses for all the plates with heat-affected-zones along all edges. Both the effects of soft zones and residual stresses are included. The maximum initial deflection amplitude was set equal to 0.005b.

In the same figure results from several design rules are plotted (British Standards Institution, 199 1; Det Norske Veritas, 1995; European Committee for Standardization, 1998).

The different DNV curves are taken from Det Norske Veritas ( 1995). These design codes are meant for steel, but the elastic modulus has been altered, and the yield strength has been replaced by the 0.2’% tensile proof stress. The effective flange capacity is the capacity of the plate when the plate is regarded as an effective flange in a beam-column analysis of a stiffened panel. The ultimate limit state capacity,

ULS, and serviceability limit state capacity, SLS, is the capacities of a pure plate respectively. British Standard 8118 and Eurocode 9 are developed directly for aluminium. Johnson-Ostenfeld is the classical linear buckling capacity with the Johnson-Ostenfeld correction (Hughes, 1988; Mazzolani, 1995).

For axially loaded plates made of alloy 6082-T6 all the design codes overestimate the buckling capacity for compact plates (~= 1). For slender plates (~=3) and very slender plates (~=5) the numerical analyses show excellent agreement with results predicted by the design codes British Standard 8118 and Eurocode 9. The DNV ULS and SLS formulas underestimate the capacity for slender plates and very slender plates as compared to the numerical analyses,

while the DNV effective flange formula shows

excellent agreement with the numerical analyses for slender plates and very slender plates.

For plates with slenderness exceeding 3.0, the breadth of the soft zones used in the analyses (25mm), is considerably larger than the breadth of the soft zones according to British Standard 8118 (see Table 4). Figure 7 shows results from analyses of plates with heat-affected-zones of a width equal to O

mm, 12.5mm and 25 mm. Results from numerical

analyses with the smaller extension of the soft zones is more representative for slender plates.

Axial Compression, Alloy 5083-0

Results from numerical analyses of plates made of alloy 5083-0 are given in Figure 8. The plates have heat-affected-zones along all edges. Both the effects of soft zones and residual stresses are included. The maximum initial deflection amplitude was set equal to 0.005b.

Axial Compression, Alloy 5083-F

Results from numerical analyses of plates made of alloy 5083-F are given in Figure 9. The plates have heat-affected-zones along all edges. Both the effects of soft zones and residual stresses are included. The maximum initial deflection amplitude was set equal to 0.005b.

Comments

Numerical analyses of axially loaded plates made of alloy 5083-F and 5083-0 show good agreement with design codes. For compact plates alloy 5083-F has higher capacities than alloy 5083-0 as compared to the design rules, but for more slender plates their relative strength is almost identical. There is a general trend, however, that the strength predicted by the numerical analyses is comparatively larger than the values of the design rules for slender plates than for compact plates.

1.2 1 0.8 I i ‘6 ?! a 0.4 0.2 0

1

Figure 6: Ultimate axial capacity obtained by numerical analyses of plates made of alloy 6082-T6, soft zones and residual stresses included, compared with several design codes

1,2 1 0.8 0,6 0.4 1 , 0.2 -0 4 1 i .5 2 2.5 9 9.5 4 4.s 5 Normalized $Iendornoss

Figure 7: Comparison between ultimate strength of plates made of base material only and plates with heat affected zones on all four edges. Results taken from the numerical analyses of alloy 6082-T6.

1.2 1 0.8 ! J 0., If z 0,4 0.2 0 -m- Num. analyses +BS 8118 +Eurocode 9 -IS-DNV, Eff. flange +DNV, ULS +DNV, SLS —Johnson-Ost.

1 1.5 2 2.5 L

Slondorrwss

Figure 8: Ultimate axial capacity obtained by numerical analyses of plates made of alloy 5083-0, soft zones and residual stresses included, compared with several design codes

1.2 1 0,8 1! 3~ 0.6 = E 3 0.4 0.2 0 _. + Num. analyaes +BS 8118 + Eurocode 9 -IS-DNV, Eff. flenge +DNV, ULS +DNV, SLS —Johneon-Ost.

1 1.5 2 2.5 0

Slenderness

Figure 9: Ultimate axial capacity obtained by numerical analyses of plates made of alloy 5083-F, soft zones and residual stresses included, compared with several design codes

Final Remarks

Plates that, in the heat-affected zones, have a 50 % reduction of 0.2 910tensile proof stress and a 50 90 reduction in the knee-factor in the Ramberg-Osgood formula (see Eq. (l)), show a large reduction in ultimate capacity as compared to numerical analyses of plates consisting of non-welded material only. The decrease can be as large as 34 percent. Plates with only 1f)Yo reduction in yield stress and knee-factor

were only moderately affected by heat-affected zones, and the reduction in strength was never larger than 1170.

