Thin-Walled Structures 61 (2012) 106–114

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Thin-Walled Structures

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Global stability of thin-walled ferritic stainless steel members

Petr Hradil n, Ludovic Fulop, Asko Talja

VTT, Technical Research Centre of Finland, P.O. Box 1000, FI-02044 VTT, Finland

a r t i c l e i n f o

Available online 24 July 2012

Keywords:

Stainless steel

Buckling

Finite element method

Stress–strain relation

Regression

31/$ - see front matter & 2012 Elsevier Ltd. A

x.doi.org/10.1016/j.tws.2012.05.006

esponding author. Tel.: þ358 400209593; fa

ail address: petr.hradil@vtt.fi (P. Hradil).

a b s t r a c t

As more metallic alloys are introduced in engineering structures, the demand for the proper utilisation

of their nonlinear stress–strain relationship is increasing. This paper discusses the inelastic buckling of

members from such materials with a special focus on ferritic stainless steels. Here we introduce an

alternative approach for the overall stability of members that considers material nonlinearity, namely

the strain hardening parameter n. The suitability of the new model is verified by regression analysis in

comparison with the commonly used standard calculations. The analysis results show that the present

approach could be applied successfully in flexural, flexural–torsional and lateral–torsional buckling.

& 2012 Elsevier Ltd. All rights reserved.

1. Introduction

The interest in corrosion resistant metallic materials forstructural applications has been constantly increasing over recentdecades. Most of these materials are stainless steel alloys, whichcombine high strength and an aesthetically clean surface withhigh corrosion resistance. Ferritic stainless steels with reducedchromium and nickel content are very competitive within thestainless steel grades due to the strong fluctuations in nickel priceand better mechanical properties than those with austeniticalloys. Such steels also have specific strain hardening behaviour,which places them between austenitic and carbon grades withregard to the stress–strain diagram. However, most design codestreat them in the same manner as austenitic steels because themajority of existing numerical and experimental results comefrom austenitic or duplex alloys. The new calculation of reductionfactors introduced in this paper offers the possibility of includingthe variation of strain hardening parameters in strength curvesand to benefit from the advantages of a typical ferritic stainlesssteel stress–strain relationship. The proposed method can, how-ever, be extended both ways, to less nonlinear materials such ashigh strength carbon steels, and to more nonlinear materials suchas austenitic grades. The method is verified by 18 referencebuckling curves created by 158 numerical calculations presentedin the paper. These results were calculated by Abaqus plug-in forvirtual testing of thin-walled structural members developed atVTT [1]. A detailed description of numerical models is followed byan overview of strength curve approximation methods that wereused in regression analysis. Finally, the curve-fitting results are

ll rights reserved.

x: þ358 207227007.

discussed in order to highlight possible advantages of the methoddescribed herein.

2. Numerical models

In order to evaluate the effect of nonlinear stress–strainbehaviour on member buckling strength, multiple series ofnumerical models have been created with variable length andmaterial parameters. The suitability of linear and quadratic shellelements was tested, as well as different shell thicknesses.Because element types and thicknesses did not significantly affectthe calculation results, we selected nine-node shells with reducedintegration (S9R5) [2], and higher material thickness to limit theeffect of local buckling in shorter members.

2.1. Loading and supports

Pinned–pinned supports and concentric axial loading wereapplied in the flexural buckling (FB) study. The single-symmetricmembers were forced to fail in torsional–flexural buckling (TFB),fixing both ends in y axis bending, torsion and warping. In thecase of lateral–torsional buckling (LTB), members were simplysupported and loaded with end-moments (see Fig. 1). Both endswere additionally restrained against torsion and warping. Theseconditions were also considered in calculation of nondimensionalslenderness.

2.2. Imperfection modelling and elastic buckling analysis

The distribution of initial imperfections was obtained fromlinear eigenvalue analysis as the first overall buckling shape withpositive critical load. In order to suppress local and distortionalbuckling modes in shorter members, each cross-section was

Fig. 1. Loading and supports in FB tests (left), TFB tests (middle) and LTB tests (right).

Fig. 2. Geometry of calculated specimen.

P. Hradil et al. / Thin-Walled Structures 61 (2012) 106–114 107

stiffened with membrane elements in eigenvalue analysis. Thismethod was verified successfully in lateral–torsional buckling [3],and our numerical results also showed a good agreement withanalytically predicted critical loads for flexural and torsional–flexural cases. The selection of imperfection amplitude usuallycorresponds to the mean geometrical imperfections and rangesfrom L/1000 to L/2000 [4–6]. For instance, European bucklingcurves were defined with imperfections L/1000 [7]. Since ournumerical models excluded the effect of residual stresses andstrains in the material, a higher amplitude of initial imperfection(L/750) was chosen to compensate for these effects. It should benoted that L/750 corresponds to the fabrication tolerances in EN1090-2 [8], where the additional deformation caused by residualstresses is expected. The lengths of tested members were selectedas an approximate match for a nondimensional slendernesssequence of 0.125, 0.25, 0.5, 0.7, 1.0, 1.4, 2.0, 2.8 and 4.0. In afew cases it was impossible to obtain ultimate loads for theshortest columns and longest beams due to the geometrical andmaterial limits. Some of the calculations were also affected bylocal and distortional buckling and were excluded from furtherstudy. All the strength curves, therefore, are based on seven tonine calculated points.

2.3. Model simplification

As it was not the goal of our numerical study to simulate anyparticular member behaviour, several simplifications were usedto increase computational efficiency and to highlight the differ-ences in specific material parameters clearly, without additionaldisturbing effects. Each of the following assumptions was care-fully studied before application:

–

Enhanced material properties in corners were included in theaverage values of the entire cross-section. This method is alsoaccepted by the Eurocode [9].

–

Residual stresses from cold-forming were not used due to theirsmall effect on the member behaviour, as concluded byGardner and Cruise [10].

–

Residual stresses from fabrication and press-braking were alsoassumed to be included in the material model and initialimperfections.

–

An isotropic material model was used with nonlinear hard-ening. This provides sufficient accuracy compared to otherpossible isotropic and anisotropic models, according to Ras-mussen et al. [11].

–

Rounded corners were ignored, giving greater flexibility forreasonable aspect ratios of flat part shell elements.

2.4. Cross-sections and material properties

For each buckling mode analysis we selected a different cross-section to demonstrate that the studied phenomena are commonto hollow sections as well as open double-symmetrical andsingle-symmetrical cross-sectional shapes. A square hollow

section (with centre-to-centre side length 72 mm, wall thickness5 mm) was used in flexural buckling analysis, while a lippedchannel (with centre-to-centre distances of web and flanges72 mm, lip length 18 mm, wall thickness 5 mm) was selected inthe case of torsional–flexural buckling. An I section (100�200 mmwith flange and web thicknesses 8.5 and 5.6 mm respectively)served as a basis for lateral–torsional buckling analyses (see Fig. 2).

The stress–strain relationship in Eqs. (1)–(3) used in thepresented study is based on Rasmussen’s modification of theMirambell–Real model [12,13]. This is also included in theexisting design rules, e.g. in Annex C of Eurocode 3, Part 1–4 [14].

e¼sE0þ0:002 s

s0:2

� �nfor srs0:2

s�s0:2E0:2þepu

s�s0:2su�s0:2

� �mþe0:2 for s4s0:2

8><>: ð1Þ

where E0 is the initial modulus of elasticity, s0.2 and su stands forthe 0.2% offset yield strength and ultimate strength respectively, n

and m are the nonlinear parameters of each segment, and thetangent modulus at 0.2% stress E0.2 can be calculated using thefollowing equation:

E0:2 ¼E0

1þ0,002nðE0=s0:2Þð2Þ

The total strain corresponding to 0.2% proof stress e0.2 and theplastic strain difference epu of the second stage are described as

e0:2 ¼s0:2

E0þ0:002 and epu ¼ eu�e0:2�

su�s0:2

E0:2ð3Þ

The first group of materials (N1–N3) represents the ferriticgrades with high su/s0.2 ratio (Table 1 and Fig. 3). The differencesin nonlinear parameter n was studied in this group, where N1 isclose to austenitic steels with low n values and N3 representsmaterials similar to carbon steel with higher nonlinear factors n.Nonlinearity of ferritic grades is usually between these values.

The second group (F1–F3) focuses on the effect of differentstrengths on materials with lower su/s0.2 ratio (Table 1 and

Table 1Material properties.