For plates with a 50 % reduction of 0.2 % tensile proof stress and knee-factor, heat-affected zones

along loaded edges or heat-affected zones

perpendicular to the loading direction were much more critical than heat-affected zones along unloaded edges, and there is almost no difference in buckling capacity if the plates contain heat-affected zones along all edges or heat-affected zones along loaded edges only. Plates with only 10% reduction in yield stress and knee-factor showed no such pattern.

The effect of residual stresses, both alone and together with soft zones, could be ignored. Residual stresses seemed to give, especially for slender plates, a minor stiffness reduction and an even less reduction of ultimate strength, when the residual stresses were introduced separately, but if residual stresses acted

together with soft zones there was no difference in ultimate strength, and the stiffness reduction was only visible for plates with slenderness, ~, larger than 2.5. Transverse Compression

Transverse compression means that the plates are strained by giving all the nodes at one of the long edges of the plate the same transverse in-plane displacement. The total force can then be recorded by adding the node reaction forces at the other long edge of the plate. The same method for calculating

normalised stresses and strains as for axial

compression is used.

Transverse Compression, Alloy 6082-T6

Figure 10 shows the maximum value of the

stresses for all the plates with heat-affected-zones along all edges. Both the effects of soft zones and residual stresses are included. The initial deflection amplitude was set equal to 0.005b. In the same figure results from three different design curves taken from DNV Classification Notes 30.1 are presented together with a classical linear buckling curve with Johnson-Ostenfeldt correction. British Standard 8118 and Eurocode 9 do not have relevant design curves for transverse compression.

c.

i

El

~ -o- DNV, ULS ~ 0.4- + DNV, SLS 3 — Johneon-Oat. 0.3- 0.2- 0.1-1 1.5 2 2.5 3 3.5 4 4.5 5 Slenderness

Figure 10: Ultimate capacity obtained by numerical analyses of plates made of alloy 6082-T6, soft zones and residual stresses included, compared with design codes

0.7-0.6I I z 0.5-j g 0.4-z 0.3-

0.2-E

Figure 11: Ultimate capacity obtained by numerical analyses of plates made of alloy 5083-0, soft zones and residual stresses included, compared with design codes

Figure 12 0.7 0.6I El +F&n-landy3e3 -lK-DNv, mffange j + w, LJLS = i? 0.4-+DNV, SLS —Johr12ml-ast. i! 0.3- 02- 0.1-04 1 i .5 2 2s 3

: Ultimate capacity obtained by numerical analyses of plates made of alloy 5083-F, soft zones and residual stresses included, compared with design codes

Transverse Compression Alloy 5083-0

Results from numerical analyses of plates made of alloy 5083-0 are given in Figure 11. The plates have heat-affected-zones along all edges. Both the effects of soft zones and residual stresses are included. The maximum initial deflection amplitude was set equal to 0.005b.

Transverse Compression, Alloy 5083-F

Results from numerical analyses of plates made of alloy 5083-F are given in Figure 12. The plates have heat-affected-zones along all edges. Both the effects of soft zones and residual stresses are included. The maximum initial deflection amplitude was set equal to 0.005b.

Comments

The DNV formula given in Buckling Notes 30.1 for effective flange of a transversely compressed plates can be used as a good approximation for all alloys and all values of the slenderness, f!; even though it will slightly overestimate the capacity of

very slender plates made of alloy 6082-T6 as

compared to the numerical analyses. The DNV ULS curve overestimates the capacity of all plates, except for very slender plates, where the agreement between numerical simulations and the design formula is good. The DNV SLS formula and classical linear buckling

theory with Johnson-Ostenfeld correction

overestimate the buckling capacity for compact plates and underestimate the capacity for slender plates.

The stiffened panels made of alloy 6082-T6 are often extruded. Then, there will be no welds along the long edges of the plate, only along the short

I

edges. The transverse buckling capacities for theses plates are higher than for plates that have been welded along all edges. The increase in ultimate strength varies between a maximum of 9.78 % for compact plates (~=1) and a minimum of 5.26 $ZOfor

plates with slenderness, ~, equal to 2.5. Biaxial Compression

General

Biaxial compression means that the plate is compressed by imposing forces both at the short and long edges simultaneously. The plates are collapsed, with the ratio between average stresses in longitudinal and transverse direction kept constant; ensuring so called proportional loading.

ov=k. ~, (4)

The stresses in the x- and y-direction are normalized with respect to the ultimate strength in the respective direction.

Rx =GrirSm, (5)

R,b,=(r,v IiTM, (6)

Results

Figure 13 shows the interaction curves for plates made of alloy 6082-T6 with heat-affected-zones along all edges. Both the effects of soft heat-affected-zones and residual stresses are included. The initial deflection amplitude was set equal to 0.005b.