Material E0 (GPa) s0.2 (MPa) n su (MPa) m eu

N1 200 300 5 600 2.75 0.50

N2 200 300 10 600 2.75 0.50

N3 200 300 25 600 2.75 0.50

F1 200 300 10 420 3.50 0.29

F2 200 400 10 560 3.50 0.29

F3 200 500 10 700 3.50 0.29

Fig. 3. Stress–strain relationship of materials with variable nonlinear factor n.

Fig. 4. Stress–strain relationship of materials with variable offset yield strength

s0.2.

P. Hradil et al. / Thin-Walled Structures 61 (2012) 106–114108

Fig. 4). The second nonlinear factors m and ultimate strains su arealso calculated using Rasmussen’s model [13].

2.5. Geometrically and materially nonlinear analysis with initial

imperfections

The loading capacities of imperfect numerical models werecalculated using the arc-length method for structural analysis,originally developed by Riks [15]. The highest load proportion-ality factor was recorded in all cases and used to determine thepredicted ultimate load (see Tables 2–4).

3. Strength curve approximation models

3.1. Existing models

The well-established Ayrton–Perry formula [16] for buckling ofcolumns with initial imperfections in Eq. (4) has been used incalculations of carbon steel members for many years, also beingadapted for stainless steel in European and Australian standards[17,18]. However, the calculation has a physical meaning only forcompressed members from linear elastic–plastic material withsinusoidal initial shape, and therefore its parameters – imperfec-tion factor a and the initial slenderness l0 – are usually selectedto match the experimental results. Many analogies of morecomplex phenomena to this model (such as lateral–torsionalbuckling, instability of tubes) have been developed later on.

w¼ 1

jþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffij2�l2

q , where j¼ 0:5ð1þZþl2Þ and Z¼ aðl�l0Þ ð4Þ

In such models, the limiting factor for the plastic collapse isusually the yield strength, which is convenient for materials witha sharp yield point. As Holmquist and Nadai noted as long ago as1939 [19], in materials without a well-defined yield point theyield strength becomes an arbitrary value, and must be substi-tuted by a different approach, for example by using reducedmodulus (or so called ‘‘double modulus’’) theory. Holmquist andNadai also laid the basis for the well-known Ramberg–Osgoodconstitutive model [20] by establishing the nonlinear factor n thatdefines the relation between stress and strain beyond the pro-portionality limit. An alternative to reduced modulus theorycould be use of the tangent modulus of material directly, asproposed by Engesser; this is the current design procedure in SEI/ASCE specification for stainless steels [21] and in the Australianand New Zealand standard [18]. Shanley showed that the trueresistance is somewhere between these models [22], meaningthat the tangent modulus provides a lower bound and reducedmodulus gives higher resistances. It should be noted that boththeories were established for geometrically perfect columns (seeFig. 5), and were in most cases replaced by the Ayrton–Perryformula that takes into account initial imperfections but leavesout the influence of material nonlinearity.

The problem of implementing material nonlinearity in evalua-tion of geometrically imperfect columns was addressed in 1997by Rasmussen and Rondal [24], who modified the imperfectionfactor formula (Eq. (5)) and parameters using Eqs. (6)–(9) asfunctions of the nonlinear n factor and with parameter e as theratio of s0.2 and E0.

Z¼ a½ðl�l1Þb�l0�Z0 ð5Þ

The curves were fitted to match finite element calculations ofcompressed rectangular hollow sections with initial geometricalimperfections of L/1500. Enhanced material properties and resi-dual stresses were included in their numerical calculations in thematerial model in the same way as in the presented study.

a¼ 1:5

ðe0:6þ0:03Þðnð0:0048=e0:55Þþ1:4þ13Þþ

0:002

e0:6ð6Þ

b¼0:36 expð�nÞ

e0:45þ0:007þtan h

n

180þ

6U10�6

e1:4þ0:04

!ð7Þ

l0 ¼ 0:82e

eþ0:0004�0:01 n

� �Z0:2 ð8Þ

l1 ¼ 0:8e

eþ0:00181�

n�5:5

nþð6e�0:0054=eþ0:0015Þ

� �1:2" #

ð9Þ

Table 2Results of FB numerical analysis.

Nondimensional slenderness l and ultimate load Fult (kN)

N1 N2 N3 F1 F2 F3

l Fult l Fult l Fult l Fult l Fult l Fult

0.125 568.9a 0.125 561.8a 0.125 548.6a 0.125 493.8a 0.125 – 0.125 –

0.25 457.3 0.25 440.1 0.25 417.8 0.25 430.1 0.25 589.8 0.25 752.6

0.5 352.7 0.5 354.7 0.5 370.3 0.5 352.1 0.5 491.0 0.5 631.6

0.7 286.0 0.7 305.4 0.7 333.6 0.7 305.6 0.7 425.9 0.7 549.0

1 217.1 1 246.7 1 274.2 1 247.1 1 341.5 1 439.5

1.4 146.7 1.4 165.4 1.4 173.2 1.4 165.5 1.4 225.3 1.4 288.0

2 86.40 2 93.49 2 95.02 2 93.52 2 125.6 2 158.6

2.8 46.54 2.8 49.07 2.8 49.75 2.8 49.08 2.8 65.63 2.8 82.85

a Ultimate loads affected by local buckling were not used in the study.

Table 3Results of TFB numerical analysis.

Nondimensional slenderness l and ultimate load Fult (kN)

N1 N2 N3 F1 F2 F3

l Fult l Fult l Fult l Fult l Fult l Fult

0.087 460.7a 0.087 455.2a 0.087 439.0a 0.087 424.6a 0.101 564.0a 0.113 701.6a

0.174 419.5a 0.174 410.0a 0.174 392.3a 0.174 396.1a 0.201 524.0a 0.225 649.5a

0.347 387.3 0.347 367.0 0.347 359.4 0.347 367.0 0.401 490.9 0.448 601.0

0.681 271.7 0.681 285.7 0.681 309.1 0.681 285.9 0.786 368.0 0.879 442.8

0.941 214.4 0.941 240.5 0.941 267.4 0.941 240.6 1.087 294.8 1.215 334.6

1.282 157.9 1.282 179.0 1.282 193.0 1.282 179.1 1.481 200.6 1.655 213.9

1.716 105.8 1.716 116.0 1.716 120.6 1.716 116.0 1.982 123.4 2.216 128.8

2.272 65.79 2.272 70.1 2.272 72.24 2.272 70.11 2.623 73.09 2.933 75.06

3.015 38.36 3.015 40.12 3.015 40.86 3.015 40.13 3.481 41.04 3.892 41.60

a Ultimate loads affected by local buckling were not used in the study.

Table 4Results of LTB numerical analysis.

Nondimensional slenderness l and ultimate load Mult (kNm)

N1 N2 N3 F1 F2 F3

l Mult l Mult l Mult l Mult l Mult l Mult

0.066 92.21a 0.066 91.66a 0.066 89.85a 0.066 80.44a 0.077 106.7a 0.086 132.24a

0.133 80.40 0.133 79.55 0.133 76.26 0.133 78.73 0.153 97.64 0.171 121.39

0.264 77.13 0.264 76.05 0.264 72.84 0.264 71.41 0.305 94.79 0.341 117.45

0.371 63.27 0.371 61.6 0.371 59.82 0.371 60.76 0.428 79.53 0.479 97.62

0.519 55.11 0.519 54.67 0.519 55.13 0.519 54.52 0.599 70.37 0.670 85.24

0.718 45.60 0.718 47.19 0.718 49.87 0.718 47.22 0.829 59.53 0.927 70.65

0.977 35.77 0.977 39.04 0.977 42.55 0.977 39.07 1.128 47.11 1.261 53.24

1.297 26.68 1.297 29.69 1.297 31.73 1.297 29.71 1.498 33.18 1.675 35.23

1.674 18.57 1.674 20.14 1.674 20.88 1.674 20.14 1.933 21.50 2.161 22.77

2.099 13.03 2.099 14.04 2.099 14.66 2.099 14.04 2.424 – 2.710 –

a Ultimate loads affected by local buckling were not used in the study.