I

E

Figure 13: Interaction curves for plates made of alloy Beta (~) is given by Eq.(2).

Comparison with design rules

Only design formulas issued by Det Norske

Veritas have been used for comparison (Det Norske Veritas, 1995, 1996). Veritasuls and Veritassls mean the ultimate limit state and the serviceability limit state interaction curves for pure plates, given in Classification Notes No. 30.1, Buckling Strength Analysis. In the same classification note there are also interaction curves for plates intended to be effective flanges in stiffened panels. Specific curves for plates with aspect ratio ah= 1 and a/b=3 are given. For plates with intermediate values of aspect ratios linear interpolation applies. Due to the fact that the plates analysed all had an aspect ratio ah equal to 3, only the curve for this value of the aspect ratio should be necessary, but the design curve for an aspect ratio equal to 1 is also given for comparison. The latter

interaction curve is found in Det Norske Veritas,

( 1996).

All the stresses in the design codes are divided by their uniaxial capacities. Only the shape of the curves can therefore be compared. For a comparison of the absolute values of the stresses, reference is made to the sections dealing with uniaxial compression. Uniaxial compression capacity given in Det Norske Veritas, (1996) is equal to linear buckling capacity with Johnson-Ostenfeld correction.

The interaction curves for plates with different values of the slenderness, ~, are given in Figures 14-22. All the plates have heat-affected-zones along all edges. Both the effects of soft zones and residual stresses are included, and the initial deflection is given by Eq. (3) with a maximum amplitude set equal to 0.005b.

t.2 T—”—— —. ——

1 4

0.2

-& 0.s

* Num. analyses, a/b=3 -=- Veritarmls 0.4 -+ Veritassls 0.2- +Vwitashighspead *Veritasatf., a/b=3 o 0.2 0.4 0.s 0.s 1 1.2 RX

Figure 14: Biaxial load, alloy 6082-T6, j3=l.O

0.s

0.4 0.2 0

i

_Num. anatyaas, alb.3 -=- Veritasula +Veritaasls +Veritaahighspead +Veritaaeff., a/b=l -+ Veritaseft., a/b=3 o 0.2 0.4 0.s 0.s 1 1.2 RX

Figure 16: Biaxial load, alloy 6082-T6, j3=2.O

1 0.s & 0.s 0,4 0.2 1

+- Num. analysas, a/b=3 + Verifasuls -+ Varitasels *Varifaahighspeed *Veritasetf., a/b.l +Veritasatf., a/b=3 o-l I o 02 0.4 0.6 0.8 1 t .2 RX

Figure 15: Biaxial load, alloy 6082-T6, ~=1.5

“2

~–-—––

11

0.s ~ 0.s

-+Num. analysas, aib=3 -=- Verltasuls

0.4-+Veritaasls *Variteshighspeed 0.2 -IS- VeritaaefL, aib=l

*VarKaseff., a/b=3

o

0 0.2 0.4 0.s 0.s 1 1.2

RX

Figure 17: Biaxial load, alloy 6082-T6, ~=2.5 29

. . 1.2 1 0.8 ~ 0.6 0.4 0.2 0

+Num. analyses, a/b=3\ ~ ~

--- Veritasuks \\! Y +Veriteaals 1 0.8 & 0.6 0.4 0,2 0 ,.4 1 0.6 & 0,8 0.4 0.2 0

+-Num. ana. a/b.3 -=- varita6u16 +Verita6ele +Veritaahighepeed +- Veritaseff., aib.1 ~ Veritaseff., e/b.3 o 0.2 0,4 0.6 0.s 1 1,2 0 0,2 0.4 0.6 0.6 1 1,2 Rx RX

Figure 18: Biaxial load, alloy 6082-T6, ~=3.O Figure 19: Biaxial load, alloy 6082-T6, ~=3.5

1,2 1 0.6 ~ 0.6 0.4 0.2 0

+Num. aria. a/b=3 + Veritasuls +Veritassls + Veritashighspaed + Veritasefi., e/b=l + Veritaseff., a/b.3 o 0.2 0,4 O.e 0.6 1 1.2 0 0.2 0.4 0.6 0,6 1 1.2 RX RX

Figure 20: Biaxial load, alloy 6082-T6, &4.O

12. 11 08 &0r2 04 02 0

Figure 21: Biaxial load, alloy 6082-T6, &4.5

\

--- Veritaaefl., a/b=3

+Num. am. alb=3

\ ~ +Veritasuls \ -- Veritassls +Veritashighspeed +Veritasetf., a/b=l 0 02 04 00 00 1 12 RX

So far only one alloy has been investigated in biaxial compression. This alloy has a 50 % reduction of 0.2 9Z0 tensile proof stress in the heat-affected-zones and a stress-strain relationship more similar to steel than an alloy with a lower relative yield strength.