P. Hradil et al. / Thin-Walled Structures 61 (2012) 106–114 109

The calculation published by Rasmussen and Rondal describesaccurately the buckling behaviour of concentrically loaded mem-bers not subjected to torsional or torsional–flexural buckling. Aset of recommended parameters for 8 basic stainless steel gradesis given in the AS/NZS standard [18] for the designer’s conve-nience. The example comparison in Fig. 6 shows the closeagreement of Rasmussen and Rondal’s model with the flexuralbuckling behaviour of rectangular hollow sections. However, themodel would require recalibrating constants in Eqs. (6)–(9) fortorsional or lateral–torsional buckling strength prediction.

3.2. Transformed Ayrton–Perry model

We propose using a similar approach to the SEI/ASCE andAS/NZS standards [18,21], where the buckling curve is calcu-lated with tangent modulus Et of the Ramberg–Osgood stress–strain relationship [20] in Eq. (10) instead of initial elasticmodulus E0.

Et ¼df

de¼

E0s0:2

s0:2þ0:002nE0ðs=s0:2Þn�1

ð10Þ

P. Hradil et al. / Thin-Walled Structures 61 (2012) 106–114110

Although these design codes are based on transformed Euler’slaw without initial imperfections [25], it is possible to extend thisidea to the Ayrton–Perry curve in Eq. (4). As a result we obtain a

Fig. 6. Existing approximation models compared to the cal

Fig. 5. The development of theories for assessment of member buckling strength

that formed a background of today’s design codes for stainless steel (based on

Maquoi and Rondal [23]).

simple recursive model in Eqs. (11)–(13), which can be solvednumerically.

w¼ 1

fþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffif2�ln2

q , where f¼ 0:5ð1þZþln2Þ and Z¼ aðln

�ln

0Þ

ð11Þ

where the transformed slenderness ln is a function of thereduction factor. Its calculation in Eq. (12) is based on the tangentmodulus approach described in [26].

ln¼ lU

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ0:002n

E0

s0:2wn�1

sð12Þ

with the following limitation of the transformed initial slender-ness l0

n:

ln

0o1:0, and therefore l0o1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ0:002nE0=s0:2

p ð13Þ

Such equations are easy to solve using the personal computerwith spreadsheet editor or any other technical computing envir-onment. In our case, the iteration script was developed in thePython programming language and integrated directly in Abaqusfinite element simulations.

The proposed model excludes several important factors ofnonlinear material behaviour: it neglects the nonlinear distribu-tion of stresses and strains over the member cross-section; theinitial imperfection shape is assumed to be sinusoidal; and thematerial stiffness reduction is constant in the entire member. Themodel is therefore unable to produce reduction factors directlywithout adjustment of its parameters to fit the real observedbuckling behaviour. However, the possibility of including theRamberg–Osgood nonlinear factor n in strength curves offers asignificant advantage compared to the standard Ayrton–Perrymodel, while the model can still be used for TFB and LTB analysesif properly calibrated parameters are provided.

culation of N material series in flexural buckling tests.

Fig. 7. Example FEM results; comparison of material models with n¼5, 10 and 25.

Fig. 8. Tangent modulus to normalised stress relationship of studied materials.

P. Hradil et al. / Thin-Walled Structures 61 (2012) 106–114 111

4. Results and discussion

The effect of the nonlinear n factor and increasing yieldstrength was studied for flexural buckling, torsional–flexuralbuckling and lateral–torsional buckling. Reduction factors w werecalculated as ratios of member loading capacities obtained byfinite element calculations (Tables 2–4) and characteristic com-pression or bending resistances according to Eurocode 3 [17].Maximum differences between reduction factors obtained at thesame nondimensional slenderness l were observed and reportedin Figs. 7 and 9.

4.1. The effect of material nonlinearity

The examples of typical strength curves are plotted in Fig. 7where the curve ‘‘Difference’’ shows the quantity wN3–wN1, reach-ing its maximum value at slenderness approximately equal to 1.

Although, the numerical studies were carried out with thecomplex two-stage material model (see Eq. (1), Figs. 3 and 4), itwas more convenient to use the simple Ramberg–Osgood equa-tion for the evaluation of strength curve approximations. In thestudied range of material strains the difference in both materialmodels is insignificant.

The effect of gradual yielding can be observed in Fig. 8 wherethe tangent modulus of elasticity is plotted versus the normalisedstress. It is clear that the most substantial changes in materialstiffness occur before the yield strength is reached, and cannot beso easily neglected in the overall buckling calculation.

4.2. The effect of material yield strength

A similar study was carried out with varying 0.2% proofstrength. As can be seen in Fig. 10, the material nonlinearityslightly increases with respect to stress normalised to the yieldstrength. Corresponding small differences in strength curvescaused by the variation of material yield point were thereforeobserved using finite element calculations (see Figs. 9 and 10).

This effect was also reported by Rasmussen and Rondal [24]for parameter e varying from 0.001 to 0.008 covering mostmetallic materials, including aluminium alloys. In our study, the

equivalent parameter would be between 0.0015 and 0.0025,representing the range of this coefficient in ferritic stainless steelgrades in current use. In the case of lateral torsional buckling, itwas not possible to find clear nondimensional slenderness corre-sponding to the maximum difference between studied yield stressvalues fy (300, 400 and 500 MPa).

4.3. Calibration of strength curve approximation models

To evaluate the suitability of the transformed Ayrton–Perrycalculation for strength curve approximation, we used nonlinearregression analysis to fit Eq. (11) to the finite element results, andcompared it with the same regression on the original Ayrton–Perry curve in Eq. (4). The reduction factors plotted in Fig. 11 arelimited to 1.0.

The outputs of nonlinear regression analysis are the unknownparameters a and l0 and the average absolute error R of the best-fitted curve to the numerical results. The Ayrton–Perry curve (AP)

Fig. 9. Example FEM results; comparison of material models with fy¼300, 400 and 500 MPa.

Fig. 10. Tangent modulus to normalised stress relationship of studied materials.

P. Hradil et al. / Thin-Walled Structures 61 (2012) 106–114112

and transformed Ayrton–Perry curve (TAP) are compared toRasmussen and Rondal’s model (R97) [24] in Tables 5–7. TheR97 model is based on the numerical analysis with lower initialimperfection amplitude (L/1500) and showing higher reductionfactors with almost constant offset to the transformed Ayrton–Perry calculation (see Fig. 11). The parameters of Rasmussen andRondal’s imperfection factor–a, b, l0 and l1–are included in thetables for comparison.

The transformed Ayrton–Perry curve has the lowest errorvalue R in 17 of 18 cases. This indicates that the proposed modelcan describe accurately the shape of the strength curve of non-linear materials taking into account the Ramberg–Osgood hard-ening parameter n. With the increasing nonlinear factor n, theinitial slenderness l0 was decreasing, which is the most visibleeffect in the transformed Ayrton–Perry results (0.36-0.27-0.18in flexural buckling, 0.35-0.27-0.18 in flexural–torsional buck-ling and 0.36-0.27-0.16 in lateral–torsional buckling).

5. Conclusions

The proposed form of transformed Ayrton–Perry curve takesinto account the effect of material gradual yielding. The mainadvantages of this alternative compared to earlier formulations isthat it keeps the basic calculation of the reduction factor in asimple, well-known form, based on the need to calibrate only twoknown parameters a and l0. Furthermore, the same formulationcan be used for both columns and beams. An obvious drawback isthe need for iterations to obtain reduction factors.

Another advantage of the TAP formulation is the way it takesinto account material nonlinearity by the material curves, usingthe concept of transformed slenderness ln. This approach replacesthe use of equivalent/imaginary imperfection to take care of thematerial nonlinear material, bringing the formulas closer to theirtrue physical meaning, even though their parameters still needcalibration by test or numerical results.

The presented strength curves are based on simplified numer-ical models with quite high initial imperfections (L/750), andtherefore cannot be used directly in a member design. However,the curves demonstrated the suitability of the transformedAyrton–Perry model as a basis of nonlinear regression analysis.Since the calculation does not include all aspects of memberbehaviour, such as nonlinear stress distribution in the cross-section, it would be necessary to adjust its parameters to the realexperimental results in the same way as for the original Ayrton–Perry curve. The proposed rule can be easily extended to allmetallic alloys and materials following the Ramberg–Osgood law[20]. This utilises a simple calculation of buckling strength takinginto account the gradual yielding of material, and can serve as abasis for further development of design recommendations.