The shape of the interaction curves are dependent on the slenderness of the plates. The effective flange

formula given in DNV buckling notes is not

dependent on the slenderness for a/b=3. The shape of the DNV buckling curves for effective flange of compact plates are believed to be conservative. For slender plates, the DNV effective flange interaction

curve meant for plates with a/b=3 is

non-conservative. Using the formula meant for plates with db= 1, when the plates have an aspect ratio a/b=3 is always conservative.

The DNV ULS interaction curve is always non-conservative when the plate is loaded in mainly transverse compression, but it can be used with good

accuracy when the loads are mainly axial

compression. The DNV SLS interaction curve is well suited for plates with slenderness in the range 2.0-3.0. The DNV interaction curve for high speed and light craft vessels can be used for plates with slenderness 2.0 or lower. For higher values of the slenderness, the interaction curve is non-conservative.

12 1 0.8 ~ 0.6 0.4 02 0 — Beta=5.o

_!

Figure 23: Numerical analyses of alloy 6082-T6

Suggestion of Interaction Formula for Design

Based on the numerical results for plates made of alloy 6082-T6 a new interaction curve will be presented. Given a transverse strength ratio of the plate equal to RY, take the maximum allowable longitudinal strength ratio of the plate, RX,equal to the lesser of l– R,* R= = (7) 1 – qRy and RX= I+ CRY (8)

By regression analyses the constants in Eqs. (7-8) have been determined to be

~=0.213-O.275(~-3) (9)

C=O.05-O. 1p (lo)

If the slenderness, ~, is larger than 3,0, Eq. (7) is the only equation that should be used.

Figures 23-28 show the interaction curves from

the numerical analyses together with the

re-1.2 1 0.8 & 0.6 0.4 0.2 0

commended design curves,

+ Beta=l.0 -m- Beta=l.5 Y \ + Bata=2.o * Beta=2.5 + Beta3.o + Beta4.5 + Beta=4.o — Ma4.5 } — Bata=5,0 o 0.2 0.4 0.6 0.6 1 1.2 Rx

1{ 0.8-~ 0.6- _{+ Beta=l.0} + DOSlgnCUNa 0,4- _{+ Beta.3.O}

I

Figure 25: Numerical results for ~=1 ,0,3.0 and

1.2 11 0.8 ~ 0.6 0.4 0.2 o

5.0 plotted together with the recommended design curves

+- Beta=2.O + Designcuws + Bets=4.O — Deaigncuws o 0.2 0.4 0.6 0.8 1 1.2 Rx

Figure 27: Numerical results for ~=2.O and 4.0 plotted together with the

recommended design curves

CONCLUSIONS AND RECOMMENDATION

OF FURTHER WORK

For axial compression, Eurocode 9 can be used for design purposes, except for plates made of alloy 6082-T6 with slenderness less than 2.5. Then a reduction of 10 $TOis needed. The effective flange formula found in Det Norske Veritas (1995) is well suited for transverse compression. The interaction

1,2 1 0.8 & 0.6 0.4 0.2 0 + Beta.1.5 + Designcutve + Beta.3.5 + Designcurve o 0.2 0.4 0,6 0,8 1 1.2 Rx

Figure 26: Numerical results for ~=1.5 and 3.5

1.2 1{ 0.8 ~ 0.8. 0.4 0.2 0,

plotted together with the recommended design curves

+ lMa=2.5 + Designcutve — Beta=4.5 —Designcurve o 0.2 0.4 0.6 0.8 1 1.2 RX

Figure 28: Numerical results for ~=2.5 and 4.5 plotted together with the

recommended design curves

curve for plates with aspect ratio equal to one, taken from Det Norske Veritas (1995) can always be used. This interaction curve is not well defined for plates with slenderness, ~, less than 1.34, but for these plates the interaction curve for plates with aspect ratio equal to three can be used instead. For plates made of alloy 6082-T6 the improved interaction formula established in this study is recommended.

Numerical analyses should be performed for plates with other aspects ratios. Other dimensions of heat-affected-zones should also be investigated. Plates subjected to multiple loads including shear-forces, lateral forces and in-plane forces in tension should also be considered.

Residual stresses have been found to have a negligible effect on the ultimate static strength, but numerical analyses of plates subjected to cyclic forces will show this even more clearly.

The edges of the plates are supposed to remain straight. This is a good assumption for plates in the middle of a stiffened panel, but at the ends of the stiffened panel, the panels will be more likely to have simply supported boundary conditions.

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