With an increasing nonlinear n parameter, initial slendernessl0 decreases. This effect implies that while initial slenderness0.4 in the Eurocode can be used for materials with a low n

parameter, it may be unconservative in combination with ferriticgrades that have generally higher n values. The effect of variationof yield strength also confirms the results of Rasmussen andRondal [24] in flexural–torsional and lateral–torsional buckling,showing that the biggest difference in studied cases occurs whenthe nondimensional slenderness l ranges from 0.5 to 1.0.

Fig. 11. Strength curves fitted for N series in flexural buckling.

Table 5Comparison of approximations of flexural buckling (FB) strength curves.

N1 (n¼5) N2 (n¼10) N3 (n¼25)

AP TAP R97 AP TAP R97 AP TAP R97

a 0.88 0.31 1.27 0.64 0.35 0.69 0.25 0.26 0.27

b – – 0.16 – – 0.15 – – 0.23

l0 0.31 0.36 0.61 0.28 0.27 0.57 0.12 0.18 0.44

l1 – – 0.35 – – 0.24 – – 0.11

R 0.030 0.016 0.059 0.046 0.010 0.051 0.044 0.011 0.061

F1 (fy¼300 MPa) F2 (fy¼400 MPa) F3 (fy¼500 MPa)

a 0.60 0.35 0.69 0.29 0.31 0.66 0.28 0.29 0.63

b – – 0.15 – – 0.13 – – 0.12

l0 0.25 0.27 0.57 0.03 0.27 0.60 0.11 0.32 0.63

l1 – – 0.24 – – 0.29 – – 0.33

R 0.029 0.007 0.030 0.018 0.011 0.017 0.018 0.010 0.020

Table 6Comparison of approximations of torsional–flexural buckling (TFB) strength

curves.

N1 (n¼5) N2 (n¼10) N3 (n¼25)

AP TAP R97a AP TAP R97a AP TAP R97a

a 0.65 0.15 1.27 0.43 0.15 0.69 0.25 0.14 0.27

b – – 0.16 – – 0.15 – – 0.23

l0 0.36 0.35 0.61 0.35 0.27 0.57 0.32 0.18 0.44

l1 – – 0.35 – – 0.24 – – 0.11

R 0.021 0.011 0.046 0.032 0.018 0.047 0.031 0.022 0.034

F1 (fy¼300 MPa) F2 (fy¼400 MPa) F3 (fy¼500 MPa)

a 0.38 0.15 0.69 0.29 0.17 0.66 0.31 0.18 0.63

b – – 0.15 – – 0.13 – – 0.12

l0 0.29 0.27 0.57 0.33 0.31 0.60 0.32 0.34 0.63

l1 – – 0.24 – – 0.29 – – 0.33

R 0.023 0.018 0.034 0.021 0.017 0.033 0.020 0.017 0.029

a R97 model was developed for flexural buckling and does not provide

accurate results in this case.

Table 7Comparison of approximations of lateral–torsional buckling (LTB) strength curves.

N1 (n¼5) N2 (n¼10) N3 (n¼25)

AP TAP R97a AP TAP R97a AP TAP R97a

a 0.79 0.22 1.27 0.72 0.28 0.69 0.56 0.26 0.27

b – – 0.16 – – 0.15 – – 0.23

l0 0.35 0.36 0.61 0.36 0.27 0.57 0.36 0.16 0.44

l1 – – 0.35 – – 0.24 – – 0.11

R 0.039 0.016 0.086 0.053 0.035 0.084 0.060 0.040 0.089

F1 (fy¼300 MPa) F2 (fy¼400 MPa) F3 (fy¼500 MPa)

a 0.64 0.28 0.69 0.52 0.22 0.66 0.46 0.26 0.63

b – – 0.15 – – 0.13 – – 0.12

l0 0.34 0.27 0.57 0.33 0.31 0.60 0.37 0.33 0.63

l1 – – 0.24 – – 0.29 – – 0.33

R 0.038 0.024 0.057 0.037 0.027 0.052 0.036 0.024 0.050

a R97 model was developed for flexural buckling and does not provide

accurate results in this case.

P. Hradil et al. / Thin-Walled Structures 61 (2012) 106–114 113

Acknowledgements

The research leading to these results has received fundingfrom the European Community’s Research Fund for Coal and Steel(RFCS) under Grant Agreement no. RFSR-CT-2010-00026, Struc-tural Applications of Ferritic Stainless Steels.

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